The Eshelby, Hill, Moment and Concentration Tensors for Ellipsoidal Inhomogeneities in the Newtonian Potential Problem and Linear Elastostatics
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Abstract
One of the most cited papers in Applied Mechanics is the work of Eshelby from 1957 who showed that a homogeneous isotropic ellipsoidal inhomogeneity embedded in an unbounded (in all directions) homogeneous isotropic host would feel uniform strains and stresses when uniform strains or tractions are applied in the farfield. Of specific importance is the uniformity of Eshelby’s tensor \(\mathbf{S}\). Following Eshelby’s seminal work, a vast literature has been generated using and developing Eshelby’s result and ideas, leading to some beautiful mathematics and extremely useful results in a wide range of application areas. In 1961 Eshelby conjectured that for anisotropic materials only ellipsoidal inhomogeneities would lead to such uniform interior fields. Although much progress has been made since then, the quest to prove this conjecture is still not complete; numerous important problems remain open. Following a different approach to that considered by Eshelby, a closely related tensor \(\mathbf{P}=\mathbf{S}\mathbf{D}^{0}\) arises, where \(\mathbf{D}^{0}\) is the host medium compliance tensor. The tensor \(\mathbf{P}\) is associated with Hill and is of course also uniform when ellipsoidal inhomogeneities are embedded in a homogeneous host phase. Two of the most fundamental and useful areas of applications of these tensors are in Newtonian potential problems such as heat conduction, electrostatics, etc. and in the vector problems of elastostatics. Knowledge of the Hill and Eshelby tensors permit a number of interesting aspects to be studied associated with inhomogeneity problems and more generally for inhomogeneous media. Micromechanical methods established mainly over the last halfcentury have enabled bounds on and predictions of the effective properties of composite media. In many cases such predictions can be explicitly written down in terms of the Hill tensor, or equivalently the Eshelby tensor and can be shown to provide excellent predictions in many cases.
Of specific interest is that a number of important limits of the ellipsoidal inhomogeneity can be taken in order to be employed in predictions of the effective properties of, for example, layered media and fibre reinforced composites and also to the cases when voids and cracks are present. In the main, results for the Hill and Eshelby tensors are distributed over a wide range of articles and books, using different notation and terminology and so it is often difficult to extract the necessary information for the tensor that one requires. The case of an anisotropic host phase is also frequently nontrivial due to the requirement of the associated Green’s tensor. Here this classical problem is revisited and a large number of results for problems that are felt to be of great utility in a wide range of disciplines are derived or recalled. A scaling argument leads to the derivation of the Eshelby tensor for potential problems where the host phase is at most orthotropic, without the requirement of using the anisotropic Green’s function. The Concentration tensor \(\boldsymbol{\mathcal{A}}\) linking interior fields to those imposed in the farfield is derived for a wide variety of problems. These results can therefore be used in the various micromechanical schemes.
Directly related to the tensors of Eshelby and Hill is the socalled Moment tensor \(\mathbf{M}\). As well as arising in the literature on micromechanics, this tensor is important in the vast area of research associated with inverse problems and specifically with the problem of identifying an object inside some domain given the application of a specific set of boundary conditions. Due to its fundamental importance and direct link to the Eshelby and Hill tensors, here we state the connection between \(\mathbf{M}, \mathbf{P}\) and \(\mathbf{S}\) in an effort to ensure that the work is of use to as wide a community as possible.
Both tensor and matrix formulations are considered and contrasted throughout. Appendices give various details that illustrate the implementation of both approaches.
Keywords
Hill tensor Eshelby tensor Moment tensor Concentration tensor Newtonian potential Elastostatics Eshelby conjecture PólyaSzegö conjecture Inclusion Inhomogeneity Ellipsoids Uniformity Micromechanics Homogenization Inverse problems Elastic modulus tensor Conductivity tensorMathematics Subject Classification
31B05 31B10 31B20 31B35 35C15 35J05 35J08 35J15 35J25 35R05 45A05 74A40 74A60 74B05 74E05 74E10 74E30 74E35 74G05 74G70 74G75 74L10 74M25 74N15 74Q05 74Q15 78A30 78M40 80A20 80M401 Introduction
The canonical isolated inhomogeneity problem has been of fundamental importance in a number of materials modelling problems now for well over a century. This problem is the following: a single inhomogeneity or particle of general shape, with different material properties to that of the surrounding material is embedded inside an unbounded (in all directions, i.e., freespace) homogeneous host medium. Given some prescribed conditions in the farfield, what form do the fields take within the inhomogeneity? As well as being interesting in its own right, this problem is of utmost importance in homogenization, micromechanics and multiscale modelling.
