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An Introduction to
Mathematical Modeling

WILEY SERIES IN COMPUTATIONAL MECHANICS
SeriesAdvisors: René de Borst, Perumal Nithiarasu,Tayfun Tezduyar,
Cenki Yagawa, Tarek Zohdi
Introduction to Finite Element Analysis: Formulation, Verification and Validation
Barna Szabó, Ivo (March 2011)
An Introduction to Mathematical Modeling A Course in Mechanics
J. Tinsley Oden (September 2011)
Computational Mechanics of Discontinua
Antonio A Munjiza, Earl Knight, Esteban Rougier (October 2011)

An Introduction to
Mathematical Modeling
A Course in Mechanics
J. Tinsley Oden
LEY
A JOHN WILEY & SONS, INC., PUBLICATION

Copyright © 2011 by John Wiley & Sons. Inc All rights reserved
Published by John Wiley & Sons. Inc. Hoboken, New Jersey
Published simultaneously in Canada
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Library of Congress Cataloging-in-Publication Data:
Oden, J. Tinsley (John Tinsley), I
An introduction to mathematical modeling a course in mechanics / J Tinsley Oden
p cm (Wiley senes in computational mechanics)
lncludes bibliographical references and index
1SBN 978-I-I (hardback)
I Mechanics, Analytic I Title
QA807 2011
2011012204
Pnnted in the United States of Amenca
oBook 978-I-I
ePDF 978-I-I 18-10576-4
ePub 978-1-118-10574-0
10 9 8 7 6 5 4 2 I

CONTENTS
Preface Xiii
I Nonlinear Continuum Mechanics 1
1 Kinematics of Deformable Bodies 3
1.1 Motion 4
1.2 Strain and Deformation Tensors . . 7
1.3 Rates of Motion . 10
1.4 Rates of Deformation 13
1.5 The Piola Transformation 15
1.6 The Polar Decomposition Theorem 19
1.7 Principal Directions and Invariants of Deformation and
Strain 20
1.8 The Reynolds' Transport Theorem 23
2 Mass and Momentum 25
2.1 Local Forms of the Principle of Conservation of Mass . . 26
2.2 Momentum 28
3 Force and Stress in Deformabte Bodies 29
4 The Principles of Balance of Linear and Angular Momentum 35
4.1 Cauchy's Theorem: The Cauchy Stress Tensor 36
VII

Viii CONTENTS
4.2 The Equations of Motion (Linear Momentum) 38
4.3 The Equations of Motion Referred to the Reference
Configuration: The Piola—Kirchhoff Stress Tensors . . . 40
4.4 Power 42
5 The Principle of Conservation of Energy 45
5.1 Energy and the Conservation of Energy 45
5.2 Local Forms of the Principle of Conservation of Energy . 47
6 Thermodynamics of Continua and the Second Law 49
7 Constitutive Equations 53
7.1 Rules and Principles for Constitutive Equations 54
7.2 Principle of Material Frame Indifference 57
7.2.1 Solids 57
7.2.2 Fluids 59
7.3 The Coleman—Noll Method: Consistency with the
Second Law of Thermodynamics 60
8 Examples and Applications 63
8.1 The Navier—Stokes Equations for Incompressible Flow . 63
8.2 Flow of Gases and Compressible Fluids:
The Compressible Navier—Stokes Equations 66
8.3 Heat Conduction 67
8.4 Theory of Elasticity 69
II Electromagnetic Field Theory and Quantum
Mechanics 73
9 Electromagnetic Waves 75
9.1 Introduction 75
9.2 Electric Fields 75
9.3 Gauss's Law 79
9.4 Electric Potential Energy 80
9.4.1 Atom Models 80
95 Magnetic Fields 81

CONTENTS ix
9.6 Some Properties of Waves 84
9.7 Maxwell's Equations 87
9.8 Electromagnetic Waves 91
10 Introduction to Quantum Mechanics 93
10.1 Introductory Comments 93
10.2 Wave and Particle Mechanics 94
10.3 Heisenberg's Uncertainty Principle 97
10.4 Schrodinger's Equation 99
10.4.1 The Case of a Free Particle 99
10.4.2 Superposition in 101
10.4.3 Hamiltonian Form 102
10.4.4 The Case of Potential Energy 102
10.4.5 Relativistic Quantum Mechanics 102
10.4.6 General Formulations of Schrodinger's Equation 103
10.4.7 The Time-Independent Schrodinger Equation . . 104
10.5 Elementary Properties of the Wave Equation 104
10.5.1 Review 104
10.5.2 Momentum 106
10.5.3 Wave Packets and Fourier Transforms 109
10.6 The Wave—Momentum Duality 110
10.7 Appendix: A Brief Review of Probability Densities . . . 111
11 Dynamical Variables and Observables in Quantum
Mechanics: The Mathematical Formalism 115
11.1 Introductory Remarks 115
11.2 The Hilbert Spaces L2(IR) (or and H'(IR) (or
Hl(IRd)) 116
11.3 Dynamical Variables and Hermitian Operators 118
11.4 Spectral Theory of Hermitian Operators: The Discrete
Spectrum 121
11.5 Observables and Statistical Distributions 125
11.6 The Continuous Spectrum 127
11.7 The Generalized Uncertainty Principle for Dynamical
Variables 128
11.7.1 Simultaneous Eigenfunctions 130

X CONTENTS
12 Applications: The Harmonic Oscillator and the Hydrogen
Atom 131
12.1
12.2
Introductory Remarks
Ground States and Energy Quanta: The Harmonic
Oscillator
12.3 The Hydrogen Atom
12.3.1 Schrodinger Equation in Spherical Coordinates
12.3.2 The Radial Equation
12.3.3 The Angular Equation
12.3.4 The Orbitals of the Hydrogen Atom
12.3.5 Spectroscopic States
13 Spin and Pauli's Principle 145
13.1 Angular Momentum and Spin 145
13.2 Extrinsic Angular Momentum 147
13.2.1 The Ladder Property: Raising and Lowering States 149
13.3 Spin 151
13.4 Identical Particles and Pauli's Principle 155
13.5 The Helium Atom 158
13.6 Variational Principle 161
14 Atomic and Molecular Structure
14.1 Introduction
14.2 Electronic Structure of Atomic Elements
14.3 The Periodic Table
14.4 Atomic Bonds and Molecules
145 Examples of Molecular Structures .
165
165
165
169
173
180
15 Ab Initio Methods: Approximate Methods and Density
Functional Theory 189
15.1 Introduction 189
15.2 The Born—Oppenheimer Approximation 190
15.3 The Hartree and the Hartree—Fock Methods 194
15.3.1 The Hartree Method 196
15.3.2 The Hartree—Fock Method 196
15.3.3 The Roothaan Equations 199
131
131
133
135
136
138
140
140

CONTENTS xi
15.4 Density Functional Theory 200
15.4.1 Electron Density 200
15.4.2 The Hohenberg—Kohn Theorem 205
15.4.3 The Kohn—Sham Theory 208
III Statistical Mechanics 213
16 Basic Concepts: Ensembles, Distribution Functions, and
Averages 215
16.2 Hamiltonian Mechanics 216
16.2.1 The Hamiltonian and the Equations of Motion. . 218
16.3 Phase Functions and Time Averages 219
16.4 Ensembles, Ensemble Averages, and Ergodic Systems. . 220
16.5 Statistical Mechanics of Isolated Systems 224
16.6 The Microcanonical Ensemble 228
16.6.1 Composite Systems 230
16.7 The Canonical Ensemble 234
16.8 The Grand Canonical Ensemble 239
16.9 Appendix: A Brief Account of Molecular Dynamics . . 240
16.9.1 Newtonian's Equations of Motion 241
16.9.2 Potential Functions 242
16.9.3 Numerical Solution of the Dynamical System . . 245
17 Statistical Mechanics Basis of Classical Thermodynamics 249
17.2 Energy and the First Law of Thermodynamics 250
17.3 Statistical Mechanics Interpretation of the Rate of Work
in Quasi-Static Processes 251
17.4 Statistical Mechanics Interpretation of the First Law of
Thermodynamics 254
17.4.1 Statistical Interpretation of Q 256
17.5 Entropy and the Partition Function 257
17.6 Conjugate Hamiltonians 259
17.7 The Gibbs Relations 261

Xii CONTENTS
17.8 Monte Carlo and Metropolis Methods 262
17.8.1 The Partition Function for a Canonical Ensemble 263
17.8.2 The Metropolis Method 264
17.9 Kinetic Theory: Boltzmann's Equation of
Nonequilibrium Statistical Mechanics 265
17.9.1 Boltzmann's Equation 265
17.9.2 Collision Invariants 268
17.9.3 The Continuum Mechanics of Compressible
Fluids and Gases: The Macroscopic Balance
Laws 269
Exercises 273
Bibliography 317
Index 325

