On the numerical integration of rate independent single crystal behavior at large strain

This paper presents a new numerical algorithm for the integration of the constitutive equations
of a single crystal for finite rate-independent elastoplastic strains. The algorithm addressed in this paper is dedicated to face-centered-cubic (FCC) crystal structures. Its first feature is a much more efficient and more accurate integration scheme of the constitutive equations compared to previous attempts. This scheme is based on a fully implicit integration procedure, yet it may be transformed easily into an explicit scheme. Determining the set of active slip systems is performed by the use of a combinatorial search procedure, and the determination of the slip rates of the different active slip systems is based on the fixed point method. The second feature of this algorithm stems from the original method used to solve the ambiguity of the possible non-uniqueness of the set of active slip systems. A robust method, based on a small positive perturbation of the critical shear stresses, is proposed to overcome this difficulty. It is worth mentioning that the algorithm developed in this paper is not limited to one particular hardening law or to FCC crystal structures. Rather, it can be used and extended to various hardening laws and crystal structures (e.g. BCC or HCP. . .) in a straightforward manner. The authors demonstrate the performance of the proposed algorithm and illustrate its accuracy and efficiency through various numerical simulations at the single crystal and polycrystal scales. The predicted results obtained from those simulations were compared with those obtained using other numerical techniques and algorithms (i.e., a pseudo-inversion technique and an explicit algorithm). Our numerical predictions are also compared with some numerical and experimental results from other papers. The response of the polycrystal was computed by using the proposed algorithm combined with Taylor’s homogenization scheme, which is used to compute the overall polycrystalline behavior. The paper ends with a statistical study of the influence of the perturbation technique on the response prediction for a single crystal and a polycrystal.