A hole of irregular shape interacting with a non-parabolic open inhomogeneity with internal uniform anti-plane stresses

Abstract

We use conformal mapping techniques together with analytic continuation to show that a non-parabolic open elastic inhomogeneity continues to admit a state of uniform internal stress when a hole with closed curvilinear traction-free boundary is placed in its vicinity and the surrounding matrix is subjected to uniform remote anti-plane stresses. The internal uniform stress field inside the inhomogeneity is found to be independent of the existence of the nearby hole and the specific non-parabolic shape of the inhomogeneity. In contrast, the non-parabolic shape of the inhomogeneity is influenced solely by the existence of the nearby hole.

Keywords

Non-parabolic inhomogeneity hole uniform stress field anti-plane elasticity conformal mapping

1. Introduction

Eshelby’s uniformity property concerning stresses and strains inside elastic inhomogeneities continues to be an intriguing topic of discussion [1 –15]. From a practical point of view, a uniform stress distribution inside an inhomogeneity is optimal in the sense that it eliminates stress peaks within the inhomogeneity, which are well known to be a primary cause of failure [9]. Recently, Wang and Schiavone [16] proved that a non-elliptical elastic inhomogeneity continues to admit an internal uniform stress state despite the presence of a nearby irregularly shaped hole when the matrix is subjected to uniform remote anti-plane shear stresses. In establishing this result the authors assumed that the boundaries of both the inhomogeneity and the hole are closed curvilinear contours. On the other hand, it is of interest to note that Wang and Schiavone [17] have also established (for both plane and anti-plane elasticity) that the internal stresses inside a parabolic elastic inhomogeneity with an open interface are unconditionally uniform when the matrix is subjected to uniform stresses at infinity. In the particular case of anti-plane elasticity, this uniformity property is in agreement with the observation that the flow velocity within a parabolic inclusion is constant under a uniform incident flow [18, 19]. The uniformity of anti-plane stresses inside a non-parabolic open elastic inhomogeneity interacting with a nearby screw dislocation or a circular Eshelby inclusion undergoing uniform anti-plane eigenstrains has also been observed [20, 21].

In this study, we examine the existence of an internal uniform stress field inside an elastic inhomogeneity with an open interface in the presence of a nearby hole with a closed curvilinear traction-free boundary when the surrounding matrix is subjected to uniform anti-plane stresses at infinity. Due to the influence of the adjacent hole, we will refer to the inhomogeneity as ‘non-parabolic’ to indicate a departure from a conventional parabolic inhomogeneity. Our results indicate that through a judicious design of the open shape of the inhomogeneity and the closed shape of the hole, the internal stress field inside the inhomogeneity can still remain uniform. Furthermore, the internal uniform stress field inside the inhomogeneity is independent of the specific open shape of the inhomogeneity and the existence of the nearby hole. The procedure used to design the shapes of the inhomogeneity and the hole is accomplished via the construction of a novel conformal mapping function for the matrix region containing first and second-order poles as well as a Laurent expansion. By satisfying the continuity conditions of traction and displacement across the inhomogeneity–matrix interface and the traction-free condition along the hole boundary, we obtain restrictions on the coefficients in the Laurent series. Satisfying these restrictions on the coefficients is sufficient to arrive at the desired configuration for the composite. In fact, we find that countless configurations for the composite can be generated in this way.

2. Complex variable formulation for anti-plane elasticity

First, we 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 $u_{3} = w$ and the 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 in Ting [6]

σ_{32} + i σ_{31} = μ f' (z),

(1)

ϕ + i μ w = μ f (z),

(2)

where µ is the shear modulus, and the two stress components can be expressed in terms of the stress function as in Ting [6]

σ_{32} = ϕ_{, 1}, σ_{31} = - ϕ_{, 2} .

(3)

Equation (2) is fundamental in describing the boundary value problem in terms of the analytic function $f (z)$ .