The first to consider this kind of inhomogeneity problem was Poisson in 1826 [103] who studied the perturbed field due to an isolated ellipsoid in the context of the Newtonian potential problem. He showed that given a uniform electric polarization (or magnetization), the induced electric (or magnetic) field inside the ellipsoid is also uniform. In 1873 Maxwell [83] derived explicit expressions for this field. Early work in linear elasticity saw a number of studies determine the field inside and around inhomogeneities, including the important case of a cavity (since this was correctly recognized as a defect or flaw). Examples of these works are those associated with the case of spheres [43, 118], spheroids [29] and ellipsoids [107, 113, 114] but all considered specific loadings, usually of the homogeneous type in the farfield, meaning uniform tractions or displacements that are linear in the independent Cartesian variable, say \(\mathbf{x}\).
The inhomogeneity problem is now usually associated with the name of Eshelby because in 1957 he showed that for general homogeneous conditions imposed in the farfield, the strain set up inside an isotropic homogeneous ellipsoid is uniform [31]. In 1961 Eshelby [32] conjectured that “…amongst closed surfaces, the ellipsoid alone has this convenient property…”. One could ask is this really true? In the sense of what it is thought that Eshelby meant when he made this conjecture (the socalled weak Eshelby conjecture, where the interior field must be uniform for any homogeneous farfield loading), this statement certainly is true although this was only proved in 2008, simultaneously by Kang and Milton [56] and Liu [71] in the case of isotropic media. There is a slightly different version (the socalled strong Eshelby conjecture), where the interior field must be uniform only for a specific, single uniform farfield loading. This strong conjecture has not been proven in the context of three dimensional isotropic linear elasticity, although significant progress has been made in the last decade, see [55] for a review. The results obtained in [4] go beyond the weak Eshelby conjecture but still do not fully prove the strong conjecture. Interestingly the weak conjecture for the associated Newtonian potential problem was proved some time before Eshelby’s 1957 elastostatics paper, by Dive in 1931 [24] and Nikliborc [97] in 1932, see also the discussion in [55, 56, 71]. In deriving these results, Dive and Nikliborc proved the converse of Newton’s theorem that if \(V_{1}\) is an ellipsoid of uniform density, the gravitational force in \(V_{1}\) is zero [58]. The strong conjecture in the context of the potential problem is true in two dimensions [111, 115] but is not true in dimensions greater than two: a nonellipsoidal counterexample associated with a specific farfield loading (equivalently a specific eigenstress) was found by Liu [71].
It is important to note that the proofs of Eshelby’s conjectures in elastostatics referred to above correspond to simply connected, isotropic inhomogeneities with Lipschitz boundaries. Eshelby’s work was followed up by numerous researchers who considered the general anisotropic case [6, 8, 32, 61, 70, 121, 122, 129, 133]. In 1974 Cherepanov [22] proved that multiple inhomogeneities of nonellipsoidal shape can interact in order to render the interior fields uniform; see also Kang and Milton [56] and Liu [71] who coined the term Einclusions for such interacting inhomogeneities. Liu and coworkers have also considered the periodic Eshelby problem in two dimensions [72, 73]. Kang and Milton [56] used their approach to prove Eshelby’s weak conjecture in the context of the fully anisotropic potential problem. Most notably however, it is stressed again that the weak Eshelby conjecture for elasticity has not yet been proved in the context of anisotropic elasticity.
Interest in deriving the Eshelby tensor for nonellipsoidal inhomogeneities has always been active in order to show that the conjecture holds for specific classes of inhomogeneities. Particular attention has been paid to polygonal and polyhedral inhomogeneities and the associated properties of Eshelby’s tensor [57, 74, 79, 80, 93, 94, 98, 108]. The supersphere case has been considered recently by [18] building on the work by [99, 100, 119]. A general method was developed by Ru [110] in order to obtain an analytical solution associated with a two dimensional inhomogeneity of arbitrary cross section and explicit forms of the stress inside hypotrochoidal and rectangular inhomogeneities were derived. Some analytical expressions have recently been derived for twodimensional problems in the Newtonian potential and plane elastostatics problems where inhomogeneities are either polygonal or their shape can be described by finite Laurent expansions [140, 141]. Additional useful properties of the Eshelby tensor have been deduced, including the relationship of the averaged Eshelby tensor for nonellipsoidal inhomogeneities to their ellipsoidal counterparts [126, 137].
More recently the inhomogeneity problem has been studied in the nonlinear elasticity context where results associated with Eshelby’s conjecture have been proved in two dimensions for socalled harmonic materials [59, 60, 112]. Although nonlinear problems are generally more difficult than linear elastostatics, the nonlinearity frees up a number of issues that are more constrained in linear problems. The study of nonlinear problems with dilatational eigenstrain was recently carried out in [135]. Giordano [41] considered the nonlinearly elastic inhomogeneity problem but where the constitutive behaviour is described via expansions in strain (Landau elasticity).
