Constitutive relationships

The stresses and strain are related through a constitutive relationship.

$$\sigma = C \epsilon$$

or in tensor notation

$$\sigma_{ij} = c_{ijkl} \: \epsilon_{kl}$$

The stress $\sigma$ and strain $\epsilon$ are symmetric, second order tensors which can be represented in matrix form

$$\sigma = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\ \sigma_{zx} & \sigma_{zy} & \sigma_{zz} \\ \end{bmatrix}$$

$$\epsilon = \begin{bmatrix} \epsilon_{xx} & \epsilon_{xy} & \epsilon_{xz} \\ \epsilon_{yx} & \epsilon_{yy} & \epsilon_{yz} \\ \epsilon_{zx} & \epsilon_{zy} & \epsilon_{zz} \\ \end{bmatrix}$$

The relationship between the stress and strain is through the constitutive relationship which is a fourth order tensor so for convenience of representing this as a matrix the stress and strain tensors are represented as vectors using modified Voigt notation:

$$\sigma = (\sigma_{xx}, \sigma_{yy}, \sigma_{zz},\sigma_{xy}, \sigma_{yz}, \sigma_{zx})$$

$$\epsilon = (\epsilon_{xx}, \epsilon_{yy}, \epsilon_{zz},\epsilon_{xy}, \epsilon_{yz}, \epsilon_{zx})$$

For a general anisotropic material the stress-strain relation ship can be expressed as

$$\begin{bmatrix} \sigma_{xx} \\ \sigma_{yy} \\ \sigma_{zz} \\ \sigma_{xy} \\ \sigma_{yz} \\ \sigma_{zx} \\ \end{bmatrix} = \begin{bmatrix} c_{xxxx} & c_{xxyy} & c_{xxzz} & c_{xxxy} & c_{xxyz} & c_{xxzx}\\ & c_{yyyy} & c_{yyzz} & c_{yyxy} & c_{yyyz} & c_{yyzx}\\ & & c_{zzzz} & c_{zzxy} & c_{zzyz} & c_{zzzx}\\ & & & c_{xyxy} & c_{xyyz} & c_{zzzx}\\ & symm & & & c_{yzyz} & c_{zzzx}\\ & & & & & c_{zxzx}\\ \end{bmatrix} \begin{bmatrix} \epsilon_{xx} \\ \epsilon_{yy} \\ \epsilon_{zz} \\ \epsilon_{xy} \\ \epsilon_{yz} \\ \epsilon_{zx} \\ \end{bmatrix}$$