Journal
JOURNAL OF ELASTICITY
Volume 106, Issue 1, Pages 1-42Publisher
SPRINGER
DOI: 10.1007/s10659-010-9284-3
Keywords
Polycrystals; Texture; Anisotropy; Material tensors; Irreducible tensor basis; Rotation group; Elasticity
Funding
- U.S. National Science Foundation [DMS-0807543]
- Natural Science Foundation of China [10972098]
- Research Fund for the Doctoral Program of Higher Education [20070403003]
- Natural Science Foundation of Jiangxi of China [2008GZW0005]
- Jiangxi Project
- Direct For Mathematical & Physical Scien [0807543] Funding Source: National Science Foundation
- Division Of Mathematical Sciences [0807543] Funding Source: National Science Foundation
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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.
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