A non-circular Eshelby inclusion near a non-parabolic inhomogeneity admitting internal uniform stresses

1. Introduction

It is well-known that a uniform stress field inside an elastic inhomogeneity is optimal in the sense that it eliminates any stress peaks within the inhomogeneity [1] and, in some cases, at the material interface between the inhomogeneity and the surrounding matrix [2]. The continuing interest in designing states of uniform stress inside elastic inhomogeneities is well-documented in the literature from the classical studies of Hardiman [3], Eshelby [4−6], Sendeckyj [7], Ru and Schiavone [8] and Ting [9] to more recent studies, including those of Liu [10], Kang et al. [11], Wang [12], Dai et al. [13,14], Wang et al. [15–17], Wang and Schiavone [18,19], Antipov [20,21], Marshall [22] and Lim and Milton [23]. Recent investigations by the current authors [16,17] have revealed that uniform stresses continue to be attainable inside an uncoated or coated non-parabolic open elastic inhomogeneity despite the presence of a nearby circular Eshelby inclusion undergoing uniform or non-uniform anti-plane eigenstrains when the surrounding matrix is subjected to uniform remote anti-plane stresses. Most recently [19], we have solved the inverse problem in anti-plane elasticity associated with the uniformity of stresses inside a non-elliptical inhomogeneity with a closed curvilinear boundary interacting with a non-circular Eshelby inclusion. The non-circular inclusion was represented by a Booth’s lemniscate inclusion. The following question arises naturally from our investigations: can the stress field inside a non-parabolic open inhomogeneity remain uniform in the presence of a non-circular inclusion? In particular, can the uniformity property continue to hold in the case of non-circular inclusions other than those in the form of a Booth’s lemniscate? This paper will address this more challenging problem.

Consequently, in this paper, we endeavor to solve the inverse problem in anti-plane elasticity associated with a non-circular Eshelby inclusion undergoing uniform anti-plane eigenstrains in the vicinity of a non-parabolic open elastic inhomogeneity with internal uniform stresses when the surrounding matrix is subjected to uniform remote stresses. The non-circular inclusion is characterized by a Booth’s lemniscate inclusion, a generalized Booth’s lemniscate inclusion or a cardioid inclusion. Using analytic continuation, the analytic function originally defined in the matrix is extended to the domain occupied by the non-circular inclusion. In doing so, this analytic function becomes analytic in the exterior of the non-parabolic inhomogeneity except at certain points where its principal or singular part can be simply and completely depicted by a number of poles of finite degree. Once this principal part has been determined, the conformal mapping function for the exterior of the non-parabolic inhomogeneity can be constructed accordingly. The presence of the non-circular inclusion is incorporated in the mapping function via the addition of a number of poles of finite degree in the image plane. We arrive at several conditions arising from the imposition of the singular behaviors at the aforementioned points. We see from these conditions that (a) the internal uniform stress field inside the non-parabolic inhomogeneity is independent of the specific open shape of the inhomogeneity and is also unaffected by the existence of the nearby non-circular inclusion, (b) the non-parabolic shape of the inhomogeneity is caused solely by the existence of the non-circular inclusion and (c) the eigenstrains imposed on the non-circular inclusion and remote loading applied in the matrix must satisfy a specific relationship for given material and geometric parameters of the composite. Numerical results are presented to demonstrate our findings.

The paper is structured as follows. The complex variable formulation for anti-plane elasticity is reviewed briefly in Section 2. The general solution corresponding to a non-circular Booth’s lemniscate inclusion is derived in Section 3. Numerical examples are presented in Section 4 in the case of a Booth’s lemniscate inclusion. We present in Section 5 the general solutions as well as numerical results for a generalized Booth’s lemniscate inclusion and a cardioid inclusion. Finally, we summarize our conclusions in Section 6.

