Exact Relations for Effective Tensors of Polycrystals. I. Necessary Conditions

The set of all effective moduli of a polycrystal usually has a nonempty interior. When it does not, we say that there is an exact relation for effective moduli. This can indeed happen as evidenced by recent results [4, 10, 12] on polycrystals. In this paper we describe a general method for finding such relations for effective moduli of laminates. The method is applicable to any physical setting that can be put into the Hilbert space framework developed by Milton[13]. The idea is to use the W-function of Milton[13] that transforms a lamination formula into a convex combination. The method reduces the problem of finding exact relations to a problem from representation theory of SO(d)(d= 2 or 3) corresponding to a particular physical setting. When this last problem is solved, there is a finite amount of calculation required to be done in order to answer the question completely. At present, each candidate relation has to be examined separately in order to confirm the stability under homogenization. We apply our general theory to the settings of conductivity and two‐dimensional elasticity. (Accepted April 4, 1997)     

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.     

Bounds for the Effective Elastic Properties of Completely Random Planar Polycrystals

Bounds have been developed for the elastic moduli of completely random planar polycrystals, the shape and crystalline orientations of the constituent grains of which are supposed to be uncorrelated. Explicit results for the aggregates of orthotropic crystals are demonstrated. This revised version was published online in August 2006 with corrections to the Cover Date.     

On the Statistical Dependence for the Components of Random Elasticity Tensors Exhibiting Material Symmetry Properties

This work is concerned with the characterization of the statistical dependence between the components of random elasticity tensors that exhibit some given material symmetries. Such an issue has historically been addressed with no particular reliance on probabilistic reasoning, ending up in almost all cases with independent (or even some deterministic) variables. Therefore, we propose a contribution to the field by having recourse to the Information Theory. Specifically, we first introduce a probabilistic methodology that allows for such a dependence to be rigorously characterized and which relies on the Maximum Entropy (MaxEnt) principle. We then discuss the induced dependence for the highest levels of elastic symmetries, ranging from isotropy to orthotropy. It is shown for instance that for the isotropic class, the bulk and shear moduli turn out to be independent Gamma-distributed random variables, whereas the associated stochastic Young modulus and Poisson ratio are statistically dependent random variables.     

The Method of Integrodifferential Relations for Linear Elasticity Problems

Some possible modifications of the governing equations of the linear theory of elasticity are considered. The stress–strain relation is specified by an integral equality instead of the local Hooke’s law. The modified integrodifferential boundary value problem is reduced to the minimization of a nonnegative functional under differential constraints. A numerical algorithm based on polynomial approximations of unknown functions (stresses and displacements) is developed and applied to linear elasticity problems. The bilateral estimation criteria of solution errors are proposed in order to analyze the algorithm convergence rate. The numerical results obtained by applying the integrodifferential relation method and the conventional variational method are compared and discussed.     

A Notion of Capacity Related to Elasticity: Applications to Homogenization

We study a notion of capacity related to elasticity which proves convenient for analyzing the concentrations of strain energy caused by rigid displacements of some infinitesimal parts of an elastic body in two or three dimensions. By way of application, we investigate the behavior of solutions to initial boundary value problems describing vibrations of periodic elastic composites with rapidly varying elastic properties. More specifically, we analyze a two-phase medium whereby a set of heavy stiff tiny particles is embedded in a softer matrix. This task is set in the context of linearized elasticity.     

Relaxation of Multiwell Energies in Linearized Elasticity and Applications to Nematic Elastomers

The relaxation of a free-energy functional which describes the order–strain interaction in nematic elastomers is obtained explicitly. We work in the regime of small strains (linearized kinematics). Adopting the uniaxial order tensor theory (Frank model) to describe the liquid crystal order, we prove that the minima of the relaxed functional exhibit an effective biaxial nematic texture, as in the de Gennes order tensor model. In particular, this implies that, at a sufficiently macroscopic scale, the response of the material is soft even if the order of the system is assumed to be fixed at the microscopic scale. The relaxed energy density satisfies a solenoidal quasiconvexification formula.     

A unified constitutive theory for polymeric liquids II. Applications to basic problems

A unified constitutive theory for polymeric liquids has been recently proposed. Its derivation is based on a combination of continuum mechanical approach, transient-network concept and thermodynamics of irreversible processes. In the resulting model, many modes may be present for each of which there are two time scales, associated with the loss rate and the nonaffine motion of transient network junctions, respectively. A single effective relaxation time, constructed from the two time scales, governs the behavior in the linear regime of deformation. Two new parameters for each mode, in comparison with other models, are introduced: (i) the ratio r of the two time scales, and (ii) the index a distinguishing the rates of loss and creation of junctions. Both are important only for the nonlinear regime of deformation. In this paper, the theory is applied to predict the following cases: (i) stress growth at constant shear strain rate, (ii) steady shear-rate-dependent viscosity and first normal-stress difference and (iii) transient elongational viscosity at constant elongational strain rate. Determination of the model parameters based on usual characterization experiments is described. Comparison of calculated and observed behavior of low-density polyethylene at 150 °C available in the literature are presented. In general, the agreement of the predictions with experiment appear gratifying even with the simplest version of the new model.     

