Circular inhomogeneity with Steigmann–Ogden interface: Local fields, neutrality, and Maxwell’s type approximation formula

Accepted Manuscript

Circular inhomogeneity with Steigmann-Ogden interface: Local fields, neutrality, and Maxwell’s type approximation formula Anna Y. Zemlyanova, Sofia G. Mogilevskaya PII: DOI: Reference:

S0020-7683(17)30516-4 10.1016/j.ijsolstr.2017.11.012 SAS 9801

To appear in:

International Journal of Solids and Structures

Received date: Revised date: Accepted date:

4 August 2017 11 October 2017 11 November 2017

Please cite this article as: Anna Y. Zemlyanova, Sofia G. Mogilevskaya, Circular inhomogeneity with Steigmann-Ogden interface: Local fields, neutrality, and Maxwell’s type approximation formula, International Journal of Solids and Structures (2017), doi: 10.1016/j.ijsolstr.2017.11.012

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ACCEPTED MANUSCRIPT

Circular inhomogeneity with Steigmann-Ogden interface: Local fields,

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neutrality, and Maxwell’s type approximation formula Anna Y. Zemlyanovaa , Sofia G. Mogilevskayab∗ November 13, 2017

of Mathematics. Kansas State University, 138 Cardwell Hall, Manhattan, Kansas,

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a) Department

66506, USA

b)

Department of Civil, Environmental, and Geo- Engineering, University of Minnesota, 500 Pillsbury

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Drive S.E., Minneapolis, MN, 55455, USA

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Abstract

The boundary conditions for the Steigmann-Ogden (1997, 1999) model are re-derived for a two dimensional surface using general expression for surface energy that include surface tension. The

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model treats the interface as a shell of vanishing thickness possessing surface tension as well as membrane and bending stiffness. The two-dimensional plane strain problem of an infinite isotropic elastic domain subjected to the uniform far-field load and containing an isotropic elastic circular

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inhomogeneity whose interface is described by the Steigmann-Ogden model is solved analytically. Closed-form expressions for all elastic fields in the domain are obtained. Dimensionless parameters that govern the problem are identified. The Maxwell type approximation formula is obtained for

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the effective plane strain properties of the macroscopically isotropic materials containing multiple inhomogeneities with the Steigmann-Ogden interfaces. The “neutrality” conditions are analyzed. It is demonstrated that while the Steigmann-Ogden model theoretically reduces to the Gurtin-Murdoch (1975, 1978) model when the bending interphase effects are neglected, the two models (for the case of zero surface tension) describe two very different interphase regimes of seven regimes proposed by Benveniste and Miloh (2001).

Keywords: Circular inhomogeneity, Surface effects, Steigmann-Ogden model, Effective properties ∗

Corresponding author. E-mail address: [email protected] (S.G. Mogilevskaya)

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1. Introduction The need to incorporate surface and interface effects in the mechanical models of nano-scale size materials and composite structures is well recognized. These effects might play a crucial role in understanding various small-scale related phenomena, e.g. torsion strength of wires (Fleck and Hutchinson, 1993), bending stiffness of plates (Miller and Shenoy, 2000), tensile strength of ultra-thin films (Judelewicz et

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al., 1994; Hong and and Weil, 1996; Read, 1998), indentation hardness of materials (Oliver and Pharr, 1992; Ma and Clarke, 1995; Nix and Gao, 1998; Horstemeyer and Baskes, 1999; Cheng and Cheng 2004; Qu et al., 2004; Gao, 2006a, 2006b). Various physical mechanisms such as self-assembly, phase transformation, thin film growth, and catalysis can also be attributed to the effects of surface energy. The study of surface tension in solids started with the work of Gibbs (1906). More general mechanical model that included the effects of surface tension and surface elasticity was developed by Gurtin and

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Murdoch (1975, 1978). In this theory, the surface of the material is modeled as a two-dimensional membrane of vanishing thickness adhering to a three-dimensional bulk solid without slipping. It is treated as an elastic surface that resists only to stretching but not to flexure. Gurtin et al. (1998) generalized the original model by allowing all the components of the displacement vector to undergo a jump across the interface. The Gurtin-Murdoch and related models of surface elasticity have been used

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to study

• nanosized inhomogenuities, e.g. Sharma and Ganti (2002, 2004), Sharma et al. (2003), Dingreville et al. (2005), Duan et al. (2005a,b,c, 2006, 2007), Lim et al. (2006), He and Li (2006), Huang

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and Wang (2006), Mi and Kouris (2006, 2015), Chen et al. (2007), Tian and Rajapakse (2007), Mogilevskaya et al. (2008, 2010), Ru (2010, 2016), Kushch et al. (2011, 2013), and many others.

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• beams, plates, and shells, e.g. Miller and Shenoy (2000), Altenbach and Eremeyev (2011), and the references therein.

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• thin films and nanowires, e.g. Diao et al. (2003), He and Lilley (2008), Cammarata (2009), Yvonnet et al. (2011), Chhapadia et al. (2011, 2012), and the references therein.

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Javili et al. (2017), analyzing the impact of Gurtin-Murdoch (1975, 1978) papers, estimated that their total citations number exceeded 1000 by 2015. Two years later that number exceeded 2500. We refer the interested reader to the papers by Chhapadia et al. (2011), .Javili et al. (2013, 2014, 2017) for the extensive review of the relevant literature. Steigmann and Ogden (1997, 1999) noted that the membrane in the Gurtin-Murdoch theory cannot support compressive stress states and suggested an improvement which includes the resistance of the membrane to both stretch and flexure. Mathematically this means that the surface energy in the Steigmann-Ogden form depends both on the surface strain tensor and the surface curvature tensor. 2

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The literature on the Steigmann-Ogden surface elasticity model is rather limited and no benchmark results are available even for the inhomogeneities of regular shapes, e.g. circular or spherical. This is despite the fact that practically all papers on surface elasticity cite the original papers by Steigmann and Ogden. The Steigmann-Ogden model has been used in Chhapadia et al. (2011, 2012) to study bending of nano-sized cantilever beams. One important result of this work is computation of the Steigmann-

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Ogden constants by using atomistic simulations. To the authors’ knowledge this is the only work where the values of the surface constants involved in the curvature-dependent term in the Steigmann-Ogden surface energy are reported up to date.

The paper by Eremeyev and Lebedev (2016) presents a mathematical study of the Steigmann-Ogden model. In particular, boundary conditions for the simplified Steigmann-Ogden model (without surface tension) are obtained using calculus of variations and the existence of weak solutions in Sobolev spaces

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is proved. Unfortunately, the detailed derivations of the governing equations are not presented in that paper and the final expressions contain misprints, which will be mentioned in Section 2. The Steigmann-Ogden theory has been used in the problem of a mixed mode plane fracture in Zemlyanova (2017a) and in the frictionless contact problem of a rigid stamp with a semi-plane, see Zemlyanova (2017b). The results in the papers Zemlyanova (2017a,b) show that curvature-dependence is increasingly important at small scales.

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The effect of curvature-dependent interfacial energy was also studied by Gao et al. (2014) where a finite deformation interface stress theory was established. The linear interface stress theory was further

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developed in Gao et al., (2017) where it was applied to study the effective modulus of nanocomposites. However, our analysis of these publications suggests that the model developed there is different from that of Steigmann and Ogden (1997, 1999). While the Steigmann-Ogden model theoretically reduces

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to the Gurtin-Murdoch one (when the bending parameters vanish), the model of Gao et al. (2014) and Gao et al., (2017) does not allow for such reduction. On another hand, it is demonstrated in Javili et

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al. (2013), Javili et al. (2017) (see also the references therein) that the Gurtin-Murdoch model could be recovered from the higher-gradient theories. The detailed comparisons of various surface elasticity models is out of scope of the present paper and could be the subject of a separate study.

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The goals of this paper are twofold. First, we re-derive the boundary conditions for the SteigmannOgden model for a two dimensional surface of any shape using more general formula of surface energy that includes surface tension. Second, we provide the benchmark solution of the plane strain problem of an infinite elastic matrix subjected to the uniform load at infinity and containing a single circular inhomogeneity with the Steigmann-Ogden interface. The paper is structured as follows. In Section 2 (with the details provided in Appendix A), the constitutive and equilibrium equations of the Steigmann-Ogden theory are presented for a two dimensional surface of any shape. The formulation of the benchmark problem is presented in Section 3. The 3

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equations for one dimensional circular surface involved in the problem are rewritten in polar coordinates in Section 4 and some details are presented in Appendix B. The complex variables solution of the plane strain problem is developed in Sections 5 with the details provided in Appendices C, D. Section 6 contains the derivations of the Maxwell type approximation formula for the effective plane strain properties of the materials containing multiple inhomogeneities with the Steigmann-Ogden interfaces.

