The Newtonian potential inhomogeneity problem: non-uniform eigenstrains in cylinders of non-elliptical cross section

Understanding the fields that are set up in and around inhomogeneities is of great importance in order to predict the manner in which heterogeneous media behave when subjected to applied loads or other fields, e.g., magnetic, electric, thermal, etc. The classical inhomogeneity problem of an ellipsoid embedded in an unbounded host or matrix medium has long been studied but is perhaps most associated with the name of Eshelby due to his seminal work in 1957, where in the context of the linear elasticity problem, he showed that for imposed far fields that correspond to uniform strains, the strain field induced inside the ellipsoid is also uniform. In Eshelby’s language, this corresponds to requiring a uniform eigenstrain in order to account for the presence of the ellipsoidal inhomogeneity, and the so-called Eshelby tensor arises, which is also uniform for ellipsoids. Since then, the Eshelby tensor has been determined by many authors for inhomogeneities of various shapes, but almost always for the case of uniform eigenstrains. In many application areas in fact, the case of non-uniform eigenstrains is of more physical significance, particularly when the inhomogeneity is non-ellipsoidal. In this article, a method is introduced, which approximates the Eshelby tensor for a variety of shaped inhomogeneities in the case of more complex eigenstrains by employing local polynomial expansions of both the eigenstrain and the resulting Eshelby tensor, in the case of the potential problem in two dimensions.     

Non-uniform eigenstrain induced anti-plane stress field in an elliptic inhomogeneity embedded in anisotropic media with a single plane of symmetry

A closed-form solution is derived for an anti-plane stress field emanating from non-uniform eigenstrains in an elliptic anisotropic inhomogeneity embedded in anisotropic media with one elastic plane of symmetry. The prescribed eigenstrains are characterized by linear functions of the inhomogeneity in Cartesian coordinates. By means of the polynomial conservation theorem, use of complex function method and conformal transformation, explicit expressions for stresses at the interior boundary of the matrix and the strain energy for the elastic inhomogeneity/matrix system are obtained in terms of coefficients in the linear functions. The coefficients are evaluated analytically using the principle of minimum potential energy of the elastic system, leading to the anti-plane stress field. The resulting solution is verified by means of the continuity condition for the shear stress at the interface between the elliptic inhomogeneity and matrix. The present solution is shown to reduce to known results for uniform eigenstrains with illustration by numerical examples.     

Determination of the scattered fields of an SH-wave by an eccentric coating-fiber ensemble using DEIM

To date, the existing theories pertinent to the determination of the scattered fields of an inhomogeneity have been limited to certain topological symmetries for which the method of wave-function expansion is widely used. In the literature the wave-function expansion method has also been employed to the case involving concentric coated fiber. An alternative approach is the dynamic equivalent inclusion method (DEIM) proposed by Fu and Mura [L.S. Fu, T. Mura, The determination of elastodynamic fields of an ellipsoidal inhomogeneity. ASME J. Appl. Mech. 50 (1983) 390-396.] who found the scattered field of a single spheroidal inhomogeneity. The pioneering work of Eshelby [J.D. Eshelby, The determination of the elastic field of an ellipsoidal inclusion, and related problems, Proc. R. Soc. London, Ser. A A241 (1957) 376-396.] on elastostatic EIM is based on polynomial form of eigenstrains which holds certain useful properties and subsequently its application is only effective for certain relevant situations and not necessarily efficient for other problems. Nevertheless, Fu and Mura’s analysis is also based on polynomial eigenstrains. It will be shown that taking the dynamic homogenizing eigenstrains in the form of the series expansion whose general term is products of functions of r and trigonometric functions of θ, is more rigorous and attractive for the problem under consideration. This natural form of solution gives very accurate result with just the first few terms of the series. Moreover, this work aims to extend the DEIM to the case of coated fiber obstacle with the rather complex topology where the coating-fiber phases are not concentric. The effect of variableness of the coating thickness on the elastodynamic fields is examined. Comparison with other analytical solutions, whenever available, establishes the remarkable accuracy and robustness of the proposed theory.     