3. Internal uniform stress field

As shown in Figure 1, we consider a domain in $ℜ^{2}$ , infinite in extent, containing an irregularly shaped hole and a non-parabolic open elastic inhomogeneity with elastic properties different from those of the surrounding matrix. We represent the matrix by the domain $S_{2}$ and assume that the inhomogeneity occupies the open region $S_{1}$ . The perfectly bonded inhomogeneity–matrix interface will be denoted by the non-parabolic open curve $L_{1}$ and the traction-free hole boundary will be denoted by the closed curvilinear contour $L_{2}$ . In addition, the matrix is subjected to uniform remote anti-plane shear stresses $σ_{32}^{\infty}$ and $σ_{31}^{\infty}$ . In what follows, the subscripts 1 and 2 will be used to identify the respective quantities in the elastic inhomogeneity $S_{1}$ and in the matrix $S_{2}$ . Our task below is to examine whether an internal uniform stress field can still be maintained inside the non-parabolic open inhomogeneity even in the presence of the nearby hole.

Figure 1.

A hole with a closed curvilinear traction-free boundary in the vicinity of a non-parabolic open elastic inhomogeneity with internal uniform stresses when the matrix is subjected to uniform remote anti-plane stresses.

By employing equation (2), the boundary value problem in the z-plane takes the form:

\begin{matrix} f_{2} (z) + \bar{f_{2} (z)} = Γ f_{1} (z) + Γ \bar{f_{1} (z)}, \\ f_{2} (z) - \bar{f_{2} (z)} = f_{1} (z) - \bar{f_{1} (z)}, z \in L_{1}; \end{matrix}

(4a)

f_{2} (z) + \bar{f_{2} (z)} = 0, z \in L_{2};

(4b)

f_{2} (z) ≅ \frac{σ_{32}^{\infty} + i σ_{31}^{\infty}}{μ_{2}} z + O (z^{\frac{1}{2}}), | z | \to \infty,

(4c)

where $Γ = μ_{1} / μ_{1} μ_{2} μ_{2}$ . Equation (4a) describes the continuity of traction and displacement across the perfect inhomogeneity–matrix interface $L_{1}$ , equation (4b) describes the traction-free condition on the hole boundary $L_{2}$ , and equation (4c) describes the remote asymptotic behaviour of $f_{2} (z)$ due to the imposed uniform anti-plane shear stresses at infinity.

To solve the boundary value problem developed above, we introduce the following conformal mapping function for the matrix region:

z = ω (ξ) = R [\frac{1}{{(ξ - 1)}^{2}} + \frac{p}{ξ - 1} + \sum_{n = 1}^{+ \infty} (a_{n} ξ^{n} + a_{- n} ξ^{- n})], ξ = ω^{- 1} (z), 1 \leq | ξ | \leq ρ^{- \frac{1}{2}},

(5)

where R is a real scaling constant; $ρ (0 < ρ < 1)$ is a given real constant; p is a given complex constant; $a_{n}$ and $a_{- n}$ are unknown complex coefficients to be determined. As shown in Figure 2, by using the mapping function in equation (5), the matrix $S_{2}$ is mapped onto the annulus $1 \leq | ξ | \leq ρ^{- \frac{1}{2}}$ ; the open non-parabolic inhomogeneity–matrix interface $L_{1}$ is mapped onto the inner unit circle $| ξ | = 1$ ; the boundary $L_{2}$ of the closed curvilinear hole is mapped onto the outer concentric circle $| ξ | = ρ^{- \frac{1}{2}}$ ; the point $z = \infty$ is mapped onto $ξ = 1$ on the unit circle. The appearance of the first and second-order poles at $ξ = 1$ in equation (5) is to account for the open shape of the inhomogeneity. When $a_{n} = a_{- n} = 0$ for $n = 1, 2, \dots, + \infty$ in equation (5), the inhomogeneity–matrix interface $L_{1}$ will take the form of a parabola. The presence of the nearby traction-free hole will induce a Laurent series in equation (5), and consequently the shape of the inhomogeneity will become non-parabolic.

Figure 2.

The problem in the ξ-plane.

To ensure that the stresses inside the non-parabolic open inhomogeneity are uniform, we assume that

f_{1} (z) = Az, z \in S_{1},

(6)

where A is a complex constant to be determined.

By enforcing the conditions of continuity of traction and displacement across the perfect inhomogeneity–matrix interface $L_{1}$ in equation (4a) with the use of $f_{1} (z)$ in equation (6), we arrive at the general expression for $f_{2} (ξ) = f_{2} (ω (ξ))$ as follows:

f_{2} (ξ) = \frac{A (Γ + 1)}{2} ω (ξ) + \frac{\bar{A} (Γ - 1)}{2} \bar{ω} (\frac{1}{ξ}), 1 \leq | ξ | \leq ρ^{- \frac{1}{2}} .