The Hill and Eshelby tensors are of great utility in a number of micromechanical methods and what is quite astonishing is that they can be evaluated analytically in a large number of very important cases. However, results are distributed over a large number of articles, reviews and textbooks, and furthermore often in articles that span a wide range of scientific fields due to the wide ranging applicability of the theory. References dealing with derivations of specific results are those of [10, 14, 28, 81, 106, 123, 132] and [67]. The field is still very much alive, pushed forward by both unresolved theoretical issues as well as applications involving not only inhomogeneities but also cracks and dislocations [95, 139] and by the desire to fully resolve the open issues described above. Recent work has focused in more detail on inhomogeneities of general shape and how these can feed into models of inhomogeneous media with distributions of noncanonical inhomogeneities [14, 15, 16, 33, 34, 35, 138, 140, 141]. Such studies are important to understand how local stress fields develop in the medium under loading. This is highly dependent upon the inhomogeneity shape. It must always be stressed that the utility of the Eshelby tensor itself for general shapes in micromechanical methods is limited by the fact that fields interior to the inhomogeneity are not generally uniform in such cases and therefore the tensor does not arise as a natural quantity from the governing integral equations ((2.7) and (2.11) below) as it does in the case of ellipsoidal inhomogeneities.
Here the objective is to gather together important results associated with the Hill and Eshelby tensors for ellipsoidal inhomogeneities in a consistent notation, derive a number of important limiting cases such as those associated with cracks and cavities, derive compact results associated with the anisotropic potential problem and finally derive and state associated Concentration tensors. This should prove useful to many who frequently require the form of the \(\mathbf{P}\) or \(\mathbf{S}\)tensors in practice but who struggle to find the appropriate reference. The emphasis here is to derive the Hill, Eshelby and Concentration tensors but as is clear from (1.3) the Moment tensor follows straightforwardly from these.
An important point to note is that using the socalled invariant notation, potential and linear elastostatics problems can be considered simultaneously, only that the latter is a higher order tensor analogue of the former. Here however the applications are made distinct in order to stress the different results and mechanisms for deriving these expressions. In particular the results from potential theory feed into those from linear elastostatics. As a result, index notation shall be used almost entirely throughout.
In much of the literature on micromechanics the terms inclusion and inhomogeneity are used interchangeably. However in some cases they are used to make an important distinction. An inhomogeneity is defined as a particle of general shape having different material properties to those of the surrounding medium in which it is embedded. On the other hand the terminology inclusion is used to represent a general shaped region within some medium that has the same properties as the surrounding medium but where this finite inclusion region has been subject to some eigenstrain (e.g., thermal strain). This differentiation is used, e.g., in Mura [92] and Qu and Cherkaoui [106] and it is also adopted here.
In Sect. 2 the integral equation formulation of the inhomogeneity problem is stated, yielding integral equations for the potential gradient and strain inside an inhomogeneity. In Sect. 3 it is illustrated that such fields are uniform when the inhomogeneity is ellipsoidal and the general expressions for the associated Hill tensors are stated. The notion of Concentration tensors is also discussed. In Sects. 4 and 5 specific results are then stated and derived for the cases of the Newtonian potential problem and elastostatics respectively. A closing discussion is given in Sect. 6 describing how the results are used in micromechanical methods together with a summary of current areas of associated research. Numerous important details and results are stated in Appendices in order for this review to be comprehensive but also to aid the flow of reading, in particular the mechanism for representing tensors of certain symmetries with respect to appropriate tensor bases, tensor contractions, and the matrix formulation of operations between tensors are covered in Appendix C.
As many pertinent references are given as possible; it is important to stress that the focus is specifically on the formulation of the Eshelby, Hill and Concentration tensors rather than articles associated with micromechanical methods, of which there are thousands. For the latter the interested reader is referred to the many textbooks that have been written over the last decade, e.g., [14, 28, 54, 67, 106].
2 Integral Equation Formulation
Index notation shall be used for tensors throughout, working in Cartesian coordinates and using repeated subscripts to imply summation. The term unbounded will be used when referring to freespace, i.e., unbounded in all directions. Although a general invariant formulation can be employed to deal with problems in the potential and linear elastostatics context simultaneously [132], this approach can obfuscate details that are important when it comes to deriving specific Hill and Eshelby tensors for given anisotropies and inhomogeneity shapes, which is the main purpose of this article.
2.1 The Potential Problem
2.2 Elastostatics
The origins of the \(\mathbf{P}\)tensor reside in the context of elastostatics rather than in potential problems even though the theory is of course analogous. The \(\mathbf{P}\)tensor originated with Hill [47] who also introduced the compact notation (now commonly referred to as Hill notation) for transversely isotropic fourth order tensors, which is summarized in Appendix C.2.3. Walpole [122], Willis [130, 131, 132] and Laws [62] amongst others followed this with influential work associated with inhomogeneities of specific shapes, paying particular attention in many cases to the scenarios of discs, fibres and cracks. A number of \(\mathbf{P}\)tensors are also stated in the excellent concise review of micromechanics by Markov [81] although unfortunately, some typographical errors are present there and these are corrected here.
