On 'strange' properties of some symmetric inhomogeneities

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On ‘strange’ properties of some symmetric inhomogeneities Sofia G. Mogilevskaya and Henryk K. Stolarski

Research Cite this article: Mogilevskaya SG, Stolarski HK. 2015 On ‘strange’ properties of some symmetric inhomogeneities. Proc. R. Soc. A 471: 20150157. http://dx.doi.org/10.1098/rspa.2015.0157 Received: 6 March 2015 Accepted: 5 June 2015

Subject Areas: materials science Keywords: elasticity problems, symmetric inhomogeneity, Eshelby inclusion Author for correspondence: Sofia G. Mogilevskaya e-mail: [email protected]

Department of Civil, Environmental, and Geo-Engineering, University of Minnesota, 500 Pillsbury Drive S.E. Minneapolis, MN 55455, USA The paper presents an analysis of elasticity problems involving a single inhomogeneity which possesses certain types of symmetries. As observed earlier, isotropic problems of that kind exhibit some ‘strange’ and remarkable properties. Under the action of uniform far-field stresses, the averages of the fields inside the inhomogeneities preserve the structure of the far-field loads. Here, it is shown that these properties are exhibited for a wider class of problems, which include anisotropic and non-uniform materials subjected to either far-field loads or constant transformational strains within the inhomogeneity. The proposed modified Eshelby technique facilitates a straightforward analysis of these problems, which is based entirely on the assumed symmetry. It is also shown that some remarkable properties of symmetric inhomogeneities discovered here are related to the so-called ‘strange’ properties of the Eshelby inclusions extensively covered in the literature. Some implications of these findings are discussed.

1. Introduction Recently, it was numerically discovered in [1] that regular polygonal and other symmetric inhomogeneities possess some remarkable properties. Under the action of uniform far-field stresses, the averages of the fields inside the inhomogeneities preserve the structure of the far-field loads: hydrostatic far-field load results in hydrostatic average stresses within the inhomogeneity and deviatoric far-field load produces deviatoric average stresses. In [1], it was suggested that those properties might be related to the properties of the corresponding Eshelby inclusions (inhomogeneity whose properties are the same as those of the matrix, see [2–4]). The latter properties were studied in [5–14] where they were referred to as ‘strange’. In this paper, we present rigorous and rather general investigation of the numerically discovered properties 2015 The Author(s) Published by the Royal Society. All rights reserved.

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(a) Problem formulation Consider an infinite two-dimensional elastic matrix containing an elastic inhomogeneity (the elastic properties of the inhomogeneity are different from those of the matrix). It is assumed that the shape of the inhomogeneity is invariant under the group of 90◦ rotations. Both the matrix and the inhomogeneity may be anisotropic and non-uniform, respectively, but they should possess the same type of symmetry as that assumed for the shape of the inhomogeneity, i.e. they are invariant under 90◦ rotations. Examples of such inhomogeneities are shown in figure 1; the lines within their areas schematically indicate anisotropy and varying material properties. The composite system is subjected to uniform far-field stresses at infinity. We introduce the Cartesian coordinate system with its centre located at the centre of the rotational symmetry. If the problem possesses the axes of geometric symmetry (e.g. figure 1a), these axes will be used as the Cartesian coordinate axes (otherwise, the axes can be chosen arbitrary provided that they are orthogonal). In any such system, the far-field load σ ∞ can be represented as follows: ∞ ∞ σ∞ = σ∞ (11) + σ (22) + σ (12) =

 ∞ σ11 0

 0 0

 +



0

0

0

∞ σ22

 +

0

∞ σ12

∞ σ12

0

 (2.1)

and the state of the load σ ∞ can be obtained by superposition of the three separate loads involved in the right-hand side of the above equation. The stresses inside the matrix/inhomogeneity system due to the unit load of the first type, ∞ = 1, may be represented as follows: σ11  σ (11) =

f11 (x1 , x2 )

f12 (x1 , x2 )

f12 (x1 , x2 )

f22 (x1 , x2 )

 ,

(2.2)

where the functions fkm (.) (k, m = 1, 2) depend on the shape of the inhomogeneity and on the elastic properties of both the inhomogeneity and the matrix.

...................................................

2. Stresses within a two-dimensional symmetric inhomogeneity under uniform far-field load

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for two- and three-dimensional elasticity problems and establish their relations with the ‘strange’ properties of the corresponding Eshelby inclusions. The analysis presented here is based entirely on the assumed symmetry of the problem and is valid for quite general classes of anisotropic matrix/inhomogeneities systems with certain position-dependent material properties. Somewhat similar symmetric considerations are also employed in [12]; however, the material considered in that paper is isotropic. In addition, the 90◦ rotational symmetry has been explicitly excluded in [12], while here all considerations are fundamentally based on that type of symmetry. The paper is structured as follows. In §2, we consider the two-dimensional elastic problem of an infinite domain containing a single anisotropic and non-uniform inhomogeneity and subjected to uniform stress fields at infinity. We show, that for certain types of symmetries in geometry and material properties, some average stresses preserve characteristics of the far-field load. In §3 (with the details provided in appendix A), we analyse similar problem in three-dimensional setting but assuming that the inhomogeneity is subjected to the constant transformational strains (Eshelby inhomogeneity). Section 4 contains a discussion of the average strains within the inhomogeneity as well as some additional comments related to the constitutive tensor of the inhomogeneity. In that section an analogue of Eshelby tensor for average strain is introduced and its structure discussed. In §5 and in appendices B and C, the special case of isotropic inclusion is considered in more details. Finally, in §6, we discuss the obtained results and their implications for a number of applications.

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(a)

(b)

x2

x2

3

Figure 1. Two-dimensional inhomogeneities with π/2 rotational symmetry. (Online version in colour.)

Similarly, the stresses in the inhomogeneity due are  f¯11 (x1 , x2 ) σ (12) = f¯12 (x1 , x2 )

∞ = 1, to the unit load of the second type, σ12

f¯12 (x1 , x2 ) f¯22 (x1 , x2 )

 ,

(2.3)

in which the functions f¯km (.) (k, m = 1, 2) also depend on the shape of the inhomogeneity and on the elastic properties of both the inhomogeneity and the matrix.

(b) Implications of assumed symmetry By the virtue of the assumed rotational symmetry, the 90◦ rotation of the problem associated with ∞ = 1 results in the problem that is geometrically and physically identical to the original the load σ12 ∞ = −1. This implies that the functions f¯ (.) must possess the following one but loaded by σ12 km properties: ⎫ f¯22 (x2 , −x1 ) = −f¯11 (x1 , x2 ),⎪ ⎪ ⎪ ⎪ ⎬ (2.4) f¯11 (x2 , −x1 ) = −f¯22 (x1 , x2 ) ⎪ ⎪ ⎪ ⎪ ⎭ and f¯12 (x2 , −x1 ) = f¯12 (x1 , x2 ). ∞ = 1 results in the problem Likewise, 90◦ rotation of the system loaded by the load σ11 ∞ = 1. Therefore, geometrically and physically identical to the original problem but loaded by σ22 the stresses inside the inhomogeneity due to the latter load are   f22 (x2 , −x1 ) −f12 (x2 , −x1 ) σ (22) = , (2.5) f11 (x2 , −x1 ) −f12 (x2 , −x1 )

in which the functions fkm (.) (k, m = 1, 2) are the same as those involved in equation (2.2). ∞ = σ ∞ = p, σ ∞ = 0, and Two different types of the far-field load: hydrostatic load, σ11 22 12 ∞ ∞ ∞  deviatoric load, σ11 = −σ22 = q, σ12 = q , will be considered next. It will be shown subsequently that hydrostatic far-field load will result in the hydrostatic average stresses within the inhomogeneity, whereas deviatoric load will yield deviatoric average stresses.

