Linear Elastic model (Automatic Differentiation)

This material is intended for demonstration purposes, to illustrate how to use the capabilities of the AD Material.

Theory

The constitutive law is given in total form as

\[\sig = \Cel : \eps\]

relating the nominal stress tensor \(\sig\) to the linearized strain tensor \(\eps\) with the fourth order stiffness tensor \(\Cel\). For the given definitions it is important to note, that the Voigt notation is used. The stiffness tensor can be specified for isotropic material behavior as follows:

Isotropic Behavior

Number of independent material parameters: 2

\[\begin{split}\Cel^{-1} = \begin{bmatrix} \frac{1}{E} & \frac{-\nu}{E} & \frac{-\nu}{E} & 0 & 0 & 0 \\ \frac{-\nu}{E} & \frac{1}{E} & \frac{-\nu}{E} & 0 & 0 & 0 \\ \frac{-\nu}{E} & \frac{-\nu}{E} & \frac{1}{E} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{G} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{G} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{G} \end{bmatrix}\end{split}\]

with

\[\displaystyle G = \frac{E}{2\,(1 + \nu)}\]

Implementation

class ADLinearElastic : public MarmotMaterialHypoElasticAD

Inheritance diagram for Marmot::Materials::ADLinearElastic:

Collaboration diagram for Marmot::Materials::ADLinearElastic:

Implementation of a isotropic linear elastic material for 3D stress states using automatic differentiation.

Public Functions

ADLinearElastic(const double *materialProperties, int nMaterialProperties, int materialNumber)

virtual double getDensity(const double *stateVars) const override

Get the mass density of the material.

Parameters:

stateVars – Pointer to the state variable array

Returns:

Mass density of the material

Public Members

const autodiff::MatrixXdual C

Elasticity tensor for isotropic materials.

Protected Functions

virtual void computeStressAD(state3DAD &state, const Marmot::Vector6dual &dStrain, const timeInfo &timeInfo) const override

Compute the Cauchy stress tensor \(\boldsymbol{\sigma}\) given an increment of the linearized strain tensor \(\Delta\boldsymbol{\varepsilon}\).

Dual numbers are used for both the Cauchy stress tensor and the linearized strain increment, such that the algorithmic tangent operator \(\frac{\partial\boldsymbol{\sigma}^{(n+1)}}{\partial\boldsymbol{\varepsilon}^{(n+1)}}\) can be obtained by means of automatic differentiation.

Parameters:

• state – [inout] State carrying the dual Cauchy stress, strain energy, and state variables

• dStrain – [in] linearized strain increment

• timeInfo – Old (pseudo-)time