Institute of Fundamental Technological Research

In the paper a new proposition of limit state criteria for anisotropic solids exhibiting different strengths at tension and compression is presented. The proposition is based on the concept of energetically orthogonal decompositions of stress state introduced by Rychlewski. The concept of stress state dependent parameters describing the influence of certain stress modes on the total measure of material effort was firstly presented by Burzynski. The both concepts are reviewed in the paper. General formulation of a new limit criterion as well as its specification for certain elastic symmetries is given. It is compared with some of the other known limit criteria for anisotropic solids. General methodology of acquiring necessary data for the criterion specification is presented. The ideas of energetic and limit state orthogonality are discussed - their application in representation of the quadratic forms of energy and limit state criterion as a sum of square terms is shown

anisotropy, strength hypothesis, material effort, yield condition, plastic potential, strength differential effect, elasticity, plasticity

In the paper a new proposition of an energy-based hypothesis of material effort is introduced. It is based on the concept of influence functions introduced by Burzyński [3] and on the concept of decomposition of elastic energy density introduced by Rychlewski [18]. A new proposition enables description of a wide class of linearly elastic materials of arbitrary symmetry exhibiting strength differential effect.

linear elasticity, anisotropy, material effort hypotheses, limit state criteria

In the paper a new proposition of an energy-based hypothesis of material effort is introduced. It is based on the concept of influence functions introduced by Burzyński [3] and on the concept of decomposition of elastic energy density introduced by Rychlewski [18]. A new proposition enables description of a wide class of linearly elastic materials of arbitrary symmetry exhibiting strength differential effect.

linear elasticity, anisotropy, material effort hypotheses, limit state criteria

The spectral decomposition of elasticity tensor for all symmetry groups of a linearly elastic material is reviewed. In the paper it has been derived in non-standard way by imposing the symmetry conditions upon the orthogonal projectors instead of the stiffness tensor itself. The numbers of independent Kelvin moduli and stiffness distributors are provided. The corresponding representation of the elasticity tensor is specified

linear elasticity, anisotropy, symmetry group, spectral decomposition

The rigid-plastic model for the single grain is developed in which the velocity gradient is split into two parts connected with crystallographic slip and micro-shear bands respectively. For crystallographic slip the regularized Schmid law proposed by Gambin is used. For the micro-shear bands the model developed by Pęcherski, which accounts for the contribution of this mechanism in the rate of plastic deformation by means of a function fms is applied. Different constitutive equations for the plastic spin due to two considered mechanisms of plastic deformation are used. The present model is applied to simulate crystallographic texture evolution in the polycrystalline element.

crystallographic texture, modelling of texture, micro-shear bands, regularized Schnid law, plastic spin

A straightforward and complete description of all possible invariant linear decompositions of the space of Hooke's tensors has been given in Part I, [1]. In this Part II we demonstrate various elaborations and consequences of these decompositions. This gives a qualitative description of the anisotropy of Hooke's tensors. In particular, we demonstrate examples A through G, not only important but also astonishing. When reference is made to the formulae in [1], we shall add "Part I" to the number. The notions and notations are the same (see Appendices 1, 2 in [1]).

The main aim of this paper is to determine all the unit stresses w (w•w = 1) for which the stored elastic energy F(w) has the local extrema in some classes of stresses. Our consideration is restricted to two classes: K1 - uniaxial tensions and then the directions for which the Young modulus assumes its extremal value are determined, and K2 - pure shears in physical space. The problem is then reduced to the determination of the planes of minimal and maximal shear modulus. The idea of a generalized proper state for Hooke's tensor is introduced. It is shown that a mathematical treatment of the considered problem comes down to the problem of the generalized proper elastic states for the compliance tensor C. The problem has been effectively solved for cubic symmetry.

In the paper the yield condition is proposed for the most general anisotropic material. It is one of the possible generalizations of the Huber-Mises-Hencky yield condition for the case of anisotropy. The body considered is anisotropic elastically as well as plastically. It is assumed that the plastic anisotropy tensor is a definite function of the elastic anisotropy tensor. The corresponding flow function is a part of the strain energy and its value remains unchanged when all normal components of stress are increased by the same value. The theory of the eigen states for fourth order tensors is used. The plastic anisotropy tensor proposed has the same deviatoric eigen states as the elastic anisotropy tensor. The proposed yield condition reduces to that of Huber-Mises-Hencky when the anisotropy is vanishingly small. The method presented in this paper can be also applied to describe other types of plastic anisotropy tensor.