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One of the key difficulties in applications of the effective media theories to real materials is the complex and "irregular" character of microstructures. However, in cases of moderate contrast between properties of the constituents, the sensitivity of the effective properties to microgeometries is much lower; it vanishes in the first approximation. The latter actually remains applicable at substantial (up to 70%–100%) contrast in properties. The effective elastic and conductive properties are considered as functions of property contrast between constituents. Whereas in the isotropic cases the Voigt–Reuss bounds provide guidance, in anisotropic cases these issues are less obvious. Two origins of effective anisotropies are discussed: (A) anisotropy of material constants and (B) geometric factors: non-random orientations of non-spherical inclusions. To the first order in contrast, the factor (A) is dominant, and factor (B) can be ignored (effective properties are independent of microgeometries, such as inhomogeneity shapes); the importance of factor (B) increases sharply with increasing contrast. For matrix composites (continuous matrices containing inhomogeneities), an interesting interplay takes place between the moderate contrast assumption and the non-interaction assumption (NIA). It shows that, in cases of moderate contrast, the NIA remains accurate at substantial concentrations. An application to monitoring fatigue of austenitic steel, whereby property contrast between the martensite and austenite phases is moderate, is discussed. Results of the second order in contrast, that are limited to the ellipsoidal inhomogeneities, are also given. [ABSTRACT FROM AUTHOR]

A 3D problem in linear elasticity is considered for a thin inhomogeneous layer subject to transverse compression. For the first time, the effect of arbitrary vertical inhomogeneity is elucidated. Two sets of boundary conditions along the faces of the layer are adapted for modelling transverse compression. Robust asymptotic formulae involving repeated integrals across the thickness are derived for displacements and stresses. As an illustration, numerical results are presented for the elastic moduli having a transverse parabolic variation. The obtained results have a potential to be implemented in modern technology, including manufacturing and design of functionally graded materials. [ABSTRACT FROM AUTHOR]

For a given class of materials, universal deformations are those that can be maintained in the absence of body forces by applying only boundary tractions. Universal deformations play a crucial role in nonlinear elasticity. To date, their classification has been accomplished for homogeneous isotropic solids following Ericksen's seminal work, and homogeneous anisotropic solids and inhomogeneous isotropic solids in our recent works. In this paper we study universal deformations for inhomogeneous anisotropic solids defined as materials whose energy function depends on position. We consider both compressible and incompressible transversely isotropic, orthotropic, and monoclinic solids. We show that the universality constraints—the constraints that are dictated by the equilibrium equations and the arbitrariness of the energy function—for inhomogeneous anisotropic solids include those of inhomogeneous isotropic and homogeneous anisotropic solids. For compressible solids, universal deformations are homogeneous and the material preferred directions are uniform. For each of the three classes of anisotropic solids we find the corresponding universal inhomogeneities—those inhomogeneities that are consistent with the universality constraints. For incompressible anisotropic solids we find the universal inhomogeneities for each of the six known families of universal deformations. This work provides a systematic approach to study analytically functionally-graded fiber-reinforced elastic solids. [ABSTRACT FROM AUTHOR]

Faber series are used extensively in the application of complex variable methods to two-dimensional elasticity theory, for example, in the mechanical analysis of composite materials where Faber series representations of complex potentials lead to convenient expressions for the corresponding displacement and stress distributions. In many cases, the use of the Faber series is combined with conformal mapping techniques which "transfer" a boundary value problem defined in the elastic body (physical plane) to a simpler problem posed in an imaginary plane characterized by the conformal mapping. In several instances in the literature, however, little attention has been paid to the domain of definition of the Faber series in the imaginary plane leading often to misunderstandings and erroneous conclusions regarding the concept and feasibility of the use of the Faber series. In this paper, we present a thorough and rigorous examination of the representation of the Faber series in both the physical (occupied by the material) and imaginary (defined by the conformal mapping) plane. In addition, we show that replacing a truncated Faber series by a truncated Taylor series does not induce any additional errors in the numerical analysis of the corresponding boundary value problem. We anticipate that the discussion in this paper will help clarify any existing misinterpretations regarding the application of the Faber series and help further extend their use to a range of problems in composite mechanics. [ABSTRACT FROM AUTHOR]

