Bulk materials

A bulk material model is assigned to a volume and defines its deformable characteristic. Available materials are summarized in the sections below. See also the BULK_MATERIAL input command.

Kirchhoff – Saint Venant

This is a simple extension of the linearly elastic material to the large deformation regime. Suitable for large rotation, small strain problems. The strain energy function \(\Psi\) of the Kirchhoff – Saint Venant materials reads

\[\Psi=\frac{1}{4}\left[\mathbf{F}^{T}\mathbf{F}-\mathbf{I}\right]:\mathbf{C}:\left[\mathbf{F}^{T}\mathbf{F}-\mathbf{I}\right]\]

where

\[C_{ijkl}=\lambda\delta_{ij}\delta_{kl}+\mu\left[\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}\right]\]

In the above \(\lambda\) and \(\mu\) are Lamé constants, while \(\delta_{ij}\) is the Kronecker delta. The Lamé constants can be expressed in terms of the Young modulus \(E\) and the Poisson ratio \(\nu\) as

\[\lambda=\frac{E\nu}{\left(1+\nu\right)\left(1-2\nu\right)}\]

\[\mu=\frac{E}{2+2\nu}\]

The first Piola stress tensor is computed as a gradient of the hyperelastic potential \(\Psi\)

\[\mathbf{P}=\partial_{\mathbf{F}}\Psi\left(\mathbf{F}\right)\]

where \(\mathbf{F}\) is the deformation gradient.