1 viscoelastic foundation on soil under vertical load

Geomechanics and Engineering, Vol. 4, No. 1 (2012) 67-78 67
Settlement analysis of viscoelastic foundation under
vertical line load using a fractional Kelvin-Voigt model
Hong-Hu Zhu*1
, Lin-Chao Liu2
, Hua-Fu Pei3
and Bin Shi1
1
School of Earth Sciences and Engineering, Nanjing University, Nanjing, China
2
School of Civil Engineering, Xinyang Normal University, Xinyang, China
3
Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hong Kong, China
(Received January 23, 2011, Revised February 19, 2012, Accepted March 05, 2012)
Abstract. Soil foundations exhibit significant creeping deformation, which may result in excessive
settlement and failure of superstructures. Based on the theory of viscoelasticity and fractional calculus, a
fractional Kelvin-Voigt model is proposed to account for the time-dependent behavior of soil foundation
under vertical line load. Analytical solution of settlements in the foundation was derived using Laplace
transforms. The influence of the model parameters on the time-dependent settlement is studied through a
parametric study. Results indicate that the settlement-time relationship can be accurately captured by
varying values of the fractional order of differential operator and the coefficient of viscosity. In comparison
with the classical Kelvin-Voigt model, the fractional model can provide a more accurate prediction of
long-term settlements of soil foundation. The determination of influential distance also affects the calculation
of settlements.
Keywords: soil foundation; fractional viscoelastic model; the Flamant-Boussinesq solution; settlement;
Laplace transform.
1. Introduction
Research results show that the soil foundation exhibits significant creeping behaviour and the
ground deformation under loading from superstructures is highly time-dependent (Bjerrum 1967).
For the estimation of the increase in stresses at various points and associated displacements caused
in soil foundations due to external loading, there have been a number of viscoelastic, elastic-visco-
plastic (EVP) or elastoplastic-viscoplastic models (e.g., Christie 1964, Kaliakin and Dafalias 1990,
Yin and Graham 1994, Justo and Durand 2000). Among them, viscoelastic foundation models can
provide satisfactory simulation of the time-dependent behaviour when the stress level in soil is low
and yielding of soil is thus not reached. The ideal assumption of the theory of viscoelasticity,
namely that the foundation is homogeneous, viscoelastic, and isotropic, is not quite true in most
cases. However, it provides a close estimation of long-term settlement and failure possibility of
superstructures and, using proper safety factors, a safe design can be developed.
Because integer-order differential operators are used in the viscoelastic constitutive equations, the
*Corresponding author, Associate Professor, E-mail: zhh@nju.edu.cn
Technical Note

68 Hong-Hu Zhu, Lin-Chao Liu, Hua-Fu Pei and Bin Shi
kernel functions (creep modulus and relaxation modulus) of viscoelastic models are normally a
combination of exponential functions. Thus these models have difficulties in predicting settlement of
soil foundation accurately and their applicable range is significantly limited. Fractional constitutive
models of viscoelastic materials were first introduced by Gemant (1936). In the constitutive equations,
integer-order differential operators were replaced by fractional-order differential operators. These models
have shown powerful capability in capturing static and dynamic stress-strain-time relationships of
viscoelastic materials and have been applied in various fields in the past few decades (Bagley and
Torvik 1983 and 1986, Welch et al. 1999, Zhang and Shimizu 1998, Li et al. 2001, Rossikhin and
Shitikova 2010). However, due to the complexity involved in obtaining the analytical solution, these
fractional models have not been applied in the field of geotechnical engineering until recent years
(Atanackovic and Stankovic 2004, Dikmen 2005, Liu et al. 2006, Paola et al. 2009, Zhu et al. 2011).
In this paper, a fractional Kelvin-Voigt viscoelastic constitutive model was proposed to study time-
dependent settlement of soil foundations. The analytical solution of a half-space under a vertical line
load is derived using Laplace transforms. The influences of the model parameters and the predetermined
influential distance on the calculated settlements at the foundation surface are further explored using
a parametric study.
2. Fractional Kelvin-Voigt model
The generalized viscoelastic constitutive equation using fractional derivatives can be described by
(Padovan 1987)
(1)
where m and n are positive integers; and are Riemann-Liouville fractional differential
operators defined by (Miller and Ross 1993)
(2)
where Γ(u) is the Gamma function, .
If m = n = 1 and b1 = 0, Eq. (1) can be expressed by
(3)
It is obvious that if α = 1, Eq. (3) represents the constitutive equation of the classical Kelvin-Voigt
viscoelastic model (KVM). Thus this new model can be called a generalized fractional Kelvin-Voigt
model (FKVM), in which the integer-order differential operator is now replaced by a fractional-
order one.
In the theory of viscoelasticity, the stress-strain relationships are
(4)
σ t( ) +
m 1=
M
∑ bm
d
βm
dt
βm
---------σ t( ) = E0ε t( ) +
n 1=
N
∑ En
d
αn
dt
αn
--------ε t( ) 0 βm 1,0 αn 1< < < <( )
d
βm
/dt
βm
d
αn
/dt
αn
d
α
f t( )
dt
α
-------------- =
1
Γ 1 α–( )
--------------------
d
dt
----
0
t
∫
f τ( )
t τ–( )
α
----------------dτ 0 α 1< <( )
Γ u( ) =
0
∞
∫ t
u 1–
e
u–
dt
σ t( ) = E0ε t( ) + E1
d
α
ε t( )
dt
α
---------------
P′Sij t( ) = Q′eij t( )

