By Wiktor Gambin (auth.)
The classical, phenomenological conception of plastically anisotropic fabrics has handed a ways: from the paintings of von Mises awarded in 1928, and the HilI formula given in 1948, to the most recent papers on huge elastic-plastic deformations of anisotropic steel sheets. A attribute function of this method is a linear circulation rule and a quadratic yield criterion. Mathematical simplicity of the speculation is a cause of its a number of functions to the research of engineering constructions throughout the onset of plastic deformations. even if, such an technique isn't adequate for description of the steel forming strategies, whilst a steel point undergoes very huge plastic traces. If we take an at the start isotropic piece of steel, it turns into plastically anisotropic in the course of the forming approach, and the caused anisotropy steadily raises. This truth strongly determines instructions of plastic circulate, and it results in an unforeseen pressure localization in sheet parts. to provide an explanation for the above, it can be crucial take into consideration a polycrystalline constitution of the steel, plastic slips on slip structures of grains, crystallographic lattice rotations, and eventually, a formation of textures and their evolution throughout the complete deformation method. briefly, it's important to introduce the plasticity of crystals and polycrystals. The polycrystal research indicates that, whilst the complicated plastic lines ensue, a few privileged crystallographic instructions, known as a crystallographic texture, ensue within the fabric. the feel formation and evolution are a chief explanation for the triggered plastic anisotropy in natural metals.
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Extra info for Plasticity and Textures
To 39 ANISOTROPIC PLASTICITY deseribe it, we must introduee a shift tensor aij' and to use as the argument deseribing the surfaee, the quantity (Tij - a i j instead of aij' Evolution laws for Uij assume that its inerement is proportional to a ij as it was proposed by MELAN , or to aeeording to Prager's suggestion , or to a ij - a i j due to ZIEGLER'S  proposition. D: - Forfinite plastie strains the inerement of derivative aij is expressed by the objective time a;. The ehoiee of this derivative in the evolution laws started a long diseussion about the nature of plastie behaviour while finite deformations take plaee.
Consider a scalar field, a vector field and a tensor field over a body B. 72). The angle between material fibres, the body force or the strain-rate tensor DIj are the examples of objective quantity. On the other hand, the particle position, its velocity or the material spin fllj is a non-objective quantity. Notice that the material derivative of an objective quantity is a non-objective one. 19), contains the terms that depend on the material spin. To avoid non-objective terms in constitutive relations, we introduce ohjective time derivatives.
The quantity 1)0 may be a function of time during the deformation process. 50) G where dV is the volume of the body in the initial configuration Co surrounded by a surface tc~"). 51) To introduce the required boundary conditions, we assume that the body surface tco , with a field ofunit normal vectors Ni' may be divided into two parts tc~T) and tc~") . Then, the boundary conditions are determined by three tractions ~. 4. PRINCIPLE OF VIRTUAL VELOCITIES The incremental description of plastic deformation processes requires a rate form of equilibrium equations and boundary conditions.