Linear elastic isotropic solid model¶

Overview¶

This model may be used to represents a solid material with a linear elastic isotropic response to loading. The relationship between stress and strain is typically represented by Hooke’s Law, which for the case of linear elasticity, may be expressed as:

$\sigma_{ij} = \lambda \epsilon_{kk} + 2 \mu \epsilon_{ij},$

where $\sigma_{ij}$ is the $ij$ component of the cauchy stress tensor, $\epsilon_{ij}$ is the $ij$ component of the the strain tensor, $\lambda$ is the Lames elastic constant, and $\mu$ is the elastic shear modulus.

Hooke’s Law may also be expressed using Voigt notation for the stress and strain tensors as:

$\tensor{\sigma} = \tensor{C} \cdot \tensor{\epsilon},$

or,

$\begin{bmatrix} \sigma_{11} \\ \sigma_{22} \\ \sigma_{33} \\ \sigma_{23} \\ \sigma_{13} \\ \sigma_{12} \end{bmatrix} = \begin{bmatrix} 2\mu+\lambda & \lambda & \lambda & 0 & 0 & 0 \\ \lambda & 2\mu+\lambda & \lambda & 0 & 0 & 0 \\ \lambda & \lambda & 2\mu+\lambda & 0 & 0 & 0 \\ 0 & 0 & 0 &\mu & 0 & 0 \\ 0 & 0 & 0 & 0 & \mu & 0 \\ 0 & 0 & 0 & 0 & 0 & \mu \end{bmatrix} \begin{bmatrix} \epsilon_{11} \\ \epsilon_{22} \\ \epsilon_{33} \\ 2\epsilon_{23} \\ 2\epsilon_{13} \\ 2\epsilon_{12} \end{bmatrix}.$

Variations¶

The application of linear elasticity as presented above is typically restricted to the case of infinitesimal strain. For the case of infinitesimal strain, the above relations are applied directly. For the case of finite strain, the above relations may be slightly modified to an incremental update and rotation:

$\Delta \tensor{\sigma} &= \tensor{C} \cdot \hat{\tensor{D}},\\ \tensor{\sigma}^{n+1} &= \hat{\tensor{R}}( \tensor{\sigma}^{n} + \Delta \tensor{\sigma} ) \hat{\tensor{R}}^T,$

where $\hat{\tensor{D}}$ is the “incremental rate of deformation tensor” and $\hat{\tensor{R}}$ is the incremental rotation tensor, which are typically calculated from the velocity gradient. This extension into finite strain constitutes a hypo-elastic update, and the choice of method to calculate $\hat{\tensor{D}}$ , and $\hat{\tensor{R}}$ determines if the update is objective. One commonly used method is the Hughes-Winget algorithm.

Parameters¶

The following attributes are supported:

Name	Type	Default	Description
defaultBulkModulus	real64	-1	Elastic Bulk Modulus Parameter
defaultDensity	real64	required	Default Material Density
defaultPoissonRatio	real64	-1	Poisson’s ratio
defaultShearModulus	real64	-1	Elastic Shear Modulus Parameter
defaultYoungsModulus	real64	-1	Elastic Young’s Modulus.
name	string	required	A name is required for any non-unique nodes

Example¶

<Constitutive>
  <LinearElasticIsotropic name="shale"
                          defaultDensity="2700"
                          defaultBulkModulus="61.9e6"
                          defaultShearModulus="28.57e6" />
</Constitutive>