Settlement analysis of viscoelastic foundation under vertical line load using a fractional Kelvin-Voigt model 69
(5)
where Sij(t) and σij(t) are deviatoric and hydrostatic stress tensors; eij(t) and εij(t) are deviatoric and
hydrostatic strain tensors; P', Q', P'' and Q'' are linear differential operators consisting of viscoelastic
parameters.
The volumetric and shear strains of foundation soils under low stress levels are assumed to be
time-dependent. The FKVM shown in Fig. 1 is thus utilized for the description of the viscoelastic
behaviour of the bulk and shear moduli,, which describe the volume and shape changes of soil mass
under external loading. Thus
P' = 1 (6)
(7)
P'' = 1 (8)
(9)
where G, K and η are the shear modulus, the bulk modulus, and the coefficient of viscosity, respectively.
Determination of the model parameters can be obtained from laboratory and field testing.
3. Analytical solutions of settlements using fractional Kelvin-Voigt model
3.1 Flamant-Boussinesq elastic solution
Fig. 2 shows the case where a line force of p0 per unit length is applied on the surface of a half-
space. The elastic solution for this plane-strain problem was developed by Flamant (1892) by
modifying the three-dimensional solution of Boussinesq (Timoshenko and Goodier 1970). The
Flamant-Boussinesq solution using an elastic model (EM) is widely used in foundation engineering
for the calculation of stresses and estimation of displacements.
In the polar coordinate system, assume that at a point P( ) on the ground surface, ,
i.e., the influential distance of ground surface settlement is assumed to be r0. The displacements at
any point in the foundation can be obtained as
P″σij t( ) = Q″εij t( )
Q′ = G + η
d
α
dt
α
-------
Q″ = K + η d
α
dt
α
-------
r0, π/2 uθ( )θ=
π
2
---,r=r0
= 0
Fig. 1 Three-parameter fractional Kelvin-Voigt model