(c) Hydrostatic far-field load In this case, the stresses inside the inhomogeneity are obtained by the superposition of the stresses of equations (2.2) and (2.5) as follows:   f11 (x1 , x2 ) + f22 (x2 , −x1 ) f12 (x1 , x2 ) − f12 (x2 , −x1 ) H (σ I ) = p . (2.6) f12 (x1 , x2 ) − f12 (x2 , −x1 ) f22 (x1 , x2 ) + f11 (x2 , −x1 )

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x1

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x1

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A

in which α, β = 1, 2 and A is the area of the inhomogeneity. Therefore, the average stresses (σ I )H  inside the inhomogeneity is of the form 2

(σ I )H  = PI,

(2.8)

2

where I is the second rank identity tensor and P is given by the following expression:  p P= [f11 (x1 , x2 ) + f22 (x1 , x2 )] dA. A A

(2.9)

Thus, it is clear that the hydrostatic far-field load produces hydrostatic average field inside the inhomogeneity.

(d) Deviatoric far-field load The stresses inside the inhomogeneity due to this load are obtained by linear combination of the stresses of equations (2.2), (2.3) and (2.5) as follows: ⎤ ⎡ f11 (x1 , x2 ) − f22 (x2 , −x1 ) f12 (x1 , x2 ) + f12 (x2 , −x1 ) ⎦ (σ I )D = q ⎣ f12 (x1 , x2 ) + f12 (x2 , −x1 ) f22 (x1 , x2 ) − f11 (x2 , −x1 ) ⎤ ⎡ f¯11 (x1 , x2 ) f¯12 (x1 , x2 ) ⎣ ⎦. +q (2.10) f¯12 (x1 , x2 ) f¯22 (x1 , x2 ) It can be shown, with the use of the conditions of equations (2.4), (2.7), that the average stresses (σ I )D  inside the inhomogeneity can be written as follows: ⎡ ⎤ Q1 + Q1 Q2 + Q2 D ⎦, (σ I )  = ⎣ (2.11) Q2 + Q2 −Q1 − Q1 where Q1 =

q A

2q Q2 = A

and

Q1 =

q A

Q2 =

q A

 A

[f11 (x1 , x2 ) − f22 (x1 , x2 )] dA,

  

A

f12 (x1 , x2 ) dA, 

q f¯11 (x1 , x2 ) dA = − A A A

f¯12 (x1 , x2 ) dA.



⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ f¯22 (x1 , x2 ) dA⎪ ⎪ ⎪ ⎪ A ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(2.12)

Thus, deviatoric load produces deviatoric average stress in the inhomogeneity.

(e) Three-dimensional problems In general, the analysis presented in the preceding sections is applicable to three-dimensional problems. However, the three-dimensional analysis must be modified to address the following issues. (1) The rotational symmetry in two-dimensional setting involved rotation only about the third axis of the system, the x3 -axis. That admitted the shapes similar to the one shown in figure 1b. In three-dimensions, the logic used in previous sections must be

...................................................

A

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The assumption of geometric and constitutive symmetry of the inhomogeneity combined with the simple change of variables leads to the following relations:   fαβ (x1 , x2 ) dA = fαβ (x2 , −x1 ) dA, (2.7)

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With a slight modification of Eshelby’s approach [2], the problem of an inhomogeneity undergoing constant transformational strain (eigenstrain) can be reduced to the problem involving mechanical load similar to that considered in §2. To illustrate the difference between two- and three-dimensional problems discussed at the end of the previous section, this approach will be presented here, and in the following section, in three-dimensional setting. A particular case of a problem that involves an isotropic inclusion (Eshelby problem [2]) will be discussed in more detail in §5.

(a) Problem formulation In this section, we assume that the matrix is isotropic and characterized by the shear modulus μ and Lamè constant λ. We also assume that a single inhomogeneity occupies the volume V that is geometrically and mechanically invariant under the group of 90◦ rotations about the assumed coordinate axes, see, e.g., the inhomogeneities presented in figure 2. The lightly shaded areas of the first shape in figure 2 may be replaced by a variety of different shapes, e.g. those shown in figure 3. This accommodates the inhomogeneity that is anisotropic and whose material properties may exhibit a special dependency on the position. In particular, for an orthotropic inhomogeneity, the principal directions of orthotropy do not have to coincide with the coordinate axes. In addition, the lines corresponding to the principal directions of orthotropy may also be curved (as shown in figure 1b for the two-dimensional case). As stated above, the inhomogeneity ∗

undergoes constant transformational strain (eigenstrain) described by the tensor . It will be shown that, for the problem under consideration, volumetric eigenstrain results in hydrostatic average stress within the inhomogeneity, while deviatoric eigenstrain yields deviatoric average stress.

(b) Analysis The following analysis will be based on a slightly modified Eshelby technique [2,15] of ‘imaginary cutting, straining and welding’. In our technique, we use ‘loading, imaginary cutting, welding and unloading’. ∗

We first subject the homogeneous matrix to the load that produces uniform eigenstrain  everywhere in the matrix. This far-field load results in the following constant stresses: ∗

∗

∗2

σ = 2μ + λ tr  I ∗

∗

(3.1)

and in the corresponding tractions t = σ · n, on the surface L that represents the trace of the boundary of the strained but stress-free inhomogeneity (n = nm is the unit normal vector to L directed away from the matrix and the notation ‘·’ here and below is used for the simple contraction of tensors). The state of deformations and of the stresses in the matrix remains unchanged when the material of the region inside of the curve L is cut out but the tractions

...................................................

3. Stresses within a symmetric inhomogeneity undergoing constant transformational strain

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applicable when the system is rotated about any of the three axes. This imposes additional restrictions on the admissible shapes of the inhomogeneity and on its properties. This will be discussed and illustrated in the following sections. The Cartesian coordinate system to be used in the analysis of three-dimensional problems consists of the three axes rotations about which leave the system unchanged. (2) As in the two-dimensional case, the matrix and the inhomogeneity may be anisotropic and inhomogeneous, as long as they exhibit the same rotational symmetry as that assumed for the shape of the inhomogeneity.

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x3

x3

x3 x2

x1

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x1

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x2

6

x2

x1

Figure 2. Three-dimensional inhomogeneities with π/2 rotational symmetry. (Online version in colour.)

x3

x3

x2

x2

x1

x1

Figure 3. Admissible replacements for lightly shaded areas of the first shape in figure 2. (Online version in colour.)

∗

∗

t = σ · n applied to the matrix along L. The stress-free inhomogeneity is then welded to the matrix. ∗

At that stage the strains everywhere in the matrix and inhomogeneity are constant and equal to , while the stresses exist only in the matrix. At the next stage, the entire system is unloaded. It is clear that the stresses within the inhomogeneity appear only during the unloading, while the (possibly highly non-uniform) strains occurring in the unloading process have to be superposed with the ∗

eigenstrain  everywhere. ∗

The stresses −σ during the unloading can be decomposed into six different parts as follows: ∗

∗

∗

∗

∗

∗

∗

− σ = σ (11) + σ (22) + σ (33) + σ (12) + σ (13) + σ (23) ,

(3.2)

∗

where the tensors σ (kj) are structured in a way illustrated for k = j = 1 and k = 2, j = 3 below, ⎤ ⎡ ∗ ⎡ ⎤ 0 0 0 σ11 0 0 ∗ ∗ ∗ ⎥ ⎢ ⎢ ⎥ (3.3) σ (11) = − ⎣ 0 σ23 ⎦ . 0 0 0⎦ and σ (23) = − ⎣0 ∗ 0 0 0 0 σ23 0 ∗

Each of the loads σ (kj) should be, of course, supplemented by the corresponding load at the matrix/inhomogeneity boundary L ∗

∗

t(kj) = σ (kj) · n,

(3.4)

in which n is defined earlier. ∗ ∗ The effects of the far-field stresses σ (kj) (accompanied by the loads t(kj) ) on the stresses inside the inhomogeneity will be analysed separately for each of the six cases. ∗

