IsotropicNonlinearElastic3D

General Nonlinear Elastic Isotropic 3D Material Framework

The IsotropicNonlinearElastic3D class is an abstract general purpose framework.

It offers an interface to allow users to define arbitrary strain energy potential based on volumetric strain and equivalent strain (squared), that is

$$W=f\left(\varepsilon_v,\varepsilon_p^2\right),$$

where

$$\varepsilon_v=\text{trace}\left(\varepsilon\right),\qquad\varepsilon_p=\sqrt{\dfrac{2}{3}\varepsilon_d: \varepsilon_d}.$$

Note it is normally expressed in terms of equivalent strain rather than its square. However, the derivation of tangent stiffness would be too cumbersome.

Overridden Method

The IsotropicNonlinearElastic3D provides a method that shall be overridden.

virtual vec compute_derivative(double, double) = 0;

The first argument is $\varepsilon_v$. The second argument is $\varepsilon_s=\varepsilon_p^2$.

The method shall return a vector of size six with following values computed.

index	value
$0$	$\dfrac{\partial{}W}{\partial\varepsilon_v}$
$1$	$\dfrac{\partial{}W}{\partial\varepsilon_s}$
$2$	$\dfrac{\partial^2W}{\partial\varepsilon_v^2}$
$3$	$\dfrac{\partial^2W}{\partial\varepsilon_s^2}$
$4$	$\dfrac{\partial^2W}{\partial\varepsilon_v\partial\varepsilon_s}$
$5$	$\dfrac{\partial^2W}{\partial\varepsilon_s\partial\varepsilon_v}$