## Abstract

We use the complex variable method and conformal mapping to derive a closed-form expression for the shear compliance parameters of some two-dimensional pores in an elastic material. The pores have an *N*-fold axis of rotational symmetry and can be represented by at most three terms in the mapping function that conformally maps the exterior of the pore into the interior of the unit circle. We validate our results against the solutions of some special cases available in the literature, and against boundary-element calculations. By extrapolation of the results for pores obtained from two and three terms of the Schwarz–Christoffel mapping function for regular polygons, we find the shear compliance of a triangle, square, pentagon and hexagon. We explicitly verify the fact that the shear compliance of a symmetric pore is independent of the orientation of the pore relative to the applied shear, for all cases except pores of fourfold symmetry. We also show that pores having fourfold symmetry, or no symmetry, will have shear compliances that vary with cos 4*θ*. An approximate scaling law for the shear compliance parameter, in terms of the ratio of perimeter squared to area, is proposed and tested.

## 1. Introduction

A basic problem in the mechanics of materials, with applications to ceramics (Rice 1998), rocks (Zimmerman 1991), and bones (Cowin 2001), lies in calculating the excess strain in a body due to the presence of a cavity or pore. Sevostianov & Kachanov (2007) have referred to this as Eshelby's second problem and developed a formalism in which the excess strain, Δ** ϵ**, due to inhomogeneity in an otherwise homogeneous medium subjected to a far-field stress

*σ*^{∞}, is expressed in terms of the fourth-order tensor, as follows:(1.1)where the colon denotes the tensor inner product. For example,

*H*

_{1212}connects the excess shear strain Δ

*ϵ*

_{12}to the applied remote shear stress, .

For ellipsoidal pores, or special cases thereof, the tensor can be expressed in terms of the Eshelby tensor, whose components are known (Eshelby 1957; Wu 1966). For non-ellipsoidal pores, however, exact known results are sparse. Three-dimensional non-ellipsoidal shapes remain generally intractable by analytical methods. In two dimensions, the complex variable methods pioneered by Kolosov and Muskhelishvili can be used, although the calculations are tedious, and essentially must be developed on a case-by-case basis. Zimmerman (1986) found the coefficient corresponding to hydrostatic compression (i.e. *H*_{iijj}), for the family of hypotrochoidal holes. Ekneligoda & Zimmerman (2006) extended this to a larger family of pore shapes that have an *N*-fold axis of rotational symmetry, and which can be represented by four terms in the mapping function from the unit circle. Jasiuk *et al.* (1994) and Kachanov *et al.* (1994) found the components of that correspond to deviatoric loading, for hypotrochoidal pores and some quasi-polygonal pores.

In this paper, we consider the family of pores treated by Ekneligoda & Zimmerman (2006) and calculate the shear-related coefficients such as *H*_{1212}. Symmetric pores provide simplified models of real pores, but are of interest in their own right, as rotationally symmetric inclusions have interesting properties (Wang & Xu 2004; Xu & Wang 2007). Moreover, our analytical solutions can be used to test our proposed scaling law for the shear compliance (see §6). In the process of deriving the solutions, we explicitly verify the fact, proved by Eroshkin & Tsukrov (2005) in a more general but somewhat abstract context, that pores (or inclusions) of *N*-fold symmetry are ‘isotropic’ with regards to their behaviour under far-field stresses, except for the anomalous cases of *N*=2 or 4. Furthermore, we show that for *N*=4, as well as for pores possessing no symmetry, the term *H*_{1212} varies as cos 4*θ*.

## 2. Formulation of the basic problem

Two-dimensional plane stress or plane strain elastostatic states can be represented in terms of two complex potentials, as follows (Sokolnikoff 1956; Muskhelishvili 1963):(2.1)where *ϕ* and *ψ* are complex-valued analytic functions, and *κ*=3−4*ν* for plane strain and (3−*ν*)/(1+*ν*) for plane stress. To formulate the problem of a body containing a cavity, with a state of pure shear stress at infinity, we use the following superposition argument (Savin 1961). First, consider a homogeneous body without a cavity, subjected to a pure shear of magnitude *τ* at infinity. For now, we take the shear to be aligned with the (*x*, *y*) axes, so that . As pointed out by Tsukrov & Novak (2002), for pores (and more generally, for elastic inclusions) possessing rotational symmetry, the coefficients of are insensitive to the orientation of the far-field stresses, except for the special cases of twofold or fourfold symmetry, which will be treated later (§5). The complex potentials corresponding to this state of stress are(2.2)We will refer to this as problem 1.

