# 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.

```cpp
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**                                                             |
| --------- | --------------------------------------------------------------------- |
| $$0$$     | $$\dfrac{\partial{}W}{\partial\varepsilon\_v}$$                       |
| $$1$$     | $$\dfrac{\partial{}W}{\partial\varepsilon\_s}$$                       |
| $$2$$     | $$\dfrac{\partial^2W}{\partial\varepsilon\_v^2}$$                     |
| $$3$$     | $$\dfrac{\partial^2W}{\partial\varepsilon\_s^2}$$                     |
| $$4$$     | $$\dfrac{\partial^2W}{\partial\varepsilon\_v\partial\varepsilon\_s}$$ |
| $$5$$     | $$\dfrac{\partial^2W}{\partial\varepsilon\_s\partial\varepsilon\_v}$$ |


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