∗ = 1 are Let us assume that the stresses inside the inhomogeneity due to the load σ (11) , with σ11 ⎡ ⎤ g11 (x1 , x2 , x3 ) g12 (x1 , x2 , x3 ) g13 (x1 , x2 , x3 ) ⎢ ⎥ ⎥ σ (11) = ⎢ (3.5) ⎣g12 (x1 , x2 , x3 ) g22 (x1 , x2 , x3 ) g23 (x1 , x2 , x3 )⎦ ,

g13 (x1 , x2 , x3 )

g23 (x1 , x2 , x3 )

g33 (x1 , x2 , x3 )

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⎫ g11 (x1 , x2 , x3 ) = g11 (x1 , x3 , −x2 ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ g22 (x1 , x2 , x3 ) = g33 (x1 , x3 , −x2 ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ g33 (x1 , x2 , x3 ) = g22 (x1 , x3 , −x2 ), ⎪ (3.6)

⎪ g12 (x1 , x2 , x3 ) = −g13 (x1 , x3 , −x2 ),⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ g13 (x1 , x2 , x3 ) = g12 (x1 , x3 , −x2 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ g23 (x1 , x2 , x3 ) = −g23 (x1 , x3 , −x2 );

and

(ii) conditions due to 180◦ rotation about x1 g23 (x1 , x2 , x3 ) = g23 (x1 , −x2 , −x3 ), g13 (x1 , x2 , x3 ) = −g12 (x1 , −x2 , −x3 ), g13 (x1 , x2 , x3 ) = −g13 (x1 , −x2 , −x3 ) and

gkk (x1 , x2 , x3 ) = gkk (x1 , −x2 , −x3 ), ∗

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ (3.7)

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ k = 1, 2, 3. ∗

∗ = 1 is identical to that with the load σ The problem under the load σ (22) , with σ22 (11) rotated by 90◦ around the x3 -axis. Therefore, the stresses inside the inhomogeneity due to that load are ⎤ ⎡ g22 (−x2 , x1 , x3 ) −g12 (−x2 , x1 , x3 ) g23 (−x2 , x1 , x3 ) ⎥ ⎢ (3.8) σ (22) = ⎢ g11 (−x2 , x1 , x3 ) −g13 (−x2 , x1 , x3 )⎥ ⎦. ⎣−g12 (−x2 , x1 , x3 ) g23 (−x2 , x1 , x3 ) −g13 (−x2 , x1 , x3 ) g33 (−x2 , x1 , x3 ) ∗

∗ = 1 is considered in similar manner The solution of the problem with the load σ (33) , with σ33 ∗

by rotation of the state associated with the load σ (11) by 90◦ around the axis x2 . As the result, the stresses inside the inhomogeneity due to that load are ⎡ ⎤ g33 (−x3 , x2 , x1 ) g23 (−x3 , x2 , x1 ) −g13 (−x3 , x2 , x1 ) ⎢ ⎥ (3.9) σ (33) = ⎢ g22 (−x3 , x2 , x1 ) −g12 (−x3 , x2 , x1 )⎥ ⎣−g23 (−x3 , x2 , x1 ) ⎦. −g13 (−x3 , x2 , x1 ) −g12 (−x3 , x2 , x1 ) g11 (−x3 , x2 , x1 ) ∗

∗ = 1 are of the following The stresses inside the inhomogeneity due to the load σ (12) , with σ12 form: ⎡ ⎤ g¯ 11 (x1 , x2 , x3 ) g¯ 12 (x1 , x2 , x3 ) g¯ 13 (x1 , x2 , x3 ) ⎢ ⎥ ⎥ (3.10) σ (12) = ⎢ ⎣g¯ 12 (x1 , x2 , x3 ) g¯ 22 (x1 , x2 , x3 ) g¯ 23 (x1 , x2 , x3 )⎦ , g¯ 13 (x1 , x2 , x3 ) g¯ 23 (x1 , x2 , x3 ) g¯ 33 (x1 , x2 , x3 )

in which the functions g¯ km (.) (k, m = 1, 2,3) also depend on the shape of the inhomogeneity and on the elastic properties of both the inhomogeneity and the matrix. In the case of the latter load, the 90◦ rotation of the composite system around the axis x3 results in the reversal of the load (the change of its sign). Consequently, the functions g¯ involved in

...................................................

(i) conditions due to 90◦ rotation about x1

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where the functions gkm (.) (k, m = 1, 2, 3) depend on the shape of the inhomogeneity and on the elastic properties of both the inhomogeneity and the matrix. Owing to the assumed rotational symmetry, the functions g involved in equation (3.5) satisfy various relations extensively used in the sequel. For example,

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equation (3.10) must also satisfy various conditions, for example:

8

and

∗

∗

∗ = 1 and load σ ∗ The corresponding stresses due to the loads load σ (13) , with σ13 (23) , with σ23 = 1 ◦ can be obtained from those given by equation (3.10) by 90 rotation around the axes x1 and x2 , respectively. They are ⎡ ⎤ −¯g13 (x1 , x3 , −x2 ) g¯ 12 (x1 , x3 , −x2 ) g¯ 11 (x1 , x3 , −x2 ) ⎢ ⎥ (3.12) σ (13) = ⎢ g¯ 33 (x1 , x3 , −x2 ) −¯g23 (x1 , x3 , −x2 )⎥ ⎣−¯g13 (x1 , x3 , −x2 ) ⎦ g¯ 12 (x1 , x3 , −x2 ) −¯g23 (x1 , x3 , −x2 ) g¯ 22 (x1 , x3 , −x2 )

⎡

and

g¯ 33 (x3 , x2 , −x1 )

−¯g23 (x3 , x2 , −x1 )

⎢ σ (23) = ⎢ ⎣−¯g23 (x3 , x2 , −x1 )

−¯g13 (x3 , x2 , −x1 )

g¯ 22 (x3 , x2 , −x1 ) g¯ 12 (x3 , x2 , −x1 )

−¯g13 (x3 , x2 , −x1 )

⎤

⎥ g¯ 12 (x3 , x2 , −x1 ) ⎥ ⎦. g¯ 11 (x3 , x2 , −x1 )

(3.13)

In the following, we consider two separate cases of prescribed uniform eigenstrain: volumetric and deviatoric.

(c) Volumetric eigenstrain In this case, it is assumed that 2

∗

 V = aI,

(3.14)

which results in the following volumetric stresses in the matrix: 2

∗

σ V = (2μ + 3λ)aI.

(3.15)

Thus, the unloading process in this case can be achieved by the superposition of only three ∗

∗

∗

loads σ (11) , σ (22) and σ (33) of those involved in equation (3.2) with ∗

∗

∗

σ11 = σ22 = σ33 = (2μ + 3λ) a.

(3.16)

Superposition of the stresses of equations (3.5), (3.8) and (3.9) yields the following stresses inside the inhomogeneity: (σ I )V = (2μ + 3λ) a [σ (11) + σ (22) + σ (33) ].

(3.17)

In appendix A, it is shown that the average stress σ I V has the following form: 2

(σ I )V  = PI,

(3.18)

 (2μ + 3λ)a [g11 (x1 , x2 , x3 ) + g22 (x1 , x2 , x3 ) + g33 (x1 , x2 , x3 )] dV (3.19) V V and the proof relies exclusively on the assumed symmetry of the problem described by equations (3.6), (3.7) and (3.11). Thus, for any inhomogeneity problem satisfying the rotational symmetry assumed here, a purely volumetric transformational strains applied inside the inhomogeneity result in hydrostatic average stress in that inhomogeneity. where

P=

...................................................

