Evaluation of the elastic moduli of a transversely isotropic aggregate of cubic crystals

The crystals and the aggregate have the same bulk modulus. The other three overall elastic moduli in the simplified stress-strain relations of Walpole (1985) are placed between upper and lower, Voigt and Reuss, bounds and some exact calculations are given for particular fibre textures.     

Bounds on the overall properties of composites with ellipsoidal inclusions

A new model is put forward to bound the effective elastic moduli of composites with ellipsoidal inclusions. In the present paper, transition layer for each ellipsoidal inclusion is introduced to make the trial displacement field for the upper bound and the trial stress field for the lower bound satisfy the continuous interface conditions which are absolutely necessary for the application of variational principles. According to the principles of minimum potential energy and minimum complementary energy, the upper and lower bounds on the effective elastic moduli of composites with ellipsoidal inclusions are rigorously derived. The effects of the distribution and geometric parameters of ellipsoidal inclusions on the bounds of the effective elastic moduli are analyzed in details. The present upper and lower bounds are still finite when the bulk and shear moduli of ellipsoidal inclusions tend to infinity and zero, respectively. It should be mentioned that the present method is simple and needs not calculate the complex integrals of multi-point correlation functions. Meanwhile, the present paper provides an entirely different way to bound the effective elastic moduli of composites with ellipsoidal inclusions, which can be developed to obtain a series of bounds by taking different trial displacement and stress fields.     

Bounds on the non-local effective elastic properties of composites

The paper deals with the effective linear elastic behaviour of random media subjected to inhomogeneous mean fields. The effective constitutive laws are known to be non-local. Therefore, the effective elastic moduli show dispersion, i.e¹ they depend on the “wave vector” k of the mean field. In this paper the well-known Hashin-Shtrikman bounds (1962) for the Lamé parameters of isotropic multi-phase mixtures are generalized to inhomogeneous mean fields k ≠ 0. The bounds involve two-point correlations of random elastic moduli. In the limit k → ∞ the bounds converge to the exact result. The interest is focussed on composites with cell structures and on binary mixtures. To illustrate the results, numerical evaluations are carried out for a binary cell material composed of nearly spherical grains of equal size.     

Rigorous bounds on the effective moduli of composites and inhomogeneous bodies with negative-stiffness phases

We review the theoretical bounds on the effective properties of linear elastic inhomogeneous solids (including composite materials) in the presence of constituents having non-positive-definite elastic moduli (so-called negative-stiffness phases). Using arguments of Hill and Koiter, we show that for statically stable bodies the classical displacement-based variational principles for Dirichlet and Neumann boundary problems hold but that the dual variational principle for traction boundary problems does not apply. We illustrate our findings by the example of a coated spherical inclusion whose stability conditions are obtained from the variational principles. We further show that the classical Voigt upper bound on the linear elastic moduli in multi-phase inhomogeneous bodies and composites applies and that it imposes a stability condition: overall stability requires that the effective moduli do not surpass the Voigt upper bound. This particularly implies that, while the geometric constraints among constituents in a composite can stabilize negative-stiffness phases, the stabilization is insufficient to allow for extreme overall static elastic moduli (exceeding those of the constituents). Stronger bounds on the effective elastic moduli of isotropic composites can be obtained from the Hashin–Shtrikman variational inequalities, which are also shown to hold in the presence of negative stiffness.     

Effective elastic moduli of triangular lattice material with defects

This paper presents an attempt to extend homogenization analysis for the effective elastic moduli of triangular lattice materials with microstructural defects. The proposed homogenization method adopts a process based on homogeneous strain boundary conditions, the micro-scale constitutive law and the micro-to-macro static operator to establish the relationship between the macroscopic properties of a given lattice material to its micro-discrete behaviors and structures. Further, the idea behind Eshelby’s equivalent eigenstrain principle is introduced to replace a defect distribution by an imagining displacement field (eigendisplacement) with the equivalent mechanical effect, and the triangular lattice Green's function technique is developed to solve the eigendisplacement field. The proposed method therefore allows handling of different types of microstructural defects as well as its arbitrary spatial distribution within a general and compact framework. Analytical closed-form estimations are derived, in the case of the dilute limit, for all the effective elastic moduli of stretch-dominated triangular lattices containing fractured cell walls and missing cells, respectively. Comparison with numerical results, the Hashin–Shtrikman upper bounds and uniform strain upper bounds are also presented to illustrate the predictive capability of the proposed method for lattice materials. Based on this work, we propose that not only the effective Young’s and shear moduli but also the effective Poisson’s ratio of triangular lattice materials depend on the number density of fractured cell walls and their spatial arrangements.     

