![16 Elastic Waves in Composite Media and Structures
Here f is the body force per unit mass. In the absence of any distributed moments in V,
the principle of balance of moment of momentum can be stated as
X f Y t X v X a
∧ + ∧ = ∧ = ∧
∫ ∫ ∫ ∫
ρ ρ ρ
V S V V
dV dS
d
dt
dV dV (2.1.26)
In the component form, equations (2.1.25) and (2.1.26) can be written as
ρ σ ρ
f dV n dS a dV i j
i
V
ji j
S
i
V
∫ ∫ ∫
+ = =
( , , , )
1 2 3 (2.1.27)
ε ρ ε σ ε ρ
ijk j k
V
ijk j lk l
S
ijk j k
V
X f dV Y n dS X a dV
∫ ∫ ∫
+ = (i, j, k = 1, 2, 3) (2.1.28)
Using the divergence theorem, the surface integrals appearing in equations (2.1.27)
and (2.1.28) can be written as
σ σ
ε σ ε
ji j
S
ji j
V
ijk j lk l
S
ijk
n dS dV
Y n dS
∫ ∫
∫
=
+
,
(
( ) ( )
, ,
X dV X dV
j lk l
V
ijk lk jl j lk l
V
σ ε σ δ σ
∫ ∫
= +
Thus, equations (2.1.27) and (2.1.28) can be rewritten respectively in the forms
( )
,
ρ σ ρ
f a dV
i ji j i
V
+ - =
∫ 0 (2.1.29)
and
ε ρ σ ρ σ
ijk j
V
k lk l k jk
X f a dV
[ ( ) ]
,
∫ + - + = 0 (2.1.30)
Since equations (2.1.29) and (2.1.30) must hold for any arbitrary volume V of the body
in the deformed state, it follows that the integrands must vanish everywhere within the
body. Hence, we obtain Cauchy’s equations of motion from equation (2.1.29) as
ρ σ ρ
f a
i ji j i
+ =
, (2.1.31)
Using equation (2.1.31), it is found from equation (2.1.30) that
εijksjk = 0
This result implies that
sjk = skj (2.1.32)
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![Fundamentals of Elastic Waves in Anisotropic Media 17
i.e., the stress tensor is symmetric. Note that for small deformation, the density r in
equation (2.1.31) can be taken as the initial density, and ai = üi.
Finally, we will give a brief outline of the principle of conservation of energy and
derive the equations of thermoelasticity. We will consider thermal energy as well as
mechanical energy. According to the first law of thermodynamics, we have
d
dt
v v U dV f v dV t v dS
i i
V
i i
V
i i
S
ρ ρ ρ
1
2
+
= + +
∫ ∫ ∫ h
hdV q n dS
V
i i
S
∫ ∫
- (2.1.33)
Here U is the internal energy and h is the rate of heat generation within V, both with
respect to unit mass, and q is the heat-flux vector representing the rate of transfer of heat
across S. The left-hand side of equation (2.1.33) represents the rate of change of kinetic
and internal energies of the material occupying V. The right-hand side is the sum of the
rate of work done by the body forces acting on V and the traction acting on S, and the
rate of change of heat energy in V.
Using the divergence theorem, the surface integrals appearing in equation (2.1.33)
can be converted into volume integrals over V. After some obvious algebraic manipula-
tion, equation (2.1.33) takes the form
[ ( ) ( ) ]
, , ,
v v f U h v q
i i
V
i ij j ij i j i i
ρ ρ σ ρ σ
� �
- - + - - +
∫ d
dV = 0 (2.1.34)
Using the equation of motion (2.1.31), it is found from equation (2.1.34) that
ρ σ
( ) , ,
�
U h v q
ij i j i i
- - + = 0 (2.1.35)
Now, we will apply the second law of thermodynamics. We assume that s is the entropy
per unit mass at (x,t). Then, the Clausius–Duhem inequality can be stated as
ρ
ρ
�
sdV
h
T
dV
q n
T
dS
i i
S
V
V
≥ -
∫
∫
∫ (2.1.36)
Here T is the absolute temperature. Applying the divergence theorem to the surface
integral in the above equation, we find
ρ
ρ
�
s
q
T
h
T
i
i
+
- ≥
,
0 (2.1.37)
Using equation (2.1.35), the h appearing in equation (2.1.37) can be eliminated, and
equation (2.1.37) can be rewritten as
ρ σ
( / ) /
,
,
� �
s U T v
q T
T
T
ij i j
i i
- + -
≥ 0 (2.1.38)
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![20 Elastic Waves in Composite Media and Structures
2.1.4 Coupled Equations of Linear Dynamic Thermoelasticity
Substitution of equation (2.1.53b) in equation (2.1.51) yields the modified equation of
heat transport as
k
t
c T u h
ij ij ij i j
θ τ ρ θ β ρ
, ,
[ ]
= +
∂
∂
+ -
1 0 0
� � (2.1.54)
Since kij is positive definite, this equation is hyperbolic, provided that t0 is positive.
Substitution of the expression for sij given by equation (2.1.49) in the equation of
motion (2.1.31) yields
c u u f
ijkl k lj i ij j i
, ,
= + -
ρ β θ ρ
�� (2.1.55)
Equations (2.1.54) and (2.1.55) constitute the coupled equations for linear thermoelas-
todynamics of a homogeneous anisotropic elastic solid including a thermal relaxation
time. These form a system of four coupled equations of the hyperbolic type. The solutions
to this system—satisfying appropriate initial and boundary conditions for the body B
occupying the volume V and bounded by the surface S—describe the thermoelastody-
namic state of B.
Under conditions of isothermal or adiabatic deformation of a homogeneous anisotro-
pic elastic medium, one needs to consider equation (2.1.55) after omitting the term on
the right-hand side involving the temperature gradient. Then, we obtain the displace-
ment equations of elasticity as
c u u f
ijkl k lj i i
, = -
ρ ρ
�� (2.1.56)
The stress, strain, and temperature relation in equation (2.1.49) becomes
σij ijkl kl
c e
= (2.1.57)
In the following section, we will consider solutions of equation (2.1.56) in the form of
plane waves when the body forces are absent, i.e., fi = 0.
2.2 Plane Waves in a Homogeneous
Anisotropic Medium
In this section, we will derive the equations and the characteristics of their solutions for
plane waves that are governed by equation (2.1.56) when the body force f is absent. For
this purpose, it will be assumed that the time dependence of u(x,t) is simple harmonic of
the form e−iwt. Thus, u(x,t) is written as
u x u x
( , ) = ( )
t e i t
- ω (2.2.1)
where w is the circular frequency and i = √−1. Then, equation (2.1.56) takes the form
c U U
ijkl k lj i
, = -ρω2
(2.2.2)
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![Fundamentals of Elastic Waves in Anisotropic Media 21
We will seek a solution of equation (2.2.2) of the form
U A e
i i
iknjx j
=
(2.2.3)
where k is the wave number, n is the unit vector representing the direction of propaga-
tion of the plane wave, and A is a constant vector amplitude of the wave. Substitution of
equation (2.2.3) in (2.2.2) gives
rv2Ai = cijklnlnj Ak (2.2.4)
where v = w/k is the phase velocity of the plane harmonic wave. Equation (2.2.4) is the
Christoffel equation for the determination of v for a wave of phase k propagating in the
direction n and its amplitude A. Equation (2.2.4) represents three linear homogeneous
equations in the three unknown components of A.
For a nontrivial solution (A ≠ 0) of equation (2.2.4), rv2 must be an eigenvalue and A
is a corresponding eigenvector of equation (2.2.4). Note that A defines the direction of
the particle velocity at x. The characteristic equation governing the eigenvalues and the
eigenvectors is
Ω( , , , )
v n n n c n n v
ijkl j l ik
1 2 3
2 0
= - =
ρ δ (2.2.5)
Here, |…| denotes a 3 × 3 determinant and dik is the Kronecker delta. Equation (2.2.5)
is cubic in v2. Thus, there are three possible values of v2 for each n. Since the Christoffel
acoustic tensor, Γik = cijklnjnl, has the property that it is symmetric (Γik = Γki) and positive
definite, it follows that the eigenvalues rv2 are positive. Thus, the phase velocities ±v are
real (∵ r > 0), with +v signifying a wave moving outward (away from the origin) and –v
moving inward. The expression v(n) for all possible directions n traces out a surface of
three sheets, called the “velocity surface.”
2.2.1 Slowness Surface
Equation (2.2.5) can be written in an alternative form in terms of 1/v as
1
0
2
v ij ij
Γ - =
ρδ (2.2.6)
Solution of this equation for v−1 as a function of n, v−1(n) traces out a surface having three
sheets called the “slowness surface.”
Once v2 has been solved from equation (2.2.5), the particle displacement vector A is
found from the matrix equation
[ ][ ]
Γik ik k
v A
- =
ρ δ
2 0 (2.2.7)
where the first matrix in the above equation is 3 × 3 and the second is 3 × 1. Since this
equation is homogeneous, A can be found except for a constant multiplying factor. Now,
it is easily shown that if vM and vN are two different eigenvalues, then the corresponding
eigenvectors AM and AN are orthogonal to each other.
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![24 Elastic Waves in Composite Media and Structures
The same contracted notation will be used to write the six components of the stress
in the form of a 6 × 1 matrix. Also, we will use the engineering shear strain components
defined as
γ γ γ
12 12 23 23 31 31
2 2 2
= = =
e e e
, , (2.2.26)
The stress strain relation in equation (2.1.57) can then be written in the matrix form
σ
σ
σ
σ
σ
σ
1
2
3
4
5
6
11 1
=
c c 2
2 13 14 15 16
22 23 2
c c c c
c c c 4
4 25 26
33 34 35
c c
c c c
symm
c
c c c
36
44 45 46
c c
55 56
c66
e
e
e
1
2
3
4
5
6
γ
γ
γ
(2.2.27)
This equation can be written in the matrix notation as
[ ] [ ][ ]
σ = c e (2.2.28)
where [s] is a 6 × 1 matrix, [c] is 6 × 6, and [e] is the second matrix (6 × 1) on the right.
The elements of the Christoffel acoustic matrix [Γ] can be written in the contracted
notation as
Γ11 11 1
2
66 2
2
55 3
2
16 1 2
2 2
= + +
+ +
c n c n c n
c n n c
c n n c n n
15 1 3 56 2 3
2
+
Γ12 16 1
2
26 2
2
45 3
2
12 66
= + +
+ +
c n c n c n
c c n
( ) 1
1 2 14 56 1 3 46 25 2 3
n c c n n c c n n
+ + + +
( ) ( )
Γ13 15 1
2
46 2
2
35 3
2
14 56
= + +
+ +
c n c n c n
c c n
( ) 1
1 2 13 55 1 3 36 45 2 3
n c c n n c c n n
+ + + +
( ) ( )
(2.2.29)
Γ22 66 1
2
22 2
2
44 3
2
26 1 2
2 2
= + +
+ +
c n c n c n
c n n c
c n n c n n
46 1 3 24 2 3
2
+
Γ23 56 1
2
24 2
2
34 3
2
46 25
= + +
+ +
c n c n c n
c c n
( ) 1
1 2 36 45 1 3 23 44 2 3
n c c n n c c n n
+ + + +
( ) ( )
Γ33 55 1
2
44 2
2
33 3
2
45 1 2
2 2
= + +
+ +
c n c n c n
c n n c
c n n c n n
35 1 3 34 2 3
2
+
The Christoffel acoustic matrix [Γ] simplifies considerably if the material has a high
level of symmetry. In the following, we will develop the equations for some special cases
of symmetry.
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![Fundamentals of Elastic Waves in Anisotropic Media 25
2.2.4.1 Isotropic Elastic Material
If the material is isotropic, then there are no preferred directions in the material, and the
elastic stiffness constants are the same independent of the choice of the coordinate axes
in which the stress and strain tensor components are expressed. It can be shown that the
stiffness tensor cijkl has the form
cijkl ij kl ik jl il jk
= + +
λδ δ µ δ δ δ δ
( ) (2.2.30)
where l and m are known as the Lamé’s constants. Thus, the stress strain relation in
equation (2.1.57) takes the form
σ λ δ µ
ij kk ij ij
e e
= + 2 (2.2.31)
The Lamé’s constants l, m have the properties
0 < 3l + 2m < ∞; 0 < m < ∞ (2.2.32)
In the matrix notation, the stiffness matrix [c] takes the form
c
=
+
+
+
λ µ λ λ
λ µ λ
λ µ
µ
µ
2 0 0 0
2 0 0 0
2 0 0 0
0 0
symmetric
0
0
µ
(2.2.33)
The Christoffel matrix elements are now given by
Γ
Γ
Γ
Γ
11 1
2
22 2
2
33 3
2
= + +
= + +
= + +
( )
( )
( )
λ µ µ
λ µ µ
λ µ µ
n
n
n
i
ij i j
n n i j
= + ≠
( ) ,
λ µ
(2.2.34)
Then, equation (2.2.4) becomes
(rv2 − m)Ai = (l + m)ninkAk (2.2.35)
This equation implies that either (a) Ai = Ani, i.e., A is parallel to n or (b) Aknk = 0, i.e., A
is perpendicular to n. In case (a), we get rv2 = (l + 2m), and in case (b) we find rv2 = m.
The wave is pure longitudinal in case (a) and it is pure shear in case (b). The three sheets
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![26 Elastic Waves in Composite Media and Structures
of the velocity surface are spherical in this case. The two sheets corresponding to the
shear waves are coincident.
The average power flow vector is obtained by using equation (2.2.30) in (2.2.12)
P A n
= + +
1
2
2
2
ω
λ µ µ
v
A n A
l l
[( ) ] (2.2.36)
Now, for the longitudinal wave, A = An and v = vl =
λ µ
ρ
+ 2
. Then, equation (2.2.36)
gives
P n
=
1
2
2 2
ρω v A
l (2.2.37)
Using equation (2.2.14), we find that the energy velocity Ve is given by
Ve = vln (2.2.38)
On the other hand, for the shear wave, Alnl = 0 and v = vs =
µ
ρ
. So, P is given by
P =
1
2
2 2
ρω v A
s n (2.2.39)
The energy velocity is then found to be
Ve = vsn (2.2.40)
Thus, in an isotropic homogeneous medium, the energy (or group) velocity associated
with a plane wave is coincident with its phase velocity.
2.2.4.2 Transversely Isotropic Material
A major emphasis of this book is on wave propagation in an infinite or finite elastic
medium having a laminated structure, where each lamina may be homogeneous isotro-
pic or is a composite made up of a homogeneous matrix reinforced by aligned fibers. In
the former case, the laminated medium can be characterized as a homogeneous aniso-
tropic medium (in the long-wavelength limit) having an axis of elastic symmetry that
is perpendicular to the plane of each lamina. In the latter instance, each lamina may be
treated as homogeneous anisotropic having the axis of symmetry parallel to the fiber
axes. Such an anisotropic medium is called transversely isotropic. Planes perpendicular
to the axis of symmetry (unique axis) are isotropic. If the x1-axis is taken parallel to the
unique axis, then the stiffness matrix [c] takes the form
c
c c c
c
=
11 12 12
22
0 0 0
c
c
23
22
0 0 0
0 0
0 0
0 0
44
symmetric c
c66 0
c66
(2.2.41)
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![28 Elastic Waves in Composite Media and Structures
ρv c n c n c D
2
2
11 1
2
33 3
2
55
1
2
= + + -
(2.2.47)
where
D c c n c c n c c n
= - + - + +
[( ) ( ) ] ( )
11 55 1
2
55 33 3
2 2
13 55
2
4 1
1
2
3
2
n
(2.2.48)
Note that D > 0 and ρ( )
v v D
1
2
2
2 0
- = > . The wave moving with the larger of the two
velocities, v1 = vqL, is called quasi-longitudinal, and the slower wave with the velocity,
v2 = vqS, is called quasi-shear. They are neither purely longitudinal nor purely shear
because the displacements associated with them are not parallel or perpendicular to the
velocity direction n.
Now, to obtain the group velocities of the three waves, we note that, for the shear wave
polarized in the x2 direction, we get from equation (2.2.43)
ΩSH( , , ) ( )
ω ρω
k k c k c k
1 3 66 1
2
44 3
2 2 0
= + - = (2.2.49)
So, it is found that
( )
Vg
SH
1
1 66
3
=
n c
v
ρ
(2.2.50a)
( )
Vg
SH
3
3 44
3
=
n c
v
ρ
(2.2.50b)
These equations show that Vg
SH is not parallel to n. Thus, the group velocity direction
deviates from the phase velocity direction. Note that V n
g
SH . is equal to v3 as per equa-
tion (2.2.15).
On the other hand, for the wave propagating and polarized in the plane of x1–x3, we
obtain the components of the group velocity of the quasi-longitudinal (qL) or quasi-
shear (qS) wave as
( )
/
/
[ { (
Vg 1
1
1 11
2
55
= -
∂ ∂
∂ ∂
=
-
Ω
Ω
k
n c v c
ω
ρ n
n c n
c v c n c n c
1
2
33 3
2
55
2
11 1
2
55 3
2
13
+ +
- + +
)}
{ ( )} (
ρ +
+
- + +
c n
v v c n c n c
55
2
3
2
2
11 1
2
33 3
2
55
2
) ]
[ ( )]
ρ ρ
(2.2.51a)
( )
/
/
[ { (
Vg 2
3
3 33
2
11
= -
∂ ∂
∂ ∂
=
-
Ω
Ω
k
n c v c
ω
ρ n
n c n
c v c n c n c
1
2
55 3
2
55
2
55 1
2
33 3
2
13
+ +
- + +
)}
{ ( )} (
ρ +
+
- + +
c n
v v c n c n c
55
2
1
2
2
11 1
2
33 3
2
55
2
) ]
[ ( )]
ρ ρ
(2.2.51b)
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![Fundamentals of Elastic Waves in Anisotropic Media 29
by taking v as v1 or v2, respectively, in equation (2.2.51). It can be shown that
�
Vg given
by equations (2.2.51a) and (2.2.51b) satisfies the equation
� �
V n
g . = v
It is found from equations (2.2.42) and (2.2.43) that, when the wave is propagating
along the x1-axis, the phase velocities and the polarizations are given by the equations
( )
c v A
11
2
1 0
- =
ρ (2.2.52a)
( )
c v A
66
2
2 0
- =
ρ (2.2.52b)
( )
c v A
55
2
3 0
- =
ρ (2.2.52c)
It is seen that a pure longitudinal wave with velocity vl = √(c11/r) propagates along the
x1-axis, and two pure shear waves, one polarized parallel to the x2-axis and the other
polarized parallel to the x3-axis, propagate with the velocities v2 = √(c66/r) and v3 =
√(c55/r), respectively. Similarly, for propagation along the x3-axis, the equations are
( )
c v A
55
2
1 0
- =
ρ (2.2.53a)
( )
c v A
44
2
2 0
- =
ρ (2.2.53b)
( )
c v A
33
2
3 0
- =
ρ (2.2.53c)
In this case, the longitudinal wave polarized in the x3 direction propagates with velocity
v3 = √(c33/r), and the two shear waves with polarizations, respectively, in the x1 and x2
directions propagate with velocities v1 = √(c55/r) and v2 = √(c44/r).
2.2.5 Transformation of Coordinates
It is sometimes convenient to choose coordinate axes that are different from the sym-
metry axes of the material. To write the stress and strain tensors in the new reference
frame, let us suppose that the new x′ y′ z′ system is obtained from the xyz system by the
orthogonal matrix
[a] =
a a a
a a a
a a a
xx xy xz
yx yy yz
zx zy zz
(2.2.54)
so that we have
xi′ = aijxj (2.2.55)
Then the stress vector transformation can be written as
sI′= MIJsJ, I,J = 1,2,3,4,5,6 (2.2.56)
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![30 Elastic Waves in Composite Media and Structures
where [M] is the 6 × 6 transformation matrix
[ ]
M
a a a
xx xy xz
=
2 2 2 2 2
a a a a
xy xz xz xx 2
2 2
a a
a a
xx xy
yx yy a a a a
yz yy yz
2 2 2 y
yz yx yx yy
zx
a a a
a
2
2 a a a a
zy zz zy zz
2 2 2 2 2
a a a a
zz zx zx zy
a
a a a a a a a a a a
yx zx yy zy yz zz yy zz yz z
+ y
y yx zz yz zx yy zx yx zy
zx xx
a a a a a a a a
a a
+ +
a a a a a a a a a
zy xy zz xz xy zz xz zy
+ x
xz zx xx zz xx zy xy zx
xx yx x
a a a a a a a
a a a
+ +
y
y yy xz yz xy yz xz yy xz yx
a a a a a a a a a
+ +a
a a a a a a
xx yz xx yy xy yx
+
(2.2.57)
In the same way, the strain vector transformation is given by
eI′ = NIJeJ (2.2.58)
where N is obtained from M by shifting the factor 2 from the upper right-hand corner to
the lower left-hand corner. It is noted that
[N]−1 = [M]T (2.2.59)
It then follows that the stiffness matrix [c] transforms as
[c′] = [M][c][M]T (2.2.60)
For future use, we will consider the transformation of the stiffness matrix given by
equation (2.2.60) when the coordinates are rotated clockwise by an angle θ about the
3-axis. The coordinate transformation matrix from x1-, x2-, and x3-axes to x1′-, x2′-, and
x3′(= x3)-axes is (see Fig. 2.2)
[a] =
cos sin
sin cos
θ θ
θ θ
- 0
0
0 0 1
(2.2.61)
x3(x3´)
–θ
α
x1
x2´
x1´
n
Fiber direction
Figure 2.2 Coordinate transformation.
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![Fundamentals of Elastic Waves in Anisotropic Media 31
The [M] matrix is then given by
[ ]
cos sin sin
sin cos sin
M =
-
2 2
2 2
0 0 0 2
0 0 0 2
θ θ θ
θ θ θ
0
0 0 1 0 0 0
0 0 0
0
cos sin
sin cos
sin s
θ θ
θ θ
θ
0
0 0 0
1
2
2
1
2
-
- i
in cos
2 0 0 0 2
θ θ
(2.2.62)
The transformed stiffness coefficients are then found to be
′ = + + +
c c c c c
11 11
4
22
4
12 66
2
1
2
2
cos sin sin
θ θ θ
θ
θ
′ = + -
+
c c c c c
12 11 22 66
2
12
4
1
4
2
( ) sin (cos θ
θ θ
θ θ
θ
+
′ = +
′ =
sin )
cos sin
sin
4
13 13
2
23
2
16
1
2
2
c c c
c [
[ cos sin ( )cos ]
c c c c
c c
11
2
22
2
12 66
22 1
2 2
θ θ θ
- - +
′ = 1
1
4
22
4
12 66
2
23
1
2
2
sin cos sin
θ θ θ
+ + +
′ =
c c c
c c
c c
c c c
13
2
23
2
26 11
2
22
1
2
2
sin cos
sin [ sin
θ θ
θ θ
+
′ = - c
cos ( )cos ]
2
12 66
2 2
θ θ
+ +
c c
(2.2.63a)
′ =
′ = -
c c
c c c
33 33
36 13 23
1
2
2
( )sin θ (2.2.63b)
′ = +
′ = -
c c c
c c c
44 44
2
55
2
45 55 44
1
2
2
cos sin
( )sin
θ θ
θ
′
′ = +
′ = + -
c c c
c c c c
55 44
2
55
2
66 11 22 12
1
4
2
sin cos
(
θ θ
)
)sin cos
2
66
2
14 15 24 25 34
2 2
θ θ
+
′ = ′ = ′ = ′ = ′ =
c
c c c c c ′
′ = ′ = ′ =
c c c
35 46 56 0
(2.2.63c)
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![38 Elastic Waves in Composite Media and Structures
media. A review of significant contributions in this important field has been presented in
Chapter 1 (Section 1.1). See also Nayfeh (1995), Lee (1972), Braga and Herrmann (1988,
1992), and Ting and Chadwick (1988).
In this chapter, we present a model for studying dispersion of elastic waves in an
infinite medium composed of periodic layers of orthotropic material. For the purpose
of generality, three-dimensional motion is dealt with first. Results for the special cases
of antiplane and plane-strain motion are given later. A stiffness method is used here for
its general applicability to the case when each layer has monoclinic symmetry about its
bounding planes. An exact treatment of the problem when each layer is transversely
isotropic is given at the end of the chapter.
In the stiffness method, an interpolation function is assumed to represent the dis-
placement within each layer in terms of a discrete set of generalized coordinates. These
generalized coordinates are the displacements and tractions at the bounding interfaces
of the layer, thus ensuring continuity of displacements and tractions across the inter-
faces between adjacent layers. By applying Hamilton’s principle and using Floquet the-
ory, the dispersion equation is obtained as a generalized algebraic eigenvalue problem
whose solution yields the frequency–wave-number relation as well as the variation of
stresses and displacements. As will be seen in later chapters, the method is adapted well
for bounded plates and cylinders.
3.2 Description of the Problem
Consider a stack of plane layers having orthotropic symmetry with a common plane of
symmetry that is taken as the x1–x2-plane. The layers are assumed to be rigidly bonded
to one another. A global Cartesian coordinates system (x1, x2, x3) is chosen so that the
x3-axis is normal to the layers. Let superscript (i) identify the variables of interest asso-
ciated with the ith layer, which is assumed to be orthotropic, having symmetry planes
parallel to the global coordinate planes. It will be convenient to use a local Cartesian
coordinate system in the ith layer with origin at the midplane of the layer and axes
parallel to the global coordinate axes. For the purpose of keeping the algebra simple, it
will be assumed that the stack is a two-layer periodically laminated body of unbounded
extent. Any two adjacent laminates in the body then comprise a unit cell (see Fig. 3.1).
We are concerned with a harmonic wave propagating in an arbitrary direction through
such a medium.
3.2.1 Stiffness Method
Let the wave-propagation direction make an angle a with the x3-axis and its projection
on the x1–x2-plane make an angle −θ with the x1-axis (see Fig. 2.2). A new global coor-
dinate system (x1′, x2′, x3′) is chosen such that the transformation from the unprimed
system to the primed system is given by the matrix [a] in equation (2.2.61). Thus, the
wave vector lies in the plane of x1′–x3′, and the displacement will be independent of x2′.
Note that the displacement will have all of the three components.
