Spectral decomposition of anisotropic elasticity

Summary A newly developed approach, based on the spectral decomposition principle, which is especially useful in crystallography, is applied in this paper. The compliance fourth-rank tensor of crystalline media belonging to the monoclinic system is spectrally decomposed, its eigenvalues are evaluated, together which its elementary idempotent tensors, which expand uniquely the fourth-rank tensor space into orthogonal subspaces. Next, the compliance tensor is spectrally analysed for anisotropic media of the orthorhombic, tetragonal, hexagonal and cubic crystal systems, by regarding these decompositions as particular cases of the spectral decomposition for monolinic media. Consequently, the characteristic values and the idempotent fourth-rank tensors are derived, as well as the stress and strain second-rank eigentensors for all the above mentioned symmetries.     

Inversions related to the stress-strain-fabric relationship

The stress-strain-fabric relationship is an extension of the anisotropic form of Hooke's law to include a dependence of the elastic coefficients upon a second-rank tensor called the fabric tensor. The fabric tensor represents features of the material microstructure associated with the type and the degree of the anisotropy. The inversion considered first in this work is that in which the stress-strain-fabric relation is constructed from the strain-stress-fabric relation and vice versa. Next, a semi-inversion of the relationship between the fourth-rank tensor of elastic coefficients and the fabric tensor is developed. This latter inversion permits the determination of the fabric tensor from a fourth-rank tensor of elastic constants. Explicit, approximate forms of these results, including a numerical example, are given for the case when the fabric tensor is normalized and terms of order three and higher in the fabric tensor are neglected.     

On an Arbitrarily Oriented Crack in a Transversely-isotropic Medium

Transversely-isotropic material with an arbitrarily oriented penny-shaped crack is considered. We calculate fourth-rank compliance contribution tensor of the crack and second-rank crack opening displacement tensor and examine their dependence on crack orientation. It is shown that this dependence for the crack opening displacement tensor is negligible if transverse isotropy has elliptic character, i.e. if material symmetry can be described in terms of a certain second rank tensor.     

Damage Constitutive Equations and its Application to Fiber Reinforced Composites under Transverse Impact

Two continuous field variables, called as continuity tensor and damage variable tensor, are used to describe the anisotropic responses of an elastic-brittle material under transverse impact load. Based on the continuum damage mechanics, anisotropic damage constitutive equations in both full and incremental forms are proposed here. The expressions of effective elastic module tensor, damage variable tensor and damage propagation force tensor are further derived, and the methods for determining the tensors are explained in detail. An example of strain and damage response of a fiber reinforced composite laminated plate under transverse impact load is employed to demonstrate the application of this theory. In the example, the damage variable coupled with geometric large deformation of laminated plate is also considered. The calculating results illustrate the influence of damage on strain field in the impacted laminated plate.     

Dual orthotropic damage–effect tensors with complementary structures

This paper deals with secant constitutive relations of orthotropic elastic damage based on the so-called damage–effect tensors, namely the fourth-order operators that define the linear transformations between nominal and effective stress and strain quantities. The damage–effect tensors are expressed by orthotropic representations in terms of symmetric second-order damage tensor variables. The paper provides a set of new dual orthotropic damage–effect tensors that possess complementary structures in the dual compliance- and stiffness-based derivations. More specifically, each orthotropic damage–effect tensor of the solution set possesses an inverse (its dual counterpart) that displays the structure of the (major) transpose of the tensor obtained by replacing the adopted second-order damage tensor variables with their inverses.     

Families of continuous spin tensors and applications in continuum mechanics

A new method is proposed for generating families of continuous spin tensors associated with families of corotational rates of second-order tensors using isotropic tensor functions of the same tensor arguments and different forms of continuous antisymmetric scalar spin functions of scalar arguments. Tensor functions are represented in terms of eigenprojections of a symmetric tensor S, which is one of the arguments of these functions. Each member of the generated family is represented as the sum of some basic spin tensor associated with the basic corotational tensor rate and the above-mentioned tensor function, whose structure is matched to the structure of the tensor function required to construct the twirl tensor of the triad of orthonormal eigenvectors of the tensor S (but this twirl tensor itself does not belong to the family of continuous spin tensors). The developed method is used in continuum mechanics to generate two families of continuous spin tensors associated with two families of objective corotational rates: Lagrangian and Eulerian. In these families, isotropic tensor functions are constructed using Lagrangian and Eulerian tensor arguments of the kinematic type, respectively. It is shown that if the same scalar spin function is used in deriving tensor functions of Lagrangian and Eulerian tensor arguments, then the corotational tensor rates associated with the generated spin tensors are objective (Lagrangian and Eulerian) counterparts of each other. It is shown that the spin tensors associated with the classical Eulerian corotational tensor rates (Zaremba–Jaumann, Green–Naghdi, d-rate) and their Lagrangian counterparts (including material rate) belong to the generated families of continuous spin tensors. It is also shown that both of these families of continuous spin tensors are subfamilies of the families of material spin tensors derived by Xiao et al. (J Elast 52:1–41, 1998). It is noted that the twirl tensors of the Lagrangian and Eulerian triads associated with the Gurtin–Spear corotational rates of tensors belong to the families of material spin tensors but do not belong to the families of continuous spin tensors. The final section gives expressions of continuous spin tensors from families associated with the families of Lagrangian and Eulerian corotational tensor rates which are appropriate for applications.     

