Elasticity/Constitutive example 4

Example 4

Given:

A monoclinic crystal has inversion symmetric about the ${\widehat {\mathbf {e} }}_{1}$-${\widehat {\mathbf {e} }}_{2}$ 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

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

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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

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

Solution

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

${\boldsymbol {\varepsilon }}={\begin{bmatrix}\varepsilon _{11}&\varepsilon _{12}&\varepsilon _{13}\\\varepsilon _{21}&\varepsilon _{22}&\varepsilon _{23}\\\varepsilon _{31}&\varepsilon _{32}&\varepsilon _{33}\end{bmatrix}}$

The transformation rule for a second order tensor ${\boldsymbol {A}}\,$ is

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

Applying this transformation to the strain tensor, we have

${\begin{aligned}{\begin{bmatrix}\varepsilon _{11}^{'}&\varepsilon _{12}^{'}&\varepsilon _{13}^{'}\\\varepsilon _{21}^{'}&\varepsilon _{22}^{'}&\varepsilon _{23}^{'}\\\varepsilon _{31}^{'}&\varepsilon _{32}^{'}&\varepsilon _{33}^{'}\end{bmatrix}}&={\begin{bmatrix}1&0&0\\0&1&0\\0&0&-1\end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}&\varepsilon _{12}&\varepsilon _{13}\\\varepsilon _{21}&\varepsilon _{22}&\varepsilon _{23}\\\varepsilon _{31}&\varepsilon _{32}&\varepsilon _{33}\end{bmatrix}}{\begin{bmatrix}1&0&0\\0&1&0\\0&0&-1\end{bmatrix}}\\&={\begin{bmatrix}1&0&0\\0&1&0\\0&0&-1\end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}&\varepsilon _{12}&-\varepsilon _{13}\\\varepsilon _{21}&\varepsilon _{22}&-\varepsilon _{23}\\\varepsilon _{31}&\varepsilon _{32}&-\varepsilon _{33}\end{bmatrix}}\\&={\begin{bmatrix}\varepsilon _{11}&\varepsilon _{12}&-\varepsilon _{13}\\\varepsilon _{21}&\varepsilon _{22}&-\varepsilon _{23}\\-\varepsilon _{31}&-\varepsilon _{32}&\varepsilon _{33}\end{bmatrix}}\end{aligned}}$

In engineering notation (Voigt notation),

${\begin{aligned}\left[{\boldsymbol {\varepsilon }}\right]&={\begin{bmatrix}\varepsilon _{11}&\varepsilon _{22}&\varepsilon _{33}&2\varepsilon _{23}&2\varepsilon _{31}&2\varepsilon _{12}\end{bmatrix}}^{T}\\&={\begin{bmatrix}\varepsilon _{1}&\varepsilon _{2}&\varepsilon _{3}&\varepsilon _{4}&\varepsilon _{5}&\varepsilon _{6}\end{bmatrix}}^{T}\end{aligned}}$

Therefore, the transformed strain tensor can be written as

${\begin{aligned}\left[{\boldsymbol {\varepsilon }}\right]^{'}&={\begin{bmatrix}\varepsilon _{1}^{'}&\varepsilon _{2}^{'}&\varepsilon _{3}^{'}&\varepsilon _{4}^{'}&\varepsilon _{5}^{'}&\varepsilon _{6}^{'}\end{bmatrix}}^{T}\\&={\begin{bmatrix}\varepsilon _{1}&\varepsilon _{2}&\varepsilon _{3}&-\varepsilon _{4}&-\varepsilon _{5}&\varepsilon _{6}\end{bmatrix}}^{T}\\\end{aligned}}$

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

$C_{ij}(\varepsilon _{i}\varepsilon _{j}-\varepsilon _{i}^{'}\varepsilon _{j}^{'})=0~~~~(i,j=1\dots 6)$

