isotropic Hencky¶

Module: solid

Category: material

Type string: "isotropic Hencky"

Parameters¶

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

Description¶

The material type for isotropic Hencky hyperelasticity ¹ is isotropic Hencky.

This material is an implementation of a hyperelastic constitutive material that reduces to the classical linear elastic material for small strains, but is objective for large deformations and rotations. The hyperelastic strain-energy function is given by:

\[ W=\frac{1}{2}\lambda\left(\tr\mathbf{H}\right)^{2}+\mu\mathbf{H}:\mathbf{H} \]

Here, \(\mathbf{H}\) is the right Hencky strain tensor and \(\lambda\) and \(\mu\) are the Lamé parameters, which are related to the more familiar Young's modulus E and Poisson's ratio \(\nu\) as follows:

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

The Cauchy stress for this material takes the form

\[ \boldsymbol{\sigma}=\frac{\lambda}{J}\left(\mathbf{I}:\mathbf{h}\right)\mathbf{I}+\frac{2\mu}{J}\mathbf{h} \]

where \(\mathbf{h}\) is the left Hencky strain tensor. In the limit of infinitesimal strains and rotations, \(J\to1\) and \(\mathbf{h}\to\boldsymbol{\varepsilon}\) where \(\boldsymbol{\varepsilon}\) is the infinitesimal strain tensor.

It is often convenient to express the material properties using the bulk modulus K and shear modulus G. To convert to Young's modulus and Poisson's ratio, use the following formulas:

\[ E=\frac{9KG}{3K+G},\quad\nu=\frac{3K-2G}{6K+2G}\,. \]

Example:

<material id="1" type="isotropic Hencky">
  <E>1000.0</E>
  <v>0.45</v>
</material>

Xiao, H and Chen, LS, "Hencky's elasticity model and linear stress-strain relations in isotropic finite hyperelasticity", Acta Mechanica 157, 1 (2002), pp. 51--60. ↩