PREFACE
This text was written for a course on An Introduction to Mathematical
Modeling for students with diverse backgrounds in science, mathemat-
ics, and engineering who enter our program in Computational Science,
Engineering, and Mathematics. It is not, however, a course on just
how to construct mathematical models of physical phenomena. It is a
course designed to survey the classical mathematical models of subjects
forming the foundations of modern science and engineering at a level ac-
cessible to students finishing undergraduate degrees or entering graduate
programs in computational science. Along the way, I develop through
examples how the most successful models in use today arise from basic
principles and modern and classical mathematics. Students are expected
to be equipped with some knowledge of linear algebra, matrix theory,
vector calculus, and introductory partial differential equations, but those
without all these prerequisites should be able to fill in some of the gaps
by doing the exercises.
I have chosen to call this a textbook on mechanics, since it cov-
ers introductions to continuum mechanics, electrodynamics, quantum
mechanics, and statistical mechanics. If mechanics is the branch of
physics and mathematical science concerned with describing the motion
of bodies, including their deformation and temperature changes, under
the action of forces, and if one adds to this the study of the propagation
of waves and the transformation of energy in physical systems, then the
term mechanics does indeed apply to everything that is covered here.
The course is divided into three parts. Part I is a short course on
XIII

Xiv PREFACE
nonlinear continuum mechanics; Part II contains a brief account of
electromagnetic wave theory and Maxwell's equations, along with an
introductory account of quantum mechanics, pitched at an undergraduate
level but aimed at students with a bit more mathematical sophistication
than many undergraduates in physics or engineering; and Part III is a
brief introduction to statistical mechanics of systems, primarily those in
thermodynamic equilibrium.
There are many good treatments of the component parts of this work
that have contributed to my understanding of these subjects and inspired
their treatment here. The books of Gurtin, Ciarlet, and Batra provide
excellent accounts of continuum mechanics at an accessible level, and
the excellent book of Griffiths on introductory quantum mechanics is a
well-crafted text on this subject. The accounts of statistical mechanics
laid out in the book of Weiner and the text of McQuarrie, among others,
provide good introductions to this subject. I hope that the short excursion
into these subjects contained in this book will inspire students to want to
learn more about these subjects and will equip them with the tools needed
to pursue deeper studies covered in more advanced texts, including some
listed in the references.
The evolution of these notes over a period of years benefited from
input from several colleagues. I am grateful to Serge Prudhomme, who
proofread early versions and made useful suggestions for improvement.
I thank Alex Demkov for reading and commenting on Part II. My sin-
cere thanks also go to Albert Romkes, who helped with early drafts, to
Ludovic Chamoin, who helped compile and type early drafts of the ma-
terial on quantum mechanics, and Kris van der Zee, who helped compile
a draft of the manuscript and devoted much time to proofreading and
helping with exercises. I am also indebted to Pablo Seleson, who made
many suggestions that improved presentations in Part II and Part III and
who was of invaluable help in putting the final draft together.
J. Tinsley Oden
Austin, Texas
June 2011

In Inirodticiion 10 .1 laiheinailcal .1 lodeling I ( in .1 lechanic
J Oden
2011 John Wiles & Sons. Inc
Part I
Nonlinear Continuum
Mechanics

In Inirothiclion 10 .1 Iaiheinaiical .1 lode/mg I ( in .1 lechanic
J Oden
Cops nghl 2011 John Wiles & Sons. Inc
CHAPTER 1
KINEMATICS OF
DEFORMABLE BODIES
Continuum mechanics models the physical universe as a collection of
"deformable bodies:' a concept that is easily accepted from our everyday
experiences with observable phenomena. Deformable bodies occupy
regions in three-dimensional Euclidean space E, and a given body will
occupy different regions at different times. The subsets of E occupied
by a body 13 are called its configurations. It is always convenient to
identify one configuration in which the geometry and physical state of
the body are known and to use that as the reference configuration; then
other configurations of the body can be characterized by comparing them
with the reference configuration (in ways we will make precise later).
For a given body, we will assume that the reference configuration is
an open, bounded, connected subset of 1R3 with a smooth boundary
The body is made up of physical points called material points.
To identify these points, we assign each a vector X and we identify
the components of X as the coordinates of the place occupied by the
material point when the body is in its reference configuration relative to
a fixed Cartesian coordinate system.
It is thus important to understand that the body 13 is a non-
denumerable set of material points X. This is the fundamental hypoth-
An Introduction to Mathematical Modeling A Course in Mechanics, First Edition By J. Tinsley Oden
© 2011 John Wiley &' 5ons, Inc. Published 2011 by John Wiley &' 5ons. Inc.
3

4 CHAPTER 1. KINEMATICS OF DEFORMABLE BODIES
Figure 1.1: Motion from the reference configuration to the current config-
uration
esis of continuum mechanics: Matter is not discrete; it is continuously
distributed in one-to-one correspondence with points in some subset of
1R3. Bodies are thus "continuous media": The components of X with
respect to some basis are real numbers. Symbolically, we could write
E
for some orthonormal basis {ei, e2, e3} and origin 0 chosen in three-
dimensional Euclidean space and, thus, identified with Hereafter,
repeated indices are summed throughout their ranges; i.e. the "summa-
tion convention" is employed.
Kinematics is the study of the motion of bodies, without regard to
the causes of the motion. It is purely a study of geometry and is an exact
science within the hypothesis of a continuum (a continuous media).
1.1 Motion
We pick a point 0 in as the origin of a fixed coordinate system
(x1, x2, x3) = x defined by orthonormal vectors e2, i = 1,2,3. The
system (x1, x2, x3) is called the spatial coordinate system. When the
physical body B occupies its reference configuration at, say, time
t = 0, the material point X occupies a position (place) corresponding
to the vector X = X2e2. The spatial coordinates (X1, X2, X3) of X
e

11 MOTION 5
0
Figure 1.2: A discrete set of material particles.
are labels that identify the material point. The coordinate labels are
sometimes called material coordinates (see Fig. 1.1).
Remark Notice that if there were a countable set of discrete material
points, such as one might use in models of molecular or atomistic dy-
namics, the particles (discrete masses) could be labeled using natural
numbers n E N, as indicated in Fig. 1.2. But the particles (material
points) in a continuum are not countable, so the use of a label of three
real numbers for each particle corresponding to the coordinates of their
position (at t = 0) in the reference configuration seems to be a very
natural way to identify such particles. 0
The body moves through E over a period of time and occupies a
configuration c at time t. Thus, material points X in (the
closure of are mapped into positions x in by a smooth vector-
valued mapping (see Fig. 1.1)
x = (1.1)
Thus, p(X, t) is the spatial position of the material point X at time t.
The one-parameter family t)} of positions is called the trajectory
of X. We demand that be differentiable, injective, and orientation
preserving. Then is called the motion of the body:
1. is called the current configuration of the body.
x3
0
0
r2 0
0
0

6 CHAPTER 1 KINEMATICS OF DEFORMABLE BODIES
2. is injective (except possibly at the boundary of no).
3. is orientation preserving (which means that the physical ma-
terial cannot penetrate itself or reverse the orientation of material
coordinates, which means that det t) > 0).
Hereafter we will not explicitly show the dependence of and other
quantities on time t unless needed; this time dependency is taken up
later.
The vector field
_________________
(1.2)
is the displacement of point X. Note that
dx = dX (i.e.. = - dX3).
The tensor
F(X) = (1.3)
is called the deformation gradient. Clearly,
F(X) = I+Vu(X), (1.4)
where I is the identity tensor and Vu is the displacement gradient.
Some Definitions
• A deformation is homogeneous if F = C = constant.
• A motion is rigid if it is the sum of a translation a and a rotation
Q:
p(X)=a+QX,
where a E IRS, Q E with the set of orthogonal matrices
of order 3 with determinant equal to + 1.

1 2 STRAIN AND DEFORMATION TENSORS 7
• As noted earlier, the fact that the motion is orientation preserving
means that
• Recall that
Cof F = cofactor matrix (tensor) of F = det F F_T.
For any matrix A = [As,] of order n and for each row i and
column j, let be the matrix of order n — 1 obtained by deleting
the ith row and jth column of A. Let = det A,.
Then the matrix
Cof A =
is the cofactor matrix of A and is the (i, j) cofactor of A. Note
that
A(CofA)T = (CofA)TA = (detA)I. (1.5)
1.2 Strain and Deformation Tensors
A differential material line segment in the reference configuration is
= dXTdX = + +
while the same material line in the current configuration is
dS2 = dxTdx = dXTFTFdX.
The tensor
C = FTF = the right Cauchy—Green deformation tensor
is thus a measure of the change in due to (gradients of) the motion
dS2 = dXTCdX — dXTdX.

