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

Consider an unbounded host medium of infinite extent into which a cylindrical inhomogeneity is embedded, which itself can be considered to be of infinite extent along its axis and place the \(\tilde{x}_1\tilde{x}_2\) plane coincident with the cross section of the cylinder. With translational invariance in the \(\tilde{x}_3\) direction then, the problem is purely two-dimensional, as depicted in Fig. 1 where the cross section of the inhomogeneity is defined as \(\tilde{D}\in {\mathbb {R}}^2.\) Since it is often useful to consider a specific physical problem, certainly in terms of language and terminology, the potential problem here is described in the context of steady-state thermal conductivity.

The equation governing the steady-state temperature distribution \(T(\tilde{{\mathbf {x}}})\) where \(\tilde{{\mathbf {x}}}=(\tilde{x}_1,\,\tilde{x}_2)\) is the Cartesian position vector in the medium described above and depicted in Fig. 1 iswhere the assumption is that no heat sources are present. The free-space Green’s function associated with the host phase satisfiesAssuming continuity of temperature and normal flux across \(\partial \tilde{D},\) the resulting temperature distribution may be straightforwardly derived in integral equation form aswhich holds for all \(\tilde{{\mathbf {x}}}.\) Here \(T^{\infty }(\tilde{{\mathbf {x}}})\) is the solution to the equivalent problem satisfying (2.1) with no inhomogeneity present (or equivalently with \(C_{ij}^1 = C_{ij}^0\)). Upon taking derivatives of (2.3) with respect to \(\tilde{x}_i\) and noting the property \(\partial \tilde{\mathcal {G}}/\partial \tilde{y}_i= {-}\partial \tilde{\mathcal {G}}/\partial \tilde{x}_i\) it is found that for all \(\tilde{{\mathbf {x}}},\) where the ith component of the temperature gradient has been defined as \(e_i=\partial T/\partial \tilde{x}_i.\)

If attention is restricted to anisotropy where the principal axes of the host and inhomogeneity are aligned, all anisotropic potential problem inhomogeneity problems can be reduced to isotropic ones by rescaling the spatial variable \(\tilde{{\mathbf {x}}}\) which modifies the inhomogeneity shape. The problem is then solved in the scaled domain and mapped back to physical space [7]. Interest here is for general shaped inhomogeneities, and therefore we consider the isotropic problem only. Therefore take \(C^0_{ij}=k_0\delta _{ij},\, C^1_{ij}=k_1\delta _{ij}\) and (2.4) becomeswhere \(\kappa =k_1/k_0\) and we have introduced the associated two-dimensional isotropic Greens function \(\tilde{G}\) satisfying \(\nabla ^2 \tilde{G} + \delta (\tilde{{\mathbf {x}}}-\tilde{\mathbf {y}}) = 0,\) i.e.It is important to note that different references place the Dirac delta function on either side of the governing equation for the Green’s function. Of course, this merely changes the sign of the Green’s function, but it is important in terms of lifting information directly from one reference to another. We follow the convention in [7, 28].

