Identifying Symmetry Classes of Elasticity Tensors Using Monoclinic Distance Function

We present a method to identify the symmetry class of an elasticity tensor whose components are given with respect to an arbitrarily oriented coordinate system. The method is based on the concept of distance in the space of tensors, and relies on the monoclinic or transversely isotropic distance function. Since the orientation of a monoclinic or transversely isotropic tensor depends on two Euler angles only, we can plot the corresponding distance functions on the unit sphere in ℝ³ and observe the symmetry pattern of the plot. In particular, the monoclinic distance function vanishes in the directions of the normals of the mirror planes, so the number and location of the zeros allows us to identify the symmetry class and the orientation of the natural coordinate system. Observing the approximate locations of the zeros on the plot, we can constrain a numerical algorithm for finding the exact orientation of the natural coordinate system.     

On Isotropic Invariants of the Elasticity Tensor

Isotropic invariants of the elasticity tensor always yield the same values no matter what coordinate system is concerned and therefore they characterize the linear elasticity of a solid material intrinsically. There exists a finite set of invariants of the elasticity tensor such that each invariant of the elasticity tensor can be expressed as a single-valued function of this set. Such a set, called a basis of invariants of the elasticity tensor, can be used to realize a parametrization of the manifold of orbits of elastic moduli, i.e. to distinguish different kinds of linear elastic materials. Seeking such a basis is an old problem in theory of invariants and seems to have been unsuccessful until now. In this paper, by means of the unique spectral decomposition of the elasticity tensor every invariant of the elasticity tensor is shown to be a joint invariant of the eigenprojections of the elasticity tensor, and then by utilizing some properties of the eigenprojections a basis for each case concerning the multiplicity of the eigenvalues of the elasticity tensor is presented in terms of joint invariants of the eigenprojections. In addition to the foregoing properties, the presented invariants may also be used to form invariant criteria for identification of elastic symmetry axes.     

Constitutive Relation of Elastic Polycrystal with Quadratic Texture Dependence

Herein we consider polycrystalline aggregates of cubic crystallites with arbitrary texture symmetry. We present a theory in which we keep track of the effects of crystallographic texture on elastic response up to terms quadratic in the texture coefficients. Under this theory, the Lamé constants pertaining to the isotropic part of the effective elasticity tensor of the polycrystal will generally depend on the texture. We introduce also two simple models, which we call HM-V and HM-R, by which we derive an explicit expression for the effective stiffness tensor and one for the effective compliance tensor. Each of these expressions contains a term quadratic in the texture coefficients and, in addition to three parameters given in terms of the single-crystal elastic constants, each carries an undetermined material coefficient. These two remaining coefficients can be determined by imposing the requirement that the expressions from models HM-V and HM-R be compatible to within terms linear in the texture coefficients.     

任意形状薄壳的弹性稳定性方程

本文用张量导出了任意形状薄壳的统一的弹性稳定性方程组,并且在正交曲线坐标系下化成了以张量的物理分量表示的方程,进而可以将正圆锥壳、球壳、圆柱壳、椭球壳、圆板、矩形板等形状的壳和板作为特例来研究。     

On the conditions for the existence of a plane of symmetry for anisotropic elastic material

We consider the problem of determining necessary and sufficient conditions for the existence of symmetry planes of an anisotropic elastic material. These conditions are given in several equivalent forms, and are used to determine special coordinate systems where the number of non-zero components in the elasticity tensor is minimized. By the method presented here it is also shown that an elastic solid has at least six coordinate systems with respect to which there are only 18 non-zero elastic constants and cannot possess more then ten traditional and distinct symmetrics by planes of symmetry.     

立方晶粒任意集合下的格林函数和有效弹性刚度张量

为了推导多晶体材料的有效弹性刚度张量,给出立方晶粒任意集合的格林函数封闭但近似的表达式,该格林函数表达式包含三个单晶弹性常数和多晶体材料五个织构系数,它考虑取向分布函数的影响直至织构系数的线性项,它适用于弱织构多晶体材料或具有弱各向异性晶粒的多晶体材料(如金属铝),它与Nishioka格林函数近似式的比较通过三个算例给出;Synge的格林函数积分式则直接通过数值计算完成,它可作为问题的精确解供参考．该文还简单介绍了多晶体材料有效弹性刚度张量的推导过程,并把所得结果和有限元计算结果进行比较。     

Some basic issues in traditional Eulerian formulations of finite elastoplasticity