C is symmetric, positive definite. Another deformation measure is
simply
dS2 — = dXT(2E) dx,
where
E = — I) = the Green—St. Venant strain (1.6)
Since F = I + Vu and C FTF, we have
E= (1.7)
The tensor
B = FFT = the left Cauchy—Green deformation tensor
is also symmetric and positive definite, and we can likewise define
dS2 — = dXT FT(2A)F dX dxT(2A) dx,
where
A = — B—') = the Almansi—Hamel strain tensor (1.8)
or
A= (1.9)
where grad u is the spatial gradient grad u = 8u/8x (i.e., (grad
see also Sec. 1.3.

1 2 STRAIN AND DEFORMATION TENSORS 9
Interpretation of E Take dS0 = (i.e.,dX (dXi, 0, O)T) Then
dS2 — = dS2 — = (1 +
so
( a measure of the stretch of a
— i) = material line originally oriented
in the direction in
We call ei the extension in the direction at X (which is a dimension-
less measure of change in length per unit length)
def dS —
dX
= + 1,
or
= (1 + ei)2 — 1.
Similar definitions apply to E22 and E33.
Now take dX = (dxi, dX2, 0)T and
C12
COSO
— IIdxiII IIdx2II = + + 2E22
(Exercise).
The shear (or shear strain) in the plane is defined by the angle
change (see Fig. 1.3),
def ir
'Yi2 =
Therefore
= I
I
(1.10)
Thus, (and, analogously, and E23) is a measure of the shear in
the Xi—X2 (or and X2—X3) plane.
Small strains The tensor
e = + vuT), (1.11)

dx1
dX1
Figure 13: Change of angle through the motion p.
is called the infinitesimal or small or engineering strain tensor. Clearly
E = e + (1.12)
Note that if E is "small" (i.e., << 1), then we obtain
(1 + 2E11)'12 — 1
= 1 + E11 — 1 +
=
that is,
dS—dX1
e11 =
= dX
, etc.,
and
2e12 = sin Y12 etc.
Thus, small strains can be given the classical textbook interpretation:
is the change in length per unit length and e12 is the change in the right
angle between material lines in the X1 and X2 directions. In the case of
small strains, the Green—St. Venant strain tensor and the Almansi—Hamel
strain tensor are indistinguishable.
1.3 Rates of Motion
If t) is the motion (of X at time t), i.e.,
x =
dX2
dx2

1 3 RATES OF MOTION 11
then
=
def
(1.13)
at
is the velocity and
= (1.14)
at2
is the acceleration. Since is (in general) bijective, we can also describe
the velocity as a function of the place x in 1R3 and time t:
v = v(x,t) =
This is called the spatial description of the velocity.
This leads to two different ways to interpret the rates of motion of
continua:
• The material description (functions are defined on material points
X in the body 13 in correspondence to points in 1R3);
• The spatial description (functions are defined on (spatial) places
x in 1R3).
When the equations of continuum mechanics are written in terms of the
material description, the collective equations are commonly referred to as
the Lagrangian form (formulation) of the equations (see Fig. 1.4). When
the spatial description is used, the term Eulerian form (formulation) is
used (see Fig. 1.5).
There are differences in the way rates of change appear in the La-
grangian and Eulerian formulations.
• In the Lagrangian case: Given a field 'ZI)m t) (the sub-
script m reminding us that we presume is a function of the
material coordinates),
dZI)m(X,t) ax
dt at
+
ax

"S —
(II
Fixed in space
)( at1 >
892(X, t0)
i)t
Figure 1.4: Lagrangian (material) description of velocity. The velocity of a
material point is the time rate of change of the position of the point as it moves
along its path (its trajectory) in R3.
with time,
hot is fixed)
-Fixed in space
Figure 1.5: Eulerian (spatial) description of velocity. The velocity at a fixed
place x in R3 is the speed and direction (at time t) of particles flowing through
the place x.
but 3X/3t = 0 because X is simply a label of a material point.
Thus,
dt,bm(X,t) —
dt —
(1.15)
dl I U

1 4 RATES OF DEFORMATION 13
• In the Eulerian case: Given a field = t),
+
dt 8t xfixed ax
but = v(x, t) is the velocity at position x and time t. Thus,
+v(x,t). (1.16)
Notation We distinguish between the gradient and divergence of fields
in the Lagrangian and Eulerian formulation as follows:
a a
Lagrangian:
a a
Eulerian: grad, v = div v.
ax ax
In classical literature, some authors write
(1.17)
as the "material time derivative" of a scalar field giving the rate of
change of at a fixed described place x at time t. Thus, in the Eulerian
formulation, the acceleration is
Dv 3v
a= =
v being the velocity.
1.4 Rates of Deformation
The spatial (Eulerian) field
L = L(x, t) t) = grad v(x, t) (1.18)

is the velocity gradient. The time rate of change of the deformation
gradient F is
F =
3 3v3x
=
=
= gradvF,
or
(1.19)
where Lm = L is written in material coordinates, so
(1.20)
It is standard practice to write L in terms of its symmetric and skew-
symmetric parts:
L=D+W. (1.21)
Here
D = (L + LT) = the deformation rate tensor,
2
(1.22)
W = — LT) = the spin tensor.
We can easily show that if v is the velocity field,
(1.23)
where w is the vorticity
w=curlv. (1.24)
Recall (cf. Exercise 2.6) that
D(detA) V = (det A)VT: A1,
for any invertible tensor A and arbitrary V C L(V, V). Also, if
f(g(t)) = f o g(t) denotes the composition of functions f and g,
the chain rule of differentiation leads to
df(g(t))
= df(g(t)).
dg(t)
= Df(g(t)) :

1.5 THE PIOLA TRANSFORMATION 15
no
Figure 1.6: Mapping from reference configuration into current configuration.
Combining these expressions, we have
detF = = D(detF) : F = detF FT : F'
=detF trLm=detF divv
(since Fr: F-' = = trLm, where trL = trgradv = divv).
Summing up:
det F det F div v. (1.25)
There is a more constructive way of deriving (1.25) using the definitions
of determinant and cofactors of F; see Exercise 4 in Set 1.2.
1.5 The Piola Transformation
The situation is this: A subdomain G0 C of the reference configura-
tion of a body, with boundary 3G0 and unit exterior vector n0 normal to
the surface-area element dAn, is mapped by the motion into a subdo-
main C = C of the current configuration with boundary ÔG
with unit exterior vector n normal to the "deformed" surface area dA
(see Fig. 1.6).
Let T = T(x) denote a tensor field defined on G and
T(x) n(x) the flux of T across 3G, n(x) being a unit normal to 3G.
Here is fixed so t is held constant and not displayed. Corresponding
to T, a tensor field T0 = T0(X) is defined on C0 that associates the

flux T0(X) no(X) through 3G0, no(X) being the unit normal to 3G0.
We seek a relationship between T0(X) and T(x) that will result in the
same totalflux through the surfaces 3G0 and 3G, so that
f To(X)no(X) dA0
= f T(x)n(x) dA, (1.26)
8G0 8G
with x = This relationship between T0 and T is called the Piola
transformation.
Proposition 1.1 (Piola Transformation) The above correspon-
dence holds if
T0(X) = detF(X) T(x) F(X)_T = T(x) CofF(X). (1.27)
Proof (This development follows that of Ciarlet [2]). We will use the
Green's formulas (divergence theorems)
f DivTodX=f
G0 8G0
and
fdivTdx=
f8G
where
DivT0=V•T0=
3X
3x3
dx = dx1dx2dx3 = det F dX = det F dX1dX2dX3.
We will also need to use the fact that
= 0.

1 5 THE PIOLA TRANSFORMATION 17
To show this, we first verify by direct calculation that
=
= ( (
axj+i axj+2
/ a / a
— ( (
J
where no summation is used. Then a direct computation shows that
a
= 0.
'-/3
Next, set
T0(X) = T(x) CofF(X).
Noting that
= F1
and
aXm — —
— _(Jz3,
3Xm 3X3
we see that
= detF = detF
Thus,
Div T0(X) 0 e3 =
aTirn(x)
.
+
aTim axr ax3
=
aXr aXm
aXr
=divT detF,

that is,
___________________________
DivT0=detF divT. (1.28)
Thus
f Div T0 dX
= f det F divTdX
= f Tono
G0 C0 8G0
T0n0 dA0
= f divT det F dX
= f div T dx
= / Tn dA,
as asserted.
Corollaries and Observations The Piola transformation provides a
means for characterizing the flux of a field through a material surface in
the current configuration in terms of the representation of the surface in
the reference configuration. It also provides fundamental relationships
between differential surface areas and their orientations in the reference
and current configurations. We list a few of these as corollaries and
observations.
• Since G0 is arbitrary (symbolically), we obtain
T0n0 dA0 = Tn dA. (1.29)
• Set T = I = identity. Then
detFFTn0dA0=ndA. (1.30)
• Sincen = . (detF)FTn0 and = 1,we have
dA = det F (Nanson's Formula), (1.31)
where . denotes the Eulerian norm. Thus
Cof Fn0
n= . (1.32)
CofFnoM