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The neutrality conditions are discussed in Section 7 and Appendix E. Finally, numerical examples are given in Section 8.

2. Basic equations of the Steigmann-Ogden (1997, 1999) model

In the case of infinitesimal deformations, the model assumes that the bulk materials are linearly elastic and governed by the standard equations of linear elasticity theory. The material surface of vanishing thickness possesses residual surface tension as well as membrane and bending stiffness. The

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equilibrium equations and the boundary conditions that describe the three-dimensional boundary value problem of that type are presented by Eqs. (11)-(15) in Eremeyev and Lebedev (2016). To obtain the equations, the authors used the variational approach, but they have not presented the detailed derivations. We re-derived the latter equations in Appendix A, as we believe that there are misprints in some equations of Eremeyev and Lebedev (2016) that are discussed at the end of this Section. In addition, Eremeyev and Lebedev (2016) used a simplified model that does not include surface tension.

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The developments of Appendix A lead to the following set of equations describing the conditions at the material surface As bounded by the curve ∂As :

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a) Continuity of the displacements

(1)

n · 4σ = ∇S · [T + (∇S · M) n] − (∇S · n) n · (∇S · M) n

(2)

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u = usur

b) Surface equilibrium conditions

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on As

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on ∂As

ν·T + ν · IS · (∇S · M) n +

∂τ · (ν·M) n + ∇S · [(ν · M) n] = 0 ∂s

(3)

at the end points (if any) of ∂As τ · (ν·M) n = 0 c) Surface constitutive equations

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(4)

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T = σ0 IS + (λ0 + σ0 ) trεsur IS + 2 (µ0 − σ0 ) εsur + σ0 ∇S u

(5)

M = 2χκ

(6)

sur

sur

+ ζtrκ

IS

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The following notations are adopted in Eqs. (1)-(6): µ0 and λ0 are the shear modulus and Lam´e parameter of the material surface, respectively; σ0 is the residual surface tension; ζ and χ are the bending stiffness parameters; 4σ is the jump of the stress tensor of the bulk material; the symbol “·” identifies the single dot product of two tensors; n, ν, τ are the unit normal, outward co-normal, and tangential vectors to As , respectively; s is the arc length parameter for ∂As ; IS = I − n ⊗ n is the unit

surface tensor; I is the three-dimensional unit tensor; “⊗” is a a dyadic product of two vectors; ∇S u is

the surface gradient of the bulk displacement vector u; usur is surface displacement vector; trεsur and

of curvature) κsur εsur =

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trκsur are the traces of the surface strain tensor εsur and the bending strain measure (tensor of changes

 1 ∇S u · IS + IS · (∇S u)T 2

 1 ∇S ϑ · IS + IS · (∇S ϑ)T 2

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κsur = −

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in which

ϑ = ∇S (n · u) + B · u

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and B = −∇S n is the curvature tensor. The comparison of Eqs. (2), (3) with Eqs. (13), (14) in Eremeyev and Lebedev (2016) pinpoints a few misprints in the latter publication (the most serious ones are the missing terms that appear as the last terms in our Eqs. (2), (3)).

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Eqs. (5), (6) are coupled as the the surface gradient ∇S u has both normal and shear components. 3. Benchmark problem formulation

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In this section we consider the benchmark problem of an infinite, isotropic elastic plane (matrix) containing a circular, isotropic, elastic inhomogeneity whose interface is described by the SteigmannOgden model.

Assume that the center of the circular inhomogeneity with the boundary L, and radius R is located

at the origin of the Cartesian coordinate system (Fig.1). The elastic properties of the inhomogeneity (shear modulus µI and Poisson’s ratio νI ) are arbitrary, and are different from those of the matrix (µ, ν). The interface between the inhomogeneity and the matrix is described by the Steigmann-Ogden model characterized by the parameters µ0 , λ0 , σ0 , ζ, and χ. The entire system is subjected to a biaxial 5

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y

L

0 μI, νI

R x

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μ, ν

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Figure 1: An inhomogeneity with the Steigmann-Ogden interface in an infinite matrix ∞ , σ ∞ , σ ∞ ). The goal is to derive closed-form expressions for the elastic fields stress field at infinity (σxx yy xy

in the composite system and to use them to obtain the Maxwell type approximation formula for the overall properties of the macroscopically isotropic materials containing multiple inhomogeneities with Steigmann-Ogden interfaces. The conditions under which the inhomogeneity does not affect the original

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elastic fields in the matrix without the inhomogeneity, the so-called “neutrality conditions,” will be also investigated.

4. Equations for the circular material surface

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For the considered here two-dimensional plane-strain problem involving a circular surface, Eqs. (1)(6) can be re-written in the polar coordinate system (r, θ) shown in Fig 2. The local components of the

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surface displacements are introduced as u` = uθ and un = ur , where n and ` identify the normal and the tangential directions, respectively. We take into account the following expressions (see Appendix B

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for the details):

IS = eθ ⊗ eθ

(7)

n = er

(8)

∇S = eθ

1 ∂ R ∂θ

∇S u = ∇S ⊗u = εsur eθ ⊗ eθ + ω sur eθ ⊗ er

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(9)

(10)

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εsur =

1 (uθ,θ + ur ) R

(11)

ω sur =

1 (ur,θ − uθ ) R

(12)

1 (ur,θθ − uθ,θ ) eθ ⊗ eθ R2

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κsur = −

(13)

where eθ , er are the unit vectors tangential and normal to the circular surface, respectively and the subscript “,” indicates differentiation, e.g. uθ,θ = ∂uθ /∂θ.

y

βt

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eθ

x

M

0

er t θt

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Figure 2: Notations related to a circular material surface Using expressions (7)-(13), one can re-write the basic equations for the problem as

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a) Continuity of the displacements

inh uinh = umat = umat r r , uθ θ

(14)

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b) Equilibrium of the material surface L

σ0 σ0 + 2 (ur,θθ − uθ,θ ) R R 1 − (λ0 + 2µ0 ) 2 (uθ,θ + ur ) R 1 − (2χ + ζ) 4 (ur,θθθθ − uθ,θθθ ) R

inh mat σrr − σrr =−

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(15)

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σ0 (ur,θ − uθ ) R2 1 + (λ0 + 2µ0 ) 2 (uθ,θθ + ur,θ ) R 1 − (2χ + ζ) 4 (ur,θθθ − uθ,θθ ) R

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inh mat σrθ − σrθ =

(c) Constitutive equations for the surface L

  1 1 T = σ0 + (2µ0 + λ0 ) (uθ,θ + ur ) eθ ⊗ eθ + σ0 (ur,θ − uθ ) eθ ⊗ er R R M = − (2χ + ζ)

1 (ur,θθ − uθ,θ ) eθ ⊗ eθ R2

(16)

(17) (18)

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The superscript inh (mat) in Eqs. (14)-(16) describes the nano-inhomogeneity (matrix); tensor T of Eq. (17) and tensor M of Eq. (18) are the surface stress and the surface couple stress tensors, respectively. Using the fact that

∂ 1 ∂ = ∂s R ∂θ

(19)

where s is the arc length of the undeformed surface, one can see that Eqs. (14)-(18) coincide with the

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equations for the Gurtin-Murdoch model (see equations of Section 3.3 in Mogilevskaya et al., 2008) when the bending interface effects are neglected, i.e. χ = ζ = 0.