Now imagine that the body contains a single cavity defined by some simple closed contour *Γ*. The region outside of *Γ* can be mapped (figure 1) from the interior of the unit circle through a conformal mapping *z*=*ω*(*ζ*), about which more will be said below. If there is no stress acting at infinity, and a complex traction vector acting along the cavity surface, then the boundary condition for the two potentials can be written along *Γ* as (Sokolnikoff 1956)(2.3)where *F* is equal to *i* times the integral of the complex traction vector, *t*_{x}*+*i*t*_{y}, along the boundary contour, starting from some arbitrary point *z*_{0} on *Γ*.

The stress state of problem 1 will lead to the correct stresses at infinity, but an unwanted, non-zero traction on *Γ*. To remove this traction, we define problem 2 for the body with the cavity as the one in which the tractions on *Γ* are the negatives of those that would occur along *Γ* in problem 1, and in which the stresses at infinity are zero. The solution for a body with a traction-free cavity and shear stress at infinity is given by the sum of the solutions for problems 1 and 2.

The traction vector ** t** on

*Γ*in problem 1 is given by

**=**

*t***, where is the stress tensor, and**

*n***is the unit normal vector to**

*n**Γ*, which must point away from the solid body, which is to say, into the cavity. This normal vector is given by (Zimmerman 1986). So, the traction on

*γ*in problem 1 is(2.4)The function

*F*that appears on the right side of (2.2) is therefore(2.5)We take the liberty of sometimes writing

**as a column vector, sometimes as a row vector and sometimes in complex form as , depending on whichever is most convenient for the given purpose.**

*t*The conformal mapping function will be of the form . If only two non-zero terms in the mapping are taken, i.e. , the hole is a hypotrochoid that is a quasi-polygon having *n*+1 equal ‘sides’ (England 1971; Zimmerman 1986). In order for the mapping to be single-valued, and for *Γ* not to contain any self-intersections, *m*_{n} must satisfy the restriction . The choice *m*_{n}=0 yields a circle, whereas the limiting value *m*_{n}=1/*n* gives a pore with *n*+1 pointed cusps. For the particular choice *m*_{n}=2/*n*(*n*+1), the mapping coincides with the first two terms of the Schwarz–Christoffel mapping for an (*n*+1)-sided equilateral polygon and resembles a polygon with slightly rounded corners (Savin 1961; Levinson & Redheffer 1970; Zimmerman 1991).

If the pore contour possesses an (*n*+1)-fold axis of symmetry, only powers that differ by (*n*+1) will appear in the mapping function, i.e.(2.6)We will focus on this family of mappings, which contains the Schwarz–Christoffel quasi-polygons as special cases. The solution will be derived in detail for the case of a two-term mapping function. For space considerations, the solution will be presented without detailed derivation for the three-term case, which poses no fundamental additional difficulties aside from increased algebraic complexity. To simplify the notation slightly, in the sequel we will write the mapping function as(2.7)

## 3. Derivation of complex potentials

We now find the stress potentials for problem 2, in which there is traction acting on the pore contour, but no traction at infinity. First, we use the chain rule to write equation (2.3) in terms of *ζ*, as follows:(3.1)where we use *σ* to represent values of *z* on the unit circle *γ* in the *ζ*-plane. We note for later use that for points on *γ*, *σ*=e^{iα}, so . For problems such as this with no far-field traction, the complex potentials can be expressed (Sokolnikoff 1956) as power series that converge for *ζ*<1, i.e.(3.2)(3.3)The first term of (3.1) takes the form(3.4)

Recalling that the mapping function has the form , we expand the second term of equation (3.1) in a power series as follows:(3.5)There is no advantage to explicitly expanding out the triple product in equation (3.5), as there is no convenient way to display all the resulting terms. Instead, when we need to isolate, say, the *σ*^{1} terms, for example, it is straightforward to see directly from equation (3.5) that the coefficient of *σ*^{1} will be .

