# For a plane strain problem, the relation between stress and strain components for an orthotropic material is σ=Cε. Which option is the correct structure of the matrix C?

For a plane strain problem, the relation between stress and strain components for an orthotropic material is σ=Cε. Which option is the correct structure of the matrix C?

(a) $\begin{pmatrix}\bar{c}_{11} & \bar{c}_{12} & 0 \\\bar{c}_{12} & \bar{c}_{22} & 0 \\0 & 0 & \bar{c}_{66} \end{pmatrix}$

(b) $\begin{pmatrix}\bar{c}_{11} & 0 & \bar{c}_{13} \\\bar{c}_{12} & \bar{c}_{22} & 0 \\0 & 0 & \bar{c}_{66} \end{pmatrix}$

(c) $\begin{pmatrix}\bar{c}_{11} & \bar{c}_{12} & 0 \\-\bar{c}_{12} & \bar{c}_{22} & 0 \\0 & 0 & \bar{c}_{66} \end{pmatrix}$

(d) $\begin{pmatrix}\bar{c}_{11} & -\bar{c}_{12} & 0 \\\bar{c}_{12} & \bar{c}_{22} & 0 \\0 & 0 & \bar{c}_{66} \end{pmatrix}$

I have been asked this question by my college professor while I was bunking the class.

I need to ask this question from Plane Elasticity topic in portion Plane Elasticity of Finite Element Method

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Right answer is (a) $\begin{pmatrix}\bar{c}_{11} & \bar{c}_{12} & 0 \\\bar{c}_{12} & \bar{c}_{22} & 0 \\0 & 0 & \bar{c}_{66} \end{pmatrix}$

The explanation: For an orthotropic material under plane strain, with principal material axes (x1, x2, x3) coinciding with the (x, y, z) coordinates, the relation between stress and strain components is $\begin{pmatrix}\sigma_{xx} \\ \sigma_{yy} \\ \sigma_{xy}\end{pmatrix} = \begin{pmatrix}\bar{c}_{11} & \bar{c}_{12} & 0 \\\bar{c}_{12} & \bar{c}_{22} & 0 \\0 & 0 & \bar{c}_{66} \end{pmatrix} = \begin{pmatrix}\varepsilon_{xx} \\ \varepsilon_{yy} \\ 2 \varepsilon_{xy}\end{pmatrix}$ where C is the elastic stiffness matrix. The state of stress is  σxz=σyz=0 and  $\sigma_{zz}=E_3(\frac{v_{13}}{E_1}\sigma_{xx}+\frac{v_{23}}{E_2}\sigma_{yy})$.

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