Strain gradient solution for a finite-domain Eshelby-type plane strain inclusion problem and Eshelby’s tensor for a cylindrical inclusion in a finite elastic matrix

A solution for the finite-domain Eshelby-type inclusion problem of a finite elastic body containing a plane strain inclusion prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is derived in a general form using a simplified strain gradient elasticity theory (SSGET). The formulation is facilitated by an extended Betti’s reciprocal theorem and an extended Somigliana’s identity based on the SSGET and suitable for plane strain problems. The disturbed displacement field is obtained in terms of the SSGET-based Green’s function for an infinite plane strain elastic body, which differs from that in earlier studies using the three-dimensional Green’s function. The solution reduces to that of the infinite-domain inclusion problem when the boundary effect is suppressed. The problem of a cylindrical inclusion embedded concentrically in a finite plane strain cylindrical elastic matrix of an enhanced continuum is analytically solved for the first time by applying the general solution, with the Eshelby tensor and its average over the circular cross section of the inclusion obtained in closed forms. This Eshelby tensor, being dependent on the position, inclusion size, matrix size, and a material length scale parameter, captures the inclusion size and boundary effects, unlike existing ones. It reduces to the classical elasticity-based Eshelby tensor for the cylindrical inclusion in an infinite matrix if both the strain gradient and boundary effects are not considered. Numerical results quantitatively show that the inclusion size effect can be quite large when the inclusion is very small and that the boundary effect can dominate when the inclusion volume fraction is very high. However, the inclusion size effect is diminishing with the increase of the inclusion size, and the boundary effect is vanishing as the inclusion volume fraction becomes sufficiently low.     

Solution of Eshelby's inclusion problem with a bounded domain and Eshelby's tensor for a spherical inclusion in a finite spherical matrix based on a simplified strain gradient elasticity theory

A solution for Eshelby's inclusion problem of a finite homogeneous isotropic elastic body containing an inclusion prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is derived in a general form using a simplified strain gradient elasticity theory (SSGET). An extended Betti's reciprocal theorem and an extended Somigliana's identity based on the SSGET are proposed and utilized to solve the finite-domain inclusion problem. The solution for the disturbed displacement field is expressed in terms of the Green's function for an infinite three-dimensional elastic body in the SSGET. It contains a volume integral term and a surface integral term. The former is the same as that for the infinite-domain inclusion problem based on the SSGET, while the latter represents the boundary effect. The solution reduces to that of the infinite-domain inclusion problem when the boundary effect is not considered. The problem of a spherical inclusion embedded concentrically in a finite spherical elastic body is analytically solved by applying the general solution, with the Eshelby tensor and its volume average obtained in closed forms. This Eshelby tensor depends on the position, inclusion size, matrix size, and material length scale parameter, and, as a result, can capture the inclusion size and boundary effects, unlike existing Eshelby tensors. It reduces to the classical Eshelby tensor for the spherical inclusion in an infinite matrix if both the strain gradient and boundary effects are suppressed. Numerical results quantitatively show that the inclusion size effect can be quite large when the inclusion is very small and that the boundary effect can dominate when the inclusion volume fraction is very high. However, the inclusion size effect is diminishing as the inclusion becomes large enough, and the boundary effect is vanishing as the inclusion volume fraction gets sufficiently low.     

Circular inclusions in anti-plane strain couple stress elasticity

The solution for a circular inclusion with a prescribed anti-plane eigenstrain is derived. It is shown that the components of the Eshelby tensor within the inclusion, corresponding to a uniform eigenstrain, can be either uniform or non-uniform, depending on the imposed interface conditions. The stress amplification factors due to circular void or rigid inclusion in an infinite medium under remote anti-plane shear stress are calculated. The failure of the couple stress elasticity to reproduce the classical elasticity solution in the limit of vanishingly small characteristic length is indicated for a particular type of boundary conditions. The solution for a circular inhomogeneity in an infinitely extended matrix subjected to remote shear stress is then derived. The effects of the imposed interface conditions, the shear stress and couple stress discontinuities, and the relationship between the inhomogeneity and its equivalent eigenstrain inclusion problem are discussed.     