The third term in (3.1) takes the form(3.6)

The function *F* on the r.h.s. of (3.1) is given in (2.5) as , and *z* is given by . So, recalling that on *γ*, the r.h.s. of (3.1) takes the form(3.7)

Inserting (3.4)–(3.7) into (3.1), and equating the coefficients of each power of *σ* on both the sides of the equation, gives us equations involving the *b*_{k} and *c*_{k} coefficients. Assuming that *n*≥2, the coefficients of the *σ*^{1} term yield the condition(3.8)The coefficients of the *σ*^{n−2} term yield the condition(3.9)Solution of these two equations gives and . The coefficients of all other terms *σ*^{q}, for *q* not equal to 1 or *n*−2, yield homogeneous algebraic equations for the *b*_{k}, showing that all other *b*_{k} are zero. This procedure also yields equations for the *c*_{k}, but these are not as easily solved; instead, the *c*_{k} are found by a different method, as shown below. Hence, the first complex potential is(3.10)

This result is valid for *n*≥2. If *n*=1, which corresponds to an elliptical pore, the term *b*_{n−2}=*b*_{−1} does not appear in series (3.2), and so the foregoing calculations do not apply. Elliptical pores require separate, although simpler, calculations. Fortunately, elliptical pores have received extensive treatment already (Savin 1961; Thorpe & Sen 1985; Kachanov *et al.* 1994), and so we will restrict our attention to the cases *n*≥2.

To find the second complex potential, we start by taking the conjugate of the boundary condition (3.1), to arrive at(3.11)We now divide each term of this equation by , where *ζ* is an arbitrary point inside the unit circle *γ*, and then integrate around *γ*. With *ϕ* given by (3.10), the first term integrates out to(3.12)since for any *ζ* inside the unit circle, and all *k*≥1 (Godfrey 1959, p. 279).

The second integral takes the form(3.13)The integrand has two poles inside the unit circle, at *σ*=*ζ* and 0. There is also a pole at , but this lies outside *γ*, due to the constraint that must hold in order for the mapping to be single-valued. The residue at *σ*=*ζ* can be read off directly from the integrand of the l.h.s. of (3.13), in the form(3.14)

To find the residue at *σ*=0, we expand the terms in the denominator as a power series, multiply the resulting series together and collect terms of similar powers, after which the residue is identified as the coefficient of *σ*^{−1}; the result is(3.15)

As *ψ* is analytic inside *γ*, the integral of the third term in (3.9) is readily found to be(3.16)

The r.h.s. of (3.11) can be integrated by expanding the integrand in partial fractions, and using the residue theorem on each term(3.17)Collecting all the integrals gives the second complex potential in the form(3.18)

The displacement is found by inserting equations (3.10) and (3.18) into equation (2.1)(3.19)Note that the second term on the r.h.s. of equation (3.18) has been cancelled out by an equivalent term that arises from .

For a pore with three terms in its mapping function, , the same procedure eventually yields the following expressions for the two complex potentials:(3.20)

(3.21)

(3.22)

(3.23)

(3.24)

(3.25)

(3.26)

(3.27)

(3.28)

(3.29)

(3.30)

## 4. Calculation of the excess strain energy and the *H*_{1212} coefficient

The stored strain energy is the sum of the energy calculated in problem 1 and the energy from problem 2. In problem 1, the strain energy per unit volume of material is *τ*^{2}/2*G*, corresponding to the energy of the material without a pore. The excess strain energy due to the pore is found from the solution to problem 2, by calculating the work done by the traction that acts over the pore surface. This traction vector is, from (2.4), given by , where we use the angular variable *α* in the *ζ*-plane to parametrize the pore boundary. Hence,(4.1)where in the last integral we treat the displacement ** u** as a complex number, .

With the displacement given by equation (3.19), and noting that on *γ*, then , and the work term for problem 2 becomes(4.2)where . Recalling that we are assuming for now that *n*≠1 or 3, only three terms in the integrand lead to non-zero contributions to the integral, and we find(4.3)But is precisely the area of the pore, *A* (Zimmerman 1986), so(4.4)We must now add the energy due to problem 1, which is *τ*^{2}(*A*_{∞}−*A*)/2*G*, where *A*_{∞} is the total area of the body, including the pore, and *A* is the area of the pore. The total energy in the body is therefore given by(4.5)We recall that , and then divide *W* by *A*_{∞}, to express the excess energy density, per unit area of the body, as(4.6)where *c*=*A*/*A*_{∞} is the volume fraction of the pores, i.e. the porosity. The excess energy is proportional to the volume fraction of pores, because we are ignoring interactions between nearby pores.

The term *τ*^{2}/2*G* is the energy density that would exist in the body, under stress, but in the absence of the pore. Hence, the second term on the right in equation (4.6) is the excess energy due to the pores, and comparison with equation (1.1) shows that, aside from the *τ*^{2} factor that represents *H*_{1212}(4.7)For plane strain, *κ*+1=4(1−*ν*), and so(4.8)

For a pore with three terms in its mapping function, the result of a lengthy calculation, analogous to that shown above for the two-term case, is(4.9)where *B*_{1} is given by equation (3.22).