2. Complex variable formulation

We first establish a Cartesian coordinate system { x i } ( i = 1 , 2 , 3 ) . Under anti-plane shear deformations of an isotropic elastic material, the two shear stress components σ 31 and σ 32 , the out-of-plane displacement w and the single stress function ϕ can be expressed in terms of a single analytic function f(z) of the complex variable z = x 1 + i x 2 as [9]

where µ is the shear modulus.

In addition, the two stress components can be expressed in terms of the single stress function as [9]

The above complex variable formulation is fundamental to the theoretical development carried out in this paper.

3. General solution for a Booth’s lemniscate inclusion

As shown in Figure 1, we consider a domain in ℜ², infinite in extent, containing both a non-parabolic elastic inhomogeneity and a non-circular Eshelby inclusion undergoing uniform stress-free anti-plane eigenstrains ( ε 31 * , ε 32 * ) . The shear modulus of the inhomogeneity is distinct from that of the surrounding matrix, whereas the shear modulus of the Eshelby inclusion is identical to that of the matrix. Let S 1 , S 2 and S 3 denote the inhomogeneity, the matrix and the Eshelby inclusion, respectively, all of which are perfectly bonded through the inhomogeneity–matrix interface L 1 and the inclusion–matrix interface L 2 . In addition, the matrix is subjected to uniform remote anti-plane stresses ( σ 31 ∞ , σ 32 ∞ ) . Throughout the paper, the subscripts 1, 2 and 3 are used to identify the respective quantities in S 1 , S 2 and S 3 .

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Figure 1.

A non-circular Eshelby inclusion undergoing uniform anti-plane eigenstrains interacting with a non-parabolic open inhomogeneity admitting an internal uniform stress field under uniform remote anti-plane stresses applied in the matrix.

The boundary value problem takes the following form in the physical z-plane

where Γ = μ 1 / μ 2 . Equations (3a) and (3b) represent the continuity conditions of traction and displacement across the two perfect interfaces L 1 and L 2 , respectively; Equation (3c) gives the asymptotic behavior of f 2 ( z ) at infinity due to the prescribed uniform remote anti-plane shear stresses in the matrix.

Equation (3b) can be rewritten in the following equivalent form

The interior of the Eshelby inclusion can be mapped onto the interior of the unit circle in the ξ-plane via the following conformal mapping function [24]

Thus, along the inclusion–matrix interface L 2 , we have

and we introduce the auxiliary function D ( z ) = m ¯ ( 1 m − 1 ( z ) ) , which is analytic in the interior of the inclusion except at certain points.

We thus introduce the following analytic continuation

As a result of the analytic continuation in Equation (7), the analytic function f 2 ( z ) originally defined in S 2 has been extended to S 2 ∪ S 3 . Now f 2 ( z ) is analytic in S 2 ∪ S 3 except at certain points within S 3 and at z = ∞ , where its asymptotic behavior is given by Equation (3c).

In this section, the non-circular inclusion is first represented by a Booth’s lemniscate inclusion, described by

where c is an arbitrary complex number characterizing the center of the Booth’s lemniscate inclusion and θ is a phase angle characterizing its inclination with respect to the x 1 -axis.

Thus, the principal part of f 2 ( z ) within S 2 ∪ S 3 , denoted as f 2 s ( z ) , is given by

where

As shown in Figure 1, both points z = z 1 and z = z 2 are located within the inclusion S 3 . Now we introduce the following conformal mapping function for the simply connected domain occupied by the matrix and the Booth’s lemniscate inclusion

where H, having the dimension of length, is taken as a scaling constant, and p 1 and p 2 are two complex constants.

