Invariant Properties for Finding Distance in Space of?Elasticity Tensors

Using orthogonal projections, we investigate distance of a given elasticity tensor to classes of elasticity tensors exhibiting particular material symmetries. These projections depend on the orientation of the elasticity tensor; hence the distance is obtained as the minimization of corresponding expressions with respect to the action of the orthogonal group. These expressions are stated in terms of the eigenvalues of both the given tensor and the projected one. The process of minimization is facilitated by the fact that, as we prove, the traces of the corresponding Voigt and dilatation tensors are invariant under these orthogonal projections. For isotropy, cubic symmetry and transverse isotropy, we formulate algorithms to find both the orientation and the eigenvalues of the elasticity tensor endowed with a particular symmetry and closest to the given elasticity tensor.      

Material symmetry and the evolution of anisotropies in a simple material—I. Change of reference configuration

Noll's rule is used to determine the structure of a material symmetry group written with respect to one reference configuration when the representation of the symmetry with respect to another configuration is the traditional material symmetry group associated with isotropy, transverse isotropy or orthotropy, and for an arbitrary deformation gradient relating the two configurations. It is shown that the former symmetry group can contain an orthogonal subgroup. It is determined whether this subgroup is that for isotropic, transversely isotropic, orthotropic, monoclinic, or triclinic response, and the preferred directions of the symmetry are determined.     

A Model of the Evolution of a Two-dimensional Defective Structure

A model of the anelastic evolution law of a two-dimensional defective solid crystal body is proposed. Assuming that the material body is made of triclinic crystals and that the evolution process does not alter the basic material symmetry group, we postulate that the evolution is driven by the present state of the density of the distribution of defects. We show that a linear relation between the inhomogeneity velocity gradient and the torsion tensor is rich enough to model such phenomena as relaxation of defects and dislocation pile-up.     

On the Material Symmetry of Elastic Rods

This paper presents a treatment of material symmetry for hyperelastic rods. The rod theory of interest is based on a Cosserat (or directed) curve with two director fields, and was developed in a series of works by Green, Naghdi and several of their co-workers. The treatment is based on Murdoch and Cohen's work on material symmetry of Cosserat surfaces. Two material symmetry groups are discussed: one pertains to the strain-energy function, while the other pertains to the response functions. The paper closes by showing how the treatment relates to the form-invariant approach used in Green and Naghdi's papers and a treatment proposed recently by Cohen. This revised version was published online in July 2006 with corrections to the Cover Date.     

An extension of the concept of material symmetry

Summary The nature of a material is in large part described by its symmetry group. It is proposed to extend this concept to a more general type of symmetry transformation, and the use of such an extension is illustrated by using it to define several classes of materials and presenting the corresponding representation theorems.

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Sommario La natura di un materiale è in gran parte descritta dal suo gruppo di simmetria. Si propone di estendere questo concetto a un tipo più generale di trasformazione simmetrica, e l'uso di questa estensione è illustrato nel definire parecchie classi di materiali e nel formulare i corrispondenti teoremi di rappresentazione.

     

Symmetry Properties of the Elastic Energy of a Woven Fabric with Bending and Twisting Resistance

A woven fabric can be described as a surface made of two families of fibers: in this work we study how the geometry of the weave pattern affects the symmetry properties of the elastic energy of the surface. Four basic symmetry classes of weave patterns are possible, depending on the angle between the fibers and their material properties. The properties of the pattern determine the material symmetry group of the network, under which the elastic energy is invariant. We derive representations for the energy of a woven fabric that are invariant under the symmetry group of the network, and discuss the relation of these invariants with the curvature and twist of the fibers.      

On the modeling of evolving anisotropy and large strains in pearlitic steel

A phenomenological model for deformation induced evolution of anisotropy at large strains in pearlitic steel is proposed. The modeled anisotropy is based on a homogenization of an ideal pearlitic microstructure. An areal affine type of reorientation is assumed for the individual grains. Furthermore, a yield criterion of the Hill type is proposed and motivated from the grain reorientation. In each pearlitic grain the cementite lamellas have a privileged direction. The symmetry group of each individual grain is therefore considered transversally isotropic. In a virgin material, the privileged directions of the different grains are randomly oriented, which allows for the interpretation that the material on the macroscopic length scale is initially isotropic. However, the cementite lamellas in the grains tend to align after large stretching or shearing deformation. The modeled evolution of anisotropy on the macroscopic length scale shows a saturation characteristics under large deformations.     

Self-similar shapes of the free boundary of a nonlinear-viscous band under uniaxial tension

An equation of evolution of small perturbations of the free boundary of a nonlinear-viscous band under quasi-static uniaxial tension is derived for studying the necking problem in metals under superplasticity conditions. It is shown that the group of symmetry of this linear parabolic equation is equivalent to the group of symmetry of the linear equation of heat conduction with an arbitrary material parameter of the model. Self-similar solutions are obtained in the form of simple and complicated steady localized structures transferred together with the material of the stretched band.     

Symmetry considerations for materials of second grade

The symmetry group associated with a material point of second grade is characterized, thereby eludicating the interplay between first-and second-order strain measures in determining its response to deformation.     