(10)
(11)
where E and μ are the modulus of elasticity and Poisson’s ratio, respectively; ur and uθ denote the
displacements in the radial and circumferential directions, respectively.
Taking E = 144 MPa, μ = 0.2, p0 = 1 MPa and r0 = 15 m, the distributions of displacements within
the soil foundation using the elastic model (EM) are depicted in Fig. 3. It is seen that the surface
settlement at the loading point approaches infinity and the settlements within the foundation reduce
sharply with increasing depth. For locations far away from the loading point, heave of the foundation
occurs.
In the case of , Eqs. (10) and (11) for radial and circumferential displacement components
at the ground surface can be given as the following simple form
(12)
(13)
ur =
2p0 1 μ
2
–( )
πE
--------------------------cosθ ln
r0
r
---- +
1 2μ–( ) 1 μ+( )p0
πE
-----------------------------------------θ sinθ −
1 μ+( )p0
πE
---------------------cosθ
uθ =
2p0 1 μ
2
–( )
πE
--------------------------– sinθ ln
r0
r
---- +
1 2μ–( ) 1 μ+( )p0
πE
-----------------------------------------θ cosθ
θ = π/2
ur( )θ=
π
2
---
=
1 2μ–( ) 1 μ+( )p0
2E
----------------------------------------- = −
3p0
4 3K G+( )
------------------------
uθ( )
θ=
π
2
---
=
2p0 1 μ
2
–( )
πE
--------------------------ln
r0
r
---- = −
p0ln
r0
r
----
2π
--------------
3
3K G+( )
--------------------- 1
G
----+
Fig. 2 Schematic illustration of a foundation subjected to vertical line load at the surface

Fig. 3 Displacement distributions calculated from the Flamant-Boussinesq elastic solution

Eqs. (12) and (13) indicate that there is a logarithmic relationship between the distance to the
loading point and the vertical displacement, and that the horizontal displacement at the ground
surface is a constant that is not affected by r.
3.2 Viscoelastic solution using classical Kelvin-Voigt model
Considering a line load applied on the foundation shown in Fig. 2, the loading condition is
specified as
(14)
where H(t) is the unit-step function defined as .
To obtain the viscoelastic solution, the volumetric and shear strains of the foundation soil satisfies
the KVM. Thus Eqs. (7) and (9) turns into
(15)
(16)
Taking the Laplace transforms of Eqs. (6), (8), (15) and (16), and combining them with Eqs. (12)
and (13), the displacements at the ground surface can be expressed by
(17)
(18)
Taking the inverse Laplace transforms of Eqs. (17) and (18), we get
(19)
(20)
where , . Eqs. (19) and (20) show that the time-dependent displacements
are exponential creep functions. It is obvious that if , the ultimate displacements equal the
Flamant-Boussinesq solution using elastic model (EM).
3.3 Viscoelastic solution using fractional Kelvin-Voigt model
Taking the Laplace transforms of Eqs. (6) to (9) and combined with Eqs. (12) and (13), the
p = p0H t( )
H t( ) =
1 t 0≥
0 t 0<⎩
⎨
⎧
Q′ = G + η
d
dt
----
Q″ = K + η
d
dt
----
ur( )θ=
π
2
---
= −
3p0
4
--------
1
3K G 4ηs+ +( )s
---------------------------------------
uθ( )θ=
π
2
---
= −
p0ln
r0
r
----
2π
--------------
3
3K G 4ηs+ +( )s
---------------------------------------
1
G ηs+( )s
-----------------------+
ur( )θ=
π
2
---
= −
3p0
4 3K G+( )
------------------------ 1 e
t
τ1
----–
–
⎝ ⎠
⎜ ⎟
⎛ ⎞
uθ( )
θ=
π
2
---
= −
p0ln
r0
r
----
2π
-------------- 3
3K G+( )
--------------------- 1 e
t
τ1
----–
–
⎝ ⎠
⎜ ⎟
⎛ ⎞ 1
G
---- 1 e
t
τ2
----–
–
⎝ ⎠
⎜ ⎟
⎛ ⎞
+
τ1 = 4η/ 3K G+( ) τ2 = η/G
t ∞→

