# 2 Elasticity

For elastic materials, the strains and effective stresses are related in a one-to-one manner — that is, the strains generated through loading along one stress path will be recovered when unloading along the same stress path.

Assuming linear elasticity, the relation between the strains and the effective stresses can be expressed as:

\boldsymbol{\varepsilon}^{e} = \mathbb{C}^{e} \boldsymbol{\sigma}' \Longleftrightarrow \boldsymbol{\sigma}' = \mathbb{D}^{e} \boldsymbol{\varepsilon}^{e} \tag{2.1}

where \mathbb{C}^{e} is the compliance modulus and \mathbb{D}^e is the stiffness modulus.

# 2.1 Isotropic Elasticity

The most common assumption regarding the stiffness of geomaterials is isotropy, i.e., that the properties are the same in all directions (vertically and horizontally).

In this case, the compliance and stiffness moduli, related in a one-to-one manner, may be expressed either in terms of Young's modulus E and Poisson's ratio \nu, or in terms of bulk modulus K and shear modulus G.

The compliance modulus is given by:

\mathbb{C}^e = \frac{1}{E}
\begin{bmatrix}
1 & -\nu & -\nu &  &  &  \\
-\nu & 1 & -\nu &  &  &  \\
-\nu & -\nu & 1 &  &  &  \\
 &  &  & 2(1+\nu) &  &  \\
 &  &  &  & 2(1+\nu) &  \\
 &  &  &  &  & 2(1+\nu)
\end{bmatrix} \tag{2.2}

or equivalently,

{\newcommand{\arraystretch}{1.3}
\mathbb{C}^e =
\begin{bmatrix}
\frac{1}{9}K^{-1}+\frac{1}{3}G^{-1} & \frac{1}{9}K^{-1}-\frac{1}{6}G^{-1} & \frac{1}{9}K^{-1}-\frac{1}{6}G^{-1} &  &  &  \\
\frac{1}{9}K^{-1}-\frac{1}{6}G^{-1} & \frac{1}{9}K^{-1}+\frac{1}{3}G^{-1} & \frac{1}{9}K^{-1}-\frac{1}{6}G^{-1} &  &  &  \\
\frac{1}{9}K^{-1}-\frac{1}{6}G^{-1} & \frac{1}{9}K^{-1}-\frac{1}{6}G^{-1} & \frac{1}{9}K^{-1}+\frac{1}{3}G^{-1} &  &  &  \\
 &  &  & G^{-1} &  &  \\
 &  &  &  & G^{-1} &  \\
 &  &  &  &  & G^{-1}
\end{bmatrix}
} \tag{2.3}

where

K = \frac{E}{3(1-2\nu)}, \quad G = \frac{E}{2(1+\nu)} \tag{2.4}

The stiffness modulus is given by:

\mathbb{D}^e = \frac{E}{(1+\nu)(1-2\nu)}
\begin{bmatrix}
1-\nu & \nu & \nu &  &  &  \\
\nu & 1-\nu & \nu &  &  &  \\
\nu & \nu & 1-\nu &  &  &  \\
 &  &  & \frac{1}{2}(1-2\nu) &  &  \\
 &  &  &  & \frac{1}{2}(1-2\nu) &  \\
 &  &  &  &  & \frac{1}{2}(1-2\nu)
\end{bmatrix}  \tag{2.5}

or equivalently,

{\newcommand{\arraystretch}{1.3}
\mathbb{D}^e =
\begin{bmatrix}
K+\frac{4}{3}G & K-\frac{2}{3}G & K-\frac{2}{3}G &  &  &  \\
K-\frac{2}{3}G & K+\frac{4}{3}G & K-\frac{2}{3}G &  &  &  \\
K-\frac{2}{3}G & K-\frac{2}{3}G & K+\frac{4}{3}G &  &  &  \\
 &  &  & G &  &  \\
 &  &  &  & G &  \\
 &  &  &  &  & G
\end{bmatrix} \tag{2.6}
}

The relations between E, \nu, K, and G are summarized in the table below.

	E =	\nu =	K =	G =
(E, \nu)	E	\nu	\frac{E}{3(1-2\nu)}	\frac{E}{2(1+\nu)}
(K, G)	\frac{9KG}{3K+G}	K - \frac{2G}{3}	K	G

Table 2.1: Relation between elastic parameters.

# 2.1.1 Undrained Conditions

Under undrained conditions, the relation between the elastic strains and the total pressures can be shown to be given by

\boldsymbol{\varepsilon}^e = \mathbb{C}^e_u \boldsymbol{\sigma} \tag{2.7}

where

{\newcommand{\arraystretch}{1.3}
\mathbb{C}^e_u = \frac{1}{E_u}
\begin{bmatrix}
1 & -\tfrac{1}{2} & -\tfrac{1}{2} &  &  &  \\
-\tfrac{1}{2} & 1 & -\tfrac{1}{2} &  &  &  \\
-\tfrac{1}{2} & -\tfrac{1}{2} & 1 &  &  &  \\
 &  &  & 3 &  &  \\
 &  &  &  & 3 &  \\
 &  &  &  &  & 3
\end{bmatrix}
= \frac{1}{3G}
\begin{bmatrix}
1 & -\tfrac{1}{2} & -\tfrac{1}{2} &  &  &  \\
-\tfrac{1}{2} & 1 & -\tfrac{1}{2} &  &  &  \\
-\tfrac{1}{2} & -\tfrac{1}{2} & 1 &  &  &  \\
 &  &  & 3 &  &  \\
 &  &  &  & 3 &  \\
 &  &  &  &  & 3
\end{bmatrix}
} \tag{2.8}

where

E_u = \frac{3E}{2(1+\nu)}\tag{2.9}

is the undrained Young’s modulus.

This is the elastic law used for the Tresca material.

# 2 Elasticity

# 2.1 Isotropic Elasticity

# 2.1.1 Undrained Conditions

See also