A circular Eshelby inclusion interacting with a coated non-elliptical inhomogeneity with internal uniform stresses in anti-plane shear

Mechanics of Materials 128 (2019) 59–63

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A circular Eshelby inclusion interacting with a coated non-elliptical inhomogeneity with internal uniform stresses in anti-plane shear Xu Wanga, Peter Schiavoneb, a b

T

⁎

School of Mechanical and Power Engineering, East China University of Science and Technology, 130 Meilong Road, Shanghai 200237, China Department of Mechanical Engineering, University of Alberta, 10-203 Donadeo Innovation Centre for Engineering Edmonton, Alberta T6G 1H9, Canada

ARTICLE INFO

ABSTRACT

Keywords: Conformal mapping Anti-plane elasticity Coated inhomogeneity Eshelby inclusion Uniform stress field

We consider a coated non-elliptical inhomogeneity interacting with a nearby circular Eshelby inclusion inside an infinite elastic matrix subjected to anti-plane shear deformations and uniform remote stresses. Using conformal mapping techniques, we prove that despite the presence of the Eshelby inclusion, it is possible to design the system to achieve a uniform stress distribution inside the inhomogeneity. The conformal mapping function used in the analysis is constructed to give rise to an infinite number of first-order poles inside the unit circle in the image plane in order to satisfy all of the conditions required by the complex potential in the matrix. Our analysis indicates that the inhomogeneity's internal uniform stress field is unaffected by the Eshelby inclusion whereas the non-elliptical shape of the coated inhomogeneity is attributed solely to the nearby Eshelby inclusion.

1. Introduction In the design and development of advanced materials, inhomogeneities (material regions with elastic properties distinct from those of the surrounding material) are often introduced into an elastic material (known as the ‘matrix’) in order to influence its mechanical and physical properties. The resulting composite material has superior properties (e.g., strength, resistance to corrosion or fatigue) not achievable by individual constituent materials alone. Eshelby (1957) pioneered the study of inhomogeneities in engineering materials by proposing the Equivalent Inclusion Method (EIM) which essentially states that an inhomogeneity can be treated as an inclusion (same elastic properties as the surrounding matrix but containing an eigenstrain). Eshelby used his EIM to show that uniform internal stress and strain distributions can be achieved inside a three-dimensional ellipsoidal inhomogeneity when the surrounding matrix is subjected to any uniform remote elastic loading (see Zhou et al., 2013 for a recent comprehensive review). Equivalent inclusions corresponding to elastic inhomogeneities with Eshelby's uniformity property are often referred to as ‘Eshelby Inclusions.’ The study of inhomogeneities which achieve uniform internal stress distributions continues to attract a great deal of attention in the literature (see for example, Hardiman, 1954; Eshelby, 1957, 1959, 1961; Sendeckyj, 1970; Ting, 1996; Markenscoff, 1998; Lubarda and Markenscoff, 1998; Ru and Schiavone, 1996; Ru, 1999; Ru et al., 1999,

⁎

2005; Liu, 2008; Kang et al., 2008; Wang and Gao, 2011; Wang, 2012; Wang et al., 2018; Dai et al., 2015). From the point of view of composite mechanics, this is mainly attributed to the desire to eliminate ‘stress peaks’ within the inhomogeneities since these are known to be a significant contributing factor to failure of the composite structure. Inhomogeneities which achieve uniform internal stress distributions not only eliminate the possibility of internal stress peaks but are also known to effectively reduce the existing stress concentration in the surrounding matrix. Eshelby's original investigations, however, concerned only a single inhomogeneity. In reality, composite materials contain a variety of multiple inhomogeneities and the effect of interaction between these inhomogeneities on the mechanical properties of the composite is of paramount importance. One such scenario arises when a circular Eshelby inclusion interacts with a not necessarily elliptical elastic inhomogeneity inside an infinite matrix subjected to uniform remote stresses. It is of great practical interest to ask whether the inhomogeneity can be designed in such a way as to guarantee an internal uniform stress field and thus gain the design advantages mentioned above. As a first step in the analysis of such a problem, Wang et al. (2018) considered the case when such a composite was subjected to anti-plane shear deformations and found that indeed an internal uniform stress field inside the inhomogeneity remains possible despite its interaction with the circular Eshelby inclusion. This result, however, was obtained under the idealized assumption that the elastic

Corresponding author. E-mail addresses: [email protected] (X. Wang), [email protected] (P. Schiavone).

https://doi.org/10.1016/j.mechmat.2018.10.005 Received 28 August 2018; Received in revised form 25 September 2018; Accepted 16 October 2018 Available online 17 October 2018 0167-6636/ © 2018 Elsevier Ltd. All rights reserved.

