Model: Elastic Isotropic¶

Overview¶

This model may be used for solid materials with a linear elastic isotropic behavior. The relationship between stress and strain is given by Hooke’s Law, 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 strain tensor, $\lambda$ is the first Lamé elastic constant, and $\mu$ is the elastic shear modulus.

Hooke’s Law may also be expressed using Voigt notation for stress and strain vectors 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¶

For finite deformation solvers, the elastic isotropic model can be called within a hypo-elastic update routine. See Finite Deformation Models with Hypo-Materials

Parameters¶

The following attributes are supported. Note that any two elastic constants can be provided, and the other two will be internally calculated. The “default” keyword in front of certain properties indicates that this is the default value adopted for a region unless the user separately specifies a heterogeneous field via the FieldSpecification mechanism.

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

Example¶

A typical Constititutive block will look like:

<Constitutive>
  <ElasticIsotropic
     name="shale"
     defaultDensity="2700"
     defaultBulkModulus="60.0e6"
     defaultShearModulus="30.0e6" />
</Constitutive>