In traditional Eulerian formulations of finite elastoplasticity, there are some basic issues that still need clarification and further investigation. Among them are the characterization of initial anisotropy for scalar- and tensor-valued constitutive functions, plastic consistency conditions involving time differentiation of anisotropic yield functions, a unified loading criterion for hardening, softening, and perfectly plastic behaviour, etc. Sometimes, it is thought that a satisfactory and complete treatment for these issues could not be achieved within a framework of traditional Eulerian formulation. In this work, efforts are made towards explaining and clarifying these basic issues. Introducing the notion of an Eulerian type rotation-conjugate group of the initial material symmetry group, we show that consistent Eulerian formulations of scalar- and tensorvalued constitutive functions may be achieved for any given type of initial anisotropy. With such consistent Eulerian formulations, we derive plastic consistency conditions in a corotating frame associated with the foregoing rotation-conjugate group. As to the loading conditions for hardening, softening, and perfectly plastic behaviour, we recall and study the unified criterion proposed by Hill [J Mech Phys Solids 6 (1958) 236]. It is pointed out that the tangential elastic stiffness tensor in this unified criterion has to fulfill rather complicated integrability conditions for the elastic rate equation with an objective stress rate. It appears to be far from being a simple or even trivial matter to give an explicit form of the tangential elastic stiffness tensor meeting these conditions. It is indicated that this issue may be resolved by using the logarithmic stress rate. With this rate, an explicit form of Hill's unified loading criterion, together with an explicit form of tangential elastic stiffness tensor, is presented in terms of a general complementary hyperelastic potential.     

Optimization of the strain energy density in linear anisotropic elasticity

The problem considered here is that of extremizing the strain energy density of a linear anisotropic material by varying the relative orientation between a fixed stress state and a fixed material symmetry. It is shown that the principal axes of stress must coincide with the principal axes of strain in order to minimize or maximize the strain energy density in this situation. Specific conditions for maxima and minima are obtained. These conditions involve the stress state and the elastic constants. It is shown that the symmetry coordinate system of cubic symmetry is the only situation in linear anisotropic elasticity for which a strain energy density extremum can exist for all stress states. The conditions for the extrema of the strain energy density for transversely isotropic and orthotropic materials with respect to uniaxial normal stress states are obtained and illustrated with data on the elastic constants of some composite materials. Not surprisingly, the results show that a uniaxial normal stress in the grain direction in wood minimizes the strain energy in the set of all uniaxial stress states. These extrema are of interest in structural and material optimization.     

Periodic Homogenization and Material Symmetry in Linear Elasticity

Here homogenization theory is used to establish a connection between the symmetries of a periodic elastic structure associated with the microscopic properties of an elastic material and the material symmetries of the effective, macroscopic elasticity tensor. Previous results of this type exist but here more general symmetries on the microscale are considered. Using an explicit example, we show that it is possible for a material to be fully anisotropic on the microscale and yet the symmetry group on the macroscale can contain elements other than plus or minus the identity. Another example demonstrates that not all material symmetries of the macroscopic elastic tensor are generated by symmetries of the periodic elastic structure.     

Coordinate-free Characterization of the Symmetry Classes of Elasticity Tensors

We formulate coordinate-free conditions for identifying all the symmetry classes of the elasticity tensor and prove that these conditions are both necessary and sufficient. Also, we construct a natural coordinate system of this tensor without the a priory knowledge of the symmetry axes.     

Perturbation approach to elastic constitutive relations of polycrystals

Herein we obtain a formula for the effective elastic stiffness tensor C^eff of an orthorhombic aggregate of cubic crystallites by the perturbation method. The effective elastic stiffness tensor of the polycrystal gives the relationship between volume average stress and volume average strain. Under Voigt's model, Reuss’ model and Man's theory, the elastic constitutive relation accounts for the effect of the orientation distribution function (ODF) up to terms linear in the texture coefficients. However, the formula derived in this paper delineates the effect of crystallographic texture on elastic response and shows quadratic texture dependence. The formula is very simple. We also consider the influence of grain shape to elastic constitutive relations of polycrystals. Some examples are given to compare computational results of the formula with those given by Voigt's model, Reuss's model, the finite element method, and the self-consistent method. In Section 3, we also present an expression of the perturbation displacement field, in which Green's function for an orthorhombic aggregate of cubic crystallites is included.     

