Linear Elasticity

The linear elastic model:

can define isotropic, orthotropic, or anisotropic material behavior;
is valid for small elastic strains (normally less than 5%);
has properties that depend on temperature or other variables.

For example, in a linear elastic model, if stress reaches 100 MPa under a load of 1,000 N, then stress will reach 1,000 MPa under a load of 10,000 N. The simplest form of linear elasticity is the isotropic case, and the stress-strain relationship is given by:

{\begin{matrix} ε_{11} \\ ε_{22} \\ ε_{33} \\ γ_{12} \\ γ_{13} \\ γ_{23} \end{matrix}} = [\begin{matrix} 1 / E & - ν / E & - ν / E & 0 & 0 & 0 \\ - ν / E & 1 / E & - ν / E & 0 & 0 & 0 \\ - ν / E & - ν / E & 1 / E & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 / G & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 / G & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 / G \end{matrix}] {\begin{matrix} σ_{11} \\ σ_{22} \\ σ_{33} \\ σ_{12} \\ σ_{13} \\ σ_{23} \end{matrix}} .

The elastic properties are completely defined by the Young's modulus, $E$ , and the Poisson's ratio, $ν$ . The shear modulus, $G$ , can be expressed in terms of $E$ and $ν$ as $G = Ε / 2 (1 + ν)$ . These parameters can be given as functions of temperature and of other predefined fields, if necessary.

The material yielding is not modeled in a linear elastic model.