Bounds on the elastic moduli of statistically isotropic multicomponent materials and random cell polycrystals

Minimum energy and complementary energy principles are used to derive the upper and lower bounds on the effective elastic moduli of statistically isotropic multicomponent materials in d ($d=2$ or 3) dimensions. The trial fields, involving harmonic and biharmonic potentials, and free parameters to be optimized, lead to the bounds containing, in addition to the properties and volume proportions of the material components, the three-point correlation information about the microgeometries of the composites. The relations and restrictions among the three-point correlation parameters are explored. The upper and lower bounds are specialized to symmetric cell materials and asymmetric multi-coated spheres, which are optimal or even converge in certain cases. New bounds for random cell polycrystals are constructed with particular results for random aggregates of cubic crystals.     

A variational approach to the theory of the elastic behaviour of polycrystals

Variational principles for anisotropic and nonhomogeneous elasticity, established by the authors in a previous paper, have been applied to the derivation of lower and upper bounds for the elastic moduli of polycrystals in terms of the moduli of the constituting crystals. The results hold for arbitrary crystal shapes. Explicit results tor cubic polycrystals showed that the present bounds are a considerable improvement of the well-known Voigt and Reuss bounds. Good agreement with experimental results has been obtained.     

On stability and instability of a compressed block pressed to a smooth basement

We study the problem on the stability of the equilibrium of a compressed homogeneous nonlinearly elastic body having the shape of a rectangular parallelepiped (block). The conditions of free sliding along the block face planes (with possible separation) are posed on all but one block faces. On the remaining face, a normal pressing “dead” load uniformly distributed over the surface is given. We obtain strict upper and lower bounds for the critical values of compression stresses, which coincide in order of magnitude with the characteristic elastic moduli of the material in the equilibrium under study; these estimates are independent of the relations between the block dimensions in the entire range of possible variation of the latter. The result indirectly confirms that the primary instability in the problem under study has a surface character (is localized near the kinematically free face with a given load) for any relations between the block dimensions and is characterized by the absence of separation from the basement even for an arbitrarily thin plate. This also implies that the “cantilever approximation” (whose application to similar problems has been attempted in the literature) cannot be used for the stability analysis in this situation in principle.     

Improved bounds for the elastic moduli of perfectly-random composites

New upper and lower bounds are constructed for the elastic moduli of a class of isotropic composites with perfectly-random microgeometries ([1–3]), which improve upon the bounds on the elastic shear modulus given in [1].     

Bounds for effective elastic moduli of inhomogeneous solid bodies

In connection with the extensive use of various kinds of inhomogeneous materials (glass, carbon and boron reinforced plastics, cermets, concrete, reinforced materials, etc.) in technology, there arises a need to calculate the elastic properties of such systems. Here in each case it is necessary to work out specific methods for finding both elastic fields and effective moduli. Since, as a rule, such methods do not take into account the character of distribution of inhomogeneities in space, which is reflected on the form of the central moment functions [1], they can be referred to a single class and, consequently, can be obtained by a common method [2], In the given paper, by means of the method of solution of stochastic problems for microinhomogeneous solid bodies proposed in the work of the author [2], we find elastic fields and effective moduli in an arbitrary approximation. Depending on the choice of parameters, the latter form bounds within which there lie the exact values of the effective moduli. It is shown that the conditions used earlier for finding these parameters [3] are not the best ones. The effective elastic moduli of an inhomogeneous medium are calculated, and bounds, narrower than the bounds formed in [3], are found for them.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhniki, No. 5, pp. 144–150, September–October, 1973.     