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5. Analytical solution of the benchmark problem As the bulk materials of the matrix and inhomogeneity are linearly elastic, we can utilize wellknown elasticity solutions for the two complementary problems: one of the circular disc subjected to

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the unknown tractions σ inh (τ ) at any boundary point τ ∈ L and another one of an infinite matrix subjected to the uniform far-field load and containing a circular hole under the action of the unknown

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boundary tractions σ mat (τ ). The solutions for both problems can be obtained by various methods, e.g. by using Airy stress functions or Kolosov-Muskhelishvili potentials combined with series expansions, see Muskhelishvili (1959). However, here we will use the complex variables approach presented in

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Mogilevskaya et al. (2008), as it allows for easier incorporation of quite complex boundary conditions of Eqs. (15), (16) and for straightforward extension to problems involving multiple inhomogeneities (that will be a subject of our future work). 5.1. Complex combinations of the elastic fields We assume that, for both problems, the unit normal n points to the right of the direction of travel (i.e. outside of the disc and inside the hole) and the unit tangent ` is directed in the direction of travel (counterclockwise for the disc and clockwise for the hole) and introduce the complex combinations of

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tractions, displacements, and far-field stresses for the inhomogeneity and the matrix as inh inh σ inh (t) = σninh (t) + iσ`inh (t) = σrr (t) + iσrθ (t)

(20)

mat mat σ mat (t) = σnmat (t) + iσ`mat (t) = σrr (t) + iσrθ (t)

where

(21)

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u(t) = ux (t) + iuy (t)   dt¯ ∞ ∞ ∞ σ (t) = − σ1 + σ2 dt

(22)

(23)

dt¯/dt = exp(−2iβt )

(24)

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∞ ∞ ∞ ∞ ∞ σ1∞ = σxx + σyy , σ2∞ = σyy − σxx − 2iσxy

in which βt is the angle between the axis Ox and tangent at the point t = x + iy ∈ L, Fig.2. Notice also that βt = θt + π/2, which leads to the following relations: exp(−2iβt ) = − exp(−2iθt ) = −

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where

R2 = −g 2 (t) t2

R t Condition (14) of continuity of the displacements has the following complex form:

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g(t) =

uinh (τ ) = umat (τ ) = u(τ ), τ ∈ L

(25)

(26)

(27)

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The complex displacement un (τ ) + iu` (τ ) = ur (τ ) + iuθ (τ ) in the local coordinate system can be

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expressed through the complex displacement u(τ ) = ux (τ ) + iuy (τ ) as follows ur (τ ) + iuθ (τ ) = iu(τ ) exp(−iβτ ) = u (τ ) exp(−iθτ ) = u (τ ) g (τ )

(28)

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Taking into account that the conjugation operation yields g(τ ) = g −1 (τ )

(29)

we obtain

i 1h u(τ )g (τ ) + u(τ )g −1 (τ ) 2 i 1 h uθ (τ ) = Im [u(τ )g (τ )] = u(τ )g (τ ) − u(τ )g −1 (τ ) 2i ur (τ ) = Re [u(τ )g (τ )] =

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(30)

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Using Eqs. (30) and performing some algebraic operations, one can rewrite formulae (15) - (16) as follows (see Appendix C for the details):

= − +

inh mat σrθ (t) − σrθ (t) =

−

   2  1 ∂u(t) ∂ u(t) −1 − σ0 + (2µ0 + λ0 ) Re − σ0 Re g (t) R ∂t ∂t2   4   3  2χ + ζ ∂ u(τ ) −3 ∂ u(τ ) −2 4 3 R Re g (τ ) + 3R Re g (τ ) R4 ∂τ 4 ∂τ 3  2  ∂ u(τ ) −1 R2 Re g (τ ) ∂τ 2   2 σ0 ∂ u(t) −1 ∂u(t) g (t) − Im − (2µ0 + λ0 ) Im ∂t2 R ∂t   3   2  2χ + ζ ∂ u(τ ) −2 ∂ u(τ ) −1 3 2 g (τ ) + R Im g (τ ) R Im R4 ∂τ 3 ∂τ 2

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−

mat σrr (t)

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inh σrr (t)

(31)

(32)

We expand the unknown tractions σ inh (τ ) at the boundary of the disc into complex Fourier series of the form σ inh (τ ) =

∞ X

inh m B−m g (τ ) +

m=1

∞ X

inh −m Bm g (τ )

(33)

m=0

inh , B inh are unknown complex coefficients. where g (τ ) is given by Eq. (26) and B−m m

(τ ) =

∞ X

mat m B−m g (τ ) m=1

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σ

mat

M

Similarly, the unknown tractions σ mat (τ ) at the boundary of the disc are represented as +

∞ X

mat −m Bm g (τ )

(34)

m=0

The displacements at both boundaries are the same and represented by the following series:

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uinh (τ ) = u(τ ) =

∞ X

A−m g m (τ ) +

m=1

∞ X

Am g −m (τ )

(35)

m=0

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5.2. Analytical solutions for the boundary displacements and tractions Following the arguments similar to those used in Mogilevskaya et al. (2008) (see Appendix D for detailed derivations), we arrive at the following system of equations that contain only three nonzero

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coefficients for the displacements: −4



 µ κ+1 ∞ µI + + η ReA1 = σ0 − Rσ1 κI − 1 2 4

h  i   − µ + κ µI + η (1) + γ A−1 + 3κ η (2) − γ A3 =

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κ+1 ∞ Rσ2 4

(36)

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η

(2)

 µI (1) − γ A−1 − µ + + 3η + 3γ A3 = 0 κI 



The solution of system (36) is

1 κ+1 1 σ0 + Rσ ∞ 4∆1 4 4∆1 1
κ + 1 µκI + µI + 3κI η (1) + γ Rσ2∞ = − 4 ∆2
κ + 1 κI η (2) − γ = − Rσ ∞ 2 4 ∆2

A−1 A3

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ReA1 = −

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where

η = (2µ0 + λ0 ) / (4R) , η (1) = η + 0.25σ0 /R, η (2) = η − 0.25σ0 /R, γ = (2χ + ζ) /R3 and

µI µ + +η κI − 1 2 = (µ + κµI ) (µκI + µI ) + η (1) [3κI (µ + κµI ) + κ (µκI + µI )] σ0 + 3κκI η + γ [κ (µκI + µI ) + 3κI (µ + κµI ) + 12κκI η] R

∆2

(38)

(39)

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∆1 =

(37)

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It can be seen from expressions (102) and (103) of Appendix D that the tractions can also be expressed through three nonzero coefficients. The coefficients for the tractions on the matrix side are

(40)

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mat B−2

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µ κ + 1 µI + η (κI − 1) ∞ σ0 + σ 2R∆1 4 ∆1 (κI − 1) 1

κ + 1 µI + η (1) + γ (µκI + µI ) + 3µI κI η (1) + γ + 3κI η (σ0 /R + 4γ) ∞ = − σ2 2 ∆2
κ + 1 3µκI η (2) − γ ∞ = σ2 2 ∆2

B0mat =

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B2mat

The coefficients for the traction on the inhomogeneity side are:

B0inh = inh B−2

B2inh

  µI σ0 κ + 1 ∞ − + σ1 ∆1 (κI − 1) R 4


κ + 1 µκI + µI + 3κI η (1) + γ ∞ = − µI σ2 2 ∆2
κ + 1 µI η (2) − γ ∞ = −3 σ2 2 ∆2 11

(41)

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The displacements and tractions at all the boundaries can be evaluated from Eqs. (33) - (35) and (37) - (41). 5.3. Calculations of the displacements, stresses and strains in the system The displacements and stresses everywhere in the system can be expressed in terms of two Kolosov-

σxx + σyy = 4Reϕ0 (z)    σ − σ + 2iσ = 2 [¯ z ϕ00 (z) + ψ 0 (z)] yy xx xy

(42)

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Muskhelishvili potentials ϕ(z) and ψ(z) by using the following formulae (Muskhelishvili, 1959):   2µu(z) = κϕ(z) − zϕ0 (z) − ψ(z)  

(43)

and the strains can be expressed via the stresses as ( εxx + εyy = 1−2ν 2µ (σxx + σyy ) εyy − εxx + 2iεxy =

1 2µ

(σyy − σxx + 2iσxy )

The expressions for the hoop stresses at the point τ on the circular boundary can be expressed via the potential ϕ0 (τ ) and the boundary tractions as follows:

(44)

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  σθθ (τ ) = Re 4ϕ0 (τ ) − σ (τ )

The general expressions for the potentials for the circular disc and the matrix in terms of unknown boundary displacements and tractions are given by Eqs. (48)-(50) in Mogilevskaya et al. (2008).