1 6 THE POLAR DECOMPOSITION THEOREM 19
1.6 The Polar Decomposition Theorem
Theorem l.A (Polar Decomposition) A real invertible matrix F
can be factored in a unique way as
F=RU=VR, (1.33)
where R is an orthogonal matrix and U and V are symmetric positive
definite matrices.
Proof (We will use as a fact the following lemma: For every symmetric
positive definite matrix A, there exists a unique symmetric positive
definite matrix B such that B2 = A.) Let us first show the existence of
the matrices U and V. Define U by
U2 = FTF = C
(which is possible by virtue of the lemma stated above). Then let
R = FU1.
Then
RTR = = U1UUU1 = I.
Thus R is a rotation. We have thus shown that there exists a U such that
F=RU.
Next, define
V = RURT.
Then
VR = RURTR = RU = F,
as asserted.
To show that U and V are unique, let F = RU, R being the rotation
matrix. Then FTF = URTRU = U2, which means U is unique by the
lemma stated. Since R = FU1, R is also uniquely defined. Finally, if
F = VR, then FFT = B = V2, so by the same lemma, V is unique. n

20 CHAPTER 1. KINEMATICS OF DEFORMA&E BODIES
I I
I I
L
L L -
Figure 1.7: The Polar Decomposition Theorem: F = RU = yR.
Summing up, if C = FTF and B = FFT are the right and left Cauchy—
Green deformation tensors and
F= RU RU = VR,
then
C = = UTU U2,
(1.34)
B=VRRTV =VVT=V2,
where U and V are the right and left stretch tensors, respectively.
Clearly, the Polar Decomposition Theorem establishes that the de-
formation gradient F can be obtained (or can be viewed) as the result of
a distortion followed by a rotation or vice versa (see Fig. 1.7).
1.7 Principal Directions and Invariants of Defor-
mation and Strain
For a given deformation tensor field C(X) and strain field E(X) (at
point X), recall that dXTCdX = 2dXTEdX — dXTdX is the square
d82 of a material line segment in the current configuration. Suppose the

1 7 PRINCIPAL DIRECTIONS AND INVARIANTS OF DEFORMATION AND STRAIN 21
material line in question is oriented in the direction of a unit vector m
in the reference configuration so that dX = m dS0. Then a measure of
the stretch or compression of a unit material element originally oriented
along a unit vector m is given by
= dS2/dS2 = mTCm,
0
(1.35)
m m = = 1.
One may ask: Of all possible directions m at X, which choice results
in the largest (or smallest) value of
This is a constrained maximization/minimization problem: Find
m = mmax (or mmjn) that makes as large (or small) as pos-
sible, subject to the constraint mTm = 1. To resolve this problem,
we use the method of Lagrange multipliers. Denote by L(m, A) =
— A(mTm — 1), A being the Lagrange multiplier. The maxima
(on minimize and maximize points) of L satisfy,
3L(m,
= 0= 2(Cm — Am).
3m
Thus, unit vectors m that maximize or minimize are associ-
ated with multipliers A and satisfy
Cm = Am, mTm = 1. (1.36)
That is, (m, A) are eigenvector/eigenvalue pairs of the deformation ten-
sor C, and m is normalized so that mTm = 1 (or = 1).
The following fundamental properties of the above eigenvalue prob-
lem can be listed.
1. There are three real eigenvalues and three eigenvectors of C (at
X); we adopt the ordering A1 A2 A3.
2. For A3, the corresponding eigenvectors are orthogonal (for
pairs (mi, and (m3, A3)), mTm3 = as can be seen as
follows:
mT(A3m3) = mTCm3 = =

so
— A3)mTmj = 0,
so
1<i,j<3
(if = we can always construct m3 so that it is orthogonal to
mi).
3. Equation (1 .36) can be written as
(C—AI)m=O. (1.37)
This equation can have nontrivial solutions only if the determinant
of C — Al is zero. This is precisely the characteristic polynomial
of C:
det(C — Al) = —A3 + I(C)A2 — 11(C)A + 11(C), (1.38)
where I, 11, 1/1 are the principal invariants of C:
=C11+C22+C33,
11(C) = — = trCofC, (1.39)
11(C) = detC = — 3trCtrC2 +2trC3).
(An invariant of a real matrix C is any real-valued function 1t(C)
with the property 1t(C) = 1t(A1CA) for all invertible matrices
A.)
4. Because the eigenvectors are all positive, it is customary to write
for the eigenvectors instead of Then (C — = 0.
Let N be the matrix with the mutually orthogonal eigenvectors as
rows. Then
0 0
NTCN = 0 0 = = 1,2,3}. (1.40)
0 0

1 8 THE REYNOLDS' TRANSPORT THEOREM 23
The coordinate system defined by the mutually orthogonal triad of
eigenvectors define the principal directions and values of C at X.
For this choice of a basis, we obtain
(1.41)
If 4 corresponds to the maximum, to the
minimum, and to a "mini-max" principal value of C (or of
Notice that the stretch along, say, m1 is (mTCmi)112 = A1, etc.
Also,
0 0
C=U2= 0 0 . (1.42)
0 0
The principal invariants are thus
1(C) = + +
11(C) = + + (1.43)
11(C) =
1.8 The Reynolds' Transport Theorem
We frequently encounter the need to evaluate the total time rate of change
of a field, either densities or measures of concentrations per unit volume,
defined over a volume w C For instance, if 'P = 'P(x, t) is a
spatial field, either scalar- or vector-valued, suppose we wish to compute
'J'dx)/dt. The following change of integration variables facilitates
such a calculation. Let w0 be the region of occupied by the material,

while in the reference configuration, that occupies w in Then,
f detF)dX
f
+1
=f +
Thus,
(1.44)
This last result is known as the Reynolds' Transport Theorem.

In Inirothiclion 10 .1 Iaiheinaiical .1 lode/mg I ( 'ourse in .1 lechanic
J Oden
CHAPTER 2
MASS AND MOMENTUM
A common dictionary definition of mass is as follows:
Mass The property of a body that is a measure of the amount of
material it contains and causes it to have weight in a gravitational
field.
In continuum mechanics, the mass of a body is continuously dis-
tributed over its volume and is an integral of a density field p: —f
called the mass density. The total mass M(B) of a body is independent
of the motion but the mass density p can, of course, change as the
volume of the body changes while in motion. Symbolically,
(2.1)
where dx = volume element in the current configuration of the body.
Given two motions and (see Fig. 2.1), let and denote
the mass densities in the configurations and respectively.
An Introduction to Mathematical Modeling A Course in Mechanics, First Edition By J Tinsley Oden
© 2011 John Wiley & Sons, Inc Published 2011 by John Wiley & 5ons, Inc
25

26 CHAPTER 2. MASS AND MOMENTUM
Figure 2.1: Two motions p and
Since the total mass is independent of the motion,
(2.2)
This fact represents the principle of conservation of mass. The mass of a
body 13 is thus an invariant property (measuring the amount of material
in 13); the weight of 13 is defined as gM (13) where g is a constant gravity
field. Thus, a body may weigh differently in different gravity fields (e.g.,
the earth's gravity as opposed to that on the moon), but its mass is the
same.
2.1 Local Forms of the Principle of Conservation
of Mass
Let po(X) be the mass density of a body in its reference configura-
tion and let Q(x, t) be the mass density in the current configuration
lit. Then
J Qo(X)dX=J
c10

2 1 LOCAL FORMS OF THE PRINCIPLE OF CONSERVATION OF MASS 27
(where the dependence of p on t has been suppressed). But dx =
det F(X) dx, so
[po(X) — det F(X)] dX = 0,
and therefore
__________________________
po(X) = Q(x)detF(X).] (2.3)
This is the material description (or the Lagrangian formulation) of the
principle of conservation of mass. To obtain the spatial description (or
Eulerian formulation), we observe that the invariance of total mass can
be expressed as
[ p(x,t)dx=0.
dt
Changing to the material coordinates gives
where (.) = d(.)/dt. Recalling that detF = detFdivv, we have
0
=
f det F div v + + v grad dX
= L0
dX
=
dx,
from which we conclude
ap + div(pv) = 0. (2.4)

28 CHAPTER 2 MASS AND MOMENTUM
2.2 Momentum
The momentum of a material body is a property the body has by virtue of
its mass and its velocity. Given a motion ip of a body B of mass density
p. the linear momentum 1(13, t) of B at time t and the angular momentum
H(B, t) of B at time t about the origin 0 of the spatial coordinate system
are defined by
_______________________
pvdx,
(2.5)
xxpvdx.
Again, dx (= dxidx2dx3) is the volume element in
The rates of change of momenta (both I and H) are of fundamental
importance. To calculate rates, first notice that for any smooth field
w = w(x,t),
L wpdx =
f w(ço(X, t), t)p(x, t) det F(X, t) dX
= L0 = f (2.6)
Thus,
dI(13,t) f dv
dt =
p-a— dx,
dH(13,t) I dv
= / xxp—dx.
dt dt