Bounds and self-consistent estimates for elastic constants of random polycrystals with hexagonal, trigonal, and tetragonal symmetries

Peselnick, Meister, and Watt have developed rigorous methods for bounding elastic constants of random polycrystals based on the Hashin-Shtrikman variational principles. In particular, a fairly complex set of equations that amounts to an algorithm has been presented previously for finding the bounds on effective elastic moduli for polycrystals having hexagonal, trigonal, and tetragonal symmetries. A more analytical approach developed here, although based on the same ideas, results in a new set of compact formulas for all the cases considered. Once these formulas have been established, it is then straightforward to perform what could be considered an analytic continuation of the formulas (into the region of parameter space between the bounds) that can subsequently be used to provide self-consistent estimates for the elastic constants in all cases. This approach is very similar in spirit but differs in its details from earlier work of Willis, showing how Hashin-Shtrikman bounds and certain classes of self-consistent estimates may be related. These self-consistent estimates always lie within the bounds for physical choices of the crystal elastic constants and for all the choices of crystal symmetry considered. For cubic symmetry, the present method reproduces the self-consistent estimates obtained earlier by various authors, but the formulas for both bounds and estimates are generated in a more symmetric form. Numerical values of the estimates obtained this way are also very comparable to those found by the Gubernatis and Krumhansl coherent potential approximation (or CPA), but do not require computations of scattering coefficients.     

Bounds on the Elastic Moduli of Completely Random Two-Dimensional Polycrystals

Explicit bounds on the elastic moduli of completely random planar polycrystals, the shape and crystalline orientations of the constituent grains of which are uncorrelated, are derived and calculated for a number of crystals of general two-dimensional anisotropy. The bounds on the elastic two-dimensional bulk modulus happen to coincide with the simple third order (in anisotropy contrast) bounds for the subclass of idealistic circular cell polycrystals. The bounds on the shear modulus are close to the much simpler bounds for circular cell polycrystals, which approximate aggregates of equiaxed grains.     

A micromechanics-based nonlocal constitutive equation incorporating three-point statistics for random linear elastic composite materials

A variational formulation employing the minimum potential and complementary energy principles is used to derive a micromechanics-based nonlocal constitutive equation for random linear elastic composite materials, relating ensemble averages of stress and strain in the most general situation when mean fields vary spatially. All information contained in the energy principles is retained; we employ stress polarization trial fields utilizing one-point statistics so that the resulting nonlocal constitutive equation incorporates up through three-point statistics. The variational structure is developed first for arbitrary heterogeneous linear elastic materials, then for randomly inhomogeneous materials, then for general n-phase composite materials, and finally for two-phase composite materials, in which case explicit variational upper and lower bounds on the nonlocal effective modulus tensor operator are derived. For statistically uniform infinite-body composites, these bounds are determined even more explicitly in Fourier transform space. We evaluate these in detail in an example case: longitudinal shear of an aligned fiber or void composite. We determine the full permissible ranges of the terms involving two- and three-point statistics in these bounds, and thereby exhibit explicit results that encompass arbitrary isotropic in-plane phase distributions; we also develop a nonlocal “Milton parameter”, the variation of whose eigenvalues throughout the interval [0, 1] describes the full permissible range of the three-point term. Example plots of the new bounds show them to provide substantial improvement over the (two-point) Hashin–Shtrikman bounds on the nonlocal operator tensor, for all permissible values of the two- and three-point parameters. We next discuss further applications of the general nonlocal operator bounds: to any three-dimensional scalar transport problem e.g. conductivity, for which explicit results are given encompassing the full permissible ranges of the two- and three-point statistics terms for arbitrary three-dimensional isotropic phase distributions; and to general three-dimensional composites, where explicit results require future research. Finally, we show how the work just summarized, treating elastostatics, can be generalized to elastodynamics, first in general, then explicitly for the longitudinal shear example.     

Bounds for effective elastic moduli of disordered materials

Recently P.H. Dederichs and R. Zeller (1973) have developed a formal theory of the bounds of odd order n for the effective elastic moduli of linearly elastic, disordered materials. The bounds are established by use of statistical information given in terms of correlation functions up to order n (= 1, 3, 5,…). This theory is extended to include the bounds of even order n. It is indicated how these bounds can be made optimum under the given statistical information. The results for bounds of even and odd order are obtained in forms which resemble Neumann series, containing multiple integrals up to order (n?1). These integrals can be calculated for certain materials which are classified in terms of a gradual statistical homogeneity, isotropy and disorder. Materials which possess these properties up to the correlation functions of nth order are called overall grade n materials. The optimum bounds for overall grade 2 and grade 3 materials are given explicitly. Optimum bounds for materials which are of grade ∞ in homogeneity and isotropy (i.e. (statistically) perfectly homogeneous and isotropic) and, at the same time, disordered of grade 2 or 3 are also derived. Those for grade 2 in disorder are the Z. Hashin and S. Shtrikman's (1963) bounds. Those for grade 3 are the narrowest, explicit bounds so far derived for random elastic materials. They contain within themselves the so-called self-consistent elastic moduli.     

