Material tangent matricies
In the solution of nonlinear problems one typically uses a Newton or quasi-Newton method to iterate
the global FEA equations to equilibrium. In this process one needs to properly compute element
tangent matricies. These can be numerically checked using the MACRo command TANG,NUMErical,1, which
solves problems by numerically differentiating (finite differencing) your element residual. This is an excellent tool
for debugging element level errors in the tangent formulation especially when you are already confident of your
gauss-point material tangent. To help check the gauss-point tangent (6 by 6 material moduli), it is useful to have a
separate tool.
To check the gauss-point level material tangent one can also employ numerical differentiation.
A typical material model returns the Cauchy stress tensor and the spatial material moduli. The input
is the deformation gradient. In order to compute the needed tangent numerically, one needs to compute
two times the derivative of the second Piola-Kirchhoff stress tensor with respect to the right Cauchy-Green deformation
tensor. Since the input to the material module is usually the deformation gradient and the output the Cauchy
stress, one needs to go through a few steps to do this properly.
The main steps are:
- Given the deformation gradient F, compute its polar decomposition F = RU, right Cauchy-Green deformation
tensor C = F^T F = U^2, the Jacobian J = det(F), and the inverse F^{-1}.
- Also call the material model to get the Cauchy stress and what it thinks the spatial moduli are.
- Pull the Cauchy-stress back to compute the 2nd Piola-Kirchhoff stress S (unperturbed).
- For each unique entry of C, perturb it and call the material model to get the Cauchy stress. For the off diagonal entries the
perturbations must be symmetric!
- Pull the Cauchy-stress back to compute the 2nd Piola-Kirchhoff stress S (perturbed).
- Difference the perturbed S with the unperturbed S, multiply by 2, and divided by the perturbation magnitude to get the
corresponding column of the reference material moduli.
- Once all 6 column are completed, this can be pushed forward to obtain the numerically computed spatial material moduli, which can
be compared to the ones originally computed by the material model. Obviously, one can also compare the pulled back moduli.
A MATLAB routine for doing so is attached along with an example material model file also in MATLAB. If you wish to run this program on
your material model written in FORTRAN or C for FEAP, then you will need to write a MEX cover function or convert this to a user MACRo
command which can be called from the command prompt in FEAP.
Source files: For dowloading
Driver file for computing numerical tangents
Example material model file