Stability and error analysis for time integrators applied to strain-softening materials

The behavior of explicit time integrators is examined for the field equations of dynamic viscoplasticity in one dimension. The analysis proceeds by linearizing the field equations about the current state and freezing coefficients so that the solution to the system of ordinary differential equations can be computed explicitly. Amplification matrices for the central difference method (with rate tangent constitutive update) and fourth-order Runge-Kutta schemes are obtained by applying these schemes to the resulting linear, constant coefficient equations. These amplification matrices lead to local truncation error estimates for the temporal integrators as well as to estimates of the resulting growth rate. Analysis of the cental difference method with a forward Euler constitutive update indicates that the local truncation error involves as the product of the third power of the strain-rate and the square of the time step. In viscoplastic models, the viscoplastic strain-rate is an increasing (decreasing) function of the viscoplastic strain in softening (hardening). Also, the eigenvalues of the rate tangent method increase with strain-rate in softening. Thus, it is recommended that the time step be reduced in localization problems to keep the strain-increment within a prescribed tolerance. As a byproduct of the analysis, conditions for numerical stability for the central difference method with a rate tangent constitutive update are obtained for strain-hardening dynamic viscoplasticity. In the softening regime, the resulting linearized differential equations admit exponentially growing solution. Thus, conventional definitions of stability such as Neumann stability and absolute stability are inappropriate. In order to characterize stability of such systems, we introduce the concept of g-rel stability which combines the concept of relative and absolute stability. We show conditions under which the resulting system is g-rel stable and thus characterize the nature of the numerical stability that can be expected in such problems.