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.     

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.      

Use of the parametrix method for estimating effective elastic moduli or randomly nonhomogeneous elastic bodies

The magnitude of the elasticity tensor of a comparison body remains unclarified if we use a singular approximation [1] to estimate the effective values of the elasticity tensor. Below we will use a parametrix method [2] to determine the first approximation of the random component of the deformation tensor and the effective values of the elasticity tensor, and will also compare the exact solution for one particular heterogeneous and a previously used approximation.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 171–175, January–February, 1976.     

弹脆性材料的损伤本构关系及应用

本文根据连续损伤力学方法,对弹脆性材料损伤的力学响应进行一般分析。理论分析中,材料与损伤都是各向异性的。还导出了计算损伤张量、有效弹性张量、真实应力张量以及损伤能耗率张量的实用表达式。     

Derivation of the general form of elasticity tensor of the transverse isotropic material by tensor derivate

SymbolsU--FunchonofstrainenergyQ--OrthonormaltensorE--StraintensorEar--ComponentsofthestraintensorE,i,j=l,2,3n--VectorofthesymmetricaamsofthetransverseisotropicmaterialU*,E.,n*--FormsofU,EandninanothercoordinatesystemJf--MaininvariantsofstraintensorE,i=l,2,3Jf'n--InvariantsofstraintensorEconnectingwithvectorn,i=4,5Ji--TheabbreviatedformsofJf,Jf,Jf,Jf,",Jf,",i=l,2,3,4,5fi--ConstantsindependentonE,n,i=l,2,3,4,5el,e"--Thecovariantandcontravariantofthonormalbasisoftheusedcoordinatesyste…     

Invariants for Rari- and Multi-Constant Theories with Generalization to Anisotropy in Biological Tissues

The rari-constant theory of linear elasticity is based on the assumption that elasticity in solids is caused by only pair potentials with coaxial forces acting between atoms. The strain energy of each pair potential depends on the square of the strain between the atoms in the pair. This strain can be determined by taking the inner product of the strain tensor with a structural tensor that is the tensor product of a unit vector with itself. It is shown that the 15 independent constants in the rari-constant theory can be generated by a complete set of 15 structural tensors. It is also shown that the 6 additional independent constants in the multi-constant theory can be generated by taking the inner product of 6 of these structural tensors with the square of the strain tensor. A generalization of these results for nonlinear elasticity is discussed with reference to recent work which compares the structural and generalized structural tensor approaches to modeling fibrous tissues.     

The evolution of Hooke's law due to texture development in FCC polycrystals

Initially isotropic aggregates of crystalline grains show a texture-induced anisotropy of both their inelastic and elastic behavior when submitted to large inelastic deformations. The latter, however, is normally neglected, although experiments as well as numerical simulations clearly show a strong alteration of the elastic properties for certain materials. The main purpose of the work is to formulate a phenomenological model for the evolution of the elastic properties of cubic crystal aggregates. The effective elastic properties are determined by orientation averages of the local elasticity tensors. Arithmetic, geometric, and harmonic averages are compared. It can be shown that for cubic crystal aggregates all of these averages depend on the same irreducible fourth-order tensor, which represents the purely anisotropic portion of the effective elasticity tensor. Coupled equations for the flow rule and the evolution of the anisotropic part of the elasticity tensor are formulated. The flow rule is based on an anisotropic norm of the stress deviator defined by means of the elastic anisotropy. In the evolution equation for the anisotropic part of the elasticity tensor the direction of the rate of change depends only on the inelastic rate of deformation. The evolution equation is derived according to the theory of isotropic tensor functions. The transition from an elastically isotropic initial state to a (path-dependent) final anisotropic state is discussed for polycrystalline copper. The predictions of the model are compared with micro–macro simulations based on the Taylor–Lin model and experimental data.     

Surface Impedance Tensor and Green’s Function for Weakly Anisotropic Elastic Materials

Recently Fu and Mielke uncovered a new identity that the surface impedance tensor of any anisotropic elastic material has to satisfy. By solving algebraically a matrix equation that follows from the new identity, we derive an explicit expression for the surface impedance tensor, which is correct up to terms linear in the components of the anisotropic part of the elasticity tensor of the material in question. From the well-known relationship between the surface impedance tensor and the Green’s function for infinite space, we obtain an explicit expression for the Green’s function, which is correct up to terms linear in the components of the anisotropic part of the elasticity tensor.      

Conditions for Strong Ellipticity of Anisotropic Elastic Materials

We consider the problem of finding the transversely isotropic elasticity tensor closest to a given elasticity tensor with respect to the Frobenius norm. A similar problem was considered by other authors and solved analytically assuming a fixed orientation of the natural coordinate system of the transversely isotropic tensor. In this paper we formulate a method for finding the optimal orientation of the coordinate system—the one that produces the shortest distance. The optimization problem reduces to finding the absolute maximum of a homogeneous eighth-degree polynomial on a two-dimensional sphere. This formulation allows us a convenient visualization of local extrema, and enables us to find the closest transversely isotropic tensor numerically.      

Compatibility conditions and equations of motion in the linear micropolar theory of elasticity

An analog of Cesàro’s formula and several compatibility conditions are given for the three-dimensional and two-dimensional linear micropolar theory of elasticity in the form different from that used in the literature. A number of formulas are obtained to determine the antisymmetric part of the strain (stress) tensor in terms of the symmetric part of the strain tensor and the symmetric part of the bending-torsion (stress and couple-stress) tensor and to determine the antisymmetric part of the bending-torsion (couple-stress) tensor in terms of the symmetric part of the bending-torsion (couplestress) tensor. Some integro-differential equations of motion expressed in terms of the symmetric parts of the stress and couple-stress tensors are proposed for the micropolar theory of elasticity.     

