Topology optimization for pressure-dependent elastoplastic structures considering a smooth hyperbolic approximation of the Drucker-Prager yield criterion

A topology optimization framework considering a smooth hyperbolic Drucker-Prager (SHDP) plasticity is presented. The smooth hyperbolic approximation is used to address the gradient-discontinuity of the Drucker-Prager yield surface at the apex, for accurate sensitivity information and improved numerical stability for both the stress integration scheme and the optimization sensitivity analysis. The return mapping algorithm for this constitutive model, based on two stress invariants, is derived and presented herein along with a discussion on the update of the plastic multiplier increment through the bisection method. In this work, pressure-dependent elastoplastic structures with high energy absorption are designed through an objective function of maximizing plastic work. The pressure-dependence of metal plasticity motivates the use of the smooth hyperbolic Drucker-Prager to model metal alloys. The path dependent sensitivity analysis was conducted using the adjoint method with all the partial derivatives computed analytically. The effectiveness of the proposed framework is demonstrated by the numerical results, where the structures significantly differ from those based on only the deviatoric stress invariant in the elastoplastic model. Lastly, we demonstrate that the proposed regularized SHDP framework allows for convergence in the forward analysis and subsequently the topology optimization problem in cases where the original Drucker-Prager could not.