Strain gradient solution for the Eshelby-type polyhedral inclusion problem

The Eshelby-type problem of an arbitrary-shape polyhedral inclusion embedded in an infinite homogeneous isotropic elastic material is analytically solved using a simplified strain gradient elasticity theory (SSGET) that contains a material length scale parameter. The Eshelby tensor for a polyhedral inclusion of arbitrary shape is obtained in a general analytical form in terms of three potential functions, two of which are the same as the ones involved in the counterpart Eshelby tensor based on classical elasticity. These potential functions, as volume integrals over the polyhedral inclusion, are evaluated by dividing the polyhedral inclusion domain into tetrahedral duplexes, with each duplex and the associated local coordinate system constructed using a procedure similar to that employed by Rodin (1996. J. Mech. Phys. Solids 44, 1977–1995). Each of the three volume integrals is first transformed to a surface integral by applying the divergence theorem, which is then transformed to a contour (line) integral based on Stokes' theorem and using an inverse approach different from those adopted in the existing studies based on classical elasticity. The newly derived SSGET-based Eshelby tensor is separated into a classical part and a gradient part. The former contains Poisson's ratio only, while the latter includes the material length scale parameter additionally, thereby enabling the interpretation of the inclusion size effect. This SSGET-based Eshelby tensor reduces to that based on classical elasticity when the strain gradient effect is not considered. For homogenization applications, the volume average of the new Eshelby tensor over the polyhedral inclusion is also provided in a general form. To illustrate the newly obtained Eshelby tensor and its volume average, three types of polyhedral inclusions – cubic, octahedral and tetrakaidecahedral – are quantitatively studied by directly using the general formulas derived. The numerical results show that the components of the SSGET-based Eshelby tensor for each of the three inclusion shapes vary with both the position and the inclusion size, while their counterparts based on classical elasticity only change with the position. It is found that when the inclusion is small, the contribution of the gradient part is significantly large and should not be neglected. It is also observed that the components of the averaged Eshelby tensor based on the SSGET change with the inclusion size: the smaller the inclusion, the smaller the components. When the inclusion size becomes sufficiently large, these components are seen to approach (from below) the values of their classical elasticity-based counterparts, which are constants independent of the inclusion size.     

Eshelby's problem of non-elliptical inclusions

The Eshelby problem consists in determining the strain field of an infinite linearly elastic homogeneous medium due to a uniform eigenstrain prescribed over a subdomain, called inclusion, of the medium. The salient feature of Eshelby's solution for an ellipsoidal inclusion is that the strain tensor field inside the latter is uniform. This uniformity has the important consequence that the solution to the fundamental problem of determination of the strain field in an infinite linearly elastic homogeneous medium containing an embedded ellipsoidal inhomogeneity and subjected to remote uniform loading can be readily deduced from Eshelby's solution for an ellipsoidal inclusion upon imposing appropriate uniform eigenstrains. Based on this result, most of the existing micromechanics schemes dedicated to estimating the effective properties of inhomogeneous materials have been nevertheless applied to a number of materials of practical interest where inhomogeneities are in reality non-ellipsoidal. Aiming to examine the validity of the ellipsoidal approximation of inhomogeneities underlying various micromechanics schemes, we first derive a new boundary integral expression for calculating Eshelby's tensor field (ETF) in the context of two-dimensional isotropic elasticity. The simple and compact structure of the new boundary integral expression leads us to obtain the explicit expressions of ETF and its average for a wide variety of non-elliptical inclusions including arbitrary polygonal ones and those characterized by the finite Laurent series. In light of these new analytical results, we show that: (i) the elliptical approximation to the average of ETF is valid for a convex non-elliptical inclusion but becomes inacceptable for a non-convex non-elliptical inclusion; (ii) in general, the Eshelby tensor field inside a non-elliptical inclusion is quite non-uniform and cannot be replaced by its average; (iii) the substitution of the generalized Eshelby tensor involved in various micromechanics schemes by the average Eshelby tensor for non-elliptical inhomogeneities is in general inadmissible.     

Some general properties of Eshelby’s tensor fields in transport phenomena and anti-plane elasticity

Consider an infinite thermally conductive medium characterized by Fourier’s law, in which a subdomain, called an inclusion, is subjected to a prescribed uniform heat flux-free temperature gradient. The second-order tensor field relating the gradient of the resulting temperature field over the medium to the uniform heat flux-free temperature gradient is referred to as Eshelby’s tensor field for conduction. The present work aims at deriving the general properties of Eshelby’s tensor field for conduction. It is found that: (i) the trace of Eshelby’s tensor field is equal to the characteristic function of the inclusion, independently of the latter’s shape; (ii) the isotropic part of Eshelby’s tensor field over the inclusion of arbitrary shape is identical to Eshelby’s tensor field over a 2D circular or 3D spherical inclusion; (iii) when the medium is made of an isotropic material and when the inclusion has some specific rotational symmetries, the value of the Eshelby’s tensor field evaluated at the inclusion gravity center and the symmetric average of Eshelby’s tensor fields are both equal to Eshelby’s tensor field for a 2D circular or 3D spherical inclusion. These results are then extended, with the help of a linear transformation, to the general case where the medium consists of an anisotropic conductive material. The method elaborated and results obtained by the present work are directly transposable to the physically analogous transport phenomena of electric conduction, dielectrics, magnetism, diffusion and flow in porous media and to the mathematically identical phenomenon of anti-plane elasticity.     

