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- Crystal plasticity (2) (entfernen)
A computationally efficient solution scheme is presented for the mechanical problems whose formulations include the Kuhn–Tucker or Signorini–Fichera conditions. It is proposed to reformulate these problems replacing inequalities in these conditions by equations with respect to new unknowns. The solutions of the modified problems have simple physical meanings and determine uniquely the unknowns of the original problems. The approach avoids application of multi-valued operators (inclusions or inequalities) in formulation of the problems. Hence, the modified formulations are suitable for numerical analysis using established powerful mathematical methods and corresponding solvers developed for solving systems of non-linear equations.
To demonstrate the advantages of the proposed approach, it is applied for solving problems in two different areas: constitutive modeling of single-crystal plasticity and mixed boundary value problems of elastic contact mechanics with free boundaries. The original formulations of these problems contain respectively the Kuhn–Tucker and Signorini–Fichera conditions. A problem of the former area is integrated using an implicit integration scheme based on the return-mapping algorithm. The derived integration scheme is free of any update procedure for identification of active slip systems. A problem of the latter area is reduced to solution of non-linear integral boundary equations (NBIEs). Numerical examples demonstrate stability and efficiency of the solution procedures and reflect the mathematical similarities between the both non-linear problems.
A crystallographic constitutive model is developed, which accounts for both rate-sensitive and rate-insensitive flow. Single-crystal plasticity and viscoplasticity are the limiting cases of the model, so that it properly reflects the material response over a wide temperature range. A non-linear dynamic recovery is included to properly describe ratchetting. We provide a robust integration scheme based on generalization of the return-mapping algorithm and of the procedure for active set search.
The implicit integration and consistent tangent are implemented through the UMAT subroutine in the ABAQUS finite element program. The capability of the model to account for both high and low strain rates is demonstrated in numerical examples. Finally, the stability of integration scheme and quadratic convergence of the global NewtonRaphson equilibrium iterations are demonstrated on the example of a notched bar under tension.