《准晶弹性的数学理论及应用》评介

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Exact static solutions to piezoelectric smart beams including peel stresses. II. Numerical results,comparison and discussion

This part presents the numerical results, comparisons and discussion for the exact static solutions of smart beams with piezoelectric (PZT) actuators and sensors including peel stresses presented in Part I. (International Journal of Solids and Structures, 39, 4677–4695) The actuated stress distributions in the adhesive and the adhesive edge stresses varying with the thickness ratios are firstly obtained and presented. The actuated internal stress resultants and displacements in the host beam are then calculated and compared with those predicted by using the shear lag model. The stresses in the adhesive caused by an applied axial force, bending moment and shear force are calculated, and then used to compute the sensing electric charges for comparison with those predicted using the shear lag model. The numerical results are given for the smart beam with (a) one bonded PZT and (b) two symmetrically bonded PZTs, with a comparison to those predicted using the shear lag model. Novel, simple and more accurate formulas for the equivalent force and bending moment induced by applied electric field are also derived for the host beam with one PZT or two symmetrically bonded PZTs. The symmetric shear stress and the anti-symmetric peel stress components caused by a shear force are discussed. In addition, in the case of PZT edge debonding, the stress redistribution in the adhesive and the self-arresting mechanism are also investigated.     

Objective Tensor Rates and Applications in Formulation of Hyperelastic Relations

Following Ogden, a class of objective (Lagrangian and Eulerian) tensors is identified among the second-rank tensors characterizing continuum deformation, but a more general definition of objectivity than that used by Ogden is introduced. Time rates of tensors are determined using convective rates. Sufficient conditions of objectivity are obtained for convective rates of objective tensors. Objective convective rates of strain tensors are used to introduce pairs of symmetric stress and strain tensors conjugate in a generalized sense. The classical definitions of conjugate Lagrangian (after Hill) and Eulerian (after Xiao et al.) stress and strain tensors are particular cases of the definition of conjugacy of stress and strain tensors in the generalized sense used in the present paper. Pairs of objective stress and strain tensors conjugate in the generalized sense are used to formulate constitutive relations for a hyperelastic medium. A family of objective generalized strain tensors is introduced, which is broader than Hill’s family of strain tensors. The basic forms of the hyperelastic constitutive relations are obtained with the aid of pairs of Lagrangian stress and strain tensors conjugate after Hill (the strain tensors in these pairs belong to the family of generalized strain tensors). A method is presented for generating reduced forms of the constitutive relations with the aid of pairs of Lagrangian and Eulerian stress and strain tensors conjugate in the generalized sense which are obtained from pairs of Lagrangian tensors conjugate after Hill by mapping tensor fields on one configuration of a deformable body to tensor fields on another configuration.      

Structural interfaces in linear elasticity. Part II: Effective properties and neutrality

The model of structural interfaces developed in Part I of this paper allows us to analytically attack and solve different problems of stress concentration and composites. In particular, (i) new formulae are given for effective properties of composite materials containing dilute suspensions of (randomly oriented) reinforced elliptical voids or inclusions; (ii) a new definition is proposed for inclusion neutrality (to account for the fact that the matrix is always ‘overstressed’, and thus non-neutral in a classical sense, at the contacts with the interfacial structure), which is shown to provide interesting stress optimality conditions. More generally, it is shown that the incorporation of an interfacial structure at the contact between two elastic solids exhibits properties that cannot be obtained using the more conventional approach of the zero-thickness, linear interface. For instance: contrary to the zero-thickness interface, both bulk and shear effective moduli can be optimized for a structural interface; effective properties higher that those possible with a perfect interface can be attained with a structural interface; and neutrality holds with a structural interface for a substantially broader range of parameters than for a zero-thickness interface.     

Damage analysis of thermopiezoelectric properties: Part II. Effective crack model

Having completed the general formulation for temperature, heat flow, displacement, electric potential and displacements and mechanical stresses of a piezoelectric material as presented in part I of this work, part II is concerned with a generalized self-consistent approximate method for determining the thermoelectroelastic properties of piezoelectric materials weakened by microcracks. A representative area element is adopted; it contains a microcrack surrounded by an elliptic matrix in a solid with effective properties. Numerical results are given for a piezoelectric BatiO₃ ceramic. The effective conductivity and effective modulus are found to decrease with increasing crack density.     

Algebra of Transversely Isotropic Sixth Order Tensors and Solution to Higher Order Inhomogeneity Problems

In this paper we provide a complete and irreducible representation for transversely isotropic sixth order tensors having minor symmetries. Such tensors appear in some practical problems of elasticity for which their inversion is required. For this kind of tensors, we provide an irreducible basis which possesses some remarkable properties, allowing us to provide a representation in a compact form which uses two scalars and three matrices of dimension 2, 3 and 4. It is shown that the calculation of sum, product and inverse of transversely isotropic sixth order tensors is greatly simplified by using this new formalism and appears to be appropriate for deriving new various solutions to some practical problems in mechanics which use such kinds of higher order tensors. For instance, we derive the fields within a cylindrical inhomogeneity submitted to remote gradient of strain. The method of resolution uses the Eshelby equivalent inclusion method extended to the case of a polynomial type eigenstrain. It is shown that the approach leads to a linear system involving a sixth order tensor whose closed form solution is derived by means of the tensorial formalism introduced in the first part of the paper.