neo-Hookean¶

Module: solid

Category: material

Type string: "neo-Hookean"

Parameters¶

Name	Description	Default	Units
`density`	density	1	[M/L^3]
`E`	Young's modulus	0	[P]
`v`	Poisson's ratio	0	[]

Description¶

This model describes an unconstrained Neo-Hookean material ¹. It has a non-linear stress-strain behavior, but reduces to the classical linear elasticity model for small strains and small rotations. It is derived from the following hyperelastic strain-energy function:

\[ \begin{equation} W=\frac{\mu}{2}\left(I_{1}-3\right)-\mu\ln J+\frac{\lambda}{2}\left(\ln J\right)^{2}. \end{equation} \]

Here, \(I_{1}\) and \(I_{2}\) are the first and second invariants of the right Cauchy-Green deformation tensor \(\mathbf{C}\) and \(J\) is the determinant of the deformation gradient tensor. The relationship between the material parameters, E, and v, and the parameters used in the strain-energy function, is as follows.

\[ \begin{equation} \mu=\dfrac{E}{2(1+\nu)},\, \lambda=\dfrac{\nu E}{(1+\nu)(1-2\nu)} \end{equation} \]

This material model uses a standard displacement-based element formulation, so care must be taken when modeling materials with nearly-incompressible material behavior to avoid element locking. For this case, use the Mooney-Rivlin material instead.

Example:

<material id="1" type="neo-Hookean">
  <E>1000.0</E>
  <v>0.45</v>
</material>

Bonet, Javier and Wood, Richard D., Nonlinear continuum mechanics for finite element analysis (Cambridge University Press, 1997). ↩