Equation (2.5) is an integral equation for the potential gradient \({\mathbf {e}}\) and determining this inside \(\tilde{D},\) which we shall denote as \({\mathbf {e}}^1\) then tells us the field everywhere. If one assumes that \({\mathbf {e}}^{\infty }\) is a uniform vector, taking \(\mathbf {y}\in \tilde{D}\) and assume that \({\mathbf {e}}^1\) is also uniform, we arrive at the expressionThis is simply (1.2), where \(\tilde{\mathbf {S}}\) is Eshelby’s tensor with componentsand where deriving (2.7) has exploited the symmetry \(S_{ij}=S_{ji}.\) Under these assumptions of uniformity, Eq. (2.7) is then only consistent if \(\tilde{\mathbf {S}}\) is a uniform tensor, which, from Eshelby’s strong conjecture in two dimensions, we know is only true in the case of elliptical inhomogeneities \(\tilde{D}.\) In two dimensions then, when the far-field temperature gradient if uniform, then one can use (2.7) in the case of isolated elliptical inhomogeneities to predict the uniform interior fields and Eshelby’s tensor takes the form ([7, (4.36)])where \(\epsilon =a_2/a_1\) where \(a_j\) is the semi-axis of the ellipse directed along the \(x_j\) axis. However, for non-elliptical inhomogeneities one must return to (2.5) and solve this integral equation. This approach also applies even for elliptical inhomogeneities when the far-field temperature gradient is non-uniform! One approach would be to pose some form for the interior strain in some basis, e.g. a polynomial expansion, e.g. for \(\tilde{{\mathbf {x}}}_1\in \tilde{D},\) for some coefficients \(A^j_{mn},\, j=1,\,2\) to be determined. Inserting these expansions in the integral equation will lead to consistency conditions on the coefficients (a linear system) with coefficients that are integrals of the formClearly such an expansion (2.10) could not be expected to converge everywhere but can certainly be of use. This method has been utilised to predict the fields interior to interacting ellipsoids in [28, Sect. 23] for example.

Consider now a different configuration to the previous section. An unbounded host medium of infinite extent contains within it an inclusion—a cylindrical region which is considered to be of infinite extent along its axis and place the \(\tilde{x}_1\tilde{x}_2\) plane coincident with the cross section of the cylinder. The conductivity tensor of the inclusion region is the same as its exterior. So-called eigenstrains (independent of \(\tilde{x}_3\)) are imposed in this region which induce a perturbed field. Note that unlike the previous section there is no imposed field at infinity. As with the previous section, with translational invariance in the \(\tilde{x}_3\) direction then, the problem is purely two-dimensional. Following Mura [28] but in the potential context, the equation governing T subject to the eigenstrain field \({\mathbf {e}}^*\) iswhere \(\chi (\tilde{{\mathbf {x}}})=1\) when \(\tilde{{\mathbf {x}}}\in \tilde{D}\) and is zero otherwise. Using the Green’s function as defined in (2.2) the induced field can be writtenwhich holds for all \(\tilde{{\mathbf {x}}}.\) As in the previous section, taking derivatives of (2.3) with respect to \(\tilde{x}_i\) and noting the property \(\partial \tilde{\mathcal {G}}/\partial \tilde{y}_i= {-}\partial \tilde{\mathcal {G}}/\partial \tilde{x}_i\) it is found that for all \(\tilde{{\mathbf {x}}},\) Following the same arguments as for the inhomogeneity problem if attention is restricted to anisotropy where the principal axes of the host and inhomogeneity are aligned, all anisotropic problems for general \(\tilde{D}\) reduce to the isotropic case due to scaling arguments and therefore set \(C^0_{ij}=k_0\delta _{ij}\) so that (2.14) becomeswhere \(\tilde{G}\) is defined in (2.6). With eigenstrain components \(e_j^*\) in the form of polynomials, the right-hand side of (2.15) can clearly then be written as linear combinations of terms in the form (2.11).

One can employ eigenstrain to ‘represent’ the effects of an inhomogeneity [28]. First add some far-field forcing to the problem so that (2.15) becomesNow comparing (2.16) with (2.5), we see that these are equivalent ifIn the case of an elliptical inhomogeneity with uniform \(e_i^{\infty },\, \tilde{S}_{ij}\) is uniform and is given by (2.9), and we havewhere \(\varvec{{\mathcal {A}}}=[\mathbf {I}+(\kappa -1)\tilde{\mathbf {S}}]^{-1}\) is known as the concentration tensor [7]. For imposed fields that are of polynomial form a polynomial eigenstrain of the same order can be employed to represent an inhomogeneity [28]. For more complex shaped domains, however, a general eigenstrain would be required.

Let us call (2.11) the generalised Eshelby tensor of order \((m,\,n).\) Upon definingwe haveDetermining explicit results for these tensors for general \(\tilde{D}\) is clearly difficult, if not impossible, unless we restrict attention to specific classes of \(\tilde{D}.\) Instead, then consider an approach that is based on polynomial expansions. The general methodology shall be developed in the next section before it is applied to specific cases.