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By substituting series expansions (33)- (35) with the coefficients of Eqs. (37)-(41) into the latter expressions and evaluating all integrals analytically (see Mogilevskaya and Crouch, 2001; Mogilevskaya

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et al., 2008), we arrive at the following expressions for the potentials in terms of the known coefficients for the displacements:

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a) potentials for the circular disc

2µI 2µI ReA1 g −1 (z) + A3 g −3 (z) κI − 1 κI 6µI ψ(z) = − A3 g −1 (z) − 2µI A−1 g −1 (z) κI ϕ(z) =

(45)

b) potentials for the matrix ϕ(z) =

   i 2 h  − µI − µ + η (1) + γ A−1 + 3 η (2) − γ A3 g(z) + ϕ∞ (z) κ+1

12

(46)

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    2 κ−1 κ−1 2 µI − µ + (κ − 1) η ReA1 + σ0 g(z) κ+1 κI − 1 2 h   i + µ − µI − η (1) + γ − κ(η (2) − γ A−1 g 3 (z)       κ + µI − µ + 3κ η (1) + γ + 3 η (2) − γ A3 g 3 (z) κI ∞ + ψ (z)

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ψ(z) =

where κ = 3 − 4ν, κI = 3 − 4νI and

σ1∞ σ∞ z, ψ ∞ (z) = 2 z (47) 4 2 Eqs. (45), (46), being used in Eqs. (42), (43), provide the expressions for the elastic fields anywhere

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ϕ∞ (z) =

in the domain. While the stresses everywhere are defined in unique way, the obtained displacements might be augmented by additional terms corresponding to the rigid body translations and rotations. These terms could be found by assuming that (i) the displacements are zero at the center of the circle and (ii) the vertical component of displacement is zero at another point located inside the circle and at the axis Ox. As the result, the missing coefficients A0 and Im A1 involved in approximation (35) can be

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defined. The additional term for the displacements from the matrix side are found from the conditions of displacement continuity along the boundary when it is approached from the matrix side.

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6. The Maxwell type approximation formulae for the effective properties The exact solution of the problem of a circular inhomogeneity derived in the preceding sections will now be used to obtain the approximation formulae for the effective elastic properties of macroscopically

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isotropic composite material containing multiple inhomogeneities with the Steigmann-Ogden interfaces. We will derive the approximation formulae of Maxwell type, see McCartney (2010) and Mogilevskaya

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et al. (2012). In these papers, it was shown that, for two-phase composites with perfectly bonded inhomogeneities, the formulae are identical to those provided by the major effective medium theories and variational bounds.

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As we assume plane strain conditions and overall isotropy in transverse plane, the effective elastic

∗ and shear properties can be described by the two elastic constants: two-dimensional bulk modulus Kef

modulus µef . To obtain these constants, we apply the following two-stage procedure that is similar to the procedure for the coated inhomogeneity suggested in Mogilevskaya et al. (2017). On the first stage we replace the composite system containing the inhomogeneity with the SteigmannOgden interface by that containing the perfectly bonded inhomogeneity of the same radius. This stage requires the solutions of two complimentary problems: i) original problem involving a single inhomogeneity with the Steigmann-Ogden interface, and ii) a problem involving an equivalent perfectly 13

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∗ , µ , which are found bonded inhomogeneity of the same radius and the unknown elastic properties Keq eq

from the comparison of the far-field responses of the two composite systems. On the second stage we use the obtained properties of the equivalent inhomogeneity in the following

∗ Kef

K∗

=

˚1 1−A ˚1 K ∗ /µ 1+A

˚−1 µef 1−A = ˚−1 µ 1 + κA ˚1 , A ˚−1 should be taken to be in which A

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Maxwell type approximation formulae, see McCartney (2010) and Mogilevskaya et al. (2012):

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 ∗  Keq 1 ˚ A1 = c 1 − ∗ ∗ /µ K 1 + Keq   1 ˚−1 = c 1 − µeq A µ 1 + κµeq /µ

(48)

(49)

involved in Eqs. (48), (49) are

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with c being the volume fraction of the equivalent inhomogeneity. The two-dimensional bulk moduli

K ∗ = 2µ/ (κ − 1)

(50)

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∗ Keq = 2µeq / (κeq − 1)

The far-field response of the first problem is described by the potentials of Eq. (46). However, to

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evaluate the effective properties correctly, the residual effects due to the presence of the surface tension need to be eliminated from the solution due to the applied load, see Mogilevskaya et al. (2010). To

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do so, the first problem must be solved twice, once with both the external load and surface tension included and the second time with just surface tension. The second solution is then subtracted from the first one. As the result, the potential ϕ (z) of Eq. (46) will remain the same, while ψ (z) will change

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as follows:

    2 κ+1 κ−1 ∞ ψ(z) = µI − µ + (κ − 1) η Rσ1 g(z) κ+1 841 κI − 1 h    i + µ − µI − η (1) + γ − κ η (2) − γ A−1 g 3 (z)       κ (1) (2) 3 + µI − µ + 3κ η + γ + 3 η − γ A3 g (z) κI +ψ ∞ (z) 14

(51)

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in which the coefficients A−1 , A3 are given by the last two expressions of Eq. (37). The corresponding potentials for the complementary problem of an equivalent inhomogeneity of a radius R can be written as follows, see e.g. Muskhelishvili (1959) and Mogilevskaya and Crouch (2001): 1 − µeq /µ σ ∞ Rg(z) + ϕ∞ (z) 2 (1 + κµeq /µ) 2 ∗ /K ∗ 1 − Keq 1 1 − µeq /µ ∞ 3
σ1∞ Rg(z) − ψeq (z) = − σ Rg (z) + ψ ∞ (z) ∗ /µ 2 (1 + κµeq /µ) 2 2 1 + Keq

(52)

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ϕeq (z) = −

The potentials for the two problems should be compared separately for the hydrostatic and deviatoric loads.

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a) Hydrostatic load

When the load is hydrostatic, the combination involved in the second expression of Eq. (23) vanishes, i.e σ2∞ = 0. Thus, it follows from Eqs. (37) that ReA1 is the only non-zero displacement coefficient. Using the relations of Eqs. (50) and comparing the potential ϕ(z) of Eqs. (46) and the potential ψ(z)

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of Eq. (51) with the potentials ϕeq (z) , ψeq (z) of Eqs. (52), one gets

(53)

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ϕ(z) = ϕeq (z) = ϕ∞ (z)  ∗  ∗ /K ∗ 1 − Keq KI K∗ κ−1

− + η = − ∗ /µ 2 2 4 KI∗ /2 + µ/2 + η 2 1 + Keq

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After some algebraic manipulation the bulk modulus of an equivalent inhomogeneity is defined as ∗ Keq = KI∗ + 2η

(54)

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Note that Eq. (54) is identical to Eq. (31) of Duan et al. (2007), if one accounts for the misprint (missing denominator in the parameter χr that, in our notations, should be equal to χr = 8η/µ) in the latter publication.

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The effective two dimensional bulk modulus of composite material containing multiple inhomo-

geneities with the Steigmann-Ogden interfaces can be obtained by using Eq. (54) in the corresponding expressions of Eqs. (48), (49). The resulting expression for the modulus is ∗ Kef = K∗ +

(1 −

c) / (K ∗

c
+ µ) + 1/ KI∗ + 2η − K ∗

(55)

It follows from Eq. (55) that neither surface tension nor the bending stiffness parameters ζ and χ ∗ and, therefore, have no influence on that modulus. The influence are involved in the expression for Kef

15

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of surface elasticity is expressed by the parameter η, which could be negative for some set of reported data, e.g. Miller and Shenoy (2000) for the data on anodic alumina. However, the overall expression in the right-hand side of Eq. (55) still remains positive for that type of material. b) Deviatoric load

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The assumption of the deviatoric load leads to the vanishing of the combination for σ1∞ , i.e. σ1∞ = 0. Thus, it follows from Eqs. (37) that ReA1 = 0. The two potentials of Eq. (46) become

ϕ(z) =

   i 2 h  − µI − µ + η (1) + γ A−1 + 3 η (2) − γ A3 g(z) + ϕ∞ (z) κ+1

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  i 2 nh µ − µI − η (1) + γ − κ(η (2) − γ A−1 κ +1      κ (1) (2) − µ + 3κ η + γ + 3 η − γ A3 g 3 (z) + µI κI + ψ ∞ (z)

ψ(z) =

(56)

(57)

These potentials need to be compared with the potentials ϕeq (z) , ψeq (z) of Eqs. (52).