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

Therefore,

${\begin{aligned}C_{11}(\varepsilon _{1}\varepsilon _{1}-\varepsilon _{1}^{'}\varepsilon _{1}^{'})+C_{12}(\varepsilon _{1}\varepsilon _{2}-\varepsilon _{1}^{'}\varepsilon _{2}^{'})+C_{13}(\varepsilon _{1}\varepsilon _{3}-\varepsilon _{1}^{'}\varepsilon _{3}^{'})&+\\C_{14}(\varepsilon _{1}\varepsilon _{4}-\varepsilon _{1}^{'}\varepsilon _{4}^{'})+C_{15}(\varepsilon _{1}\varepsilon _{5}-\varepsilon _{1}^{'}\varepsilon _{5}^{'})+C_{16}(\varepsilon _{1}\varepsilon _{6}-\varepsilon _{1}^{'}\varepsilon _{6}^{'})&+\\C_{21}(\varepsilon _{2}\varepsilon _{1}-\varepsilon _{2}^{'}\varepsilon _{1}^{'})+C_{22}(\varepsilon _{2}\varepsilon _{2}-\varepsilon _{2}^{'}\varepsilon _{2}^{'})+C_{23}(\varepsilon _{2}\varepsilon _{3}-\varepsilon _{2}^{'}\varepsilon _{3}^{'})&+\\C_{24}(\varepsilon _{2}\varepsilon _{4}-\varepsilon _{2}^{'}\varepsilon _{4}^{'})+C_{25}(\varepsilon _{2}\varepsilon _{5}-\varepsilon _{2}^{'}\varepsilon _{5}^{'})+C_{26}(\varepsilon _{2}\varepsilon _{6}-\varepsilon _{2}^{'}\varepsilon _{6}^{'})&+\\C_{31}(\varepsilon _{3}\varepsilon _{1}-\varepsilon _{3}^{'}\varepsilon _{1}^{'})+C_{32}(\varepsilon _{3}\varepsilon _{2}-\varepsilon _{3}^{'}\varepsilon _{2}^{'})+C_{33}(\varepsilon _{3}\varepsilon _{3}-\varepsilon _{3}^{'}\varepsilon _{3}^{'})&+\\C_{34}(\varepsilon _{3}\varepsilon _{4}-\varepsilon _{3}^{'}\varepsilon _{4}^{'})+C_{35}(\varepsilon _{3}\varepsilon _{5}-\varepsilon _{3}^{'}\varepsilon _{5}^{'})+C_{36}(\varepsilon _{3}\varepsilon _{6}-\varepsilon _{3}^{'}\varepsilon _{6}^{'})&+\\C_{41}(\varepsilon _{4}\varepsilon _{1}-\varepsilon _{4}^{'}\varepsilon _{1}^{'})+C_{42}(\varepsilon _{4}\varepsilon _{2}-\varepsilon _{4}^{'}\varepsilon _{2}^{'})+C_{43}(\varepsilon _{4}\varepsilon _{3}-\varepsilon _{4}^{'}\varepsilon _{3}^{'})&+\\C_{44}(\varepsilon _{4}\varepsilon _{4}-\varepsilon _{4}^{'}\varepsilon _{4}^{'})+C_{45}(\varepsilon _{4}\varepsilon _{5}-\varepsilon _{4}^{'}\varepsilon _{5}^{'})+C_{46}(\varepsilon _{4}\varepsilon _{6}-\varepsilon _{4}^{'}\varepsilon _{6}^{'})&+\\C_{51}(\varepsilon _{5}\varepsilon _{1}-\varepsilon _{5}^{'}\varepsilon _{1}^{'})+C_{52}(\varepsilon _{5}\varepsilon _{2}-\varepsilon _{5}^{'}\varepsilon _{2}^{'})+C_{53}(\varepsilon _{5}\varepsilon _{3}-\varepsilon _{5}^{'}\varepsilon _{3}^{'})&+\\C_{54}(\varepsilon _{5}\varepsilon _{4}-\varepsilon _{5}^{'}\varepsilon _{4}^{'})+C_{55}(\varepsilon _{5}\varepsilon _{5}-\varepsilon _{5}^{'}\varepsilon _{5}^{'})+C_{56}(\varepsilon _{5}\varepsilon _{6}-\varepsilon _{5}^{'}\varepsilon _{6}^{'})&+\\C_{61}(\varepsilon _{6}\varepsilon _{1}-\varepsilon _{6}^{'}\varepsilon _{1}^{'})+C_{62}(\varepsilon _{6}\varepsilon _{2}-\varepsilon _{6}^{'}\varepsilon _{2}^{'})+C_{63}(\varepsilon _{6}\varepsilon _{3}-\varepsilon _{6}^{'}\varepsilon _{3}^{'})&+\\C_{64}(\varepsilon _{6}\varepsilon _{4}-\varepsilon _{6}^{'}\varepsilon _{4}^{'})+C_{65}(\varepsilon _{6}\varepsilon _{5}-\varepsilon _{6}^{'}\varepsilon _{5}^{'})+C_{66}(\varepsilon _{6}\varepsilon _{6}-\varepsilon _{6}^{'}\varepsilon _{6}^{'})&=0\end{aligned}}$

