Elasticity/Constitutive relations

In the above matrix equation we have used ${\displaystyle \gamma _{ij}}$ to represent the shear strains rather than ${\displaystyle \varepsilon _{ij}}$. You should keep in mind that

${\displaystyle \gamma _{ij}=2~\varepsilon _{ij}}$

An alternative form of the isotropic linear elastic stress-strain relation that is easier to work with is:

${\displaystyle {\boldsymbol {\sigma }}=\lambda ~{\text{tr}}({\boldsymbol {\varepsilon }})~{\boldsymbol {\mathit {1}}}+2~\mu ~{\boldsymbol {\varepsilon }}}$

or, in index notation,

${\displaystyle \sigma _{ij}=\lambda ~\varepsilon _{kk}~\delta _{ij}+2~\mu ~\varepsilon _{ij}}$

Here ${\displaystyle \lambda }$ is the Lamé modulus and ${\displaystyle \mu }$ is the shear modulus. In terms of ${\displaystyle E}$ and ${\displaystyle \nu }$:

${\displaystyle \lambda ={\cfrac {E~\nu }{(1+\nu )(1-2\nu )}}~;~~\mu ={\cfrac {E}{2(1+\nu )}}}$

The inverse relationship is

${\displaystyle {\boldsymbol {\varepsilon }}=-{\cfrac {\nu }{E}}~{\text{tr}}({\boldsymbol {\sigma }})~{\boldsymbol {\mathit {1}}}+{\cfrac {1+\nu }{E}}~{\boldsymbol {\sigma }}}$

or, in index notation,

${\displaystyle \varepsilon _{ij}=-{\cfrac {\nu }{E}}~\sigma _{kk}~\delta _{ij}+{\cfrac {1+\nu }{E}}~\sigma _{ij}}$

To express the general stress-strain relation for a linear elastic material in terms of matrices (as we did for the isotropic elastic material) we use what is called the Voigt notation.

In this notation, the stress and strain are expressed as ${\displaystyle 6\times 1}$ column vectors and the elasticity tensor is expressed as a symmetric ${\displaystyle 6\times 6}$ matrix as shown below.

${\displaystyle {\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\\sigma _{23}\\\sigma _{31}\\\sigma _{12}\end{bmatrix}}={\begin{bmatrix}C_{1111}&C_{1122}&C_{1133}&C_{1123}&C_{1131}&C_{1112}\\C_{2211}&C_{2222}&C_{2233}&C_{2223}&C_{2231}&C_{2212}\\C_{3311}&C_{3322}&C_{3333}&C_{3323}&C_{3331}&C_{3312}\\C_{2311}&C_{2322}&C_{2333}&C_{2323}&C_{2331}&C_{2312}\\C_{3111}&C_{3122}&C_{3133}&C_{3123}&C_{3131}&C_{3112}\\C_{1211}&C_{1222}&C_{1233}&C_{1223}&C_{1231}&C_{1212}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\2~\varepsilon _{23}\\2~\varepsilon _{31}\\2~\varepsilon _{12}\end{bmatrix}}}$

or

${\displaystyle \mathbf {\sigma } =\mathbf {C} ~\mathbf {\varepsilon } }$

The inverse relation is

${\displaystyle {\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\2~\varepsilon _{23}\\2~\varepsilon _{31}\\2~\varepsilon _{12}\end{bmatrix}}={\begin{bmatrix}S_{1111}&S_{1122}&S_{1133}&2~S_{1123}&2~S_{1131}&2~S_{1112}\\S_{2211}&S_{2222}&S_{2233}&2~S_{2223}&2~S_{2231}&2~S_{2212}\\S_{3311}&S_{3322}&S_{3333}&2~S_{3323}&2~S_{3331}&2~S_{3312}\\2~S_{2311}&2~S_{2322}&2~S_{2333}&4~S_{2323}&4~S_{2331}&4~S_{2312}\\2~S_{3111}&2~S_{3122}&2~S_{3133}&4~S_{3123}&4~S_{3131}&4~S_{3112}\\2~S_{1211}&2~S_{1222}&2~S_{1233}&4~S_{1223}&4~S_{1231}&4~S_{1212}\end{bmatrix}}{\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\\sigma _{23}\\\sigma _{31}\\\sigma _{12}\end{bmatrix}}}$

or

${\displaystyle \mathbf {\varepsilon } =\mathbf {S} ~\mathbf {\sigma } }$

We can show that

${\displaystyle {\mathbf {S} }={\mathbf {C} }^{-1}}$

We have already seen the matrix form of the stress-strain equation for isotropic linear elastic materials. In this case the stiffness tensor has only two independent components because every plane is a plane of elastic symmetry. In direct tensor notation

${\displaystyle {\boldsymbol {\mathsf {C}}}=\lambda ~{\boldsymbol {\mathit {1}}}\otimes {\boldsymbol {\mathit {1}}}+2~\mu ~{\boldsymbol {\mathsf {I}}}}$

where ${\displaystyle \lambda }$ and ${\displaystyle \mu }$ are the elastic constants that we defined before, ${\displaystyle {\boldsymbol {\mathit {1}}}}$ is the second-order identity tensor, and ${\displaystyle {\boldsymbol {\mathsf {I}}}}$ is the symmetric fourth-order identity tensor. In index notation

${\displaystyle C_{ijkl}=\lambda ~\delta _{ij}~\delta _{kl}+2~\mu ~{\cfrac {1}{2}}(\delta _{ik}~\delta _{jl}+\delta _{il}~\delta _{jk})}$

You could alternatively express this equation in terms of the Young's modulus (${\displaystyle E}$) and the Poisson's ratio (${\displaystyle \nu }$) or in terms of the bulk modulus (${\displaystyle K}$) and the shear modulus (${\displaystyle \mu }$) or any other combination of two independent elastic parameters.