As shown in Figure 2, using the mapping function in Equation (11), the exterior of the non-parabolic inhomogeneity is mapped onto the right half-plane Re { ξ } ≥ H 1 2 ; the inhomogeneity–matrix interface L 1 is mapped onto the vertical straight line: { Re { ξ } = H 1 2 , − ∞ < Im { ξ } < + ∞ } ; the point z = z 1 is mapped onto the point ξ = ξ 1 = ω − 1 ( z 1 ) , while the point z = z 2 is mapped onto the point ξ = ξ 2 = ω − 1 ( z 2 ) . The existence of the Booth’s lemniscate inclusion has been incorporated into the mapping function in Equation (11) by the presence of the two first-order poles at ξ = 2 H 1 2 − ξ ¯ 1 and ξ = 2 H 1 2 − ξ ¯ 2 .

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Figure 2.

The image ξ-plane.

In order to ensure that the stress field inside the non-parabolic inhomogeneity is uniform, the analytic function f 1 ( z ) defined in the inhomogeneity should take the following form

where k is an unknown complex number to be determined.

By enforcing the continuity conditions in Equation (3a) with the use of Equations (11) and (12), we arrive at

or more explicitly

By using Equation (14) to satisfy the singular behaviors of f 2 ( z ) in S 2 ∪ S 3 at the three points z = ∞ , z 1 , z 2 in Equation (9), we arrive at the following conditions

where

It follows from Equation (15) that

Thus, the internal uniform stress field within the non-parabolic inhomogeneity can be obtained from Equations (1), (12) and (17) as follows

which is, in fact, independent of the non-parabolic open shape of the inhomogeneity and is also unaffected by the existence of the nearby Booth’s lemniscate inclusion.

The parameter p 2 can be uniquely obtained from Equation (18) using the iteration method for given values of ξ 1 , ξ 2 and p 1 . Thus, the mapping function in Equation (11) has been completely determined. In addition, Equation (19) can be considered as a relationship between the eigenstrains imposed on the Booth’s lemniscate inclusion and the remote loading for given material and geometric parameters of the composite.

The two parameters z 1 and z 2 can be uniquely determined from Equation (11) as follows

A comparison of Equation (21) with Equation (10) leads to

where z 1 and z 2 are determined by Equation (21). The two parameters c and θ can be uniquely determined from Equation (22)_1,2, whilst the other two parameters a and b are required to satisfy the relationship in Equation (22)₃.

If the eigenstrains imposed on the Booth’s lemniscate inclusion are zero (i.e., ε 31 * = ε 32 * = 0 ), we have from Equation (15)_2,3 that p 1 = p 2 = 0 . In this case, the mapping function in Equation (11) simply describes a parabolic interface L 1 . This fact implies that the non-parabolic shape of the inhomogeneity is caused solely by the presence of the nearby Booth’s lemniscate inclusion.

4. Numerical examples

In this section, several numerical examples will be presented to demonstrate the general solution for a Booth’s lemniscate inclusion obtained in the previous section.

In the first example, we choose

The parameter p 2 can be uniquely determined from Equation (18) via iteration as

Consequently, we obtain from Equation (22) that

The non-parabolic shape of the inhomogeneity–matrix interface L 1 and the Booth’s lemniscate shapes of L 2 for different values of the parameter b are illustrated in Figure 3. In this example, the interface L 1 has a non-convex portion due to the presence of the nearby Booth’s lemniscate inclusion.

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Figure 3.

The non-parabolic open shape of L 1 and the Booth’s lemniscate shapes of L 2 for different values of b with the three parameters ξ 1 , ξ 2 and p 1 given by Equation (23).

In the second example, we choose

The parameter p 2 can again be uniquely determined from Equation (18) via iteration as

Consequently, we obtain from Equation (22) that

The non-parabolic shape of the inhomogeneity–matrix interface L 1 and the Booth’s lemniscate shapes of L 2 for different values of the parameter b are illustrated in Figure 4. In this example, the non-convex portion of the interface L 1 just contacts L 2 when b = 0 . 25 .

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Figure 4.

The non-parabolic open shape of L 1 and the Booth’s lemniscate shapes of L 2 for different values of b with the three parameters ξ 1 , ξ 2 and p 1 given by Equation (26).

