2- Elastic Material Models

The general equation for the elastic constitutive models in incremental form is

𝑒𝑒
𝜎𝜎𝑖𝑖𝑖𝑖 = 𝐷𝐷𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝜀𝜀𝑘𝑘𝑘𝑘 (2.1)

In above 𝐷𝐷𝑒𝑒𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 is the fourth order elastic constitutive tensor. The elastic constitutive equations can
also be presented using stress and strain vectors:

𝜎𝜎11 1 − 𝜈𝜈 𝜈𝜈 𝜈𝜈 0 0 0 𝜀𝜀11
⎧𝜎𝜎22 ⎫ ⎡ 𝜈𝜈 1 − 𝜈𝜈 𝜈𝜈 0 0 0 ⎤ ⎧ 𝜀𝜀22 ⎫
⎪𝜎𝜎 ⎪ 𝐸𝐸 ⎢ 𝜈𝜈 𝜈𝜈 1 − 𝜈𝜈 0 0 0 ⎥ ⎪ 𝜀𝜀 ⎪
33 = ⎢ ⎥ 33
(1 − 2𝜈𝜈)⁄2 0 0
⎨𝜎𝜎12 ⎬ (1 − 2𝜈𝜈)(1 + 𝜈𝜈) ⎢ 0 0 0 ⎥ 𝛾𝛾
⎨ 12 ⎬
⎪𝜎𝜎23 ⎪ ⎢ 0 0 0 0 (1 − 2𝜈𝜈)⁄2 0 ⎥ ⎪𝛾𝛾23 ⎪
⎩𝜎𝜎31 ⎭ ⎣ 0 0 0 0 0 (1 − 2𝜈𝜈)⁄2 ⎦ ⎩𝛾𝛾31 ⎭

(2.2)
In above, 𝐸𝐸 is the elastic modulus and 𝜈𝜈 is the Poisson’s ratio.
Other elastic properties that could be useful are the shear modulus, 𝐺𝐺, and bulk modulus 𝐾𝐾.

𝐸𝐸 𝐸𝐸
𝐺𝐺 = ; 𝐾𝐾 =
2(1 + 𝜈𝜈) 3(1 − 2𝜈𝜈)

The elastic behavior can be linear or nonlinear. In linear elasticity the elastic material properties are
constants, but in nonlinear elastic models they change based on some assumptions. There are different
options for linear and nonlinear elasticity in RS2 and RS3, the following sections will present all these
options. There are also a few constitutive models that have their own specific formulation for elastic
behavior. For those special cases the elastic behavior is discussed in their corresponding sections.

2.1- Linear Elasticity

In linear elastic material models, the elastic properties are constants. The options in these family of models
are isotropic, transversely isotropic and orthotropic elastic behavior.

2.1.1- Linear Isotropic Elastic Model

The constitutive equations for linear isotropic elastic behavior is presented in equation (2.2).
This equation is for the three dimensional situations as in RS3. RS2 is a 2 dimensional program and it is
limited to plane-strain and axisymmetric configurations. For the reference, the elastic constitutive equations
for plane-strain and axisymmetric cases are presented in equation (2.3).

𝜎𝜎11 1 − 𝜈𝜈 𝜈𝜈 0 𝜈𝜈 𝜀𝜀11
𝜎𝜎22 𝐸𝐸 𝜈𝜈 1 − 𝜈𝜈 0 𝜈𝜈 𝜀𝜀
�𝜎𝜎 � = (1−2𝜈𝜈)(1+𝜈𝜈) � � �𝛾𝛾22 � (2.3)
12 0 0 (1 − 2𝜈𝜈)⁄2 0 12
𝜎𝜎33 𝜈𝜈 𝜈𝜈 0 1 − 𝜈𝜈 𝜀𝜀33

2.1.2- Linear Transversely Isotropic Elastic Model

The constitutive equations for linear transversely isotropic elastic material in the material coordinated,
(𝑒𝑒̂1 , 𝑒𝑒̂2 , 𝑒𝑒̂3 ) is presented in equation (2.4). It is assumed that 𝑒𝑒̂3 is the unit vector normal to the plane of
isotropy.

�𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
𝜎𝜎�𝑖𝑖𝑖𝑖 = 𝐷𝐷 𝑒𝑒
𝜀𝜀̂𝑘𝑘𝑙𝑙 (2.4)

The constitutive equation in the material coordinate system can be formulated in terms of 5 independent
elastic parameters, E1 , E3 ,ν 12 ,ν 13 , G13 (e.g. Timoshenko and Goodier 1970, Saada 1993)

The components of the transversely isotropic constitutive tensor are:

Dˆ 1111
e
=Dˆ e2222 =E1 (1 −ν 31ν 13 ) ϒ
Dˆ e = E (1 −ν 2 ) ϒ
3333 3 12

Dˆ 1122
e
=Dˆ e2211 =E1 (ν 1 +ν 31ν 13 ) ϒ
Dˆ=e
1133 Dˆ=e
3311 Dˆ=
e
2233 Dˆ= e
3322 E1 (ν 31 +ν 31ν 12=
) ϒ E3 (ν 13 +ν 13ν 12 ) ϒ (2.5)
Dˆ=e
1212 Dˆ=e
2121 Dˆ=
e
1221Dˆ=e
2112G 12

Dˆ= Dˆ= Dˆ= Dˆ

e
1313
e
3131
e
1331= G13 e
3113

Dˆ=
e
2323 Dˆ=
e
3232 Dˆ=
e
2332 Dˆ=
e
3223 G13

where

ϒ= (1 −ν 122 − 2ν 13ν 31 − 2ν 12ν 13ν 31 ) −1

1 E1
G12 = ( D1111 − D1122 )= (2.6)
2 2(1 +ν 12 )
E1 ν 31 = E3 ν 13
In RS2 and RS3, for using this option of constitutive equations the increment of strain is first transformed
from the global coordinate to the material coordinate. The increment of stress is then calculated based on
equation 2.4 in the material coordinate, and finally the increment of stress is transformed from the material
coordinate to the global coordinate and added to the state of stress in global coordinate.

2.1.3- Linear Orthotropic Elastic Model

The constitutive equations for linear transversely isotropic elastic material in the material coordinated,
(𝑒𝑒̂1 , 𝑒𝑒̂2 , 𝑒𝑒̂3 ) is presented in equation (2.7).

�𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
𝜎𝜎�𝑖𝑖𝑖𝑖 = 𝐷𝐷 𝑒𝑒
𝜀𝜀̂𝑘𝑘𝑘𝑘 (2.7)

The constitutive equation in the material coordinate system can be formulated in terms of 9 independent
elastic parameters, E1 , E2 , E3 ,ν 12 ,ν 13 ,ν 23 , G12 , G13 , G23 .

The components of the transversely isotropic constitutive tensor are:

Dˆ 1111
e
=E1 (1 −ν 23ν 32 ) ϒ
Dˆ e
2222 =E (1 −ν ν ) ϒ
2 31 13

Dˆ 3333
e
=E3 (1 −ν 21ν 12 ) ϒ
Dˆ=e
1122 Dˆ=
e
2211 E1 (ν 21 +ν 31ν 23=
) ϒ E2 (ν 12 +ν 13ν 32 ) ϒ
Dˆ=e
1133 Dˆ=
e
3311 E1 (ν 31 +ν 21ν 32=
) ϒ E3 (ν 13 +ν 12ν 23 ) ϒ (2.8)
Dˆ=e
2233 Dˆ=
e
3322 E (ν +ν ν =
2 32 ) ϒ E (ν +ν ν ) ϒ
12 31 3 23 21 13