The two adjacent laminae comprising a typical unit cell Cn have elastic constants
given, respectively, by cij
(i) and cij
(i+1), thicknesses 2h(i) and 2h(i+1), and densities r(i) and
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![often concealed by quick and complete naturalization their
foreignness to English remains none the less obvious. I should
worry,[46] in its way, is correct English, but in essence it is as
completely Yiddish as kosher, ganof, schadchen, oi-yoi, matzoh or
mazuma.[47] Black-hand, too, is English in form, but it is
nevertheless as plainly an Italian loan-word as spaghetti, mafia or
padrone.
The extent of such influences upon American, and particularly
upon spoken American, remains to be studied; in the whole
literature I can find but one formal article upon the subject. That
article[48] deals specifically with the suffix -fest, which came into
American from the German and was probably suggested by
familiarity with sängerfest. There is no mention of it in any of the
dictionaries of Americanisms, and yet, in such forms as talk-fest and
gabfest it is met with almost daily. So with -heimer, -inski and -bund.
Several years ago -heimer had a great vogue in slang, and was
rapidly done to death. But wiseheimer remains [Pg152] in colloquial use
as a facetious synonym for smart-aleck, and after awhile it may
gradually acquire dignity. Far lowlier words, in fact, have worked
their way in. Buttinski, perhaps, is going the same route. As for the
words in -bund, many of them are already almost accepted. Plunder-
bund is now at least as good as pork-barrel and slush-fund, and
money-bund is frequently heard in Congress.[49] Such locutions
creep in stealthily, and are secure before they are suspected. Current
slang, out of which the more decorous language dredges a large
part of its raw materials, is full of them. Nix and nixy, for no, are
debased forms of the German nichts; aber nit, once as popular as
camouflage, is obviously aber nicht. And a steady flow of nouns, all
needed to designate objects introduced by immigrants, enriches the
vocabulary. The Hungarians not only brought their national
condiment with them; they also brought its name, paprika, and that
name is now thoroughly American.[50] In the same way the Italians
brought in camorra, padrone, spaghetti and a score of other
substantives, and the Jews made contributions from Yiddish and
Hebrew and greatly reinforced certain old borrowings from German.](https://image.slidesharecdn.com/2210806-250606054719-6c0d384e/85/Elastic-Waves-In-Composite-Media-And-Structures-With-Applications-To-Ultrasonic-Nondestructive-Evaluation-1st-Subhendu-K-Datta-56-320.jpg)
![Once such a loan-word gets in it takes firm root. During the first
year of American participation in the World War an effort was made,
on patriotic grounds, to substitute liberty-cabbage for sour-kraut, but
it quickly failed, for the name had become as completely
Americanized as the thing itself, and so liberty-cabbage seemed
affected and absurd. In the same way a great many other German
words survived the passions of the time. Nor could all the influence
of the professional patriots obliterate that German influence which
has fastened upon the American yes something of the quality of ja.
Constant familiarity with such contributions from foreign
languages and with the general speech habits of foreign peoples has
made American a good deal more hospitable to loan-words than
English, even in the absence of special pressure. Let the same [Pg153]
word knock at the gates of the two languages, and American will
admit it more readily, and give it at once a wider and more intimate
currency. Examples are afforded by café, vaudeville, employé,
boulevard, cabaret, toilette, exposé, kindergarten, dépôt, fête and
menu. Café, in American, is a word of much larger and more varied
meaning than in English and is used much more frequently, and by
many more persons. So is employé, in the naturalized form of
employee. So is toilet: we have even seen it as a euphemism for
native terms that otherwise would be in daily use. So is
kindergarten: I read lately of a kindergarten for the elementary
instruction of conscripts. Such words are not unknown to the
Englishman, but when he uses them it is with a plain sense of their
foreignness. In American they are completely naturalized, as is
shown by the spelling and pronunciation of most of them. An
American would no more think of attempting the French
pronunciation of depot or of putting the French accents upon it than
he would think of spelling toilet with the final te or of essaying to
pronounce Anheuser in the German manner. Often curious battles go
on between such loan-words and their English equivalents, and with
varying fortunes. In 1895 Weber and Fields tried to establish music-
hall in New York, but it quickly succumbed to vaudeville-theatre, as
variety had succumbed to vaudeville before it. In the same way](https://image.slidesharecdn.com/2210806-250606054719-6c0d384e/85/Elastic-Waves-In-Composite-Media-And-Structures-With-Applications-To-Ultrasonic-Nondestructive-Evaluation-1st-Subhendu-K-Datta-57-320.jpg)
![lawn-fete (without the circumflex accent, and commonly pronounced
feet) has elbowed out the English garden-party. But now and then,
when the competing loan-word happens to violate American speech
habits, a native term ousts it. The French crèche offers an example;
it has been entirely displaced by day-nursery.
The English, in this matter, display their greater conservatism very
plainly. Even when a loan-word enters both English and American
simultaneously a sense of foreignness lingers about it on the other
side of the Atlantic much longer than on this side, and it is used with
far more self-consciousness. The word matinée offers a convenient
example. To this day the English commonly print it in italics, give it
its French accent, and pronounce it with some attempt at the French
manner. But in America it is entirely naturalized, and the most
ignorant man [Pg154] uses it without any feeling that it is strange. The
same lack of any sense of linguistic integrity is to be noticed in many
other directions—for example, in the freedom with which the Latin
per is used with native nouns. One constantly sees per day, per
dozen, per hundred, per mile, etc., in American newspapers, even
the most careful, but in England the more seemly a is almost always
used, or the noun itself is made Latin, as in per diem. Per, in fact, is
fast becoming an everyday American word. Such phrases as "as per
your letter (or order) of the 15th inst." are incessantly met with in
business correspondence. The same greater hospitality is shown by
the readiness with which various un-English prefixes and affixes
come into fashion, for example, super- and -itis. The English accept
them gingerly; the Americans take them in with enthusiasm, and
naturalize them instanter.[51]
The same deficiency in reserve is to be noted in nearly all other
colonialized dialects. The Latin-American variants of Spanish, for
example, have adopted a great many words which appear in true
Castilian only as occasional guests. Thus in Argentina matinée,
menu, début, toilette and femme de chambre are perfectly good
Argentine, and in Mexico sandwich and club have been thoroughly
naturalized. The same thing is to be noted in the French of Haiti, in
the Portuguese of Brazil, and even in the Danish of Norway. Once a](https://image.slidesharecdn.com/2210806-250606054719-6c0d384e/85/Elastic-Waves-In-Composite-Media-And-Structures-With-Applications-To-Ultrasonic-Nondestructive-Evaluation-1st-Subhendu-K-Datta-58-320.jpg)
![language spreads beyond the country of its origin and begins to be
used by people born, in the German phrase, to a different
Sprachgefühl, the sense of loyalty to its vocabulary is lost, along with
the instinctive feeling for its idiomatic habits. How far this
destruction of its forms may go in the absence of strong contrary
influences is exhibited by the rise of the Romance languages from
the vulgar Latin of the Roman provinces, and, here at home, by the
decay of foreign languages in competition with English. The Yiddish
that the Jews from Russia bring in is German debased with Russian,
Polish and [Pg155] Hebrew; in America, it quickly absorbs hundreds of
words and idioms from the speech of the streets. Various conflicting
German dialects, among the so-called Pennsylvania Dutch and in the
German areas of the Northwest, combine in a patois that, in its end
forms, shows almost as much English as German. Classical examples
of it are "es giebt gar kein use," "Ich kann es nicht ständen" and
"mein stallion hat über die fenz gescheumpt und dem nachbar sein
whiet abscheulich gedämätscht."[52] The use of gleiche for to like, by
false analogy from gleich (=like, similar) is characteristic. In the
same way the Scandinavians in the Northwest corrupt their native
Swedish and Dano-Norwegian. Thus, American-Norwegian is heavy
with such forms as strit-kar, reit-evé, nekk-töi and staits-pruessen,
for street-car, right away, necktie and states-prison, and admits such
phrases as "det meka ingen difrens."[53]
The changes that Yiddish has undergone in America, though
rather foreign to the present inquiry, are interesting enough to be
noticed. First of all, it has admitted into its vocabulary a large
number of everyday substantives, among them boy, chair, window,
carpet, floor, dress, hat, watch, ceiling, consumption, property,
trouble, bother, match, change, party, birthday, picture, paper (only
in the sense of newspaper), gambler, show, hall, kitchen, store,
bedroom, key, mantelpiece, closet, lounge, broom, tablecloth, paint,
landlord, fellow, tenant, shop, wages, foreman, sleeve, collar, cuff,
button, cotton, thimble, needle, pocket, bargain, sale, remnant,
sample, haircut, razor, waist, basket, school, scholar, teacher, baby,
mustache, butcher, grocery, dinner, street and walk. And with them](https://image.slidesharecdn.com/2210806-250606054719-6c0d384e/85/Elastic-Waves-In-Composite-Media-And-Structures-With-Applications-To-Ultrasonic-Nondestructive-Evaluation-1st-Subhendu-K-Datta-59-320.jpg)
![many characteristic Americanisms, [Pg156] for example, bluffer, faker,
boodler, grafter, gangster, crook, guy, kike, piker, squealer, bum,
cadet, boom, bunch, pants, vest, loafer, jumper, stoop, saleslady,
ice-box and raise, with their attendant verbs and adjectives. These
words are used constantly; many of them have quite crowded out
the corresponding Yiddish words. For example, ingel, meaning boy
(it is a Slavic loan-word in Yiddish), has been obliterated by the
English word. A Jewish immigrant almost invariably refers to his son
as his boy, though strangely enough he calls his daughter his
meidel. "Die boys mit die meidlach haben a good time" is excellent
American Yiddish. In the same way fenster has been completely
displaced by window, though tür (=door) has been left intact. Tisch
(=table) also remains, but chair is always used, probably because
few of the Jews had chairs in the old country. There the beinkel, a
bench without a back, was in use; chairs were only for the well-to-
do. Floor has apparently prevailed because no invariable
corresponding word was employed at home: in various parts of
Russia and Poland a floor is a dill, a podlogé, or a bricke. So with
ceiling. There were six different words for it.
Yiddish inflections have been fastened upon most of these loan-
words. Thus, "er hat ihm abgefaked" is "he cheated him," zubumt is
the American gone to the bad, fix'n is to fix, usen is to use, and so
on. The feminine and diminutive suffix -ké is often added to nouns.
Thus bluffer gives rise to blufferké (=hypocrite), and one also notes
dresské, hatké, watchké and bummerké. "Oi! is sie a blufferké!" is
good American Yiddish for "isn't she a hypocrite!" The suffix -nick,
signifying agency, is also freely applied. Allrightnick means an
upstart, an offensive boaster, one of whom his fellows would say "He
is all right" with a sneer. Similarly, consumptionick means a victim of
tuberculosis. Other suffixes are -chick and -ige, the first exemplified
in boychick, a diminutive of boy, and the second in next-doorige,
meaning the woman next-door, an important person in ghetto social
life. Some of the loan-words, of course, undergo changes on
Yiddish-speaking lips. Thus, landlord becomes lendler, lounge
becomes lunch, tenant becomes tenner, and whiskers loses its final](https://image.slidesharecdn.com/2210806-250606054719-6c0d384e/85/Elastic-Waves-In-Composite-Media-And-Structures-With-Applications-To-Ultrasonic-Nondestructive-Evaluation-1st-Subhendu-K-Datta-60-320.jpg)
![s. "Wie gefällt dir sein whisker?" (=how do you like his beard?) [Pg157]
is good Yiddish, ironically intended. Fellow, of course, changes to the
American feller, as in "Rosie hat schon a feller" (=Rosie has got a
feller, i. e., a sweetheart). Show, in the sense of chance, is used
constantly, as in "git ihm a show" (=give him a chance). Bad boy is
adopted bodily, as in "er is a bad boy." To shut up is inflected as one
word, as in "er hat nit gewolt shutup'n" (=he wouldn't shut up). To
catch is used in the sense of to obtain, as in "catch'n a gmilath
chesed" (=to raise a loan). Here, by the way, gmilath chesed is
excellent Biblical Hebrew. To bluff, unchanged in form, takes on the
new meaning of to lie: a bluffer is a liar. Scores of American phrases
are in constant use, among them, all right, never mind, I bet you, no
sir and I'll fix you. It is curious to note that sure Mike, borrowed by
the American vulgate from Irish English, has gone over into
American Yiddish. Finally, to make an end, here are two complete
and characteristic American Yiddish sentences: "Sie wet clean'n die
rooms, scrub'n dem floor, wash'n die windows, dress'n dem boy und
gehn in butcher-store und in grocery. Dernoch vet sie machen
dinner und gehn in street für a walk."[54]
American itself, in the Philippines, and to a lesser extent in Porto
Rico and on the Isthmus, has undergone similar changes under the
influence of Spanish and the native dialects. Maurice P. Dunlap[55]
offers the following specimen of a conversation between two
Americans long resident in Manila:](https://image.slidesharecdn.com/2210806-250606054719-6c0d384e/85/Elastic-Waves-In-Composite-Media-And-Structures-With-Applications-To-Ultrasonic-Nondestructive-Evaluation-1st-Subhendu-K-Datta-61-320.jpg)
![Hola, amigo.
Komusta kayo.
Porque were you hablaing with ese señorita?
She wanted a job as lavandera.
Cuanto?
Ten cents, conant, a piece, so I told her no kerry.
Have you had chow? Well, spera till I sign this chit and I'll take a
paseo with you.
[Pg158]
Here we have an example of Philippine American that shows all
the tendencies of American Yiddish. It retains the general forms of
American, but in the short conversation, embracing but 41 different
words, there are eight loan-words from the Spanish (hola, amigo,
porque, ese, señorita, lavandera, cuanto and paseo), two Spanish
locutions in a debased form (spera for espera and no kerry for no
quiro), two loan-words from the Taglog (komusta and kayo), two
from Pigeon English (chow and chit), one Philippine-American
localism (conant), and a Spanish verb with an English inflection
(hablaing).
The immigrant in the midst of a large native population, of course,
exerts no such pressure upon the national language as that exerted
upon an immigrant language by the native, but nevertheless his
linguistic habits and limitations have to be reckoned with in dealing
with him, and the concessions thus made necessary have a very
ponderable influence upon the general speech. In the usual sense,
as we have seen, there are no dialects in American; two natives,
however widely their birthplaces may be separated, never have any
practical difficulty understanding each other. But there are at least
quasi-dialects among the immigrants—the Irish, the German, the
Scandinavian, the Italian, the Jewish, and so on—and these quasi-
dialects undoubtedly leave occasional marks, not only upon the
national vocabulary, but also upon the general speech habits of the](https://image.slidesharecdn.com/2210806-250606054719-6c0d384e/85/Elastic-Waves-In-Composite-Media-And-Structures-With-Applications-To-Ultrasonic-Nondestructive-Evaluation-1st-Subhendu-K-Datta-62-320.jpg)
![country, as in the case, for example, of the pronunciation of yes,
already mentioned, and in that of the substitution of the diphthong
oi for the ur-sound in such words as world, journal and burn—a
Yiddishism now almost universal among the lower classes of New
York, and threatening to spread.[56] More important, however, is the
support given to a native tendency by the foreigner's incapacity for
employing (or even comprehending) syntax of any complexity, or
words not of the simplest. This is the tendency toward succinctness
[Pg159] and clarity, at whatever sacrifice of grace. One English
observer, Sidney Low, puts the chief blame for the general
explosiveness of American upon the immigrant, who must be
communicated with in the plainest words available, and is not
socially worthy of the suavity of circumlocution anyhow.[57] In his
turn the immigrant seizes upon these plainest words as upon a sort
of convenient Lingua Franca—his quick adoption of damn as a
universal adjective is traditional—and throws his influence upon the
side of the underlying speech habit when he gets on in the vulgate.
Many characteristic Americanisms of the sort to stagger
lexicographers—for example, near-silk—have come from the Jews,
whose progress in business is a good deal faster than their progress
in English. Others, as we have seen, have come from the German
immigrants of half a century ago, from the so-called Pennsylvania
Dutch (who are notoriously ignorant and uncouth), and from the
Irish, who brought with them a form of English already very corrupt.
The same and similar elements greatly reinforce the congenital
tendencies of the dialect—toward the facile manufacture of
compounds, toward a disregard of the distinctions between parts of
speech, and, above all, toward the throwing off of all etymological
restraints.
§ 5
Processes of Word Formation](https://image.slidesharecdn.com/2210806-250606054719-6c0d384e/85/Elastic-Waves-In-Composite-Media-And-Structures-With-Applications-To-Ultrasonic-Nondestructive-Evaluation-1st-Subhendu-K-Datta-63-320.jpg)
![—Some of these tendencies, it has been pointed out, go back to the
period of the first growth of American, and were inherited from the
English of the time. They are the products of a movement which,
reaching its height in the English of Elizabeth, was dammed up at
home, so to speak, by the rise of linguistic self-consciousness toward
the end of the reign of Anne, but continued almost unobstructed in
the colonies. For example, there is what philologists call the habit of
back-formation—a sort of instinctive search, etymologically unsound,
for short roots in long words. This habit, in Restoration days,
precipitated a quasi-English word, mobile, from the Latin [Pg160]
mobile vulgus, and in the days of William and Mary it went a step
further by precipitating mob from mobile. Mob is now sound English,
but in the eighteenth century it was violently attacked by the new
sect of purists,[58] and though it survived their onslaught they
undoubtedly greatly impeded the formation and adoption of other
words of the same category. But in the colonies the process went on
unimpeded, save for the feeble protests of such stray pedants as
Witherspoon and Boucher. Rattler for rattlesnake, pike for turnpike,
draw for drawbridge, coon for raccoon, possum for opossum, cuss
for customer, cute for acute, squash for askutasquash—these
American back-formations are already antique; Sabbaday for
Sabbath-day has actually reached the dignity of an archaism. To this
day they are formed in great numbers; scarcely a new substantive of
more than two syllables comes in without bringing one in its wake.
We have thus witnessed, within the past two years, the genesis of
scores now in wide use and fast taking on respectability; phone for
telephone, gas for gasoline, co-ed for co-educational, pop for
populist, frat for fraternity, gym for gymnasium, movie for moving-
picture, prep-school for preparatory-school, auto for automobile,
aero for aeroplane. Some linger on the edge of vulgarity: pep for
pepper, flu for influenza, plute for plutocrat, pen for penitentiary,
con for confidence (as in con-man, con-game and to con), convict
and consumption, defi for defiance, beaut for beauty, rep for
reputation, stenog for stenographer, ambish for ambition, vag for
vagrant, champ for champion, pard for partner, coke for cocaine,](https://image.slidesharecdn.com/2210806-250606054719-6c0d384e/85/Elastic-Waves-In-Composite-Media-And-Structures-With-Applications-To-Ultrasonic-Nondestructive-Evaluation-1st-Subhendu-K-Datta-64-320.jpg)
![simp for simpleton, diff for difference. Others are already in perfectly
good usage: smoker for smoking-car, diner for dining-car, sleeper
for sleeping-car, oleo for oleomargarine, hypo for hyposulphite of
soda, Yank for Yankee, confab for confabulation, memo for
memorandum, pop-concert for popular-concert. Ad for
advertisement is struggling hard for recognition; some of its
compounds, e. g., ad-writer, want-ad, display-ad, ad-card, ad-rate,
column-ad and ad-man, are already accepted in technical
terminology. Boob for booby promises to become sound American in
a few years; its synonyms are no more respectable than it is. At
[Pg161] its heels is bo for hobo, an altogether fit successor to bum for
bummer.[59]
A parallel movement shows itself in the great multiplication of
common abbreviations. "Americans, as a rule," says Farmer, "employ
abbreviations to an extent unknown in Europe.... This trait of the
American character is discernible in every department of the national
life and thought."[60] O. K., C. O. D., N. G., G. O. P. (get out and
push) and P. D. Q., are almost national hall-marks; the immigrant
learns them immediately after damn and go to hell. Thornton traces
N. G. to 1840; C. O. D. and P. D. Q. are probably as old. As for O. K.,
it was in use so early as 1790, but it apparently did not acquire its
present significance until the 20's; originally it seems to have meant
"ordered recorded."[11] During the presidential campaign of 1828
Jackson's enemies, seeking to prove his illiteracy, alleged that he
used it for "oll korrect." Of late the theory has been put forward that
it is derived from an Indian word, okeh, signifying "so be it," and Dr.
Woodrow Wilson is said to support this theory and to use okeh in
endorsing government papers, but I am unaware of the authority
upon which the etymology is based. Bartlett says that the figurative
use of A No. 1, as in an A No. 1 man, also originated in America, but
this may not be true. There can be little doubt, however, about T. B.
(for tuberculosis), G. B. (for grand bounce), 23, on the Q. T., and D.
& D. (drunk and disorderly). The language breeds such short forms
of speech prodigiously; every trade and profession has a host of
them; they are innumerable in the slang of sport.[61]](https://image.slidesharecdn.com/2210806-250606054719-6c0d384e/85/Elastic-Waves-In-Composite-Media-And-Structures-With-Applications-To-Ultrasonic-Nondestructive-Evaluation-1st-Subhendu-K-Datta-65-320.jpg)
![What one sees under all this, account for it as one will, is a double
habit, the which is, at bottom, sufficient explanation of the gap
which begins to yawn between English and American, particularly on
the spoken plane. On the one hand it is a habit of verbal economy—
a jealous disinclination to waste two words on what can be put into
one, a natural taste for the brilliant and [Pg162] succinct, a disdain of
all grammatical and lexicographical daintiness, born partly, perhaps,
of ignorance, but also in part of a sound sense of their imbecility.
And on the other hand there is a high relish and talent for metaphor
—in Brander Matthews' phrase, "a figurative vigor that the
Elizabethans would have realized and understood." Just as the
American rebels instinctively against such parliamentary
circumlocutions as "I am not prepared to say" and "so much by way
of being,"[62] just as he would fret under the forms of English
journalism, with its reporting empty of drama, its third-person
smothering of speeches and its complex and unintelligible jargon,[63]
just so, in his daily speech and writing he chooses terseness and
vividness whenever there is any choice, and seeks to make one
when it doesn't exist. There is more than mere humorous contrast
between the famous placard in the wash-room of the British
Museum: "These Basins Are For Casual Ablutions Only," and the
familiar sign at American railroad-crossings: "Stop! Look! Listen!"
Between the two lies an abyss separating two cultures, two habits of
mind, two diverging tongues. It is almost unimaginable that
Englishmen, journeying up and down in elevators, would ever have
stricken the teens out of their speech, turning sixteenth into simple
six and twenty-fourth into four; the clipping is almost as far from
their way of doing things as the climbing so high in the air. Nor have
they the brilliant facility of Americans for making new words of
grotesque but penetrating tropes, as in corn-fed, tight-wad, bone-
head, bleachers and juice (for electricity); when they attempt such
things the result is often lugubrious; two hundred years of
schoolmastering has dried up their inspiration. Nor have they the
fine American hand for devising new verbs; to maffick and to
limehouse are their best specimens in twenty years, and both have](https://image.slidesharecdn.com/2210806-250606054719-6c0d384e/85/Elastic-Waves-In-Composite-Media-And-Structures-With-Applications-To-Ultrasonic-Nondestructive-Evaluation-1st-Subhendu-K-Datta-66-320.jpg)
![an almost pathetic flatness. Their business with the language,
indeed, is not in this department. They are [Pg163] not charged with its
raids and scoutings, but with the organization of its conquests and
the guarding of its accumulated stores.
For the student interested in the biology of language, as opposed
to its paleontology, there is endless material in the racy neologisms
of American, and particularly in its new compounds and novel verbs.
Nothing could exceed the brilliancy of such inventions as joy-ride,
high-brow, road-louse, sob-sister, nature-faker, stand-patter,
lounge-lizard, hash-foundry, buzz-wagon, has-been, end-seat-hog,
shoot-the-chutes and grape-juice-diplomacy. They are bold; they are
vivid; they have humor; they meet genuine needs. Joy-ride, I note,
is already going over into English, and no wonder. There is
absolutely no synonym for it; to convey its idea in orthodox English
would take a whole sentence. And so, too, with certain single words
of metaphorical origin: barrel for large and illicit wealth, pork for
unnecessary and dishonest appropriations of public money, joint for
illegal liquor-house, tenderloin for gay and dubious neighborhood.[64]
Most of these, and of the new compounds with them, belong to the
vocabulary of disparagement. Here an essential character of the
American shows itself: his tendency to combat the disagreeable with
irony, to heap ridicule upon what he is suspicious of or doesn't
understand.
The rapidity with which new verbs are made in the United States
is really quite amazing. Two days after the first regulations of the
Food Administration were announced, to hooverize appeared
spontaneously in scores of newspapers, and a week later it was
employed without any visible sense of its novelty in the debates of
Congress and had taken on a respectability equal to that of to
bryanize, to fletcherize and to oslerize. To electrocute appeared
inevitably in the first public discussion of capital [Pg164] punishment by
electricity; to taxi came in with the first taxi-cabs; to commute no
doubt accompanied the first commutation ticket; to insurge attended
the birth of the Progressive balderdash. Of late the old affix -ize,
once fecund of such monsters as to funeralize, has come into favor](https://image.slidesharecdn.com/2210806-250606054719-6c0d384e/85/Elastic-Waves-In-Composite-Media-And-Structures-With-Applications-To-Ultrasonic-Nondestructive-Evaluation-1st-Subhendu-K-Datta-67-320.jpg)
![again, and I note, among its other products, to belgiumize, to
vacationize, to picturize and to scenarioize. In a newspaper headline
I even find to s o s, in the form of its gerund.[65] Many characteristic
American verbs are compounds of common verbs and prepositions
or adverbs, with new meanings imposed. Compare, for example, to
give and to give out, to go back and to go back on, to beat and to
beat it, to light and to light out, to butt and to butt in, to turn and to
turn down, to show and to show up, to put and to put over, to wind
and to wind up. Sometimes, however, the addition seems to be
merely rhetorical, as in to start off, to finish up, to open up and to
hurry up. To hurry up is so commonplace in America that everyone
uses it and no one notices it, but it remains rare in England. Up
seems to be essential to many of these latter-day verbs, e. g., to
pony up, to doll up, to ball up; without it they are without
significance. Nearly all of them are attended by derivative adjectives
or nouns; cut-up, show-down, kick-in, come-down, hang-out, start-
off, run-in, balled-up, dolled-up, wind-up, bang-up, turn-down,
jump-off.