Dislocations in second strain gradient elasticity

A second strain gradient elasticity theory is proposed based on first and second gradients of the strain tensor. Such a theory is an extension of first strain gradient elasticity with double stresses. In particular, the strain energy depends on the strain tensor and on the first and second gradient terms of it. Using a simplified but straightforward version of this gradient theory, we can connect it with a static version of Eringen’s nonlocal elasticity. For the first time, it is used to study a screw dislocation and an edge dislocation in second strain gradient elasticity. By means of this second gradient theory it is possible to eliminate both strain and stress singularities. Another important result is that we obtain nonsingular expressions for the force stresses, double stresses and triple stresses produced by a straight screw dislocation and a straight edge dislocation. The components of the force stresses and of the triple stresses have maximum values near the dislocation line and are zero there. On the other hand, the double stresses have maximum values at the dislocation line. The main feature is that it is possible to eliminate all unphysical singularities of physical fields, e.g., dislocation density tensor and elastic bend-twist tensor which are still singular in the first strain gradient elasticity.     

Anisotropy of plane complex elastic bodies

The problem of the search of the invariants of an anisotropic elastic tensor representing the mechanical response of a complex elastic body in a two dimensional space is addressed, in particular for a tensor that does not possess all the tensor symmetries typical of classical elasticity. The invariants of the stiffness tensor are found and all the possible types of orthotropy are discussed.     

Averaging Anisotropic Elastic Constant Data

A method of averaging the data on the anisotropic elastic constants of a material is presented. The anisotropic elastic constants are represented by the elasticity tensor which is expressed as a second rank tensor in a space of six dimensions. The method consists of averaging eigenbases of different measurements of the elasticity tensor, then averaging the eigenvalues referred to the average eigenbasis. The eigenvalues and eigenvectors are obtained by using a representation of the stress-strain relations due, in principle, to Kelvin [17, 18]. The formulas for the representation of the averaged elasticity tensor are simple and concise. The applications of these formulas are illustrated using previously reported data, and are contrasted with the traditional analysis of the same data by Hearmon [9]. An interesting result that emerges from this analysis is a method dealing with variable composition anisotropic elastic materials whose elastic constants depend upon the particular composition. In the case of porous isotropic materials, for example, it is customary to regress the Young's modulus against porosity. The results of this paper suggest a structure or paradigm for extending to anisotropic materials this empirical method of regressing elastic constant data against composition or porosity.     

A new uniqueness theorem of the linear thermo-elastic dynamics on unbounded domains

In the present paper, the uniqueness of the solution to the initial boundary value problem of the linear thermo-elastic dynamics on unbounded domains is obtained under less restrictive conditions, including abandoning the positive semi-definiteness of the elasticity tensor and boundness of the material tensor and restrictions on the acoustic tensor and the coupled tensor, and the results in [1] are refined. The conclusion here is valid for the case on bounded domains and the linear elastic dynamics on unbounded domains, hence the results in [2–4] are refined too. Abandoning the positive semi-definiteness of elasticity tensor permits that the uniqueness of the kinetic process is still valid for deformation of the wider materials, especially for the case that there are phase-transition during deformation process provided that the constitutive equations are unchanged in forms. The project is partially supported by The Youth Foundtion of Science of the Higher-Education of Shanghai and YFNSC (No. 19802012)     

Laminations in Linearized Elasticity: The Isotropic Non-very Strongly Elliptic Case

The explicit computation of the effective elasticity tensor of the material produced by laminating two homogeneous elastic media is used to show that, in 2-dimensional and 3-dimensional linear elasticity, for any isotropic material a whose elasticity tensor is strongly elliptic, but not semipositive definite, we can select very strongly elliptic materials, so that through laminations between these with material a, we can create a nonstrongly elliptic media, whose existence contradicts properties concerning the propagation of elastic waves. This revised version was published online in August 2006 with corrections to the Cover Date.     

Modeling of deformation induced anisotropy in free-end torsion

The main purpose of this work is to develop a phenomenological model, which accounts for the evolution of the elastic and plastic properties of fcc polycrystals due to a crystallographic texture development and predicts the axial effects in torsion experiments. The anisotropic portion of the effective elasticity tensor is modeled by a growth law. The flow rule depends on the anisotropic part of the elasticity tensor. The normalized anisotropic part of the effective elasticity tensor is equal to the 4th-order coefficient of a tensorial Fourier expansion of the crystal orientation distribution function. Hence, the evolution of elastic and viscoplastic properties is modeled by an evolution equation for the 4th-order moment tensor of the orientation distribution function of an aggregate of cubic crystals. It is shown that the model is able to predict the plastic anisotropy that leads to the monotonic and cyclic Swift effect. The predictions are compared to those of the Taylor–Lin polycrystal model and to experimental data. In contrast to other phenomenological models proposed in the literature, the present model predicts the axial effects even if the initial state of the material is isotropic.     

The Eshelby stress tensor,angular momentum tensor and dilatation flux in gradient elasticity

The Eshelby (static energy momentum) stress tensor, the angular momentum tensor and the dilatation flux are derived for anisotropic linear gradient elasticity in non-homogeneous materials. The divergence of these tensors gives the configurational forces, moments and work terms in gradient elasticity. There are several types of configurational forces, acting on the dislocation density and its gradient, on the inhomogeneities, proportional to the distortion, and linear and quadratic in the distortion gradient, and on the body force.     

Wave Propagation along Axes of Symmetry in Linearly Elastic Media with Initial Stress

The proofs of the Fedorov–Stippes and Fedorov's theorems, which hold for linearly elastic homogeneous bodies in natural configurations, remain valid for any linearly elastic medium with initial stress provided that Hooke's tensor be replaced by a suitable elasticity tensor.