Dynamic stability analysis of an elastic composite material having a negative-stiffness phase

The rigorous classical bounds of elastic composite materials theory provide limits on the achievable composite stiffnesses in terms of the properties and arrangements of the composite's constituents. These bounds result from the assumption, presumably made for stability reasons, that each constituent material must have positive-definite elastic moduli. If this assumption is relaxed, recently published elasticity analyses and experimental measurements show these bounds can be greatly exceeded, resulting in new materials of enormous potential.The key question is whether a composite material having a non-positive-definite constituent can be stable overall in the practically useful situation of applied traction boundary conditions. Drugan [2007. Elastic composite materials having a negative-stiffness phase can be stable. Phys. Rev. Lett. 98 (5), article no. 055502] first proved the answer is yes, by applying the energy criterion of elastic stability to the basic two- and three-dimensional composites consisting of a cylinder or sphere having non-positive-definite (but strongly elliptic) moduli with a thin positive-definite coating and proving overall stability provided the coating is sufficiently stiff.Here, we perform a complete and direct dynamic stability analysis of the plane strain fundamental elastic composite consisting of a circular cylinder of non-positive-definite material firmly bonded to a positive-definite concentric coating, for the full range of coating thicknesses (i.e., volume fractions). We determine quantitatively the full permissible range of inclusion and coating moduli, as a function of coating thickness, for which the overall composite is stable under dead traction boundary conditions. Among the results, we show that in the thin-coating case, the present dynamic stability analysis leads to precisely the same analytical stability requirements as those derived via the energy criterion by Drugan [2007. Elastic composite materials having a negative-stiffness phase can be stable. Phys. Rev. Lett. 98 (5), article no. 055502], and we derive new analytical stability requirements that are valid for a wider range of coating thickness. At the other extreme, we show that in the case of very thick coatings (corresponding to the dilute case of a matrix-inclusion composite), even an inclusion with merely strongly elliptic moduli can be stabilized by a positive-definite matrix satisfying weak requirements, for which we derive analytical expressions. Overall, our results show that surprisingly weak restrictions on the moduli and thickness of the positive-definite coating are sufficient to stabilize a non-positive-definite inclusion, even one whose moduli are merely strongly elliptic. These results legitimize expanding the search for novel materials with extreme properties to those incorporating a non-positive-definite constituent, and they provide quantitative restrictions on the constituent materials’ moduli and volume fractions, for the geometry examined here, that ensure overall stability of such composite materials.     

Grain size effects in multiphase steels assisted by transformation-induced plasticity

The influence of the austenitic grain size on the overall stress–strain behavior in a multiphase carbon steel is analyzed through three-dimensional finite element simulations. A recently developed multiscale martensitic transformation model is combined with a plasticity model to simulate the transformation-induced plasticity effects of a grain of retained austenite embedded in a ferrite-based matrix. Grain size effects are included via a surface energy term in the Helmholtz energy. Tensile simulations for representative orientations of the grain of retained austenite show that the initial stability of the austenite increases as the grain size decreases. Consequently, the effective strength is initially higher for smaller grains. The influence of the grain size on the evolution of the transformation process strongly depends on the grain orientation. For “hard” orientations, the transformation rate is higher for larger grains. In addition, the phase transformation is partially suppressed as the grain size decreases. In contrast, for “soft” orientations, the transformation rate is lower for larger grains. The phase transformation is more homogeneous for smaller grains and, consequently, the effective transformation strain is larger. Nevertheless, in multiphase carbon steels with a relatively low percentage of retained austenite, the influence of the austenitic grain size on the overall constitutive response is smaller than the influence of the austenitic grain orientation.     