In Voigt notation the expression for the stress-strain law for isotropic materials can be written as

${\displaystyle {\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\\sigma _{23}\\\sigma _{31}\\\sigma _{12}\end{bmatrix}}={\begin{bmatrix}C_{11}&C_{12}&C_{12}&0&0&0\\C_{12}&C_{11}&C_{12}&0&0&0\\C_{12}&C_{12}&C_{11}&0&0&0\\0&0&0&(C_{11}-C_{12})/2&0&0\\0&0&0&0&(C_{11}-C_{12})/2&0\\0&0&0&0&0&(C_{11}-C_{12})/2\end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\2~\varepsilon _{23}\\2~\varepsilon _{31}\\2~\varepsilon _{12}\end{bmatrix}}}$

where

${\displaystyle C_{11}={\cfrac {E~(1-\nu )}{(1+\nu )(1-2\nu )}}~;~~C_{12}={\cfrac {E~\nu }{(1+\nu )(1-2\nu )}}~;~~(C_{11}-C_{12})/2={\cfrac {E}{2(1+\nu )}}=\mu ~.}$

The Voigt form of the strain-stress relation can be written as

${\displaystyle {\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\2~\varepsilon _{23}\\2~\varepsilon _{31}\\2~\varepsilon _{12}\end{bmatrix}}={\begin{bmatrix}S_{11}&S_{12}&S_{12}&0&0&0\\S_{12}&S_{11}&S_{12}&0&0&0\\S_{12}&S_{12}&S_{11}&0&0&0\\0&0&0&2~(S_{11}-S_{12})&0&0\\0&0&0&0&2~(S_{11}-S_{12})&0\\0&0&0&0&0&2~(S_{11}-S_{12})\end{bmatrix}}{\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\\sigma _{23}\\\sigma _{31}\\\sigma _{12}\end{bmatrix}}}$

where

${\displaystyle S_{11}={\cfrac {1}{E}}~;~~S_{12}=-{\cfrac {\nu }{E}}~;~~2~(S_{11}-S_{12})={\cfrac {2(1+\nu )}{E}}={\cfrac {1}{\mu }}~.}$

The relations between various moduli are shown in the table below:

	${\displaystyle \mu ,\nu }$	${\displaystyle \nu ,\lambda }$	${\displaystyle \mu ,\lambda }$	${\displaystyle K,\lambda }$	${\displaystyle \mu ,E}$	${\displaystyle \mu ,K}$	${\displaystyle \nu ,E}$	${\displaystyle \nu ,K}$	${\displaystyle K,E}$
${\displaystyle \lambda }$	${\displaystyle {\cfrac {2~\mu ~\nu }{1-2\nu }}}$	-	-	-	${\displaystyle {\cfrac {\mu ~(E-2\mu )}{3\mu -E}}}$	${\displaystyle K-{\cfrac {2}{3}}~\mu }$	${\displaystyle {\cfrac {E~\nu }{(1+\nu )(1-2\nu )}}}$	${\displaystyle {\cfrac {3~K~\nu }{1+\nu }}}$	${\displaystyle {\cfrac {3~K~(3K-E)}{9K-E}}}$
${\displaystyle \mu }$	-	${\displaystyle {\cfrac {\lambda ~(1-2\nu )}{2\nu }}}$	-	${\displaystyle {\cfrac {3}{2}}(K-\lambda )}$	-	-	${\displaystyle {\cfrac {E}{2(1+\nu )}}}$	${\displaystyle {\cfrac {3K(1-2\nu )}{2(1+\nu )}}}$	${\displaystyle {\cfrac {3KE}{9K-E}}}$
${\displaystyle \nu }$	-	-	${\displaystyle {\cfrac {\lambda }{2(\lambda +\mu )}}}$	${\displaystyle {\cfrac {\lambda }{3K-\lambda }}}$	${\displaystyle {\cfrac {E-2\mu }{2\mu }}}$	${\displaystyle {\cfrac {3K-2\mu }{2(3K+\mu )}}}$	-	-	${\displaystyle {\cfrac {3K-E}{6K}}}$
${\displaystyle E}$	${\displaystyle 2\mu (1+\nu )}$	${\displaystyle {\cfrac {\lambda (1+\nu )(1-2\nu )}{\nu }}}$	${\displaystyle {\cfrac {\mu (3\lambda +2\mu )}{\lambda +\mu }}}$	${\displaystyle {\cfrac {9K(K-\lambda )}{3K-\lambda }}}$	-	${\displaystyle {\cfrac {9K\mu }{3K+\mu }}}$	-	${\displaystyle 3K(1-2\nu )}$	-
${\displaystyle K}$	${\displaystyle {\cfrac {2\mu (1+\nu )}{3(1-2\nu )}}}$	${\displaystyle {\cfrac {\lambda (1+\nu )}{3\nu }}}$	${\displaystyle \lambda +{\cfrac {2}{3}}~\mu }$	-	${\displaystyle {\cfrac {\mu E}{3(3\mu -E)}}}$	-	${\displaystyle {\cfrac {E}{3(1-2\nu )}}}$	-	-