A Formulation of Anisotropic Plasticity at Large Strains
A description of an appearance and development of a plastic anisotropy in strongly deformed elastic-plastic materials is proposed within a modified continuum plasticity approach. The polycrystalline structure of the materials and crystallographic texture development are assumed as the main reason of the anisotropy. A continuous model of the elastic-plastic material with a texture is introduced. The basic idea is the following: to consider continuum of lattice frames attached to a material particle instead of a finite number of grains in a small volume of the material. The above concept leads to the notion of a “textured material body” ℒ, which may be embodied into a six-dimensional space which is the Cartesian product of the Euclidean point-space and the Riemanian orientation-space. A motion of C determines texture changes. The behaviour of the model is described on two levels. On the local (micro-) level, an evolution equation for the orientation distribution function and constitutive relations for the single crystal plasticity are used. Here, all considered fields are functions of particle positions, lattice orientations and time. On the global (macro-) level, the virtual work principle (VWP) in the extended (six-dimensional) physical space is assumed. Due to the local constitutive relations, the VWP may be expressed in terms of a velocity field only. It leads to the finite element formulation of elastic-plastic analysis of textured materials. Because the form of VWP is the same as in the classical continuum plasticity, one can adapt the standard FEM procedures to obtain an adequate numerical code.
KeywordsAnisotropy Manifold Stein Triad
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