(3.11)

⎪ g¯ 33 (x1 , x2 , x3 ) = −¯g33 (x2 , −x1 , x3 ),⎪ ⎪ ⎪ ⎪ ⎪ ⎪ g¯ 23 (x1 , x2 , x3 ) = −¯g13 (x2 , −x1 , x3 ) ⎪ ⎪ ⎪ ⎪ ⎪ g¯ 13 (x1 , x2 , x3 ) = −¯g23 (x2 , −x1 , x3 ).⎭

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⎫ g¯ 22 (x2 , −x1 , x3 ) = −¯g11 (x1 , x2 , x3 ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ g¯ 11 (x2 , −x1 , x3 ) = −¯g22 (x1 , x2 , x3 ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ g¯ 12 (x2 , −x1 , x3 ) = g¯ 12 (x1 , x2 , x3 ),

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(d) Deviatoric eigenstrain b

⎢ ∗ ()D = ⎢ ⎣c

c

d

⎥ ⎥, ⎦ −(1 − α)b

−αb

d

⎤ (3.20)

e

e ∗

which results in the following deviatoric stresses (σ )D in the matrix: ∗

∗

(σ )D = 2μ()D .

(3.21)

The unloading process in this case can be realized by the superposition of all six loads defined by equation (3.2). Thus, the stresses inside the inhomogeneity are obtained by superposing the stresses given by equations (3.5), (3.8)–(3.10), (3.12) and (3.13). As a result, the stresses within the inhomogeneity have the following form: (σ I )D = −2μ[bσ (11) − αbσ (22) − (1 − α)bσ (33) + cσ (12) + dσ (13) + eσ (23) ].

(3.22)

It is shown in appendix A that the average stresses within the inhomogeneity are deviatoric and have the following structure: ⎤ ⎡ C D B11 ⎥ ⎢ (3.23) (σ I )D  = ⎢ E ⎥ ⎦ ⎣ C B22 D E B33 with B11 + B22 + B33 = 0. Finally, we note that the average stress field within the inhomogeneity due to any load, including constant far-filed load or arbitrary eigenstrain can, of course, be always decomposed into its volumetric and deviatoric parts. However, the fact that the hydrostatic (deviatoric) part of it is entirely associated with volumetric (deviatoric) eigenstrains (or far-field load) is less obvious. It is particularly so, considering that, as stated above, the analysis adopted here is valid for a non-isotropic, and even inhomogeneous, material and for inhomogeneity of quite complex geometry.

4. Strains within a symmetric inhomogeneity The preceding symmetry-based analysis can be used also to evaluate the average strains inside the inhomogeneity. In a manner analogous to that used in the preceding sections, it can be shown that hydrostatic load (or volumetric eigenstrain) leads to volumetric average strains within the inhomogeneity, while deviatoric load (eigenstrain) yields deviatoric average strains. In what follows, we employ constitutive relations to discuss the implications of the above conclusions on other relevant tensors such as constitutive tensor and the Eshelby tensor for average strains. The approach will be illustrated in the context of transformational strain only. The total strains at the point x inside the inhomogeneity due to the action of arbitrary constant ∗

 have the following form:

∗

 I (x) =  + S(x) : σ I (x)

(4.1)

in which S is the elasticity compliance tensor, and the standard notation ‘:’ is used for the double contraction of tensors. The average strains (x) can, therefore, be written as follows: ∗

 I (x) =  + S(x) : σ I (x),

(4.2)

which shows that, in the case of non-uniform material, the average strain cannot be related to the ∗

∗

average stress in a standard linear way. It is interesting to observe, however, that when  = ()V the left-hand side of equation (4.2) is volumetric, which implies that the average strain related

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⎡

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In this case, it is assumed that

9

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to the stress σ (involved in the same equation) must also be volumetric. This should be the case regardless of the functional form of the compliance tensor S(x), as long as the required symmetry ∗

∗

(4.3)

Note that the above equation should remain valid for volumetric and deviatoric eigenstrains

 considered separately. However, whether or not the hydrostatic average stress is related to the volumetric average strain depends on the nature of anisotropy descried by the constant tensor S, which implies that some restrictions on tensor S should be imposed. From the preceding sections and from the comment at the beginning of this section, it is known ∗

that volumetric (deviatoric) eigenstrain  results in the hydrostatic (deviatoric) average stresses σ I  and volumetric (deviatoric) average strains σ I  when the constant tensor S satisfies the required rotational symmetry. Below we discuss the general form of such tensor S. To study the range of admissible anisotropic tensors S (the tensors that satisfy the required symmetry), we assume that principal directions of anisotropy are defined by constant orthonormal vectors ak with k = 1, 2 in two dimensions and k = 1, 2, 3 in three dimensions. Then any allowed 90◦ rotation of the inhomogeneity exchanges the positions of these vectors. The requirement that under such group of rotations the system should remain invariant leads to the conclusion that relative to vectors ak the material should posses cubic symmetry, that is S should be of the form 4

2

2

4

S = α I + β I ⊗ I + γ J,

(4.4)

where α, β, γ are arbitrary constants and ⎫ ⎪ ⎪ I= aj ⊗ a ⊗ aj ⊗ a ,⎪ ⎪ ⎪ ⎪ ⎪ ⎪ =1 j=1 ⎪ ⎪ ⎪ ⎪ ⎪ k ⎬  2 I= a ⊗ a ⎪ ⎪ ⎪ =1 ⎪ ⎪ ⎪ ⎪ ⎪ k ⎪  4 ⎪ ⎪ ⎪ J= aj ⊗ aj ⊗ aj ⊗ aj ⎪ ⎭ 4

and

k k  

(4.5)

j=1

and standard notation ⊗ is used for the dyadic product of vectors. For two-dimensional problems, where the only relevant rotation is about the normal to the plane of the system, vectors ak may be aligned arbitrarily, e.g. along the straight lines shading the area in figure 1a. In three-dimensional setting, the system should remain invariant under 90◦ rotation about all three axes. In this case, the physical invariance is preserved only if ak are aligned with the three orthogonal axes around which the rotation takes place. ∗

∗

2

Consider again two cases of the prescribed eigenstrains: volumetric  = ()V = aI and ∗

∗

deviatoric  = ()D to verify that tensor S given by equation (4.4) connects volumetric (deviatoric) strains to hydrostatic (deviatoric) load. In the case of volumetric eigenstrain, the average stress inside the inhomogeneity is hydrostatic and given by equation (3.18). By substituting this stress in equation (4.3) with S given by

...................................................

∗

 I (x) =  + S : σ I (x).

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∗

conditions are fulfilled. The same logic applies to the case when  = ()D . When the material of the inhomogeneity is uniform (even for an orthotropic material), the relationship between the average stresses and strains is of the following linear form:

10

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equation (4.4), one can obtain the average strains  I  as 2

4

2

11

2

4

2

2

2

2

2

= (a + αP + 3βP + γ P)I.

(4.6)

Thus, that volumetric eigenstrain, which leads to hydrostatic average stress, produces volumetric average strains within the inhomogeneity when S is given by equation (4.4). ∗

Similarly, assuming that the eigenstrain is deviatoric, i.e. tr = 0, we have shown that the average stress inside the inhomogeneity is deviatoric, i.e. trσ I  = 0. From equations (4.3) and (4.4), it follows that 4

∗

2

2

4

 I  =  + (α I + β I ⊗ I + γ J) : σ I  2

∗

=  + ασ I  + β trσ I I + γ

k 

σ I  a ⊗ a .