Elastic stability of inhomogeneous thin plates on an elastic foundation

We study the elastic stability of infinite inhomogeneous thin plates on an elastic foundation under in-plane compression. The elastic stiffness constants depend on the coordinate variable in the thickness direction of the plate. The elastic foundation is represented as a Winkler-type model characterized by linear and nonlinear spring constants. First we derive the Föppl–von Kármán equations by taking variations of the elastic strain energy. Next we develop the linear stability analysis of the plate under uniform in-plane compression and explicitly derive the critical loads and wave numbers for particular three cases. The effects of the material inhomogeneity, material orthotropy and loading orthotropy on the critical states are examined independently. Finally, we perform a weakly nonlinear analysis of the plate at the onset of the buckling instability. With the multiple scales method, the amplitude equations for the unstable modes that provide insight into the mode type and its amplitude are derived and then the effect of the material inhomogeneity on buckling modes are evaluated qualitatively.     

Features of the stress field near tunnel inclusions in an inhomogeneous anisotropic space

This paper proposes a method to solve problems for interface tunnel defects in a piecewise-homogeneous elastic material that is under generalized plane strain and has no planes of elastic symmetry. The method is based on integral relations between the discontinuities and sums of the components of the displacement vector and stress tensor at the interface. Closed-form solutions are obtained for a system of interface tunnel inclusions with mixed contact conditions between the space and the inclusions. The dependences of the indices of singularity of the solutions on orthogonal coordinate transformation are established for different combinations of materials of monoclinic and orthorhombic systems. The effect of the antiplane component on the behavior of the solutions is revealed __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 6, pp. 36–45, June 2008.     

On free energy-based formulations for kinematic hardening and the decomposition F = fpfe

Within the framework of linear plasticity, based on additive decomposition of the linear strain tensor, kinematical hardening can be described by means of extended potentials. The method is elegant and avoids the need for evolution equations. The extension of small strain formulations to the finite strain case, which is based on the multiplicative decomposition of the deformation gradient into elastic and inelastic parts, proved not straight forward. Specifically, the symmetry of the resulting back stress remained elusive. In this paper, a free energy-based formulation incorporating the effect of kinematic hardening is proposed. The formulation is able to reproduce symmetric expressions for the back stress while incorporating the multiplicative decomposition of the deformation gradient. Kinematic hardening is combined with isotropic hardening where an associative flow rule and von Mises yield criterion are applied. It is shown that the symmetry of the back stress is strongly related to its treatment as a truly spatial tensor, where contraction operations are to be conducted using the current metric. The latter depends naturally on the deformation gradient itself. Various numerical examples are presented.     

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.     

Degradation of elastic modulus in elastoplastic coupling with finite strains

An elastic potential W is postulated for the case of the finite-strain theory of elastoplastic coupling with damage effects. The potential is defined in terms of the invariants of two internal variables p and q. The internal variables are used to express the degradation of the elastic stiffness tensor due to the accumulation of plastic strains. The material damage is independently introduced to both Lame's coefficients H and G. The physical significance of this softening of the elastic stiffness is demonstrated experimentally in uniaxial loading, reverse loading of metals at finite strains.

For elastoplastic coupling, the Il'Iushin postulate does not yield normality of the plastic strain increment. An associative flow rule is postulated in this work for the combined components of the plastic strain increment and the elastic coupling strain increment.

The formulation is implemented in the Langrangian coordinate system. Through the use of the Oldroyd or Truesdell stress rate, the equivalent consistent spatial coordinate formulation is presented.     

Anholonomic configuration spaces and metric tensors in finite elastoplasticity

Deformation mappings are considered that correspond to the motions of lattice defects, elastic stretch and rotation of the lattice, and initial defect distributions. Intermediate (i.e., relaxed) configuration spaces associated with these deformation maps are identified and then classified from the differential-geometric point of view. A fundamental issue is the proper selection of coordinate systems and metric tensors in these configurations when such configurations are classified as anholonomic. The particular choice of a global, external Cartesian coordinate system and corresponding covariant identity tensor as a metric on an intermediate configuration space is shown to be a constitutive assumption often made regardless of the existence of geometrically necessary crystal defects associated with the anholonomicity (i.e., the non-Euclidean nature) of the space. Since the metric tensor on the anholonomic configuration emerges necessarily in the definitions of scalar products, certain transpose maps, tensorial symmetry operations, and Jacobian invariants, its selection should not be trivialized. Several alternative (i.e., non-Euclidean) representations proposed in the literature for the metric tensor on anholonomic spaces are critically examined.     

具有非轴对称刚度转轴的分岔

研究具有非轴对称刚度转轴的１／２亚谐共振和分岔,首先用Ｈａｍｉｌｔｏｎ原理导出运动微分方程,这是刚度系数周期性变化的参数激励方程,然后用多尺度法求得平均方程分岔响应方程和定常解,最后用奇异性理论分析分岔响应方程和定常解的稳定性,得到了局部分岔集和不同区域的不同分岔响应曲线。