3 Uniformity of the Hill and Eshelby Tensors
3.1 The Potential Problem
Clearly since the integral is over the surface of the unit sphere \(S^{2}\), it is sensible to resolve \(\overline{\xi}_{i}\) into spherical polar coordinates for the purposes of evaluating this integral. The form of (3.3) illustrates the important general result that the \(\mathbf{P}\)tensor is uniform for an arbitrarily anisotropic ellipsoidal inhomogeneity embedded inside an arbitrarily anisotropic host phase. That Eshelby’s (weak) conjecture is true for anisotropic potential problems [56, 71], means that the ellipsoid is the only shaped inhomogeneity for which the interior temperature gradient is uniform under any such farfield condition of the form (3.1). The fact that the strong conjecture is not true in the Newtonian potential problem means that there exists shapes where the interior temperature gradient field can be uniform for specifically chosen homogeneous farfield conditions [71].
To determine the appropriate \(\mathbf{P}\)tensor in any circumstance then one can appeal to (3.3) and carry out the necessary integration. Alternatively, as shall be shown in Sect. 4, in many cases it is relatively straightforward to use symmetry arguments and results from potential theory in the isotropic host case together with scalings in some cases of host anisotropy, in order to derive explicit results, often in a more straightforward manner than by directly evaluating the general expression (3.3). In fact in the potential problem context, symmetry arguments and results from potential theory [58] are often sufficient to derive results for many special cases of ellipsoids in host media that are at most orthotropic. The general result (3.3) is thus suitable for more complex anisotropies than orthotropy or for example if the semiaxes of the ellipsoid are not aligned with the axes of symmetry of host anisotropy.

In the host region \(V_{0}\) the temperature gradient is generally not uniform.

For nonhomogeneous temperature gradient conditions in the farfield, the temperature gradient field inside an ellipsoidal inhomogeneity is generally not uniform. However if the prescribed temperature gradient is a polynomial of order \(n\), then so is the field inside an ellipsoidal inhomogeneity, see [6]. This is known as the polynomial conservation property for ellipsoids.

Generally for nonellipsoidal inhomogeneities in unbounded domains and general shaped inhomogeneities in bounded host domains \(V\), the temperature gradient inside the inhomogeneities is not uniform, although interacting Einclusions [71] can lead to uniform interior temperature gradients and for specific loadings, nonellipsoidal inhomogeneities can yield uniform interior temperature gradients, e.g., the counterexample of the Strong Eshelby conjecture given by Liu [71].
3.2 Elastostatics
That Eshelby’s (weak) conjecture is true for isotropic elastostatics problems [56, 71], means that the ellipsoid is the only shaped inhomogeneity for which the interior temperature gradient is uniform under any such farfield condition of the form (3.7). It is stressed however that it is not yet clear whether the weak conjecture is true in the context of anisotropic problems.
To determine the \(\mathbf{P}\)tensor for an ellipsoid for a given host anisotropy one merely has to evaluate the surface integral in (3.9) and this can be done numerically very efficiently. For host anisotropies more complex than transversely isotropic it is generally recommended that the form (3.9) be employed and integrals are evaluated numerically. In what follows here the \(\mathbf{P}\)tensor shall be determined in the case of an isotropic host phase by appealing to various symmetries and potential theory. An important result derived by Withers [133] associated with a transversely isotropic host phase is also stated.
3.3 The Potential Gradient Tensor and Strain Concentration Tensor
Note that \(\mathcal{A}_{ij}\) is the Concentration tensor associated with an isolated inhomogeneity inside an unbounded host medium. The calligraphic notation \(\mathcal{A}_{ij}\) has been used to stress the link with (and distinguish from) the exact Concentration tensor whose components are usually defined as \(A_{ij}\), and which links the phase average of the true temperature gradient inside an inhomogeneity to that in the farfield in a complex inhomogeneous medium, which may consist of interacting inhomogeneities. For a dilute medium where interaction effects can be neglected, \(A_{ij}=\mathcal{A}_{ij}\).
4 The Potential Problem: Specific Cases
4.1 Isotropic Host Phase
Once \(P_{ij}\) is determined for an isotropic host phase the associated Concentration tensor for an isolated inhomogeneity may then be found from (3.19) as is now illustrated in a number of special cases of specific inhomogeneities with given shape and anisotropy.
4.1.1 Sphere in an Isotropic Host Phase
Isotropic Sphere
Anisotropic Sphere
4.1.2 Circular Cylinder in an Isotropic Host Phase
4.1.3 Ellipsoid in an Isotropic Host Phase
4.1.4 Spheroid in an Isotropic Host Phase
 (i)
a sphere, i.e., \(\varepsilon \rightarrow1\), it is deduced that \(\gamma =\gamma _{3} = \frac{1}{3}\),
 (ii)
a cylinder, i.e., \(\varepsilon \rightarrow\infty\), it is deduced that \(\gamma _{3}= 0\), \(\gamma =\frac{1}{2}\),
 (iii)
a disc or layer, i.e., \(\varepsilon \rightarrow0\), it is deduced that \(\gamma _{3}=1\), \(\gamma =0\).