In the third example, we choose

The parameter p 2 can be uniquely determined from Equation (18) via iteration as

Consequently, it is obtained from Equation (22) that

The non-parabolic shape of the inhomogeneity–matrix interface L 1 and the Booth’s lemniscate shapes of L 2 for different values of the parameter b are illustrated in Figure 5. In this example, ξ 2 = ξ ¯ 1 and the value of the parameter p 1 have been adjusted so that p 2 = p ¯ 1 . Consequently, both L 1 and L 2 in Figure 5 are symmetric with respect to the x 1 -axis.

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Figure 5.

The non-parabolic open shape of L 1 and the Booth’s lemniscate shapes of L 2 for different values of b with the three parameters ξ 1 , ξ 2 and p 1 given by Equation (29).

In the fourth example, we choose

The parameter p 2 can again be uniquely determined from Equation (18) via iteration as

Consequently, from Equation (22) we find that

The non-parabolic shape of the inhomogeneity–matrix interface L 1 and the Booth’s lemniscate shapes of L 2 for different values of the parameter b are illustrated in Figure 6. In this example, a sharp corner appears on the interface L 1 due to the presence of the nearby Booth’s lemniscate inclusion.

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Figure 6.

The non-parabolic open shape of L 1 and the Booth’s lemniscate shapes of L 2 for different values of b with the three parameters ξ 1 , ξ 2 and p 1 given by Equation (32).

5. Further discussion

In the analysis and numerical results in Sections 3 and 4, the non-circular inclusion is represented by a Booth’s lemniscate inclusion. Do other non-circular shapes of the nearby inclusion similarly permit an internal uniform field inside the non-parabolic open inhomogeneity? In this section, we will discuss in detail the two cases of a generalized Booth’s lemniscate inclusion and a cardioid inclusion.

5.1. A generalized Booth’s lemniscate inclusion

The domain occupied by the generalized Booth’s lemniscate inclusion is given by [25]

z = m ( ξ ) = c + a e i θ ξ 1 + b ξ 3 , a > 0 , 0 < b ≤ 0 . 5 , | ξ | ≤ 1 ,

(35)

where c is a complex number characterizing the center of the inclusion and θ is a phase angle characterizing its orientation.

Thus, the principal part of f 2 ( z ) within S 2 ∪ S 3 , denoted as f 2 s ( z ) , is determined by

f 2 s ( z ) = σ 32 ∞ + i σ 31 ∞ μ 2 z − a 2 ( 1 + 2 b 2 ) ( ε 32 * − i ε 31 * ) 3 ( 1 − b 2 ) 2 ( 1 z − z 1 + 1 z − z 2 + 1 z − z 3 ) ,

(36)

where

z 1 = c − e i θ a b 1 3 1 − b 2 , z 2 = c + e i θ a b 1 3 1 − b 2 1 − i 3 2 , z 3 = c + e i θ a b 1 3 1 − b 2 1 + i 3 2 .

(37)

Now we introduce the following conformal mapping function for the simply connected domain occupied by the matrix and the generalized Booth’s lemniscate inclusion

z = ω ( ξ ) = ξ 2 + p 1 ξ + ξ ¯ 1 − 2 H 1 2 + p 2 ξ + ξ ¯ 2 − 2 H 1 2 + p 3 ξ + ξ ¯ 3 − 2 H 1 2 , ξ = ω − 1 ( z ) , H > 0 , Re { ξ } ≥ H 1 2 ,