An example of seven seismogenic regions of the world with different conditions of tectonic deformation confirms the reliability of confinedness of weak earthquakes to two horizons in the Earth's continental crust at depths of 5 and 10 km, regardless of the region. The width of extrema at the level of 0.7 from the maximum does not exceed 2–3 km in most cases. This markedly pronounced bimodal depth distribution of the numbers of weak earthquakes is interpreted from the standpoint of increasing strength and decreasing inhomogeneity of the material in the Earth's crust with depth (Mogi, 1962; Scholz, 1968; Mori and Abercombie, 1997). It is assumed that the concentration of weak earthquakes near a depth of 5 km is determined by the fact that, due to a relatively high degree of inhomogeneity and low brittle fracture strength of the material in the crust, the fracture process stops before it develops into a major seismic event because of an obstacle to its development in the form of a site with increased strength. At greater depths (9–15 km), an initiation of rupture more likely develops into a larger event, since the rock material at these depths becomes more homogeneous and more durable due to the growth of confining pressure and temperature within the brittle–ductile transition zone. This leads to a relative increase in the number of larger earthquakes near a depth of 10 km, providing the second extremum in the depth distribution of the number of earthquakes. Such implications also explain the confinedness of strong earthquakes observed in seismology to the bottom rather than to the roof of the seismogenic layer. At the same time, the almost complete absence of relatively strong earthquakes at depths greater than 15–16 km may indicate a sharp change in elastic properties of the material and in the deformation nature in the lower part of the brittle–ductile transition zone and deeper. Here, brittle seismogenic deformation gives way to a plastic aseismic flow. [ABSTRACT FROM AUTHOR]

Due to the large surface/interface-to-volume ratio, surface/interface effect in nanomaterials begins to play an important role in changing the constitutive law seen in classical elasticity theory. This work investigates the size-dependent elastic field of nano-inhomogeneity due to interface and interphase effect, respectively. Closed-form solution to the elastic field of nano-inhomogeneity with interface and interphase effect is found. For comparison, the numerical result of InAs/GaAs system is provided as an example, which demonstrates that interface and interphase effect could have significant influence on the elastic field of nano-inhomogeneity. The continuity/discontinuity of the elastic field of nano-inhomogeneity with interface and interphase effect is discussed in detail. It is verified both numerically and analytically that the elastic field from interphase effect tends to converge to that from interface effect when the interphase thickness shrinks to zero. Furthermore, all results converge to the classical solution at positions far away from the interfacial region, so it is unnecessary to distinguish interface or interphase effect at positions far away from the interfacial region in practical situations. [ABSTRACT FROM AUTHOR]

We show that, in general, the translational average over a spatial variable—discussed by Backus (J. Geophys. Res. 67(11):4427–4440, 1962), and referred to as the equivalent-medium average—and the rotational average over a symmetry group at a point—discussed by Gazis et al. (Acta Crystallogr. 16(9):917–922, 1963), and referred to as the effective-medium average—do not commute. However, they do commute in special cases of particular symmetry classes, which correspond to special relations among the elasticity parameters. We also show that this noncommutativity is a function of the strength of anisotropy. Surprisingly, a perturbation of the elasticity parameters about a point of weak anisotropy results in the commutator of the two types of averaging being of the order of the square of this perturbation. Thus, these averages nearly commute in the case of weak anisotropy, which is of interest in such disciplines as quantitative seismology, where the weak-anisotropy assumption results in empirically adequate models. [ABSTRACT FROM AUTHOR]

Abstract In this study, three-dimensional problem of the theory of elasticity (3DPTE) for radially inhomogeneous (INH) transversally-isotropic thin hollow spheres is investigated using the asymptotic integration method. The basic relations and equilibrium equations for radially inhomogeneous transversally-isotropic thin hollow spheres are formed and inhomogeneous solutions (INHSs) and homogeneous solutions (HSs) are constructed. The built solutions completely reveal the qualitative structure of a three-dimensional stress-strain state of radially inhomogeneous transversely-isotropic spheres of small thickness and serve as an effective apparatus for solving boundary value problems, the basis for evaluating existing applied theories and for creating new, more refined applied theories. Asymptotic formulas are obtained that allow the calculation of the three-dimensional stress-strain state of spheres. New solution groups (boundary layer solutions) have been found that are absent in applied theories. The behavior of homogeneous solutions in the inner parts of the sphere and in the vicinity of conical sections has been studied, when the thin-walled parameter of the sphere tends to zero. The nature of the constructed homogeneous solutions is clarified. On the basis of the theoretical analysis, three types of the stress-strain state (SSS) in the radially inhomogeneous transversally-isotropic hollow spheres (RINHTIHSs) are considered: a penetrating stress state, a simple edge effect, and a boundary layer. Finally numerical calculations are made and the influences of inhomogeneity on the stress distributions are investigated. Highlights • 3D problem for radially inhomogeneous transversally-isotropic spheres (RIHTIHSs) is studied. • Solution of 3D problem of elasticity theory for RIHTIHSs used asymptotic integration method. • Basic equations for thin hollow RIHTIHSs in the spherical coordinates are formed. • Stress-strain state are considered: internal stress state, simple edge effect, and boundary layer. • The influences of inhomogeneity on the stress distributions are investigated. [ABSTRACT FROM AUTHOR]