4.1.5 Limiting Case of an Elliptical Cylinder
4.1.6 Limiting Cases of a Cavity, PennyShaped Crack and RibbonCrack
It does not make sense to define a temperature gradient Concentration tensor in the context of cracks or cavities because clearly there is no interior field. However it turns out that this concept is useful and can be interpreted as linking the farfield to the field on the surface of such inhomogeneities [50] with an appropriate definition of cavity temperature gradient. Here the results above are used in order to derive associated Concentration tensors for cracks and cavities.
The coefficient of \(\delta_{i3}\delta_{j3}\) in (4.40) involves an apparently singular limit as \(\varepsilon \rightarrow0\). That this is not a problem arises from the fact that this expression is used in formulae for effective properties of cracked media where this term is always multiplied by a volumefraction term (or rather a crackdensity term) that is proportional to \(\varepsilon \) [48, 49]. Note that taking the limits in the opposite order, i.e., \(\varepsilon \rightarrow0\) and then \(k_{1}\rightarrow0\) yields an inconsistent result, giving rise to singular effective material behaviour in the crack limit, which cannot be correct.
4.2 Anisotropic Host Phase
The general form (3.3) for the \(\mathbf{P}\)tensor associated with arbitrary host anisotropy requires the necessary surface integral to be evaluated. In the case of transversely isotropic and orthotropic media however, where principal axes are aligned with the semiaxes of the ellipsoid, the problem can be simplified significantly by employing a scaling of the Cartesian variables in order to reduce the isolated ellipsoidal inhomogeneity problem in an anisotropic medium to the case of an ellipsoidal inhomogeneity (with different semiaxes) in an isotropic medium. Therefore the results derived above for the isotropic host phase case can be used in the scaled domain and then mapped back to the physical domain in order to obtain the appropriate physical Hill and Eshelby tensors.
4.2.1 Spheroid in a Transversely Isotropic Host Phase
4.2.2 Circular Cylinder in a Transversely Isotropic Host Phase
An interesting nonstandard example is the case of a spheroid embedded inside a transversely isotropic host phase where the axes of symmetry and semiaxes are not coincident. In this case the general (surface integral) form of the \(\mathbf{P}\) and \(\mathbf{S}\)tensors must be used with the semiaxes aligned with the \(\mathbf{x}\) axes but with all components of the modulus tensor being generally nonzero.
4.2.3 Ellipsoid in an Orthotropic Host Phase
Summary of results for the \(\mathbf{P}\)tensor associated with ellipsoidal inhomogeneities for potential problems
Host anisotropy  Inhomogeneity shape  Ptensor 

Isotropic  Ellipsoid \(a_{1}\neq a_{2}\neq a_{3}\) \(\varepsilon _{n}=a_{3}/a_{n}\)  Use potential theory: \(P_{ij} = \frac{1}{k_{0}}\sum_{n=1}^{3}\mathcal{E}(\varepsilon _{n};\varepsilon _{1},\varepsilon _{2})\delta_{in}\delta_{jn}\) 
Spheroid \(a_{1}=a_{2}=a\neq a_{3}\) \(\varepsilon =a_{3}/a\)  Use potential theory: \(P_{ij} = \frac{1}{k_{0}} (\gamma \varTheta_{ij}+\gamma _{3}\delta_{i3}\delta _{j3} )\) \(\gamma =\frac{1}{2}(1\gamma _{3}), \gamma _{3} = \mathcal{S}(\varepsilon )\)  
Sphere  Use symmetry: \(P_{ij} = \frac{1}{3k_{0}}\delta_{ij}\)  
Transversely isotropic \(\upsilon_{1}=\upsilon_{2}=1\neq\upsilon_{3}=\upsilon\)  Ellipsoid \(a_{1}\neq a_{2}\neq a_{3}\) \(\varepsilon _{n}=a_{3}/a_{n}\)  Use scalings and potential theory: \(P_{ij} = \frac{1}{k_{0}}\sum_{n=1}^{3}\frac{1}{\upsilon_{n}}\mathcal {E}(\hat{\varepsilon }_{n}; \hat{\varepsilon }_{1},\hat{\varepsilon }_{2})\delta_{in}\delta_{jn}\) \(\hat{\varepsilon }_{n} = \hat{a}_{3}/\hat{a}_{n}\) and \(\hat{a}_{n} = a_{n}/\sqrt {\upsilon_{n}}\) 