displacements at the ground surface can be expressed by
(21)
(22)
In order to take the inverse Laplace transform, note the following characteristics of the fractional
derivative in the Laplace transform for zero initial condition
(23)
where Eα is the Mittag-Leffler function defined as .
In view of Eq. (23), taking the inverse Laplace transforms of Eqs. (21) and (22), we get
(24)
(25)
where , .
It is obvious that when , , .
Thus the ultimate displacements equal the elastic solution.
If , . Thus Eqs. (24) and (25) turn into
(26)
(27)
Eqs. (26) and (27) indicate that the displacements at the ground surface are not time-dependent
any more and the displacement values are half of those from the elastic solution. Therefore, the
FKVM collapses to be a ideal elastic model as .
ur( )
θ=
π
2
---
= −
3p0
4
-------- 1
3K G 4ηs
α
+ +( )s
-----------------------------------------
uθ( )θ=
π
2
---
= −
p0ln
r0
r
----
2π
--------------
3
3K G 4ηs
α
+ +( )s
-----------------------------------------
1
G ηs
α
+( )s
-------------------------+
L
1– 1
G ηs
α
+( )s
------------------------- =
1
G
---- 1 Eα–
t
τ
---
⎝ ⎠
⎛ ⎞
α
–
⎩ ⎭
⎨ ⎬
⎧ ⎫
Eα t( ) =
0
∞
∑ t
n
Γ 1 α n+( )
-------------------------; τ =
η
G
----
ur( )
θ=
π
2
---
= −
3p0
4 3K G+( )
------------------------ 1 Eα–
t
τ 1
-----
⎝ ⎠
⎛ ⎞
α
–
⎩ ⎭
⎨ ⎬
⎧ ⎫
uθ( )
θ=
π
2
---
= −
p0ln
r0
r
----
2π
--------------
3 3Eα
t
τ 1
------
⎝ ⎠
⎛ ⎞
α
––
3K G+( )
----------------------------------------
1 Eα–
t
τ 2
------
⎝ ⎠
⎛ ⎞
α
–
G
-------------------------------------+
⎩ ⎭
⎪ ⎪
⎨ ⎬
⎪ ⎪
⎧ ⎫
τ1 =
4η
3K G+
------------------ τ2 =
η
G
----
0 α< 1≤ limt ∞→ ur( )
θ=
π
2
---
=
3p0
4 3K G+( )
--------------------------– limt ∞→ uθ( )
θ=
π
2
---
=
p0ln
r0
r
----
2π
---------------–
3
3K G+( )
----------------------
1
G
----+
α 0→ Eα t( )
0
∞
∑ t
n
Γ 1( )
----------- =
0
∞
∑ t
n
=
1
1 t–
---------- 1– t 1< <( )→
ur( )
θ=π
2
---
= −
3p0
8 3K G+( )
------------------------
uθ( )
θ=
π
2
---
=
p0ln
r0
r
----
4π
--------------–
3
3K G+
-----------------
1
G
----+
α 0→

Table 1 Parameters of the FKVM
Item Value
Young’s modulus E (MPa) 144
Poisson’s ratio μ 0.2
Shear modulus G (MPa) 60
Bulk modulus K (MPa) 80
Coefficient of viscosity η (MPa·d) 1000
Fractional differential order α 0.5
r0 (m) 15
If α = 1, . Obviously, Eqs. (23) and (24) currently become the
viscoelastic solution, which means that the FKVM turns into a classical KVM.
4. Numerical example and analysis
4.1 Comparison of elastic and viscoelastic models
How to determine the fractional model parameters is a major concern in practical applications. To
carry out a preliminary investigation of these parameters’ influence on the calculation results, a
parametric study is presented as following. Here, we analyze the quasistatic problem of a viscoelastic
foundation subjected to line loading p0 = 1 MPa using the FKVM. The basic parameters used in this
analysis are listed in Table 1.
Fig. 4 shows the surface settlement results using the fractional model at t = 10 d and 50 d,
together with the results from the classical KVM solution and the Flamant-Boussinesq elastic
solution. The present results show that the discrepancy between the results from FKVM and KVM
increases as t increases. The calculated settlements using KVM develops rapidly with the elapsed
time while those using FKVM will take much more time.
4.2 Parametric study of the FKVM
To study the influence of the coefficient of viscosity in the FKVM, η is set to 500, 1000 and
1500 MPa·d. Fig. 5 depicts the settlement-time curves at r = 1 m over time using the FKVM, the
KVM and the EM. It is shown that the coefficient of viscosity mainly affects the overall creeping
rate of the foundation soil. For the same time point, the settlements increase with the decrease of η.
Note that the ultimate settlement in the viscoelastic solution is equal to that in the elastic solution, η
may determine the magnitude of creeping by controlling the time duration to achieve the ultimate
settlement. If η decreases, the creeping effect will function more slowly and the ultimate settlement
will be obtained for a much longer time. Results from the classical Kelvin-Voigt model (α = 1),
which is shown in Fig. 6 by a dashed line, shows larger settlements and less time to get to the
ultimate settlement.
Eα t( ) =
0
∞
∑ t
n
Γ 1 αn+( )
------------------------- =
0
∞
∑ t
n
n!
----- = e
t