Abstract This paper presents an approach for modelling three-dimensional soft elastohydrodynamic lubrication (EHL) contact of heterogeneous materials. In this approach, a hydrodynamic interface element is developed to achieve the coupling of elastic deformation of heterogeneous contacting bodies and the steady-state hydrodynamic lubrication at the contact interface. The deformation behaviors are simulated by the finite element method, while the hydrodynamic lubrication is dealt with by solving the Reynolds equation. This approach can enable the simulation of soft EHL contact between bodies with heterogeneous materials and complicated geometric configurations. Predicted lubrication performances of soft EHL contacts show good agreement with the experimental measurements reported in literature. Analysis and discussion are also conducted on various cases of soft EHL contact between an elastic ball and a coating-substrate system with or without inhomogeneities. Highlights • An approach for modelling three-dimensional soft elastohydrodynamic lubrication contacts of heterogeneous materials is proposed. • A hydrodynamic interface element is developed to achieve the coupling of elastic deformations of heterogeneous contacting bodies and the steady-state hydrodynamic lubrication at the contact interface. • Various cases of soft EHL contact between an elastic ball and a coating-substrate system with heterogeneous materials are investigated. [ABSTRACT FROM AUTHOR]

An axisymmetric contact problem of the elasticity theory of the rigid rough-surfaced spherical punch indentation into an elastic transversely isotropic half-space with the functionally-graded transversely isotropic coating is considered. Elastic moduli of the coating vary with depth according to the arbitrary continuous or piecewise constant independent functions. The technique based on the integral transformations is used to reduce the problem to the integral equation. The punch roughness is modeled by the Fourier-Bessel series. Special approximation for the kernel transform is used to obtain the approximated analytical solution to the integral equation. The resulting solution is asymptotically exact for both small and large values of the relative thickness of the coating. A method of construction of the compliance functions is presented for the case of the simultaneous action of the arbitrary axisymmetric normal and tangential loadings.

A stress-gradient material model was recently proposed by Forest and Sab ( Mech. Res. Comm. 40 , 16–25, 2012) as an alternative to the well-known strain-gradient model introduced in the mid 60s. We propose a theoretical framework for the homogenization of stress-gradient materials. We derive suitable boundary conditions ensuring that Hill–Mandel’s lemma holds. As a first result, we show that stress-gradient materials exhibit a softening size-effect (to be defined more precisely in this paper), while strain-gradient materials exhibit a stiffening size-effect. This demonstrates that the stress-gradient and strain-gradient models are not equivalent as intuition would have it, but rather complementary. Using the solution to Eshelby’s spherical inhomogeneity problem that we derive in this paper, we propose Mori–Tanaka estimates of the effective properties of stress-gradient composites with spherical inclusions, thus opening the way to more advanced multi-scale analyses of stress-gradient materials. [ABSTRACT FROM AUTHOR]

We derive a rigorous solution to the anti-plane shear problem of a circular elastic inhomogeneity with an imperfect interface described by surface strain gradient elasticity. The loadings considered include a uniform remote stress field and a screw dislocation applied either in the matrix or in the inhomogeneity. By using analytical continuation, the problem is reduced to a third-order differential equation for a single analytic function. The analysis indicates that in the absence of the screw dislocation, the internal stress field is uniform and is reliant on two size-dependent parameters arising from surface strain gradient elasticity. The size-dependent image force acting on the screw dislocation is derived. The conditions for the existence of equilibrium positions for the dislocation in the matrix or in the inhomogeneity are obtained. Several numerical examples are presented to illustrate the solution. As an application of the solution, the stabilities of a misfit screw dislocation dipole in the matrix and a screw dislocation pair in the inhomogeneity are also discussed. [ABSTRACT FROM AUTHOR]

This paper describes the immediate settlement of uniformly loaded rough ring footings with any stiffness on an inhomogeneous finite layer overlying a rough rigid base, which is not yet covered in the literature. Numerical solutions for a wide range of geometric and material combinations are obtained by finite element method. The effects of dimensionless parameters related to footing internal opening, compressibility, footing stiffness, finite layer thickness and soil inhomogeneity are examined. Based on the results, design charts are presented in the form of settlement influence factors that can be used to calculate the immediate settlements at the inner and outer points of ring footings. [ABSTRACT FROM AUTHOR]

In this study, the coupled effects of gravity and nonhomogeneity on Rayleigh waves in an anisotropic layer placed over an isotropic viscoelastic half-space of higher order are discussed. The dispersion properties of waves are derived. The numerical solutions are also discussed for the Rayleigh waves. It is observed that the phase velocity of Rayleigh waves is influenced quite remarkably by the presence of inhomogeneity, gravity, and strain rates of the strain parameters in the half-space.

