Elasticity/Constitutive example 4

Example 4

Given:

A monoclinic crystal has inversion symmetric about the ${\widehat{𝐞}}_1$-${\widehat{𝐞}}_2$ plane. Therefore, the material properties do not change for a mirror-reflection through this plane. The stress-strain relations must therefore remain unchanged under this transformation. The transformation matrix $\left[L\right]$ for this for the mirror inversion is given by

$\left[L\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& -1\end{array}\right]$

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If we apply this transformation to the stress and strain tensors, then the stiffness matrix of the material (in Voigt notation) is

$\left[C\right]=\left[\begin{array}{cccccc}{C}_{11}& C_{12}& C_{13}& 0& 0& C_{16}\\ C_{21}& C_{22}& C_{23}& 0& 0& C_{26}\\ C_{31}& C_{32}& C_{33}& 0& 0& C_{36}\\ 0& 0& 0& C_{44}& C_{45}& 0\\ 0& 0& 0& C_{54}& C_{55}& 0\\ C_{61}& C_{62}& C_{63}& 0& 0& C_{66}\end{array}\right]$

Solution

In 3$\times$3 matrix form, the strain tensor is given by

$\mathit{\epsilon }=\left[\begin{array}{ccc}{\epsilon }_{11}& {\epsilon }_{12}& {\epsilon }_{13}\\ {\epsilon }_{21}& {\epsilon }_{22}& {\epsilon }_{23}\\ {\epsilon }_{31}& {\epsilon }_{32}& {\epsilon }_{33}\end{array}\right]$

The transformation rule for a second order tensor $𝑨$ is

${\left[A\right]}^{\text{'}}=\left[L\right]\left[A\right]{\left[L\right]}^T$

Applying this transformation to the strain tensor, we have

$\begin{array}{cc}\left[\begin{array}{ccc}{\epsilon }_{11}^{\text{'}}& {\epsilon }_{12}^{\text{'}}& {\epsilon }_{13}^{\text{'}}\\ {\epsilon }_{21}^{\text{'}}& {\epsilon }_{22}^{\text{'}}& {\epsilon }_{23}^{\text{'}}\\ {\epsilon }_{31}^{\text{'}}& {\epsilon }_{32}^{\text{'}}& {\epsilon }_{33}^{\text{'}}\end{array}\right]& =\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& -1\end{array}\right]\left[\begin{array}{ccc}{\epsilon }_{11}& {\epsilon }_{12}& {\epsilon }_{13}\\ {\epsilon }_{21}& {\epsilon }_{22}& {\epsilon }_{23}\\ {\epsilon }_{31}& {\epsilon }_{32}& {\epsilon }_{33}\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& -1\end{array}\right]\\ & =\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& -1\end{array}\right]\left[\begin{array}{ccc}{\epsilon }_{11}& {\epsilon }_{12}& -{\epsilon }_{13}\\ {\epsilon }_{21}& {\epsilon }_{22}& -{\epsilon }_{23}\\ {\epsilon }_{31}& {\epsilon }_{32}& -{\epsilon }_{33}\end{array}\right]\\ & =\left[\begin{array}{ccc}{\epsilon }_{11}& {\epsilon }_{12}& -{\epsilon }_{13}\\ {\epsilon }_{21}& {\epsilon }_{22}& -{\epsilon }_{23}\\ -{\epsilon }_{31}& -{\epsilon }_{32}& {\epsilon }_{33}\end{array}\right]\end{array}$

In engineering notation (Voigt notation),

$\begin{array}{cc}\left[\mathit{\epsilon }\right]& ={\left[\begin{array}{cccccc}{\epsilon }_{11}& {\epsilon }_{22}& {\epsilon }_{33}& 2{\epsilon }_{23}& 2{\epsilon }_{31}& 2{\epsilon }_{12}\end{array}\right]}^T\\ & ={\left[\begin{array}{cccccc}{\epsilon }_1& {\epsilon }_2& {\epsilon }_3& {\epsilon }_4& {\epsilon }_5& {\epsilon }_6\end{array}\right]}^T\end{array}$

Therefore, the transformed strain tensor can be written as

$\begin{array}{cc}{\left[\mathit{\epsilon }\right]}^{\text{'}}& ={\left[\begin{array}{cccccc}{\epsilon }_1^{\text{'}}& {\epsilon }_2^{\text{'}}& {\epsilon }_3^{\text{'}}& {\epsilon }_4^{\text{'}}& {\epsilon }_5^{\text{'}}& {\epsilon }_6^{\text{'}}\end{array}\right]}^T\\ & ={\left[\begin{array}{cccccc}{\epsilon }_1& {\epsilon }_2& {\epsilon }_3& -{\epsilon }_4& -{\epsilon }_5& {\epsilon }_6\end{array}\right]}^T\\ \end{array}$

The expression for the strain energy density of a linear elastic material imposes a constraint on the components of the stiffness tensor in the presence of planes of material symmetry. This constraint is

$C_{ij}({\epsilon }_i{\epsilon }_j-{\epsilon }_i^{\text{'}}{\epsilon }_j^{\text{'}})=0(i,j=1\dots 6)$

where $C_{ij}$ are the components of the 6$\times$6 matrix that contains the independent components of the stiffness tensor.