Stress fields and Eshelby forces on half-plane inhomogeneities with eigenstrains and strip inclusions meeting a free surface

The stress field due to a half-plane inhomogeneity with plane eigenstrain is obtained by a limiting procedure from the one of a circular Eshelby inhomogeneity/inclusion. This field, which requires tractions to be applied at infinity to be sustained, has minimum strain energy versus any other superposed homogeneous one, and is the Eshelby solution inside plus the Hill jump conditions. By superposition, the stresses due to an infinite strip (Eshelby property domain) inhomogeneity with eigenstrain are obtained, and, by superposition periodic strips or laminates can be obtained. By cancelling the stresses on a free-surface, strips of inclusions meeting a free surface are solved. They exhibit tensile stresses under the free surface, and logarithmic singularities in the tensile stress at the vertex, which may initiate cracking. The Eshelby self-forces on the boundary of circular and half-plane inhomogeneities are computed.     

On the Solution of an Elliptical Inhomogeneity in Plane Elasticity by the Equivalent Inclusion Method

Stress analysis of an elliptical inhomogeneity in an infinite isotropic elastic plane is a classical elasticity problem, which is usually solved by means of the complex variable formulation. In this work, we demonstrate that an alternative method of solution for such a problem, via the equivalent inclusion method, may be more convenient and straightforward without recourse to complex potentials or curvilinear coordinates. The explicit analytical solution can be derived through simple algebraic manipulation, although the longitudinal eigenstrain component should be handled with care in the case of plane strain. Since the exterior Eshelby tensor for an elliptical inclusion is available in closed-form, the present study provides a full field stress solution expressed in Cartesian coordinates. Furthermore, the in-plane stress components are represented in terms of Dundurs’ parameters. The solution methodology and the convenient formulae of the stress concentration may be of practical use to the engineers in developing benchmarks for design evaluation.     

Micromechanics-based constitutive modeling for unidirectional laminated composites

A micromechanics-based constitutive model is developed to predict the effective mechanical behavior of unidirectional laminated composites. A newly developed Eshelby’s tensor for an infinite circular cylindrical inclusion [Cheng, Z.Q., Batra, R.C., 1999. Exact Eshelby tensor for a dynamic circular cylindrical inclusion. J. Appl. Mech. 66, 563–565] is adopted to model the unidirectional fibers and is incorporated into the micromechanical framework. The progressive loss of strength resulting from the partial fiber debonding and the nucleation of microcracks is incorporated into the constitutive model. To validate the proposed model, the predicted effective stiffness of transversely isotropic composites under far field loading conditions is compared with analytical solutions. The constitutive model incorporating the damage models is then implemented into a finite element code to numerically characterize the elastic behavior of laminated composites. Finally, the present predictions on the stress–strain behavior of laminated composite plate containing an open hole is compared with experimental data to verify the predictive capability of the model.     

Size dependent,non-uniform elastic field inside a nano-scale spherical inclusion due to interface stress

The primary objective of the present paper is to analyze the influence of interface stress on the elastic field within a nano-scale inclusion. Special attention is focused on the case of non-hydrostatic eigenstrain. From the viewpoint of practicality, it is assumed that the inclusion is spherically shaped and embedded into an infinite solid, within which an axisymmetric eigenstrain is prescribed. Following Goodier’s work, the elastic fields inside and outside the inclusion are obtained analytically. It is found that the presence of interface stress leads to conclusion that the elastic field in the inclusion is not only dependent on inclusion size but also on non-uniformity. The result is in strong contrast to Eshelby’s solution based on classical elasticity, and it is helpful in the understanding of relevant physical phenomena in nano-structured solids.     