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

${\begin{aligned}C_{14}(\varepsilon _{1}\varepsilon _{4}+\varepsilon _{1}\varepsilon _{4})+C_{15}(\varepsilon _{1}\varepsilon _{5}+\varepsilon _{1}\varepsilon _{5})+C_{24}(\varepsilon _{2}\varepsilon _{4}+\varepsilon _{2}\varepsilon _{4})&+\\C_{25}(\varepsilon _{2}\varepsilon _{5}+\varepsilon _{2}\varepsilon _{5})+C_{34}(\varepsilon _{3}\varepsilon _{4}+\varepsilon _{3}\varepsilon _{4})+C_{35}(\varepsilon _{3}\varepsilon _{5}+\varepsilon _{3}\varepsilon _{5})&+\\C_{41}(\varepsilon _{4}\varepsilon _{1}+\varepsilon _{4}\varepsilon _{1})+C_{42}(\varepsilon _{4}\varepsilon _{2}+\varepsilon _{4}\varepsilon _{2})+C_{43}(\varepsilon _{4}\varepsilon _{3}+\varepsilon _{4}\varepsilon _{3})&+\\C_{46}(\varepsilon _{4}\varepsilon _{6}+\varepsilon _{4}\varepsilon _{6})+C_{51}(\varepsilon _{5}\varepsilon _{1}+\varepsilon _{5}\varepsilon _{1})+C_{52}(\varepsilon _{5}\varepsilon _{2}+\varepsilon _{5}\varepsilon _{2})&+\\C_{53}(\varepsilon _{5}\varepsilon _{3}+\varepsilon _{5}\varepsilon _{3})+C_{56}(\varepsilon _{5}\varepsilon _{6}+\varepsilon _{5}\varepsilon _{6})+C_{64}(\varepsilon _{6}\varepsilon _{4}+\varepsilon _{6}\varepsilon _{4})&+\\C_{65}(\varepsilon _{6}\varepsilon _{5}+\varepsilon _{6}\varepsilon _{5})&=0\end{aligned}}$

or,

${\begin{aligned}2~C_{14}\varepsilon _{1}\varepsilon _{4}+2~C_{15}\varepsilon _{1}\varepsilon _{5}+2~C_{24}\varepsilon _{2}\varepsilon _{4}+2~C_{25}\varepsilon _{2}\varepsilon _{5}+2~C_{34}\varepsilon _{3}\varepsilon _{4}+2~C_{35}\varepsilon _{3}\varepsilon _{5}&+\\2~C_{41}\varepsilon _{4}\varepsilon _{1}+2~C_{42}\varepsilon _{4}\varepsilon _{2}+2~C_{43}\varepsilon _{4}\varepsilon _{3}+2~C_{46}\varepsilon _{4}\varepsilon _{6}+2~C_{51}\varepsilon _{5}\varepsilon _{1}+2~C_{52}\varepsilon _{5}\varepsilon _{2}&+\\2~C_{53}\varepsilon _{5}\varepsilon _{3}+2~C_{56}\varepsilon _{5}\varepsilon _{6}+2~C_{64}\varepsilon _{6}\varepsilon _{4}+2~C_{65}\varepsilon _{6}\varepsilon _{5}&=0\end{aligned}}$

Using the symmetry of the stiffness matrix, we have

${\begin{aligned}4~C_{14}\varepsilon _{1}\varepsilon _{4}+4~C_{15}\varepsilon _{1}\varepsilon _{5}+4~C_{24}\varepsilon _{2}\varepsilon _{4}+4~C_{25}\varepsilon _{2}\varepsilon _{5}+4~C_{34}\varepsilon _{3}\varepsilon _{4}+4~C_{35}\varepsilon _{3}\varepsilon _{5}&+\\4~C_{46}\varepsilon _{4}\varepsilon _{6}+4~C_{56}\varepsilon _{5}\varepsilon _{6}&=0\end{aligned}}$

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

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

Hence shown.

Elasticity