Effects of interfacial excess energy on the elastic field of a nano-inhomogeneity

A semi-analytical approach based on a variational framework is developed to obtain the three-dimensional solution for a nano-scale inhomogeneity with arbitrary eigenstrains embedded in a matrix of infinite extent. Both the inhomogeneity and the matrix can be elastically anisotropic. The Gurtin–Murdoch surface/interface model is used to describe the elastic behavior of the inhomogeneity/matrix interface. The displacement fields in the inhomogeneity and the matrix are represented, respectively, by two sets of polynomials. Coefficients of these polynomials are determined by solving a system of linear algebraic equations that are derived from minimizing the total potential energy of the system. In the case of an isotropic spherical inhomogeneity with dilatational eigenstrain in an isotropic matrix, our solution shows an excellent agreement with the corresponding analytical solution available in the literature. To demonstrate the capabilities of the method developed here and to investigate the effect of interfacial excess energy, numerical examples are also presented when the inhomogeneity and matrix are both elastically anisotropic. Both dilatational and pure shear eigenstrains are considered in these examples.     

Implementation of the transformation field analysis for inelastic composite materials

Conclusions The transformation field analysis is a general method for solving inelastic deformation and other incremental problems in heterogeneous media with many interacting inhomogeneities. The various unit cell models, or the corrected inelastic self-consistent or Mori-Tanaka fomulations, the so-called Eshelby method, and the Eshelby tensor itself are all seen as special cases of this more general approach. The method easily accommodates any uniform overall loading path, inelastic constitutive equation and micromechanical model. The model geometries are incorporated through the mechanical transformation influence functions or concentration factor tensors which are derived from elastic solutions for the chosen model and phase elastic moduli. Thus, there is no need to solve inelastic boundary value or inclusion problems, indeed such solutions are typically associated with erroneous procedures that violate (6₂); this was discussed by Dvorak (1992). In comparison with the finite element method in unit cell model solutions, the present method is more efficient for cruder mesches. Moreover, there is no need to implement inelastic constitutive equations into a finite element program. An addition to the examples shown herein, the method can be applied to many other problems, such as those arising in active materials with eigenstrains induced by components made of shape memory alloys or other actuators. Progress has also been made in applications to electroelastic composites, and to problems involving damage development in multiphase solids. Finally, there is no conceptural obstacle to extending the approach beyond the analysis of representative volumes of composite materials, to arbitrarily loaded structures.This work was supported by the Air Force Office of Scienctific Research, and by the Office of Naval Research     

Elastoplastic behavior of metal matrix composites based on incremental plasticity and the Mori-Tanaka averaging scheme

The applicability of the Mori-Tanaka averaging method for the prediction of the response of binary composites loaded in the plastic range is investigated. The applied loading is subdivided into small increments and the Eshelby solution for the inhomogeneity problem is used in conjunction with the Mori-Tanaka averaging scheme to obtain the load increments in the various phases. Since the Eshelby solution depends on the instantaneous matrix material properties and these are updated at the end of each load increment by using the backward difference scheme, an iterative procedure is necessary for the calculation of the correct load increments in the phases (concentration factors). The performance of the Mori-Tanaka method is compared with results obtained using the periodic hexagonal array (PHA) finite element model and experimental results for a B-Al unidirectional fibrous composite; it is also compared with numerical simulations obtained from the modified PHA model for a SiC _w -Al particulate composite.     

Exact solution of Eshelby–Christensen problem in gradient elasticity for composites with spherical inclusions

The gradient elasticity theory is employed to solve exactly the problem of Eshelby–Christensen for filled composites with spherical inclusions across length scales. Relying on the fundamental symmetry considerations and using Lagrange’s variational formalism, we derive the governing relations of linear isotropic gradient elasticity. We demonstrate that to avoid spurious solutions, one should necessarily impose some additional symmetry restrictions on the operational strain gradient elastic constants that can be considered as a new correctness condition. By enforcing the “strain gradient” symmetry condition, we offer the variant of the correct applied one-parametric gradient theory of interfacial layer model. To solve the Eshelby–Christensen problem, we employ the generalized Eshelby’s integral representations for the gradient elasticity models that allow to formulate the closing equations in a self-consistent three-phase method, and we also use the generalized Papkovich–Neuber representation to determine the general form of the gradient solution and the structure of the scale effects. As a result, we obtained for the first time an exact solution of Eshelby–Christensen problem for composites reinforced with spherical inclusions in framework of the gradient interfacial layer model. There are known analogs of fundamental results for gradient models related to closed solution for composites with spherical inclusions obtained by R.M. Christensen and K.H. Lo in 1976. The obtained analytical solution of Eshelby–Christensen problem for correct gradient theory is used to determine the stress–strain state and the effective properties of dispersed composites. The analysis of the effect of scale factors is given; the error associated with the use of gradient theories that do not obey the proposed condition of correctness is estimated.     