This work aims at designing a numerical strategy towards assessing the nocivity of a small defect in terms of its size and position in a structure, at low computational cost, using only a mesh of the defect-free reference structure. The modification of the fields induced by the presence of a small defect is taken into account by using asymptotic corrections of displacements or stresses. This approach helps determining the potential criticality of defects by considering trial micro-defects with varying positions, sizes and mechanical properties, taking advantage of the fact that parametric studies on defect characteristics become feasible at virtually no extra computational cost. The proposed treatment is validated and demonstrated on two numerical examples involving 2D elastic configurations. [ABSTRACT FROM AUTHOR]

This article deals with the determination of temperature distribution, displacement, and thermal stresses in a rectangular plate with inhomogeneous material properties. All the material properties except Poisson’s ratio and density are assumed to be given by a simple power law inycoordinate. The system of fundamental equations formulated has been solved by integral transform method. The effects of thermal and mechanical inhomogeneity on temperature and thermal stress distributions are examined. Numerical calculations are performed by taking into account the variation in inhomogeneous property of shear modulus of elasticity and depicted graphically. [ABSTRACT FROM AUTHOR]

The present study deals with the propagation of Love wave (a type of surface wave) in crustal layer having temperature dependent inhomogeneity. It is assumed that the inhomogeneity in the cmstal layer arises due to linear temperature variation in rigidity and density. The upper boundary of the cmstal layer is traction free. Numerical results for Love wave are discussed by plotting analytical curves between phase velocity against wave number and stress against depth in the presence of inhomogeneity and temperature parameters. The effects boundary condition on the Love wave propagation in the crustal layer is also analyzed. The results presented in this study would be useful for seismologists and geologists. [ABSTRACT FROM AUTHOR]

We study the transient elastic field induced by an edge dislocation near a nanosized circular elastic inhomogeneity in which the effects of interface slip and diffusion are incorporated into the model of deformation. Separate Gurtin-Murdoch surface elasticities are specified on the surface of the inhomogeneity and on the adjoining surface of the surrounding matrix. In addition, rate-dependent interface slip and diffusion are assumed to occur concurrently on the inhomogeneity-matrix interface. The ensuing interaction problem is solved using a simple yet effective method based on analytic continuation and a convenient decomposition of the proposed solution. In particular, our method allows us to circumvent the second-order tangential derivative taken with respect to the interfacial normal stress, typically a source of additional complication and often an obstacle to the solution of such problems. The original problem is reduced to two coupled linear algebraic equations and a number of mutually independent sets of state-space equations, the general solutions of which can be obtained by solving the associated generalized eigenvalue problem. The image force acting on the edge dislocation is derived using the Peach-Koehler formula. Corresponding stress and displacement fields as well as the image force are found to be dependent on four size-dependent dimensionless parameters (arising from the surface elasticities) and on two size-dependent parameters (having the dimension of time) arising from the incorporation of interface slip and diffusion and they evolve with an infinite number of size-dependent relaxation times. [ABSTRACT FROM AUTHOR]

This paper describes the immediate settlement of uniformly loaded rough ring footings with any stiffness on an inhomogeneous finite layer overlying a rough rigid base, which is not yet covered in the literature. Numerical solutions for a wide range of geometric and material combinations are obtained by finite element method. The effects of dimensionless parameters related to footing internal opening, compressibility, footing stiffness, finite layer thickness and soil inhomogeneity are examined. Based on the results, design charts are presented in the form of settlement influence factors that can be used to calculate the immediate settlements at the inner and outer points of ring footings.

Inverse problems of the theory of elasticity for inhomogeneous bodies, as a rule, are used to identify the dependence of the mechanical properties of the material from the coordinates. It is known that in a thick-walled cylindrical or spherical shell under internal or external pressure the highest stresses are occurred close to the internal surface of the shell. The article presents several solutions of inverse problems, which are defined dependence of the elastic modulus of the radius, at which the equivalent stress in the shell is constant. Corresponding shells can be called equal stress shells. In most cases the change of the module of elasticity of a material leads to changes in its strength. It is shown that for some materials it is possible to create model of equal strength shell in which equivalent stress in each point is equal to material strength. In the article is also considered the way of modification material's deformation properties at which beams and rods, working under combined stress state (for example, the eccentric compression of columns, the strip foundation and so forth), are also will be equal stress. [ABSTRACT FROM AUTHOR]

This paper provides a solution for stress intensity factors for a crack embedded in an elliptic inclusion. The inclusion is bonded to an infinite matrix. The inclusion and the infinite matrix have different elastic properties. The crack is replaced by a very slender elliptic configuration. After this replacement, the stress concentration factor can be evaluated. The stress intensity factor at the crack tip can be evaluated from the stress concentration factor by using an available equation. In the solution, the complex variable function method, conformal mapping and the matrix transfer method are used. [ABSTRACT FROM AUTHOR]

