The constitutive relation (relation between stress and strain) in linear elasticity (known as the Hook’s law) is given by the proportionality between stress and strain, \(\mathbf{\sigma} = \mathbf{D}\mathbf{\varepsilon}\), where \(\mathbf{\sigma}\) is a second-rank stress tensor, \(\mathbf{\varepsilon}\) is a second-rank strain tensor and \(\mathbf{D}\) is a fourth-rank tensor of elastic constants.
The simplest but most often implemented tensor of elastic constants (mainly for polycrystalline materials) is that of isotropic elasticity.
\[\mathbf{D} = \begin{bmatrix} D_{11} & D_{12} & D_{12} & 0 & 0 & 0 \\ D_{12} & D_{11} & D_{12} & 0 & 0 & 0 \\ D_{12} & D_{12} & D_{11} & 0 & 0 & 0 \\ 0 & 0 & 0 & D_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & D_{44} & 0 \\ 0 & 0 & 0 & 0 & 0 & D_{44} \\ \end{bmatrix}, \label{}\tag{1}\]
in which
\[\begin{array}{l} D_{11} = \frac{E(1-\nu)}{(1+\nu)(1-2\nu)} \\ D_{12} = \frac{E\nu}{(1+\nu)(1-2\nu)} \\ D_{44} = \frac{E}{2(1+\nu)} \\ \end{array}, \label{}\tag{2}\]
where \(E\) is the Young’s modulus and \(\mu\) is the Poisson’s ratio. It is only these two constants sufficient for describing the isotropic behaviour.
This constitutive relation in 2D is simplified into
\[\begin{bmatrix} \sigma_{xx} \\ \sigma_{yy} \\ \tau_{xy} \\ \end{bmatrix} = \begin{bmatrix} C_{11} & C_{12} & 0 \\ C_{12} & C_{11} & 0 \\ 0 & 0 & C_{44} \\ \end{bmatrix} \begin{bmatrix} \varepsilon_{xx} \\ \varepsilon_{yy} \\ \gamma_{xy} \\ \end{bmatrix}, \label{}\tag{3}\]
where the elastic constants \(C_{11}\), \(C_{12}\), and \(C_{44}\) are defined for
- plane strain
\[\begin{array}{l} C_{11} = \frac{E(1-\mu)}{(1+\mu)(1-2\mu)} \\ C_{12} = \frac{E\mu}{(1+\mu)(1-2\mu)} \\ C_{44} = \frac{E}{2(1+\mu)} \\ \end{array}. \label{}\tag{4}\]
- plane stress
\[\begin{array}{l} C_{11} = \frac{E}{1-\mu^2} \\ C_{12} = \frac{E\mu}{1-\mu^2} \\ C_{44} = \frac{E}{2(1+\mu)} \\ \end{array}. \label{}\tag{5}\]
For an elastically isotropic material the following condition holds: \(C_{44}=\frac{C_{11}-C_{12}}{2}\). There is no directionality in the elastic properties. Polycrystalline materials may be considered isotropic as the sufficient amount of grains with a random (non-textured) grain orientation will average the elastic properties.