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In Appendix E we prove that this comparison produces no admissible solutions, except for one special case. However, it can be seen from Eqs. (52), (56), (57) that the potentials ψ (z) contain only

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high order terms (z −3 ), while the potentials ϕ (z) contain the low order terms (z −1 ) only. Defining equivalency as an assumption that the z −1 terms for the potentials for the equivalent inhomogeneity and those for the coated inhomogeneity are the same, we arrive at the following equation:

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   i 1 − µeq /µ 2 h  − µI − µ + η (1) + γ A−1 + 3 η (2) − γ A3 = − σ∞R κ+1 2 (1 + κµeq /µ) 2

(58)

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From Eqs. (37), (58) it follows that the shear modulus of the equivalent inhomogeneity can be

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written as

µeq



µI + η (1) + γ (µκI + µI ) + 3κI η (1) + γ µI + η 4γ +
= µκI + µI + 3κI η (1) + γ

σ0 R




(59)

Note that, in the special case of σ0 = γ = 0, Eq. (59) is identical to Eq. (32) of Duan et al. (2007), if one again accounts for the misprint in the parameter χr of that publication. In another special case of η (2) = γ, see Eq. (110) of Appendix E, the shear modulus of the equivalent

inhomogeneity is µeq = µI + 2γ + 16

σ0 2R

(60)

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The effective shear modulus of composite material containing multiple inhomogeneities with the Steigmann-Ogden interfaces can be obtained by using Eq. (59) in the corresponding expressions of Eqs. (48), (49). The resulting expression for the modulus is c =µ 1+ (1 − c) κ/ (κ + 1) − 1/ (1 − µeq /µ)



(61)

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µef



It is clear from the analysis of Eqs. (59), (61) that, in general case, all surface parameters of the Steigmann-Ogden model should affect the overall shear modulus of the material described by that model. 7. “Neutrality” conditions

It is long been known in the literature (e.g. Ru, 1998; Milton and Serkov, 2001; Benveniste and

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Miloh, 2007; Bertoldi et al., 2007; Wang and Schiavone, 2012 a,b, 2015, and from the references therein) that certain inhomogeneities do not disturb the fields in the original homogeneous body and thus become “invisible.” Typically, neutrality is achieved by the modification of the interfaces between the inhomogeneities and the matrix, e.g. by applying the coatings with the specially designed properties. It is of interest, therefore, to study the “neutrality” conditions for the inhomogeneity problem described by the Steigmann-Ogden model. This study can be done by using the idea of equivalent inhomogeneity

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used in the previous section. Here, too we consider the cases of the hydrostatic and deviatoric load

a) Hydrostatic load

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separately.

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∗ = K ∗ , i.e. when From Eqs. (52)-(54) it follows that the original fields are not disturbed when Keq

2η = K ∗ − KI∗

(62)

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So, while it is possible to achieve neutrality by maintaining certain mismatch in the bulk moduli of the matrix and inhomogeneity, the mismatch could be quite small for the reported values of the surface parameters.

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b) Deviatoric load

It follows from Eqs. (52) that neutrality for the equivalent perfectly bonded inhomogeneity can only

be achieved when µeq = µ. However, as was shown in Appendix E, it is in general impossible to find an equivalent inhomogeneity such that it induces the same far fields as those due to the inhomogeneity with the Steigmann-Ogden type interface. Therefore, neutrality for the deviatoric load can not be, in general, achieved. However, it follows from Eqs. (58), (59) that there could be the inhomogeneities that induce “weak” disturbance in the original fields. Namely, when 17

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η (1) + γ (µκI + µI ) + 3κI η 4γ +
µ = µI + µκI + µI + 3κI η (1) + γ

σ0 R




(63)

the only disturbance in the displacements is that of the order of z −3 , while that for the stresses has the order of z −4 .

µI = µ − 2γ −

σ0 2R

8. Examples 8.1. Verification of the approach

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In special case of η (2) = γ, see Eq. (110) of Appendix E, the “strong” neutrality is achieved when (64)

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We are not aware of any benchmark solution for the Steigmann-Ogden model with a circular inhomogeneity. However, this model theoretically reduces to the Gurtin and Murdoch model when the bending effects are neglected. The benchmark solution for a circular inhomogeneity described by the latter model is presented in Mogilevskaya et al. (2008), where it was shown that the solution is identical to those by Duan et al. (2005a) and Chen et al. (2007) who obtained it for the less general case of σ0 = 0. It was also shown that benchmark solution of Mogilevskaya et al. (2008) is identical to the

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axisymmetric solution given in the papers by Gurtin and Murdoch (1975, 1978) for the case of zero external loading. Therefore, we only need to verify our solution for the case when γ 6= 0.

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To do so, we show that, when σ0 = 0, the boundary conditions of the Steigmann-Ogden model can

be simulated by those for the problem of a coated inhomogeneity with an appropriate choice of the properties of all three components of the composite system. The benchmark solution for the problem of

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a circular inhomogeneity with an uniform interphase layer is given in Mogilevskaya and Crouch (2004); it is identical to previously obtained solution by Ru (1999). 8.2. Comparison between the Steigmann and Ogden model and the model of an interphase layer

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Benveniste and Miloh (2001) showed that the problem of an inhomogeneity with a thin interphase layer can be reduced to that of an imperfectly bonded inhomogeneity with the interface conditions

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described by one of seven distinct regimes that depend on the elastic properties of all three components of composite system. In Mogilevskaya et al. (2008), it was shown that one such regime — a membranetype interphase layer, N = 1 — could simulate the Gurtin and Murdoch model with σ0 = 0 and

a suitable choice for the elastic properties and the thickness of the layer. Here, we show that the Steigmann-Ogden model with σ0 = 0 can be simulated by another Benveniste and Miloh (2001) regime — inextensible shell-type interphase layer, N = 3. This regime is characterized, see Eqs. (2.15) (2.16)

of Benveniste and Miloh (2001), by the continuity of the displacements, and the following conditions (rewritten in our notations): 18

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a) inextensibility condition uθ,θ = −ur

(65)

b) jump conditions for the tractions

where

D=

µint (λint + µint ) 3 h 3 (λint + 2µint )

and h is the constant thickness of the interphase.

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D in mat in mat σrθ − σrθ + σrr,θ − σrr,θ = − 4 [ur,θθθθθ + 2ur,θθθ + ur,θ ] R

(66)

(67)

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Using condition of Eq. (65) in Eqs. (15), (16) with σ0 = 0, we arrive at the equation

γ mat in mat in = − [ur,θθθθθ + 2ur,θθθ + ur,θ ] − σrr,θ + σrr,θ − σrθ σrθ R

(68)

Thus, the boundary conditions for the problem of the inhomogeneity with inextensible shell-type interphase can simulate the boundary conditions of the Steigmann-Ogden model with σ0 = 0, if µint (λint + µint ) h3 3 (λint + 2µint ) R3

M γ=

(69)

ED

and the parameters µ0 , λ0 are such that the inextensibility condition of Eq. (65) is satisfied. Note that the analysis of Eqs. (38), (67), (69) suggests that the physical meaning of the combination 2χ + ζ is that of bending (flexural) rigidity.

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It follows from Eq. (65) and from the second equation of Eq. (99) of Appendix C that this condition

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leads to the following requirements on the coefficients of Eq. (37): ReA1 = 0, A−1 = 3A3

(70)

that could be met if η = (2µ0 + λ0 ) / (4R) is very large.

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Keeping this in mind, we compare the solutions of the two problems (Fig. 3) for the case when the

elastic properties of the matrix and the inhomogeneity are as follows: µI /µ = 0.5, ν = νI = 0.3

(71)

In the case of Steigmann-Ogden’s model, we assume that the inhomogeneity has a unit radius R = 1, while in the case of Benveniste and Miloh’s inextensible shell type interphase layer model, we assume

19

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that the thickness h of the layer is such that  = h/R = 10−2 and the dimensionless Lam´e parameters of the interface, introduced by Benveniste and Miloh, are of the following orders λint ∼

O(1) O(1) , µint ∼ 3 3 

(72)

λint =

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where  = h/R and λint µint , µint = 2µav + λav 2µav + λav

(73)

λav = (λI + λ) /2, µav = (µI + µ) /2

To satisfy (72) we adopt the following elastic properties of the interface

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µint /µ = 2625000, νint = 0.3

(74)

For a thin interphase layer with elastic parameters (74) to simulate the interface conditions employed in Steigmann-Ogden’s model, the value of γ according to Eq. (67) should be γ/µ = 0.625

(75)

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We choose the parameters µ0 , λ0 to be (see Eq. (64) in Mogilevskaya et al., 2008): 2µint λint h = 22500µ, µ0 = µint h = 26250µ λint + 2µint which make them to be sufficiently large.