## 5. Special case of fourfold symmetry

As discussed in several previous papers (Kachanov *et al.* 1994; Eroshkin & Tsukrov 2005), pores with fourfold rotational symmetry are not isotropic with respect to a far-field shear stress. The fact that *n*=3 is a special case can be anticipated from equation (4.2), where additional terms are seen to arise in the stored strain energy expression.

To analyse this case, we need to consider a more general applied shear stress that may be oriented at an arbitrary angle. If the far-field shear stresses of equation (2.4) are rotated by an angle *θ*, then by applying the usual rules of stress transformation and performing calculations analogous to those given in equations (2.4) and (2.5), we find that the function *F* in equation (2.5) will be multiplied by e^{−2iθ}. This multiplicative factor of e^{−2iθ} then carries through to the r.h.s. of equations (3.8) and (3.9), having the effect of multiplying *b*_{1} by e^{−2iθ} and multiplying *b*_{n−2} by e^{2iθ}. Hence, the first complex potential is(5.1)Similarly, the second complex potential takes the form(5.2)The displacement is found by inserting the potentials (5.1) and (5.2) into equation (2.1)(5.3)Calculations similar to those that led to equation (4.2) show that the work term gains an additional multiplicative factor of e^{2iθ}. Hence, for the case where the far-field stresses are rotated by an angle *θ*,

(5.4)

This new factor of e^{4iθ} survives the integration only if *n*=3 (recalling that the above equations do not apply for *n*=1). Hence, the result obtained in §4 for the cases *n*=2, 4, 5, … but *θ*=0, in fact holds for arbitrary *θ*, thus explicitly verifying that these pores are isotropic with respect to shear.

For *n*=3, the integral in (5.4) gives, aside from the terms already found in §4, an additional term . Again recalling that , we find(5.5)The shear compliance therefore contains a small term that varies as cos 4*θ*. Owing to the constraint , the relative amplitude of this term cannot exceed 1/3. Equation (5.5) also shows that expression found in §4 for *θ*=0 in fact represents the average value of *H*_{1212} over all angles, as would apply to a collection of such pores whose orientation angles are randomly distributed. Note that equation (5.5) holds only for *n*=3, but we write it with *n* appearing explicitly so as to allow easy comparison with equation (4.4), which holds for *n*=2 or *n*≥4.

## 6. Results and discussion

To validate our results, we compare them against the results for several simple geometrical shapes that have been obtained previously, and also compare them with the values obtained by boundary-element calculations. The boundary-element calculations were performed using a code described by Martel & Muller (2000), which is a simplified version of the more general two-dimensional BEM code from Crouch & Starfield (1983) which is based on the displacement discontinuity method.

The boundary-element calculations correspond to problem 2, in which the tractions are applied to the boundary of the pore in an infinite region, with no stresses at infinity. The pore boundary is discretized into a number of equal length elements. The number of elements is always taken to be a multiple of *n+*1, the number of sides of the pore, to ensure that two boundary elements meet precisely at each corner or cusp, so that the corners are not chopped off. After the stress and displacement fields are calculated, the strain energy is calculated by numerically performing the integral specified in equation (4.1). We generally found that roughly 300 boundary elements were sufficient to achieve convergence of the computed shear compliance.

Our analytical expressions for the shear compliance parameter *H*_{1212} showed that for plane strain this parameter is proportional to (1−*ν*)/*G*, as well as to the porosity, *c*. Hence, we can define a dimensionless shear compliance as *h*=*H*_{1212}*G*/(1−*ν*)*c*. For a circular hole, equation (4.8) recovers the well-known result *h*=2.

The normalized shear compliances of some quasi-polygonal holes represented by two or three terms in the Schwarz–Christoffel mapping function are shown in table 1. The present results agree with those cases previously derived by Jasiuk *et al.* (1994) and Kachanov *et al.* (1994). By extrapolating our results out to *N*→∞, where *N* represents the number of terms taken in the mapping function, we can find the shear compliance of regular polygons, which are represented by an infinite number of terms in the Schwarz–Christoffel mapping. As shown by Ekneligoda & Zimmerman (2006) for the case of the pore compressibility, this can be done by linear extrapolation, using *N*^{−2} as the independent variable, and extrapolating the results for 1/4 and 1/9 down to 1/*N*^{2}=0. The results are *h*=2.667, 2.359, 2.143 and 2.100, for triangles, squares, pentagons and hexagons, respectively. (This result for a square represents an average over all angles of the applied shear stress, relative to the pore.) For the case of pore compressibility, these values were 3.250, 2.414, 2.192 and 2.105, showing that the shearability of a polygonal pore varies with the number of sides less drastically than does the compressibility.