A thin-walled structure is homogeneously embeddable if it can be obtained by carving it out of a homogeneous material block or, in other words, if it is materially defect-free. Explicit analytic and geometric conditions for the embedded homogeneity of plane linearly elastic beams are derived and discussed. In the geometric setting, a prominent role is played by the properties of the hodograph of uniaxial material tensor fields defined on the beam axis. [ABSTRACT FROM AUTHOR]

This paper provides a general solution for the problem of multiple confocally elliptical layers in antiplane elasticity. In the problem, the elastic medium is composed of many confocally elliptical layers with different elastic properties, and it is assumed to be isotropic. The eigenstrain is placed in the inclusion. The complex potentials for the inclusion and many layers are assumed in a particular form with two undetermined coefficients. The continuity conditions for the traction and the displacement along the interfaces are formulated exactly. From those conditions, the relation between two sets for two undetermined coefficients in two adjacent layers can be evaluated. By using the above-mentioned relation, a definite relation among (1) the remote loading, (2) the stress in inclusion and (3) the eigenstrian in inclusion can be evaluated. Within the three components, two of them are independent. The proposed condition of the uniform remote stress and the eigenstrain admits an internal uniform stress field. Many computed results are provided, which give a relation among the remote loading, the eigenstrain and stress in inclusion. [ABSTRACT FROM AUTHOR]

An in-depth study has been carried out for the dispersion of Love waves in an isotropic elastic layer sandwiched between orthotropic and prestressed inhomogeneous elastic half-spaces. The inhomogeneities in density and rigidity of the lower half-space are space dependent and an arbitrary function of depth. Simple mathematical techniques are used to obtain dispersion relation for Love wave propagation in an isotropic layer. An extensive analysis is carried out through numerical computation to explore the effect of inhomogeneity and initial stress the lower half on the phase velocity of the Love waves. The numerical analysis of dispersion equation manifests that the phase velocity of the Love wave increases with the increase of stress parameter. The results further indicate that the inhomogeneity of the half space affect the wave velocity significantly. These results can be useful to study geophysical prospecting and understanding the cause and estimation of damage due to earthquakes. [ABSTRACT FROM AUTHOR]

The notion of embedded homogeneity of thin-walled structures is introduced as the property characterizing the provenance of such a structure from a homogeneous material. This property needs to be distinguished from other definitions of homogeneity formulated exclusively in terms of a purely structural constitutive equation. Necessary conditions for embedded homogeneity are derived for planar beams and their geometric interpretation is expressed as the condition for the elastic hodograph to lie on a hypersphere containing the origin of a six-dimensional space of tensors. [ABSTRACT FROM AUTHOR]

We developed a method and presented it as a Mathematica code to calculate the stress and strain fields inside and outside of two interacting ellipsoidal inhomogeneities with arbitrary orientation with respect to each other, using the Eshelby technique. The Eshelby technique can be used to determine the elastic fields in and around these inhomogeneities. Assuming same material properties for one of the inclusions and the surrounding matrix, this code can be also used for the single inhomogeneity problem. Different geological features like faults and aquifers can be modeled as inhomogeneous inclusions. We start by reviewing Eshelby's solution for a single inclusion, a single inhomogeneity and a double inhomogeneity problem with the required formulation to calculate Eshelby tensors. Then, we describe our code structure and validate it with existing solutions in the literature and present numerical solutions. [ABSTRACT FROM AUTHOR]

A simple method of estimating the effect of inhomogeneity interactions on the overall properties (elastic and conductive) is developed. It is formulated in terms of property contribution tensors that give the contribution of an inhomogeneity to the overall properties. The method can be viewed as further development of the approach of Rodin and Hwang (1991) and Rodin (1993) that generalized the method of analysis of crack interactions (Kachanov, 1987) to inhomogeneities. We also extend the method to the conductive properties. Considering the effect of interactions on the property contribution tensors on the example of pores we find that this effect is generally moderate, at most (even when pores touch one another) – in contrast with the effect on local fields. On example of two spheres, we compare the interaction effects on the elastic and the conductive properties, and discuss the impact of interactions on the cross-property connections. [ABSTRACT FROM AUTHOR]