J Oden
CHAPTER 3
FORCE AND STRESS IN
DEFORMABLE BODIES
The concept of force is used to characterize the interaction of the motion
of a material body with its environment. More generally, as will be seen
later, force is a characterization of interactions of the body with agents
that cause a change in its momentum. In continuum mechanics, there
are basically two types of forces: (1) contact forces, representing the
contact of the boundary surfaces of the body with the exterior universe
(i.e., its exterior environment) or the contact of internal parts of the body
on surfaces that separate them, and (2) body forces, acting on material
points of the body by its environment.
Body Forces Examples of body forces are the weight-per-unit volume
exerted by the body by gravity or forces per unit volume exerted by
an external magnetic field. Body forces are a type of external force,
naturally characterized by a given vector-valued field f called the body
force density per unit volume. The total body force is then
L f(x, t) dx.
An Introduction to Mathematical Modeling A Course in Me hank s. First Edition By J Tinsley Oden
© 2011 John Wiley & Sons, Inc Published 2011 by John Wiley & Sons. Inc
29

30 CHAPTER 3. FORCE AND STRESS IN DEFORMA&E BODIES
.' 11
B I
exterior
I
Figure 3.1: External contact forces (bottom left) and internal contact forces
(bottom right).
Alternatively, we can measure the body force with a density b per unit
mass: f = Then
I fdx= I (3.1)
Contact Forces Contact forces are also called surface forces because
the contact of one body with another or with its surroundings must take
place on a material surface. Contact forces fall into two categories (see
Fig. 3.1):
• External contact forces representing the contact of the exterior
boundary surface of the body with the environment outside the
body, and
• Internal contact forces representing the contact of arbitrary parts
of the body that touch one another on parts of internal surfaces
they share on their common boundary.
The Concept of Stress There is essentially no difference between the
structure of external or internal contact forces; they differ only in what

CHAPTER 3. FORCE AND STRESS IN DEFORMABL.E BODIES 31
Figure 3.2: Illustrative example of the stress concept.
is interpreted as the boundary that separates a material body from its
surroundings. Portion I of the partitioned body in Fig. 3.1 could just as
well be defined as body B and portion II would then be part of its exterior
environment.
Figure 3.2 is an illustration of the discrete version of the various
forces: a collection of rigid spherical balls of weight W, each resting in
a rigid bowl and pushed downward by balancing a book of weight P on
the top balls. Explode the collection of balls into free bodies as shown.
The five balls are the body B. The exterior contact with the outside
environment is represented by the force of magnitude P distributed into
two equal parts of magnitude P and the contact forces representing
the fact that the balls press against the bowl and the bowl against the
balls in an equal and opposite way. Then (2x), N1, N2, and N3 are
external contact forces. The weights W are the body forces. Internally,
the balls touch one another on exterior surfaces of each ball. The action
of a given ball on another is equal and opposite to the action of the other
balls on the given ball. These contact forces are internal. They cancel
out (balance) when the balls are reassembled into the whole body B.

32 CHAPTER 3. FORCE AND STRESS IN DEFORMABLE BODIES
H
Figure 33: The Cauchy hypothesis.
In the case of a continuous body, the same idea applies, except that the
contact of any part of the body (part I say) with the complement (part II)
is continuous (as there are now a continuum of material particles in
contact along the contact surface) and the nature of these contact forces
depends upon how (we visualize) the body is partitioned. Thus, at a point
x, if we separate 13 (conceptually) into bodies I and II with a surface AA
defined with an orientation given by a unit vector n, the distribution of
contact forces at a point x on the surface will be quite different than that
produced by a different partitioning of the body defined by a different
surface BB though the same point x but with orientation defined by a
different unit vector m (see Fig. 3.3).
These various possibilities are captured by the so-called Cauchy
hypothesis: There exists a vector-valued surface (contact) force density
a(n, x, t)
giving the force per unit area on an oriented surface F through x with
unit normal n, at time t. The convention is that a(n, x, t) defines the
force per unit area on the "negative" side of the material (n is a unit
exterior or outward normal) exerted by the material on the opposite side
(thus, the direction of a on body II is opposite to that on I because the
exterior normals are in opposite directions; see Fig. 3.4).
Thus, if the vector field a(n, x, t) were known, one could pick an
arbitrary point x in the body (or, equivalently, in the current configuration
uìt) at time t, and pass a surface through x with orientation given by the
unit normal n. The vector cr(n, x, t) would then represent the contact
force per unit area on this surface at point x at time t. The surface F
through x partitions the body into two parts: The orientation of the

CHAPTER 3. FORCE AND STRESS IN DEFORMABLE BODIES 33
U
.)
Figure 3.4: The stress vector a.
vector c(n, x, t) on one part (at x) is opposite to that on the other part.
The vector field a is called the stress vector field and o(n, x, t) is the
stress vector at x and t for orientation n. The total force on surface F is
dA being the surface area element.
The total force acting on body B and the total moment about the
origin 0, at time t when the body occupies the current configuration
are, respectively,
fdx+/ a(n)dA,
M(B, t) = I x x f dx + I x x a(n) dA,
Jilt Jaclt
where we have suppressed the dependence of f and a on x and t.

J Oden
CHAPTER 4
THE PRINCIPLES OF
BALANCE OF LINEAR AND
ANGULAR MOMENTUM
The momentum balance laws are the fundamental axioms of mechanics
that connect motion and force:
The Principle of Balance of Linear Momentum The time rate of
change of linear momentum 1(13, t) of a body B at time t equals (or
is balanced by) the total force F(B, t) acting on the body:
dI(13,t)
(4.1)
An Introduction to Mathematkal Modeling A Course in Mechanics. First Edition By J Tinsley Oden
© 2011 John Wiley & Sons. Inc Published 2011 by John Wiley & Sons. Inc
35

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Transcriber's Note:
Every effort has been made to replicate this text as faithfully as possible, including
inconsistent hyphenation. Some changes have been made. They are listed at the end of
the text.
STORIES OF THE SHIPS
DEDICATED
TO
CAPT. ELLERTON
STORIES OF THE SHIPS
BY
LIEUT LEWIS R. FREEMAN, R.N.V.R.
OFFICIAL PRESS REPRESENTATIVE WITH THE GRAND FLEET
LONDON
JOHN MURRAY, ALBEMARLE STREET, W.
1919
All rights reserved

CONTENTS
I. STORIES OF THE SHIPS PAGE
The Story of the Cornwall 3
i. Plymouth to the Falklands 3
ii. The Battle of the Falklands 28
The Story of the Sydney 53
i. The Signalman's Tale 53
ii. Naval Hunnism 85
II. LIFE IN THE FLEET
A Battleship at Sea 107
A North Sea Sweep 130
A Visit to the British Fleet 157
The Health of the Fleet 169
Economy in the Grand Fleet 178
Christmas in a "Happy" Ship 194
In a Balloon Ship 203

Coaling the Grand Fleet 216
The Stokers 232
III. AMERICA ARRIVES
The U.S. Navy 247
"Getting Together" 259
i. How the Officers of the British and American Ships
that are working together in European Waters are
making each other's acquaintance 259
ii. What the British Bluejacket thinks of the American 268
iii. What the American Bluejacket thinks of Britain and the
British 278

THE STORY OF THE CORNWALL
I. Plymouth to the Falklands
Of the countless stories of naval action which I have listened to in
the course of the months I have spent with the Grand Fleet, I
cannot recall a single one which was told as the consequence of
being asked for with malice aforethought. I have never yet found a
man of action who was enamoured of the sound of his own voice
raised in the recital of his own exploits, and if there is one thing
more than another calculated to throw an otherwise not untalkative
British Naval Officer into a state of uncommunicativeness, in
comparison with which the traditional silence of the sphinx or the
proverbial close-mouthedness of the clam are alike sheer garrulity, it
is to ask him, point blank, to tell you (for instance) how he took his
submarine into the Baltic, or what his destroyer did at Jutland, or
how he fought his cruiser at Dogger Bank, or something similar.
The quiet-voiced but always interesting and often dramatic recitals
of such things as these which I have heard have invariably been led
up to quite incidentally—at dinner, on the bridge or quarter-deck,
around the wardroom fire, or through something else that has just
been told. Several times I have found in officers' diaries—little
records never meant for other eyes than those of the writers' own
friends or families—which have been turned over to me to verify
some point regarding which I had inquired, laconic references to
incidents and events of great human and even historic interest, and
one of the most amusing and dramatic yarns I have ever listened to
was told me in a "kite" balloon—plunging in the forty-mile wind
against which it was being towed like a hooked salmon—by a man
who had assured me before we went up that nothing really exciting
had ever fallen to his experience.