A continuum damage mechanics approach to crack tip shielding in brittle solids

This paper focuses firstly on the development of a comprehensive anisotropic theory of continuum damage mechanics for brittle solids suffering progressive deterioration. The basic concept of damage parameterization is re-examined and a new set of damage variables introduced yielding a new damage effect tensor through which the effective stress and strain tensors are defined. Constitutive equations of the damaged material are established incorporating a new hypothesis on equivalence between damaged and undamaged responses of the material. The model is completed by introduction of a general damage characteristic tensor which accounts for the experimentally observed fact that the rate of damage growth depends nonlinearly on applied external loads. The established damage model is next applied to investigate the crack-tip shielding effect due to anisotropic microcracking. The ratio of near-tip to remote stress intensity factors is obtained in closed form. A moderate but definite effect of anisotropy of microcracking is observed. The case of isotropic damage is found to be the least effective in screening remote external loads and is in accord with the results obtained by other researchers using different approaches.     

Application of higher-order tensor theory for formulating enhanced continuum models

Summary In this paper attention is focussed on the derivation of higher-order isotropic tensors and their application in the formulation of enhanced continuum models. A mathematical theory will be discussed which relates formal orthogonal invariant polynomial functions to isotropic tensors. Using this theory, the second-order to the sixth-order isotropic tensor will be derived. When the tensor order increases, the derivation procedure clearly reveals a repeatable character. Thereafter, an example will be given of how the higher-order isotropic tensors can be used in the formulation of an enhanced continuum model. It will be demonstrated that symmetry conditions significantly reduce the number of material parameters.     

Eshelby tensors for an ellipsoidal inclusion in a micropolar material

Micropolar Eshelby tensors for an ellipsoidal inclusion are derived in an analytical form, which involves only one-dimensional integral. The numerical evaluation of the Eshelby tensors are also performed, it is found that the micropolar Eshelby tensors are not uniform in the ellipsoidal inclusion, however, their variations over the ellipsoidal domain are not significant. When size of inclusion is large compared to the characteristic length of the micropolar material, the micropolar Eshelby tensor is reduced to the classical one. It is also demonstrated that for a general ellipsoidal inclusion a uniform eigenstrain or eigentorsion produces on average only nonzero strain or torsion, and the average Eshelby relations are uncoupled.     

Generalized radial‐return mapping algorithm for anisotropic von Mises plasticity framed in material eigenspace

A computationally efficient integration algorithm for anisotropic plasticity is proposed, which is identified as a generalization of the radial‐return mapping algorithm to anisotropy. The algorithm is based upon formulation within the eigenspace of a material anisotropy tensor associated with anisotropic quadratic von Mises (J₂) plasticity (also called Hill plasticity), for which it is shown to ensure that the flow rule remains associative, ie, the normality condition is satisfied. Extension of the algorithm to include anisotropic elasticity (anisotropic elastoplasticity) is further provided, made possible by the identification of a certain fourth‐order material tensor dependent on both the elastic and plastic anisotropy. The derivation of the fully elastoplastically anisotropic algorithm involves further complexity, but the resulting algorithm is shown to closely resemble the purely plastically anisotropic one once the appropriate eigenspace is identified. The proposed generalized radial‐return algorithm is compared to a classical closest‐point projection algorithm, for which it is shown to provide considerable advantage in computational cost. The efficiency, accuracy, and robustness of the algorithm are demonstrated through various illustrative test cases and in the finite element simulation of Taylor impact tests on tantalum.     

