A Representation Theorem for Material Tensors of Weakly-Textured Polycrystals and Its Applications in Elasticity
A Representation Theorem for Material Tensors of Weakly-Textured Polycrystals and Its Applications in Elasticity
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DOI：
10.1007/s10659-010-9284-3
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发表时间：
2010-11
影响因子：
2
通讯作者：
C. Man;Mojia Huang
中科院分区：
文献类型：
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作者：
C. Man;Mojia Huang
与多晶聚集体有关的材料张量也应表明晶体织构对所讨论的材料性能的影响。本文利用构成旋转群不可约表示基的张量，证明了一个表示定理，该定理证明了弱织构多晶体的给定材料张量是一组标准正交的不可约基张量的线性组合，这些张量的分量以织构系数和一组待定材料参数显式给出。一旦公式中出现的不可约基张量被确定，这个对所有织构和晶体对称性都有效的表示公式将定量地描述晶体织构对所讨论的材料张量的影响。我们给出了一个积分公式和一个标准正交化过程，作为显式确定表示公式中所需的不可约基张量的过程的基础。对于应用，我们确定了弹性张量的一组不可约基张量和四阶张量的一组，这些张量定义了不可压缩弹性中的本构方程和塑性中的希尔二次屈服函数。我们证明了张量的方向平均可以很容易地完成，如果我们手头有一组不可约的基张量来分解这个张量。作为说明，我们导出了一个公式，适用于所有纹理和晶体对称，在Voigt模型下的弹性张量。
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.