Abstract: Many materials contain inhomogeneities or inclusions that may greatly affect their mechanical properties. Such inhomogeneities are for example encountered in the case of composite materials or materials containing precipitates. This paper presents an analysis of contact pressure and subsurface stress field for contact problems in the presence of anisotropic elastic inhomogeneities of ellipsoidal shape. Accounting for any orientation and material properties of the inhomogeneities are the major novelties of this work. The semi-analytical method proposed to solve the contact problem is based on Eshelby’s formalism and uses 2D and 3D Fast Fourier Transforms to speed up the computation. The time and memory necessary are greatly reduced in comparison with the classical finite element method. The model can be seen as an enrichment technique where the enrichment fields from the heterogeneous solution are superimposed to the homogeneous problem. The definition of complex geometries made by combination of inclusions can easily be achieved. A parametric analysis on the effect of elastic properties and geometrical features of the inhomogeneity (size, depth and orientation) is proposed. The model allows to obtain the contact pressure distribution – disturbed by the presence of inhomogeneities – as well as subsurface and matrix/inhomogeneity interface stresses. It is shown that the presence of an inclusion below the contact surface affects significantly the contact pressure and subsurfaces stress distributions when located at a depth lower than 0.7 times the contact radius. The anisotropy directions and material data are also key elements that strongly affect the elastic contact solution. In the case of normal contact between a spherical indenter and an elastic half space containing a single inhomogeneity whose center is located straight below the contact center, the normal stress at the inhomogeneity/matrix interface is mostly compressive. Finally when the axes of the ellipsoidal inclusion do not coincide with the contact problem axes, the pressure distribution is not symmetrical. [Copyright &y& Elsevier]

A particular case of the theory of growing thin-walled structures, namely the theory of transversely accreted shells is considered. A correspondence between equilibrium equations in terms of rates for transversally accreted shells and equilibrium equations in terms of displacements for permanent shells is established. The equilibrium equations for accreted shells that obey the kinematics hypothesis of Mindlin type are derived in terms of displacements as well as in terms of rates. [ABSTRACT FROM AUTHOR]

A scheme is proposed for reconstructing inhomogeneously thick elasticity modules of an orthotropic layer from acoustic sounding data. The problem of reconstruction has been reduced to stepwise reconstruction of the functions characterizing the elasticity modules, based on iteration regularization and A.N. Tikhonov’s regularization method, which utilize analysis of the averaged characteristics. A computer experiment is performed for different inhomogeneity layers, the effective frequency sounding regions for identification are revealed, and various aspects of numerical realization are discussed. [ABSTRACT FROM AUTHOR]

Abstract: The study of inclusions is of significance to the development of advanced materials for aerospace, marine, automotive and many other applications. This is because the presence of inclusions in materials affects their elastic field at the local and the global scale and thus greatly influences their mechanical and physical properties. Since the pioneering work on an ellipsoidal inclusion in an infinite space by Eshelby (1957), extensive research has been devoted to this area. This paper provides a comprehensive survey of recent works on inclusion in an infinite space, a half-space under prescribed surface loading, a half-space under surface contact loading or a finite space, and the Eshelby’s conjecture. The problems of a single inclusion, two inclusions, and multiple inclusions, dislocations and cracks as well as various methods used to address these problems are discussed. The review concludes with an outlook on future research directions. [Copyright &y& Elsevier]

Abstract: The present work aims at studying Bleustein–Gulyaev (BG) waves in a specific class of functionally graded materials. The inhomogeneity under question requires that the elastic stiffness, the piezoelectric term and the dielectric term all vary proportionally to the same inhomogeneity function. However, the density is allowed here to vary in a different manner. As a result we are able to analyse BG waves for media with variable bulk Shear Horizontal (SH) wave velocity profiles. We make use of the Liouville–Green approximation for second order Matrix Differential Equations, with explicit computable error bounds. Our analysis is limited to a certain integrability condition and to the numerical evaluation of a series of improper integrals. We present dispersion curves for BG waves for the metalized boundary condition for different inhomogeneity profiles. The results show that variations in the SH bulk wave velocities have a profound effect on the BG dispersion curves for the metalized boundary condition. [Copyright &y& Elsevier]

Abstract: In heterogeneous materials, inhomogeneities (pores, cracks, particles) usually represent a mixture of diverse shapes. In modeling the effective elastic and conductive properties, they are often replaced by identical spheroidal shapes – for which exact solutions are available – that intend to represent certain “average shape” (for cracks, for example, the circular shape is routinely used). We find that, in anisotropic cases of non-random orientations, such a replacement cannot generally be done (with the exception of flat cracks). In other words, the concept of “average spheroid’ is not legitimate, and its use may lead to large errors in predicting the effective properties – information on shape distribution is needed. In cases of overall isotropy, the replacement is possible in certain cases; even in these cases, however, the aspect ratio of the “average shape” may not be intuitively obvious. [Copyright &y& Elsevier]

Abstract: In this paper we consider an application of the Eshelby theory concerning the elastic behavior of prestrained or prestressed inhomogeneities. The theory, in its original version, deals with a configuration where both the ellipsoidal particle and the surrounding matrix are in elastostatic equilibrium if no external loads are applied to the system. Here, we consider slightly different shapes and sizes for the particle and the hosting cavity (whose surfaces are firmly bonded together) and, therefore, we observe a given state of strain (or stress) even without externally applied loads. We develop a complete procedure able to determine the uniform elastic field induced in an arbitrarily prestrained particle subjected to arbitrary remote loadings. [Copyright &y& Elsevier]