The present results are more general than previous available results such as those shown in table 1, as we have derived closed-form expressions that are valid for all values of the parameters *m*_{1} and *m*_{2}, not necessarily restricted to be those corresponding to the Schwarz–Christoffel quasi-polygons. Explicit results for some additional shapes are shown in tables 2 and 3.

The complex variable method can in principle be used to compute the shear compliance of a pore of any shape. However, calculation of the mapping coefficients for the sort of complex pore shapes that are observed in rocks or ceramics is extremely tedious, and computationally non-trivial (Sisavath *et al.* 2001; Tsukrov & Novak 2002). Moreover, the additional complexity exhibited by the analytic solution for the three-term mapping as opposed to the two-term mapping indicates that analytical treatment of pores whose mappings contain more than three terms is not feasible. Hence, it would be useful if the pore shear compliance could be calculated from simple geometric attributes of the pore shape. Such a capability would be useful in attempts to estimate elastic moduli from images of heterogeneous media, as attempted, for example, by Tsukrov *et al.* (2005).

Zimmerman (1991) proposed a scaling law in which the normalized pore compressibility is proportional to the dimensionless geometric parameter *P*^{2}/2*πA*, where *P* is the perimeter and *A* is the area of the pore. This parameter measures deviations from circularity, and in fact 4*πA*/*P*^{2} is often referred to as the ‘roundness’ in particle technology literature. The multiplicative factor 1/2*π* in the scaling law was chosen so that the correlation is exact for a circular pore. This law was tested on various pore shapes by Zimmerman (1991), Tsukrov & Novak (2002) and Ekneligoda & Zimmerman (2006), with errors that were always less than 22%, and usually less than 10%.

We have found that the shear compliance also seems to scale linearly with the geometrical parameter *P*^{2}/2*πA*, according to the following expression:(6.1)Results for a few symmetric pores are shown in tables 2 and 3, where it is seen that the scaling law (6.1) is very accurate in these cases. These results are also plotted in figure 2, along with the results for a few other symmetric pores not shown in the tables. Figure 2 shows graphically that the correlation between *h* and *P*^{2}/2*πA* is strong and non-trivial. Equation (6.1) is not the best-fitting line through the data, but represents a compromise between accuracy and elegance, in the sense that it is exact for a circle, and the coefficients of the constant term and the *P*^{2}/2*πA* term are round numbers. It is interesting that the slope of the normalized shearability, with respect to the geometric parameter *P*^{2}/2*πA*, is *half* of the corresponding slope in the scaling law for the normalized hydrostatic compressibility.

Validation of the scaling law for symmetric pores does not guarantee that it will be useful for ‘realistic’ pore shapes, so we have also tested it for pores observed in scanning electron micrograph (SEM) images of a SiC ceramic (Reynaud *et al.* 2005; figure 3). For the present purposes, we ignore any issues related to the fact that these are two-dimensional slices of three-dimensional pores (Lock *et al.* 2002), and treat the pores as two-dimensional. Again, the ‘exact’ values were assumed to be those obtained from the BEM calculations. The image analysis software provides a digitized representation of the pore boundary in the form of an irregular polygon with a large number *M* of sides (typically several dozen). The computational boundary elements were nominally equally spaced, with the spacings modified slightly so that two elements met precisely at each corner of the *M*-sided polygon. As with the regular shapes obtained from conformal mapping, convergence of the shear compliance was usually achieved with approximately 300 elements.

A mean value of the shear compliance of each pore is found by first calculating *H*_{1212} as a function of *θ*, where *θ* is the angle between the pore orientation and the far-field shear. The calculations were done for an increment in *θ* of 5°, i.e. for *θ*=0°, 5°, 10°, etc. and then taking the average value. This averaging process would have been appropriate, if the ceramic had contained a collection of pores whose angular orientations were randomly distributed. It is encouraging that the mean shear compliances of the ceramic pores tend to fall close to the line of the scaling law (6.1).