It was in this way—an anecdote now and then as this or that
incident of the day recalled it to his mind—that Captain —— came to
tell me the story of the Cornwall during those eventful early months
of the war when he commanded that now famous cruiser. He
mentioned her first, I believe, one night in his cabin when, referring
to a stormy midwinter month, most of which had been spent by his
Division of the Grand Fleet on some sort of work at sea, I spoke of
the "rather strenuous interval" we had experienced.
"If you think life in a battleship of the Grand Fleet strenuous,"
laughed the Captain, extending himself comfortably in his armchair
before the glowing grate, "I wonder what you would have thought of
the life we led in the old Cornwall. Not very far from a hundred and
twenty thousand miles of steaming was her record for the first two
years of the war, and in that time she ploughed most of the Seven
Seas and coasted in the waters of all but one of the Six Continents.
Always on the lookout for something or other, coaling as we could,
provisioning as we might—let me tell you that coming to the Grand
Fleet after that (at least until a few months had elapsed and the
contrast wore off) was like retiring on a pension in comparison."
He settled himself deeper into the soft upholstery, extended his feet
nearer the fire, lighted a fresh cigar, and, in the hour which elapsed
before the evening mail came aboard, told me of the work of the
Cornwall in those first chaotic weeks of the war which preceded the
battle of the Falklands.
"We were at Plymouth when the war began," said he, "and our first
work was in connexion with protecting and 'shepherding' safely to
port several British ships carrying bullion which were still on the high
seas. It was a baffling sort of a job, especially as there were two or
three German raiders loose in the North Atlantic, the favourite ruse
of each of which was to represent itself as a British cruiser engaged
in the same benevolent work the Cornwall was on. Warned of these
'wolves-in-shepherds'-clothing,' the merchantmen we sought to
protect were afraid to reveal their whereabouts by wireless, the
consequence being that our first forerunning efforts to safeguard the

seas resolved themselves into a sort of marine combination of 'Blind-
Man's Buff' and 'Hide-and-Seek,' played pretty well all over the
Atlantic. All the ships with especially valuable cargoes got safely to
port ultimately, though none of them, that I recall, directly under the
wing of the Cornwall. It was our first taste of the 'So-near-and-yet-
so-far' kind of life that is the inevitable lot of the cruiser which
essays the dual rôle of 'Commerce Protector' and 'Raider Chaser.'
"After a few hours at 'Gib,' we were next sent across to Casa Blanca,
where the appearance of the Cornwall was about the first tangible
evidence that French Africa had of the fact that England was really
coming into the war in earnest. There was a good deal of unrest in
Morocco at the time, for the Germans were even then at work upon
their insidious propaganda among the Moslems of all the colonies of
the Allies. The 'buzz' in the bazaars that the appearance of a British
warship started must have served a very useful purpose at this
critical juncture in carrying to the Arabs of the interior word that
France was not going to have to stand alone against Germany. Our
reception by both the French and native population of Casa Blanca
was most enthusiastic, and during all of our stay a cheering
procession followed in the wake of every party of officers or men
who went ashore.
"Leaving Casa Blanca, we were sent back to the Atlantic to search
for commerce destroyers, ultimately working south by the Canaries
and Cape Verde Islands to South American waters, where the
Karlsruhe was then at the zenith of her activities. The chase of this
enterprising and elusive raider, whose career was finally brought to
an inglorious end by her going aground on a West Indian Island,
kept the Cornwall—along with a number of other British cruisers—
steadily on the move, until the ominous and painful news of the
destruction of Craddock's fleet off Coronel suddenly brought us face
to face with the fact that there was soon going to be bigger game
than a lone pirate to be stalked.
"We never had the luck to sight even so much as the smoke of the
Karlsruhe, although—as I only learned too late to take advantage of

the information—the Cornwall was within an hour or two's steaming
of her on one occasion. I did think we had her once, though—a jolly
amusing incident it was, too. I was getting uncomfortably short of
food at the time—a very common experience in the 'here-to-day-
and-gone-to-morrow' sort of life we were leading;—so that when the
welcome news reached me by wireless one morning that a British
ship—Buenos Aires to New York with frozen beef—was due to pass
through the waters we were then patrolling, I lost no time in
heading over to intercept her on the chance of doing a bit of
marketing.
"We picked her up promptly as reckoned, but, while she was still hull
down on the horizon, her skipper began to signal frantically, 'I am
being chased by the "Karlsruhe"!' Here was luck indeed. I ordered
'Action Stations' to be sounded, and the course of the ship to be
altered toward the point where I figured the smoke of the pursuing
pirate would begin to smudge the sky-line as she came swooping
down upon her prey. Sighting nothing after holding on this course
for a while, I came to the conclusion that the raider must be hidden
by the impenetrable smoke-pall with which the flying beef-ship had
masked a wide arc of the western horizon, and headed up in that
direction, begging the fugitive in the meantime to give me the
bearing of her pursuer as accurately as possible.
"Her only reply to this, however, was to belch out 'smoke-screen'
faster than ever and continue rending the empyrean ether with
renewed 'I am being chased by the "Karlsruhe"!' In vain I assured
her that we were the H.M.S. Cornwall, and would take the greatest
delight in seeing that the chase was put an end to, if she would only
tell us from which direction the Karlsruhe was coming, and cease to
throw out a bituminous blanket for the enemy to hide behind.
Blacker and blacker rolled the smoke, heavier and heavier piled the
screen to leeward, and still more frantically shrilled the appeals for
help. At the end of my patience at actions which it now began to
dawn upon me looked more than a little suspicious, I headed the
Cornwall straight after the runaway and soon reduced the interval

separating us sufficiently to reach her with 'Visual.' She brought up
sharp at my 'Stop instantly!' and a quarter of an hour later my
boarding party was clambering over her side.
"'Where's the Karlsruhe?' I shouted impatiently to the Boarding
Officer as his boat came back alongside again. I knew something of
the accuracy of German long range naval gunnery, and was far from
being easy in mind regarding the kind of surprise packet that might
at any moment be wafted out of that slowly thinning smoke-blur to
leeward.
"'There,' he replied with a comprehensive sweep of the arm in quite
the opposite direction from the one I had been expecting the enemy.
'Right there, Sir.' That old lunatic of a skipper thought the Cornwall
was the Karlsruhe!"
"Did you get your frozen beef?" I asked.
The Captain smiled the pleased smile of one who recalls something
that has given him great satisfaction.
"I think that afternoon marked the beginning of the 'Food Economy'
campaign in the Navy," he replied. "If the Admiralty had been able to
continue buying frozen beef at the rate that crestfallen but highly
relieved skipper—quite of his own free will—charged for the lot we
loaded up after he had found it was not to be his fate to be sunk by
the Karlsruhe,—well, the Government could have probably built a
new battleship or two and never missed the money out of the
saving."
The recollection of the treat that fresh meat was after a long period
on "bully beef" ration turned the Captain's thoughts to another time
of plenty he had experienced after the Cornwall had helped the
wounded Carmania limp back to Base following her successful
engagement with the Cap Trafalgar.
"In these times of food economy and restricted rations," he said, "it
fairly makes my mouth water to think of the feasts Captain G——

spread for us during the days we were devising a way to get the
battered Carmania back to England. You see, when the war started
she was just about to sail on one of her transatlantic voyages with
the usual midsummer cargo of American millionaires, and her cuisine
was of a character calculated to satisfy their Epicurean tastes. When
they converted her to an auxiliary cruiser, it was the usual sledge-
hammer, crow-bar, and over-the-side procedure with the mirrors, the
upholsteries, and the mahoganies, but they left the stores, God bless
them, they left the stores. Can you fancy how things such as truffled
quail, and asparagus tips with mayonnaise—iced—and café parfait,
and Muscat dates, and California oranges—with the big gold labels
on—tasted to men who had been for weeks pretty nearly down to
the classic old wind-jammer ration of 'lobscouse' and 'dog's-body'?
And those plump, black, five-inch-long Havanas in the silver foil (I
can smell the soothing fumes of them yet), and that rarely blended
Mocha, and those bottles of 1835 Cognac—the pungent bouquet of
them scents the memories of the long evenings I sat with G—— in
the wreck of his fire-swept cabin while he yarned to me of the
ripping fight he had just come out of. And how we all envied G——
his luck—getting as sporting a show as a man could ask for in that
half-converted liner while we cruisers were vainly chasing smoke and
rumours over most of the South Atlantic. Nothing less than the
banquets he gave us would have salved our heart-burnings."
And so it was that the Captain was led on to speak of what he had
heard—from those who took part in it, and only a few hours from
the time it happened—of the first great duel ever fought between
modern armed merchantmen, a conflict, indeed, which is still
practically unique in naval history.
"There was not much to choose between the ships," he said. "The
Cap Trafalgar—one of the latest of the Hamburg Sud Amerika liners
—had a good deal the best of it on the score of age, and the
Carmania probably something on the score of size. The latter had
been hastily converted at Liverpool immediately after the outbreak of
the war, while the former turned herself from sheep into wolf about