In many directions the same prodigal fancy shows itself—for
example, in the free interchange of parts of speech, in the bold
inflection of words not inflected in sound English, and in the
invention of wholly artificial words. The first phenomenon has
already concerned us. Would an English literary critic of any
pretensions employ such a locution as "all by her lonesome"? I have
a doubt of it—and yet I find that phrase in a serious book by the
critic of the New Republic.[66] Would an English M. P. use "he has
another think coming" in debate? Again I doubt it—but even more
anarchistic dedications of verbs and adjectives to substantival use
are to be found in the Congressional Record every day. Jitney is an
old American substantive lately [Pg165] revived; a month after its
revival it was also an adjective, and before long it may also be a
verb and even an adverb. To lift up was turned tail first and made a
substantive, and is now also an adjective and a verb. Joy-ride
became a verb the day after it was born as a noun. And what of
livest? An astounding inflection, indeed—but with quite sound](https://image.slidesharecdn.com/2210806-250606054719-6c0d384e/85/Elastic-Waves-In-Composite-Media-And-Structures-With-Applications-To-Ultrasonic-Nondestructive-Evaluation-1st-Subhendu-K-Datta-68-320.jpg)
![American usage behind it. The Metropolitan Magazine, of which Col.
Roosevelt is an editor, announces on its letter paper that it is "the
livest magazine in America," and Poetry, the organ of the new poetry
movement, prints at the head of its contents page the following
encomium from the New York Tribune: "the livest art in America
today is poetry, and the livest expression of that art is in this little
Chicago monthly."
Now and then the spirit of American shows a transient faltering,
and its inventiveness is displaced by a banal extension of meaning,
so that a single noun comes to signify discrete things. Thus laundry,
meaning originally a place where linen is washed, has come to mean
also the linen itself. So, again, gun has come to mean fire-arms of all
sorts, and has entered into such compounds as gun-man and gun-
play. And in the same way party has been borrowed from the
terminology of the law and made to do colloquial duty as a synonym
for person. But such evidences of poverty are rare and abnormal;
the whole movement of the language is toward the multiplication of
substantives. A new object gets a new name, and that new name
enters into the common vocabulary at once. Sundae and hokum are
late examples; their origin is dubious and disputed, but they met
genuine needs and so they seem to be secure. A great many more
such substantives are deliberate inventions, for example, kodak,
protectograph, conductorette, bevo, klaxon, vaseline, jap-a-lac,
resinol, autocar, postum, crisco, electrolier, addressograph,
alabastine, orangeade, pianola, victrola, dictagraph, kitchenette,
crispette, cellarette, uneeda, triscuit and peptomint. Some of these
indicate attempts at description: oleomargarine, phonograph and
gasoline are older examples of that class. Others represent efforts to
devise designations that will meet the conditions of advertising
psychology and the trade-marks law, to wit, that they [Pg166] be (a)
new, (b) easily remembered, and (c) not directly descriptive.
Probably the most successful invention of this sort is kodak, which
was devised by George Eastman, inventor of the portable camera so
called. Kodak has so far won acceptance as a common noun that
Eastman is often forced to assert his proprietary right to it.[67]](https://image.slidesharecdn.com/2210806-250606054719-6c0d384e/85/Elastic-Waves-In-Composite-Media-And-Structures-With-Applications-To-Ultrasonic-Nondestructive-Evaluation-1st-Subhendu-K-Datta-69-320.jpg)
![Vaseline is in the same position. The annual crop of such inventions
in the United States is enormous.[68] The majority die, but a hearty
few always survive.
Of analogous character are artificial words of the scalawag and
rambunctious class, the formation of which constantly goes on.
Some of them are shortened compounds: grandificent (from grand
and magnificent), sodalicious (from soda and delicious) and
warphan(age) (from war and orphan(age)).[69] Others are made up
of common roots and grotesque affixes: swelldoodle, splendiferous
and peacharino. Yet others are mere extravagant inventions:
scallywampus, supergobsloptious and floozy. Most of these are
devised by advertisement writers or college students, and belong
properly to slang, but there is a steady movement of selected
specimens into the common vocabulary. The words in -doodle hint at
German influences, and those in -ino owe something to Italian, or at
least to popular burlesques of what is conceived to be Italian.
§ 6
Pronunciation
—"Language," said Sayce, in 1879, "does not consist of letters, but
of sounds, and until this fact has been brought home to us our study
of it will be little better than an [Pg167] exercise of memory."[70] The
theory, at that time, was somewhat strange to English grammarians
and etymologists, despite the investigations of A. J. Ellis and the
massive lesson of Grimm's law; their labors were largely wasted
upon deductions from the written word. But since then, chiefly under
the influence of Continental philologists, and particularly of the
Dane, J. O. H. Jespersen, they have turned from orthographical
futilities to the actual sounds of the tongue, and the latest and best
grammar of it, that of Sweet, is frankly based upon the spoken
English of educated Englishmen—not, remember, of conscious
purists, but of the general body of cultivated folk. Unluckily, this new
method also has its disadvantages. The men of a given race and](https://image.slidesharecdn.com/2210806-250606054719-6c0d384e/85/Elastic-Waves-In-Composite-Media-And-Structures-With-Applications-To-Ultrasonic-Nondestructive-Evaluation-1st-Subhendu-K-Datta-70-320.jpg)
![time usually write a good deal alike, or, at all events, attempt to
write alike, but in their oral speech there are wide variations. "No
two persons," says a leading contemporary authority upon English
phonetics,[71] "pronounce exactly alike." Moreover, "even the best
speaker commonly uses more than one style." The result is that it is
extremely difficult to determine the prevailing pronunciation of a
given combination of letters at any time and place. The persons
whose speech is studied pronounce it with minute shades of
difference, and admit other differences according as they are
conversing naturally or endeavoring to exhibit their pronunciation.
Worse, it is impossible to represent a great many of these shades in
print. Sweet, trying to do it,[72] found himself, in the end, with a
preposterous alphabet of 125 letters. Prince L.-L. Bonaparte more
than doubled this number, and Ellis brought it to 390.[73] Other
phonologists, English and Continental, have gone floundering into
the same bog. The dictionary-makers, forced to a far greater
economy of means, are brought into obscurity. The difficulties of the
enterprise, in fact, are probably unsurmountable. It is, as White
says, "almost impossible for one person to express to another by
signs the [Pg168] sound of any word." "Only the voice," he goes on, "is
capable of that; for the moment a sign is used the question arises,
What is the value of that sign? The sounds of words are the most
delicate, fleeting and inapprehensible things in nature.... Moreover,
the question arises as to the capability to apprehend and distinguish
sounds on the part of the person whose evidence is given."[74]
Certain German orthoepists, despairing of the printed page, have
turned to the phonograph, and there is a Deutsche Grammophon-
Gesellschaft in Berlin which offers records of specimen speeches in a
great many languages and dialects, including English. The
phonograph has also been put to successful use in language
teaching by various American correspondence schools.
In view of all this it would be hopeless to attempt to exhibit in
print the numerous small differences between English and American
pronunciation, for many of them are extremely delicate and subtle,
and only their aggregation makes them plain. According to a recent](https://image.slidesharecdn.com/2210806-250606054719-6c0d384e/85/Elastic-Waves-In-Composite-Media-And-Structures-With-Applications-To-Ultrasonic-Nondestructive-Evaluation-1st-Subhendu-K-Datta-71-320.jpg)
![and very careful observer,[75] the most important of them do not lie
in pronunciation at all, properly so called, but in intonation. In this
direction, he says, one must look for the true characters "of the
English accent." I incline to agree with White,[76] that the pitch of
the English voice is somewhat higher than that of the American, and
that it is thus more penetrating. The nasal twang which Englishmen
observe in the vox Americana, though it has high overtones, is itself
not high pitched, but rather low pitched, as all constrained and
muffled tones are apt to be. The causes of that twang have long
engaged phonologists, and in the main they agree that there is a
physical basis for it—that our generally dry climate and rapid
changes of temperature produce an actual thickening of the
membranes concerned in the production of sound.[77] We are, in
brief, a somewhat snuffling [Pg169] people, and much more given to
catarrhs and coryzas than the inhabitants of damp Britain. Perhaps
this general impediment to free and easy utterance, subconsciously
apprehended, is responsible for the American tendency to pronounce
the separate syllables of a word with much more care than an
Englishman bestows upon them; the American, in giving
extraordinary six distinct syllables instead of the Englishman's
grudging four, may be seeking to make up for his natural disability.
Marsh, in his "Lectures on the English Language,"[78] sought two
other explanations of the fact. On the one hand, he argued that the
Americans of his day read a great deal more than the English, and
were thus much more influenced by the spelling of words, and on
the other hand he pointed out that "our flora shows that the climate
of even our Northern States belongs ... to a more Southern type
than that of England," and that "in Southern latitudes ... articulation
is generally much more distinct than in Northern regions." In support
of the latter proposition he cited the pronunciation of Spanish,
Italian and Turkish, as compared with that of English, Danish and
German—rather unfortunate examples, for the pronunciation of
German is at least as clear as that of Italian. Swedish would have
supported his case far better: the Swedes debase their vowels and
slide over their consonants even more markedly than the English.](https://image.slidesharecdn.com/2210806-250606054719-6c0d384e/85/Elastic-Waves-In-Composite-Media-And-Structures-With-Applications-To-Ultrasonic-Nondestructive-Evaluation-1st-Subhendu-K-Datta-72-320.jpg)
![Marsh believed that there was a tendency among Southern peoples
to throw the accent back, and that this helped to "bring out all the
syllables." One finds a certain support for this notion in various
American peculiarities of stress. Advertisement offers an example.
The prevailing American pronunciation, despite incessant
pedagogical counterblasts, puts the accent on the penult, whereas
the English pronunciation stresses the second syllable. Paresis
illustrates the same tendency. The English accent the first syllable,
but, as Krapp says, American usage clings to the [Pg170] accent on the
second syllable.[79] There are, again, pianist, primarily and
telegrapher. The English accent the first syllable of each; we
commonly accent the second. In temporarily they also accent the
first; we accent the third. Various other examples might be cited. But
when one had marshalled them their significance would be at once
set at naught by four very familiar words, mamma, papa, inquiry and
ally. Americans almost invariably accent each on the first syllable;
Englishmen stress the second. For months, during 1918, the
publishers of the Standard Dictionary, advertising that work in the
street-cars, explained that ally should be accented on the second
syllable, and pointed out that owners of their dictionary were
safeguarded against the vulgarism of accenting it on the first.
Nevertheless, this free and highly public instruction did not suffice to
exterminate al´ly. I made note of the pronunciations overheard, with
the word constantly on all lips. But one man of my acquaintance
regularly accented the second syllable, and he was an eminent
scholar, professionally devoted to the study of language.
Thus it is unsafe, here as elsewhere, to generalize too facilely, and
particularly unsafe to exhibit causes with too much assurance. "Man
frage nicht warum," says Philipp Karl Buttmann. "Der
Sprachgebrauch lässt sich nur beobachten."[80] But the greater
distinctness of American utterance, whatever its genesis and
machinery, is palpable enough in many familiar situations. "The
typical American accent," says Vizetelly, "is often harsh and
unmusical, but it sounds all of the letters to be sounded, and slurs,
but does not distort, the rest."[81] An American, for example, almost](https://image.slidesharecdn.com/2210806-250606054719-6c0d384e/85/Elastic-Waves-In-Composite-Media-And-Structures-With-Applications-To-Ultrasonic-Nondestructive-Evaluation-1st-Subhendu-K-Datta-73-320.jpg)
![always sounds the first l in fulfill; an Englishman makes the first
syllable foo. An American sounds every syllable in extraordinary,
literary, military, secretary and the other words of the -ary-group; an
Englishman never pronounces the a of the penultimate syllable.
Kindness, with the d silent, would attract notice in the United States;
in England, according to [Pg171] Jones,[82] the d is "very commonly, if
not usually" omitted. Often, in America, commonly retains a full t; in
England it is actually and officially offen. Let an American and an
Englishman pronounce program (me). Though the Englishman
retains the long form of the last syllable in writing, he reduces it in
speaking to a thick triple consonant, grm; the American enunciates it
clearly, rhyming it with damn. Or try the two with any word ending
in -g, say sporting or ripping. Or with any word having r before a
consonant, say card, harbor, lord or preferred. "The majority of
Englishmen," says Menner, "certainly do not pronounce the r ...; just
as certainly the majority of educated Americans pronounce it
distinctly."[83] Henry James, visiting the United States after many
years of residence in England, was much harassed by this persistent
r-sound, which seemed to him to resemble "a sort of morose
grinding of the back teeth."[84] So sensitive to it did he become that
he began to hear where it was actually non-existent, save as an
occasional barbarism, for example, in Cuba-r, vanilla-r and
California-r. He put the blame for it, and for various other departures
from the strict canon of contemporary English, upon "the American
common school, the American newspaper, and the American
Dutchman and Dago." Unluckily for his case, the full voicing of the r
came into American long before the appearance of any of these
influences. The early colonists, in fact, brought it with them from
England, and it still prevailed there in Dr. Johnson's day, for he
protested publicly against the "rough snarling sound" and led the
movement which finally resulted in its extinction.[85] Today, extinct, it
is mourned by English purists, and the Poet Laureate denounces the
clergy of the Established Church for saying "the sawed of the Laud"
instead of "the sword of the Lord."[86]](https://image.slidesharecdn.com/2210806-250606054719-6c0d384e/85/Elastic-Waves-In-Composite-Media-And-Structures-With-Applications-To-Ultrasonic-Nondestructive-Evaluation-1st-Subhendu-K-Datta-74-320.jpg)
![But even in the matter of elided consonants American is not
always the conservator. We cling to the r, we preserve the final [Pg172]
g, we give nephew a clear f-sound instead of the clouded English v-
sound, and we boldly nationalize trait and pronounce its final t, but
we drop the second p from pumpkin and change the m to n, we
change the ph(=f)-sound to plain p in diphtheria, diphthong and
naphtha,[87] we relieve rind of its final d, and, in the complete
sentence, we slaughter consonants by assimilation. I have heard
Englishmen say brand-new, but on American lips it is almost
invariably bran-new. So nearly universal is this nasalization in the
United States that certain American lexicographers have sought to
found the term upon bran and not upon brand. Here the national
speech is powerfully influenced by Southern dialectical variations,
which in turn probably derive partly from French example and partly
from the linguistic limitations of the negro. The latter, even after two
hundred years, has great difficulties with our consonants, and often
drops them. A familiar anecdote well illustrates his speech habit. On
a train stopping at a small station in Georgia a darkey threw up a
window and yelled "Wah ee?" The reply from a black on the platform
was "Wah oo?" A Northerner aboard the train, puzzled by this
inarticulate dialogue, sought light from a Southern passenger, who
promptly translated the first question as "Where is he?" and the
second as "Where is who?" A recent viewer with alarm[88] argues
that this conspiracy against the consonants is spreading, and that
English printed words no longer represent the actual sounds of the
American language. "Like the French," he says, "we have a marked
liaison—the borrowing of a letter from the preceding word. We invite
one another to 'c'meer' (=come here) ... 'Hoo-zat?' (=who is that?)
has as good a liaison as the French vois avez." This critic believes
that American tends to abandon t for d, as in Sadd'y (=Saturday)
and siddup (=sit up), and to get rid of h, as in "ware-zee?" (=where
is he?). But here we invade the vulgar speech, which belongs to the
next chapter. [Pg173]
Among the vowels the most salient difference between English
and American pronunciation, of course, is marked off by the flat](https://image.slidesharecdn.com/2210806-250606054719-6c0d384e/85/Elastic-Waves-In-Composite-Media-And-Structures-With-Applications-To-Ultrasonic-Nondestructive-Evaluation-1st-Subhendu-K-Datta-75-320.jpg)
![American a. This flat a, as we have seen, has been under attack at
home for nearly a century. The New Englanders, very sensitive to
English example, substitute a broad a that is even broader than the
English, and an a of the same sort survives in the South in a few
words, e. g., master, tomato and tassel, but everywhere else in the
country the flat a prevails. Fashion and the example of the stage
oppose it,[89] and it is under the ban of an active wing of
schoolmasters, but it will not down. To the average American,
indeed, the broad a is a banner of affectation, and he associates it
unpleasantly with spats, Harvard, male tea-drinking, wrist watches
and all the other objects of his social suspicion. He gets the flat
sound, not only into such words as last, calf, dance and pastor, but
even into piano and drama. Drama is sometimes drayma west of
Connecticut, but almost never drahma or drawma. Tomato with the
a of bat, may sometimes borrow the a of plate, but tomahto is
confined to New England and the South. Hurrah, in American, has
also borrowed the a of plate; one hears hurray much oftener than
hurraw. Even amen frequently shows that a, though not when sung.
Curiously enough, it is displaced in patent by the true flat a. The
English rhyme the first syllable of the word with rate; in America it
always rhymes with rat.
The broad a is not only almost extinct outside of New England; it
begins to show signs of decay even there. At all events, it has
gradually disappeared from many words, and is measurably less
sonorous in those in which it survives than it used to be. A century
ago it appeared, not only in dance, aunt, glass, past, etc., but also in
Daniel, imagine, rational and travel.[90] And in 1857 Oliver Wendell
Holmes reported it in matter, handsome, caterpillar, apple and
satisfaction. It has been displaced in virtually all of these, even in
the most remote reaches of the back country, [Pg174] by the national
flat a. Grandgent[91] says that the broad a is now restricted in New
England to the following situations:
1. when followed by s or ns, as in last and dance.
2. when followed by r preceding another consonant, as in cart.](https://image.slidesharecdn.com/2210806-250606054719-6c0d384e/85/Elastic-Waves-In-Composite-Media-And-Structures-With-Applications-To-Ultrasonic-Nondestructive-Evaluation-1st-Subhendu-K-Datta-76-320.jpg)
![3. when followed by lm, as in calm.
4. when followed by f, s or th, as in laugh, pass and path.
The u-sound also shows certain differences between English and
American usage. The English reduce the last syllable of figure to ger;
the educated American preserves the u-sound as in nature. The
English make the first syllable of courteous rhyme with fort; the
American standard rhymes it with hurt. The English give an oo-
sound to the u of brusque; in America the word commonly rhymes
with tusk. A u-sound, as everyone knows, gets into the American
pronunciation of clerk, by analogy with insert; the English cling to a
broad a-sound, by analogy with hearth. Even the latter, in the United
States, is often pronounced to rhyme with dearth. The American, in
general, is much less careful than the Englishman to preserve the
shadowy y-sound before u in words of the duke-class. He retains it
in few, but surely not in new. Nor in duke, blue, stew, due, duty and
true. Nor even in Tuesday. Purists often attack the simple oo-sound.
In 1912, for example, the Department of Education of New York City
warned all the municipal high-school teachers to combat it.[92] But it
is doubtful that one pupil in a hundred was thereby induced to insert
the y in induced. Finally there is lieutenant. The Englishman
pronounces the first syllable left; the American invariably makes it
loot. White says that the prevailing American pronunciation is
relatively recent. "I never heard it," he reports, "in my boyhood."[93]
He was born in New York in 1821.
The i-sound presents several curious differences. The English
make it long in all words of the hostile-class; in America it is
commonly short, even in puerile. The English also lengthen it in
sliver; in America the word usually rhymes with liver. The [Pg175] short
i, in England, is almost universally substituted for the e in pretty, and
this pronunciation is also inculcated in most American schools, but I
often hear an unmistakable e-sound in the United States, making the
first syllable rhyme with bet. Contrariwise, most Americans put the
short i into been, making it rhyme with sin. In England it shows a
long e-sound, as in seen. A recent poem by an English poet makes](https://image.slidesharecdn.com/2210806-250606054719-6c0d384e/85/Elastic-Waves-In-Composite-Media-And-Structures-With-Applications-To-Ultrasonic-Nondestructive-Evaluation-1st-Subhendu-K-Datta-77-320.jpg)
![the word rhyme with submarine, queen and unseen.[94] The o-
sound, in American, tends to convert itself into an aw-sound. Cog
still retains a pure o, but one seldom hears it in log or dog. Henry
James denounces this "flatly-drawling group" in "The Question of
Our Speech,"[95] and cites gawd, dawg, sawft, lawft, gawne, lawst
and frawst as horrible examples. But the English themselves are not
guiltless of the same fault. Many of the accusations that James levels
at American, in truth, are echoed by Robert Bridges in "A Tract on
the Present State of English Pronunciation." Both spend themselves
upon opposing what, at bottom, are probably natural and inevitable
movements—for example, the gradual decay of all the vowels to one
of neutral color, represented by the e of danger, the u of suggest,
the second o of common and the a of prevalent. This decay shows
itself in many languages. In both English and High German, during
their middle periods, all the terminal vowels degenerated to e—now
sunk to the aforesaid neutral vowel in many German words, and
expunged from English altogether. The same sound is encountered
in languages so widely differing otherwise as Arabic, French and
Swedish. "Its existence," says Sayce, "is a sign of age and decay;
meaning has become more important than outward form, and the
educated intelligence no longer demands a clear pronunciation in
order to understand what is said."[96]
All these differences between English and American pronunciation,
separately considered, seem slight, but in the aggregate they are
sufficient to place serious impediments between mutual [Pg176]
comprehension. Let an Englishman and an American (not of New
England) speak a quite ordinary sentence, "My aunt can't answer for
my dancing the lancers even passably," and at once the gap
separating the two pronunciations will be manifest. Here only the a
is involved. Add a dozen everyday words—military, schedule, trait,
hostile, been, lieutenant, patent, nephew, secretary, advertisement,
and so on—and the strangeness of one to the other is augmented.