(76)

ED

λ0 =

Using parameters (71) and (74) and assuming unit shear modulus for the matrix and unit uniaxial

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tension along the x axis, we solved the problem of a single inhomogeneity with the uniform interface layer bounded by the boundaries of the circles L1 of the radius R − h/2 and L2 of the radius R + h/2

using equations of Section 5 in Mogilevskaya and Crouch (2004). The companion problem of a single

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inhomogeneity of the radius R with the material surface defined by parameters of Eqs. (71), (75), (76) under unit uniaxial tension along the x axis was solved by using equations of Section 5.

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First, we compared the results for the jumps in the components of tractions along the boundary of the inhomogeneity. For the Steigmann-Ogden model these are defined by formula (104) of Appendix D and for the uniform interphase layer model they are defined as ∆σrr = σrr |L1 − σrr |L2 , ∆σrθ = σrθ |L1 − σrθ |L2

(77)

We can conclude from the results shown in Fig. 4 that the agreement between these two solutions is excellent. The agreement of the results for the stresses inside the inhomogeneity and the matrix (Fig. 5) and for the hoop stresses at the boundary of the inhomogeneity (Fig. 6) is also very good. 20

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I

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I

CR IP T

σ0, μ0, λ0, ζ, χ

I

ED

M

I

Figure 3: (a) An inhomogeneity with the Steigmann-Ogden interface. (b) An inhomogeneity with the

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uniform interphase layer.

It is interesting to compare the stresses due to the Gurtin-Murdoch and Steigmann-Ogden models. In both models, the stresses σxx inside the inhomogeneity are constant along the axis Ox (Figs. 5a in

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Mogilevskaya et al., 2008 and in the present paper). However, the values of the stresses are different for both models. While the stresses σxx along the axis Ox (Figs. 5a in Mogilevskaya et al., 2008) are continuous for the Gurtin-Murdoch model, the corresponding stresses for the Steigmann-Ogden model

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(Fig. 5a) undergo jump along the same axis. Another interesting observation is that the jumps in the shear components of tractions along the boundary are about the same, while the the jumps in the normal components are drastically different for the two models (Figs. 4 in Mogilevskaya et al., 2008 and in the present paper). While the comparisons of this subsection were performed primarily for the verification purposes, the outcome of these comparisons has additional significance. We have shown that the Steigmann-Ogden model cannot be automatically reduced to the Gurtin-Murdoch model by just neglecting the bending

21

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0.8 0.6

xxx

0.4

Steigmann-Ogden model Uniform interphase layer

∆σl

0.2 0

-0.4 -0.6

∆σn

-0.8 -1 -1.2

0

30

60

90

120

θ

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-0.2

150

180

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Figure 4: Jumps in the normal and shear tractions along the boundary. (a) 1.5

(b) 0.6

Steigmann-Ogden approach x x x Uniform interphase layer

Steigmann-Ogden approach x x x Uniform interphase layer

0.4

σ yy

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σ xx

1

0

0

0.5

ED

0.5

1

1.5

0.2

0

-0.2

2

0

0.5

1

1.5

2

x

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x

Figure 5: Distribution of the stresses σxx (a) and σyy (b) along the axis Ox.

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interphase effects, as the surface elasticity parameters µ0 , λ0 for the two models could be completely different. It is due to the fact that they are obtained, see Eq. (76), from the elastic moduli of the coatings

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that reproduce jump conditions for the models and the moduli for the Steigmann-Ogden model differ in a few orders of magnitude from those for the Gurtin-Murdoch model. Thus, the Steigmann-Ogden and Gurtin-Murdoch models describe two very different interphase regimes. 8.3. Illustrative example The analysis of the relevant literature reveals the lack of reliable data on surface parameters involved in the Steigmann-Ogden model. It can be seen from Eqs. (36), (45), (46) that the distributions of the normalized stresses σxx /µ, σyy /µ, σxy /µ, and σrr /µ are governed by the following seven dimensionless parameters: 22

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(a) 0.6

(b) 0.08

0.5

0.06

Steigmann-Ogden model Uniform interphase layer

xxx

0.04

0.4

Steigmann-Ogden model Uniform interphase layer

0.02

0.3

0 -0.02

0.2

-0.04 0.1

-0.06 -0.08

0 0

30

60

90

120

150

0

180

30

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σθθ

σθθ

xxx

60

90

120

150

180

θ

θ

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Figure 6: Hoop stresses along the boundary (a) on the matrix side, and (b) on the inhomogeneity side.

µI /µ, ν, η/µ, σ0 /µR, γ/µ, σ1∞ /µ, σ2∞ /µ

(78)

The first four parameters of Eq. (78) are reported in the literature that is related to the GurtinMurdoch model, see e.g. Gurtin and Murdoch (1978); Miller and Shenoy (2000); Sharma and Ganti

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(2002, 2004); Duan et al. (2005c, 2006); Chen et al. (2007). However, these parameters have been obtained numerically for that specific model and, therefore, they might not be valid for the SteigmannOgden model, as explained at the end of Section 8.2. So, here we present the example for the hypothetical

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dimensionless parameters (some within a range of available data) just to demonstrate the difference between the classical results and those for the Steigmann-Ogden model. We assumed the case of an

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infinite plane with the cavity µI /µ = 0 of radius R = 5 nm under the deviatoric load σ1∞ /µ = 0, σ2∞ /µ = 1 and took Poisson’s ratio of the matrix to be ν = 0.3. Other parameters adopted in our computations

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are

η/µ = 0.030156, σ0 /µR = 0.048892, γ/µ = 0.00028382

(79)

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Fig. 7 shows the normalized stresses σxx /µ (a) and σyy /µ (b) along the axis Ox, while Fig. 8 shows the corresponding normalized stresses along axis Oy. The difference with the classical case is

localized near the circular boundary and is more significant on Fig. 7b and more so on Fig. 8b. Interestingly, unlike in the case of Gurtin-Murdoch model (see Mogilevskaya et al., 2008), the results for the Steigmann-Ogden model with and without the surface tension are very close to each other (except for the results of Fig. 8b). The distribution of the normalized hoop stresses on the boundary of the cavity is plotted on Fig. 9. Again, unlike in the case of the Gurtin-Murdoch model, the results for the Steigmann-Ogden model 23

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(a) 0.2 0.1

Steigmann-Ogden model with surface tension σ 0

Steigmann-Ogden model with σ 0 =0

0

Steigmann-Ogden model with σ 0 =0

1.5

Steigmann-Ogden model with surface tension σ 0

Classical case without surface energy

-0.2

1

0.5

-0.3 -0.4

0

-0.5 0

1

2

3

4

5

0

x/R

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-0.1

σyy/µ

σxx/µ

2

(b) Classical case without surface energy

1

2

3

4

5

x/R

(a)

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Figure 7: Distribution of the stresses σxx /µ (a) and σyy /µ (b) along the axis Ox. (b) 0.5

0

0.4 0.3

σyy/µ

σxx/µ

-0.5

-1 Classical case without surface energy

Steigmann-Ogden model with σ 0 =0

0.1

Steigmann-Ogden model with surface tension σ 0

0

M

-1.5

0.2

Steigmann-Ogden model with surface tension σ 0

Steigmann-Ogden model with σ 0 =0

-0.1

-2

Classical case without surface tension

-0.2 1

2

3

4

ED

0

5

0

y/R

1

2

3

4

5

y/R

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Figure 8: Distribution of the stresses σxx (a) and σyy (b) along the axis Oy. with and without surface tension are close to each other.

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Finally, the normalized effective shear modulus of the composite material that exhibit overall isotropy (hexagonal or random arrangement of fibers) is plotted on Figure 10 as a function of the volume fraction of the fibers. Here too, the surface tension plays less dominant role as compared with the Gurtin-

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Murdoch model.