Therefore,

$\begin{array}{cc}{C}_{11}({\epsilon }_1{\epsilon }_1-{\epsilon }_1^{\text{'}}{\epsilon }_1^{\text{'}})+C_{12}({\epsilon }_1{\epsilon }_2-{\epsilon }_1^{\text{'}}{\epsilon }_2^{\text{'}})+C_{13}({\epsilon }_1{\epsilon }_3-{\epsilon }_1^{\text{'}}{\epsilon }_3^{\text{'}})& +\\ C_{14}({\epsilon }_1{\epsilon }_4-{\epsilon }_1^{\text{'}}{\epsilon }_4^{\text{'}})+C_{15}({\epsilon }_1{\epsilon }_5-{\epsilon }_1^{\text{'}}{\epsilon }_5^{\text{'}})+C_{16}({\epsilon }_1{\epsilon }_6-{\epsilon }_1^{\text{'}}{\epsilon }_6^{\text{'}})& +\\ C_{21}({\epsilon }_2{\epsilon }_1-{\epsilon }_2^{\text{'}}{\epsilon }_1^{\text{'}})+C_{22}({\epsilon }_2{\epsilon }_2-{\epsilon }_2^{\text{'}}{\epsilon }_2^{\text{'}})+C_{23}({\epsilon }_2{\epsilon }_3-{\epsilon }_2^{\text{'}}{\epsilon }_3^{\text{'}})& +\\ C_{24}({\epsilon }_2{\epsilon }_4-{\epsilon }_2^{\text{'}}{\epsilon }_4^{\text{'}})+C_{25}({\epsilon }_2{\epsilon }_5-{\epsilon }_2^{\text{'}}{\epsilon }_5^{\text{'}})+C_{26}({\epsilon }_2{\epsilon }_6-{\epsilon }_2^{\text{'}}{\epsilon }_6^{\text{'}})& +\\ C_{31}({\epsilon }_3{\epsilon }_1-{\epsilon }_3^{\text{'}}{\epsilon }_1^{\text{'}})+C_{32}({\epsilon }_3{\epsilon }_2-{\epsilon }_3^{\text{'}}{\epsilon }_2^{\text{'}})+C_{33}({\epsilon }_3{\epsilon }_3-{\epsilon }_3^{\text{'}}{\epsilon }_3^{\text{'}})& +\\ C_{34}({\epsilon }_3{\epsilon }_4-{\epsilon }_3^{\text{'}}{\epsilon }_4^{\text{'}})+C_{35}({\epsilon }_3{\epsilon }_5-{\epsilon }_3^{\text{'}}{\epsilon }_5^{\text{'}})+C_{36}({\epsilon }_3{\epsilon }_6-{\epsilon }_3^{\text{'}}{\epsilon }_6^{\text{'}})& +\\ C_{41}({\epsilon }_4{\epsilon }_1-{\epsilon }_4^{\text{'}}{\epsilon }_1^{\text{'}})+C_{42}({\epsilon }_4{\epsilon }_2-{\epsilon }_4^{\text{'}}{\epsilon }_2^{\text{'}})+C_{43}({\epsilon }_4{\epsilon }_3-{\epsilon }_4^{\text{'}}{\epsilon }_3^{\text{'}})& +\\ C_{44}({\epsilon }_4{\epsilon }_4-{\epsilon }_4^{\text{'}}{\epsilon }_4^{\text{'}})+C_{45}({\epsilon }_4{\epsilon }_5-{\epsilon }_4^{\text{'}}{\epsilon }_5^{\text{'}})+C_{46}({\epsilon }_4{\epsilon }_6-{\epsilon }_4^{\text{'}}{\epsilon }_6^{\text{'}})& +\\ C_{51}({\epsilon }_5{\epsilon }_1-{\epsilon }_5^{\text{'}}{\epsilon }_1^{\text{'}})+C_{52}({\epsilon }_5{\epsilon }_2-{\epsilon }_5^{\text{'}}{\epsilon }_2^{\text{'}})+C_{53}({\epsilon }_5{\epsilon }_3-{\epsilon }_5^{\text{'}}{\epsilon }_3^{\text{'}})& +\\ C_{54}({\epsilon }_5{\epsilon }_4-{\epsilon }_5^{\text{'}}{\epsilon }_4^{\text{'}})+C_{55}({\epsilon }_5{\epsilon }_5-{\epsilon }_5^{\text{'}}{\epsilon }_5^{\text{'}})+C_{56}({\epsilon }_5{\epsilon }_6-{\epsilon }_5^{\text{'}}{\epsilon }_6^{\text{'}})& +\\ C_{61}({\epsilon }_6{\epsilon }_1-{\epsilon }_6^{\text{'}}{\epsilon }_1^{\text{'}})+C_{62}({\epsilon }_6{\epsilon }_2-{\epsilon }_6^{\text{'}}{\epsilon }_2^{\text{'}})+C_{63}({\epsilon }_6{\epsilon }_3-{\epsilon }_6^{\text{'}}{\epsilon }_3^{\text{'}})& +\\ C_{64}({\epsilon }_6{\epsilon }_4-{\epsilon }_6^{\text{'}}{\epsilon }_4^{\text{'}})+C_{65}({\epsilon }_6{\epsilon }_5-{\epsilon }_6^{\text{'}}{\epsilon }_5^{\text{'}})+C_{66}({\epsilon }_6{\epsilon }_6-{\epsilon }_6^{\text{'}}{\epsilon }_6^{\text{'}})& =0\end{array}$

