About Material Stability

From a numerical simulation standpoint, it is critical that the material model you are using is stable. If the material becomes unstable, it will often cause convergence trouble and negatively affect the accuracy of your results.

An important consideration in judging the quality of the fit to experimental data is the concept of material or Drucker stability. The Drucker stability condition for a material requires that the infinitesimal change in the stress, $d σ$ , following from any infinitesimal change in the logarithmic strain, $d ε$ , satisfies the inequality

d σ : d ε > 0.

Using $d σ = D : ε$ , where $D$ is the tangent material stiffness, the inequality becomes, $d ε : D : d ε > 0$ . Therefore, the tangential material stiffness has to be positive-definite.

Material Stability in the Analytical Execution Mode

When the app is in analytical execution mode (see About Material Models and Execution Modes) you can both check the stability of the hyperelastic or hyperfoam behavior of the current material model and you can enforce a set of constraints designed to return a stable material definition during a calibration.

When you run a calibration, the app attempts to minimize the deviation between the chosen test data sets and the computed response curves. If you do not request that the app enforce material stability during a calibration, the app chooses the calibrated constants to minimize the deviation between the chosen test data sets and the computed response curves. After the calibration is complete, you can plot the stability of the calibrated material for all the supported deformation modes over the strain ranges you specify. If your calibrated material is unstable within a strain range that you expect to reach during a simulation, you can rerun the calibration after setting the app to enforce material stability. The app then attempts to minimize the deviation between the chosen test data sets and the computed response curves while at the same time enforcing the Drucker stability condition with a penalty function. In general, this approach will modify the calibrated material constants and the quality of the fit will be inferior to the fit when you do not enforce the stability constraint. However, in many cases you should be able to achieve a good fit to the test data while at the same time ensuring a more stable result.

The particle swarm algorithm (see Calibration Algorithms and Parameter Sensitivity) can be particularly effective if you are enforcing stability during a calibration. This is because as a global method it is less likely to get stuck at a non-optimal local minimum as compared to local minimization algorithms.

Material Stability in the Numerical Execution Mode

When the app is in numerical execution mode (see About Material Models and Execution Modes) and either a hyperelastic or hyperfoam material is active you can check the stability of the hyperelastic or hyperfoam behavior. You cannot enforce stability during calibration.