Spheroid \(a_{1}=a_{2}=a\neq a_{3}\) and \(a_{3}\) is aligned with axis \(x_{3}\) of transverse isotropy  Use scalings and potential theory: \(P_{ij} = \frac{1}{k_{0}} (\gamma \varTheta _{ij}+\gamma _{3}\delta_{i3}\delta_{j3} )\) \(\gamma =\frac{1}{2}(1\upsilon \gamma _{3})\), \(\gamma _{3} = \frac{1}{\upsilon }\mathcal{S} (\frac{\varepsilon }{\sqrt {\upsilon}} )\)  
Sphere  Special case of spheroid result above: \(P_{ij} = \frac{1}{k_{0}} (\gamma \varTheta_{ij}+\gamma _{3}\delta_{i3}\delta _{j3} )\) \(\gamma =\frac{1}{2}(1\upsilon \gamma _{3}), \gamma _{3} = \frac{1}{\upsilon }\mathcal{S} (\frac{1}{\sqrt{\upsilon}} )\)  
Orthotropic \(\upsilon_{1}=1\neq\upsilon_{2}\neq\upsilon_{3}\)  Ellipsoid \(a_{1}\neq a_{2}\neq a_{3}\)  Use scalings and potential theory: \(P_{ij} = \frac{1}{k_{0}}\sum_{n=1}^{3}\frac{1}{\upsilon_{n}}\mathcal {E}(\hat {\varepsilon }_{n};\hat{\varepsilon }_{1},\hat{\varepsilon }_{2})\delta_{in}\delta _{jn}\) \(\hat{\varepsilon }_{n} = \hat{a}_{3}/\hat{a}_{n}\) and \(\hat{a}_{n} = a_{n}/\sqrt {\upsilon_{n}}\) 
Worse than orthotropic or semiaxes of ellipsoids not aligned with axes of anisotropy  Use general integral form: \(P_{ij}=\frac{\mathrm {det}(\mathbf{a})}{4\pi} \int _{S^{2}}\frac {\varPhi_{ij}\,dS(\overline{\boldsymbol{\xi }})}{(\overline{\xi}_{k} a_{k\ell}a_{\ell m}\overline{\xi }_{m})^{3/2}}\) \(\varPhi_{ij}= \overline{\xi}_{i}\overline{\xi}_{j}/(C_{k\ell }^{0}\overline{\xi}_{k}\overline{\xi}_{\ell})\) 
5 Elastostatics: Specific Cases
5.1 Isotropic Host Phase
Once \(P_{ijk\ell}\) is determined, the components of the Eshelby tensor can be calculated from (1.2) and the associated Concentration tensor can be found from (3.23). Recall that no assumptions have been made regarding the anisotropy of the inhomogeneity. This is not required in order for the \(\mathbf{P}\)tensor to be determined. The only aspects of the inhomogeneity that influence the \(\mathbf{P}\)tensor are its shape and, for anisotropic host phases, its orientation with respect to the axes of anisotropy of the host phase.
5.1.1 Sphere in an Isotropic Host Phase
Isotropic Sphere
Cubic Sphere
5.1.2 Circular Cylinder in an Isotropic Host Phase
Shorthand notation for contractions of the basis tensors \(\mathcal {H}_{ijkl}^{n}\)
\(\mathcal{H}^{1}\)  \(\mathcal{H}^{2}\)  \(\mathcal{H}^{3}\)  \(\mathcal{H}^{4}\)  \(\mathcal{H}^{5}\)  \(\mathcal{H}^{6}\)  

\(\mathcal{H}^{1}\)  \(\mathcal{H}^{1}\)  \(\mathcal{H}^{2}\)  0  0  0  0 
\(\mathcal{H}^{2}\)  0  0  2\(\mathcal{H}^{1}\)  \(\mathcal {H}^{2}\)  0  0 
\(\mathcal{H}^{3}\)  \(\mathcal{H}^{3}\)  2\(\mathcal{H}^{4}\)  0  0  0  0 
\(\mathcal{H}^{4}\)  0  0  \(\mathcal{H}^{3}\)  \(\mathcal{H}^{4}\)  0  0 
\(\mathcal{H}^{5}\)  0  0  0  0  \(\mathcal{H}^{5}\)  0 
\(\mathcal{H}^{6}\)  0  0  0  0  0  \(\mathcal{H}^{6}\) 
5.1.3 Spheroid in an Isotropic Host Phase
5.1.4 Elastic Layer
5.1.5 Limiting Case of a PennyShaped Crack
The case of an ellipsoid in an isotropic medium shall now be considered. In order to deal with this generally in a tensor setting, ideally an orthotropic tensor basis should be used. Although it is possible to write down such a basis, details are rather lengthy and in fact for practical computation, it is perhaps most sensible to write down the nine independent components of the \(\mathbf{P}\)tensor and use matrix computations in the manner described after Sect. 5.1.2 above.