On self-similarly expanding Eshelby inclusions: Spherical inclusion with dilatational eigenstrain

For a subsonically self-similarly expanding spherical inclusion with dilatational transformation strain in a linear elastic solid, the governing system of partial differential equations is shown to be elliptic under scaling of uniform stretching of the variables, and the resulting elliptic equation is solved by satisfying the Hadamard jump conditions on the moving boundary. The solution has the Eshelby constant stress property for the interior domain, and can thus be used for the expanding inhomogeneity with transformation strain according to Eshelby (1957). The driving force on the moving boundary is also obtained.     

Homogenization of fibrous piezoelectric composites with general imperfect interfaces under anti-plane mechanical and in-plane electrical loadings

The present paper aims mainly to estimate the size-dependent effective properties of fibrous piezoelectric composites with general imperfect interfaces. The interface model used states that the displacement, traction, electric potential, and normal electric displacement all suffer jumps across an interface. In addition, it can degenerate into the well-known special ones by employing appropriate high-contrast interfacial parameters. To achieve our objective, an auxiliary inhomogeneity problem of a circular fiber embedded in an infinite cylindrical reference phase via general imperfect interface under anti-plane mechanical and in-plane electrical boundary conditions is analytically solved. This solution allows us to apply the well-known micromechanical schemes such as the dilute, Mori–Tanaka to obtain the closed-form expressions for the size-dependent overall properties of composites under consideration. Some numerical examples are provided for illustrating the features of the obtained general results.     

Micromechanics and effective elastic moduli of particle-reinforced composites with near-field particle interactions

A micromechanical framework is proposed to predict effective elastic moduli of particle-reinforced composites. First, the interacting eigenstrain is derived by making use of the exterior-point Eshelby tensor and the equivalence principle associated with the pairwise particle interactions. Then, the near-field particle interactions are accounted for in the effective elastic moduli of spherical-particle-reinforced composites. On the foundation of the proposed interacting solution, the consistent versus simplified micromechanical field equations are systematically presented and discussed. Specifically, the focus is upon the effective elastic moduli of two-phase composites containing randomly distributed isotropic spherical particles. To demonstrate the predictive capability of the proposed micromechanical framework, comparisons between the theoretical predictions and the available experimental data on effective elastic moduli are rendered. In contrast to higher-order formulations in the literature, the proposed micromechanical formulation can accommodate the anisotropy of reinforcing particles and can be readily extended to multi-phase composites.     

Effective porothermoelastic properties of transversely isotropic rock-like composites

The present work is devoted to the determination of the overall porothermoelastic properties of transversely isotropic rock-like composites with transversely isotropic matrix and randomly oriented ellipsoidal inhomogeneities and/or pores. By using the solution of a single ellipsoidal inhomogeneity arbitrarily oriented in a transversely isotropic matrix presented by Giraud et al. [A. Giraud, Q.V. Huynh, D. Hoxha, D. Kondo, Effective poroelastic properties of transversely isotropic rocks-like composites with arbitrarily oriented ellipsoidal inclusions, Mechanics of Materials 39 (11) (2007) 1006-1024], it is possible to observe the effect of the shape and orientation distribution of inhomogeneities on the effective porothermoelastic properties. Based on recent works on porous rock-like composites such as shales or argillites, an application of the developed solution to a two-level microporomechanics model is presented. The microporosity is homogenized at the first level, and multiple solid mineral phase inclusions are added at the second level. The overall porothermoelastic coefficients are estimated in the particular context of heterogeneous solid matrix. The present model generalizes to transversely isotropic media a recently developed two-level model in the simpler case of isotropic media (see Giraud et al. [A. Giraud, D. Hoxha, D.P. Do, V. Magnenet, Effect of pore shape on effective thermoporoelastic properties of isotropic rocks, International Journal of Solids and Structures 45 (2008) 1-23]). Numerical results are presented for data representative of transversely isotropic rock-like composites.     