the same time by arming herself with the guns of a small German
gun-boat. This craft, by the way, steamed to the nearest Brazilian
port and, with true Hunnish logic, claimed the right to intern as a
peaceful German Merchantman on the strength of the fact that it
was no longer armed! The largest guns that either ship had were
four-inch, the Carmania having slight advantage on the score of
number. The Carmania would have been no match for the Karlsruhe,
just as the Cap Trafalgar would have fallen easy prey to the Cornwall
or another of the British cruisers in those waters. Under the
circumstances, it was a happy fatality that let these two ex-floating
palaces fight with each other and in their own class.
"The first word we had of the engagement was a wireless Captain G
—— sent out saying, in effect, that he had sunk the Trafalgar, but,
as his bridge was burned up, his steering gear shot away, and all his
navigating instruments destroyed, that he would be glad to have
some one come and tell him where he was and lead him to a place
where he could, so to speak, lie down and lick his wounds for a
while. It took a jolly good bit of searching to find a ship that couldn't
tell any more about itself than that, but we finally sighted her ragged
silhouette and gave her a lead to such a haven as the practically
open seas of our rendezvous afforded.
"Poor G—— had lost a good deal more than his steering gear it soon
transpired, for the fire which had consumed his bridge had also
gutted his cabin, and reduced everything in it to cinders except an
old Norfolk jacket. How that escaped we never could figure out, for
of garments hanging on pegs to the left and right of it no trace was
left. As G—— was of about three times the girth of any other British
officer in those waters at the time, the wardrobe we tried to get
together for him was a grotesque combination; indeed, so far as I
recall now, the old Norfolk had to serve him as everything from
pyjamas and bath-robe to dinner-jacket and great-coat during that
trying period. It was a weird figure he cut presiding at those
Gargantuan feasts he spread for us on the bruised and battered old
Carmania, but there wasn't a one of us who wouldn't have changed

places with him—Norfolk and all—for the assurance of half his luck.
Such is the monotony of this patrol work in the outer seas, that,
after your first enthusiasm wears off, you get into a state of mind in
which you can never conceive that anything is ever going to happen.
That we had the one most decisive naval battle of the war just
ahead of us, no one dreamed at this time.
"The fight between the Carmania and Cap Trafalgar," he continued,
"has well been called 'The Battle of the Haystacks,' for never before
(or since, for that matter) have two ships with such towering upper
works stood off and tried to batter each other to pieces with gunfire.
Indeed, I well recall G——'s saying that, up to the very end, he could
not conceive that either ship could sink the other, and of how—even
after the Carmania had been struck three or four-score times and a
raging fire forward had driven him from the bridge—he kept
wondering in the back of his brain what sort of a fight the duel
would resolve itself into when both had exhausted their shells.
Luckily, he did not have to face that problem.
"Both ships, according to G——'s account, began blazing at each
other as soon as they came in range, and, as each was eager to
fight it out to a finish, the distance separating them was, for a while,
reduced as rapidly as possible. At something like three thousand
yards, however, some sort of a rapid-fire gun burst into action on the
Trafalgar. 'It didn't appear to be doing me much harm,' said G—— in
telling of it, 'but the incessant "pom-pom" of the accursed thing got
so much on my nerves that I drew off far enough to dull the edge of
its infernal yapping.'
"A thing which came near to putting the Carmania out of the running
before she had completed the polishing off of her opponent was the
shell which I have spoken of as violating the sanctity of the Captain's
cabin—the one that burned everything but the Norfolk jacket. This
projectile—a four-inch—though (probably owing to the small
resistance offered by the light upper works) it did not explode,
generated enough heat in its passage to start a fire. Beginning on G
——'s personal effects, this conflagration spread to the bridge,

destroying the navigating instruments and ultimately making it
impossible to remain there—the latter a serious blow in itself. What
made this fire especially troublesome was the difficulty, because of
the cutting of the main, of bringing water to bear upon it. As it was,
it was necessary to head the Carmania 'down the wind' to reduce
the draught fanning the flames. Nothing else would have saved her.
Except for one thing, this expedient would have enabled the now
thoroughly worsted (though G—— didn't know it) Trafalgar to
withdraw from the action, and this was that the latter was herself on
fire and had to take the same course willy-nilly. From that moment
on the battle was as irretrievably joined as one of those old Spanish
knife-duels in which the opponents were locked together in a room
to fight to a finish. Often as not, so they say, the victor in one of
these fights only survived the loser by minutes or hours, and so
would it have been in this instance had they not finally been able to
extinguish the fire on the Carmania.
"G——'s account of the way he had to carry on after being driven
from the bridge—it was really a splendid bit of seamanship—was
funny in the extreme, but the reality must have been funnier still,
that is, if that term can be applied to anything happening while
shells are bursting and blowing men to bits every few seconds. G
—— is one of the biggest men in the Navy—around the waist, I
mean—so it wasn't to be expected of him to be very shifty on his
feet. And yet, by the irony of Fate, it was he of all men who was
suddenly confronted with a task that required only less 'foot-work'
than it did 'head-work.' With the battle going on all the time, they
rigged up some sort of a 'jury' steering gear, or it may be that they
steered her by her screws. At any rate, G—— had to con her from
the most commanding position he could find on one of the after
decks, or rather, as he had no longer voice-pipe communication with
the engine-room, he had to keep dashing back and forth (it must
have been for all the world like a batsman running in cricket)
between two or three commanding positions. 'If I wanted to open
the range a bit,' he said, 'I had to nip for'ard, wait till there was an
interval in both gun-fire and shell-burst, and yell down a hatchway'

(or was it a ventilator?) 'to the engine-room to "Slow port!" or if I
suddenly found it imperative to open the distance, I had to make the
same journey and pass the word down to "Stop starboard!" The very
thought of that mad shuttling back and forth under the equatorial
sun used to make poor G—— mop his forehead and pour himself a
fresh drink every time he told the story.
"Battered and burning fiercely as both ships were, G—— confessed
that even at this juncture he could not rid himself of the feeling that
neither of them had enough shells to sink the other. 'I was racking
my brain for some plan of action to follow when that moment
arrived,' he said, 'when suddenly the Trafalgar began to heel sharply
and started to sink. It was our second or third salvo, which had
holed her badly at the water-line, that did the business. She had
kept steaming and fighting for close to an hour and a quarter
afterwards, though.'
"G—— told us one very good story about his Gunnery officer. 'It was
just before the shell which started the fire struck us,' he said, 'that Y
——'s sun helmet was knocked off—I don't remember whether it was
by the wind or the concussion of the firing. Seeing it fall to the deck
below, he ran to the rail of the bridge and began shouting for some
one to bring it back to him. Before long, luckily, a seaman who had
heard the shouting in a lull of the firing, poked his head out to see
what it was about, and presently came puffing up the ladder with
the fugitive head-piece. I say luckily, because the gun-control for the
whole ship was suspended while Y—— waited for that infernal
helmet. And the funniest thing about it all was that, when I ventured
to suggest a few days later that it might be well if he made use of
the chin-strap of his helmet the next time he was in action, he
claimed to have no recollection whatever of the incident—thought he
had been "sticking to his guns" all the time. Just shows how a man's
brain works in air-tight compartments when he is really busy.'
"The Surgeon of the Carmania (continued the Captain)—a splendid
chap who had given up a lucrative West-end practice and sworn he
was under forty (although he was really fifty-two) in order to get a

chance to do something for his country—told me many stories to
prove the splendid spirit of the men that passed under his hands
during and after the fight. Though most of the crew were only Royal
Naval Reservists, with no experience of and but little training for
fighting, it appears that they stood what is perhaps the hardest of all
trials—that of seeing their mates wounded and killed beside them—
like seasoned veterans.
"'There was one stout-hearted young Cockney,' said the Surgeon,
'whom, after I had finished removing a number of shell fragments
from various parts of his anatomy, I asked what he thought of the
fight. "Rippin', Sir," he replied, grinning ecstatically through the
bandage that held up the flap of a torn cheek; "rippin', never been in
one like it before." Then, as his eye caught the smile which I could
not quite repress at the lifetime of naval battling suggested by that
"nev'r afore," he concluded with "Not ev'n in Whitechapel."'
"The Surgeon came across one man who insisted that the blood
flowing from a ragged tear in his arm was really spattered there
when one of his mates—whose mangled body he bestrode—had
been decapitated by a shell a few minutes before; and there was
one lot of youngsters who went on cheerily 'Yo-heave-ho-ing' in
hoisting some badly needed shells which were so slippery with blood
that they had to be sanded before they could be handled. Grimly
pathetic was the story he told me of a gunner whose torn hand he
had just finished amputating and bandaging when some one
shouted into the door of the dressing station that the Trafalgar was
going down.
"'He crowded to a port I had had opened,' said the Surgeon, 'just in
time to see one of the last salvoes from the Carmania go crashing
into the side of the heeling enemy. "Huroor, boys," he shouted; "give
'em beans," and as he cheered he started (what had evidently been
a favourite gesture of approval and excitement with him) to smite
mightily with his right fist into the palm of his left hand. But the blow
fell upon air; there was no answering thwack. The gnarled, weather-
beaten fist shot past a bandaged stump. He drew back with surprise

