A Representation Theorem for Material Tensors of Weakly-Textured Polycrystals and Its Applications in Elasticity

Abstract

Material tensors pertaining to polycrystalline aggregates should manifest also the influence of crystallographic texture on the material properties in question. In this paper we make use of tensors which form bases of irreducible representations of the rotation group and prove a representation theorem by which a given material tensor of a weakly-textured polycrystal is expressed as a linear combination of an orthonormal set of irreducible basis tensors, with the components given explicitly in terms of texture coefficients and a set of undetermined material parameters. Once the irreducible basis tensors that appear in the formula are determined, the representation formula, which is valid for all texture and crystal symmetries, will delineate quantitatively the effect of crystallographic texture on the material tensor in question. We present an integral formula and an orthonormalization process which serve as the basis for a procedure to determine explicitly the irreducible basis tensors required in the representation formula. For applications we determine a set of irreducible basis tensors for the elasticity tensor and a set for fourth-order tensors that define constitutive equations in incompressible elasticity and Hill's quadratic yield functions in plasticity. We show that orientation averaging of a tensor can be done easily if we have in hand a set of irreducible basis tensors for the decomposition of the tensor in question. As illustration we derive a formula, which is valid for all texture and crystal symmetries, for the elasticity tensor under the Voigt model.

A Simple Explicit Formula for the Voigt-Reuss-Hill Average of Elastic Polycrystals with Arbitrary Crystal and Texture Symmetries

“…To find an explicit formula for the VRH average, it is convenient to expand the ODF, which is a real-valued function defined on the rotation group SO(3), as an infinite series in terms of the (complex-valued) Wigner D-functions D l mn (·) [6][7][8]:…”

“…We write w iso = 1/(8π 2 ), which is the ODF for all isotropic polycrystals. Since both C V (w) and K V (w) are obtained from orientation averaging of fourth-order tensors with major and minor symmetries, it is well known [8,9] that they can depend only on those texture coefficients c l mn with l = 2 and l = 4. Thus the VRH average C H (w) can depend only on those same texture coefficients.…”

“…Recently we have derived [8] a general formula for orientation averaging of tensors, which is based on the determination of a set of irreducible basis tensors (see Sect. 2.3) for the tensor space in question.…”

“…We will follow suit in the present paper. In the language of [8], we will restrict our attention to "weakly-textured polycrystals". In fact, in the same paper [8] that we obtain our formula for C V (w), we have also derived a general representation formula (see (127) there) for the elasticity tensor C(w) of any weakly-textured polycrystal, which reads:…”

“…2.3 below) in the space of fourth-order tensors with major and minor symmetries, and a specific set of these tensors has been determined explicitly in [8]; α k,s n are material parameters to be determined empirically or through micromechanical modeling. All micromechanical models of the weakly-textured polycrystal, including the Voigt and Reuss models, will lead to an elasticity tensor C(w) which can be put in this form.…”