The Moving Plane Inhomogeneity Boundary with?Transformation Strain

Within the context of linear elastodynamics, the radiated fields (including inertia) for a plane inhomogeneous inclusion boundary with transformation strain (or eigenstrain), moving in general motion under applied loading, have been obtained on the basis of Eshelby??s equivalent inclusion method, by using the strain field of a moving homogeneous inclusion boundary previously obtained. This dynamic strain field, obtained from the dynamic Green??s function (for an isotropic material), is unique, and has as initial condition the limit of the spherical Eshelby inclusion, as the radius tends to infinity, which is the minimum energy solution for the half-space inclusion. With the equivalent dynamic eigenstrain (which is dependent on the velocity of the boundary), the radiated fields for the inhomogeneous plane inclusion boundary can be obtained, and from them the driving force on the moving boundary can be computed, consisting of a self-force (which is the rate of mechanical work (including inertia) required to create an incremental region of inhomogeneity with eigenstrain), and of a Peach-Koehler force associated with the external loading. While for an expanding plane homogeneous inclusion boundary the Peach-Koehler force is independent of the boundary velocity, in the case of an inhomogeneous one it is not.     

Inclusion of arbitrary polygon with graded eigenstrain in an anisotropic piezoelectric full plane

In this paper, an exact closed-form solution for the Eshelby problem of a polygonal inclusion with a graded eigenstrain in an anisotropic piezoelectric full plane is presented. For this electromechanical coupling problem, by virtue of Green’s function solutions, the induced elastic and piezoelectric fields are first expressed in terms of line integrals on the boundary of the inclusion. Using the line-source Green’s function, the line integral is then carried out analytically for the linear eigenstrain case, with the final expression involving only elementary functions. Finally, the solution is applied to the semiconductor quantum wire (QWR) of square, triangle, circle and ellipse shapes within the GaAs (0 0 1) substrate. It is demonstrated that there exists significant difference between the induced field by the uniform eigenstrain and that by the linear eigenstrain. Since the misfit eigenstrain in most QWR structures is actually non-uniform, the present solution should be particularly appealing to nanoscale QWR structure analysis where strain and electric fields are coupled and are affected by the non-uniform misfit strain.     

Eshelby problem of polygonal inclusions in anisotropic piezoelectric full- and half-planes

This paper presents an exact closed-form solution for the Eshelby problem of polygonal inclusion in anisotropic piezoelectric full- and half-planes. Based on the equivalent body-force concept of eigenstrain, the induced elastic and piezoelectric fields are first expressed in terms of line integral on the boundary of the inclusion with the integrand being the Green's function. Using the recently derived exact closed-form line-source Green's function, the line integral is then carried out analytically, with the final expression involving only elementary functions. The exact closed-form solution is applied to a square-shaped quantum wire within semiconductor GaAs full- and half-planes, with results clearly showing the importance of material orientation and piezoelectric coupling. While the elastic and piezoelectric fields within the square-shaped quantum wire could serve as benchmarks to other numerical methods, the exact closed-form solution should be useful to the analysis of nanoscale quantum-wire structures where large strain and electric fields could be induced by the misfit strain.     

A circular inclusion with a nonuniform interphase layer in anti-plane shear

An analytical solution is derived for the problem of a nonuniformly coated circular inclusion in an unbounded matrix under anti-plane deformations. The inclusion/interphase/matrix system is subject to (1) remote uniform shear and uniform eigenstrain imposed on the circular inclusion, and (2) a screw dislocation or a point force in the matrix. It is found that the varying interphase thickness will exert a significant influence on the nonuniform stress field within the circular inclusion, and on the direction and magnitude of the image force acting on a screw dislocation. In the course of development, it is found that the presence of certain coated inclusions, which are termed stealth, will not cause change of elastic energy in the body. The derived analytical solution for a screw dislocation is then employed as Green’s function to investigate a radial matrix crack interacting with the nonuniformly coated inclusion. The numerical results show that the varying interphase thickness will also affect the stress intensity factors.     

Inclusions in a finite elastic body

Within the framework of 2D or 3D linear elasticity, a general approach based on the superposition principle is proposed to study the problem of a finite elastic body with an arbitrarily shaped and located inclusion. The proposed approach consists in decomposing the initial inclusion problem into the problem of the inclusion embedded in the corresponding infinite body and the auxiliary problem of the finite body subjected to the appropriate boundary loading provided by solving the former problem. Thus, our approach renders it possible to circumvent the difficulty due to the unavailability of the relevant Green function, use various existing solutions for the problem of an inclusion inside an unbounded body and clearly makes appear the finite boundary effects. The general approach is applied and specified in the context of 2D isotropic elasticity. The complex potentials for the problem of an inclusion in an infinite body are given as two boundary integrals, and the boundary integral equation governing the complex potentials for the auxiliary problem is provided. In the important particular situation where a finite body with an arbitrarily shaped and located inclusion is circular, the exact explicit expressions for the complex potentials are derived, leading to those for the strain, stress and Eshelby’s tensor fields inside and outside the inclusion. These results are analytically detailed and numerically illustrated for the cases of a square inclusion placed concentrically, and a circular inclusion located eccentrically, inside a circular body.     