Bounds for the non-local effective properties of random media

The paper deals with a random medium subjected to a static scalar field with inhomogeneous mean values. Then, effective linear material parameters show dispersion, i.e. they depend on the “wave vector” k of the mean field. The variational methods of P.H. Dederichs and R. Zeller (1973) are generalized to derive upper and lower bounds for scalar effective material parameters as functions of k. In the limit k → 0 (homogeneous mean fields), bounds of the Hashin-Shtrikman type are reproduced. For k → ∞, the bounds coincide with the exact result. In the general case, a two-point moment of the stochastic material parameter is involved. Especially, composites with cell structure and binary mixtures are considered. Detailed calculations are carried out for effective dielectricity, relating mean electric displacement to the mean electric field (which is mathematically equivalent to electrical and thermal conductivities and other scalar parameters), of a binary system composed of nearly spherical grains of equal size.     

Differential geometrical method in elastic composite with imperfect interfaces

I.IntroductionWhethertheinterfacesofcompositematerialsareperfectornotwillaffectitsmacromechanicaloreffectivepropertiesimportantly.Butsofar,almostallofthestudiesontheeffectivepropertiesofcompositematerialsarebasedontheassumptionthattheinterfacesareperfectl"2].Infact,thisisnotappropriateforallinterfaces[31.Thusthestudiesonmechanicalpropertyofcompositematerialswithimperfaceintert'acehavebeenconsideredrecentlyinsomeliteratures.Hashin16]hasextendedtheelasticextremumprinciplesofminimumpotentialandm…     

Theory of the elastic moduli of a heterogeneous medium

It is shown that the theory of random functions permits the expansion of the effective tensor X~jkl for the elastic moduli with respect to correlation functions and that it leads in the second approximation in the Voigt-Reuss scheme to values that lie to one side of the Xijkl, while in the third approximation it brackets the latter. The analysis is used to refine the Hashin limits to the elastic moduli for a mechanical mixture of isotrcpic components and polycrystalline aggregates of cubic structure.There are two methods for calculating the effective elastic moduli of heterogeneous solids: virial expansion [2] (as a power series in the concentration of one of the components) and the method of correlation functions [2] (expansion with respect to relative fluctuation of the elastic moduli). Identical results should be obtained in the two cases if all terms are incorporated, but great mathematical difficulties restrict one to the lowest approximations. The first approximation in the virial method gives better results when the concentration of one component is low, while the method of correlation functions gives better results when the fluctuations in the elastic moduli are small and the concentrations are similar.Methods have been developed for determining the upper and lower bounds in both approaches, and various schemes of averaging are used for this purpose in the correlation-function method. The upper bound is established by renormalizing the equation of equilibrium, while the lower one is found by renormalizing the equation of incompatibility. The range of the bracketing can be reduced by means of higher approximations. The range can be reduced in the limit to zero, which implies passing from an approximate effective tensor to the true one, which relates the means in stress and strain over the material. Here we show that the two methods of renormalization give identical results when all terms of the series are summed.If the tensor has a Gaussian distribution, the moment functions of odd order are zero, while the even ones are expressed via combinations of the binary functions [3]. However, a mechanical mixture of several components is not Gaussian, and the odd moments are not zero. Splitting of the higher-order correlation functions is possible also for mechanical mixtures having determinate phase interfaces, but this involves various simplifying assumptions. A derivation is given for a moment of arbitrary order, which allows one to formulate the conditions under which such splitting is possible. The results are used in calculating the exact value of the effective bulk modulus for a medium with a homogeneous shear modulus.We are indebted to V. V. Bolotin for a discussion.     

Interval analysis of nonlinear frames with uncertain connection properties

Semi-rigid connections can often be a more economical solution for a framing system than one with either fully fixed connections or fully pinned ones. In view of the fact that the properties of such ductile and partial-strength connections are not known accurately, this paper presents a method for the obtention of both upper and lower bound responses of semi-rigid frames for possible variations in their moment-rotation properties. The latter are thus assumed to be known within some key upper and lower bound values, namely a constitutive law that is still deterministic but is described in terms of a so-called “interval” model. A mathematical programming approach is used to formulate and solve the problem. In particular, for each load level, a pair of nonstandard optimization problems known as interval mathematical programs with equilibrium constraints (or interval MPECs) are solved to provide the required bounds. A number of examples are provided to highlight the important effects of considering uncertainties in semi-rigid connection properties.