Explicit and implicit methods for shear band modeling at high strain rates

High strain rate loading of metals typically leads to material instabilities known as shear bands. These are narrow bands of intense plastic deformation that weaken the load bearing capacity of the material and serve as a precursor to fracture. Shear bands are modeled as a coupled thermo-mechanical set of nonlinear PDEs and a plastic flow rule that depends on strain rate, strain hardening and thermal softening is used to describe their behavior. In addition, thermal conductivity, which counterbalances thermal heating and creates a weak length scale, must also be employed in order to regularize the PDE system and yield mesh insensitive results. In this paper, we employ mixed finite element formulations to discretize the system in space, and investigate the performance of implicit, explicit and semi-explicit time integration schemes. Small deformation kinematics is assumed in the current work. Stability, accuracy and computational efficiency of the integration schemes are studied on two benchmark examples: a plate in tension with a shear band formed in \(45^{\circ }\) and an impact onto a notched steel plate. Our findings confirm that implicit methods with larger time steps outperform explicit schemes in terms of cpu modeling time and also yield orders of magnitude more accurate results.