(7)

In writing the expression for $f_{2} (ξ)$ in equation (7), the technique of analytic continuation [8] has been employed. By imposing the traction-free condition along the hole boundary $L_{2}$ in equation (4b) with the aid of equation (7), we finally arrive at the following relationship:

a_{- n} = - \frac{\bar{A}}{A} \frac{K + ρ^{- n}}{1 + K ρ^{- n}} {\bar{a}}_{n} - \frac{n - 1 + p}{1 + K ρ^{- n}} - \frac{\bar{A}}{A} \frac{K (n + 1 - \bar{p})}{1 + K ρ^{- n}}, n = 1, 2, \dots, + \infty,

(8)

where the mismatch parameter K is defined by

K = \frac{Γ - 1}{Γ + 1}, - 1 \leq K \leq 1 .

(9)

For given values of the coefficients $a_{n}$ for $n = 1, 2, \dots, + \infty$ , the remaining coefficients $a_{- n}$ can be uniquely determined from equation (8). This fact implies that we have the freedom to arbitrarily select $a_{n}$ as long as the conformal mapping function in equation (5) is one-to-one for the matrix region. In doing so, countless configurations for the composite satisfying our design requirement can be generated. Apparently, the shapes of the inhomogeneity and the matrix are dependent on the phase angle of the complex number A.

By using equation (7) to satisfy the prescribed remote asymptotic behaviour of $f_{2} (z)$ in equation (4c), we obtain

\frac{A (Γ + 1) + \bar{A} (Γ - 1)}{2} = \frac{σ_{32}^{\infty} + i σ_{31}^{\infty}}{μ_{2}},

(10)

from which the complex constant A can be uniquely determined quite simply as

A = \frac{σ_{32}^{\infty} + i Γ σ_{31}^{\infty}}{μ_{1}} .

(11)

Thus, the internal uniform stresses inside the non-parabolic inhomogeneity are explicitly given by

σ_{32} = σ_{32}^{\infty}, σ_{31} = Γ σ_{31}^{\infty}, z \in S_{1} .

(12)

It is clear from the above that the internal uniform stress field inside the inhomogeneity is independent of the existence of the nearby hole and the specific non-parabolic shape of the inhomogeneity. In fact, the uniform stress field in equation (12) is identical to that inside a parabolic inhomogeneity in the absence of the nearby hole [17].

Furthermore, the stresses are non-uniformly distributed in the matrix as follows:

σ_{32} + i σ_{31} = \frac{(Γ + 1) (σ_{32}^{\infty} + i Γ σ_{31}^{\infty})}{2 Γ} + \frac{(1 - Γ) (σ_{32}^{\infty} - i Γ σ_{31}^{\infty})}{2 Γ} \frac{\bar{ω}' (\frac{1}{ξ})}{ξ^{2} ω' (ξ)}, 1 \leq | ξ | \leq ρ^{- \frac{1}{2}} .

(13)

It is easily verified from the above that $σ_{32} + i σ_{31} ≅ σ_{32}^{\infty} + i σ_{31}^{\infty}$ as $ξ \to 1$ (or equivalently as $z \to \infty$ ).

4. Numerical results and discussions

In this section, detailed numerical results will be presented to demonstrate the feasibility of the theory proposed in the previous section.