Probabilistic modeling of apparent tensors in elastostatics: A MaxEnt approach under material symmetry and stochastic boundedness constraints

In this work, we address the stochastic modeling of apparent elasticity tensors, for which both material symmetry and stochastic boundedness constraints have to be taken into account, in addition to the classical constraint of invertibility. We first introduce a stochastic measure of anisotropy, which is defined using metrics in the set of elasticity tensors and used for quantitatively characterizing the fulfillment of material symmetry constraints. After having defined a numerical approximation for the stochastic boundedness constraint, we then propose a methodology allowing one to unify maximum entropy based models that have been previously derived by considering some of these constraints and which consists in constructing a probabilistic model for an auxiliary random variable. The latter can be interpreted as a stochastic compliance tensor, for which the available information to be used in the maximum entropy formulation can be readily deduced from the one considered for the elasticity tensor. A numerical illustration of the approach to an elastic microstructure is finally provided.     

Micromechanical approach to damage mechanics of composite materials with fabric tensors

The purpose of this study is to apply continuum damage mechanics – introduced through the concept of fabric tensors – to composite materials within the framework of the theory of elasticity. A directional data model of damage mechanics for composite materials will be developed using fabric tensors. The introduction of fabric tensors into the analysis of damage of composite materials will allow for an enhanced and better understood physical meaning of damage. The micromechanical approach will be used here to relate the damage effect through fabric tensors to the behavior of composite materials. In this approach, damage mechanics is introduced separately to the constituents of the composite material through different constituents’ damage effect tensors. The damaged properties of the composite system as a whole can then be obtained by proper homogenization of the damaged properties of the constituents.

The derivation of a generalized formulation of damage evolution will be shown here in a mathematically consistent manner that is based on sound thermodynamic principles. Numerical examples will be presented to show applicability. In addition, damage evolution for the one-dimensional tension case is also illustrated.     

On stiffness and strength reduction of solids due to crack development

Modification of the effective elastic and plastic constants of initially homogeneous and isotropic material with regularly distributed cracks is considered in the paper. The stress-strain relation for linearly elastic range is formulated as a tensor function with two independent variables: the stress tensor and damage tensor describing the current state of the cracked solid. This equation made it possible to evaluate all the elastic constants and is a starting point in the analysis of the plastic behavior of the damaged material. The appropriate yield criterion is derived in the form of an isotropic scalar function with the same variables as in the elastic range. To choose the most important terms of the general representation of this function, the energy of the elastic strain was calculated for homogenized equivalent material. This was done employing the stress-strain relation of elasticity for damaged solid proposed in the paper. The theoretical considerations were verified experimentally. To this end the material constants determined theoretically in the elastic and plastic ranges were compared with those measured experimentally for the models simulating the damaged material.     

张量分析中简化记法在公式推导中的应用及张量分量的计算

张量分析在计算力学中应用广泛,但其理论比较复杂,较难掌握和熟练运用。为此,给出了张量简化记法,这种记法与Fortran程序中或商业软件Matlab中的数组形似。将简化后张量的表达式应用于公式推导,可快速准确地得出结论。同时,通过引入矩阵的矢量化概念,将四阶张量的分量用一个方阵表示出来,这有利于符号运算和数值计算。以上两者结合可使得张量分析变得易于掌握,也有利于张量运算的推广。     

Characterization of the anisotropic materials capable of exhibiting an isotropic Young or shear or area modulus

By means of the decomposition of an anisotropic elastic tensor into symmetric traceless tensors, the general intrinsic expressions involving no redundant elastic coefficients are obtained for the orientation distribution functions of the Young, shear and area moduli of an anisotropic material. Necessary and sufficient conditions are established for each of these moduli to be isotropic. It is found that an anisotropic material exhibiting an isotropic Young or shear or area modulus can be only either transversely isotropic or orthotropic.     

Expression of third-order effective nonlinear susceptibility for third-harmonic generation in crystals

Third-harmonic-generation processes in crystals are governed by the fourth-rank tensor ((3))(Xijkl), which reflects the crystal symmetry. In this case, the third-order nonlinear susceptibility tensor can be contracted to the compact matrix form ((3))(Xim). The matrices ((3))(Xim) for isotropic media and all 32 crystallographic point groups are presented. With these matrices, the analytic expressions of third-order effective nonlinear susceptibility can be easily derived.     

Characterization of microstructural anisotropy in orthotropic materials using a second rank tensor

A second rank symmetric tensor which describes the degree of orientation in orthotropic materials is presented and shown to reflect accurately patterns of experimental data. The use of this tensor to describe microstructural anisotropy is compared to currently accepted methods and is found to be more useful and accurate in experimental studies. A method for determining the anisotropy tensor in a material is given, based on measurements on any three mutually perpendicular planes, and the fundamental restriction of this method to orthotropic materials is discussed. Experimentally determined anisotropy tensors in five specimens of cancellous bone from five different human bones are given.