Abstract: A semi-analytic solution is given for multiple three-dimensional inhomogeneous inclusions of arbitrary shape in an isotropic half space under contact loading. The solution takes into account interactions between all the inhomogeneous inclusions as well as the interaction between the inhomogeneous inclusions and the loading indenter. In formulating the governing equations for the inhomogeneous inclusion problem, the inhomogeneous inclusions are treated as homogenous inclusions with initial eigenstrains plus unknown equivalent eigenstrains, according to Eshelby’s equivalent inclusion method. Such a treatment converts the original contact problem concerning an inhomogeneous half space into a homogeneous half-space contact problem, for which governing equations with unknown contact load distribution can be conveniently formulated. All the governing equations are solved iteratively using the Conjugate Gradient Method. The iterative process is performed until the convergence of the half-space surface displacements, which are the sum of the displacements due to the contact load and the inhomogeneous inclusions, is achieved. Finally, the obtained solution is applied to two example cases: a single inhomogeneity in a half space subjected to indentation and a stringer of inhomogeneities in an indented half-space. The validation of the solution is done by modeling a layer of film as an inhomogeneity and comparing the present solution with the analytic solution for elastic indentation of thin films. This general solution is expected to have wide applications in addressing engineering problems concerning inelastic deformation and material dissimilarity as well as contact loading. [Copyright &y& Elsevier]

Abstract: We discuss connections between the effective elastic properties of a solid with inhomogeneities and the far-field asymptotics of the elastic fields generated by them. We focus on the dependence of the far-field asymptotics on the inhomogeneity shape. This shape dependence in the inhomogeneity problem is in contrast with shape independence of the far field in the eigenstrain problem. For the latter, the far field applies to inclusions of any shape. We show that the external fields in the eigenstrain – and the inhomogeneity problems are interrelated by the same tensor that characterizes the compliance contribution of an inhomogeneity. [Copyright &y& Elsevier]

a preliminary study to more complex situations of interest in small-scale technology, this paper envisages the elementary propagation properties of elastic waves in one-spatial dimension when some of the properties (mass density, elasticity) may vary suddenly in space or in time, the second case being of course more original. Combination of the two may be of even greater interest. Toward this goal, a critical examination of what happens to solutions at the crossing of pure space-like and time-like material discontinuities is given together with simple solutions for smooth transitions and numerical simulations in the discontinuous case. The effects on amplitude, speed of propagation, frequency changes and the appearance of a Doppler-like effect are demonstrated although the whole physical system remains linear. [ABSTRACT FROM AUTHOR]

We consider a problem on an ellipsoidal inhomogeneity in an infinitely extended homogeneous isotropic elastic medium. The inhomogeneity differs from the ambient body in the elastic moduli (Poisson's ratio ν and shear modulus µ) and in that it has intrinsic strains. We use the equivalent inclusion method to write out expressions for the Helmholtz and Gibbs free energy of the inhomogeneity as quadratic forms in the intrinsic strains and strains at infinity. The general expressions for the coefficients of these quadratic forms are written out as three rank four tensors characterizing the contribution to the energy by the plastic strain ( ɛ), by the strain at infinity ( ɛ), and (only for the Gibbs energy) by the cross term ɛ ɛ. The solutions for some important particular cases are written out by asymptotically expanding the general solution in small parameters. The parameters determining the type of the asymptotic behavior are the inclusion semiaxis ratio and the inclusion-to-matrix elastic modulus ratio. We consider various physical situations arising when studying needle- and disk-shaped inclusions and characterized by various order of passage to the limit. We analyze the applicability domains of the asymptotics by reduction to a single small parameter. For the case in which the parameters considerably differ from unity simultaneously, we single out seven domains corresponding to various relations between these parameters. We construct the diagrams of relative error isolines for the resulting tensors corresponding to the energy contribution from various strain tensor components. [ABSTRACT FROM AUTHOR]

