Variational formulations for non-associated plasticity

Develop generalized variational or hybrid Deep Energy Method formulations for classes of non-associated plasticity in which the yield function differs from the plastic potential, including bipotential and augmented-Lagrangian formulations that can represent the constitutive flow while preserving the relevant constraints.

Background

The paper explains that the incremental energetic formulation is natural for associated plasticity because the yield function and plastic potential coincide, allowing the flow rule to follow from a single convex dissipation potential. For non-associated plasticity, the yield function and plastic potential are different, so the constitutive flow does not arise from the same scalar potential and a straightforward monolithic incremental energy formulation is unavailable.

The authors identify bipotential formulations and hybrid DEM formulations as possible routes forward. Bipotentials may represent certain non-associated constitutive laws through generalized variational principles, while hybrid methods could combine DEM-based elastic-energy minimization with penalty or augmented terms enforcing yield admissibility and a separate flow rule. The incorporation of these approaches into DEM is described as less mature than the associated case.

References

For certain classes of non-associated constitutive models, such as bipotential formulations, generalized variational structures can still be constructed. Their incorporation into DEM remains an active area of research and is generally less mature than the associated case.

— Beyond Residuals: Energy based solutions of partial differential equations using scientific machine learning  (2609.08239 - Rabczuk et al., 8 Sep 2026) in Section 2, subsection “Plasticity,” paragraph (C) “DEM (incremental energetic formulation)”