Dˆ= Dˆ=
e
1212
e
2121 Dˆ= Dˆ= G12
e
1221
e
2112

Dˆ=
e
1313 Dˆ=
e
3131 Dˆ=
e
1331 Dˆ=e
3113 G13
Dˆ=
e
2323 Dˆ=
e
3232 Dˆ=e
2332 Dˆ=e
3223 G13

where

ϒ= (1 −ν 12ν 21 −ν 13ν 31 −ν 23ν 32 − 2ν 21ν 32ν 13 ) −1

(2.9)
Ei ν ji = E j ν ij

In RS2 and RS3, for using this option of constitutive equations the increment of strain is first transformed
from the global coordinate to the material coordinate. The increment of stress is then calculated based on
equation 2.4 in the material coordinate, and finally the increment of stress is transformed from the material
coordinate to the global coordinate and added to the state of stress in global coordinate.
2.2- Nonlinear Elasticity
In nonlinear elastic material models the elastic properties are dependent on some measures of stress or strain
tensor. The nonlinear elasticity in RS2 and RS3 is isotropic.
The first two options for nonlinear elasticity is based on the fact that the elastic modulus of porous materials
is proportional to the confinement.

𝛼𝛼
𝑝𝑝
𝐸𝐸 = 𝐸𝐸0 �𝑝𝑝 � (2.10)
𝑟𝑟𝑟𝑟𝑟𝑟

𝛼𝛼
𝑏𝑏𝑏𝑏+𝑎𝑎
𝐸𝐸 = 𝐸𝐸0 �𝑝𝑝 +𝑎𝑎
� (2.11)
𝑟𝑟𝑟𝑟𝑟𝑟

In above 𝐸𝐸0 is the elastic modulus at reference pressure. 𝑝𝑝𝑟𝑟𝑟𝑟𝑟𝑟 is the reference pressure, and 𝑎𝑎, 𝑏𝑏 and 𝛼𝛼 are
material parameters. 𝑝𝑝 is the mean stress, assuming compression positive. The value of 𝛼𝛼 is usually between
0.5 and 1.0.
Laboratory data suggests degradation of shear modulus with increase in the deviatoric strain. To capture
such an effect there is another option for nonlinear elasticity in the RS2 and RS3

𝑟𝑟
𝛾𝛾
𝐺𝐺 = 𝐺𝐺𝑚𝑚𝑚𝑚𝑚𝑚 �1 + 𝑎𝑎 𝛾𝛾 � (2.12)
𝑦𝑦

In above 𝐺𝐺𝑚𝑚𝑚𝑚𝑚𝑚 is the maximum shear modulus and, 𝑎𝑎, 𝛾𝛾𝑦𝑦 and 𝑟𝑟 are material parameters. To simulate
degradation parameter 𝑟𝑟 should be less than zero. The deviatoric strain, 𝛾𝛾, is reset to zero every time the
direction of loading changes.
To combine the effect of confinement and deviatoric strain, one can use the combination of equations (2.13)
and (2.14). In the first equation the maximum elastic modulus is calculated based on the level of
confinement and in the second one the degradation because of deviatoric strain is taken into account.

𝛼𝛼
𝑏𝑏𝑏𝑏+𝑎𝑎
𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚 = 𝐸𝐸0 �𝑝𝑝 +𝑎𝑎
� (2.13)
𝑟𝑟𝑟𝑟𝑟𝑟

𝑟𝑟
𝛾𝛾
𝐸𝐸 = 𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚 �1 + 𝑎𝑎 𝛾𝛾 � (2.14)
𝑦𝑦

For this last case, the deviatoric strain, 𝛾𝛾, depends on the loading history. Once the direction of loading is
changed the stiffness regains a maximum recoverable value in the order of its initial value, 𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚 . When an
increment of strains is applied to the material, each principal direction is checked for a possible change in
the loading direction. This option can be used to mimic the hysteretic behavior of soils in dynamic loading,
but it is not a robust constitutive model for this purpose, since this phenomenon is best described by using
deviatoric hardening plasticity models.
The Duncan-Chang model (Duncan and Chang, 1970) is also a nonlinear elastic model that is included in
RS2 and RS3, but his model will be presented in a separate chapter. There are other models that take
advantage of nonlinear elastic behavior that is embedded in their formulations, these models will also be
explained in details in in their own chapters.

References
Timoshenko S.P. and Goodier J.N. (1970), Theory of Elasticity, McGraw-Hill.
Saada A.S. (1993), Elasticity; Theory and Applications, 2nd edition, published by Krieger Publishing
Company, Pergamon Press Inc.