9. Conclusions In this paper we re-derived the governing equations for the Steigmann-Ogden model and presented

the closed-form analytical solution of the plane strain problem of an infinite isotropic elastic domain containing an isotropic elastic circular inhomogeneity whose interface is described by that model. Dimensionless parameters that govern the problem are fully identified. To the best of our knowledge, this is the first benchmark solution both for the elastic fields everywhere in the domain as well as for the 24

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2 Steigmann-Ogden model with surface tension σ 0

1.5

Steigmann-Ogden model with σ 0 =0

1

Classical case without surface energy

0 -0.5 -1 -1.5 -2 0

20

40

60

80

100

θ

120

140

CR IP T

σθθ /µ

0.5

160

180

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Figure 9: Normalized hoop stresses along the boundary of the cavity

effective properties of macroscopically isotropic materials containing circular inhomogeneities of that type. We also presented the analysis of so-called “neutrality” conditions under which the presence of the inhomogeneity does not disturb the elastic fields in the matrix. An important new finding is the demonstration of the fact that, while the Steigmann-Ogden model theoretically reduces to the Gurtin-

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Murdoch model when the bending interphase effects are neglected, the two models (for the case of zero surface tension) describe two very different interphase regimes of seven regimes proposed by Benveniste

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and Miloh. While it is out of scope of the present paper, we believe that this finding could be useful in understanding the results of atomistic calculations, see e.g. Chhapadia et al. (2011, 2012), that predict dramatically different elastic modulus of nanostructures under bending and under tension. We also

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believe that the presented theoretical developments could be important for the studies of various surface elasticity models. As we demonstrated here, the analytical technique used for the circular surface

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allows for accurate handling of jump conditions described by high order differential equations; similar technique (based on vectorial spherical harmonics) could be used to analyze the corresponding problems involving spherical surfaces. We intent to study these problems in our future publications. However,

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the mathematical derivations related to three-dimensional study could be quite involved, as they would require to use differential calculus of vectorial spherical harmonics. The extension to the problems involving interacting circular/eliptical inhomogeneities should be more straightforward, as it was for the Gurtin-Murdoch model, and could be also a subject of future studies. However, it is clear that any meaningful application of the Steigmann-Ogden model will only be possible when reliable data on surface parameters that are involved in the model will be obtained by either physical or computational experiments. We believe that the presented solutions could be useful in designing such experiments in somewhat similar manner as proposed in Cordero et al. (2016) for the case of nanowires. As have 25

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0.7

Steigmann-Ogden model with surface tension σ 0

0.6

Steigmann-Ogden model with σ 0 =0

0.5

Classical case without surface energy

0.4 0.3 0.2 0.1 0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

c

CR IP T

µ eff /µ

0.8

0.8

0.9

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Figure 10: Normalized effective shear modulus for the composite material with overall isotropy been demonstrated here, the expressions for the fields, e.g. strains, far away from the inhomogeneity and for the effective properties of the material contain all surface parameters involved in the model. As the strains and effective properties are measurable quantities (see Pyatigorets, et al. (2010), for the experiments with the conventional elastic porous materials), the use of the combined measurements

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of these fields and properties for e.g. porous nanomaterials could provide some estimates of surface

Acknowledgements

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parameters.

The first author (A.Z.) gratefully acknowledges the support from Simons Collaboration grant, award number 319217. The second author (S.M.) gratefully acknowledges the support provided by

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the Theodore W. Bennett Chair, University of Minnesota. Appendix A. Equilibrium equations and the boundary conditions of the Steigmann-Ogden

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(1997) model

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Following Eremeyev and Lebedev (2016), consider the virtual work principle in the form δJ = δE − δP = 0,

where

E=

ZZZ

V

WdV +

ZZ

As

(80)

UdS

(81)

is the sum of the volume and surface energies, with W and U being correspondingly the volume and surface energy densities, and

P =

ZZZ

V

f · udV + 26

ZZ

As

t0 · udS

(82)

ACCEPTED MANUSCRIPT

is the work of external forces, with f being a given volume force and t0 being a given surface traction. We will consider the energy densities W and U of the following forms: 1 W = µe : e + λtr2 e, 2

(83)

CR IP T

1 1 U = σ0 (1 + trεsur ) + (λ0 + σ0 )(trεsur )2 + (µ0 − σ0 )εsur : εsur + σ0 |∇s u|2 2 2 1 + ζ(trκsur )2 + χκsur : κsur . 2 Taking the first variation of the functional J = E − P , obtain: ZZ ZZZ (Ts : δεsur + σ0 ∇s u : ∇s u + M : δκsur )dS− σ : δe dV + δJ = As

ZZZ

V

f · δudV −

ZZ

(85)

t0 · δudS,

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V

(84)

As

where the symbol ”:” denotes the double dot product of two tensors, and the tensor Ts is defined by Ts = σ0 IS + (λ0 + σ0 ) trεsur IS + 2 (µ0 − σ0 ) εsur while the tensor M is given by the formula (6).

M

Using the tensor identity:

∇ · (σ · δu) = (∇ · σ) · δu + σ : (∇u)T ,

ED

(86)

PT

and applying the divergence theorem to the first term in a standard way, obtain: ZZZ ZZ ZZZ σ : δe dV = n · σ · δudS − (∇ · σ) · δudV. V

As

CE

Following Gurtin and Murdoch [25], it is possible to obtain ZZ Z ZZ s sur (T : δε + σ0 ∇s u : ∇s u)dS = ν · T · δuds − (∇s · T) · δudS, As

(87)

V

∂As

(88)

As

AC

where ∂As is a curve bounding the surface As , and the tensor T is given by the formula (5). Then the third term in (85) can be written as ZZ Z sur M : δκ dS = − As

∂As

ν · M · δϑds +

ZZ

As

(∇s · M) · δϑdS.

The last integral here can be further rewritten as ZZ ZZ ZZ (∇s · M) · δϑdS = (∇s · M) · (n · (∇s δu)T )dS = (∇s · M)n : (∇s δu)T dS. As

As

As

27

(89)

(90)

ACCEPTED MANUSCRIPT

Applying the surface tensor identity and the surface divergence theorem again, obtain: ZZ Z ZZ (∇s · n)n · (∇s · M)n · δudS− ν · IS · (∇s · M)n · δuds + (∇s · M) · δϑdS =

(91)

As

∂As

ZZ

As

∇s · ((∇s · M)n) · δudS.

Finally, Z Z B ν · M · δϑds = τ · (ν · M)n · δu|A −

∂As

∂As

CR IP T

As

∂τ · (ν · M)n · δuds + ∂s

where A, B denote the end-points (if any) of the curve ∂As .

Z

∂As

∇s · ((ν · M)n) · δuds, (92)

Collecting all the terms and setting the first variation δJ to zero for all the virtuals δu, obtain the

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boundary equations (2), (3), (4).

Appendix B. Derivations of the equations for the circular material surface Observe that, in general, the formulas (2)-(6) describe a three-dimensional object. Here, we consider a cylindrical surface of the same circular cross-section independent of the variable z. The construction is assumed to be in a plane strain state with the stresses and displacements dependent only on the first two coordinates x, y. To make the formulas less bulky, we will drop all the components related to the

M

z coordinate similarly to the formulas (7)-(13) since ultimately they do not affect the stressed state of the object.

ED

To derive the boundary conditions (15), (16), substitute the formulas (7)-(13), (17), (18) into the boundary condition (2). Obtain:

1 1 (ur,θθθ − uθ,θθ )eθ ⊗ er + (2χ + ζ) 3 (ur,θθ − uθ,θ )er ⊗ er ; 3 R R  σ0 σ0 1 ∇s · (T + (∇s · M)n) = − + 2 (ur,θθ − uθ,θ ) − (λ0 + 2µ0 ) 2 (uθ,θ + ur )+ R R R  1 1 (2χ + ζ) 4 (ur,θθ − uθ,θ ) − (2χ + ζ) 4 (ur,θθθθ − uθ,θθθ ) er + R R  σ0 1 (ur,θ − uθ ) + (λ0 + 2µ0 ) 2 (uθ,θθ + ur,θ )− 2 R R  1 (2χ + ζ) 4 (ur,θθθ − uθ,θθ ) eθ ; R

(93) (94)

AC

CE

PT

(∇s · M)n = −(2χ + ζ)

(∇s · n)n · (∇s · M)n = (2χ + ζ)

1 (ur,θθ − uθ,θ )er . R4

(95)

Substituting the formulas (94), (95) into the conditions (2) produces then the boundary conditions (15), (16).