Spiral patterns of color symmetry from dynamics

This paper explores the esthetics of a kind of logarithmic spiral tilings that has not been investigated. It possesses the form similar to the structure of spiral galaxies, which globally displays the cyclic symmetry. The paper first studies the symmetry group associated with the spiral tiling. Then, using the generators of the group, the construction method of such tilings is analyzed in detail. To create colorful patterns on spiral tilings, a special dynamical system which is compatible with its symmetry group is designed. To promote the esthetic appeal of spiral patterns, a simple but practical strategy for generating patterns of color symmetry is presented. Based on the resulting patterns, several interesting methods are proposed to construct more types of derived spiral patterns.     

蜂窝夹芯圆环的拓扑优化设计及尺度效应研究

采用尺度关联的一体化设计方法开展了旋转周期圆环结构的拓扑优化设计研究，以宏观 结构的最大刚度为目标，研究了材料表征体胞尺度、构型以及不同载荷作用形式对蜂窝夹芯 圆环结构优化结果的影响. 所提出的无量纲结构构型因子实现了优化结构的结构效率量化评 估. 结合SIMP材料模型和周长控制方法，实现了宏观结构和细观表征体胞的优化设计，获得 清晰的材料分布. 数值算例表明，尺度关联的一体化设计方法能有效地完成圆环结构的拓扑 优化设计，设计结果充分反映体胞尺度效应对旋转周期圆环结构夹芯构型的影响.     

基于拉格朗日乘子法和群论的具有任意位移边界条件的旋转周期对称结构有限元分析

对任意位移边界条件下的旋转周期对称结构，由拉格朗日乘子法建立有限元方程。在对称适应的坐标系下，由结构刚度矩阵的块循环性质，利用群变换给出一种新的求解方法。数值验证给出令人满意的结果。     

Material Tensors and Pseudotensors of Weakly-Textured Polycrystals with Orientation Distribution Function Defined on the Orthogonal Group

A representation theorem on material tensors of weakly-textured polycrystals was recently established by Man and Huang, which delineates quantitatively the effect of crystallographic texture on the material tensor in question. Man and Huang’s theorem is based on the classical assumption in texture analysis that ODFs are defined on the rotation group SO(3), which strictly speaking makes it applicable only to polycrystals with crystallite symmetry defined by one of the 11 proper point groups. In this paper we let ODFs be defined on the orthogonal group O(3) and extend the representation theorem of Man and Huang to cover pseudotensors and polycrystals with crystallite symmetry defined by any of the 21 improper point groups. In SO(3)-based texture analysis, as a result of the inherent limitation imposed by the restricted definition of the ODF, each improper group of crystallite symmetry in question is routinely treated as if it were its peer proper group in the same Laue class. In light of the extended representation theorem, we examine the conditions under which this ad hoc practice will still work as far as effects of texture on material tensors and pseudotensors are concerned and the circumstances under which it won’t.     

Replacing ordinary derivatives by gauge derivatives in the continuum theory of dislocations to compensate the action of the symmetry group

When the symmetry group of a body is continuous it plays a fundamental role on the nonlinear continuum theory of dislocations: it induces a non-uniqueness to the field that describes the defects – the uniform reference – and affects also other fundamental ingredients of the theory. The purpose of the present paper is to examine how certain important quantities of the dislocation theory are affected from symmetry's group action. Apart from the uniform reference we study how the deformation gradient, the first and second Piola–Kirchhoff stress tensors, the elasticities of the material and the momentum equation are affected from the action of the symmetry group. This action is inhomogeneous, namely, differs from point to point. A similar inhomogeneous action of a group may be found in gauge theories. Prompt by the gauge approach, we propose the use of the gauge covariant exterior derivative to compensate for the action of the symmetry group on the uniform reference. The main advantage of using this derivative is that the momentum equation for the static case retains its divergence form. It remains an open question how the Yang–Mills potentials may be determined for the present theory.     

Deformation of brittle materials with scattered cracking

A structural approach combined with the energy method is used to construct a continuum model of the joint process of deformation and scattered cracking in elastic-brittle isotropic materials. The proposed approach allows one to match the form of the symmetry of the effective deformation properties and damageability of the material depending on the stress state character.     

Symmetry conditions for third order elastic moduli and implications in nonlinear wave theory

Several results are presented concerning symmetry properties of the tensor of third order elastic moduli. It is proven that a set of conditions upon the components of the modulus tensor are both necessary and sufficient for a given direction to be normal to a plane of material symmetry. This leads to a systematic procedure by which the underlying symmetry of a material can be calculated from the 56 third order moduli. One implication of the symmetry conditions is that the nonlinearity parameter governing the evolution of acceleration waves and nonlinear wave phenomena is identically zero for all transverse waves associated with a plane of material symmetry.     

On the restrictions imposed by non-affine material symmetry groups for elastic rods: application to helical substructures

Wire ropes, DNA strands and helical springs are among those bodies which can be modeled as an elastic rod with a helical substructure. The resulting form of the strain-energy function is a matter of material symmetry. This symmetry is explored using a novel treatment which combines non-affine transformations and a relabeling of the material coordinates. The restrictions this treatment imposes on the strain-energy function include a periodic dependency on torsional strain. In addition, comparisons are made with results from a recent treatment of helical symmetry by Healey. Finally, conclusions applicable to material symmetry restrictions for other polar elastic continua are presented.     

Material Symmetry in a Theory of Continuously Defective Crystals

We propose that a particular group, which depends on the dislocation density tensor, be adopted as material symmetry group for some classes of defective crystals, and give motivation for this proposal.