For a monoclinic material, replacing the transformed strain components by the equivalent original strain components, we get

$\begin{array}{cc}{C}_{14}({\epsilon }_1{\epsilon }_4+{\epsilon }_1{\epsilon }_4)+C_{15}({\epsilon }_1{\epsilon }_5+{\epsilon }_1{\epsilon }_5)+C_{24}({\epsilon }_2{\epsilon }_4+{\epsilon }_2{\epsilon }_4)& +\\ C_{25}({\epsilon }_2{\epsilon }_5+{\epsilon }_2{\epsilon }_5)+C_{34}({\epsilon }_3{\epsilon }_4+{\epsilon }_3{\epsilon }_4)+C_{35}({\epsilon }_3{\epsilon }_5+{\epsilon }_3{\epsilon }_5)& +\\ C_{41}({\epsilon }_4{\epsilon }_1+{\epsilon }_4{\epsilon }_1)+C_{42}({\epsilon }_4{\epsilon }_2+{\epsilon }_4{\epsilon }_2)+C_{43}({\epsilon }_4{\epsilon }_3+{\epsilon }_4{\epsilon }_3)& +\\ C_{46}({\epsilon }_4{\epsilon }_6+{\epsilon }_4{\epsilon }_6)+C_{51}({\epsilon }_5{\epsilon }_1+{\epsilon }_5{\epsilon }_1)+C_{52}({\epsilon }_5{\epsilon }_2+{\epsilon }_5{\epsilon }_2)& +\\ C_{53}({\epsilon }_5{\epsilon }_3+{\epsilon }_5{\epsilon }_3)+C_{56}({\epsilon }_5{\epsilon }_6+{\epsilon }_5{\epsilon }_6)+C_{64}({\epsilon }_6{\epsilon }_4+{\epsilon }_6{\epsilon }_4)& +\\ C_{65}({\epsilon }_6{\epsilon }_5+{\epsilon }_6{\epsilon }_5)& =0\end{array}$

or,

$\begin{array}{cc}2C_{14}{\epsilon }_1{\epsilon }_4+2C_{15}{\epsilon }_1{\epsilon }_5+2C_{24}{\epsilon }_2{\epsilon }_4+2C_{25}{\epsilon }_2{\epsilon }_5+2C_{34}{\epsilon }_3{\epsilon }_4+2C_{35}{\epsilon }_3{\epsilon }_5& +\\ 2C_{41}{\epsilon }_4{\epsilon }_1+2C_{42}{\epsilon }_4{\epsilon }_2+2C_{43}{\epsilon }_4{\epsilon }_3+2C_{46}{\epsilon }_4{\epsilon }_6+2C_{51}{\epsilon }_5{\epsilon }_1+2C_{52}{\epsilon }_5{\epsilon }_2& +\\ 2C_{53}{\epsilon }_5{\epsilon }_3+2C_{56}{\epsilon }_5{\epsilon }_6+2C_{64}{\epsilon }_6{\epsilon }_4+2C_{65}{\epsilon }_6{\epsilon }_5& =0\end{array}$

Using the symmetry of the stiffness matrix, we have

$\begin{array}{cc}4C_{14}{\epsilon }_1{\epsilon }_4+4C_{15}{\epsilon }_1{\epsilon }_5+4C_{24}{\epsilon }_2{\epsilon }_4+4C_{25}{\epsilon }_2{\epsilon }_5+4C_{34}{\epsilon }_3{\epsilon }_4+4C_{35}{\epsilon }_3{\epsilon }_5& +\\ 4C_{46}{\epsilon }_4{\epsilon }_6+4C_{56}{\epsilon }_5{\epsilon }_6& =0\end{array}$

Since the strains can be arbitrary, the above condition is satisfied only if

$C_{14}=C_{15}=C_{24}=C_{25}=C_{34}=C_{35}=C_{46}=C_{56}=0$

Therefore, the stiffness matrix is given by

$\left[C\right]=\left[\begin{array}{cccccc}{C}_{11}& C_{12}& C_{13}& 0& 0& C_{16}\\ C_{21}& C_{22}& C_{23}& 0& 0& C_{26}\\ C_{31}& C_{32}& C_{33}& 0& 0& C_{36}\\ 0& 0& 0& C_{44}& C_{45}& 0\\ 0& 0& 0& C_{54}& C_{55}& 0\\ C_{61}& C_{62}& C_{63}& 0& 0& C_{66}\end{array}\right]$

Hence shown.