Our numerical calculations showed that, for irregular pores having no axis of symmetry, *H*_{1212} always varies with cos 4*θ*, aside from some phase shift in the angle. One particular example is shown in figure 4, for a pore in Berea sandstone. This result can be explained as follows. Anti-clockwise rotation of the far-field shear stresses by 90° is equivalent to changing the sign of the stresses, since, for example, the component *τ*_{xy}=*τ* that points in the positive *y*-direction gets rotated into a component *τ*_{yx}=−*τ* that points in the negative *x*-direction. But the strain energy is a quadratic form in the stresses, so multiplying the stresses by −1 cannot change the energy, and so cannot change *H*_{1212}. Hence, *H*_{1212} can only vary with angle according to cos 4*θ*, cos 8*θ*, etc. Allowing for a possible phase shift, the term that varies trigonometrically with 4*θ* would also yield a sin 4*θ* term, etc. But the transformation law for fourth-order tensors contains only trigonometric terms such as sin *kθ* or cos *kθ*, where *k cannot exceed* 4 (Eroshkin & Tsukrov 2005). Hence, the only angular-dependent terms that can appear in *H*_{1212} are those containing cos 4*θ* and sin 4*θ*, or, equivalently, cos(4*θ*+*δ*), where the constant *δ* represents the phase shift. Hence, a cos(4*θ*+*δ*) term should be expected to appear, but vanishes when the pore has *N*-fold rotational symmetry of order *N*=3, 5, 6, etc. Alternatively, by retracing the steps that led to equation (5.4), one can show that in the general case of an arbitrary number of terms in the mapping function, the angle *θ* enters the integral that expresses the excess energy only in the form of e^{4iθ}. These results are consistent with those of Wang & Xu (2004), who showed that the average value of the Eshelby tensor of an *N*-fold rotationally symmetric elastic inclusion is independent of the inclusion orientation, for all *N* except 2 and 4.

## 7. Conclusions

We have obtained a closed-form solution for the shear compliance of some symmetric pores, as quantified by the coefficient *H*_{1212} that was defined by Kachanov *et al.* (1994). This parameter *H*_{1212} was found to be proportional to (1−*ν*)/*G*, where *G* and *ν* are the elastic parameters of the host material, and to the porosity, *c*. Hence, a normalized shear compliance can be defined as *h*=*H*_{1212}*G*/(1−*ν*)*c*. Our analytical results were validated against some previously computed special cases and against BEM calculations. It was found that the shear compliance tends to increase with macroscopic ‘non-circularity’ of the pore and also increases if the pore has cusps. Perhaps unexpectedly, we found that the shear compliance of a non-circular pore differs from that of a circular pore less drastically than does the analogous variation in hydrostatic pore compressibility.

Our analytical expressions explicitly verify the general theorem of Eroshkin & Tsukrov (2005), which specified that for pores (or inclusions) having *N*-fold rotational symmetry, *H*_{1212} is independent of the orientation of the pore with respect to the shear direction, except for the case of *N*=4. For *N*=4, we found explicitly that *H*_{1212} varies with cos 4*θ*, where *θ* quantifies the orientation of the applied shear with respect to the pore, i.e. , for two constants *H*_{0} and *H*_{1}. More generally, we showed that this type of variation, involving *θ* only through terms such as cos 4*θ* or sin 4*θ*, also occurs for pores possessing no rotational symmetry.

We mention as an aside, both for completeness and to avoid any possible confusion, that the case of *N*=2 is anomalous yet again. For such pores containing only two terms in the mapping function (2.7), i.e. ellipses, calculations such as those described above eventually lead to the result , which contains no dependence on *θ*! At first, this seems to contradict the assertion by Eroshkin & Tsukrov (2005) that pores with 180° rotational symmetry will not be elastically ‘isotropic’. This paradox is removed by noting that the coefficient *H*_{1111} of an elliptical pore, for example, *will* contain angular-dependent terms (Tsukrov & Novak 2002), and so the overall elastic moduli tensor of a body containing aligned elliptical pores will indeed be anisotropic.

Finally, we proposed a scaling law, , where *P* is the perimeter of the pore. We tested this law against our analytical solutions, and on some pores observed in SEM images of a porous ceramic. The agreement was quite good, with an average error, over 17 ceramic pores, of only −1.9%. We are currently using this scaling law, along with an earlier scaling law for the hydrostatic pore compressibility (Zimmerman 1991), to estimate the elastic moduli of porous materials from SEM images.

## Footnotes

- Received October 13, 2007.
- Accepted November 29, 2007.

- © 2008 The Royal Society