Mechanics of Materials 128 (2019) 59–63

X. Wang, P. Schiavone

inhomogeneity was perfectly bonded to the matrix, that is, in the absence of any coating between the inhomogeneity and the matrix. In the design of composite structures, however, the more physically realistic scenario does indeed involve the placing of a layer of separate elastic material (a coating or interphase layer e.g., glue) between the inhomogeneity and the matrix (see, for example, Markenscoff and Dundurs, 2014 and the references contained therein). In addition to offering several other important advantages, such a layer plays an effective role in reducing residual stresses (Ru, 1998; Ru et al., 1999). In this paper, we advance the analysis of the ‘interaction problem’ begun in Wang et al. (2018) one step further by considering the more practical and physically realistic case when a coating phase is placed between the inhomogeneity and the matrix. We remain within the context of the ‘simpler’ anti-plane deformations of the composite since this allows us to use specialized conformal mapping techniques which are central to our analysis. Intuitively, one would expect that when a circular Eshelby inclusion is located near a coated inhomogeneity, the stress field inside the inhomogeneity will become intrinsically nonuniform. In fact, we find that the internal stresses inside the inhomogeneity can indeed be maintained uniform by making a judicious choice of the non-elliptical shape of the coated inhomogeneity. Furthermore, our analysis reveals that the (close) spacing between the circular Eshelby inclusion and the coated inhomogeneity exerts no influence whatsoever on the uniform stress field inside the inhomogeneity, that is, the internal (uniform) field is independent of the presence of the nearby circular Eshelby inclusion. In direct contrast, we find that the non-elliptical shape of the coated inhomogeneity depends exclusively on the interaction with the circular Eshelby inclusion itself. Both the shape of the coated inhomogeneity and the location of the center of the circular Eshelby inclusion can be conveniently determined once the five non-trivial parameters describing the mapping function are given. Finally, we mention that the use of D'Alembert's ratio test allows us to establish a criterion for the convergence of the series appearing in the corresponding conformal mapping function. This is a critical step underpinning the analysis used to establish our conclusions.

Fig. 1. A coated non-elliptical elastic inhomogeneity interacting with a circular Eshelby inclusion in anti-plane shear.

Our objective is to analyse whether the non-elliptical inhomogeneity can still admit a uniform internal stress field even in the presence of the intermediate coating and the nearby circular Eshelby inclusion. 3. The internal uniform stresses In the physical z-plane, the boundary value problem for the fourphase composite consisting of the inhomogeneity, coating, matrix and the circular Eshelby inclusion takes the following form:

f2 (z ) + f2 ¯(z ) = 1f1 (z ) + 1f1 ¯(z ) , * )z f2 (z ) f2 ¯(z ) = f1 (z ) f1 ¯(z ) + 2( *32 + i 31

31

= µf (z ),

+ iµw = µf (z ),

=

,1,

31

=

,2 .

(3b)

f3 (z ) + f3 ¯(z ) = f4 (z ) + f4 ¯(z ) , ** + i 31 **) z f3 (z ) f3 ¯(z ) = f4 (z ) f4 ¯(z ) + 2( 32

2( ** 32

i ** 31 ) z¯, z

L3 ; (3c)

32

f3 (z )

+i µ3

31

z + O (1), z

,

(3d)

where Γ1 and Γ2 are two stiffness ratios defined by

(1)

where μ is the shear modulus, and the two stress components can be expressed in terms of the stress function as (Ting, 1996) 32

L1 ; (3a)

Under anti-plane shear deformations of an isotropic elastic material, the two shear stress components σ31 and σ32, the out-of-plane displacement w and the stress function ϕ can be expressed in terms of a single analytic function f(z) of the complex variable z = x1 + ix2 as (Ting, 1996)

+i

i *31) z¯, z

f3 (z ) + f3 ¯(z ) = 2 f2 (z ) + 2 f2 ¯(z ) , f3 (z ) f3 ¯(z ) = f2 (z ) f2 ¯(z ) , z L2 ;

2. Problem formulation

32

* 2( 32

1

=

µ1 , µ2

2

=

µ2 . µ3

(4)