As we shall discuss in the Sect. 3, we will seek polynomial expansions as approximations to \(\tilde{J}^{mn}\) for a given \(\tilde{D}.\) As should be expected, it transpires that the choice of location about which the polynomial is expanded is extremely influential as to the domain of validity of the polynomial expansion. Suppose therefore that the polynomial expansion is centred on \(\tilde{{\mathbf {x}}}=\tilde{{\mathbf {r}}}.\) The methodology to follow relies upon orthogonality and therefore \(\tilde{J}^{mn}(\tilde{{\mathbf {x}}};\,\tilde{\mathbf {s}})\) shall first be written as a linear combination of \(\tilde{J}^{mn}(\tilde{{\mathbf {x}}};\,\tilde{\mathbf {s}}),\) i.e. we write Further, in an effort to yield a polynomial expansion that is convergent over a larger domain, one can expand about distinct points \(\tilde{{\mathbf {r}}}^j,\, j=1,\,2,\ldots ,p\) and then split the domain into non-intersecting support regions say \(\tilde{D}_j\) (such that \(\tilde{D}=\tilde{D}_1\cup \tilde{D}_2\cup \cdots \cup \tilde{D}_{p}\)) over which the expansion is locally valid. For example, with reference to Fig. 2 write (2.23) aswhere

Note that in principal more complex eigenstrain fields can be considered in (2.15); particularly in the context of the local expansion approach (2.24). One would Taylor expand the eigenstrain about \(\tilde{{\mathbf {r}}}^j,\, j=1,\,2,\ldots ,p\) inside \(\tilde{D}\) as above, which leads to generalised Eshelby tensors (polynomial eigenstrains). For example, if we havethen in \(\tilde{D}_j,\) Taylor expand \(f(\tilde{\mathbf {y}})\) above \(\tilde{\mathbf {y}}=\tilde{{\mathbf {r}}}\) to giveso that upon truncating

Since we are in two dimensions, we introduce the planar polar coordinate systemswith \(\tilde{\rho },\,\tilde{\zeta }\ge 0\) and \(0\le \theta ,\,\phi <2\pi .\) Define the boundary of the inhomogeneity as \(\tilde{\rho }=\tilde{f}_j(\theta )=L_jf_j(\theta ),\) where \(L_j=\min \tilde{f}_j(\theta )\) so that \(f_j(\theta )\ge 1,\) noting the subscript j that indicates dependence on \({\mathbf {r}}^j\) since this will affect the definition of \(\tilde{f},\, f\) and \(L_j.\) Finally, employ the scalings \(\tilde{\rho }=L_j\rho ,\, \tilde{\zeta }=L_j\zeta ,\) and defineand D as the scaled domain over which we integrate. We can write where

For general f (i.e. for a general shaped inhomogeneity D), it is clearly not possible to obtain an analytical form for the expression (3.6); generally, it must be evaluated numerically for any point \({\mathbf {x}}\in D.\) Given its importance in a number of inhomogeneity and inclusion problems as indicated above, it would therefore be useful if approximate schemes can be developed in order to generate expressions for the right-hand side. Let us suppose that we may expand J in powers of its argument, e.g. in the formfor non-negative integers \(p,\,q\) and some positive integer P,  where the notation \({\mathcal {J}}^{mn}\) has been used to indicate that this is an approximation to \(J^{mn}.\) We will see in fact that this approach is useful for many practical cases, especially with an appropriate choice of \({\mathbf {r}}^j.\) Note that the upper limit \(P\rightarrow \infty \) in general.