(38)

where H is a scaling constant and p 1 , p 2 and p 3 are three complex constants. By using the mapping function in Equation (38), the exterior of the non-parabolic inhomogeneity is mapped onto the right half-plane Re { ξ } ≥ H 1 2 ; the inhomogeneity–matrix interface L 1 is mapped onto the vertical straight line: { Re { ξ } = H 1 2 , − ∞ < Im { ξ } < + ∞ } ; the point z = z 1 is mapped onto the point ξ = ξ 1 = ω − 1 ( z 1 ) , the point z = z 2 is mapped onto the point ξ = ξ 2 = ω − 1 ( z 2 ) and the point z = z 3 is mapped onto the point ξ = ξ 3 = ω − 1 ( z 3 ) . The existence of the generalized Booth’s lemniscate inclusion has been incorporated into the mapping function in Equation (38) via the three first-order poles at ξ = 2 H 1 2 − ξ ¯ 1 , ξ = 2 H 1 2 − ξ ¯ 2 and ξ = 2 H 1 2 − ξ ¯ 3 .

By enforcing the continuity conditions in Equation (3a) with the use of Equations (12) and (38), we arrive at

f 2 ( ξ ) = k ( Γ + 1 ) 2 [ ξ 2 + p 1 ξ + ξ ¯ 1 − 2 H 1 2 + p 2 ξ + ξ ¯ 2 − 2 H 1 2 + p 3 ξ + ξ ¯ 3 − 2 H 1 2 ] + k ¯ ( Γ − 1 ) 2 [ ( 2 H 1 2 − ξ ) 2 − p ¯ 1 ξ − ξ 1 − p ¯ 2 ξ − ξ 2 − p ¯ 3 ξ − ξ 3 ] , Re { ξ } ≥ H 1 2

(39)

By using Equation (39) to satisfy the singular behaviors of f 2 ( z ) in S 2 ∪ S 3 at the four points z = ∞ , z 1 , z 2 , z 3 in Equation (36), we arrive at the following conditions

k ( Γ + 1 ) + k ¯ ( Γ − 1 ) 2 = σ 32 ∞ + i σ 31 ∞ μ 2 , p 1 = 2 a 2 ( 1 + 2 b 2 ) ( ε 32 * + i ε 31 * ) 3 k ( Γ − 1 ) ( 1 − b 2 ) 2 1 ω ′ ( ξ 1 ) ¯ , p 2 = 2 a 2 ( 1 + 2 b 2 ) ( ε 32 * + i ε 31 * ) 3 k ( Γ − 1 ) ( 1 − b 2 ) 2 1 ω ′ ( ξ 2 ) ¯ , p 3 = 2 a 2 ( 1 + 2 b 2 ) ( ε 32 * + i ε 31 * ) 3 k ( Γ − 1 ) ( 1 − b 2 ) 2 1 ω ′ ( ξ 3 ) ¯ ,

(40)

where

ω ′ ( ξ ) = 2 ξ − p 1 ( ξ + ξ ¯ 1 − 2 H 1 2 ) 2 − p 2 ( ξ + ξ ¯ 2 − 2 H 1 2 ) 2 − p 3 ( ξ + ξ ¯ 3 − 2 H 1 2 ) 2 .

(41)

It follows from Equation (40) that the complex number k is still determined by Equation (17) and that

p 2 p 1 = ω ′ ( ξ 1 ) ¯ ω ′ ( ξ 2 ) ¯ , p 3 p 1 = ω ′ ( ξ 1 ) ¯ ω ′ ( ξ 3 ) ¯ ,

(42)

μ 1 ( ε 32 * + i ε 31 * ) ( Γ − 1 ) ( σ 32 ∞ + i Γ σ 31 ∞ ) = 3 p 1 ( 1 − b 2 ) 2 2 a 2 ( 1 + 2 b 2 ) [ 2 ξ ¯ 1 − p ¯ 1 ( ξ 1 + ξ ¯ 1 − 2 H 1 2 ) 2 − p ¯ 2 ( ξ ¯ 1 + ξ 2 − 2 H 1 2 ) 2 − p ¯ 3 ( ξ ¯ 1 + ξ 3 − 2 H 1 2 ) 2 ] .