It is readily deduced from Eq. (3c) that

(2)

f3 (z ) +

Consider a domain in ℜ2, infinite in extent, containing a non-elliptical coated elastic inhomogeneity with elastic properties different from those of the surrounding matrix and a circular Eshelby inclusion with elastic properties identical to those of the surrounding matrix (see Fig. 1). The center of the circular Eshelby inclusion is located at z = z 0 and its radius is a. Let S1, S2, S3 and S4 denote the inhomogeneity, the coating, the matrix and the circular Eshelby inclusion, respectively, all of which are perfectly bonded through the inhomogeneity-coating nonelliptical interface L1, the coating-matrix non-elliptical interface L2 and the inclusion-matrix circular interface L3. There is no intersection among any of the three interfaces. The matrix is subjected to uniform remote anti-plane shear stresses ( 31 , 32 ) , uniform anti-plane shear ei* ) are imposed on the non-elliptical inhomogeneity S1, genstrains ( *31, 32 **) are imposed on the and uniform anti-plane shear eigenstrains ( ** 31 , 32 circular Eshelby inclusion S4. In what follows, the subscripts 1, 2, 3 and 4 will be used to identify the respective quantities in S1, S2, S3 and S4.

** i 31 **) a2 ( 32 ** + i 31 **) z = f4 (z ) + ( 32 z z0

** ( 32

i ** 31 ) z¯ 0 , z

L3 , (5)

from which the following auxiliary function g(z) is constructed

g (z ) =

f3 (z ) +

a2 ( 32 ** i ** 31 ) , z z0

** + i 31 **) z f4 (z ) + ( 32

** ( 32

z

S3; **) z¯0 , z i 31

S4.

(6)

It is clearly seen from the above definition that g(z) is continuous across L3 and analytic in S3∪S4 except at the point at infinity where its asymptotic behavior is the same as that of f3(z). Now consider the following conformal mapping function

z=

( ),

=

1 (z ),

1,

(7)

which maps the exterior of the simply-connected inhomogeneity onto the exterior of the unit circle in the ξ-plane (see Fig. 2). More 60

Mechanics of Materials 128 (2019) 59–63

X. Wang, P. Schiavone

and arrive at the following expression for q1

** + i 31 **) 4a2 ( 32

q1 = q =

R

2 0

( 0 )[k (

1)(

1

+ 1)

2

2(

2

* + i 31 * )] + 1)( 32

.

(14)

By utilizing Eqs. (12) and (14), the mapping function in Eq. (11) can be further expressed as

z=

( )=R

p

+

+

j

j=0

j¯ 1 0

+q

,

1.

(15)

By enforcing the remote asymptotic condition in Eq. (3d), we arrive at the following relationship

k [(

+ 1)( ¯ + kp¯ [( 1

= Fig. 2. The problem in the image ξ-plane.

1

||

1 2,

and | |

k(

respectively; the two interfaces L1 and L2 1

k¯ (

+

+ +

1)

1

( *32

2

*) ¯ i 31

k( 1

1)( 2 4

k¯ ( 1

1)( 2 + 1) 4

( 2

1)

k¯ ( 1 + 1)( 2 4

1)

+

1)( 32 * + i 31 * ) 2

( 2 + 1)( 32 * 2

i 31 * )

( 2

i 31 * )

1)( 32 * 2

1

1 2.

,1

¯

32

z=

( )=R

+

+

(9)

+

j 1¯ 1 0

j =1

1 2.

,

(10)

**

i

1,

qj ,

2, …,

+

=

(1 (1 +

2)[k ( 1 2)[k ( 1

,

* + i 31 * )] + 1) + 2( 32 , * + i *31)] 1) 2( 32

(

1)],

2

(16)

1)( 2

1)]

1)] .

1)]}

= µ1 k , z

(18)

S1,

p 2 0

+

j

q j =0

(

j ¯ 1) 2 0

0

.