(4.7)

=1

Thus, using the fact that, ak · ak = 1, for any k = 1, 2, 3, we obtain ∗

tr I  = tr + α trσ I  + 3β trσ I  + γ trσ I  = 0,

(4.8)

which proves that the average strain related to average stresses via equations (4.3) and (4.4) is deviatoric. An interesting observation is made when an analogue of the Eshelby tensor is developed for average strains within the inhomogeneity. To this end, it is first noted that the stresses within the ∗

inhomogeneity can be related to the transformational strains  as follows: ∗

σ I (x) = T(x) : ,

(4.9)

where T(x) is a suitable tensor field of rank four. Substituting the stresses of equation (4.9) into equation (4.3), one gets ∗

∗

∗

 I (x) =  + S(x) : T(x) :  = E : , in which

(4.10)

4

E = I + S(x) : T(x)

(4.11)

is a constant tensor of rank four, equivalent to that introduced by Eshelby [2] in his analysis of ellipsoidal inclusion problem, but this time related to the average strain within the inhomogeneity. ∗

Based on the preceding discussion, we know that volumetric  yields volumetric average ∗

strains within the inhomogeneity, while deviatoric  yields deviatoric average strains. In addition, because of the assumed rotational symmetry, the same conclusion should be valid after any 90◦ rotation. Thus, the arguments leading to equation (4.4) also imply that tensor E has to have structure given by equation (4.4), independently of the allowed anisotropy and spacial variation of inhomogeneity’s properties. Remark. While the previous analysis of the average stresses was done for the region occupied by the inhomogeneity, the results of that analysis are applicable to any other region that is located either inside of the inhomogeneity or surrounds it, as long as the required symmetry of the region is preserved. In addition, the regions fulfilling those symmetry requirements can also be multiply connected.

5. Isotropic inclusion The problem was extensively studied in a number of previous publications (see [5–14] and references therein). Owing to isotropy of the material involved, it allows for direct representations

...................................................

2

= aI + αPI + 3βPI + γ PI

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 I  = aI + (α I + β I ⊗ I + γ J) : PI

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As discussed in §3b, the stresses within the inclusion only appear during the unloading. Therefore, they can be evaluated from the solution of the problem of an infinite medium containing a single inclusion of the same properties as those of the matrix and subjected to the far-field ∗

load−σ given by equation (3.1). The boundary conditions on the matrix–inclusion interface are ∗

those of continuity of the corresponding displacements and of −σ nm magnitude jumps in the corresponding tractions. For an inclusion of arbitrary shape, this problem can be effectively solved with the use of Kolosov–Muschelishvili formalism, and the solution is briefly described in appendix B. √ The stresses at any point z = x1 + ix2 , i = −1 within the inclusion and in the matrix can be derived (see appendix B) from the complex variables integral representations proposed in [17]. The stresses within the inclusion have the following form: ⎫ κ −1 ∗ 2 ∗ ∗ ∗ ∗ ⎪ (σ11 + σ22 ) + Re[(σ22 − σ11 − 2iσ12 )F(z)]⎪ ⎪ ⎪ κ +1 κ +1 ⎪ ⎪ ⎪ ⎬ 1 ∗ ∗ ∗ (σ22 − σ11 + 2iσ12 ) σ22 (z) − σ11 (z) + 2iσ12 (z) = − ⎪ κ +1 ⎪ ⎪ ⎪ ⎪ ⎪ 1 ∗ ∗ ∗ ∗ ∗ ⎭ + [(κ − 1)(σ11 + σ22 ) + (σ22 − σ11 − 2iσ12 )]F(z), ⎪ κ +1 σ11 (z) + σ22 (z) = −

and

(5.1)

in which κ = 3 − 4ν, ν is the Poisson ratio of the material, and 1 F(z) = 2π i

 L

dτ¯ , τ −z

(5.2)

where L is the boundary of the inclusion, τ = τ1 + iτ2 ∈ L, a bar over a symbol denotes complex conjugation, the direction of travel along L is counterclockwise. The stresses in the matrix are obtained by superposing the stresses during the unloading with ∗

those due to the applied load σ . They are ⎫ 2 ∗ ∗ ∗ ⎪ ⎪ Re[(σ22 − σ11 − 2iσ12 )F(z)] ⎬ κ +1 ⎪ 1 ∗ ∗ ∗ ∗ ∗ ⎪ [(κ − 1)(σ11 + σ22 ) + (σ22 − σ11 − 2iσ12 )]F(z).⎭ σ22 (z) − σ11 (z) + 2iσ12 (z) = κ +1 σ11 (z) + σ22 (z) =

and

(5.3)

The strains within the inclusion and the matrix can be obtained from the stresses of equations (5.1) and (5.3) as follows: ⎫ κ −1 ∗ ∗ ⎪ [σ11 (z) + σ22 (z)] + χA (z)(ε11 + ε22 )⎪ ⎪ ⎪ 4μ ⎪ ⎪ ⎬ 1 [σ22 (z) − σ11 (z) + 2iσ12 (z)] ε22 (z) − ε11 (z) + 2iε12 (z) = ⎪ ⎪ 2μ ⎪ ⎪ ⎪ ⎪ ∗ ∗ ∗ ⎭ + χA (z)(ε22 − ε11 + 2iε12 ), ε11 (z) + ε22 (z) =

and

(5.4)

...................................................

(a) Two-dimensional complex variables analysis

12

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of all elastic fields. As a result, the so-called ‘strange’ properties of symmetric inclusions can be studied in more details. In this section, we first use the proposed ‘loading, imaginary cutting, welding and unloading’ technique in combination with complex variables formalism to reproduce the results reported in [12,14] for two-dimensional problems. In addition, we obtain two-dimensional analogues of some shape-independent properties of the Eshelby tensor reported for more general case of prescribed eigenstrains and material properties in [16]. The extension to three-dimensional problems is discussed subsequently.

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where

 χA (z) =

13

1 z∈A

∗

∗

∗

∗

∗

∗

∗

(i) for the volumetric eigenstrain (ε11 = ε22 = ε, ε12 = 0 or σ11 = σ22 = 2(μ + λ)ε = 4μ/ ∗

(κ − 1)ε,

∗ σ12

= 0)

⎫ C ∗ ∗ ⎪ (σ11 + σ22 ) ⎪ ⎬ κ +1 ⎪ κ −1 ∗ ∗ ⎪ (σ11 + σ22 )F(z),⎭ σ22 (z) − σ11 (z) + 2iσ12 (z) = κ +1 σ11 (z) + σ22 (z) = −

and

(5.5)

where C = κ − 1 within the inclusion and C = 0 in the matrix. For inclusions possessing 90◦ rotational symmetry, considerations similar to those used in §§2 and 3 lead to the conclusion that σ11 (0) = σ22 (0), σ12 (0) = 0 (where the origin of the coordinate system coincides with the centre of rotational symmetry). Thus, at that centre, the state of stresses is hydrostatic. This implies that the value of the left-hand side of the second equation in equation (5.5) vanishes at z = 0 and F(z). This is in agreement with findings presented in [12,14]. ∗

∗

∗

∗

∗

∗

∗

∗

∗

(ii) for the deviatoric eigenstrain (ε11 = −ε22 = ε, ε12 = 0 or σ11 = −σ22 = 2με, σ12 = 2με12 ) ⎫ 2 ∗ ∗ ∗ ⎪ Re[(σ22 − σ11 − 2iσ12 )F(z)]⎪ σ11 (z) + σ22 (z) = ⎪ ⎪ κ +1 ⎪ ⎪ ⎬ 1 ∗ ∗ ∗ (5.6) [−C1 (σ22 − σ11 + 2iσ12 ) ⎪ and σ22 (z) − σ11 (z) + 2iσ12 (z) = ⎪ κ +1 ⎪ ⎪ ⎪ ⎪ ∗ ∗ ∗ ⎭ + (σ22 − σ11 − 2iσ12 )F(z)], where C1 = 1 within the inclusion and C1 = 0 in the matrix. For this type of eigenstrain, and 90◦ rotational symmetry of the inclusion, the stress at the centre must be σ11 (0) = −σ22 (0), σ12 (0) = 0. This implies that the hydrostatic stress at the centre described by the first equation in (5.6) vanishes, while the deviatoric part defined by the second equation in (5.6) does not. So, the state of stresses at the centre is deviatoric. Vanishing of the hydrostatic part of stress at the inclusion centre associated with deviatoric eigenstrain implies vanishing of integral (5.2) at z = 0. This condition is in agreement with the one drawn in the case of hydrostatic load, and it should be so as the value of that integral is entirely related to the geometry of the inclusion. Summarizing these results, we can conclude that the stresses within the inclusion are (i) for the volumetric eigenstrain and inclusion of any shape σ11 (z) + σ22 (z) = −

κ −1 ∗ ∗ (σ11 + σ22 ), κ +1

(5.7)

(ii) for the volumetric eigenstrain and rotationally 90◦ symmetric inclusion σ22 (0) − σ11 (0) = 0, (iii) for the deviatoric eigenstrain and rotationally

σ11 (0) + σ22 (0) = 0 and

σ12 (0) = 0,

90◦

σ22 (0) − σ11 (0) + 2iσ12 (0) = −

symmetric inclusion

(5.8) ⎫ ⎪ ⎬

∗ 1 ∗ ∗ ⎭ (σ22 −σ11 +2iσ12 ).⎪ κ +1

(5.9)

...................................................

and A is the area occupied by the inclusion. The following expressions for the stresses are obtained from equations (5.1)–(5.4) when volumetric and deviatoric eigenstrains are considered separately.