"Every Englishman visiting the States for the first time," said an
English dramatist some time ago, "has a difficulty in making himself
understood. He often has to repeat a remark or a request two or](https://image.slidesharecdn.com/2210806-250606054719-6c0d384e/85/Elastic-Waves-In-Composite-Media-And-Structures-With-Applications-To-Ultrasonic-Nondestructive-Evaluation-1st-Subhendu-K-Datta-78-320.jpg)
![three times to make his meaning clear, especially on railroads, in
hotels and at bars. The American visiting England for the first time
has the same trouble."[97] Despite the fact that American actors
imitate English pronunciation to the best of their skill, this visiting
Englishman asserted that the average American audience is
incapable of understanding a genuinely English company, at least
"when the speeches are rattled off in conversational style." When he
presented one of his own plays with an English company, he said,
many American acquaintances, after witnessing the performance,
asked him to lend them the manuscript, "that they might visit it
again with some understanding of the dialogue."[98]](https://image.slidesharecdn.com/2210806-250606054719-6c0d384e/85/Elastic-Waves-In-Composite-Media-And-Structures-With-Applications-To-Ultrasonic-Nondestructive-Evaluation-1st-Subhendu-K-Datta-79-320.jpg)
Elastic Waves In Composite Media And Structures With Applications To Ultrasonic Nondestructive Evaluation 1st Subhendu K Datta
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- 5. Elastic Waves in Composite Media and Structures With Applications to Ultrasonic Nondestructive Evaluation 53388_C000.indd 1 10/14/08 3:52:54 PM © 2009 by Taylor & Francis Group, LLC
- 6. Mechanical Engineering Series Frank Kreith & Roop Mahajan - Series Editors Published Titles Computer Techniques in Vibration Clarence W. de Silva Distributed Generation: The Power Paradigm for the New Millennium Anne-Marie Borbely & Jan F. Kreider Elastic Waves in Composite Media and Structures: With Applications to Ultrasonic Nondestructive Evaluation Subhendu K. Datta and Arvind H. Shah Elastoplasticity Theory Vlado A. Lubarda Energy Audit of Building Systems: An Engineering Approach Moncef Krarti Energy Converstion D. Yogi Goswami and Frank Kreith Energy Management and Conservation Handbook Frank Kreith and D. Yogi Goswami Finite Element Method Using MATLAB, 2nd Edition Young W. Kwon & Hyochoong Bang Fluid Power Circuits and Controls: Fundamentals and Applications John S. Cundiff Fundamentals of Environmental Discharge Modeling Lorin R. Davis Handbook of Energy Efficiency and Renewable Energy Frank Kreith and D. Yogi Goswami Heat Transfer in Single and Multiphase Systems Greg F. Naterer Introductory Finite Element Method Chandrakant S. Desai & Tribikram Kundu Intelligent Transportation Systems: New Principles and Architectures Sumit Ghosh & Tony Lee Machine Elements: Life and Design Boris M. Klebanov, David M. Barlam, and Frederic E. Nystrom Mathematical & Physical Modeling of Materials Processing Operations Olusegun Johnson Ilegbusi, Manabu Iguchi & Walter E. Wahnsiedler Mechanics of Composite Materials Autar K. Kaw Mechanics of Fatigue Vladimir V. Bolotin Mechanism Design: Enumeration of Kinematic Structures According to Function Lung-Wen Tsai Mechatronic Systems: Devices, Design, Control, Operation and Monitoring Clarence W. de Silva MEMS: Applications Mohamed Gad-el-Hak 53388_C000.indd 2 10/14/08 3:52:54 PM © 2009 by Taylor & Francis Group, LLC
- 7. MEMS: Design and Fabrication Mohamed Gad-el-Hak The MEMS Handbook, Second Edition Mohamed Gad-el-Hak MEMS: Introduction and Fundamentals Mohamed Gad-el-Hak Multiphase Flow Handbook Clayton T. Crowe Nanotechnology: Understanding Small Systems Ben Rogers, Sumita Pennathur, and Jesse Adams Optomechatronics: Fusion of Optical and Mechatronic Engineering Hyungsuck Cho Practical Inverse Analysis in Engineering David M. Trujillo & Henry R. Busby Pressure Vessels: Design and Practice Somnath Chattopadhyay Principles of Solid Mechanics Rowland Richards, Jr. Thermodynamics for Engineers Kau-Fui Wong Vibration Damping, Control, and Design Clarence W. de Silva Vibration and Shock Handbook Clarence W. de Silva Viscoelastic Solids Roderic S. Lakes 53388_C000.indd 3 10/14/08 3:52:54 PM © 2009 by Taylor & Francis Group, LLC
- 8. Elastic Waves in Composite Media and Structures With Applications to Ultrasonic Nondestructive Evaluation Subhendu K. Datta Arvind H. Shah CRC Press is an imprint of the Taylor & Francis Group, an informa business Boca Raton London New York 53388_C000.indd 5 10/14/08 3:52:54 PM © 2009 by Taylor & Francis Group, LLC
- 9. CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2009 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-4200-5338-8 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher can- not assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copy- right.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that pro- vides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Datta, S. K. (Subhendu K.) Elastic waves in composite media and structures : with applications to ultrasonic nondestructive evaluation / Subhendu K. Datta and Arvind H. Shah. p. cm. -- (Mechanical engineering series) Includes bibliographical references and index. ISBN 978-1-4200-5338-8 (alk. paper) 1. Fibrous composites--Testing. 2. Ultrasonic testing. 3. Elastic waves. I. Shah, Arvind H. II. Title. III. Series. TA418.9.C6D2886 2009 620.1’187--dc22 2008040950 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com 53388_C000.indd 6 10/14/08 3:52:54 PM © 2009 by Taylor & Francis Group, LLC
- 10. vii Dedication Dedicated to our wives Bishakha and Ranjan, and to our children Kinshuk, Ketki, and Seema. 53388_C000.indd 7 10/14/08 3:52:54 PM © 2009 by Taylor & Francis Group, LLC
- 11. ix Table of Contents Preface...................................................................................................................... xi 1 Introduction 1.1 Historical Background...........................................................................................1 1.2 Scope of the Book...................................................................................................8 2 Fundamentals of Elastic Waves in Anisotropic Media 2.1 Waves in Homogeneous Elastic Media: Formulation of Field Equations...............................................................................................................11 2.2 Plane Waves in a Homogeneous Anisotropic Medium................................. 20 2.3 Numerical Results and Discussion....................................................................33 3 Periodic Layered Media 3.1 Introduction..........................................................................................................37 3.2 Description of the Problem................................................................................ 38 3.3 Numerical Results................................................................................................51 3.4 Remarks. ................................................................................................................ 56 4 Guided Waves in Fiber-Reinforced Composite Plates 4.1 Introduction......................................................................................................... 64 4.2 Governing Equations.......................................................................................... 64 4.3 Numerical Results................................................................................................76 4.4 Application to Materials Characterization...................................................... 85 4.5 Thin Layers........................................................................................................... 88 4.6 Guided Waves in Plates with Thin Coating and Interface Layers. ............... 98 4.7 Transient Response due to a Concentrated Source of Excitation...............114 4.8 Laminated Plate with Interface Layers........................................................... 142 4.9 Remarks. .............................................................................................................. 146 4.10 Laser-Generated Thermoelastic Waves...........................................................147 4.11 Results for Thermoelastic Dispersion and Laser-Generated Waves.......... 157 53388_C000.indd 9 10/14/08 3:52:54 PM © 2009 by Taylor & Francis Group, LLC
- 12. x 5 Wave Propagation in Composite Cylinders 5.1 Introduction........................................................................................................165 5.2 Governing Equations........................................................................................ 166 5.3 Analytical Solution for Transversely Isotropic Composite Cylinder........ 168 5.4 Stiffness Method I...............................................................................................174 5.5 Stiffness Method II.............................................................................................179 5.6 Numerical Results—Circular Cylinder..........................................................181 5.7 Guided Waves in a Cylinder of Arbitrary Cross Section............................ 186 5.8 Numerical Results—Cylinder with Rectangular and Trapezoidal Cross Sections.................................................................................................... 195 5.9 Harmonic Response of a Composite Circular Cylinder due to a Point Force.......................................................................................................... 205 5.10 Forced Motion of Finite-Width Plate..............................................................213 6 Scattering of Guided Waves in Plates and Cylinders 6.1 Introduction....................................................................................................... 223 6.2 Scattering in a Plate........................................................................................... 224 6.3 Scattering in a Pipe............................................................................................ 257 Appendix A: Computer Programs A.1 Programs............................................................................................................. 283 A.2 Executing Programs.......................................................................................... 286 References......................................................................................................... 289 53388_C000.indd 10 10/14/08 3:52:54 PM © 2009 by Taylor & Francis Group, LLC
- 13. xi Preface This book is an outgrowth of over 30 years of research and teaching done by our groups at the Department of Mechanical Engineering at the University of Colorado at Boulder and the Department of Civil Engineering at the University of Manitoba at Winnipeg. The book deals with the fundamentals of waves in fiber-reinforced laminated plates and shells. Composite materials have mechanical and other physical properties that are often superior to traditional monolithic metallic or polymeric materials for use in civil, mechanical, and aerospace structures. They are also generally lighter and more econom- ical to use. They can be tailored to the needs of particular structural applications. For these reasons, composite materials have found widespread use in many structural appli- cations. These materials have complex microstructures and are, in general, anisotropic and inhomogeneous. Thus, they present considerable challenges for characterization of their mechanical properties and prediction of dynamic response. Understanding of elastic wave propagation characteristics in such materials and structures is essential for prediction and interpretation of their dynamic response, and for ultrasonic nondestruc- tive evaluation of mechanical properties of, and defects in, these structures. The goal of this book is to present analytical and numerical techniques that have been found to be effective in solving a wide class of problems involving wave propagation and scattering by defects in anisotropic layered plates and shells. It contains a systematic treatment of elastic waves in unbounded and bounded layered media composed of fiber-reinforced materials using analytical and numerical tools. In addition, extensive numerical results and key executable computer programs are included in the accompanying CD. While an introductory knowledge of elastic waves is desirable when reading this book, essential concepts and equations governing elastic waves in unbounded and bounded anisotropic media are discussed in sufficient detail so that the readers should be able to follow and use the material presented. They will get a good understanding of the char- acteristics of ultrasonic wave propagation in composite structures. Also, the book will help practitioners simulate and interpret measured dynamic data. The goal here is to provide the reader with theoretical tools to perform tasks such as ultrasonic nondestruc- tive material characterization, nondestructive testing, and impact response of aircraft components, pipelines, coatings, interfaces, and other layered structures. It is believed 53388_C000.indd 11 10/14/08 3:52:54 PM © 2009 by Taylor & Francis Group, LLC
- 14. xii that the book will be found useful by the beginning graduate students, experienced researchers, and practitioners as a source for rigorous mathematical models of dispersive wave propagation in laminated structures. In addition, the reader will be able to use the computer programs to solve many of the problems that arise in practical applications. Elastic waves propagating in composite media and structures are significantly influ- enced by the elastic properties and geometry of the reinforcing phase (fiber, particle, coating, or interface/interphase), layers or laminae, defects (cracks, pores, cavities), and overall geometry. Thus, they provide an effective means of nondestructive characteriza- tion of these properties. For this purpose, it is necessary to understand clearly the salient features of dispersive guided modes and how they are modified by defects, inhomogene- ities, and boundaries. These features are carefully examined in this book. Topics covered include: waves in unbounded periodically layered media and modeling of effective qua- sistatic elastic properties; guided waves in laminated plates and shells; Green’s functions and transient response of plates and shells; thermal effects on guided waves in plates; scattering of waves by cracks, delaminations, and joints; and reflection from the edges of plates and cylinders. Treatments of these topics presented here are believed to be suf- ficiently complete, and extensive references are provided so that the book can be used by students, researchers, and practitioners to solve problems or to use it as a reference. The authors have benefited immensely from their interactions over the years with many graduate students and professional colleagues. In particular, they would like to acknowledge the help they have received from their former and present graduate stu- dents Drs. O. M. Mukdadi, H. Al-Qahtani, T. H. Ju, S. W. Liu, R. L. Bratton, W. M. Karunasena, N. Rattanawangcharoen, H. Bai, J. Zhu, W. Zhuang, and Mr. J. K. T. Yeo. They are especially indebted to Dr. M. J. Frye for his untiring work in drawing all of the figures appearing in this book, to Mr. Faisal Shibley for making the computer programs interactive, to Dr. H. Bai for his unhesitating help with preparing the programs, and to Mr. D. Stoyko for help in the preparation of the manuals and the programs. The authors would like to express special thanks to the Natural Science and Engineering Research Council of Canada (NSERC) and the Manitoba Hydro for their continued support of much of the work that appears in this book. Support from the National Science Foundation, the Office of the Basic Energy Sciences (DOE), and the Office of Naval Research is also gratefully acknowledged. 53388_C000.indd 12 10/14/08 3:52:54 PM © 2009 by Taylor & Francis Group, LLC
- 15. 1 1 Introduction 1.1 Historical Background ...............................................................1 Elastic Waves in Layered Media with Isotropic Layers • Waves in Fiber-Reinforced Media • Waves in Anisotropic Layered Media • Guided Waves in Laminated Plates and Circular Cylindrical Shells • Cylinders of Arbitrary Cross Section • Particle-Reinforced Composites 1.2 Scope of the Book........................................................................8 1.1 Historical Background 1.1.1 Elastic Waves in Layered Media with Isotropic Layers Wave propagation in elastic composite media has been studied extensively since the 1950s. Early investigations dealt with plane-layered composite media and were moti- vated by seismic exploration. Postma (1955), White and Angona (1955), and Rytov (1956) derived expressions for effective wave speeds of plane waves propagating in a periodi- cally laminated medium. They assumed that wavelengths of such waves are much longer than the thicknesses of the layers. In this limit, the wave speeds are independent of frequency, and the layered medium can be modeled as a homogeneous transversely iso- tropic medium with the symmetry axis parallel to the direction of layering. Expressions for the effective static elastic moduli of the anisotropic medium were derived in these papers. Carcione et al. (1991) carried out one-dimensional and two-dimensional numer- ical simulations of longitudinal and shear wave propagation in periodically bilayered elastic media, and they showed that, for sufficiently long wavelengths, dispersion was negligible and that the effective homogeneous transversely isotropic approximation was valid. When the wavelengths became comparable with the layer thicknesses, strong dispersion (frequency-dependent wave speeds) was observed. This would be expected because, at short wavelengths, refraction and reflection of waves at the interfaces of the layers give rise to dispersive behavior. Various approximate theories that account for dispersion of elastic waves in a periodi- cally laminated medium were proposed in the 1960s and 1970s. Among these were the 53388_C001.indd 1 10/8/08 10:45:22 AM © 2009 by Taylor & Francis Group, LLC
- 16. 2 Elastic Waves in Composite Media and Structures effective stiffness theory proposed by Sun et al. (1968b) and Achenbach et al. (1968) (see also Achenbach and Sun 1972), the mixture theory used by Bedford and Stern (1971, 1972) and McNiven and Mengi (1979), and the interacting continuum theory (Hegemier 1972; Hegemier and Bache 1974; and Nayfeh and Gurtman 1974). In addition to the approximate theories mentioned above, exact solutions for har- monicwavepropagationinalaminatedmediumhavebeenpresentedbySunetal.(1968a) and Lee and Yang (1973) for antiplane strain problems, and by Sve (1971) for the plane strain case. Later, Delph et al. (1978, 1979, 1980) presented exact solutions for harmonic wave propagation using Floquet’s theory. In all of these papers mentioned here, each layer material was considered to be linearly elastic, homogeneous, and isotropic. During the last half century, there has been a continued interest in the dynamic behavior of layered anisotropic medium due to the increasing use of advanced compos- ite materials in aerospace, naval, and civil structures. Such structural components are typically made of fibers of high strength and stiffness-reinforcing plastics, metals, or ceramics. The morphology of these materials makes their mechanical response much more complicated than the (usually) isotropic homogeneous matrix materials. In gen- eral, these are anisotropic and inhomogeneous. Furthermore, the mechanical properties are strongly dependent upon the properties of the interfaces between the constituent phases. Composite structural components are usually made up of a stack of layers (plies or laminae), i.e., they are laminated. This adds another layer of complexity to their dynamic behavior. 1.1.2 Waves in Fiber-Reinforced Media To model ultrasonic wave propagation in a composite laminate, it is necessary to model the dynamic properties of a lamina that is made of a matrix reinforced by fibers. Wave propagation in a homogeneous elastic medium reinforced by aligned continuous fibers has been investigated by Achenbach (1976) and Hlavacek (1975) when the fiber distribu- tion is periodic. Other studies on periodic distribution of fibers include Hegemier et al. (1973), Nemat-Nasser and Yamada (1981a), and Nayfeh (1995). Wave propagation in the presence of a random distribution of aligned fibers has been studied by Bose and Mal (1973, 1974), Datta (1975), Datta and Ledbetter (1983), Datta et al. (1984), Willis (1983), Varadan et al. (1978, 1986), and Beltzer and Brauner (1985), among others. When the wavelength of the propagating plane waves is much longer than the fiber diameter, effec- tive wave speeds are found to lead to the static effective elastic constants. Static effective elastic properties of aligned, continuous, fiber-reinforced composites have been studied in great detail since the early 1960s. Among the pioneering works are those by Hashin and Rosen (1964), Hill (1964), Hashin (1979), and Christensen and Lo (1979). References to many other works on static effective thermoelastic properties can be found in the NASA report by Hashin (1972) and in the monograph by Christensen (1991). There are now well-established theories for the modeling of effective anisotropic thermoelastic properties of aligned fiber-reinforced composite media. These anisotropic properties are used to then model the dynamic behavior of laminated media. 53388_C001.indd 2 10/8/08 10:45:22 AM © 2009 by Taylor & Francis Group, LLC
- 17. Introduction 3 1.1.3 Waves in Anisotropic Layered Media Wave propagation in an infinitely layered medium when each layer (lamina) is anisotropic has been studied by Hegemier (1972), Hegemier et al. (1973), and Hegemier and Nayfeh (1973) using mixed spatial and asymptotic expansions as well as a mixture theory. Exact dispersion relations for propagating waves in periodic anisotropic layered media have been studied by Nemat-Nasser and Yamada (1981b), Yamada and Nemat-Nasser (1981), Shah and Datta (1982), Braga and Herrmann (1992), Nayfeh (1995), and Datta (2000), who have used the concept of Floquet waves. Approximate solutions to the Floquet waves using variational principles have also been presented in Nemat-Nasser (1972) and Nemat-Nasser and Minagawa (1975) (see also Minagawa and Nemat-Nasser 1977). 1.1.4 Guided Waves in Laminated Plates and Circular Cylindrical Shells Dispersive behavior of guided waves in laminated plates and shells has been studied extensively in the last 20 years or so by many investigators. Dispersive modal propaga- tion behavior is strongly influenced by the anisotropic properties of each lamina and the stacking sequence used. Thus, this can be exploited to determine material properties of each lamina. Early studies of the propagation of free guided waves (Lamb waves) in an anisotropic plate were reported by Ekstein (1945), Newman and Mindlin (1957), and Kaul and Mindlin (1962a, 1962b). Later, Solie and Auld (1973) gave a detailed description of guided waves in cubic plates, and Li and Thompson (1990) studied the dispersion charac- teristics of orthotropic plates. Nayfeh and Chimenti (1989) developed the equations for a generally anisotropic plate. Extensive numerical results presented in these papers showed many characteristic features that arise due to anisotropy and that can be exploited to determine the anisotropic properties of the material composition of the plate. Guided waves in plates composed of uniaxial or multidirectional fiber-reinforced laminates show very complex behavior because of the complicated reflection and refrac- tion phenomena arising at the interfaces between the anisotropic laminae (layers). Various schemes have been developed for the theoretical studies of this problem. One of these is the method of partial waves that was developed by Rayleigh (1885, 1889) and was used by Lamb (1917) to study guided waves. Many problems of guided wave propa- gation in free and fluid-loaded plates and layered semi-infinite spaces have been solved by this method and by its extension using the transfer-matrix approach (Thomson 1950; Haskell 1953). Mention may be made of the works by Bogy and Gracewski (1983), Nayfeh and Chimenti (1988, 1989), Mal (1988a, 1988b), Chimenti and Nayfeh (1990a, 1990b), Nayfeh and Chimenti (1991), Nayfeh (1991), Karunasena et al. (1991b, 1991c), and Pan and Datta (1999). Nayfeh (1995) gives a good exposition of this method and its applica- tions. Comparison of theoretical and experimental results has led to efficient techniques for the inverse characterization of individual lamina properties. Although this method has been widely used since the early works of Thomson and Haskell, it has been found to have precision problems at high frequencies and for thick layers. There have been several different modifications proposed by many investigators to overcome these instability problems. Among these, mention may be made of the 53388_C001.indd 3 10/8/08 10:45:22 AM © 2009 by Taylor & Francis Group, LLC
- 18. 4 Elastic Waves in Composite Media and Structures delta matrix approach (Dunkin 1965; Kundu and Mal 1985; Castaings and Hosten 1994; Zhu et al. 1995a); global matrix method (Knopoff 1964; Schmidt and Jensen 1985; Mal 1988a; Mal 1988b; Mukdadi et al. 2001); reflectivity method (Kennett and Kerry 1979; Kennett 1983; Fryer and Frazer 1984); and the stiffness matrix method (Kausel and Roesset 1981; Wang and Rajapakse 1994; Wang and Rokhlin 2001, 2002a). Comprehensive reviews of guided waves in composite plates and their use in material characterization have been published by Chimenti (1997) and Datta (2000) (see also Rokhlin and Wang 2002). In all the matrix methods mentioned above, finding the roots of the transcendental dispersion equations is quite cumbersome and time consuming, especially when the number of layers is large. Also, addition of each layer involves a new equation and a new search. Additional complications arise when it is necessary to obtain not only the roots corresponding to the propagating modes, but also those corresponding to the evanescent (and nonpropagating) modes. The latter are needed to study scattering by defects in the plate or reflection of waves from the edges. They are also needed to express the Green’s function for the plate as a modal sum. For reasons of numerical efficacy and general applicability, an alternative procedure was developed by Datta et al. (1988c), Karunasena et al. (1991a, 1991b, 1991c), and Karunasena (1992) to model dispersion of guided waves in single- as well as many-layered plates. This was a stiffness method that was originally proposed by Dong and Nelson (1972). In this approach, each lamina was divided into several sublayers. The variation of the displacement through the thickness of each sub- layer was approximated by polynomials in a thickness variable with coefficients chosen such that the displacement (or displacement and traction) continuity was maintained at the interfaces between sublayers. Then, using Hamilton’s principle, the dispersion equation was obtained as a standard algebraic eigenvalue problem. Eigenvalues of this equation yielded the wave numbers (real and complex) corresponding to different fre- quencies for the guided modes. Corresponding eigenvectors were the displacements (or, displacements and tractions) at the nodes. Discussion of other numerical methods can be found in Liu and Xi (2002). In many applications, laminated composite plates are composed of periodic layers, where the layering is the repetition of unit cells, each cell being made up of uniaxial fiber-reinforced plies oriented in different in-plane directions. Lamb wave propagation in such a periodically laminated plate can be analyzed in terms of Floquet modes in an infinite medium having the same periodic structure. Dispersion of Lamb waves in such a periodic laminated plate was studied by Shull et al. (1994), and they found interesting features of mode clustering and gaps in the dispersion behavior. It was suggested that these unusual features were related to the Floquet wave pass and no-pass zones. In this paper, the stiffness method described above was used to obtain the dispersion curves. Safaeinili and Chimenti (1995) and Safaeinili et al. (1995) used the Floquet wave analy- sis to simplify significantly the solution to the dispersion equation for a multilayered plate. Wang and Rokhlin (2002b, 2002c) used the Floquet wave method to derive the homogenized properties of a multilayered cross-ply composite plate and for the deter- mination of the single-ply properties of a multidirectional composite. Homogenized effective properties of a multilayered cross-ply composite plate were derived earlier by Karunasena et al. (1991b) and Datta et al. (1992) using the stiffness method. 53388_C001.indd 4 10/8/08 10:45:22 AM © 2009 by Taylor & Francis Group, LLC
