## Abstract

This paper presents a mathematical formulation and implicit numerical algorithm for solving the integral of a three-dimensional momentum balance based on the inelastic evolution of microstructural vectors for thin plates in Eulerian formulation. A recent theoretical discussion (Lee and Rubin, 2020, “Modeling Anisotropic Inelastic Effects in Sheet Metal Forming Using Microstructural Vectors—Part I: Theory,” Int. J. Plast., 134, p. 102783. 10.1016/j.ijplas.2020.102783) showed that Eulerian constitutive equation based on microstructural vectors for thin plates has the advantage of capturing the anisotropic behavior of the material axis with insensitivity to the randomness of the reference configuration. However, all the discussions were theoretically conducted only at a local material point in homogeneous deformation conditions, which do not require consideration of the momentum balance with flexible velocity gradients in a three-dimensional volume. For usability, numerical algorithms are needed to solve evolution of the microstructural vectors in the three-dimensional space. This paper presents the first numerical algorithm to solve the inelastic evolution of microstructural vectors in the Eulerian formulation. A generalized material coordinated system is matched to the microstructural vectors in a three-dimensional space by considering the Eulerian constitutive equations insensitive to the superposed rigid body motions (SRBM). Numerical algorithms were then introduced to implicitly solve the nonlinear momentum balance, evolution of the microstructural vectors, and tangent modulus. The formula and numerical algorithms were validated by predicting the tension tests when the principal loading angle varied from the reference axis. The results show that the proposed numerical algorithm can describe the evolution of the microstructure based on the Eulerian formulation.

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