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0 z∈ /A

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∗ εk (z) = Dkmn (z)εmn

and re-writing equations (5.1) and (5.4) as   ⎫ κ +2 κ 1 ∗ ⎪ ⎪ − Re F(z) ε11 ε11 (z) = ⎪ ⎪ κ +1 2 2 ⎪ ⎪   ⎪  ⎪ ⎪ ⎪ κ −2 κ −4 ∗ ∗ ⎪ + Re F(z) ε22 ,⎪ + − + (κ − 2)Im F(z)ε12 ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ ⎪   ⎬ κ −2 κ −2 1 ∗ − ε22 (z) = − Re F(z) ε11 ⎪ κ +1 2 2 ⎪ ⎪ ⎪    ⎪ ⎪ κ +2 κ +2 ⎪ ∗ ∗ ⎪ + Re F(z) ε22 + κIm F(z)ε12 + ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ⎪  ∗  ⎪ ∗ ⎪ 3ε22 ε11 ⎪ 1 ∗ ⎭ Im F(z) + Im F(z) + [κ − Re F(z)]ε12 ⎪ and ε12 (z) = κ +1 2 2

(5.11)

(5.12)

the following shape-independent properties of two-dimensional tensor can be easily proved (note that κ − Re F(z) = D1212 (z) + D1221 (z)): ⎫ Dkk (z) = 2, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ , Dkk (z) = ⎬ 1−v (5.13) ⎪ ⎪ D12 (z) + 2D1112 (z) − 2D2212 (z) = 0 ⎪ ⎪ ⎪ ⎪ ⎪ κ ⎪ D1112 .⎭ and D2212 = κ −2

...................................................

In the view of equations (5.7) and (5.9), it means that the averages of equation (5.10) coincide with the values of the corresponding stress combinations at the centre of the inclusion. This is another yet agreement with the finding presented in [12,14]. The formalism presented above can be also used to obtain additional shape-independed properties similar to those reported in [16] and references therein. For example, by introducing the Eshelby tensor with the components Dkmn (z) as follows:

14

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We emphasize again, that equations (5.1)–(5.6) are valid for the inclusion of arbitrary shape. It is shown in appendix B that, for the case of circular boundary, the integral of equation (5.2) vanishes for any point z within the inclusion. The integral-independent terms involved in equations (5.1), (5.5) and (5.6) are the stresses related to the isotropic part of Eshelby’s tensor introduced in [12]. They are, of course, uniform and coincide with those for the circular inclusion, which is in accordance with the result presented in [12]. Here, we can make even a stronger conclusion that, under the volumetric eigenstrain, the hydrostatic stresses and strains within the inclusion of arbitrary shape are constant and the hydrostatic stresses within the matrix containing an inclusion of arbitrary shape vanish. Shape-dependency of the stresses within the inclusion manifests itself exclusively via the integral of equation (5.2). This integral is the complex conjugate of the integral used in the basic integral representation of [14]. Thus, equations (5.5) and (5.6) provide the essential link with the results of [12,14] related to the ‘strange’ properties of symmetric inhomogeneities. The derivations presented in [14] are based on conformal mapping and Laurent series expansions for the mapping functions and are quite involved. In appendix B, we show in that, in many cases of the boundary L, the main results of [14] could be obtained by performing direct evaluation of the basic integral. Some properties can even be proved without the need to calculate the basic integral. For example, in appendix C, we show that, for inclusions possessing 90◦ rotational symmetry, the average of the stresses given by equation (5.1) are ⎫ κ −1 ∗ ∗ ⎪ ⎪ (σ11 + σ22 ) σ11 (z) + σ22 (z) = − ⎬ κ +1 (5.10) 1 ∗ ∗ ∗ ⎪ ⎪ ⎭ and σ22 (z) − σ11 (z) + 2iσ12 (z) = − (σ22 − σ11 + 2iσ12 ). κ +1

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(b) Two- and three-dimensional real variables analysis

15

L

∗ σjm nm (ξ )Gkj (x,ξ ) dLξ ,

(5.14)

where L is the boundary of the inclusion (curve in two dimensions and surface in three dimensions), uk (x) is the kth component of the displacement vector, nm (ξ ) is mth component of the normal vector at the point ξ ∈ L pointing away from the inclusion and Gkj (x, ξ ) is the Kelvin fundamental solution (Green’s function). The explicit expressions for the stresses within the inclusion and in the matrix are  σkj (x) =

L

∗ σm n (ξ )Smkj (x, ξ ) dLξ − χV (z)σkj∗ ,

(5.15)

where Smkj (x, ξ ) =

1 [(1 − 2ν)(δkm r,j + δjm r,k − δkj r,m ) + dr,m r,k r,j ] 4π (1 − ν)(d − 1)rd−1

(5.16)

in which d = 2 or d = 3 for two- and three-dimensional problem, respectively; r = |x − ξ |, r,k = ∂r/∂xk . In two-dimensional case, equations (5.15) are real-variables analogues of equations (5.1). It is clear that real variables formalism leads to more integrals than its complex variables counterpart; evaluation of some of these integrals could be quite involved. However, the real variables formalism can be used in three-dimensional problems where some ‘strange’ properties of symmetric inclusions can be obtained even without the need to evaluate three-dimensional integrals. For example, the analysis of the following hydrostatic stresses in the matrix: ⎧ ⎪ 3 ⎪ ⎨ 

⎫ ⎪ ⎪    3 ⎬  1 1+ν ∂ ∂ 1 ∗ ∗ dLξ + dLξ σjj nj (ξ ) σjm nm (ξ ) σ11 (x) + σ22 (x) + σ33 (x) = − ⎪ L ⎪ 4π (1 − ν) ∂ξj r ∂ξj r L ⎪ ⎩ ⎭ m=1 j=1 ⎪ m=j

∗ − χV (z)(σ11

∗ + σ22

∗ + σ33 )

(5.17)

suggests that under the action of the volumetric eigenstrain (resulting in hydrostatic stresses σjj∗ = p; σkj∗ = 0, k = j) and inclusion of any shape, the hydrostatic stress in the matrix vanishes and has a constant value inside the inclusion, exactly as in the two-dimensional case. That can be easily seen from the fact that 3  

∂ σjj∗ nj (ξ ) ∂ξj L j=1

      0 1 1 ∂ dLξ = p dLξ = r r −4π L ∂nξ

x∈ /V x ∈ V.

(5.18)

The proof of some other three-dimensional shape-independent properties without explicit integration can be found in [16] and references therein. Some results for symmetric inhomogeneities might require explicit evaluation of all integrals involved in equation (5.15). However, for the inclusions of some regular shapes, integration can still be performed analytically. For example, the technique presented in [18] can be used for polyhedral inclusions.