For this local expansion to be useful in (2.24), we need to re-introduce the dimensional variables, so that (3.7) becomeswhere \(\tilde{D}^{mn}_{pq}=D^{mn}_{pq}/L_j^{p+q}.\)

In order to evaluate the coefficients \(D_{pq}^{mn}\) it is then useful to write (3.7) in terms of the orthogonal functions \(\cos (n\theta ),\, \sin (n\theta ){\text {:}}\) where the coefficients \(a_j^{mn},\, b_j^{mn}\) are determined explicitly in terms of \(\rho \) and \(D^{mn}_{pq}\) by employing the plane polar form (3.2)\(_{I}\) for \({\mathbf {x}}-{\mathbf {r}}^j\) and then expand the powers of the resulting trigonometric functions in their multiple angle forms (this is straightforwardly done algorithmically in a symbolic package such as mathematica) so that for exampleand analogously for \(a_j^{mn},\, b_j^{mn}.\)

Next note that the \(D_{pq}^{mn}\) are not independent. We can see this by applying the Laplacian operator to (3.7) and exploiting the form of (3.6) with \(\nabla ^2 G + \delta (\mathbf {y}-{\mathbf {x}})=0\) to findComparing coefficients of \(x_1^a x_2^b\) provides equations linking the coefficients \(D_{pq}^{\delta \xi },\) so that for a fixed pair \(m,\,n\) we have for \(p,\,q\ge 0\) Next introduce the operators \({\mathcal {C}}_k(*)\) and \({\mathcal {S}}_k(*)\) as those that multiply \(*\) by \(\cos (k\theta )\) and \(\sin (k\theta ),\) respectively and then integrate over \(\theta \in [0,\,2\pi ).\) Equate (3.3) and (3.9) and apply the operators \({\mathcal {C}}_m\) and \({\mathcal {S}}_m\) in turn to each side of the resulting equations to yieldwhere \(\delta _{m0}\) is the Kronecker delta. Without loss of generality, set \(\rho =1,\) to find that where Note again here that by construction, \(\zeta =f_j(\phi )\) prescribes the boundary of the fibre with \(f_j(\phi )\ge 1\) so that \(\zeta \ge 1.\) By employing some exact results for certain integrals, we now show that we are able to write (3.14)–(3.16) in terms of a single integral in \(\phi \) involving the boundary function \(f_j(\phi ).\) To proceed, defineand this substitution allows us to simplify the expression with various terms reducing to zero by employing double angle formulae for the trigonometric functions. We then obtain whereNext we use the results (Gradshteyn and Ryzhik [34], Eq. 15 of Sect.  4.224 and Eq. 6 of Sect. 4.397)where \(k\ge 1\) in the latter. The \(\zeta \) integration can then be straightforwardly carried through using integration by parts, to determine that noting that we have introduced the notation \(F_0^{mn}\) for this integral form of \(a_0^{mn}\) so that we can refer to it later on. Proceeding analogously, definingwhereit can be straightforwardly shown that, for \(k\ge 1,\) We now employ the equations relating \(a_m,\, b_m\) and \(D_{pq}\) and equate these to the integral forms just developed above, so that, e.g. These equations coupled with the conditions (3.12) appropriately truncated gives a closed system from which the coefficients \(D_{pq}^{mn}\) are determined, i.e. this can be writtenwhere \({\mathbf {D}}\) is a vector of the unknowns \(D_{pq}^{mn}\) and \({\mathbf {F}}\) is a vector of the right-hand sides associated with (3.29)–(3.33) and (3.12). It should be noted that the matrix \({\mathbf {A}}\) is identical regardless of the shape of the inhomogeneity and the expansion point \({\mathbf {r}}.\) The latter affect only the right-hand side \({\mathbf {F}}.\) Therefore one can immediately write the explicit formand the only computation in order to form the polynomial approximation are the integrals in \({\mathbf {F}}.\) In the next section several examples are considered with various eigenstrains and a number of inhomogeneities of distinct shape, in order to illustrate the efficacy of this scheme.