Abstract: Bone quality is significantly correlated with the inhomogeneous distribution of material and ultrastructural properties (e.g., modulus and mineralization) of the tissue. Current techniques for quantifying inhomogeneity consist of descriptive statistics such as mean, standard deviation and coefficient of variation. However, these parameters do not describe the spatial variations of bone properties. The objective of this study was to develop a novel statistical method to characterize and quantitatively describe the spatial variation of bone properties at ultrastructural levels. To do so, a random field defined by an exponential covariance function was used to represent the spatial uncertainty of elastic modulus by delineating the correlation of the modulus at different locations in bone lamellae. The correlation length, a characteristic parameter of the covariance function, was employed to estimate the fluctuation of the elastic modulus in the random field. Using this approach, two distribution maps of the elastic modulus within bone lamellae were generated using simulation and compared with those obtained experimentally by a combination of atomic force microscopy and nanoindentation techniques. The simulation-generated maps of elastic modulus were in close agreement with the experimental ones, thus validating the random field approach in defining the inhomogeneity of elastic modulus in lamellae of bone. Indeed, generation of such random fields will facilitate multi-scale modeling of bone in more pragmatic details. [ABSTRACT FROM AUTHOR]

By passing to the limit in the general solution, we obtain several exact asymptotic formulas for the energy of an inhomogeneity in a body located in an external stress field. The parameters determining the type of the asymptotic behavior are the inclusion semiaxis ratio and the inclusion-matrix elastic modulus ratio. In the case of a large simultaneous deviation of the parameters from unity, we distinguish seven regions corresponding to various successive passages to the limit as these parameters approach zero (or infinity). [ABSTRACT FROM AUTHOR]

The heat dipole consists of a heat source and a heat sink. The problem of an interfacial crack of a composite containing a circular inclusion under a heat dipole is investigated by using the analytical extension technique, the generalized Liouville theorem, and the Muskhelishvili boundary value theory. Temperature and stress fields are formulated. The effects of the temperature field and the inhomogeneity on the interfacial fracture are analyzed. As a numerical illustration, the thermal stress intensity factors of the interfacial crack are presented for various material combinations and different positions of the heat dipole. The characteristics of the interfacial crack depend on the elasticity, the thermal property of the composite, and the condition of the dipole. [ABSTRACT FROM AUTHOR]

Abstract: We use molecular statics simulations with the embedded atom method potential to delineate yielding (material instability) and buckling (structural instability) in gold nanowires deformed axially in compression. It is found that both local (stacking faults) and global instabilities occur when the gold nanowire yields but only global instabilities occur when the nanowire buckles. Furthermore strong surface effects reorient the lattice structure which significantly increases Young’s modulus in the axial direction and cause a nanowire of relatively small slenderness ratio (e.g., 14) to buckle. Upon complete unloading of the nanowires, the average axial stress and the total potential energy revert to their values in the reference configuration for the nanowires that buckled but not for the one that yielded. [Copyright &y& Elsevier]

Abstract: The stress and electric fields in electrostrictive materials under general electric loading at infinity are obtained in this paper. It is shown that the pseudo total stresses are continuous in the whole body. The elliptic inhomogeneity problem is first discussed in this paper and its solution is also given. The results show that the stress in the inhomogeneity is not uniform which is different from the solution of Eshelby theory for elastic materials. When the inhomogeneity and matrix have the same dielectric permittivity or the matrix is a non-electrostrictive material, the stress field is uniform in the inhomogeneity. The form of stress function is simple when the inhomogeneity degenerates to a circle. [Copyright &y& Elsevier]

Abstract: This article is concerned with the relaxed Saint-Venant’s problem in the case when the body forces and surface tractions on the lateral surface are polynomials in the axial coordinate. A new method of solving this problem is presented. The method is applied to study the problem of a uniformly loaded cylinder. [Copyright &y& Elsevier]

Abstract: The theory of material inhomogeneities, according to Noll’s approach, is revisited in detail and its interpretations in the recent literature on material anelastic behaviors, growth and phase transition phenomena are critically discussed. [Copyright &y& Elsevier]

Abstract: Network microstructures of poly(acrylamide) (PAAm) and poly(N,N-dimethylacrylamide) (PDMA) hydrogels were compared by static light scattering and elasticity measurements. The hydrogels were prepared by free-radical crosslinking copolymerization of the monomers acrylamide (AAm) or N,N-dimethylacrylamide (DMA) with N,N′-methylenebis(acrylamide) as a crosslinker. During the formation of PAAm gels, the reaction time dependence of the scattered light intensity exhibits a maximum at a critical reaction time, while in case of PDMA gels, both a maximum and a minimum were observed, corresponding to the chain overlap threshold and the gel point, respectively. This difference in the time-course between the two gelling systems is due to the late onset of gelation in the DMA system with respect to the critical overlap concentration. Compared to the AAm system, no significant scattered light intensity rise was observed during the crosslinking polymerization of DMA. It was shown that, regardless of the crosslinker ratio and of the initial monomer concentration, PDMA gel is much more homogeneous than the corresponding PAAm gel due to the shift of the gelation threshold to the semidilute regime of the reaction system. The results suggest that the spatial gel inhomogeneity can be controlled by varying the gel point with respect to the critical overlap concentration during the preparation of gels by free-radical mechanism. [Copyright &y& Elsevier]