To investigate the influence of the fractional order of differential operator in the FKVM, α is set
to 0.1, 0.3, 0.5, 0.7 and 0.9 while η is fixed to 1000 MPa·d. Fig. 6 illustrates the settlement at r = 1
m over time. It is shown that the fractional order α has a significant influence on the long-term
performance of the foundation surface settlements. At t1/2 = 10 d, the settlements are half of the
ultimate settlement (5.75 mm) for any value of α. t1/2 is found to have an approximately linear
relationship with η and 1/E. After that time point, with the increase of α, the foundation settlement
and corresponding settlement rate increase. Therefore, α may determine the functioning law of
creeping effect by separating the time-dependent deformation of soil foundation into two stages. The
first stage may associate with the consolidation process of the foundation soil and the second stage
is mainly due to secondary compression. Therefore, a small α value is fit for sand foundation,
which has a small permeability coefficient. While for soft clay foundation, the α value should be
Fig. 4 Comparsion of settlements calculated using elastic, Kelvin-Voigt and fractional Kelvin-Voigt models

increased. In the KVM where α = 1 (the dashed line in Fig. 6), a minimum settlement rate at stage
I and a maximum settlement rate at stage II is obtained.
The key feature of the FKVM illustrated in Figs. 5 and 6 is the capability to represent generalised
viscoelastic behaviour over many decades of time with only three parameters. It is also indicated
from the above parametric study that the classical viscoelastic model may overestimate the
development of long-term settlement (Stage II) and fail to describe the attenuation creep of soil
mass accurately. Using the fractional viscoelastic models, we can obtain the best fitting effect of the
actual soil deformation characteristics by selecting appropriate values of the fractional differential
order and the coefficient of viscosity. In geotechnical design, this model can provide more accurate
settlement prediction and thus save construction costs of foundations or embankments.
Fig. 6 Settlement-time relationship at r = 1 m under different α values
Fig. 5 Settlement-time relationship at r = 1 m under different η values

4.3 Determination of the influential distance
The determination of influential distance due to loading is another major engineering concern. Fig.
7 shows the settlement results when the value of r0 is set to 10, 15 and 20 m while α is fixed to be
0.5. It is shown that with the increase of r0, the time-dependent and ultimate settlement (t = 1000 d)
will increase but the settlement profile is not affected significantly. If the influential distance of
settlement is overestimated (like r0 = 20 m), a larger settlements will be predicted.
5. Conclusions
Based on the theory of viscoelasticity and fractional calculus, a fractional Kelvin-Voigt viscoelastic
model is developed to account for the time-dependent behaviour of soil foundations under
concentrated line load. Analytical solution of displacements in the foundation was derived using
Laplace transforms. The fractional theory shows its potential in modeling the long-term foundation
deformation. The presented analytical procedure can be extended to solve other geotechnical problems.
From a parametric study, the fractional order of derivative differential operator and the coefficient of
viscosity exert a significant influence on the time-dependent settlements of foundation. Compared
with the KVM, the FKVM can accurately capture the actual deformation of soil foundations under
loading.
However, the empirical relationship between the model parameters and soil properties is an
important issue to be solved, which requires the support of abundant measurement data from
laboratory and field experiments. Careful attention should be paid to the predetermined influential
distance, as well.
Fig. 7 Settlement distributions at the foundation surface under different predetermined r0 values

Acknowledgements
The financial supports provided by the National Basic Research Program of China (973 Program)
(Grant No. 2011CB710605) and National Key Technology R&D Program of China (2012BAK10B05)
are gratefully acknowledged.
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