On time integration of viscoplastic constitutive models suitable for creep

Creep of critical components such as electrical solder connections may occur over long periods of time. E6cient numerical simulations of such problems generally require the use of quasi-static formulations with conjugate-gradient techniques for solving the large number of algebraic equations. Implicit in the approach is the need to solve the constitutive equation several times for large time steps and for loading directions that may have no resemblance to the actual solution. Therefore, an unconditionally stable and e6cient algorithmfor solving the constitutive equation is essential for the overall e6ciency of the solution procedure. Unfortunately, constitutive equations suitable for simulating the materials of interest are notoriously di6cult to solve numerically and most existing algorithms have a stability limit on the time step which may be several orders of magnitude smaller than the desired time step. Here an algorithmis proposed which is a combination of the use of a trapezoidal rule and an iterative Newton–Raphson method for solving implicitly the non-linear equations. The key to the success of the proposed approach is to always use an initial guess based on the steady-state solution to the constitutive equation. A representative viscoplastic constitutive equation is used as a model for illustrating the approach. The algorithmis developed and typical numerical results are provided to substantiate the claimthat stability has been achieved