Multiple 3D inhomogeneous inclusions in a half space under contact loading

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.     

Progressive failure of triaxial woven fabric (TWF) composites with open holes

A computational approach to the investigation of crack evolution and interaction effects of microcracks and particles on the overall behavior of particle-reinforced brittle composites (PRBCs) is presented. To account for interactions of microcracks and particles, and their effects on the overall mechanical behavior, approximate solutions of a micromechanical model considering second-order, ensemble-volume averaged perturbations are employed. By combining the micromechanical framework with a fracture-mechanics based damage model, an evolutionary damage model of PRBCs is subsequently developed and the evolutionary damage model is implemented into a finite element code. The proposed computational damage model is exercised from benchmark examples on PRBCs to illustrate the capability of the proposed damage models for predicting the progressive crack evolution in PRBCs.     

A micromechanical model for linear homogenization of brick masonry

A micromechanical model, originally developed for long-fiber composites, is applied to determination of the overall linear-elastic mechanical properties of simple-texture brick masonry. The model relies upon exact solution after Eshelby and describes brickwork as a mortar matrix with insertions of elliptic cylinder-shaped bricks. The macroscopic elastic constants are derived from the mechanical properties of the constituent materials and the phase volume ratios. The ability of the suggested model to predict the behavior of real brickwork has been checked by performing uniaxial compression tests on brick masonry panels of two types, with cement mortar and lime mortar. The results obtained through the proposed model fit experimental data more closely than other models selected from the literature for the sake of comparison.     

Evaluating microstructural and damage effects in rule-of-mixtures predictions of the mechanical properties of Ni-Al2O3 composites

Ni-Al₂O₃ composites containing 0 to 100 vol % Ni were fabricated using powder processing techniques. By varying the metal : ceramic particle size ratio, either particle-reinforced or interpenetrating-phase microstructures were obtained. The mechanical properties of the composites were characterized and compared with rule-of-mixtures (ROM) predictions. For certain particle-reinforced composites, the elastic moduli measured ultrasonically did not obey the ROM. This result was attributed to the presence of damage that could be accounted for using existing models. In four-point bending, most composites exhibited linear elastic behavior, however significant inelastic deformation was observed for composites containing 60 and 80 vol % Ni. The inelastic deformation was reasonably well described using ROM models, except when substantial damage was present. Damaged materials were modeled as two phase composites containing one damage-free phase and one completely damaged phase that was assumed to behave like a porous material. The failure strains of composites with continuous ceramic phases were explained using a semi-empirical model that included both damage and residual stress effects. Fracture stresses were calculated from predicted fracture strains using a new ROM deformation model. The model was modified to include constraint effects in order to accurately describe the deformation behavior of the ductile continuous-ceramic composites.     

Mean-field based micro-mechanical modelling of short wavy fiber reinforced composites

Eshelby based Mean-Field homogenization is an effective method for modelling the mechanical response of short fiber reinforced composites. These models and especially the Mori–Tanaka model have been successfully used in previous studies for predicting the overall composite mechanical response. The present work describes a method for extending Mean-Field methods to discontinuous wavy fiber reinforced composites, including calculating local stresses in the fibers. The method involves discretization of a wavy fiber into smaller segments and replacing the original segments with an equivalent ellipsoids system which can be solved with Eshelby concept. The focus of this work is the validation of the local stress fields in fibers using Finite Element benchmarks of original Volume Elements (VEs) of wavy fibers. This validation is an essential basis for further accurate modelling of the damage behavior (i.e. debonding, fiber fracture) of discontinuous wavy fiber composites.     