(43)

In this case, the internal uniform stress field within the non-parabolic inhomogeneity is also determined by Equation (20), which is independent of the non-parabolic shape of the inhomogeneity and is also unaffected by the existence of the nearby non-circular generalized Booth’s lemniscate inclusion described by Equation (35). The parameters p 2 and p 3 can be determined from Equation (42) using iteration for given values of the four parameters ξ 1 , ξ 2 , ξ 3 and p 1 . Thus, the mapping function in Equation (38) has been completely determined. In addition, Equation (43) can be considered as a relationship between the imposed eigenstrains and remote loading for given material and geometric parameters of the composite.

The three parameters z 1 , z 2 and z 3 can be uniquely determined from Equation (38) as follows

z 1 = ξ 1 2 + p 1 ξ 1 + ξ ¯ 1 − 2 H 1 2 + p 2 ξ 1 + ξ ¯ 2 − 2 H 1 2 + p 3 ξ 1 + ξ ¯ 3 − 2 H 1 2 , z 2 = ξ 2 2 + p 1 ξ 2 + ξ ¯ 1 − 2 H 1 2 + p 2 ξ 2 + ξ ¯ 2 − 2 H 1 2 + p 3 ξ 2 + ξ ¯ 3 − 2 H 1 2 , z 3 = ξ 3 2 + p 1 ξ 3 + ξ ¯ 1 − 2 H 1 2 + p 2 ξ 3 + ξ ¯ 2 − 2 H 1 2 + p 3 ξ 3 + ξ ¯ 3 − 2 H 1 2 .

(44)

A comparison of Equation (44) with Equation (37) leads to

c = z 1 + z 2 + z 3 3 , θ = π 2 + arg ( z 2 − z 3 ) , 3 a b 1 3 1 − b 2 = | z 2 − z 3 | = | z 1 − z 3 | = | z 1 − z 2 | ,

(45)

where z 1 , z 2 and z 3 are determined by Equation (44). Thus, the three parameters ξ 1 , ξ 2 and ξ 3 should be adjusted so that z 1 , z 2 and z 3 determined from Equation (44) satisfy the necessary constraint that | z 1 − z 2 | = | z 1 − z 3 | = | z 2 − z 3 | . If the eigenstrains imposed on the generalized Booth’s lemniscate inclusion are zero, we have from Equation (40)_2,3,4 that p 1 = p 2 = p 3 = 0 . In this case, the mapping function in Equation (38) simply describes a parabolic interface L 1 . This fact implies that the non-parabolic shape of the inhomogeneity is caused solely by the presence of the nearby generalized Booth’s lemniscate inclusion.

For example, the four parameters ξ 1 , ξ 2 , ξ 3 and p 1 are chosen as

ξ 1 = 2 H 1 2 , ξ 2 = 3 H 1 2 , ξ 3 = ( 2 . 5946 + 0 . 8505 i ) H 1 2 , p 1 = − 4 H 3 2 .

(46)

The two parameters p 2 and p 3 can be uniquely determined from Equation (42) via iteration as

p 2 = ( − 3 . 3294 + 0 . 1214 i ) H 3 2 , p 3 = ( − 3 . 5760 − 0 . 5497 i ) H 3 2 .

(47)

Consequently, we obtain from Equation (45) that

c = ( 2 . 7535 + 1 . 3469 i ) H , θ = 0 . 5592 , a b 1 3 1 − b 2 = 3 . 5923 H .

(48)

The non-parabolic shape of the inhomogeneity–matrix interface L 1 and the generalized Booth’s lemniscate shapes of L 2 for different values of the parameter b are illustrated in Figure 7. The value of the parameter ξ 3 has been carefully adjusted so that | z 1 − z 2 | = | z 1 − z 3 | = | z 2 − z 3 | = 6 . 2220 H in Figure 7.

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Figure 7.

The non-parabolic open shape of L 1 and the generalized Booth’s lemniscate shapes of L 2 for different values of b with the four parameters ξ 1 , ξ 2 , ξ 3 and p 1 given by Equation (46).