Compliance and Hill polarization tensor of a crack in an anisotropic matrix

This work aims at developing an efficient method to compute the compliance due to a crack modeled as a flat ellipsoid of any shape in an infinite elastic matrix of arbitrary anisotropy (Eshelby problem) when no closed-form solution seems currently available. Whereas the solution of this problem usually requires the calculation of the so-called fourth-order Hill polarization tensor if the ellipsoid is not singular, it is shown that the crack compliance can be derived from the first-order term in the Taylor expansion of the Hill tensor with respect to the smallest aspect ratio of the ellipsoidal inclusion. For a 3D ellipsoidal crack model, this first-order term is expressed as a simple integral thanks to the Cauchy residue theorem. A similar method allows to express the same term in the case of a cylindrical crack model without any integral. A numerical example is finally treated.     

On the anti-plane shear of an elliptic nano inhomogeneity

The elastic field of an elliptic nano inhomogeneity embedded in an infinite matrix under anti-plane shear is studied with the complex variable method. The interface stress effects of the nano inhomogeneity are accounted for with the Gurtin–Murdoch model. The conformal mapping method is then applied to solve the formulated boundary value problem. The obtained numerical results are compared with the existing closed form solutions for a circular nano inhomogeneity and a traditional elliptic inhomogeneity under anti-plane. It shows that the proposed semi-analytic method is effective and accurate. The stress fields inside the inhomogeneity and matrix are then systematically studied for different interfacial and geometrical parameters. It is found that the stress field inside the elliptic nano inhomogeneity is no longer uniform due to the interface effects. The shear stress distributions inside the inhomogeneity and matrix are size dependent when the size of the inhomogeneity is on the order of nanometers. The numerical results also show that the interface effects are highly influenced by the local curvature of the interface. The elastic field around an elliptic nano hole is also investigated in this paper. It is found that the traction free boundary condition breaks down at the elliptic nano hole surface. As the aspect ratio of the elliptic hole increases, it can be seen as a Mode-III blunt crack. Even for long blunt cracks, the surface effects can still be significant around the blunt crack tip. Finally, the equivalence between the uniform eigenstrain inside the inhomogeneity and the remote loading is discussed.     

Computational model for short-fiber composites with eigenstrain formulation of boundary integral equations

A computational model is proposed for short-fiber reinforced materials with the eigenstrain formulation of the boundary integral equations （BIE） and solved with the newly developed boundary point method （BPM）. The model is closely derived from the concept of the equivalent inclusion Of Eshelby tensors. Eigenstrains are iteratively determined for each short-fiber embedded in the matrix with various properties via the Eshelby tensors, which can be readily obtained beforehand either through analytical or numerical means. As unknown variables appear only on the boundary of the solution domain, the solution scale of the inhomogeneity problem with the model is greatly reduced. This feature is considered significant because such a traditionally time-consuming problem with inhomogeneity can be solved most cost-effectively compared with existing numerical models of the FEM or the BEM. The numerical examples are presented to compute the overall elastic properties for various short-fiber reinforced composites over a representative volume element （RVE）, showing the validity and the effectiveness of the proposed computational modal and the solution procedure.     

Computational model for short-fiber composites with eigenstrain formulation of boundary integral equations

A computational model is proposed for short-fiber reinforced materials with the eigenstrain formulation of the boundary integral equations(BIE)and solved with the newly developed boundary point method(BPM).The model is closely derived from the concept of the equivalent inclusion of Eshelby tensors.Eigenstrains are iteratively determined for each short.fiber embedded in the matrix with various properties via the Eshelby tensors,which can be readily obtained beforehand either through analytical or numerical means.As unknown variables appear only on the boundary of the solution domain,the solution scale of the inhomogeneity problem with the model is greatly reduced.This feature is considered significant because such a traditionally time-consuming problem with inhomogeneity can be solved most cost-effectively compared with existing numerical models of the FEM or the BEM.The numerical examples are presented to compute the overall elastic properties for various short-fiber reinforced composites over a representative volume element(RVE),showing the validity and the effectiveness of the proposed computational modal and the solution procedure.