Modeling of the contact between a rigid indenter and a heterogeneous viscoelastic material

In this paper, the contact problem between a rigid indenter and a viscoelastic half space containing either isotropic or anisotropic elastic inhomogeneities is solved. The model presented here is 3D and based on semi-analytical methods. To take into account the viscoelastic properties of the matrix, contact and subsurface problem equations are discretized in the spatial and temporal dimensions. A conjugate gradient method and the fast Fourier transform are used to solve the normal problem, contact pressure, subsurface problem and real contact area simultaneously. The Eshelby’s formalism is applied at each step of the temporal discretization to account for the effect of the inhomogeneity on pressure distribution and subsurface stresses. This method can be seen as an enrichment technique where the enrichment fields from heterogeneous solutions are superimposed to the homogeneous viscoelastic problem solution. Note that both problems are fully coupled. The model is validated by comparison with a Finite Element Model. A parametric analysis of the effect of elastic properties and geometrical features of the inhomogeneity 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 at every step of the temporal discretization.     

Stress in particulate reinforcements and overall stress response on aluminum alloy matrix composites during straining by analytical and numerical modeling

SiC and Al₂O₃ (10–20v%) particle-reinforced Al-2618 matrix composites subjected to tensile loading were selected to simulate stress–strain curves and average stress in particles, and to examine mechanical properties experimentally in comparison. A particle-compounded mechanical model was established based on Eshelby equivalent inclusion approach to simulate stress–strain curves by introducing secant modulus and tangent modulus techniques, and to calculate stress in particles and in matrices. The same modeling work was carried out by FEM analysis based on the unit cell model using a commercial ANSYS code. The modeling and experiment were also applied to compare the mechanical behaviors between hard matrix and soft matrix, which were produced under different heat treatments. Through the comparison of the results between simulations and experiment, it is shown that Eshelby particle-compounded mechanical model can predict the stress–strain curve of the composites with both hard matrix and soft matrix, while the FEM model can match the experimental data with only hard matrix. The modeling was also carried out to study the influence of different volume fractions and aspect ratios on elastic modulus and yield strength of the composites with different reinforcing particles to get a better understanding of strengthening mechanisms of the composites.     

A new homogenization method based on a simplified strain gradient elasticity theory

Homogenization methods utilizing classical elasticity-based Eshelby tensors cannot capture the particle size effect experimentally observed in particle–matrix composites at the micron and nanometer scales. In this paper, a new homogenization method for predicting effective elastic properties of multiphase composites is developed using Eshelby tensors based on a simplified strain gradient elasticity theory (SSGET), which contains a material length scale parameter and can account for the size effect. Based on the strain energy equivalence, a homogeneous comparison material obeying the SSGET is constructed, and two sets of equations for determining an effective elastic stiffness tensor and an effective material length scale parameter for the composite are derived. By using Eshelby’s eigenstrain method and the Mori–Tanaka averaging scheme, the effective stiffness tensor based on the SSGET is analytically obtained, which depends not only on the volume fractions and shapes of the inhomogeneities (i.e., phases other than the matrix) but also on the inhomogeneity sizes, unlike what is predicted by the existing homogenization methods based on classical elasticity. To illustrate the newly developed homogenization method, sample cases are quantitatively studied for a two-phase composite filled with spherical, cylindrical, or ellipsoidal inhomogeneities (particles) using the averaged Eshelby tensors based on the SSGET that were derived earlier by the authors. Numerical results reveal that the particle size has a large influence on the effective Young’s moduli when the particles are sufficiently small. In addition, the results show that the composite becomes stiffer when the particles get smaller, thereby capturing the particle size effect.     

Micromechanical effective elastic moduli of continuous fiber-reinforced composites with near-field fiber interactions

A higher-order micromechanical framework is presented to predict the overall elastic deformation behavior of continuous fiber-reinforced composites with high-volume fractions and random-fiber distributions. By taking advantage of the probabilistic pair-wise near-field interaction solution, the interacting eigenstrain is analytically derived. Subsequently, by making use of the Eshelby equivalence principle, the perturbed strain within a continuous circular fiber is accounted for. Further, based on the general micromechanical field equations, effective elastic moduli of continuous fiber-reinforced composites are constructed. An advantage of the present framework is that the higher-order effective elastic moduli of composites can be analytically predicted with relative simplicity, requiring only material properties of the matrix and fibers, the fiber?Cvolume fraction and the microstructural parameter ??. Moreover, no Monte Carlo simulation is needed for the proposed methodology. A series of comparisons between the analytical predictions and the available experimental data for isotropic and anisotropic fiber reinforced composites illustrate the predictive capability of the proposed framework.