We first illustrate in Figures 3 and 4 the non-parabolic open shape of the elastic inhomogeneity and the irregular closed shape of the hole for different values of $a_{n}$ with $σ_{31}^{\infty} = 0, K = \pm 0.5, p = 3, ρ = 0.3$ . The values of $a_{n}$ not shown in Figures 3 and 4 are zero. In this case, the internal uniform stress field can be simply determined from equation (12) as $σ_{32} = σ_{32}^{\infty}, σ_{31} = 0, z \in S_{1}$ . It is seen from equation (9) that $K = 0.5$ in Figure 3 means that the inhomogeneity is harder than the matrix ( $Γ > 1$ ) while $K = - 0.5$ in Figure 4 means that the inhomogeneity is softer than the matrix ( $Γ < 1$ ). It is clearly observed from Figures 3 and 4 that both the open shape of the inhomogeneity and the closed shape of the hole are affected by the specific values of $a_{n}$ . To ensure that the mapping in equation (5) is one-to-one, the greater the order m of $a_{m}$ , the smaller the value of $| a_{m} |$ is allowed. This rule is consistent with that observed by Milton and Serkov [22]. In Figures 3 and 4, some values of $a_{n}$ are non-zero. In fact, all the values of $a_{n}$ can be set to zero and the corresponding shapes of the inhomogeneity and hole are illustrated in Figures 5 and 6 for different values of the mismatch parameter K. It is seen from Figures 5 and 6 that $| K |$ should be large enough in order to ensure that the mapping in equation (5) is one-to-one: if a positive K attains its critical value, one portion of the interface $L_{1}$ just touches the other portion of $L_{1}$ (see the bottom and right subplot in Figure 5); if a negative K attains its critical value, two sharp corners will appear on the interface $L_{1}$ (see the bottom and right subplot in Figure 6). This fact can be more clearly observed in Figure 7 for the two disjointed permissible regions of the pair $(K, p)$ when $σ_{31}^{\infty} = 0, Im {p} = 0, ρ = 0.3, a_{n} \equiv 0, n = 1, 2, \dots, + \infty$ . The result in Figure 7 suggests that it is necessary that $p \geq 1$ .

Figure 3.

The non-parabolic open shape of the inhomogeneity and the irregular closed shape of the hole for different values of $a_{n}$ with $σ_{31}^{\infty} = 0, K = 0.5, p = 3, ρ = 0.3$ .

Figure 4.

The non-parabolic open shape of the inhomogeneity and the irregular closed shape of the hole for different values of $a_{n}$ with $σ_{31}^{\infty} = 0, K = - 0.5, p = 3, ρ = 0.3$ .

Figure 5.

The non-parabolic open shape of the inhomogeneity and the irregular closed shape of the hole for different positive values of K with $σ_{31}^{\infty} = 0, p = 3, ρ = 0.3, a_{n} \equiv 0, n = 1, 2, \dots, + \infty$ .

Figure 6.

The non-parabolic open shape of the inhomogeneity and the irregular closed shape of the hole for different negative values of K with $σ_{31}^{\infty} = 0, p = 3, ρ = 0.3, a_{n} \equiv 0, n = 1, 2, \dots, + \infty$ .

Figure 7.

Condition for the mapping in equation (5) to be one-to-one for the matrix region when $σ_{31}^{\infty} = 0, Im {p} = 0, ρ = 0.3, a_{n} \equiv 0, n = 1, 2, \dots, + \infty$ .

We can also conveniently discuss the uniformity of anti-plane stresses inside a non-parabolic open elastic inhomogeneity interacting with a rigid inhomogeneity. In this case, instead of equation (4b) for a traction-free hole, the fixed condition on the boundary of the rigid inhomogeneity is given by

f_{2} (z) - \bar{f_{2} (z)} = 0, z \in L_{2} .

(14)

Consequently, to satisfy equation (14), it is sufficient to replace equation (8) by the following relationship:

a_{- n} = - \frac{\bar{A}}{A} \frac{K - ρ^{- n}}{1 - K ρ^{- n}} {\bar{a}}_{n} - \frac{n - 1 + p}{1 - K ρ^{- n}} - \frac{\bar{A}}{A} \frac{K (n + 1 - \bar{p})}{1 - K ρ^{- n}}, n = 1, 2, \dots, + \infty,

(15)

and the mapping function in equation (5) is still valid. A comparison of equation (8) with equation (15) reveals the fact that the result for the case of a hole can be adapted to that for a rigid inhomogeneity by using the following substitution in equation (8):

K \to - K, A \to i A .

(16)

An example is illustrated in Figure 8. In this case, the internal uniform stress field is determined from equation (12) as $σ_{31} = Γ σ_{31}^{\infty} = \frac{1}{3} σ_{31}^{\infty}, σ_{32} = 0, z \in S_{1}$ . In Figure 8, the non-zero value of $a_{8}$ will cause the shape of the rigid inhomogeneity to be ‘wavy’ in the portion close to the elastic inhomogeneity.

Figure 8.

The non-parabolic open shape of the elastic inhomogeneity and the irregular closed shape of the rigid inhomogeneity when choosing $σ_{32}^{\infty} = 0, K = - 0.5, p = 3, ρ = 0.3, a_{1} = - 0.3, a_{8} = - 0.0009$ .