- 19. Introduction 5 Guided elastic waves in composite circular cylinders have many similarities with those in composite plates and have been studied analytically as well as numerically. Propagation of elastic waves in hollow circular cylinders has been the subject of exten- sive investigations in the past. Gazis (1959a, 1959b) presented exact three-dimensional solutions to the problem of waves in hollow isotropic cylinders. Extensive numerical results were presented in Gazis (1959b) for the flexural wave modes (see also Armenakas et al. 1969). Three-dimensional wave propagation in hollow cylinders was also studied by Greenspon (1959). McNiven et al. (1966a, 1966b) presented exact and approximate solutions for axisymmetric waves in hollow cylinders. Guided waves in a composite cir- cular cylindrical shell were studied by Armenakas (1967, 1971). An early study of longitudinal waves in a solid cylinder having transversely isotro- pic properties was published by Chree (1890), who derived solutions to the governing equations using a power-series method. Morse (1954) obtained the solutions in terms of Bessel functions for axisymmetric guided waves in solid, transversely isotropic cylin- ders. Einspruch and Truell (1959) also derived the dispersion equations for this case and for the torsional waves. Wave propagation in a hollow, transversely isotropic cylinder was studied by Mirsky (1965a, 1965b) using a displacement potential representation due to Buchwald (1961). Eliot and Mott (1968) studied solid and hollow cylinders having transversely isotropic symmetry. McNiven and Mengi (1971) analyzed in detail axisym- metric modes in transversely isotropic rods (including a uniaxial fiber-reinforced com- posite rod). Recently, Berliner and Solecki (1996a, 1996b) and Honarvar et al. (2007) have studied, respectively, dispersion of guided waves in fluid-loaded hollow and solid transversely isotropic cylinders. For a cylindrically orthotropic symmetry, Mirsky (1964a, 1964b) obtained the solu- tion to the axisymmetric motion of a cylindrical shell using an approximate theory. A similar approximate theory was also used by Mengi and McNiven (1971) to study axially symmetric waves in transversely isotropic rods. For axisymmetric waves in orthotropic cylinders, solutions were obtained by Mirsky (1964b), who used the Frobenius method. Chou and Achenbach (1972) and Armenakas and Reitz (1973) studied the flexural motion of orthotropic cylinders, also using the Frobenius method. Dispersion of guided waves in composite rods (having a solid core bonded to an outside shell, both isotropic) was studied by McNiven et al. (1963), Whittier and Jones (1967), Armenakas (1970), and Lai et al. (1971). Clad rods or wires have been inves- tigated for use as acoustic delay lines and fiber acoustic waveguides. There have been several investigations dealing with guided waves in isotropic clad rods (fibers). A sur- vey of the early literature was given by Thurston (1978). More recently, fiber acous- tic waveguides having isotropic properties have been investigated by Safaai-Jazi et al. (1986) and Jen et al. (1986) under the assumption of weak guidance and by Dai et al. (1992a, 1992b) for transversely isotropic materials. The equation governing disper- sion of guided waves in a clad cylinder is complicated, even when both materials are isotropic. They are much more complicated if the materials are anisotropic, as in the case of fiber-wound tubes. As mentioned above, for isotropic homogeneous cylinders, extensive analytical, numerical, and experimental research on vibration of guided waves has been reported since the early work of Pochhammer (1876). Early studies of dispersion of waves in elastic 53388_C001.indd 5 10/8/08 10:45:23 AM © 2009 by Taylor & Francis Group, LLC
- 20. 6 Elastic Waves in Composite Media and Structures circular cylinders include Onoe et al. (1962), Pao and Mindlin (1960), and Mindlin and McNiven (1960). Detailed accounts can be found in Meeker and Meitzler (1964), Achenbach (1973), Miklowitz (1978), Graff (1991), and Rose (1999). In the past, there were very few detailed studies of waves in composite cylinders of general anisotropy. Even though, for the transversely isotropic materials, it is possible to obtain analytical solutions, the dispersion equations for nonaxisymmetric motion are rather complicated, and solutions to these equations representing not only propagating waves, but also evanescent waves, are best accomplished by some efficient numerical schemes. Nelson et al. (1971) and Huang and Dong (1984) developed a stiffness method to study propagation of guided waves in laminated anisotropic cylinders with arbitrary lamina layup. This is a generalization of the stiffness method used for laminated plates (see the foregoing discussion). In this, the cylinder is discretized into coaxial cylinders, and radial variation is accounted for by an appropriate polynomial interpolation func- tion in each subcylinder (sublayer). As in the case of laminated plates, quadratic and cubic interpolation functions have been used, and both show excellent agreement with exact solutions when available. In regard to experimental work, the resonance method was used by Zemanek (1972) to verify theoretical predictions of dispersion of waves in a cylinder. Curtis (1982) reviewed wave-propagation techniques available for the determination of material properties of cylinders. Oblique insonification of circular cylinders immersed in fluid has been used to generate and measure guided waves in isotropic cylinders (Flax et al. 1980; Maze et al. 1985; Molinero and de Billy 1988; Li and Ueda 1989). Acoustic scattering by a transversely isotropic cylinder has been studied recently by Honarvar and Sinclair (1995) and Ahmad and Rahman (2000). Guided waves in a trans- versely isotropic cylinder in a fluid medium has been investigated by Nagy (1995) (see also Berliner and Solecki 1996a, 1996b; Ahmad 2001). Guided waves in a solid cylinder having a transversely isotropic core with an interface layer lying between the core and the outer layer were studied by Xu and Datta (1991). Nayfeh and Nagy (1996) analyzed axisymmetric waves in multilayered transversely isotropic cylinders. It may be noted that similar problems for isotropic cylinders were studied by Huang et al. (1995). Huang et al. (1996) investigated both theoretically and experimentally scattering by multilay- ered isotropic cylinders in fiber-reinforced composite media. Niklasson and Datta (1998) reported wave scattering and propagation in a transversely isotropic medium contain- ing a transversely isotropic cylinder. Xu and Datta (1991) used a hybrid method that combined finite-element representa- tion of the core and exact eigenfunction expansion for the isotropic outer cylinder. They also used the exact solution to obtain dispersion curves for axisymmetric motion of a transversely isotropic cylinder. Since the analytical formulation of wave propagation in a laminated cylinder with arbitrarily oriented fiber layups in the laminae is intrac- table, several approximate schemes have been proposed. The most common ones are the various shell theories in which the constitutive relations for the radially inhomogeneous cylinders are replaced by integral forms to reduce the problem to that of homogeneous equivalent cylinders (Tsai and Roy 1971; Sun and Whitney 1974). Other references to approximate shell theories can be found in the work by Barbero et al. (1990). The stiffness method used by Nelson et al. (1971) and Huang and Dong (1984) was generalized by Kohl 53388_C001.indd 6 10/8/08 10:45:23 AM © 2009 by Taylor & Francis Group, LLC
- 21. Introduction 7 et al. (1992a, 1992b), Rattanawangcharoen et al. (1992, 1994), and Rattanawangcharoen (1993) to study dispersive guided waves in laminated cylinders. 1.1.5 Cylinders of Arbitrary Cross Section As evidenced from the above review, the vast majority of the studies have been concerned with cylinders of circular cross sections. However, there have been a few studies that have dealt with cylinders of noncircular cross sections. In most of these studies, the material has been taken to be homogeneous and isotropic. For particular ratios of width to depth of a rectangular bar, Mindlin and Fox (1960) found a discrete set of points on the branches of the frequency spectra. Later, general solutions were presented by Morse (1948, 1950), Kynch (1957), Nigro (1966), Fraser (1969), Aalami (1973), Nagaya (1981), SeGi et al. (1994), and Taweel et al. (2000). Recently, attention has been given to composite and anisotropic materials that have applications to aerospace structures and quantum wires. Volovoi et al. (1998) considered beams of fiber-reinforced composite materials, and Nishiguchi et al. (1997), Mukdadi et al. (2002a, 2002b, 2005), and Mukdadi and Datta (2003) considered anisotropic and layered plates having rectangular as well as more general cross sections. 1.1.6 Particle-Reinforced Composites The above brief historical review of waves in composite media has been confined to lam- inated and fiber-reinforced composites. There have also been significant research studies on wave propagation in particle-reinforced media. Early studies of wave propagation in a two-phase medium were motivated by the need to model seismic wave velocities in rocks permeated by fluids. These include wave propagation in a fluid medium with a suspen- sion of spherical particles by Ament (1953) and elastic wave propagation in a medium with spherical inclusions by Yamakawa (1962) and Mal and Knopoff (1967). These were limited to low concentrations of particles and long wavelengths compared with the dimensions of the scatterers. For arbitrary concentrations, long-wavelength propaga- tion of elastic waves in a medium containing spherical inclusions has been studied by Waterman and Truell (1961), Mal and Bose (1974), and Berryman (1980a). In this limit, one obtains effective static elastic properties of an elastic medium containing spherical inclusions. In recent years, there have appeared several publications that contain various approximate theories for dispersion and attenuation of effective plane waves propagat- ing in an elastic medium containing spherical inclusions. Among these are Beltzer et al. (1983), Beltzer and Brauner (1986, 1987), Sabina and Willis (1988), Shindo et al. (1995), Kim et al. (1995), Yang (2003), and Aggelis et al. (2004). Compared with the large volume of work dealing with wave propagation in a medium containing spherical inclusions that has accumulated over the last 40 years, there are very few studies that have been concerned with the effect of inclusion shape on the effec- tive dynamic properties of particle-reinforced composites. Kuster and Toksöz (1974) and Berryman (1986) presented long-wavelength results when the inclusions were randomly orientedspheroids.LedbetterandDatta(1986)consideredbothalignedandrandomlyori- ented ellipsoids (see also Datta 1977). Recently, Sabina et al. (1993) and Smyshlayaev et al. (1993) have considered effective wave speeds and attenuation in a medium containing 53388_C001.indd 7 10/8/08 10:45:23 AM © 2009 by Taylor & Francis Group, LLC
- 22. 8 Elastic Waves in Composite Media and Structures aligned and randomly oriented spheroids. There are very few experimental investiga- tions of wave propagation in a particle-reinforced composite. Some of these are: Kinra et al. (1980, 1982), Kinra and Anand (1982), Ledbetter and Datta (1986, 2000), Datta and Ledbetter (1986a), and Ledbetter et al. (1984, 1995). In the above, we have mentioned major works dealing with waves in composite mate- rials having elastic constituents. However, anelastic effects have a large influence on wave speeds and attenuation. Wave propagation in a composite with viscoelastic constituents can be obtained essentially along the same lines if the elastic properties are treated as complex functions of frequency, and there have been many works in the literature. In this book, we will restrict our attention mostly to elastic composites, with one exception. This is the case of thermoelastic wave propagation in an anisotropic plate. 1.2 Scope of the Book In Chapter 2, we present the fundamental equations and their solutions for plane wave propagation in an infinite homogeneous anisotropic elastic material. Because composite materials are in most cases anisotropic, it is necessary to present this background mate- rial for the convenience of the readers and users of this book as well as for establishing notations that are used throughout this book. Chapter 3 deals with wave propagation in a periodically laminated infinite medium. Here, Floquet theory together with a stiffness method is presented. The stiffness method used is quite versatile and is applicable to multiple laminae in a cell having general anisotropic properties. Results are presented showing the anisotropic and dispersive characteristics of plane waves in such media. Wherever possible, results are compared with experiments and with predictions obtained by other modeling techniques. In Chapter 4, we discuss harmonic and transient guided (Lamb) wave propagation in a free-free multilayered fiber-reinforced laminated plate. The stiffness method discussed in Chapter 3 has been used to obtain frequency–wave-number (dispersion) relations for propagating and evanescent modes. A systematic investigation of the effects of different fiber layups and increasing number of laminae on the dispersion characteristics reveals features that are useful for experimental determination of lamina properties. For this study, each lamina has been treated as a transversely isotropic medium with effective properties that are determined by a wave-scattering theory (Datta and Ledbetter 1983; Datta et al. 1984). In addition to the stiffness method, this chapter also includes an exact analytical treatment of the guided-wave problem. The exact treatment leads to refine- ments of the modal frequencies and mode shapes that are needed to study transient wave propagation and scattering by defects. This chapter also includes some results of com- parison between model predictions and experimental observations for a particular case. In addition, the dynamic response of a composite plate to applied external forces and to laser thermal excitation is analyzed. Both time-harmonic and transient waves are treated analytically as well as numerically. The stiffness method that has been dis- cussed thoroughly in this chapter for the isothermal case is generalized to treat the cou- pled problem of thermoelastic waves. In this case, the thermoelasticity theory, which includes a relaxation time in the heat-conduction equation, has been used. This makes the thermal transport equation hyperbolic. Because of thermal-diffusion effects, the 53388_C001.indd 8 10/8/08 10:45:23 AM © 2009 by Taylor & Francis Group, LLC
- 23. Introduction 9 frequency–wave-number relation is now complex, and the thermal waves are found to have high attenuation. However, the primarily elastic waves suffer small attenuation. Guided wave propagation in composite cylinders and cylindrical shells is the subject of Chapter 5. Here again, as in the case of plates, exact solutions are presented for circu- lar cylinders composed of transversely isotropic materials with the symmetry axis par- allel to the axis of the cylinder. Also, stiffness methods (similar to that used for plates), in which the cylinder is subdivided into a number of coaxial cylinders and radial interpo- lation functions, are used to approximate radial variations of displacements (and trac- tions) to obtain the dispersion relation showing the frequency–wave-number behavior of wave propagation along the axis. Cylinders of noncircular cross sections composed of homogeneous and layered anisotropic materials are also considered in this chapter. In the stiffness method that is employed here, the finite element discretization is used to approximate the displace- ment (and traction) over the cross section. Whereas shape functions involve one spatial (radial) variable for circular cylinders, they involve two spatial variables in the plane of noncircular cross sections. Scattering of guided waves by cracks in composite plates and cylinders is treated in Chapter 6. A hybrid method that combines the finite element discretization of a finite region containing the cracks or joints and wave function expansion of the fields in the exterior regions is used. Continuity of displacements and tractions at the (artificial) boundaries between the finite region and the exterior semi-infinite regions is enforced to obtain the coefficients of the modal sum as well as the nodal displacements in the finite region. The second is a combined boundary integral and finite element method in which the defects are again enclosed in a finite region, which is modeled by finite ele- ments. The wave field exterior to the finite region is represented by a boundary integral using the Green’s functions for the composite plate. Both methods are shown to give convergent results. Comparison with some available experimental results shows good agreement between the model results and observations. In addition, a boundary integral method combined with a multidomain decomposition is presented for the analysis of scattering in a composite plate. 53388_C001.indd 9 10/8/08 10:45:23 AM © 2009 by Taylor & Francis Group, LLC
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- 25. 11 2 Fundamentals of Elastic Waves in Anisotropic Media 2.1 Waves in Homogeneous Elastic Media: Formulation of Field Equations......................................................................11 Deformation • Balance Laws • Constitutive Equations • Coupled Equations of Linear Dynamic Thermoelasticity 2.2 Plane Waves in a Homogeneous Anisotropic Medium...... 20 Slowness Surface • Energy Transport • Group Velocity • Special Cases • Transformation of Coordinates 2.3 Numerical Results and Discussion.........................................33 2.1 Waves in Homogeneous Elastic Media: Formulation of Field Equations In this section, we will be concerned with the field equations governing thermoelastic time-dependent deformation of homogeneous elastic media. A brief derivation of the linearized equations of motion is presented first. The readers are referred to many excel- lent textbooks on this subject, e.g., Sokolnikoff (1956), Mal and Singh (1991), and many others. 2.1.1 Deformation An elastic medium has a natural undeformed state in the absence of any external or internal mechanical or other (thermal, electromechanical) sources of disturbance acting on it. We will choose this as the reference state of the body in which a particle in the body is located at a point x. Here, x is the position vector of the particle and can be written as x = x x x 1 2 3 e e e 1 2 3 + + (2.1.1) where x1, x2, x3 are the components of the vector x referred to a fixed Cartesian coor- dinates system and e1, e2, e3 are the unit vectors along the 1-, 2-, 3-axes, respectively 53388_C002.indd 11 10/8/08 11:04:38 AM © 2009 by Taylor & Francis Group, LLC
- 26. 12 Elastic Waves in Composite Media and Structures (Fig. 2.1). Using the summation convention, equation (2.1.1) can be written as x e = = 1, 2, 3 x i i i , (2.1.2) When a source of disturbance acts on the body to change the reference state to a deformed state, the deformation is measured by the displacements of the particles in the body. Let X denote the new position of the particle that was at x in the reference state initially. The displacement u of the particle is given by u X x = - (2.1.3) The new position of the particle, X, is a function of x and time t (observation time). In general, u can be large and the deformation will be nonlinear. Here, we will restrict ourselves to small deformations such that the spatial gradients of u are much smaller than unity. Let x + dx be the position of a particle in a neighborhood of the particle at x. In the deformed position, the position of this particle may be denoted by X + dX. The length of the line element dx is ds dx dx i i = (2.1.4) In the deformed state, its length is dS dX dX i i = (2.1.5) Thus, the change in length of the line element dx is dS ds dX dX dx dx X i i i i k 2 2 - = - = ∂ ∂ ∂ ∂ ∂ - x X x dx dx i k j ij i j δ x1 x2 x3 e1 e2 e3 Figure 2.1 Reference coordinates in the undeformed state. 53388_C002.indd 12 10/8/08 11:04:40 AM © 2009 by Taylor & Francis Group, LLC
- 27. Fundamentals of Elastic Waves in Anisotropic Media 13 This relation can be written as dS ds E dx dx ij i j 2 2 2 - = (2.1.6) where Eij is the Lagrangian or the Green strain tensor and is given by E X x X x ij k i k j ij = ∂ ∂ ∂ ∂ - 1 2 δ (2.1.7) Note that equation (2.1.6) can be written in the alternative form dS ds dX dX x X x X dX dX i i k i k j i j 2 2 - = - ∂ ∂ ∂ ∂ = 2e dX dX ij i j (2.1.8) where the Eulerian or Alamansi strain tensor eij is e x X x X ij ij k i k j = - ∂ ∂ ∂ ∂ 1 2 δ (2.1.9) Using equation (2.1.3) in (2.1.7) gives Eij = 1 2 ∂ ∂ + ∂ ∂ + ∂ ∂ ∂ ∂ u x u x u x u x i j j i k i k j (2.1.10) As mentioned above, the displacement gradients will be assumed to be small. So, keeping only the first-order terms in equation (2.1.10), we obtain E u x u x ij i j j i ≈ ∂ ∂ + ∂ ∂ 1 2 (2.1.11) Similarly, using equation (2.1.3) in (2.1.9) and keeping only first-order terms, there results e u X u X ij i j j i ≈ ∂ ∂ + ∂ ∂ 1 2 (2.1.12) It is easily shown that, to the first order of approximation, the linearized strain tensor is εij ij ij i j j i e E u x u x ≈ ≈ = ∂ ∂ + ∂ ∂ 1 2 (2.1.13) 53388_C002.indd 13 10/8/08 11:04:43 AM © 2009 by Taylor & Francis Group, LLC
- 28. 14 Elastic Waves in Composite Media and Structures Thus, for small deformations in which the displacement gradients are small, the infini- tesimal strain tensor is the same in both Eulerian and Lagrangian descriptions and may be written as e t u x u x ij i j j i ( , x ) = 1 2 ∂ ∂ + ∂ ∂ (2.1.14) In the rest of this book, we will use the following equation to express the linearized strain tensor as e u u ij i j j i = + 1 2 ( ) , , (2.1.15) where a comma denotes derivative with respect to a coordinate xi (i = 1, 2, 3). It would be convenient to introduce the infinitesimal rotation tensor as ωij i j j i u u = - 1 2 ( ) , , (2.1.16) The velocity of a point X is v X = ∂ ∂t (2.1.17) and the acceleration is a v X v x v X X v = ) = , ) = D t Dt t t t t i i ( , ( ∂ ∂ + ∂ ∂ ∂ ∂ ∂ ∂ + + ∂ ∂ vi i v X (2.1.18) Now, keeping only the first-order terms, we obtain from equations (2.1.17) and (2.1.18) v u u = = ∂ ∂t � (2.1.19) and a v v = = ∂ ∂t � (2.1.20) where the superimposed dot (.) denotes a derivative with respect to time. 2.1.2 Balance Laws In this section, we develop the equations governing conservation of mass, linear and angular momenta, and energy. First, let us consider the principle of conservation of mass. For this purpose, let us denote the density of the material at (a) X at time t in the 53388_C002.indd 14 10/8/08 11:04:45 AM © 2009 by Taylor & Francis Group, LLC
- 29. Fundamentals of Elastic Waves in Anisotropic Media 15 deformed state by r(X, t) and (b) x and initial time t0 at the reference state by r0(x, t0). Let dV be a volume element containing the point X, with dV0 as its initial value. Then, by the principle of conservation of mass, we have rdV = r0dV0 (2.1.21a) Because dV = JdV0, where J is the Jacobian matrix of transformation from x to X, equation (2.1.21a) gives rJ = r0 (2.1.21b) Differentiating equation (2.1.21b) with respect to t keeping x fixed, we find D J Dt ( ) ρ = 0 or ρ ρ DJ Dt J D Dt + = 0 (2.1.22) Using the property DJ/Dt = Jvk,k, one obtains from equation (2.1.22) the equation of continuity D Dt vk k ρ ρ + = , 0 (2.1.23) Note that, correct to the first order, we have 1/J ≈ 1 - uk,k So, equation (2.1.21) gives r = r0 (1 - uk,k) (2.1.24) Next we will consider the principle of balance of linear momentum. Here, we will be concerned with mechanical forces acting on the interior of the body B as well as on its boundary, ∂B, causing it to deform. Let V be a part of the solid enclosed by a surface S in the deformed state at time t. There are forces acting on S caused by the action of the material outside S on that within V. Consider a plane element dS of the surface S. Let n be a unit normal to dS pointing outward. It is postulated that the elementary force of action on dS by the material outside S on that in the interior is tdS and that the elemen- tary moment of action on dS is zero. The force t is called the traction. It depends on n and the position Y of dS. The force t has the properties: (a) t(n, Y) = −t(−n, Y) and (b) t(n, Y) = sijniej. The tensor s is the Cauchy stress tensor. Now, the principle of balance of linear momentum can be stated in the form of the following equation: ρ ρ ρ f t v a dV dS d dt dV dV V S V V ∫ ∫ ∫ ∫ + = = (2.1.25) 53388_C002.indd 15 10/8/08 11:04:47 AM © 2009 by Taylor & Francis Group, LLC
- 30. 16 Elastic Waves in Composite Media and Structures Here f is the body force per unit mass. In the absence of any distributed moments in V, the principle of balance of moment of momentum can be stated as X f Y t X v X a ∧ + ∧ = ∧ = ∧ ∫ ∫ ∫ ∫ ρ ρ ρ V S V V dV dS d dt dV dV (2.1.26) In the component form, equations (2.1.25) and (2.1.26) can be written as ρ σ ρ f dV n dS a dV i j i V ji j S i V ∫ ∫ ∫ + = = ( , , , ) 1 2 3 (2.1.27) ε ρ ε σ ε ρ ijk j k V ijk j lk l S ijk j k V X f dV Y n dS X a dV ∫ ∫ ∫ + = (i, j, k = 1, 2, 3) (2.1.28) Using the divergence theorem, the surface integrals appearing in equations (2.1.27) and (2.1.28) can be written as σ σ ε σ ε ji j S ji j V ijk j lk l S ijk n dS dV Y n dS ∫ ∫ ∫ = + , ( ( ) ( ) , , X dV X dV j lk l V ijk lk jl j lk l V σ ε σ δ σ ∫ ∫ = + Thus, equations (2.1.27) and (2.1.28) can be rewritten respectively in the forms ( ) , ρ σ ρ f a dV i ji j i V + - = ∫ 0 (2.1.29) and ε ρ σ ρ σ ijk j V k lk l k jk X f a dV [ ( ) ] , ∫ + - + = 0 (2.1.30) Since equations (2.1.29) and (2.1.30) must hold for any arbitrary volume V of the body in the deformed state, it follows that the integrands must vanish everywhere within the body. Hence, we obtain Cauchy’s equations of motion from equation (2.1.29) as ρ σ ρ f a i ji j i + = , (2.1.31) Using equation (2.1.31), it is found from equation (2.1.30) that εijksjk = 0 This result implies that sjk = skj (2.1.32) 53388_C002.indd 16 10/8/08 11:04:49 AM © 2009 by Taylor & Francis Group, LLC
- 31. Fundamentals of Elastic Waves in Anisotropic Media 17 i.e., the stress tensor is symmetric. Note that for small deformation, the density r in equation (2.1.31) can be taken as the initial density, and ai = üi. Finally, we will give a brief outline of the principle of conservation of energy and derive the equations of thermoelasticity. We will consider thermal energy as well as mechanical energy. According to the first law of thermodynamics, we have d dt v v U dV f v dV t v dS i i V i i V i i S ρ ρ ρ 1 2 + = + + ∫ ∫ ∫ h hdV q n dS V i i S ∫ ∫ - (2.1.33) Here U is the internal energy and h is the rate of heat generation within V, both with respect to unit mass, and q is the heat-flux vector representing the rate of transfer of heat across S. The left-hand side of equation (2.1.33) represents the rate of change of kinetic and internal energies of the material occupying V. The right-hand side is the sum of the rate of work done by the body forces acting on V and the traction acting on S, and the rate of change of heat energy in V. Using the divergence theorem, the surface integrals appearing in equation (2.1.33) can be converted into volume integrals over V. After some obvious algebraic manipula- tion, equation (2.1.33) takes the form [ ( ) ( ) ] , , , v v f U h v q i i V i ij j ij i j i i ρ ρ σ ρ σ � � - - + - - + ∫ d dV = 0 (2.1.34) Using the equation of motion (2.1.31), it is found from equation (2.1.34) that ρ σ ( ) , , � U h v q ij i j i i - - + = 0 (2.1.35) Now, we will apply the second law of thermodynamics. We assume that s is the entropy per unit mass at (x,t). Then, the Clausius–Duhem inequality can be stated as ρ ρ � sdV h T dV q n T dS i i S V V ≥ - ∫ ∫ ∫ (2.1.36) Here T is the absolute temperature. Applying the divergence theorem to the surface integral in the above equation, we find ρ ρ � s q T h T i i + - ≥ , 0 (2.1.37) Using equation (2.1.35), the h appearing in equation (2.1.37) can be eliminated, and equation (2.1.37) can be rewritten as ρ σ ( / ) / , , � � s U T v q T T T ij i j i i - + - ≥ 0 (2.1.38) 53388_C002.indd 17 10/8/08 11:04:51 AM © 2009 by Taylor & Francis Group, LLC
- 32. 18 Elastic Waves in Composite Media and Structures 2.1.3 Constitutive Equations The Helmholtz free energy is defined by the equation F U Ts = - ρ( ) (2.1.39) For an elastic material, F is a single-valued function of the strain tensor, eij, the absolute tem- perature T, and its gradient, T,i. The energy equation (2.1.35) can be expressed in the form � F d Ts dt h v q ij i j i i + - - + = ρ σ ( ) , , 0 (2.1.40) The inequality in equation (2.1.38) can be written as ( / ) ( ) , , σ ρ ij i j i i v qT T F sT - - + ≥ � � 0 (2.1.41) Furthermore, since � � � � F F e e F T T F T T ij ij i i = ∂ ∂ + ∂ ∂ + ∂ ∂ , , , the inequality in equation (2.1.41) takes the form σ ρ ij ij i j i F e v s F T T F T - ∂ ∂ - + ∂ ∂ - ∂ ∂ , , � � � T q T T i i i , , - ≥ 0 (2.1.42) Note that the dependent variables sij, s, and F are determined by the values of the inde- pendent variables eij, T, and T,i. On the other hand, their rates given by vi,j, � T, and � Ti , can be chosen arbitrarily so that the inequality will be violated for every possible choice of the independent variables. Thus, for the inequality to hold for all choices of the independent variables, the coefficients of vi,j, � T, and � Ti , must vanish. We then have the equations: σij ij F e = ∂ ∂ (2.1.43) s F T = - ∂ ∂ 1 ρ (2.1.44) 0 = ∂ ∂ F Ti , (2.1.45) The inequality in equation (2.1.42) then becomes qiT,i ≤ 0 (2.1.46) Using equations (2.1.43)–(2.1.45) in the energy equation (2.1.40), we obtain ρ( ) , � sT h qi i 0 0 - + = (2.1.47) 53388_C002.indd 18 10/8/08 11:04:55 AM © 2009 by Taylor & Francis Group, LLC
