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The Sensitivity Is in the Details
Original Post - 28 Nov 2023 - Michael H. Scott
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Although the Hardening and Steel01 uniaxial materials can be calibrated to give the same response, the DDM response sensitivity with respect to the same parameter can be different due to how the material models are implemented.
Consider the truss model from a previous post on minimal DDM examples. The stress-strain response shows the elastoplastic tangent for the Hardening uniaxial material in terms of the elastic modulus, E, and the hardening modulus, H.
As shown in another previous post, the hardening ratio, b, of the Steel01 material can be computed in order to give the same post-yield stiffness as the Hardening material. With calibrated parameters, the two material models lead to the same truss load-displacement response.
The above results can be obtained via either load control or displacement control. But that’s not the issue.
When we compute the DDM response sensitivity using either load control or displacement control, the post-yield response sensitivities with respect to the elastic modulus, E, are different between the two material models.
Post-yield, the sensitivity with respect to E continues to increase for Steel01 and remains more or less unchanged for Hardening material.
The sensitivities with respect to E are different due to how the post-yield response is computed in the two models. In Steel01, the post-yield response is computed directly from bE, i.e., the strain-hardening ratio times the elastic modulus. For the Hardening material, the post-yield response is computed via plasticity according to the simple one-dimensional return mapping algorithm described in Simo and Hughes (1998).
Differing response sensitivities with respect to the same parameter of different models can be significant for gradient-based algorithms like structural reliability, optimization, and machine learning.
As always, DDM response sensitivity or not, it’s important to be aware of modeling choices and their ramifications.