...................................................

 uk (x) =

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The fields within the inclusion of arbitrary shape and those in the matrix can also be explicitly written in real variables (both for two- and three-dimensional problems) using classical Eshelby formalism [2,3]. Thus, the displacements at the point x within the inclusion and in the matrix are

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6. Discussions and implications

16

1 − μI /μ π i(κ + 1)μI /μ

     1 1 1 , (6.1) σ (τ ) dτ + 2 τ σ (τ ) dτ + O 3 z L z L z

where z = x1 + ix2 is the point outside of the inhomogeneity with the boundary L, μI is the shear modulus of the inhomogeneity, and σ (τ ) = σn (τ ) + iσs (τ ). The coefficient in the first term of expansion of equation (6.1) represents the resultant force acting on the boundary of the inhomogeneity, while the coefficient in the second term (dipole coefficient) of that expansion is equal to [1] 

 L

τ σ (τ ) dτ = −i

A

[σ22 (z) − σ11 (z) − 2iσ12 (z)] dA,

(6.2)

in which A is the area occupied by the inhomogeneity, and z ∈ A. Similarly, the asymptotic expansions for the deviatoric stresses in the matrix involve dipole coefficient that represents average hydrostatic stresses within the inhomogeneity. For three-dimensional problems, the analogous conditions can be deduced from the analysis of the coefficients in asymptotic expansions of the fields presented in [19]. Therefore, in the view of the above results, vanishing of some coefficients in asymptotic expansions of far-fields may shed light on the type of symmetry of the problem when the geometry of the inhomogeneity is not known a priori and the fields due to that inhomogeneity are available in limited numbers of points located on one line far away from that inhomogeneity. Many interesting properties were discovered in this paper due to the separation of the volumetric (hydrostatic) and deviatoric far-fields. While, for isotropic materials, these properties can be implicitly obtained from the results reported in existing literature, e.g. Theorem 3 in [12] or eqn (3.24) in [19], the authors of these papers have never articulated those results. However, the implications of the discovered properties can be significant. For example, the fact that under the hydrostatic load, the hydrostatic stresses and strains within the inclusion of arbitrary shape are constant and vanish outside of the inclusion may explain why various homogenization techniques based on the concept of Eshelby’s equivalent inclusion work well for the estimates of the effective bulk modulus of two-phase composite materials. It may have to do with the fact that the solution for the problem of multiple inclusions subjected to identical volumetric eigenstrains can be obtained by direct superposition of the solutions for each inclusion considered separately. Our technique provides natural explanation of this fact, as the constant volumetric eigenstrain within the inclusion of any shape results in constant hydrostatic stress. Finally, we note that the modification of the Eshelby technique proposed here, allows for simpler and more straightforward derivations of many interesting results reported in the literature and, in addition, leads to new solutions for anisotropic and non-uniform inhomogeneities. Data accessibility. The manuscript is self-contained; all information needed for reproducibility of the results is represented in the text.

...................................................

∞ ∞ + σ22 −2 σ11 (z) + σ22 (z) = σ11

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While the results obtained in the previous sections describe some peculiar phenomena of symmetric problems (which are interesting on their own), their value may become even more significant in some applications. One example of that kind is given in [1], where it was shown that, for the two-dimensional plane strain and antiplane problems of a homogeneous isotropic elastic plane containing a single, isotropic, perfectly bonded inhomogeneity of arbitrary shape and subjected to the uniform farfield load σ ∞ , the volumetric averages of the stresses inside the inhomogeneity are involved in coefficients of the leading terms in the Laurent series expansions of the fields outside the inhomogeneity. For example, the hydrostatic stress in the matrix admits the following Laurent series representation in complex variables:

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Authors’ contributions. Both authors substantially contributed to conception, design, analysis and interpretation

(a) Volumetric eigenstrain Here we prove the result of equation (3.18) using assumed symmetry conditions. First, the symmetry guarantees that the range of values for all arguments x1 , x2 , x3 is identical. This property, combined with the standard change of variables in the integration procedure applied to equation (3.17) together with equations (3.5), (3.8) and (3.9), yields the following average of the stresses (σ I )V (11) :  (σ I )V (11)  = P = =

V

[g11 (x1 , x2 , x3 ) + g22 (−x2 , x1 , x3 ) + g33 (−x3 , x2 , x1 )] dV

(2μ + 3λ)a V

 V

[g11 (x1 , x2 , x3 ) + g22 (x1 , x2 , x3 ) + g33 (x1 , x2 , x3 )] dV.

(A 1)

V We now show that the averages of the stresses (σ I )V (22) and (σ I )(33) are also equal to P, and that V V the averages of the stresses (σ I )V (12) , (σ I )(13) and (σ I )(23) vanish. It can be seen that, due to the assumed symmetry, the 90◦ rotation of the problem with the load ∗

∗ = 1) around x -axis should lead to the following relationships between the functions of σ (11) (σ11 1 equation (3.5):

g22 (x1 , x2 . x3 ) = g33 (x1 , −x3 , x2 ).

(A 2)

The relation of the above equation and the integration procedure analogous to the one of equation (A 1) used to evaluate equation (3.17) lead to the following result: V V (σ I )V (22) = (σ I )(33) = (σ I )(33) = P.

(A 3)

The average (σ I )V (12) can be evaluated as follows: (σ I )V (12)  =

(2μ + 3λ)a V

=

(2μ + 3λ)a V

=

(2μ + 3λ)a V

 

V



V

V

[g12 (x1 , x2 , x3 ) − g12 (−x1 , x1 , x3 ) − g23 (−x3 , x2 , x1 )] dV [g12 (x1 , x2 , x3 ) − g12 (x1 , x2 , x3 ) − g23 (−x1 , x2 , x3 )] dV [−g23 (x1 , x2 , x3 )] dV.

(A 4) ∗

∗ = 1) around the x -axis We note also that the 180◦ rotation of the system loaded by σ (11) (σ11 3 moves the point (x1 , x2 , x3 ) ∈ V to the point (−x1 , −x2 , x3 ) ∈ V and

g23 (x1 , x2 , x3 ) = −g23 (−x1 , −x2 , x3 ).

(A 5)

Therefore, the last integral of equation (A 1) vanishes due to the fact that integration over the part with, e.g. x2 > 0 cancels the integral over the part with x2 < 0. Thus, (σ I )V (12)  = 0.

(A 6)

...................................................

Appendix A. Evaluation of average stress within three-dimensional inhomogeneity

17

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of the results. It was truly a collaborative effort. Competing interests. We have no competing interests. Funding. No external funding was involved. Acknowledgements. The first author (S.M.) gratefully acknowledges the support from the Theodore W. Bennett Chair, University of Minnesota. Special thanks to Dmitry Nikolskiy for help in preparing the manuscript. We also wish to thank the anonymous reviewers for interesting and useful suggestions.

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Similar arguments can be used to prove that

18 (A 7)

(b) Deviatoric eigenstrain The essential steps of the analysis of this case are identical to the corresponding steps for the case of volumetric eigenstrain and involve relationships resulting from the assumed symmetry. However, in this case all six loading cases of equation (3.2) need to be superposed. This time, however, it is sufficient to analyse only the diagonal terms of equation (3.23), as the symmetry of the off-diagonal terms is automatically guaranteed. ∗

∗

∗

To this end, it is first observed that the contributions of the loads σ (12) , σ (13) and σ (23) to the averages of the diagonal terms of equation (3.23) vanish. This results from the fact that, as seen from equations (3.10), (3.12) and (3.13), the contributions to the diagonal terms are described by the functions g¯ kk , k = 1, 2, 3 with various permutations (and signs) of their arguments. However, as explained in connection with equation (A 1), the difference in the arguments can be readily taken care of by the change of variables during the integration procedure, so all integrals of interests are of the following form:  (A 8) J¯kk = [¯gkk (x1 , x2 , x3 )] dV. V

The vanishing of these integrals comes from the fact that all functions g¯ kk are associated with the ∗

load σ (12) (see equation (3.10)). Under the adopted symmetry assumptions, rotation of this load by 180◦ around x1 -axis results in re-location of the point (x1 , x2 , x3 ) ∈ V to the point (x1 , −x2 , −x3 ) ∈ V and in the change in the direction of the applied load (its sign) as compared to the direction of the load prior to rotation. As a result, the functions g¯ kk obey the following relations: g¯ kk (x1 , x2 , x3 ) = −¯gkk (x1 , −x2 , −x3 ),

