Topology optimization for designing periodic microstructures based on finite strain visco-plasticity

In the current work, we present a topology optimization framework for designing periodic microstructures with viscoplastic constitutive behavior under finite deformation. Materials with tailored macroscopic mechanical properties, i.e. maximum viscoplastic energy absorp- tion and prescribed Poisson's ratio, are designed by performing numerical tests of a single unit cell subjected to periodic boundary conditions. The kinematic and constitutive models are based on finite strain isotropic hardening viscoplasticity, and the mechanical balance laws are formulated in a total Lagrangian finite element setting. To solve the coupled momentum balance equation and constitutive equations, a nested Newton method is used together with an adaptive time-stepping scheme. The optimization problem is iteratively solved using the method of moving asymptotes (MMA), where path-dependent sensitivities are derived using the adjoint method. The applicability of the framework is demonstrated by numerical examples of optimized continuum structures exposed to multiple load cases over a wide macroscopic strain range.