5.1.6 Ellipsoid in an Isotropic Host Phase
The nine independent components of the \(\mathbf{P}\)tensor for an ellipsoid can be defined in terms of the function \(\mathcal{E}(\varepsilon _{n};\varepsilon _{1},\varepsilon _{2})\) and the semiaxes ratios \(\varepsilon _{n}\).
5.1.7 Elliptical Cylinder and RibbonCrack Limit
5.1.8 Flat Ellipsoid
5.1.9 Spheroid Limit Check
5.2 Anisotropic Host Phase
In the potential problem case, scaling coordinate systems assisted in the derivation of results associated with anisotropic media. Although such methods can sometimes lead to modest simplifications in elasticity, the general theory does not lead to any significant advances, certainly for the problems that are of greatest interest in micromechanics. In particular such methods do not lead to significant simplifications for generally transversely isotropic media which is a material symmetry of great importance. Therefore to derive the \(\mathbf{P}\)tensors associated with inhomogeneities in anisotropic host phases, it is best to work with the integral form of the \(\mathbf{P}\)tensor as defined in (3.9) for an ellipsoid.
Few explicit results are available in general however since the Green’s tensor cannot generally be determined analytically. One of the few that can be determined is that associated with transversely isotropic media. Withers derived the associated Eshelby tensor for a spheroid [133] using the form of the Green’s function determined by Pan and Chou [101] and we state his result here where the semimajor or minor axis of the spheroid is aligned with the axis of transverse isotropy. This result shall then be validated by employing the general integral form (3.9). Only in the last decade have articles started to appear that compute effective properties of composite media with anisotropic inhomogeneities via micromechanical methods (e.g., [42, 116]). It is also important to note specific results for the Eshelby and Hill tensors associated with cracks in anisotropic media, e.g., Gruescu et al. [44] and Barthélémy [9].
5.2.1 Spheroid in a Transversely Isotropic Host Phase
Derivation from Withers’ Eshelby Tensor
Derivation from the Direct Integral Form
Concentration Tensor
Alternatively suppose that the inhomogeneity is transversely isotropic with the same symmetry axis as the host so that it possesses the elastic modulus tensor of the form (5.145) but where now the constants \(c^{1}_{n}\) are defined in terms of the 5 independent components of this tensor. Then the Concentration tensor is again defined by (5.151) but of course now with the \(c^{1}_{n}\) associated with the transversely isotropic cylinder. This indicates the merit of using the above notation since one can still use (5.151)–(5.153) in this case, merely modifying the \(c^{1}_{n}\) to account for the transverse isotropy of the cylinder.
As usual, the matrix form of these fourth order tensors can be employed for computational efficiency when the problems lack simple symmetries.
5.2.2 Circular Cylinder in a Transversely Isotropic Host
6 Discussion
6.1 Association with Micromechanics
Many micromechanical methods use a more sophisticated approximation that can account, in an approximate manner at least, for interactions. One of the most commonly employed methods is the socalled classical self consistent method [54]. Interaction is approximated in this most simple selfconsistent scheme by taking the host medium in the determination of \(\boldsymbol{\mathcal{A}}\) to be the unknown effective medium. In general then (6.1) gives rise to a nonlinear system of equations for the determination of effective properties. In many cases these are not even algebraic equations. Furthermore for the self consistent method, one has to make an assumption in advance of the symmetry properties of the effective tensor. For example in the case of aligned spheroids \(\mathbf{C}^{*}\) will be transversely isotropic.
The textbooks referred to at the end of Sect. 1 provide an excellent introduction to the numerous micromechanical methods, many of which are based on the form of effective modulus tensor defined in (6.1). A similar form can be deduced for media where there is no distinguishable host phase (for example the case of polycrystals) and also for media where multiphysics effects are important as described in the next section.
6.2 Beyond the Potential Problem and Elastostatics
A large number of explicit, compact results associated with the Hill and Eshelby tensors for ellipsoidal inhomogeneities, as well as their associated Concentration tensors have been collected, stated and in some cases derived. The intention is that this will be of great utility to a large number of researchers for implementation in micromechanical and bounding schemes. A thorough discussion of both matrix and tensor (where possible due to space limitations) formulations has been carried out. Typographical errors in past articles and reviews have been corrected and a common notation has been employed.
Although the general integral forms (3.3) and (3.9) are useful they should generally be avoided where explicit forms are available. Gavazzi and Lagoudas [40] described a numerical implementation for elasticity. It should be noted that recently Masson [82] derived a new form of the \(\mathbf{P}\)tensor in terms of a single integral, although the integrand is inevitably more complex than that in the surface integral in (3.9).