for a moment, and then, grinning a bit sheepishly, like a boy
surprised in some foolish action, edged back beside me at the port.
"Quite forgot there was su'thin' missin'," he said half apologetically,
trying to wriggle the elbow of the maimed arm back into the sling
from which it had slipped. "S'pose I'll be havin' to get used to it,
won't I?" As the Trafalgar took a new list and began rapidly to settle
he burst into renewed "Huroors." "By Gawd, Sir," he cried, when she
had finally gone, "if I 'ad as many 'ands as an oktypuss, I'd 'a giv'n
'em all fer the joy o' puttin' that blinkin' pyrit down to Davy Jones."'"
The Captain gazed long at the coals of the grate, on his face the
pleased smile of one who recalls treasured memories. "I can't tell
you how sorry we were to see the Carmania go," he said finally. "My
word, how we did enjoy those feasts good old G—— spread for us!"
With a laugh he roused himself from the pleasant reverie and took
up again the narrative of the Cornwall.
"The first intimation we had" (he resumed) "of the sinking of Admiral
Craddock's fleet came in the form of a wireless from the Defence
asking if I had heard of the disaster at Coronel. Details which came
in the course of the next day or two brought home to us the
astonishing change in the whole situation which the appearance of
Von Spee in South American waters had wrought. The blow fell like a
bolt from the blue.
"As rapidly as possible the various British warships in the South
Atlantic rendezvoused off Montevideo to discuss a plan of action.
What the next move of the victorious Von Spee would be we could
only surmise. German prisoners picked up after the Falklands battle
said his ultimate plan—after seizing Port Stanley for a base, and
undergoing such a refit there as was practicable with the means at
his disposal—was to scatter his ships as commerce raiders all over
the Atlantic, cutting, if possible, the main sea arteries of England to
North America. The Germans figured, according to these prisoners,
that the suspension of the North Atlantic traffic for even a month (no
impossible thing for five speedy cruisers in the light of the delays to
sailings caused by the Emden and Karlsruhe working alone) would

practically paralyse England's war efforts and reduce her military
effort in France to almost negligible proportions. I am much more
inclined to believe that this—rather than escorting a fleet of German
merchantmen, bearing German reservists from Argentina, Uruguay,
and Southern Brazil, to South-West Africa from Buenos Aires and
Montevideo—was the real plan of Von Spee.
However, it was the immediate rather than the ultimate plans of the
Germans that was our chief—in fact, our only—concern. Whether
Von Spee intended heading for the North Atlantic later or South
Africa, or up the Thames—the only way he could clear the road to
any of these objectives was by first destroying such British warships
as still remained in South American waters. It was these ships which
had hurried to get together off Montevideo, in order to make the
path of the enemy as thorny and full of pitfalls as possible.
"They had no illusions respecting what the immediate future held for
them, that little group of cruiser captains that gathered in the
Admiral's cabin of the Defence to formulate a plan of action. We
knew nothing at that time of what had been decided upon at the
Admiralty; indeed, we were quite in agreement that it would be
deemed inexpedient to send any battle cruisers away from the North
Sea, where they might be imperatively needed any day, on a voyage
to the South Atlantic that might easily resolve itself into a months'-
long wild-goose chase. Our plans, therefore, were laid entirely on
the assumption that we should have to do the best we could with
the ships already available.
"There was not a man of us who was not keen on the chance of a
fight at even the prohibitive odds under which it appeared inevitable
that the one ahead of us must be fought, but the prospects of
success were anything but alluring. Every day that passed had
brought reports revealing the completeness of the enemy's victory at
Coronel, and all of these were more than confirmed when the
Glasgow—whose captain had had the good sense to retire from a
battle in which there was no longer a chance for him to be of any
use—came in and joined us.

"It would be easy to suggest conditions under which one naval force,
faced by another as much stronger than itself as the Germans were
than the British at this time, would be justified in avoiding an action.
The present was not such an occasion, however; in fact, I don't
think it ever occurred to any of us to bring up a discussion of that
phase of the question at all. This, briefly, was the way the matter
presented itself to us: The measure of the power of the Germans to
inflict harm to the Allies was their supply of shells. These gone—
always provided no new supply reached them—the menace, even
though the ships were yet unsunk, was practically at an end. We
knew that they had already used up a considerable quantity of their
munition in a foolish bombardment of the little tropical port of
Papeete, in the French Societies, and we knew that a very large
amount had been expended at Coronel. They still probably had
enough, we figured, to see them through many months of
commerce raiding if only they could avoid another general action
against warships, and such an action, even if it was a losing one
from our standpoint, it was our manifest rôle to provoke, and at the
earliest possible moment.
"This point decided, about all that remained to be considered was
how to make the most effective disposition of such ships as we had
at our disposal when once the enemy was in sight. We knew just
what ships we would have to meet. We also knew, practically to a
gun, how they were armed. Moreover, with Coronel as an object
lesson, we knew how well those ships were handled, and with what
deadly effectiveness those guns were served. Now that it is all
ancient history, I think there is no reason why I should not tell you
how we arranged that our ships should 'take partners' for the little
'sea-dance' they were expecting to shake their heels at.
"The Defence—an armoured cruiser of the Minotaur type,
subsequently sunk at Jutland—was to tackle the Scharnhorst, Von
Spee's flagship. The former was the only ship we had that was
anywhere nearly a match for either of the larger German cruisers.
She exceeded them in displacement by several thousand tons, and

her four nine-point-twos and ten seven-point-fives had a comfortable
margin of metal over that fired from the Scharnhorst's eight eight-
point-twos and six five-point-nines. In a fair duel with either of the
larger Germans, I think there is little doubt she would have had the
best of it. In the battle we expected to go into, however, there could
be no certainty that she was going to be able to give her undivided
attention to the vis-à-vis we had picked for her during a sufficient
interval to finish up the job.
"The Carnarvon and the Cornwall were to be given the formidable
task of keeping the Gneisenau so busy that she could not help her
sister fight the Defence. Our combined displacement was about
equal to that of our prospective opponent, but the four seven-point-
fives and twenty six-inch (all we had between us) could hardly have
prevented her pounding us to pieces with her eight-point-twos, in
the event that she elected to use her speed to keep beyond the
effective range of our lighter guns. By dashing into close range we
might have had a chance with her, or, again there was the possibility
we might lead her a dance that would take her out of the way long
enough to give the Defence time to finish polishing off the
Scharnhorst, in which event the former might have been able to
intervene in our favour.
"Small as would have been our chance of carrying through our part
of the programme successfully, the Gneisenau was the one opponent
I desired above all the others, on account of the way I knew it would
buck up the ship's company to feel that they were having a whack at
the ship that sunk the Monmouth. There were a good many men in
the Monmouth who had gone to her from the Cornwall, and our men
never tired cursing the Hun for letting their mates drown at Coronel
without making any effort to save them. They had something to say
on that score when their turn came at the Falklands.
"The Glasgow we were going to give a chance to wipe out her
Coronel score by sending her in against the Nürnberg. With her
superior speed, and her two six-inch and ten four-inch guns against
the latter's ten four-point ones, she would probably have had the

best of what could not but have been a very pretty fight if no one
had interfered with it. Here again, unluckily, the chances were
against a duel to the finish. Against the Dresden—a very worthy
sister of the Emden—the very best we could muster was the armed
merchantman, Orama. This (unless another armed merchantman—
the Otranto, which had escaped with the Glasgow from Coronel—
became available) left us nothing to oppose to the Leipzig, which, in
that event, would have been a sort of a 'rover,' free to bestow her
attention and shells wherever they appeared likely to do the most
harm. And (from the way she was fought at the Falklands, where
she was my 'opposite number') let me tell you that a jolly
troublesome 'rover' she would have been.
"That, in a few words, was our little plan for making Von Spee use
up the remainder of his ammunition. That was our principal object,
and there can be no doubt that we would have come pretty near
complete success in attaining it. For the rest, you can judge for
yourself what our chances would have been. As the Fates would
have it, however, that battle was never to be fought, save on paper
in the Admiral's cabin of the old Defence. Before ever we had
completed preparations for our 'magazine-emptying' sally against
Von Spee, word was winged to us that the Admiralty had a plan of
its own in process of incubation, and that we were to standby to co-
operate.
"Sturdee and his battle cruisers were well on their way to the South
Atlantic, however, before even an inkling of what was afoot was
vouchsafed us, and even then my orders were simply to rendezvous
with him at the 'Base' I have spoken of before—the one where we
foregathered and feasted with the Carmania. I breathed no word of
where and why we were going until the muddy waters of the Plate
estuary were left behind and the last least possibility of a 'leak' to
the shore was out of the question. Then I simply passed it on to the
men by posting some word of it on the notice-board. There was no
cheering, either then or even a few days later, when the Inflexible
and the Invincible, the latter flying Admiral Sturdee's flag, came

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