First consider the case when \(\mathbf {s}=\mathbf {0}\) and the inhomogeneity is elliptical, withwhere \(a_1\) and \(a_2\) are the semi-axes of the ellipse in the \(x_{1}\) and \(x_{2}\) directions, respectively. Note that for convenience, we choose \(\min (a_1,\,a_2)=1.\) As stated above, for elliptical inhomogeneities, polynomial eigenstrains of order \(m+n\) will give rise to \(J^{mn}\) integrals that are polynomials of order \(m+n+2.\) Our first check is therefore to ensure that the method above reproduces this so-called property of polynomial conservation. Let us compute \(J^{30}\) which corresponds to a cubic eigenstrain. The method yields \(J^{30}\) as a fifth-order polynomial and gives rise to the three components of the Eshelby tensor \(S_{20}^{30},\, S_{11}^{30}\) and \(S_{02}^{30}.\) With all coefficients taken to five significant figures, for \(a_1=2,\, a_2=1,\) these polynomials take the formand surface plots of these components are provided in Fig. 4.

Now consider domains of more complex form than elliptical and for now take \(\mathbf {s}=\mathbf {0}.\) Gielis [35] utilised what is referred to as the superellipse shape function to reproduce many shapes commonly occurring in nature. The function is defined asHere \(\gamma \) can be considered a parameter associated with rotational symmetry, so for a given \(\gamma \) the shape produced will have \(\gamma \)-fold rotational symmetry. Further, a and b are associated with the aspect ratio of the shape, and \(\delta _{1},\, \delta _{2}\) and \(\delta _{3}\) help to further control the form of the shape. For instance, to reproduce an asymmetric shape, \(\delta _{1},\, \delta _{2}\) and \(\delta _{3}\) should be chosen to have distinct values and to produce ‘star’ shapes, \(\delta _{1}\) should be chosen less than \(\delta _{2}\) and \(\delta _{3}.\) Note that choosing \(\delta _{1}=\delta _{2}=\delta _{3}=2\) and \(\gamma =4\) returns the shape function governing an ellipse, with \(a=b\) further reducing the equation to that of a circle or radius a. The function (4.1) is incredibly versatile, and can be used to produce close approximations to many boundary shapes, including polygons, with the essential rule of thumb being to choose \(a=b=1,\, \gamma \) to be the degree of polygon sought, \(\delta _{2}\) and \(\delta _{3}\) to be large and equal, and \(\delta _{1}\) to be larger still—except for the square case, where all three \(\delta _j\) parameters are chosen to be equally large.

Figures 5, 6 and 7 illustrate how the polynomial approximation \({\mathcal {J}}^{mn}\) for square (with \(m=2,\,n=0\)), pentagonal (with \(m=0,\, n=3\)) and hexagonal (with \(m=1,\, n=1\)) inclusions tends to the numerical evaluation of the integral \(J^{mn}\) as the order of the polynomial increases by plotting the following measure of the error,across the whole of the domain. In each example \(a=b=1\) and \(\delta _{2}=\delta _{3}=\delta \) have been used in the superellipse definition, with the choices of the other parameters listed in the appropriate captions of the figures. Note that each shape features a different choice of m and n in order to illustrate that the scheme works equally well regardless of this choice.

These plots illustrate that as the order of polynomial approximation P increases, the blue regions (representing very small error) become more prominent and the yellow regions (representing moderate errors) become less so. In the case of the pentagon, both the polynomial approximation and the numerical evaluation are exactly zero when \(x_{2}=0\) (due to the choice of m and n), hence the dark blue line in the direction of this axis. The same is true for the hexagonal case when \(x_{1}=0\) and \(x_{2}=0.\) Note that if better representations are required in regions of moderate error, the polynomial expansion could be taken about a point close to this location in order to obtain ‘patched’ expansions in different regions of space.