- 33. Fundamentals of Elastic Waves in Anisotropic Media 19 In writing equation (2.1.47), we have replaced T by the initial temperature T0 because s, h, and q are small. In the linearized approximation, F can be expanded in the form, keeping terms up to the second order F c e e e c T ijkl ij kl ij ij = - - 1 2 1 2 2 0 β θ ρ θ (2.1.48) where q = T – T0. Here cijkl is a constant fourth-order tensor (the elasticity tensor), βij is a constant second-rank symmetric tensor (the thermoelasticity tensor), and c is the spe- cific heat at constant deformation. It is assumed that q is much smaller than T0. Using equation (2.1.48) in (2.1.43) and (2.1.44), we obtain the constitutive relations for linear thermoelasticity as σ β θ ij ijkl kl ij c e = - (2.1.49) ρ β ρ θ s e c T ij ij = + 0 (2.1.50) Using equation (2.1.50), the energy equation (2.1.47) can then be written as q T v c h i i ij i j , , = - - + β ρ θ ρ 0 � (2.1.51) Because sij and eij are symmetric, it follows that the elasticity tensor, cijkl, has the follow- ing symmetry properties c c c c ijkl jikl ijlk klij = = = (2.1.52) In the usual coupled theory of thermoelasticity, qi is assumed to be a linear function of q,i, in the form q k i ij j = - θ, (2.1.53a) where kij is the thermal conductivity tensor. The condition given in equation (2.1.46) implies that kij is positive-definite symmetric. Substitution of this in equation (2.1.51) leads to the parabolic equation for heat conduction, giving the speed of travel for the thermal disturbance to be infinite. This is not physically realistic, and several theo- ries have been proposed to remove this paradox. Among these are Lord–Shulman (LS), Green–Lindsay (GL), and other theories. The reader is referred to the paper by Chandrasekharaiah (1986) for a thorough review of the literature. The simplest of these is the LS theory, in which a relaxation time τ0 (time lag needed to establish steady-state heat conduction in an element of volume when a temperature gradient is suddenly applied on the volume) is introduced in equation (2.1.53a) (Lord and Shulman 1967). In this case, the modified form of equation (2.1.53a) is 1 0 + ∂ ∂ = - τ θ t q k i ij j (2.1.53b) 53388_C002.indd 19 10/8/08 11:04:58 AM © 2009 by Taylor & Francis Group, LLC
- 34. 20 Elastic Waves in Composite Media and Structures 2.1.4 Coupled Equations of Linear Dynamic Thermoelasticity Substitution of equation (2.1.53b) in equation (2.1.51) yields the modified equation of heat transport as k t c T u h ij ij ij i j θ τ ρ θ β ρ , , [ ] = + ∂ ∂ + - 1 0 0 � � (2.1.54) Since kij is positive definite, this equation is hyperbolic, provided that t0 is positive. Substitution of the expression for sij given by equation (2.1.49) in the equation of motion (2.1.31) yields c u u f ijkl k lj i ij j i , , = + - ρ β θ ρ �� (2.1.55) Equations (2.1.54) and (2.1.55) constitute the coupled equations for linear thermoelas- todynamics of a homogeneous anisotropic elastic solid including a thermal relaxation time. These form a system of four coupled equations of the hyperbolic type. The solutions to this system—satisfying appropriate initial and boundary conditions for the body B occupying the volume V and bounded by the surface S—describe the thermoelastody- namic state of B. Under conditions of isothermal or adiabatic deformation of a homogeneous anisotro- pic elastic medium, one needs to consider equation (2.1.55) after omitting the term on the right-hand side involving the temperature gradient. Then, we obtain the displace- ment equations of elasticity as c u u f ijkl k lj i i , = - ρ ρ �� (2.1.56) The stress, strain, and temperature relation in equation (2.1.49) becomes σij ijkl kl c e = (2.1.57) In the following section, we will consider solutions of equation (2.1.56) in the form of plane waves when the body forces are absent, i.e., fi = 0. 2.2 Plane Waves in a Homogeneous Anisotropic Medium In this section, we will derive the equations and the characteristics of their solutions for plane waves that are governed by equation (2.1.56) when the body force f is absent. For this purpose, it will be assumed that the time dependence of u(x,t) is simple harmonic of the form e−iwt. Thus, u(x,t) is written as u x u x ( , ) = ( ) t e i t - ω (2.2.1) where w is the circular frequency and i = √−1. Then, equation (2.1.56) takes the form c U U ijkl k lj i , = -ρω2 (2.2.2) 53388_C002.indd 20 10/8/08 11:05:00 AM © 2009 by Taylor & Francis Group, LLC
- 35. Fundamentals of Elastic Waves in Anisotropic Media 21 We will seek a solution of equation (2.2.2) of the form U A e i i iknjx j = (2.2.3) where k is the wave number, n is the unit vector representing the direction of propaga- tion of the plane wave, and A is a constant vector amplitude of the wave. Substitution of equation (2.2.3) in (2.2.2) gives rv2Ai = cijklnlnj Ak (2.2.4) where v = w/k is the phase velocity of the plane harmonic wave. Equation (2.2.4) is the Christoffel equation for the determination of v for a wave of phase k propagating in the direction n and its amplitude A. Equation (2.2.4) represents three linear homogeneous equations in the three unknown components of A. For a nontrivial solution (A ≠ 0) of equation (2.2.4), rv2 must be an eigenvalue and A is a corresponding eigenvector of equation (2.2.4). Note that A defines the direction of the particle velocity at x. The characteristic equation governing the eigenvalues and the eigenvectors is Ω( , , , ) v n n n c n n v ijkl j l ik 1 2 3 2 0 = - = ρ δ (2.2.5) Here, |…| denotes a 3 × 3 determinant and dik is the Kronecker delta. Equation (2.2.5) is cubic in v2. Thus, there are three possible values of v2 for each n. Since the Christoffel acoustic tensor, Γik = cijklnjnl, has the property that it is symmetric (Γik = Γki) and positive definite, it follows that the eigenvalues rv2 are positive. Thus, the phase velocities ±v are real (∵ r > 0), with +v signifying a wave moving outward (away from the origin) and –v moving inward. The expression v(n) for all possible directions n traces out a surface of three sheets, called the “velocity surface.” 2.2.1 Slowness Surface Equation (2.2.5) can be written in an alternative form in terms of 1/v as 1 0 2 v ij ij Γ - = ρδ (2.2.6) Solution of this equation for v−1 as a function of n, v−1(n) traces out a surface having three sheets called the “slowness surface.” Once v2 has been solved from equation (2.2.5), the particle displacement vector A is found from the matrix equation [ ][ ] Γik ik k v A - = ρ δ 2 0 (2.2.7) where the first matrix in the above equation is 3 × 3 and the second is 3 × 1. Since this equation is homogeneous, A can be found except for a constant multiplying factor. Now, it is easily shown that if vM and vN are two different eigenvalues, then the corresponding eigenvectors AM and AN are orthogonal to each other. 53388_C002.indd 21 10/8/08 11:05:01 AM © 2009 by Taylor & Francis Group, LLC
- 36. 22 Elastic Waves in Composite Media and Structures 2.2.2 Energy Transport For the complex exponential form of the displacement given by equations (2.2.1) and (2.2.3), the stress tensor sij takes the form σ ω ij l k ijkl i klxl t ik A c e = - ( ) (2.2.8) The strain tensor eij is given by e i k A k A e ij i j j i i klxl t = + - 2 ( ) ( ) ω (2.2.9) Here we have written kl = knl. Now, the flux of energy across a small element of surface due to the propagating plane wave can be calculated in the following manner. We have, from equation (2.1.56), the time-averaged power flow per unit area as P = - 1 2 σij j i n v* (2.2.10) where * denotes a complex conjugate. Using equations (2.2.1) and (2.2.8) in the above equation, we obtain P = 1 2 2 ρω vA A k k (2.2.11) The average power flow vector (Poynting vector) (Auld 1990) is given by P e = - = 1 2 1 2 2 〈 〉 v v c A A n i ij ijkl i k l j *σ ω (2.2.12) Thus, we have the relation n P . = P (2.2.13) The energy velocity is defined as V P P e = = 〈 〉 e 1 2 2 ρω A A k k (2.2.14) where 〈E〉 is the average total energy density. It is seen that n.Ve = v (2.2.15) 2.2.3 Group Velocity The group velocity of the plane wave is given by Auld (1990) Vg = ∇ = ∂ ∂ = ∂ ∂ k i i i i k v n ω ω e e (2.2.16) 53388_C002.indd 22 10/8/08 11:05:04 AM © 2009 by Taylor & Francis Group, LLC
- 37. Fundamentals of Elastic Waves in Anisotropic Media 23 where we have used the relations v = w/k and k = kn. Now, using equation (2.2.7), we get ρv A A c n n A A i i ikjl k l i j 2 = (2.2.17) Differentiating both sides of this equation with respect to nm, we obtain ∂ ∂ = v n v c n A A A A m imjl l i j i i 1 ρ (2.2.18) Thus, it is found that V e g = 1 ρv c n A A A A imjl l i j i i m (2.2.19) Comparison of this equation with (2.2.14) shows the important identity Vg = Ve (2.2.20) Now, equation (2.2.15) can be written as dv dn i i = ( ) Vg (2.2.21) Also, the slowness vector is given by s n = v (2.2.22) Taking the differential of both sides, we get vds + dvs = dn (2.2.23) Taking the “dot” product of this with Vg gives, using equations (2.2.21) and (2.2.22), Vg.ds = 0 (2.2.24) This shows that the group velocity vector (or the energy velocity vector) is normal to the slowness surface. 2.2.4 Special Cases In this section, we will consider simplifications that occur when the material possesses some symmetry properties. For convenience of subsequent discussion the four-suffix elements of the stiffness tensor cijkl will be represented by two-suffix elements of a 6 × 6 matrix. This is done by the following identification scheme: 11 1 → → → → → → , 22 2, 33 3 23,32 4, 31,13 5, 12,21 6 (2.2.25) 53388_C002.indd 23 10/8/08 11:05:06 AM © 2009 by Taylor & Francis Group, LLC
- 38. 24 Elastic Waves in Composite Media and Structures The same contracted notation will be used to write the six components of the stress in the form of a 6 × 1 matrix. Also, we will use the engineering shear strain components defined as γ γ γ 12 12 23 23 31 31 2 2 2 = = = e e e , , (2.2.26) The stress strain relation in equation (2.1.57) can then be written in the matrix form σ σ σ σ σ σ 1 2 3 4 5 6 11 1 = c c 2 2 13 14 15 16 22 23 2 c c c c c c c 4 4 25 26 33 34 35 c c c c c symm c c c c 36 44 45 46 c c 55 56 c66 e e e 1 2 3 4 5 6 γ γ γ (2.2.27) This equation can be written in the matrix notation as [ ] [ ][ ] σ = c e (2.2.28) where [s] is a 6 × 1 matrix, [c] is 6 × 6, and [e] is the second matrix (6 × 1) on the right. The elements of the Christoffel acoustic matrix [Γ] can be written in the contracted notation as Γ11 11 1 2 66 2 2 55 3 2 16 1 2 2 2 = + + + + c n c n c n c n n c c n n c n n 15 1 3 56 2 3 2 + Γ12 16 1 2 26 2 2 45 3 2 12 66 = + + + + c n c n c n c c n ( ) 1 1 2 14 56 1 3 46 25 2 3 n c c n n c c n n + + + + ( ) ( ) Γ13 15 1 2 46 2 2 35 3 2 14 56 = + + + + c n c n c n c c n ( ) 1 1 2 13 55 1 3 36 45 2 3 n c c n n c c n n + + + + ( ) ( ) (2.2.29) Γ22 66 1 2 22 2 2 44 3 2 26 1 2 2 2 = + + + + c n c n c n c n n c c n n c n n 46 1 3 24 2 3 2 + Γ23 56 1 2 24 2 2 34 3 2 46 25 = + + + + c n c n c n c c n ( ) 1 1 2 36 45 1 3 23 44 2 3 n c c n n c c n n + + + + ( ) ( ) Γ33 55 1 2 44 2 2 33 3 2 45 1 2 2 2 = + + + + c n c n c n c n n c c n n c n n 35 1 3 34 2 3 2 + The Christoffel acoustic matrix [Γ] simplifies considerably if the material has a high level of symmetry. In the following, we will develop the equations for some special cases of symmetry. 53388_C002.indd 24 10/8/08 11:05:09 AM © 2009 by Taylor & Francis Group, LLC
- 39. Fundamentals of Elastic Waves in Anisotropic Media 25 2.2.4.1 Isotropic Elastic Material If the material is isotropic, then there are no preferred directions in the material, and the elastic stiffness constants are the same independent of the choice of the coordinate axes in which the stress and strain tensor components are expressed. It can be shown that the stiffness tensor cijkl has the form cijkl ij kl ik jl il jk = + + λδ δ µ δ δ δ δ ( ) (2.2.30) where l and m are known as the Lamé’s constants. Thus, the stress strain relation in equation (2.1.57) takes the form σ λ δ µ ij kk ij ij e e = + 2 (2.2.31) The Lamé’s constants l, m have the properties 0 < 3l + 2m < ∞; 0 < m < ∞ (2.2.32) In the matrix notation, the stiffness matrix [c] takes the form c = + + + λ µ λ λ λ µ λ λ µ µ µ 2 0 0 0 2 0 0 0 2 0 0 0 0 0 symmetric 0 0 µ (2.2.33) The Christoffel matrix elements are now given by Γ Γ Γ Γ 11 1 2 22 2 2 33 3 2 = + + = + + = + + ( ) ( ) ( ) λ µ µ λ µ µ λ µ µ n n n i ij i j n n i j = + ≠ ( ) , λ µ (2.2.34) Then, equation (2.2.4) becomes (rv2 − m)Ai = (l + m)ninkAk (2.2.35) This equation implies that either (a) Ai = Ani, i.e., A is parallel to n or (b) Aknk = 0, i.e., A is perpendicular to n. In case (a), we get rv2 = (l + 2m), and in case (b) we find rv2 = m. The wave is pure longitudinal in case (a) and it is pure shear in case (b). The three sheets 53388_C002.indd 25 10/8/08 11:05:10 AM © 2009 by Taylor & Francis Group, LLC
- 40. 26 Elastic Waves in Composite Media and Structures of the velocity surface are spherical in this case. The two sheets corresponding to the shear waves are coincident. The average power flow vector is obtained by using equation (2.2.30) in (2.2.12) P A n = + + 1 2 2 2 ω λ µ µ v A n A l l [( ) ] (2.2.36) Now, for the longitudinal wave, A = An and v = vl = λ µ ρ + 2 . Then, equation (2.2.36) gives P n = 1 2 2 2 ρω v A l (2.2.37) Using equation (2.2.14), we find that the energy velocity Ve is given by Ve = vln (2.2.38) On the other hand, for the shear wave, Alnl = 0 and v = vs = µ ρ . So, P is given by P = 1 2 2 2 ρω v A s n (2.2.39) The energy velocity is then found to be Ve = vsn (2.2.40) Thus, in an isotropic homogeneous medium, the energy (or group) velocity associated with a plane wave is coincident with its phase velocity. 2.2.4.2 Transversely Isotropic Material A major emphasis of this book is on wave propagation in an infinite or finite elastic medium having a laminated structure, where each lamina may be homogeneous isotro- pic or is a composite made up of a homogeneous matrix reinforced by aligned fibers. In the former case, the laminated medium can be characterized as a homogeneous aniso- tropic medium (in the long-wavelength limit) having an axis of elastic symmetry that is perpendicular to the plane of each lamina. In the latter instance, each lamina may be treated as homogeneous anisotropic having the axis of symmetry parallel to the fiber axes. Such an anisotropic medium is called transversely isotropic. Planes perpendicular to the axis of symmetry (unique axis) are isotropic. If the x1-axis is taken parallel to the unique axis, then the stiffness matrix [c] takes the form c c c c c = 11 12 12 22 0 0 0 c c 23 22 0 0 0 0 0 0 0 0 0 44 symmetric c c66 0 c66 (2.2.41) 53388_C002.indd 26 10/8/08 11:05:12 AM © 2009 by Taylor & Francis Group, LLC
- 41. Fundamentals of Elastic Waves in Anisotropic Media 27 Here c44 = 1 2 22 23 ( ) c c - . In this case, there are five independent elastic constants. For an orthotropic material with the 12-, 23-, and 31-planes as symmetry planes, the Christoffel matrix can be written as Γ Γ Γ 11 11 1 2 66 2 2 55 3 2 12 12 66 1 2 1 = + + = + c n c n c n c c n n ( ) 3 3 13 55 1 3 22 66 1 2 22 2 2 44 3 2 23 = + = + + ( ) c c n n c n c n c n Γ Γ = = + = + + ( ) c c n n c n c n c n 44 23 2 3 33 55 1 2 44 2 2 33 3 2 Γ (2.2.42) For transverse isotropy, we have c12 = c13, c22 = c33, and c55 = c66, and c44 is given by the relation given above. In the case of transverse isotropy, since the x1-axis is the axis of symmetry, the x1–x3- plane can be chosen as the plane containing the direction of propagation of the plane wave. In that case, n2 = 0. Then, equation (2.2.7) takes the form Γ Γ 11 2 13 0 0 - ρv Γ Γ Γ 22 2 13 3 0 0 - ρv 3 3 2 1 2 3 0 - = ρv A A A (2.2.43) where Γ Γ Γ 11 11 1 2 55 3 2 33 55 1 2 33 3 2 22 = + = + = c n c n c n c n c , , 6 66 1 2 44 3 2 13 13 55 1 3 n c n c c n n + = + , ( ) and Γ It follows from equation (2.2.43) that, of the three sheets of the velocity surface, there is one with displacement polarized perpendicular to the plane of propagation contain- ing the axis of symmetry. This is a pure shear motion with the phase velocity given by v c n c n 3 66 1 2 44 3 2 = + ρ (2.2.44) The other two sheets are traced out by the velocities v1 and v2, where v1 2 and v2 2 are the roots of the quadratic equation Ω Γ ( , , ) ( ) ( ) ω ρ ρ k k v v c n c n c 1 3 2 2 2 11 1 2 33 3 2 55 1 = - + + + 1 1 33 13 2 0 Γ Γ - = (2.2.45) The velocities of the two waves with displacement polarizations in the plane of x1–x3 are then given by ρv c n c n c D 1 2 11 1 2 33 3 2 55 1 2 = + + + (2.2.46) 53388_C002.indd 27 10/8/08 11:05:15 AM © 2009 by Taylor & Francis Group, LLC
- 42. 28 Elastic Waves in Composite Media and Structures ρv c n c n c D 2 2 11 1 2 33 3 2 55 1 2 = + + - (2.2.47) where D c c n c c n c c n = - + - + + [( ) ( ) ] ( ) 11 55 1 2 55 33 3 2 2 13 55 2 4 1 1 2 3 2 n (2.2.48) Note that D > 0 and ρ( ) v v D 1 2 2 2 0 - = > . The wave moving with the larger of the two velocities, v1 = vqL, is called quasi-longitudinal, and the slower wave with the velocity, v2 = vqS, is called quasi-shear. They are neither purely longitudinal nor purely shear because the displacements associated with them are not parallel or perpendicular to the velocity direction n. Now, to obtain the group velocities of the three waves, we note that, for the shear wave polarized in the x2 direction, we get from equation (2.2.43) ΩSH( , , ) ( ) ω ρω k k c k c k 1 3 66 1 2 44 3 2 2 0 = + - = (2.2.49) So, it is found that ( ) Vg SH 1 1 66 3 = n c v ρ (2.2.50a) ( ) Vg SH 3 3 44 3 = n c v ρ (2.2.50b) These equations show that Vg SH is not parallel to n. Thus, the group velocity direction deviates from the phase velocity direction. Note that V n g SH . is equal to v3 as per equa- tion (2.2.15). On the other hand, for the wave propagating and polarized in the plane of x1–x3, we obtain the components of the group velocity of the quasi-longitudinal (qL) or quasi- shear (qS) wave as ( ) / / [ { ( Vg 1 1 1 11 2 55 = - ∂ ∂ ∂ ∂ = - Ω Ω k n c v c ω ρ n n c n c v c n c n c 1 2 33 3 2 55 2 11 1 2 55 3 2 13 + + - + + )} { ( )} ( ρ + + - + + c n v v c n c n c 55 2 3 2 2 11 1 2 33 3 2 55 2 ) ] [ ( )] ρ ρ (2.2.51a) ( ) / / [ { ( Vg 2 3 3 33 2 11 = - ∂ ∂ ∂ ∂ = - Ω Ω k n c v c ω ρ n n c n c v c n c n c 1 2 55 3 2 55 2 55 1 2 33 3 2 13 + + - + + )} { ( )} ( ρ + + - + + c n v v c n c n c 55 2 1 2 2 11 1 2 33 3 2 55 2 ) ] [ ( )] ρ ρ (2.2.51b) 53388_C002.indd 28 10/8/08 11:05:18 AM © 2009 by Taylor & Francis Group, LLC
- 43. Fundamentals of Elastic Waves in Anisotropic Media 29 by taking v as v1 or v2, respectively, in equation (2.2.51). It can be shown that � Vg given by equations (2.2.51a) and (2.2.51b) satisfies the equation � � V n g . = v It is found from equations (2.2.42) and (2.2.43) that, when the wave is propagating along the x1-axis, the phase velocities and the polarizations are given by the equations ( ) c v A 11 2 1 0 - = ρ (2.2.52a) ( ) c v A 66 2 2 0 - = ρ (2.2.52b) ( ) c v A 55 2 3 0 - = ρ (2.2.52c) It is seen that a pure longitudinal wave with velocity vl = √(c11/r) propagates along the x1-axis, and two pure shear waves, one polarized parallel to the x2-axis and the other polarized parallel to the x3-axis, propagate with the velocities v2 = √(c66/r) and v3 = √(c55/r), respectively. Similarly, for propagation along the x3-axis, the equations are ( ) c v A 55 2 1 0 - = ρ (2.2.53a) ( ) c v A 44 2 2 0 - = ρ (2.2.53b) ( ) c v A 33 2 3 0 - = ρ (2.2.53c) In this case, the longitudinal wave polarized in the x3 direction propagates with velocity v3 = √(c33/r), and the two shear waves with polarizations, respectively, in the x1 and x2 directions propagate with velocities v1 = √(c55/r) and v2 = √(c44/r). 2.2.5 Transformation of Coordinates It is sometimes convenient to choose coordinate axes that are different from the sym- metry axes of the material. To write the stress and strain tensors in the new reference frame, let us suppose that the new x′ y′ z′ system is obtained from the xyz system by the orthogonal matrix [a] = a a a a a a a a a xx xy xz yx yy yz zx zy zz (2.2.54) so that we have xi′ = aijxj (2.2.55) Then the stress vector transformation can be written as sI′= MIJsJ, I,J = 1,2,3,4,5,6 (2.2.56) 53388_C002.indd 29 10/8/08 11:05:21 AM © 2009 by Taylor & Francis Group, LLC
- 44. 30 Elastic Waves in Composite Media and Structures where [M] is the 6 × 6 transformation matrix [ ] M a a a xx xy xz = 2 2 2 2 2 a a a a xy xz xz xx 2 2 2 a a a a xx xy yx yy a a a a yz yy yz 2 2 2 y yz yx yx yy zx a a a a 2 2 a a a a zy zz zy zz 2 2 2 2 2 a a a a zz zx zx zy a a a a a a a a a a a yx zx yy zy yz zz yy zz yz z + y y yx zz yz zx yy zx yx zy zx xx a a a a a a a a a a + + a a a a a a a a a zy xy zz xz xy zz xz zy + x xz zx xx zz xx zy xy zx xx yx x a a a a a a a a a a + + y y yy xz yz xy yz xz yy xz yx a a a a a a a a a + +a a a a a a a xx yz xx yy xy yx + (2.2.57) In the same way, the strain vector transformation is given by eI′ = NIJeJ (2.2.58) where N is obtained from M by shifting the factor 2 from the upper right-hand corner to the lower left-hand corner. It is noted that [N]−1 = [M]T (2.2.59) It then follows that the stiffness matrix [c] transforms as [c′] = [M][c][M]T (2.2.60) For future use, we will consider the transformation of the stiffness matrix given by equation (2.2.60) when the coordinates are rotated clockwise by an angle θ about the 3-axis. The coordinate transformation matrix from x1-, x2-, and x3-axes to x1′-, x2′-, and x3′(= x3)-axes is (see Fig. 2.2) [a] = cos sin sin cos θ θ θ θ - 0 0 0 0 1 (2.2.61) x3(x3´) –θ α x1 x2´ x1´ n Fiber direction Figure 2.2 Coordinate transformation. 53388_C002.indd 30 10/8/08 11:05:22 AM © 2009 by Taylor & Francis Group, LLC
- 45. Fundamentals of Elastic Waves in Anisotropic Media 31 The [M] matrix is then given by [ ] cos sin sin sin cos sin M = - 2 2 2 2 0 0 0 2 0 0 0 2 θ θ θ θ θ θ 0 0 0 1 0 0 0 0 0 0 0 cos sin sin cos sin s θ θ θ θ θ 0 0 0 0 1 2 2 1 2 - - i in cos 2 0 0 0 2 θ θ (2.2.62) The transformed stiffness coefficients are then found to be ′ = + + + c c c c c 11 11 4 22 4 12 66 2 1 2 2 cos sin sin θ θ θ θ θ ′ = + - + c c c c c 12 11 22 66 2 12 4 1 4 2 ( ) sin (cos θ θ θ θ θ θ + ′ = + ′ = sin ) cos sin sin 4 13 13 2 23 2 16 1 2 2 c c c c [ [ cos sin ( )cos ] c c c c c c 11 2 22 2 12 66 22 1 2 2 θ θ θ - - + ′ = 1 1 4 22 4 12 66 2 23 1 2 2 sin cos sin θ θ θ + + + ′ = c c c c c c c c c c 13 2 23 2 26 11 2 22 1 2 2 sin cos sin [ sin θ θ θ θ + ′ = - c cos ( )cos ] 2 12 66 2 2 θ θ + + c c (2.2.63a) ′ = ′ = - c c c c c 33 33 36 13 23 1 2 2 ( )sin θ (2.2.63b) ′ = + ′ = - c c c c c c 44 44 2 55 2 45 55 44 1 2 2 cos sin ( )sin θ θ θ ′ ′ = + ′ = + - c c c c c c c 55 44 2 55 2 66 11 22 12 1 4 2 sin cos ( θ θ ) )sin cos 2 66 2 14 15 24 25 34 2 2 θ θ + ′ = ′ = ′ = ′ = ′ = c c c c c c ′ ′ = ′ = ′ = c c c 35 46 56 0 (2.2.63c) 53388_C002.indd 31 10/8/08 11:05:24 AM © 2009 by Taylor & Francis Group, LLC
- 46. 32 Elastic Waves in Composite Media and Structures It may be noted that the stiffness matrix in this rotated coordinate system has a similar structure as that of a monoclinic material having the 12-plane as the plane of symme- try. Relation (2.2.63) holds for an orthotropic material having the 12- and 13-planes as planes of symmetry. For a plane wave propagating in the plane of x1′–x3′, let a be the angle made by the direction of propagation n with the x3-axis. Then, n1 = sina cosq, n2 = −sina sinq, n3 = cosa (2.2.64) The velocities of the three propagating waves can be obtained from equations (2.2.44)– (2.2.47) in the following manner. Let the angle between the wave propagation direction and the x1-axis be β and the projection of n on the x2–x3-plane be taken as the x3′-axis. Then, the direction cosines of the wave-propagation direction referred to the x1-, x2′-, and x3′-axes are given by n n n 1 2 3 0 = ′′= ′′= cos , , sin β β (2.2.65) These can be expressed in terms of a and q as n n n n 1 2 3 1 2 0 1 = ′′= ′′= - sin cos , , α θ (2.2.66) The elements of the Christoffel matrix are given by Γ Γ Γ 11 11 1 2 55 3 2 12 13 13 55 1 3 0 = + ′′ = = + ′′ c n c n c c n n ( ) Γ Γ Γ Γ 22 66 1 2 44 3 2 23 33 55 1 2 33 3 0 = + ′′ = = + ′′ c n c n c n c n 2 2 (2.2.67) Equation (2.2.7) then takes the form Γ Γ Γ Γ Γ 11 2 13 22 2 13 33 2 1 0 0 0 0 - - - ρ ρ ρ v v v A ′ ′′ ′′ = A A 2 3 0 (2.2.68) As seen before, the wave with displacement polarized in the x2″ direction is uncou- pled from the quasi-longitudinal and quasi-shear waves polarized in the x1–x3″-plane. The velocities of the latter two waves are obtained from equations (2.2.46)–(2.2.48) after replacing n3 by n3″. The polarizations of these waves are given by the eigenvectors of equation (2.2.68). As expected, for a transversely isotropic medium, quasi-longitudinal and quasi-shear waves have displacement polarizations in the plane containing the sym- metry axis and the direction of propagation of the wave, whereas the pure shear wave is polarized perpendicular to this plane. 53388_C002.indd 32 10/8/08 11:05:25 AM © 2009 by Taylor & Francis Group, LLC
- 47. Fundamentals of Elastic Waves in Anisotropic Media 33 2.3 Numerical Results and Discussion As an illustrative example, we now consider a plane wave propagating in a fiber- reinforced composite medium. Because continuous graphite-fiber-reinforced polymeric matrix materials are widely used in applications, numerical results will be presented for a particular case of graphite–epoxy-composite medium. It is now well established that an elastic homogeneous matrix reinforced by a random distribution of aligned continu- ous cylindrical elastic fibers can be modeled as a transversely isotropic homogeneous elastic medium when the wavelength of a propagating wave is much longer than the fiber diameter. The effective properties of such a medium are the static effective properties. For a typical graphite-fiber-reinforced epoxy matrix, these anisotropic elastic constants are as follows, in units of 109 N/m2, c11 = 160.73, c12 = c13 = 6.44, c22 = c33 = 13.92, c55 = c66 = 7.07, c44 = 3.5 (2.3.1) Here, the x-axis is taken to be parallel to the fiber axis. The density r = 1578 kg/m3. Thus, the phase velocity of the longitudinal wave propagating in the fiber direction is v1 L = 10.1 km/s. On the other hand, phase velocities of shear waves polarized in the 3- or 2- directions (S or SH directions) and propagating along the 1-direction (fiber direction) are v v 1 S 1 SH = = 2.12 km/s—see equations (2.2.52a,b,c). The phase velocities of the longitudi- nal (L), S, and SH waves propagating along the 3-direction are v3 L = 2.97 km/s, v3 S = 2.12 km/s, and v3 SH = 1.49 km/s, respectively—see equations (2.2.53a,b,c). Equations (2.2.44), (2.2.46), and (2.2.47) can be used to calculate the phase velocities of SH, qL, and qS waves propagating along the direction making an angle a with the x3-axis in the 13-plane. Figure 2.3 shows the polar plot of normalized values of the phase velocities for dif- ferent angles of propagation ax = 90° − a. Velocities are normalized with respect to v1 S . 1 3 2 4 0 90 180 270 5 qL qS SH Figure 2.3 Polar plot of normalized phase-velocity surfaces versus propagation angle (in degrees). 53388_C002.indd 33 10/8/08 11:05:27 AM © 2009 by Taylor & Francis Group, LLC
- 48. 34 Elastic Waves in Composite Media and Structures The slowness s n = v v / 1 S is shown as a polar plot in Fig. 2.4. The x1 and x3 components of the group velocities of the qL, qS, and SH waves are calculated using equations (2.2.50) and (2.2.51) and are plotted in Fig. 2.5. Note the characteristic cusps appearing in the group velocity plot of the qS wave. It is noted that the directions of group-velocity prop- agation are different than the phase-velocity propagation directions, a. As discussed above, the group-velocity direction is normal to the slowness surface at the point. The skew angle measuring the difference between the angles made by the phase and group 270 90 180 0 0.5 1.0 1.5 qL qS SH Figure 2.4 Polar plot of normalized slowness surfaces versus propagation angle (in degrees). Normalized VgSH-z, -VgQL-z, -VgQS-z Normalized VgSH-x, -VgQL-x, -VgQS-x 0.0 0.5 1.0 1.5 0 1 2 3 4 5 SH qS qL Figure 2.5 x–z plot of normalized group velocities. 53388_C002.indd 34 10/8/08 11:05:29 AM © 2009 by Taylor & Francis Group, LLC
- 49. Fundamentals of Elastic Waves in Anisotropic Media 35 velocities with the x3-direction is shown in Fig. 2.6. It can be seen that this can be quite large in this particular case of strong anisotropy. A program for calculating the results shown graphically in Figs. 2.3–2.6 is included in the disc accompanying this book. The readers are encouraged to use the data for the material properties for their particular applications in this program to generate the plots for their use. It should be emphasized that this program can be used for orthotropic materials as well for propagation in a plane of symmetry. Furthermore, for a transversely isotropic material, the phase velocity, slowness, and group velocity can be obtained by using equations (2.2.66)–(2.2.68) for propagation in an arbitrary plane. –20 0 20 40 60 80 10 20 30 40 50 Propagation Angle Skew Angle 60 70 80 90 qL qS SH Figure 2.6 Skew angle versus propagation angle (in degrees). 53388_C002.indd 35 10/8/08 11:05:29 AM © 2009 by Taylor & Francis Group, LLC
- 50. 53388_C002.indd 36 10/8/08 11:05:29 AM © 2009 by Taylor & Francis Group, LLC