(A 9)

which show that the integral over the part with, e.g. x2 > 0 cancels the integral over the part with x2 < 0 leading to the conclusion that  (A 10) J¯kk = [¯gkk (x1 , x2 , x3 )] dV = 0. V

The result of the above equation combined with the expressions of equations (3.5), (3.8) and (3.9) implies that the entries on the main diagonal of tensor (σ I )D , equation (3.22), are ⎫ (σ )D ⎪ 11 = 2μ[g11 (x1 , x2 , x3 ) − αg22 (−x2 , x1 , x3 ) − (1 − α)g33 (−x3 , x2 , x1 )],⎪ ⎪ ⎬ D (A 11) (σ )22 = 2μ[g22 (x1 , x2 , x3 ) − αg11 (−x2 , x1 , x3 ) − (1 − α)g22 (−x3 , x2 , x1 )] ⎪ ⎪ ⎪ ⎭ and (σ )D 33 = 2μ[g33 (x1 , x2 , x3 ) − αg33 (−x2 , x1 , x3 ) − (1 − α)g11 (−x3 , x2 , x1 )]. Consequently, the integration procedure accompanied by standard change of variables results in ⎫ (σ )D ⎪ 11  = B11 = −2μ[J11 − αJ22 − (1 − α)J33 ],⎪ ⎪ ⎬ D (A 12) (σ )22  = B22 = −2μ[J22 − αJ11 − (1 − α)J22 ] ⎪ ⎪ ⎪ ⎭ D and (σ )33  = B33 = −2μ[J33 − αJ33 − (1 − α)J11 ], where

 Jkk =

V

[gkk (x1 , x2 , x3 )] dV.

(A 13)

...................................................

This concludes the proof of the results of equation (3.18).

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V (σ I )V (13)  = (σ I )(23)  = 0.

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The summation of the expressions involved in equation (A 12) results in tr(σ I )D = B11 + B22 + B33 = 0.

19 (A 14)

(a) Derivation of equation (5.1) For the problems with traction boundary discontinuities, the expressions for the Kolosov– Muschelishvili potentials [20] are presented in §30 of [17]. Using these expressions, the stresses (during the unloading) within the inclusion and in the matrix can be represented as  1 σ (τ )dτ 4 ∗ ∗ Re σ11 (z) + σ22 (z) = −(σ11 + σ22 ) + κ + 1 2π i L τ − z and

∗

∗

∗

σ22 (z) − σ11 (z) + 2iσ12 (z) = −(σ22 − σ11 + 2iσ12 )     σ (τ )dτ¯ 1 τ¯ − z¯ − σ (τ ) + dτ , κ π i(κ + 1) (τ − z)2 L τ −z L

(B 1)

where σ = σn + iσs , σn , σs are the normal and shear tractions, respectively (the normal nI is directed away from the inclusion, i.e. nI = −nm ), σ = σ I − σ m , and the superscript m identifies the matrix material. We take into account that σ = i t exp(−iα), (B 2) where α is is the angle between the axis Ox1 and the tangent to L at the corresponding point, t is the following complex combination: (B 3) t = t1 + it2 in which tk is the kth Cartesian component of the traction vector t introduced in §3 and m t = (tI1 + itI2 ) − (tm 1 + it2 ).

(B 4)

Introducing the following complex combination associated with the normal vector nI nI = nI1 + inI2

(B 5)

and using equations (B 4) and (B 3), one can re-write equation (B 2) as follows: ∗

∗

∗

∗

σ = i{Re[(σ11 + iσ12 )nI ] + i Re[(σ12 + iσ22 )nI ]} exp(−iα).

(B 6)

Taking into account that nI = i exp(−iα)

and

exp(−2iα) =

dτ¯ dτ

(B 7)

the following expression is obtained from equation (B 6): 1 ∗ ∗ ∗ ∗ ∗ σ = [(σ11 + σ22 ) + (σ22 − σ11 − 2iσ12 ) exp(−2iα)] 2   1 ∗ ∗ ∗ ∗ ∗ dτ¯ (σ11 + σ22 ) + (σ22 − σ11 − 2iσ12 ) . = 2 dτ

(B 8)

Substituting the expression of equation (B 8) into the integrals of equations (B 1) and performing simple algebraic manipulations, one arrives at the expressions given by equations (5.1) and (5.3). During these manipulation, we use the fact that   1 z∈A 1 dτ (B 9) = 2π i L τ − z 0 z∈ / A.

...................................................

Appendix B. Two-dimensional Eshelby problem for inclusions of arbitrary shapes

rspa.royalsocietypublishing.org Proc. R. Soc. A 471: 20150157

This concludes the proof of the results of equation (3.23).

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(b) Evaluation of the integral involved in equation (5.1) for the circular arcs and straight lines

20

F(z) =

1 2π i

 L

R2 dτ¯ =− τ −z 2π i

 L

dτ . (τ − z)(τ − zc )2

(B 11)

It can be easily verified that the integral in the right-hand side of the above equation vanishes when L is a complete circle and z located within the circle. (ii) A part of L is a straight segment with the beginning at the point a = a1 + ib1 and the end at the point b = b1 + ib1 . In this case. b¯ − a¯ (τ − a) (B 12) τ¯ = a¯ + b−a and F(z) =

1 2π i

b a

dτ¯ b¯ − a¯ 1 b−z = ln . τ − z b − a 2π i a − z

(B 13)

In problems of polygonal inclusions that possess certain rotational symmetry, the use of equation (B 13) allows for simpler and more straightforward derivations of the results such as those reported in [14].

Appendix C. Average stress in symmetric inclusions Consider the inclusion that possesses 90◦ rotational symmetry, e.g. the one shown in figure 1. To prove the property presented in equation (5.10), it will be demonstrated that, for the inclusion with the assumed symmetry F(z) = 0. (C 1) The point −iz is the image of point z that results from 90◦ counterclockwise rotation. The integral involved in equation (C 1) evaluated at those two points can be evaluated as follows: F(z) =

4  dτ¯ 1  2π i τ −z Lk k=1

1  2π i 4

and F(−iz) =

k=1

 Lk

dτ¯ , τ − iz

(C 2)

where Lk is the part of the contour located in the kth quadrant of the coordinate system. Assumed rotational symmetry guarantees that the contours Lk , k = 2, 3, 4 the image of the contour L1 that result from 90◦ rotation. Therefore, any τ ∈ Lk , k = 2, 3, 4 can be obtained from τ ∈ L1 as follows:   i(k − 1)π . (C 3) τLk = τL1 exp 2 Substituting expressions of equation (C 3) into equations (C 2) leads to the following expressions:  ⎫   1 1 1 1 1 ⎪ ⎪ − + − F(z) = dτ¯ ⎪ 2π i L1 τ − z τ + iz τ + z τ − iz ⎬ (C 4)    ⎪ 1 1 1 1 1 ⎪ ⎪ .⎭ dτ¯ and F(−iz) = − + − 2π i L1 τ + iz τ + z τ − iz τ − z

...................................................

and

rspa.royalsocietypublishing.org Proc. R. Soc. A 471: 20150157

In the cases when τ¯ can be expressed via τ as an elementary function, the integral involved in equations (5.1) can be evaluated analytically. Two special cases are [21,22]: (i) L or its part is a circular arc located on the circle of radius R and the centre zc (in our case zc = 0). In this case R2 (B 10) τ¯ = z¯ c + τ − zc

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References

21

rspa.royalsocietypublishing.org Proc. R. Soc. A 471: 20150157

Summation of two equations of the above equations is equal to zero. Thus, the contribution of any two points related to each other by 90◦ rotation to the left-hand side of equation (C 1) vanishes. Taking into account that, for inclusions invariant under the group of 90◦ rotations, there are two pairs of such points, the average involved in equation (C 1) must be zero.