In the literature many of the cases described above are considered as approximations to more complicated shaped inhomogeneities. In terms of the derivation of overall effective properties this is extremely useful, certainly as a first approximation, since it avoids complex computational simulations. However it must be stressed that more advanced analysis is required if detailed micromechanical information such as stress concentration calculations close to inhomogeneities of a complex shape is required [15]. For finite domains, provided the host phase is in some sense much larger than the inhomogeneity, if the inhomogeneity is ellipsoidal then the temperature gradient or strain field inside the ellipsoid is well approximated as being uniform. The inhomogeneity problem associated with bounded domains is described in the book by Li and Wang [67] which summarizes the work in [68, 69].
Continuing with discussion associated with the potential problem and elastostatics first, an important extension of the inhomogeneity problem is that of the coated inhomogeneity. This problem is popular, not least because it arises as a micromechanics problem in the generalized selfconsistent method (GSCM) [23]. The socalled double inclusion problem dates back many decades and was solved approximately by Hori and NematNasser [50] although the approximations involved lead to some rather counterintuitive predictions when used in the GSCM [51]. Exact solutions in the case of concentric spheroids or ellipsoids have been derived by Hatta and Taya [46] in the thermal context and Jiang et al. [52] in twodimensional elasticity. See also [75, 76] for elasticity problems involving coated spheres and cylinders respectively. The case of inhomogeneities with radially dependent material properties has been considered in Chap. 3 of [54] amongst others. The coated inhomogeneity is also of interest due to its association with the neutral inclusion problem [89]. Associated with the coated inhomogeneity is the scenario when the interface of an inhomogeneity with the host phase is imperfect [37, 63]. This imperfection can itself be used as the basis for a neutral inclusion [12, 109]. Moment tensors have recently been discussed in the context of cloaking [5].
It is important to note that when the inhomogeneity becomes very small, i.e., the case of a nanoinhomogeneity, then surface energies become nonnegligible. This problem has been considered by Sharmi and Ganti [117] and Duan et al. [25] for example. Including surface energies is important in order to incorporate sizedependent effects in effective properties. These are absent in classical micromechanical methods that use the standard Eshelby or Hill tensors.
Eshelby’s problem has also been considered in the context of microcontinuum elasticity models, which themselves were introduced in order to bridge the gap between continuum and atomistic/molecular models [30]. Micropolar (Cosserat) theory has been considered by Cheng and He [20, 21] and Ma and Hu [77]. Microstretch theory has been developed by Ma and Hu [78]. Strain gradient constitutive behaviour was studied by Gao and Ma [38, 39].
The dynamic problem was considered for spheres and cylinders by Mikata and NematNasser [87, 88] and more generally in [19, 84]. Rate dependence of the Hill and Eshelby tensors has been considered by Suvarov and Dvorak [120] and viscoelastic properties have been studied by Wang and Weng [125] by using transform techniques and correspondence principles. Nguyen et al. [96] studied cracked viscoelastic solids using the appropriate Eshelby tensor. Extensions to plasticity have been considered, see [36, 53, 64] for example.
The Newtonian potential and elastostatics problems are canonical problems that can assist with the development of coupled (multiphysics) problems. Dunn and Taya [26], Dunn and Wienecke [27] and Mikata [85, 86] considered the case of piezoelectricity and the prediction of the electroelastic moduli. Li and Dunn [66] and Zhang and Soh [136] considered full coupling and the resulting effective moduli associated with piezoelectromagnetic media. The theory associated with poroelastic and thermoelastic behaviour was developed by Berryman [11] and extended to the anisotropic case by Levin and AlvarezTostado [65].
Upon closing it should be noted that it is very fortuitous that such elegant and concise uniformity results hold for ellipsoidal inhomogeneities. These results allow a large number of expressions to be derived analytically and therefore the results have been utilized a great deal. Having said that there is much work to be done. As has been noted, the Eshelby conjecture is still not fully resolved [4], analysis for general shaped inhomogenities continues [16], specifically in the context of stress analysis and resulting effective properties and although computational methods are powerful, they are still only able to solve elasticity problems for inhomogeneous media with an order of 1000 inhomogeneities in “reasonable” times. For use in MonteCarlo schemes this is therefore still computationally expensive. Nonlinear problems in the context of finite elasticity still require attention [135] and this applies to coupled problems as well.
Footnotes
 1.
This latter point does not appear to have been recognized in the original papers on this subject, see [133]. An example of a transversely isotropic material for which \(v_{2}=\overline{v_{1}}\in\mathbb{C}\) is zinc with (all in GPa) \(K=80\), \(\ell=33\), \(n=50\), \(m=63\), \(g=40\), for which \(v_{1}=1.1284+0.6465 i\) to 4dp. (Private communication with P.J. Withers and T. Mori.)
 2.
This material is Lead Zirconate Titanate, a material frequently used in piezoelectric composites
Notes
Acknowledgements
The author is grateful to the Engineering and Physical Sciences Research Council for funding his research fellowship (EP/L018039/1). He is also thankful to Prof. J. Berger (Colorado School of Mines) and Prof. H. Ammari (ETH, Zurich) for helpful comments.
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