- 51. 37 3 Periodic Layered Media 3.1 Introduction...............................................................................37 3.2 Description of the Problem..................................................... 38 Stiffness Method • Effective Modulus Method 3.3 Numerical Results.....................................................................51 Isotropic Laminates • Anisotropic Laminates 3.4 Remarks ..................................................................................... 56 3.1 Introduction In Chapter 2 (Section 2.2), plane waves in a homogeneous anisotropic elastic medium were considered. Equations governing the phase and group velocities were derived. Since the medium is homogeneous, these velocities are independent of frequency, i.e., waves are nondispersive. If the medium is nonhomogeneous, e.g., a composite medium made up of a distribution of fibers or particles of different material properties embedded in a matrix of some other material properties, then an incident plane wave propagating in the medium will be scattered, and constructive and destructive interferences will take place. If the wavelength is much longer than the characteristic dimensions of the fibers or inclusions, then a coherent plane wave will emerge with frequency-dependent phase velocity and with amplitude decaying with distance of travel. Thus, the wave will be dispersive and attenuative. A similar phenomenon takes place when a plane wave propagates through a plane- layered medium with layers having different material properties. Here, an incident plane wave will be reflected and refracted (scattered) at the plane interfaces of adjacent layers. Again, if the wavelength is large compared with the thicknesses of the layers, then a plane wave will propagate through the composite medium, having velocity that will be dependent upon frequency and amplitude that will be decaying. If the medium has a periodic layered structure, Floquet wave theory leads to a dispersion equation governing various modes (propagating and evanescent) of propagation of harmonic waves. In the long-wavelength limit, this equation yields three speeds of wave propagation that corre- spond to those for an effective homogeneous anisotropic medium. The dispersion equa- tion defines a surface in the frequency–wave-number space that has the characteristic feature of exhibiting passing and stopping bands found in wave propagation in periodic 53388_C003.indd 37 10/8/08 11:07:36 AM © 2009 by Taylor & Francis Group, LLC
- 52. 38 Elastic Waves in Composite Media and Structures media. A review of significant contributions in this important field has been presented in Chapter 1 (Section 1.1). See also Nayfeh (1995), Lee (1972), Braga and Herrmann (1988, 1992), and Ting and Chadwick (1988). In this chapter, we present a model for studying dispersion of elastic waves in an infinite medium composed of periodic layers of orthotropic material. For the purpose of generality, three-dimensional motion is dealt with first. Results for the special cases of antiplane and plane-strain motion are given later. A stiffness method is used here for its general applicability to the case when each layer has monoclinic symmetry about its bounding planes. An exact treatment of the problem when each layer is transversely isotropic is given at the end of the chapter. In the stiffness method, an interpolation function is assumed to represent the dis- placement within each layer in terms of a discrete set of generalized coordinates. These generalized coordinates are the displacements and tractions at the bounding interfaces of the layer, thus ensuring continuity of displacements and tractions across the inter- faces between adjacent layers. By applying Hamilton’s principle and using Floquet the- ory, the dispersion equation is obtained as a generalized algebraic eigenvalue problem whose solution yields the frequency–wave-number relation as well as the variation of stresses and displacements. As will be seen in later chapters, the method is adapted well for bounded plates and cylinders. 3.2 Description of the Problem Consider a stack of plane layers having orthotropic symmetry with a common plane of symmetry that is taken as the x1–x2-plane. The layers are assumed to be rigidly bonded to one another. A global Cartesian coordinates system (x1, x2, x3) is chosen so that the x3-axis is normal to the layers. Let superscript (i) identify the variables of interest asso- ciated with the ith layer, which is assumed to be orthotropic, having symmetry planes parallel to the global coordinate planes. It will be convenient to use a local Cartesian coordinate system in the ith layer with origin at the midplane of the layer and axes parallel to the global coordinate axes. For the purpose of keeping the algebra simple, it will be assumed that the stack is a two-layer periodically laminated body of unbounded extent. Any two adjacent laminates in the body then comprise a unit cell (see Fig. 3.1). We are concerned with a harmonic wave propagating in an arbitrary direction through such a medium. 3.2.1 Stiffness Method Let the wave-propagation direction make an angle a with the x3-axis and its projection on the x1–x2-plane make an angle −θ with the x1-axis (see Fig. 2.2). A new global coor- dinate system (x1′, x2′, x3′) is chosen such that the transformation from the unprimed system to the primed system is given by the matrix [a] in equation (2.2.61). Thus, the wave vector lies in the plane of x1′–x3′, and the displacement will be independent of x2′. Note that the displacement will have all of the three components. The two adjacent laminae comprising a typical unit cell Cn have elastic constants given, respectively, by cij (i) and cij (i+1), thicknesses 2h(i) and 2h(i+1), and densities r(i) and 53388_C003.indd 38 10/8/08 11:07:36 AM © 2009 by Taylor & Francis Group, LLC
- 53. Periodic Layered Media 39 r(i+1). For the ith and (i + 1)th laminae, assuming orthotropic symmetry, the stress rela- tions are given by σ σ σ σ σ σ xx j yy j zz j yz j zx j xy j ( ) ( ) ( ) ( ) ( ) ( ) = c c c j j 11 12 13 ( ) ( ) ( j j j j c c c ) ( ) ( ) 0 0 0 12 22 23 3 13 23 0 0 0 ( ) ( ) ( ) j j j c c c c j 33 0 0 0 0 0 ( ) 0 0 0 0 0 44 c j ( ) 0 0 0 0 0 55 c j ( ) 0 0 0 66 c j ( ) e e e xx j yy j zz ( ) ( ) ( j j yz j zx j xy j ) ( ) ( ) ( ) γ γ γ (j = i, i + 1) (3.2.1) where sij (j), eij (j), and 1 2 (γij (j)) are the stress and strain components. In the transformed (primed) coordinates system, the stiffness coefficients cij (j) appearing in equation (3.2.1) will be transformed as given by equation (2.2.60). We will write the transformed stress– strain relation for the jth layer as σij ijkl kl D e ′ = ′ (3.2.2) where Dijkl = cijkl′. Note that the stiffness matrix Dijkl has the same form as that for a monoclinic material having the x1 – x2-plane as the symmetry plane. Thus, the disper- sion equation derived in the following holds also for layers of monoclinic materials. The displacement components in the ith and (i + 1)th laminae will be approximated by interpolation polynomials. To this end, it will be convenient to divide each laminate into N sublayers of equal thickness 2h without loss of generality, where N is a positive B L d1 = 2h(1) d2 = 2h(2) x y z Figure 3.1 Geometry of the unit cell. 53388_C003.indd 39 10/8/08 11:07:37 AM © 2009 by Taylor & Francis Group, LLC
- 54. 40 Elastic Waves in Composite Media and Structures integer. The displacement components in the mth sublayer will be approximated by the following cubic polynomials (omitting the prime.) u u f u f D D w x f m m m m m = + + − − ∂ ∂ + 1 1 2 44 45 3 ∆ ∆ χ τ + D D w x f m m m 44 1 45 1 1 4 ∆ ∆ χ τ + + + − − ∂ ∂ (3.2.3a) υ υ υ τ χ = + + − + m m m m f f D D f 1 1 2 55 45 3 ∆ ∆ + D D D f m m 55 1 45 1 4 ∆ ∆ τ χ + + − (3.2.3b) w w f w f D D D u x D D x m m m m m = + + − ∂ ∂ − ∂ ∂ + 1 1 2 33 13 33 36 33 σ υ − ∂ ∂ − + + f D D D u x D D m m 3 1 33 13 33 1 36 + σ 3 33 1 4 ∂ ∂ + υm x f (3.2.3c) where fn (n = 1, 2, 3, 4) are cubic polynomials in the local coordinate z and are given by f f f h 1 3 2 3 3 2 3 1 4 2 3 1 4 2 3 4 1 = − + = + − = − − + ( ) ( ) ( η η η η η η η ) ) ( ) f h 4 2 3 4 1 = − − + + η η η (3.2.4) and ∆ = − = − = ′ − ′ + + D D D h z z z z m m m m 44 55 45 2 1 1 1 2 1 2 ( ) ( ) (3.2.5) In the above, h = z/h, z = x3′, and um, υm, wm, χm, tm, sm are the values of u′, υ′, w′, szx′, syz′, szz′, respectively, at the mth interface (m = 1, 2, … , N). Note that 2Nh = 2h(i). This choice of interpolation preserves the continuity of the displacement and traction at the interface between adjacent layers. Similarly, the (i + 1)th laminate is divided into N′ sublayers of equal thickness, and the displacement components within each sublayer are expressed in terms of the same polynomials of the local variable h = z/h′, with ′ = ′ + h h N i ( ) 1 . 53388_C003.indd 40 10/8/08 11:07:39 AM © 2009 by Taylor & Francis Group, LLC
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- 56. often concealed by quick and complete naturalization their foreignness to English remains none the less obvious. I should worry,[46] in its way, is correct English, but in essence it is as completely Yiddish as kosher, ganof, schadchen, oi-yoi, matzoh or mazuma.[47] Black-hand, too, is English in form, but it is nevertheless as plainly an Italian loan-word as spaghetti, mafia or padrone. The extent of such influences upon American, and particularly upon spoken American, remains to be studied; in the whole literature I can find but one formal article upon the subject. That article[48] deals specifically with the suffix -fest, which came into American from the German and was probably suggested by familiarity with sängerfest. There is no mention of it in any of the dictionaries of Americanisms, and yet, in such forms as talk-fest and gabfest it is met with almost daily. So with -heimer, -inski and -bund. Several years ago -heimer had a great vogue in slang, and was rapidly done to death. But wiseheimer remains [Pg152] in colloquial use as a facetious synonym for smart-aleck, and after awhile it may gradually acquire dignity. Far lowlier words, in fact, have worked their way in. Buttinski, perhaps, is going the same route. As for the words in -bund, many of them are already almost accepted. Plunder- bund is now at least as good as pork-barrel and slush-fund, and money-bund is frequently heard in Congress.[49] Such locutions creep in stealthily, and are secure before they are suspected. Current slang, out of which the more decorous language dredges a large part of its raw materials, is full of them. Nix and nixy, for no, are debased forms of the German nichts; aber nit, once as popular as camouflage, is obviously aber nicht. And a steady flow of nouns, all needed to designate objects introduced by immigrants, enriches the vocabulary. The Hungarians not only brought their national condiment with them; they also brought its name, paprika, and that name is now thoroughly American.[50] In the same way the Italians brought in camorra, padrone, spaghetti and a score of other substantives, and the Jews made contributions from Yiddish and Hebrew and greatly reinforced certain old borrowings from German.
- 57. Once such a loan-word gets in it takes firm root. During the first year of American participation in the World War an effort was made, on patriotic grounds, to substitute liberty-cabbage for sour-kraut, but it quickly failed, for the name had become as completely Americanized as the thing itself, and so liberty-cabbage seemed affected and absurd. In the same way a great many other German words survived the passions of the time. Nor could all the influence of the professional patriots obliterate that German influence which has fastened upon the American yes something of the quality of ja. Constant familiarity with such contributions from foreign languages and with the general speech habits of foreign peoples has made American a good deal more hospitable to loan-words than English, even in the absence of special pressure. Let the same [Pg153] word knock at the gates of the two languages, and American will admit it more readily, and give it at once a wider and more intimate currency. Examples are afforded by café, vaudeville, employé, boulevard, cabaret, toilette, exposé, kindergarten, dépôt, fête and menu. Café, in American, is a word of much larger and more varied meaning than in English and is used much more frequently, and by many more persons. So is employé, in the naturalized form of employee. So is toilet: we have even seen it as a euphemism for native terms that otherwise would be in daily use. So is kindergarten: I read lately of a kindergarten for the elementary instruction of conscripts. Such words are not unknown to the Englishman, but when he uses them it is with a plain sense of their foreignness. In American they are completely naturalized, as is shown by the spelling and pronunciation of most of them. An American would no more think of attempting the French pronunciation of depot or of putting the French accents upon it than he would think of spelling toilet with the final te or of essaying to pronounce Anheuser in the German manner. Often curious battles go on between such loan-words and their English equivalents, and with varying fortunes. In 1895 Weber and Fields tried to establish music- hall in New York, but it quickly succumbed to vaudeville-theatre, as variety had succumbed to vaudeville before it. In the same way
- 58. lawn-fete (without the circumflex accent, and commonly pronounced feet) has elbowed out the English garden-party. But now and then, when the competing loan-word happens to violate American speech habits, a native term ousts it. The French crèche offers an example; it has been entirely displaced by day-nursery. The English, in this matter, display their greater conservatism very plainly. Even when a loan-word enters both English and American simultaneously a sense of foreignness lingers about it on the other side of the Atlantic much longer than on this side, and it is used with far more self-consciousness. The word matinée offers a convenient example. To this day the English commonly print it in italics, give it its French accent, and pronounce it with some attempt at the French manner. But in America it is entirely naturalized, and the most ignorant man [Pg154] uses it without any feeling that it is strange. The same lack of any sense of linguistic integrity is to be noticed in many other directions—for example, in the freedom with which the Latin per is used with native nouns. One constantly sees per day, per dozen, per hundred, per mile, etc., in American newspapers, even the most careful, but in England the more seemly a is almost always used, or the noun itself is made Latin, as in per diem. Per, in fact, is fast becoming an everyday American word. Such phrases as "as per your letter (or order) of the 15th inst." are incessantly met with in business correspondence. The same greater hospitality is shown by the readiness with which various un-English prefixes and affixes come into fashion, for example, super- and -itis. The English accept them gingerly; the Americans take them in with enthusiasm, and naturalize them instanter.[51] The same deficiency in reserve is to be noted in nearly all other colonialized dialects. The Latin-American variants of Spanish, for example, have adopted a great many words which appear in true Castilian only as occasional guests. Thus in Argentina matinée, menu, début, toilette and femme de chambre are perfectly good Argentine, and in Mexico sandwich and club have been thoroughly naturalized. The same thing is to be noted in the French of Haiti, in the Portuguese of Brazil, and even in the Danish of Norway. Once a
- 59. language spreads beyond the country of its origin and begins to be used by people born, in the German phrase, to a different Sprachgefühl, the sense of loyalty to its vocabulary is lost, along with the instinctive feeling for its idiomatic habits. How far this destruction of its forms may go in the absence of strong contrary influences is exhibited by the rise of the Romance languages from the vulgar Latin of the Roman provinces, and, here at home, by the decay of foreign languages in competition with English. The Yiddish that the Jews from Russia bring in is German debased with Russian, Polish and [Pg155] Hebrew; in America, it quickly absorbs hundreds of words and idioms from the speech of the streets. Various conflicting German dialects, among the so-called Pennsylvania Dutch and in the German areas of the Northwest, combine in a patois that, in its end forms, shows almost as much English as German. Classical examples of it are "es giebt gar kein use," "Ich kann es nicht ständen" and "mein stallion hat über die fenz gescheumpt und dem nachbar sein whiet abscheulich gedämätscht."[52] The use of gleiche for to like, by false analogy from gleich (=like, similar) is characteristic. In the same way the Scandinavians in the Northwest corrupt their native Swedish and Dano-Norwegian. Thus, American-Norwegian is heavy with such forms as strit-kar, reit-evé, nekk-töi and staits-pruessen, for street-car, right away, necktie and states-prison, and admits such phrases as "det meka ingen difrens."[53] The changes that Yiddish has undergone in America, though rather foreign to the present inquiry, are interesting enough to be noticed. First of all, it has admitted into its vocabulary a large number of everyday substantives, among them boy, chair, window, carpet, floor, dress, hat, watch, ceiling, consumption, property, trouble, bother, match, change, party, birthday, picture, paper (only in the sense of newspaper), gambler, show, hall, kitchen, store, bedroom, key, mantelpiece, closet, lounge, broom, tablecloth, paint, landlord, fellow, tenant, shop, wages, foreman, sleeve, collar, cuff, button, cotton, thimble, needle, pocket, bargain, sale, remnant, sample, haircut, razor, waist, basket, school, scholar, teacher, baby, mustache, butcher, grocery, dinner, street and walk. And with them
- 60. many characteristic Americanisms, [Pg156] for example, bluffer, faker, boodler, grafter, gangster, crook, guy, kike, piker, squealer, bum, cadet, boom, bunch, pants, vest, loafer, jumper, stoop, saleslady, ice-box and raise, with their attendant verbs and adjectives. These words are used constantly; many of them have quite crowded out the corresponding Yiddish words. For example, ingel, meaning boy (it is a Slavic loan-word in Yiddish), has been obliterated by the English word. A Jewish immigrant almost invariably refers to his son as his boy, though strangely enough he calls his daughter his meidel. "Die boys mit die meidlach haben a good time" is excellent American Yiddish. In the same way fenster has been completely displaced by window, though tür (=door) has been left intact. Tisch (=table) also remains, but chair is always used, probably because few of the Jews had chairs in the old country. There the beinkel, a bench without a back, was in use; chairs were only for the well-to- do. Floor has apparently prevailed because no invariable corresponding word was employed at home: in various parts of Russia and Poland a floor is a dill, a podlogé, or a bricke. So with ceiling. There were six different words for it. Yiddish inflections have been fastened upon most of these loan- words. Thus, "er hat ihm abgefaked" is "he cheated him," zubumt is the American gone to the bad, fix'n is to fix, usen is to use, and so on. The feminine and diminutive suffix -ké is often added to nouns. Thus bluffer gives rise to blufferké (=hypocrite), and one also notes dresské, hatké, watchké and bummerké. "Oi! is sie a blufferké!" is good American Yiddish for "isn't she a hypocrite!" The suffix -nick, signifying agency, is also freely applied. Allrightnick means an upstart, an offensive boaster, one of whom his fellows would say "He is all right" with a sneer. Similarly, consumptionick means a victim of tuberculosis. Other suffixes are -chick and -ige, the first exemplified in boychick, a diminutive of boy, and the second in next-doorige, meaning the woman next-door, an important person in ghetto social life. Some of the loan-words, of course, undergo changes on Yiddish-speaking lips. Thus, landlord becomes lendler, lounge becomes lunch, tenant becomes tenner, and whiskers loses its final
- 61. s. "Wie gefällt dir sein whisker?" (=how do you like his beard?) [Pg157] is good Yiddish, ironically intended. Fellow, of course, changes to the American feller, as in "Rosie hat schon a feller" (=Rosie has got a feller, i. e., a sweetheart). Show, in the sense of chance, is used constantly, as in "git ihm a show" (=give him a chance). Bad boy is adopted bodily, as in "er is a bad boy." To shut up is inflected as one word, as in "er hat nit gewolt shutup'n" (=he wouldn't shut up). To catch is used in the sense of to obtain, as in "catch'n a gmilath chesed" (=to raise a loan). Here, by the way, gmilath chesed is excellent Biblical Hebrew. To bluff, unchanged in form, takes on the new meaning of to lie: a bluffer is a liar. Scores of American phrases are in constant use, among them, all right, never mind, I bet you, no sir and I'll fix you. It is curious to note that sure Mike, borrowed by the American vulgate from Irish English, has gone over into American Yiddish. Finally, to make an end, here are two complete and characteristic American Yiddish sentences: "Sie wet clean'n die rooms, scrub'n dem floor, wash'n die windows, dress'n dem boy und gehn in butcher-store und in grocery. Dernoch vet sie machen dinner und gehn in street für a walk."[54] American itself, in the Philippines, and to a lesser extent in Porto Rico and on the Isthmus, has undergone similar changes under the influence of Spanish and the native dialects. Maurice P. Dunlap[55] offers the following specimen of a conversation between two Americans long resident in Manila:
- 62. Hola, amigo. Komusta kayo. Porque were you hablaing with ese señorita? She wanted a job as lavandera. Cuanto? Ten cents, conant, a piece, so I told her no kerry. Have you had chow? Well, spera till I sign this chit and I'll take a paseo with you. [Pg158] Here we have an example of Philippine American that shows all the tendencies of American Yiddish. It retains the general forms of American, but in the short conversation, embracing but 41 different words, there are eight loan-words from the Spanish (hola, amigo, porque, ese, señorita, lavandera, cuanto and paseo), two Spanish locutions in a debased form (spera for espera and no kerry for no quiro), two loan-words from the Taglog (komusta and kayo), two from Pigeon English (chow and chit), one Philippine-American localism (conant), and a Spanish verb with an English inflection (hablaing). The immigrant in the midst of a large native population, of course, exerts no such pressure upon the national language as that exerted upon an immigrant language by the native, but nevertheless his linguistic habits and limitations have to be reckoned with in dealing with him, and the concessions thus made necessary have a very ponderable influence upon the general speech. In the usual sense, as we have seen, there are no dialects in American; two natives, however widely their birthplaces may be separated, never have any practical difficulty understanding each other. But there are at least quasi-dialects among the immigrants—the Irish, the German, the Scandinavian, the Italian, the Jewish, and so on—and these quasi- dialects undoubtedly leave occasional marks, not only upon the national vocabulary, but also upon the general speech habits of the
- 63. country, as in the case, for example, of the pronunciation of yes, already mentioned, and in that of the substitution of the diphthong oi for the ur-sound in such words as world, journal and burn—a Yiddishism now almost universal among the lower classes of New York, and threatening to spread.[56] More important, however, is the support given to a native tendency by the foreigner's incapacity for employing (or even comprehending) syntax of any complexity, or words not of the simplest. This is the tendency toward succinctness [Pg159] and clarity, at whatever sacrifice of grace. One English observer, Sidney Low, puts the chief blame for the general explosiveness of American upon the immigrant, who must be communicated with in the plainest words available, and is not socially worthy of the suavity of circumlocution anyhow.[57] In his turn the immigrant seizes upon these plainest words as upon a sort of convenient Lingua Franca—his quick adoption of damn as a universal adjective is traditional—and throws his influence upon the side of the underlying speech habit when he gets on in the vulgate. Many characteristic Americanisms of the sort to stagger lexicographers—for example, near-silk—have come from the Jews, whose progress in business is a good deal faster than their progress in English. Others, as we have seen, have come from the German immigrants of half a century ago, from the so-called Pennsylvania Dutch (who are notoriously ignorant and uncouth), and from the Irish, who brought with them a form of English already very corrupt. The same and similar elements greatly reinforce the congenital tendencies of the dialect—toward the facile manufacture of compounds, toward a disregard of the distinctions between parts of speech, and, above all, toward the throwing off of all etymological restraints. § 5 Processes of Word Formation
- 64. —Some of these tendencies, it has been pointed out, go back to the period of the first growth of American, and were inherited from the English of the time. They are the products of a movement which, reaching its height in the English of Elizabeth, was dammed up at home, so to speak, by the rise of linguistic self-consciousness toward the end of the reign of Anne, but continued almost unobstructed in the colonies. For example, there is what philologists call the habit of back-formation—a sort of instinctive search, etymologically unsound, for short roots in long words. This habit, in Restoration days, precipitated a quasi-English word, mobile, from the Latin [Pg160] mobile vulgus, and in the days of William and Mary it went a step further by precipitating mob from mobile. Mob is now sound English, but in the eighteenth century it was violently attacked by the new sect of purists,[58] and though it survived their onslaught they undoubtedly greatly impeded the formation and adoption of other words of the same category. But in the colonies the process went on unimpeded, save for the feeble protests of such stray pedants as Witherspoon and Boucher. Rattler for rattlesnake, pike for turnpike, draw for drawbridge, coon for raccoon, possum for opossum, cuss for customer, cute for acute, squash for askutasquash—these American back-formations are already antique; Sabbaday for Sabbath-day has actually reached the dignity of an archaism. To this day they are formed in great numbers; scarcely a new substantive of more than two syllables comes in without bringing one in its wake. We have thus witnessed, within the past two years, the genesis of scores now in wide use and fast taking on respectability; phone for telephone, gas for gasoline, co-ed for co-educational, pop for populist, frat for fraternity, gym for gymnasium, movie for moving- picture, prep-school for preparatory-school, auto for automobile, aero for aeroplane. Some linger on the edge of vulgarity: pep for pepper, flu for influenza, plute for plutocrat, pen for penitentiary, con for confidence (as in con-man, con-game and to con), convict and consumption, defi for defiance, beaut for beauty, rep for reputation, stenog for stenographer, ambish for ambition, vag for vagrant, champ for champion, pard for partner, coke for cocaine,
- 65. simp for simpleton, diff for difference. Others are already in perfectly good usage: smoker for smoking-car, diner for dining-car, sleeper for sleeping-car, oleo for oleomargarine, hypo for hyposulphite of soda, Yank for Yankee, confab for confabulation, memo for memorandum, pop-concert for popular-concert. Ad for advertisement is struggling hard for recognition; some of its compounds, e. g., ad-writer, want-ad, display-ad, ad-card, ad-rate, column-ad and ad-man, are already accepted in technical terminology. Boob for booby promises to become sound American in a few years; its synonyms are no more respectable than it is. At [Pg161] its heels is bo for hobo, an altogether fit successor to bum for bummer.[59] A parallel movement shows itself in the great multiplication of common abbreviations. "Americans, as a rule," says Farmer, "employ abbreviations to an extent unknown in Europe.... This trait of the American character is discernible in every department of the national life and thought."[60] O. K., C. O. D., N. G., G. O. P. (get out and push) and P. D. Q., are almost national hall-marks; the immigrant learns them immediately after damn and go to hell. Thornton traces N. G. to 1840; C. O. D. and P. D. Q. are probably as old. As for O. K., it was in use so early as 1790, but it apparently did not acquire its present significance until the 20's; originally it seems to have meant "ordered recorded."[11] During the presidential campaign of 1828 Jackson's enemies, seeking to prove his illiteracy, alleged that he used it for "oll korrect." Of late the theory has been put forward that it is derived from an Indian word, okeh, signifying "so be it," and Dr. Woodrow Wilson is said to support this theory and to use okeh in endorsing government papers, but I am unaware of the authority upon which the etymology is based. Bartlett says that the figurative use of A No. 1, as in an A No. 1 man, also originated in America, but this may not be true. There can be little doubt, however, about T. B. (for tuberculosis), G. B. (for grand bounce), 23, on the Q. T., and D. & D. (drunk and disorderly). The language breeds such short forms of speech prodigiously; every trade and profession has a host of them; they are innumerable in the slang of sport.[61]