So far no “re-expansion” has been required in the sense that the natural “origin” of the domains considered was the origin itself. Let us now consider the case of the so-called limacon, whose boundary is governed by the polar functionFigure 8a considers the error in approximating \(J^{02}\) for this shape with \(a=1\) and \(b=2\) by a 24th-order polynomial approximation with no re-expansion, i.e. \({\mathbf {r}}=\mathbf {0}.\) Notably, the error is particularly significant (bordering on order of magnitude 1000) in the farthest right-hand regions of the inclusion. This suggests that the centre of re-expansion should be positioned further to the right. Furthermore, the symmetry of the inclusion suggests the expansion point should remain along the \(x_{1}\) axis. In fact, the most obvious way of maximising the radius of convergence would be to move the basis to \({\mathbf {r}}=(1,\,0),\) as a circle of radius 2 based at this point will not only fit inside the limacon, but also cover the vast majority of the area of this domain. Figure 8b illustrates how the error improves when this re-expansion point is chosen, with the regions previously featuring the most extreme errors now featuring errors in the region of \(10^{-7}.\)

Let us now consider some more complex shapes of non-polygonal form. By choosing \(\gamma =4,\, a=1,\, b=10,\, \delta _{1}=4.2,\, \delta _{2}=17\) and \(\delta _{3}=1.5\) in the superellipse expression (4.1), we obtained what we shall term the ‘hourglass’ due to its resemblance to the traditional time-keeping device, as can be seen in Fig. 9a and take \(\mathbf {s}=\mathbf {0}.\)

As one would expect, basing an expansion of the polynomial approximation at the origin yields large errors in the extreme north and south regions of the domain, as illustrated in Fig. 10a, focusing on the \(x_{2}\ge 0\) region and illustrated in the case of \(J^{12}.\) The symmetry of the domain is such that it makes sense to split it into two domains: \(x_{2}\ge 0\) and \(x_{2} < 0,\) and use expansions about two points on the \(x_{2}\) axis equidistant from the origin in each. Once again, focusing on \(x_{2}\ge 0,\) Fig. 10b shows how the error improves upon centering the expansion about \({\mathbf {r}}=(0,\,1.13).\)

An alternative to consider is the hypogenis, as depicted in Fig. 9b, boundary of which is described bywith the factor of 3/2 giving rise to \(f(\theta ) \ge 1.\) In a similar vein to the ‘hourglass’, here it is advantageous to split the domain into the sub-domains \(x_{1}\ge 0\) and \(x_{1} < 0\) and use expansions about two points on the \(x_{1}\) axis equidistant from the origin for each. Figure 11 focuses on the \(x_{1}\ge 0\) region and shows that an expansion about \({\mathbf {r}}=(0.82,\,0)\) improves upon an expansion about the origin.

In all previous examples, \(\mathbf {s}=\mathbf {0}\) and expansions were taken based on an appropriate choice of \({\mathbf {r}}\) when it would improve the convergence of the scheme in various subdomains of the inhomogeneity or inclusion region. Let us now consider an example case where \(\mathbf {s} = (1,\,0),\) but the inclusion shape remains centred at the origin. Here the domain to be examined is once again a square. In the case of, for instance, \(J^{20}({\mathbf {x}};\,(1,\,0)),\) the expansion formula (2.23) will need to be utilised for \({\mathbf {r}}=\mathbf {0}\) in order to express \(J^{20}({\mathbf {x}};\,(1,\,0))\) in terms of \(J^{mn}({\mathbf {x}};\,\mathbf {0})\) integrals—terms which themselves shall need to be approximated by polynomial expansions about the origin. Figure 12 illustrates how increasing the order of these expansions improves the overall accuracy of the approximation of \(J^{20}({\mathbf {x}};\,(1,\,0)).\)

Finally, in order to illustrate the principle outlined in Sect. 2.4, (2.26) shall be examined for the case of a non-polynomial eigenstrain, e.g. when \(f(\mathbf {y})=\sin {y_{1}}\) and in the case of a circular domain D. In this case, the eigenstrain may be expanded about \(\mathbf {y}=\mathbf {0}\) to generate a sum of polynomial eigenstrain terms. The error in approximating the integral (2.26) by the truncated series (2.29) is defined asFigure 13 shows the error (4.4) when the Taylor expansion of the eigenstrain is truncated to the eleventh order (\(M=11,\, N=0\)).