5. Conclusions

Within the framework of anti-plane elasticity, we have achieved the uniformity of stresses inside a non-parabolic open inhomogeneity in the vicinity of a hole with a closed curvilinear traction-free boundary when the matrix is subjected to uniform stresses at infinity. The non-parabolic open interface $L_{1}$ and the irregular closed hole boundary $L_{2}$ can be expediently determined from the mapping function in equation (5) with $| ξ | = 1$ and $| ξ | = ρ^{- \frac{1}{2}}$ , respectively. The internal uniform stresses inside the inhomogeneity are simply given by equation (12), and the non-uniform stresses in the matrix are determined by equation (13).

Footnotes

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the National Natural Science Foundation of China (Grant No. 11272121) and through a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada (Grant No: RGPIN – 2017 - 03716115112).

ORCID iD

Peter Schiavone

References

Eshelby

JD.

The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc R Soc London A 1957; 241: 376–396.

Eshelby

JD.

The elastic field outside an ellipsoidal inclusion. Proc Roy Soc London A 1959; 252: 561–569.

Eshelby

JD.

Elastic inclusions and inhomogeneities. Prog Solid Mech 1961; II: 89–140.

Sendeckyj

GP.

Elastic inclusion problem in plane elastostatics. Int J Solids Struct 1970; 6: 1535–1543.

Gong

Meguid

SA.

A general treatment of the elastic field of an elliptic inhomogeneity under anti-plane shear. ASME J Appl Mech 1992; 59: S131–S135.

Ting

TCT.

Anisotropic elasticity-theory and applications. New York: Oxford University Press, 1996.

Schiavone

On the elliptical inclusion in anti-plane shear. Math Mech Solids 1996; 1: 327–333.

CQ.

Three-phase elliptical inclusions with internal uniform hydrostatic stresses. J Mech Phys Solids 1999; 47: 259–273.

Schiavone

Mioduchowski

Uniformity of stresses within a three-phase elliptic inclusion in anti-plane shear. J Elasticity 1999; 52: 121–128.

10.

Lubarda

Markenscoff

On the absence of Eshelby property for non-ellipsoidal inclusions. Int J Solids Struct 1998; 35: 3405–3411.

11.

Antipov

Schiavone

On the uniformity of stresses inside an inhomogeneity of arbitrary shape. IMA J Appl Math 2003; 68: 299–311.

12.

Liu

LP.

Solution to the Eshelby conjectures. Proc R Soc London A 2008; 464: 573–594.

13.

Kang

Kim

Milton

GW.

Inclusion pairs satisfying Eshelby’s uniformity property. SIAM J Appl Math 2008; 69, 577–595.

14.

Wang

Uniform fields inside two non-elliptical inclusions. Math Mech Solids 2012; 17: 736–761.

15.

Dai

Gao

CQ.

Uniform stress fields inside multiple inclusions in an elastic infinite plane under plane deformation. Proc R Soc London A 2015; 471(2177): 20140933.

16.

Wang

Schiavone

Uniform stress state inside a non-elliptical inhomogeneity near an irregularly shaped hole in anti-plane shear. Math Mech Solids 2020; 25(2): 362–373.

17.

Wang

Schiavone

Uniformity of stresses inside a parabolic inhomogeneity. Z Angew Math Phys 2020; 71(2): 48.

18.

Philip

JR.

Seepage shedding by parabolic capillary barriers and cavities. Water Resour Res 1998; 34: 2827–2835.

19.

Kacimov

Obnosov

Yu V

. Steady water flow around parabolic cavities and through parabolic inclusions in unsaturated and saturated soils. J Hydrol 2000; 238: 65–77.

20.

Wang

Schiavone

A screw dislocation near a non-parabolic open inhomogeneity with internal uniform stresses. Comptes Rendus Mecanique 2019; 347(12): 967–972.

21.

Wang

Yang

Schiavone

A circular Eshelby inclusion interacting with a non-parabolic open inhomogeneity with internal uniform anti-plane stresses. Math Mech Solids 2020; 25(3): 573–581.

22.

Milton

Serkov

SK.

Neutral coated inclusions in conductivity and anti-plane elasticity. Proc R Soc London A 2001; 457: 1973–1997.