- 66. What one sees under all this, account for it as one will, is a double habit, the which is, at bottom, sufficient explanation of the gap which begins to yawn between English and American, particularly on the spoken plane. On the one hand it is a habit of verbal economy— a jealous disinclination to waste two words on what can be put into one, a natural taste for the brilliant and [Pg162] succinct, a disdain of all grammatical and lexicographical daintiness, born partly, perhaps, of ignorance, but also in part of a sound sense of their imbecility. And on the other hand there is a high relish and talent for metaphor —in Brander Matthews' phrase, "a figurative vigor that the Elizabethans would have realized and understood." Just as the American rebels instinctively against such parliamentary circumlocutions as "I am not prepared to say" and "so much by way of being,"[62] just as he would fret under the forms of English journalism, with its reporting empty of drama, its third-person smothering of speeches and its complex and unintelligible jargon,[63] just so, in his daily speech and writing he chooses terseness and vividness whenever there is any choice, and seeks to make one when it doesn't exist. There is more than mere humorous contrast between the famous placard in the wash-room of the British Museum: "These Basins Are For Casual Ablutions Only," and the familiar sign at American railroad-crossings: "Stop! Look! Listen!" Between the two lies an abyss separating two cultures, two habits of mind, two diverging tongues. It is almost unimaginable that Englishmen, journeying up and down in elevators, would ever have stricken the teens out of their speech, turning sixteenth into simple six and twenty-fourth into four; the clipping is almost as far from their way of doing things as the climbing so high in the air. Nor have they the brilliant facility of Americans for making new words of grotesque but penetrating tropes, as in corn-fed, tight-wad, bone- head, bleachers and juice (for electricity); when they attempt such things the result is often lugubrious; two hundred years of schoolmastering has dried up their inspiration. Nor have they the fine American hand for devising new verbs; to maffick and to limehouse are their best specimens in twenty years, and both have
- 67. an almost pathetic flatness. Their business with the language, indeed, is not in this department. They are [Pg163] not charged with its raids and scoutings, but with the organization of its conquests and the guarding of its accumulated stores. For the student interested in the biology of language, as opposed to its paleontology, there is endless material in the racy neologisms of American, and particularly in its new compounds and novel verbs. Nothing could exceed the brilliancy of such inventions as joy-ride, high-brow, road-louse, sob-sister, nature-faker, stand-patter, lounge-lizard, hash-foundry, buzz-wagon, has-been, end-seat-hog, shoot-the-chutes and grape-juice-diplomacy. They are bold; they are vivid; they have humor; they meet genuine needs. Joy-ride, I note, is already going over into English, and no wonder. There is absolutely no synonym for it; to convey its idea in orthodox English would take a whole sentence. And so, too, with certain single words of metaphorical origin: barrel for large and illicit wealth, pork for unnecessary and dishonest appropriations of public money, joint for illegal liquor-house, tenderloin for gay and dubious neighborhood.[64] Most of these, and of the new compounds with them, belong to the vocabulary of disparagement. Here an essential character of the American shows itself: his tendency to combat the disagreeable with irony, to heap ridicule upon what he is suspicious of or doesn't understand. The rapidity with which new verbs are made in the United States is really quite amazing. Two days after the first regulations of the Food Administration were announced, to hooverize appeared spontaneously in scores of newspapers, and a week later it was employed without any visible sense of its novelty in the debates of Congress and had taken on a respectability equal to that of to bryanize, to fletcherize and to oslerize. To electrocute appeared inevitably in the first public discussion of capital [Pg164] punishment by electricity; to taxi came in with the first taxi-cabs; to commute no doubt accompanied the first commutation ticket; to insurge attended the birth of the Progressive balderdash. Of late the old affix -ize, once fecund of such monsters as to funeralize, has come into favor
- 68. again, and I note, among its other products, to belgiumize, to vacationize, to picturize and to scenarioize. In a newspaper headline I even find to s o s, in the form of its gerund.[65] Many characteristic American verbs are compounds of common verbs and prepositions or adverbs, with new meanings imposed. Compare, for example, to give and to give out, to go back and to go back on, to beat and to beat it, to light and to light out, to butt and to butt in, to turn and to turn down, to show and to show up, to put and to put over, to wind and to wind up. Sometimes, however, the addition seems to be merely rhetorical, as in to start off, to finish up, to open up and to hurry up. To hurry up is so commonplace in America that everyone uses it and no one notices it, but it remains rare in England. Up seems to be essential to many of these latter-day verbs, e. g., to pony up, to doll up, to ball up; without it they are without significance. Nearly all of them are attended by derivative adjectives or nouns; cut-up, show-down, kick-in, come-down, hang-out, start- off, run-in, balled-up, dolled-up, wind-up, bang-up, turn-down, jump-off. In many directions the same prodigal fancy shows itself—for example, in the free interchange of parts of speech, in the bold inflection of words not inflected in sound English, and in the invention of wholly artificial words. The first phenomenon has already concerned us. Would an English literary critic of any pretensions employ such a locution as "all by her lonesome"? I have a doubt of it—and yet I find that phrase in a serious book by the critic of the New Republic.[66] Would an English M. P. use "he has another think coming" in debate? Again I doubt it—but even more anarchistic dedications of verbs and adjectives to substantival use are to be found in the Congressional Record every day. Jitney is an old American substantive lately [Pg165] revived; a month after its revival it was also an adjective, and before long it may also be a verb and even an adverb. To lift up was turned tail first and made a substantive, and is now also an adjective and a verb. Joy-ride became a verb the day after it was born as a noun. And what of livest? An astounding inflection, indeed—but with quite sound
- 69. American usage behind it. The Metropolitan Magazine, of which Col. Roosevelt is an editor, announces on its letter paper that it is "the livest magazine in America," and Poetry, the organ of the new poetry movement, prints at the head of its contents page the following encomium from the New York Tribune: "the livest art in America today is poetry, and the livest expression of that art is in this little Chicago monthly." Now and then the spirit of American shows a transient faltering, and its inventiveness is displaced by a banal extension of meaning, so that a single noun comes to signify discrete things. Thus laundry, meaning originally a place where linen is washed, has come to mean also the linen itself. So, again, gun has come to mean fire-arms of all sorts, and has entered into such compounds as gun-man and gun- play. And in the same way party has been borrowed from the terminology of the law and made to do colloquial duty as a synonym for person. But such evidences of poverty are rare and abnormal; the whole movement of the language is toward the multiplication of substantives. A new object gets a new name, and that new name enters into the common vocabulary at once. Sundae and hokum are late examples; their origin is dubious and disputed, but they met genuine needs and so they seem to be secure. A great many more such substantives are deliberate inventions, for example, kodak, protectograph, conductorette, bevo, klaxon, vaseline, jap-a-lac, resinol, autocar, postum, crisco, electrolier, addressograph, alabastine, orangeade, pianola, victrola, dictagraph, kitchenette, crispette, cellarette, uneeda, triscuit and peptomint. Some of these indicate attempts at description: oleomargarine, phonograph and gasoline are older examples of that class. Others represent efforts to devise designations that will meet the conditions of advertising psychology and the trade-marks law, to wit, that they [Pg166] be (a) new, (b) easily remembered, and (c) not directly descriptive. Probably the most successful invention of this sort is kodak, which was devised by George Eastman, inventor of the portable camera so called. Kodak has so far won acceptance as a common noun that Eastman is often forced to assert his proprietary right to it.[67]
- 70. Vaseline is in the same position. The annual crop of such inventions in the United States is enormous.[68] The majority die, but a hearty few always survive. Of analogous character are artificial words of the scalawag and rambunctious class, the formation of which constantly goes on. Some of them are shortened compounds: grandificent (from grand and magnificent), sodalicious (from soda and delicious) and warphan(age) (from war and orphan(age)).[69] Others are made up of common roots and grotesque affixes: swelldoodle, splendiferous and peacharino. Yet others are mere extravagant inventions: scallywampus, supergobsloptious and floozy. Most of these are devised by advertisement writers or college students, and belong properly to slang, but there is a steady movement of selected specimens into the common vocabulary. The words in -doodle hint at German influences, and those in -ino owe something to Italian, or at least to popular burlesques of what is conceived to be Italian. § 6 Pronunciation —"Language," said Sayce, in 1879, "does not consist of letters, but of sounds, and until this fact has been brought home to us our study of it will be little better than an [Pg167] exercise of memory."[70] The theory, at that time, was somewhat strange to English grammarians and etymologists, despite the investigations of A. J. Ellis and the massive lesson of Grimm's law; their labors were largely wasted upon deductions from the written word. But since then, chiefly under the influence of Continental philologists, and particularly of the Dane, J. O. H. Jespersen, they have turned from orthographical futilities to the actual sounds of the tongue, and the latest and best grammar of it, that of Sweet, is frankly based upon the spoken English of educated Englishmen—not, remember, of conscious purists, but of the general body of cultivated folk. Unluckily, this new method also has its disadvantages. The men of a given race and
- 71. time usually write a good deal alike, or, at all events, attempt to write alike, but in their oral speech there are wide variations. "No two persons," says a leading contemporary authority upon English phonetics,[71] "pronounce exactly alike." Moreover, "even the best speaker commonly uses more than one style." The result is that it is extremely difficult to determine the prevailing pronunciation of a given combination of letters at any time and place. The persons whose speech is studied pronounce it with minute shades of difference, and admit other differences according as they are conversing naturally or endeavoring to exhibit their pronunciation. Worse, it is impossible to represent a great many of these shades in print. Sweet, trying to do it,[72] found himself, in the end, with a preposterous alphabet of 125 letters. Prince L.-L. Bonaparte more than doubled this number, and Ellis brought it to 390.[73] Other phonologists, English and Continental, have gone floundering into the same bog. The dictionary-makers, forced to a far greater economy of means, are brought into obscurity. The difficulties of the enterprise, in fact, are probably unsurmountable. It is, as White says, "almost impossible for one person to express to another by signs the [Pg168] sound of any word." "Only the voice," he goes on, "is capable of that; for the moment a sign is used the question arises, What is the value of that sign? The sounds of words are the most delicate, fleeting and inapprehensible things in nature.... Moreover, the question arises as to the capability to apprehend and distinguish sounds on the part of the person whose evidence is given."[74] Certain German orthoepists, despairing of the printed page, have turned to the phonograph, and there is a Deutsche Grammophon- Gesellschaft in Berlin which offers records of specimen speeches in a great many languages and dialects, including English. The phonograph has also been put to successful use in language teaching by various American correspondence schools. In view of all this it would be hopeless to attempt to exhibit in print the numerous small differences between English and American pronunciation, for many of them are extremely delicate and subtle, and only their aggregation makes them plain. According to a recent
- 72. and very careful observer,[75] the most important of them do not lie in pronunciation at all, properly so called, but in intonation. In this direction, he says, one must look for the true characters "of the English accent." I incline to agree with White,[76] that the pitch of the English voice is somewhat higher than that of the American, and that it is thus more penetrating. The nasal twang which Englishmen observe in the vox Americana, though it has high overtones, is itself not high pitched, but rather low pitched, as all constrained and muffled tones are apt to be. The causes of that twang have long engaged phonologists, and in the main they agree that there is a physical basis for it—that our generally dry climate and rapid changes of temperature produce an actual thickening of the membranes concerned in the production of sound.[77] We are, in brief, a somewhat snuffling [Pg169] people, and much more given to catarrhs and coryzas than the inhabitants of damp Britain. Perhaps this general impediment to free and easy utterance, subconsciously apprehended, is responsible for the American tendency to pronounce the separate syllables of a word with much more care than an Englishman bestows upon them; the American, in giving extraordinary six distinct syllables instead of the Englishman's grudging four, may be seeking to make up for his natural disability. Marsh, in his "Lectures on the English Language,"[78] sought two other explanations of the fact. On the one hand, he argued that the Americans of his day read a great deal more than the English, and were thus much more influenced by the spelling of words, and on the other hand he pointed out that "our flora shows that the climate of even our Northern States belongs ... to a more Southern type than that of England," and that "in Southern latitudes ... articulation is generally much more distinct than in Northern regions." In support of the latter proposition he cited the pronunciation of Spanish, Italian and Turkish, as compared with that of English, Danish and German—rather unfortunate examples, for the pronunciation of German is at least as clear as that of Italian. Swedish would have supported his case far better: the Swedes debase their vowels and slide over their consonants even more markedly than the English.
- 73. Marsh believed that there was a tendency among Southern peoples to throw the accent back, and that this helped to "bring out all the syllables." One finds a certain support for this notion in various American peculiarities of stress. Advertisement offers an example. The prevailing American pronunciation, despite incessant pedagogical counterblasts, puts the accent on the penult, whereas the English pronunciation stresses the second syllable. Paresis illustrates the same tendency. The English accent the first syllable, but, as Krapp says, American usage clings to the [Pg170] accent on the second syllable.[79] There are, again, pianist, primarily and telegrapher. The English accent the first syllable of each; we commonly accent the second. In temporarily they also accent the first; we accent the third. Various other examples might be cited. But when one had marshalled them their significance would be at once set at naught by four very familiar words, mamma, papa, inquiry and ally. Americans almost invariably accent each on the first syllable; Englishmen stress the second. For months, during 1918, the publishers of the Standard Dictionary, advertising that work in the street-cars, explained that ally should be accented on the second syllable, and pointed out that owners of their dictionary were safeguarded against the vulgarism of accenting it on the first. Nevertheless, this free and highly public instruction did not suffice to exterminate al´ly. I made note of the pronunciations overheard, with the word constantly on all lips. But one man of my acquaintance regularly accented the second syllable, and he was an eminent scholar, professionally devoted to the study of language. Thus it is unsafe, here as elsewhere, to generalize too facilely, and particularly unsafe to exhibit causes with too much assurance. "Man frage nicht warum," says Philipp Karl Buttmann. "Der Sprachgebrauch lässt sich nur beobachten."[80] But the greater distinctness of American utterance, whatever its genesis and machinery, is palpable enough in many familiar situations. "The typical American accent," says Vizetelly, "is often harsh and unmusical, but it sounds all of the letters to be sounded, and slurs, but does not distort, the rest."[81] An American, for example, almost
- 74. always sounds the first l in fulfill; an Englishman makes the first syllable foo. An American sounds every syllable in extraordinary, literary, military, secretary and the other words of the -ary-group; an Englishman never pronounces the a of the penultimate syllable. Kindness, with the d silent, would attract notice in the United States; in England, according to [Pg171] Jones,[82] the d is "very commonly, if not usually" omitted. Often, in America, commonly retains a full t; in England it is actually and officially offen. Let an American and an Englishman pronounce program (me). Though the Englishman retains the long form of the last syllable in writing, he reduces it in speaking to a thick triple consonant, grm; the American enunciates it clearly, rhyming it with damn. Or try the two with any word ending in -g, say sporting or ripping. Or with any word having r before a consonant, say card, harbor, lord or preferred. "The majority of Englishmen," says Menner, "certainly do not pronounce the r ...; just as certainly the majority of educated Americans pronounce it distinctly."[83] Henry James, visiting the United States after many years of residence in England, was much harassed by this persistent r-sound, which seemed to him to resemble "a sort of morose grinding of the back teeth."[84] So sensitive to it did he become that he began to hear where it was actually non-existent, save as an occasional barbarism, for example, in Cuba-r, vanilla-r and California-r. He put the blame for it, and for various other departures from the strict canon of contemporary English, upon "the American common school, the American newspaper, and the American Dutchman and Dago." Unluckily for his case, the full voicing of the r came into American long before the appearance of any of these influences. The early colonists, in fact, brought it with them from England, and it still prevailed there in Dr. Johnson's day, for he protested publicly against the "rough snarling sound" and led the movement which finally resulted in its extinction.[85] Today, extinct, it is mourned by English purists, and the Poet Laureate denounces the clergy of the Established Church for saying "the sawed of the Laud" instead of "the sword of the Lord."[86]
- 75. But even in the matter of elided consonants American is not always the conservator. We cling to the r, we preserve the final [Pg172] g, we give nephew a clear f-sound instead of the clouded English v- sound, and we boldly nationalize trait and pronounce its final t, but we drop the second p from pumpkin and change the m to n, we change the ph(=f)-sound to plain p in diphtheria, diphthong and naphtha,[87] we relieve rind of its final d, and, in the complete sentence, we slaughter consonants by assimilation. I have heard Englishmen say brand-new, but on American lips it is almost invariably bran-new. So nearly universal is this nasalization in the United States that certain American lexicographers have sought to found the term upon bran and not upon brand. Here the national speech is powerfully influenced by Southern dialectical variations, which in turn probably derive partly from French example and partly from the linguistic limitations of the negro. The latter, even after two hundred years, has great difficulties with our consonants, and often drops them. A familiar anecdote well illustrates his speech habit. On a train stopping at a small station in Georgia a darkey threw up a window and yelled "Wah ee?" The reply from a black on the platform was "Wah oo?" A Northerner aboard the train, puzzled by this inarticulate dialogue, sought light from a Southern passenger, who promptly translated the first question as "Where is he?" and the second as "Where is who?" A recent viewer with alarm[88] argues that this conspiracy against the consonants is spreading, and that English printed words no longer represent the actual sounds of the American language. "Like the French," he says, "we have a marked liaison—the borrowing of a letter from the preceding word. We invite one another to 'c'meer' (=come here) ... 'Hoo-zat?' (=who is that?) has as good a liaison as the French vois avez." This critic believes that American tends to abandon t for d, as in Sadd'y (=Saturday) and siddup (=sit up), and to get rid of h, as in "ware-zee?" (=where is he?). But here we invade the vulgar speech, which belongs to the next chapter. [Pg173] Among the vowels the most salient difference between English and American pronunciation, of course, is marked off by the flat
- 76. American a. This flat a, as we have seen, has been under attack at home for nearly a century. The New Englanders, very sensitive to English example, substitute a broad a that is even broader than the English, and an a of the same sort survives in the South in a few words, e. g., master, tomato and tassel, but everywhere else in the country the flat a prevails. Fashion and the example of the stage oppose it,[89] and it is under the ban of an active wing of schoolmasters, but it will not down. To the average American, indeed, the broad a is a banner of affectation, and he associates it unpleasantly with spats, Harvard, male tea-drinking, wrist watches and all the other objects of his social suspicion. He gets the flat sound, not only into such words as last, calf, dance and pastor, but even into piano and drama. Drama is sometimes drayma west of Connecticut, but almost never drahma or drawma. Tomato with the a of bat, may sometimes borrow the a of plate, but tomahto is confined to New England and the South. Hurrah, in American, has also borrowed the a of plate; one hears hurray much oftener than hurraw. Even amen frequently shows that a, though not when sung. Curiously enough, it is displaced in patent by the true flat a. The English rhyme the first syllable of the word with rate; in America it always rhymes with rat. The broad a is not only almost extinct outside of New England; it begins to show signs of decay even there. At all events, it has gradually disappeared from many words, and is measurably less sonorous in those in which it survives than it used to be. A century ago it appeared, not only in dance, aunt, glass, past, etc., but also in Daniel, imagine, rational and travel.[90] And in 1857 Oliver Wendell Holmes reported it in matter, handsome, caterpillar, apple and satisfaction. It has been displaced in virtually all of these, even in the most remote reaches of the back country, [Pg174] by the national flat a. Grandgent[91] says that the broad a is now restricted in New England to the following situations: 1. when followed by s or ns, as in last and dance. 2. when followed by r preceding another consonant, as in cart.
- 77. 3. when followed by lm, as in calm. 4. when followed by f, s or th, as in laugh, pass and path. The u-sound also shows certain differences between English and American usage. The English reduce the last syllable of figure to ger; the educated American preserves the u-sound as in nature. The English make the first syllable of courteous rhyme with fort; the American standard rhymes it with hurt. The English give an oo- sound to the u of brusque; in America the word commonly rhymes with tusk. A u-sound, as everyone knows, gets into the American pronunciation of clerk, by analogy with insert; the English cling to a broad a-sound, by analogy with hearth. Even the latter, in the United States, is often pronounced to rhyme with dearth. The American, in general, is much less careful than the Englishman to preserve the shadowy y-sound before u in words of the duke-class. He retains it in few, but surely not in new. Nor in duke, blue, stew, due, duty and true. Nor even in Tuesday. Purists often attack the simple oo-sound. In 1912, for example, the Department of Education of New York City warned all the municipal high-school teachers to combat it.[92] But it is doubtful that one pupil in a hundred was thereby induced to insert the y in induced. Finally there is lieutenant. The Englishman pronounces the first syllable left; the American invariably makes it loot. White says that the prevailing American pronunciation is relatively recent. "I never heard it," he reports, "in my boyhood."[93] He was born in New York in 1821. The i-sound presents several curious differences. The English make it long in all words of the hostile-class; in America it is commonly short, even in puerile. The English also lengthen it in sliver; in America the word usually rhymes with liver. The [Pg175] short i, in England, is almost universally substituted for the e in pretty, and this pronunciation is also inculcated in most American schools, but I often hear an unmistakable e-sound in the United States, making the first syllable rhyme with bet. Contrariwise, most Americans put the short i into been, making it rhyme with sin. In England it shows a long e-sound, as in seen. A recent poem by an English poet makes
- 78. the word rhyme with submarine, queen and unseen.[94] The o- sound, in American, tends to convert itself into an aw-sound. Cog still retains a pure o, but one seldom hears it in log or dog. Henry James denounces this "flatly-drawling group" in "The Question of Our Speech,"[95] and cites gawd, dawg, sawft, lawft, gawne, lawst and frawst as horrible examples. But the English themselves are not guiltless of the same fault. Many of the accusations that James levels at American, in truth, are echoed by Robert Bridges in "A Tract on the Present State of English Pronunciation." Both spend themselves upon opposing what, at bottom, are probably natural and inevitable movements—for example, the gradual decay of all the vowels to one of neutral color, represented by the e of danger, the u of suggest, the second o of common and the a of prevalent. This decay shows itself in many languages. In both English and High German, during their middle periods, all the terminal vowels degenerated to e—now sunk to the aforesaid neutral vowel in many German words, and expunged from English altogether. The same sound is encountered in languages so widely differing otherwise as Arabic, French and Swedish. "Its existence," says Sayce, "is a sign of age and decay; meaning has become more important than outward form, and the educated intelligence no longer demands a clear pronunciation in order to understand what is said."[96] All these differences between English and American pronunciation, separately considered, seem slight, but in the aggregate they are sufficient to place serious impediments between mutual [Pg176] comprehension. Let an Englishman and an American (not of New England) speak a quite ordinary sentence, "My aunt can't answer for my dancing the lancers even passably," and at once the gap separating the two pronunciations will be manifest. Here only the a is involved. Add a dozen everyday words—military, schedule, trait, hostile, been, lieutenant, patent, nephew, secretary, advertisement, and so on—and the strangeness of one to the other is augmented. "Every Englishman visiting the States for the first time," said an English dramatist some time ago, "has a difficulty in making himself understood. He often has to repeat a remark or a request two or
- 79. three times to make his meaning clear, especially on railroads, in hotels and at bars. The American visiting England for the first time has the same trouble."[97] Despite the fact that American actors imitate English pronunciation to the best of their skill, this visiting Englishman asserted that the average American audience is incapable of understanding a genuinely English company, at least "when the speeches are rattled off in conversational style." When he presented one of his own plays with an English company, he said, many American acquaintances, after witnessing the performance, asked him to lend them the manuscript, "that they might visit it again with some understanding of the dialogue."[98]
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