(19)

j

4 j =0 2 0

(

0

2

j )2

+

4 q¯ [k¯ (

1)(

1

2

( ) ( ** a 2 R

32

+ 1)

2(

**) i 31 2

* + 1)( 32

* )] i 31 (20)

p.

a R

2

** ( 32

**)/[k¯ ( i 31

1

1)(

2

+ 1)

2(

2

+ 1)( *32

* )] i 31

(21)

can be uniquely determined. Note that the complex parameter k given by Eq. (17) is independent of the existence of the circular Eshelby inclusion. Consequently, a fixed value of the ratio γ means that the eigenstrains on the circular Eshelby inclusion should be properly imposed in contrast to the remote loading and the eigenstrains imposed on the non-elliptical inhomogeneity. Thus γ can be considered as a (complex) loading ratio. It can be easily verified that the following expression for f4 ( ) = f4 ( ( )) is analytic in its region of definition including at the point = 0

(12)

where Λ is a dimensionless complex parameter defined by

=

+1

2

The above expression implies that for given values of the five parameters p, q, ρ, Λ and ξ0, the ratio γ defined by

(11)

**)

j = 1,

0

=

32 31 f3 (z ) = + g (z ) which is deduced from Eq. (6), we arrive at z z0 the following recurrence relation

qj + 1 =

31

+

q

where R is a real scaling constant, p is a given complex constant, qj , j = 1, 2, …, + are complex constants to be determined. Inserting Eq. (11) into Eq. (10) and noting that f3(ξ) should be = j 0, j = 1, 2, …, + regular at the points and that a2 (

* )][ i 31

Consequently, the following relationship can be derived from Eqs. (14) and (19)

1

qj

1)]

2

* p¯ ( 32

is unaffected by the presence of the circular Eshelby inclusion embedded in the matrix. Thus, the internal uniform stress field in Eq. (18) coincides with that for a confocally coated elliptical inhomogeneity characterized by q = 0 in Eq. (15) in Ru et al. (1999). It is also seen from Eqs. (13) and (17) that the complex parameter Λ is dependent on the remote loading and the eigenstrains imposed on the non-elliptical inhomogeneity. It is then quite simply deduced from the mapping function in Eq. (15) that

An examination of the above expression for f3(ξ) suggests that the mapping function should take the following specialized form

p

+i

( 0) = R 1

,

+ 1)(

It is seen from the above analysis that the internal uniform stress field inside the inhomogeneity given by

( )

1

1

+ p 2 [( 1 1)( 2 + 1) + ( 1 + 1)( 2 * * )[ 2 + 1 + ( 2 1)] 2p¯ 1 ( 32 i 31

( ) ¯

+ 1) + (

*) 2[( *32 + i 31

1)]

2

(17)

( )

( 2 + 1)( 32 * + i 31 * ) k ( 1 + 1)( 2 + 1) + 4 2

f3 ( ) = +

+ 1) * + i 31 *) + ( 32 2

2

* + i 31 * ){[( 1 + 1)( 2 + 1) + ( 1 ( 32 ×

(8)

S1,

1

1)(

4( 32 + i 31 ) µ3

1)(

1

4( 32 + i 31 )[( 1 + 1)( 2 + 1) + ( 1 1)( 2 1)] 4p¯ ( 32 i 31 )[( 1 1)( 2 + 1) + ( 1 + 1)( 2 µ3 {[( 1 + 1)( 2 + 1) + ( 1 1)( 2 1)]2 p 2 [( 1 1)( 2 + 1) + ( 1 + 1)( 2 1)]2 } 2[ 2 + 1 ( 2 1)] [( 1 + 1)( 2 + 1) + ( 1 1)( 2 1)]2 p 2 [( 1 1)( 2 + 1) + ( 1 + 1)( 2 1)]2

where k is a complex number to be determined. By enforcing the continuity conditions in Eqs. (3a) and (3b) across the two perfect interfaces L1 and L2, we arrive at the following expressions for f2 ( ) = f2 ( ( )) and f3 ( ) = f3 ( ( ))

f2 ( ) =

+ 1) + (

k=

are mapped onto the two concentric circles | | = 1 and | | = 2 ; the center of the circular Eshelby inclusion z = z 0 is mapped onto = 0 (i.e., z 0 = ( 0 ) ). Here ρ(0 < ρ < 1) is a dimensionless parameter measuring the relative thickness of the coating. In order to ensure that the internal stresses inside the inhomogeneity are uniform, f1(z) defined in the inhomogeneity must take the following form:

f1 (z ) = kz , z

2

from which the complex number k can be uniquely determined as

specifically, as shown in Fig. 2, the regions S2 and S3 are mapped onto 1 2

1

(13) 61

Mechanics of Materials 128 (2019) 59–63

X. Wang, P. Schiavone

f4 ( ) = f3 ( ) +

** a2 ( 32 ( )

**) i 31 ( 0)

** + i 31 **) ( ) + ( 32 ** ( 32

**) (¯ 0 ) . i 31 (22)

Using D'Alembert's ratio test, it is not difficult to verify that the series in the mapping function (15) is convergent when |Λ| < 1. 4. Illustrative examples In this section, several specific numerical examples will be presented to demonstrate the theoretical result obtained in the previous section. In performing the calculations, the series in Eq. (15) is truncated at j = 100 to arrive at satisfactory results. As a first example, the configuration in Fig. 1 is obtained by using the following parameters

p=

0.5, q = 0.2,

=

= 0.5,

0

= 2 exp

i . 6

(23)

Fig. 4. The non-elliptical shape of the coated inhomogeneity by choosing the parameters in Eq. (25). The star is the center of the circular Eshelby inclusion.

The non-elliptical shape of the coated inhomogeneity and the center of the circular Eshelby inclusion are illustrated in Figs. 3–6 for the following respective sets of parameters

p=

0.5, q = 0.2,

p = 0.5, q =

0.2,

=

= 0.5,

0

= 2.

(24)

=

= 0.5,

0

= 2.

(25)

p=

0.5, q =

0.2,

p=

0.5, q = 0.2733,

=

= 0.5, =

= 0.5,

0

= 1.8.

(26)

= 2.

(27)

0

It is deduced from Eq. (13) that = in the above five examples * = 32 * = 0 or when μ1 ≠ μ3 can be satisfied either when µ1 = µ3 and 31 * + i *31)/( 1 2 1) , which can be considered as a relationand k = 2( 32 ship between the remote loading and the eigenstrains imposed on the * = 11 31 + 12 32 , µ3 *32 = 21 31 + 22 32 with inhomogeneity (i.e., µ3 31 λ11, λ12, λ21 and λ22 being four real dimensionless coefficients). It is seen from Fig. 5 that two portions of the inner interface L1 are almost in contact with each other, and from Fig. 6 that there is a sharp corner on the inner interface L1. It is numerically verified that the existence of the sharp corner in Fig. 6 is due to the fact that (1) = 0 . It is readily deduced from Eqs. (1) and (9) that the stresses in the coating remain bounded at the sharp corner on L1. The specific values of the loading ratio γ defined by Eq. (21) obtained by using the five sets of parameters in Eqs. (23)–(27) are respectively

= 0.0952 + 0.1732i, 0.1952,

0.2048,

0.2205, 0.2519.

Fig. 5. The non-elliptical shape of the coated inhomogeneity by choosing the parameters in Eq. (26). The star is the center of the circular Eshelby inclusion.

(28)

Fig. 6. The non-elliptical shape of the coated inhomogeneity by choosing the parameters in Eq. (27). The star is the center of the circular Eshelby inclusion.

Note that the radius of the circular Eshelby inclusion in Figs. 1, 3–6 can be set to a value at which the inclusion-matrix circular interface L3 is barely in touch with the coating-matrix non-elliptical interface L2.

Fig. 3. The non-elliptical shape of the coated inhomogeneity by choosing the parameters in Eq. (24). The star is the center of the circular Eshelby inclusion. 62

Mechanics of Materials 128 (2019) 59–63

X. Wang, P. Schiavone

5. Conclusions

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Within the framework of anti-plane elasticity, we have proved that a coated non-elliptical inhomogeneity continues to admit an internal uniform stress field despite interaction with a circular Eshelby inclusion in a matrix subjected to uniform stresses at infinity. This interesting result is obtained via the introduction of the mapping function in Eq. (15) which contains five non-trivial parameters p, q, ρ, Λ and ξ0. An infinite number of first-order poles within the unit circle in the mapping function (15) are present in order to completely remove the unnecessary first-order poles at = j 0, j = 1, 2, …, + while retaining the first-order pole at = 0 for f3(ξ). The theory is validated through several typical examples. Acknowledgements 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). References Dai, M., Gao, C.F., Ru, C.Q., 2015. Uniform stress fields inside multiple inclusions in an elastic infinite plane under plane deformation. Proc. R. Soc. London A 471 (2177), 20140933. Eshelby, J.D., 1957. The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc. R. Soc. Lond. A 241, 376−396. Eshelby, J.D., 1959. The elastic field outside an ellipsoidal inclusion. Proc. R. Soc. Lond. A 252, 561−569.

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