28

ACCEPTED MANUSCRIPT

Appendix C. Derivations of Eqs. (31), (32) and of the derivatives of complex displacements The derivatives over θ involved in Eqs. (15), (16) can be expressed via those over τ using expressions (19), (26), and the following relations: R dτ dτ = exp (−iθτ ) , = iexp (iθτ ) = ig −1 (τ ) , = −ig (τ ) τ ds ds 1 g 0 (τ ) = − g 2 (τ ) R For example, it follows from expression (30) that

CR IP T

g (τ ) =

(98)

∂u(τ ) ∂u(τ ) + Im [u(τ )g (τ )] = −R Im + uθ ∂τ ∂τ

M

= − R Im

(97)

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( " # )  u(τ ) −2 ∂u(τ ) u(τ ) 2 ∂u(τ ) −1 ∂τ R ∂τ g (τ ) − g (τ ) + g (τ ) − g (τ ) ur,θ (τ ) = 2 ∂τ R ∂s ∂τ R ∂s ( " # )  u(τ ) 2 ∂u(τ ) −1 u(τ ) −2 Ri ∂u(τ ) −1 g (τ ) − g (τ ) g (τ ) − g (τ ) − g (τ ) g (τ ) = 2 ∂τ R ∂τ R #) (  " ∂u(τ ) u(τ ) −1 Ri ∂u(τ ) u(τ ) = − g (τ ) − − g (τ ) 2 ∂τ R ∂τ R

(96)

ED

Using similar arguments for all remaining derivatives involved in Eqs. (15), (16), one can arrive to the following complex combinations:

PT

∂u(τ ) ∂τ ∂u(τ ) uθ,θ + ur = R Re ∂τ  ∂ 2 u(τ ) −1 2 ur,θθ − uθ,θ = −R Re g (τ ) ∂τ 2  2  ∂ u(τ ) −1 2 uθ,θθ + ur,θ = −R Im g (τ ) ∂τ 2  3   2  ∂ u(τ ) −2 ∂ u(τ ) −1 3 2 ur,θθθ − uθ,θθ = R Im g (τ ) + R Im g (τ ) ∂τ 3 ∂τ 2  4   3  ∂ u(τ ) −3 ∂ u(τ ) −2 4 3 ur,θθθθ − uθ,θθθ = R Re g (τ ) + 3R Re g (τ ) ∂τ 4 ∂τ 3  2  ∂ u(τ ) −1 2 +R Re g (τ ) ∂τ 2

AC

CE

ur,θ − uθ = −R Im

29

(99)

ACCEPTED MANUSCRIPT

The complex derivatives involved in the boundary conditions of Eqs. (31), (32) can be obtained by differentiating the series of Eq. (35) and using Eq. (97). The resulting expression for the kth derivative of complex displacement is

u

k

(τ ) = (−1)

∞ X m (m + 1) . . . (m + k − 1)

Rk

m=1

+

∞ X m (m − 1) . . . (m − k + 1)

Rk

m=k

A−m g m+k (τ )

(100)

CR IP T

(k)

Am g −(m−k) (τ )

Taking the complex conjugation of expression (100) and using Eq. (29), one obtains the following

u(k) (τ ) = (−1)k

∞ X m (m + 1) . . . (m + k − 1)

m=1

+

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expressions for the conjugates of kth derivative of complex displacement:

Rk

∞ X m (m − 1) . . . (m − k + 1)

Rk

(101)

Am g (m−k) (τ )

M

m=k

A−m g −(m+k) (τ )

Appendix D. Derivations of Eqs. (36)

It is shown in Mogilevskaya et al. (2008) that the coefficients for the tractions at the inhomogeneity

ED

boundary can be expressed in terms of those for the displacements as follows: (102)

CE

PT

inh B−1 = 0 κI − 1 inh 2 B0 = ReA1 2µI R 1 inh m−1 B−m = A−(m−1) , m ≥ 2 2µI R κI inh m+1 Bm = A(m+1) , m ≥ 1 2µI R

AC

inh indicate that the disc is in equilibrium. Also, the The expressions for the coefficients B0inh and B−1

coefficients A0 and imaginary parts of the coefficient A1 do not appear in the system as these terms of the boundary displacements define the rigid body translation and rotation and can be recovered later.

Complementary problem for the hole can be reduced, see Mogilevskaya et al. (2008), to the solution

of the following equation:

30

ACCEPTED MANUSCRIPT

(103)

CR IP T

∞ ∞ 1 X mat −m 1 κ X mat m B−m g (t) − Bm g (t) − B0mat − 2µ 2µ µ m=2 m=1 " # ∞ ∞ X X 1 − 2ReA1 + mA−m g m+1 (t) + mAm g −(m−1) (t) R m=1 m=2  κ+1 ∞ =− σ1 − g 2 (t)σ2∞ 4µ

The complex coefficients for the tractions involved in Eq. (103) can be expressed via the coefficients for the boundary displacements by using expressions of Eqs. (102) and the following interface conditions written in the complex form (see Eqs. (31), (32) and Appendix C):

h 2Xn σ0 4η (m − 1) (m + 1) η (2) Am+1 (τ ) = − − ReA1 + (104) R R R m=1 i h − (m − 1) η (1) A−(m−1) g m (τ ) + (m + 1) − (m + 1) η (1) Am+1 i o  γ  (m − 1) m2 (m − 1) A−(m−1) g m (τ ) + (m − 1) η (2) A−(m−1) g −m (τ ) − 2R   + (m + 1) A−(m−1) g −m (τ ) + m2 (m + 1) (m + 1) Am+1 g −m (τ )  + (m − 1) Am+1 g m (τ ) ∞

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(τ ) − σ

mat

M

σ

inh

ED

where η, η (1) , η (2) , γ are given by Eq. (38).

The resulting expressions for the coefficients for the tractions in terms of those for the displacements

PT

are

CE

B0mat =

mat B−m =

AC

+

mat Bm =

+

    µI 1 4 + η ReA1 + σ0 R κI − 1 i o m − 1 nh 2 µI + (m − 1) η (1) A−(m−1) − (m + 1) η (2) A(m+1) R   γ 2 m (m − 1) (m − 1) A−(m−1) + (m + 1) A(m+1) , m ≥ 1 2R     m+1 µI (2) (1) 2 − (m − 1) η A−(m−1) + + (m + 1) η A(m+1) R κI   γ 2 m (m + 1) (m + 1) A(m+1) + (m − 1) A−(m−1) , m ≥ 1 2R

(105)

mat = 0, and the coefficients A and imaginary Note that, similar to the case of the disc, B0mat is real, B−1 0

parts of the coefficient A1 do not appear in system (105).

After substituting expressions (105) into Eq. (103) and using the orthogonality properties of Fourier series, we arrive at Eqs. (36). 31

ACCEPTED MANUSCRIPT

Appendix E. Existence of an equivalent inhomogeneity for the deviatoric load Consider the case of deviatoric load. From the analysis of Eqs. (52), (56), (57), it follows that if the perfectly bonded inhomogeneity induces the same far-fields as the inhomogeneity of the same radius but with the Steigmann-Ogden type interface, then

CR IP T

  h i κ (1) (1) (2) (2) µ − µI − η − κη + (κ − 1) γ A−1 + 3η + µI − µ + 3κη + 3 (κ − 1) γ A3 κI h     i = − µI − µ + η (1) + γ A−1 + 3 η (2) − γ A3 Eq. (106) can be simplified as κ η

(2)



− γ A−1

   κ (1) − µ + 3κ η + γ A3 = µI κI

From Eqs. (37) and (107), it follows that

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 h  i κ η (2) − γ µκI + µI + 3κI η (1) + γ

(106)

(107)

(108)

h  i = η (2) − γ µI κ − µκI + 3κ κI η (1) + γ

M



η (2) = γ κI µ (κ + 1) = 0

(109)

PT

ED

The only solutions of this equation are

The second equation of Eqs. (109) produces no admissible solutions. The first equation produces the

2χ + ζ 2µ0 + λ0 − σ0 = 4 R2

(110)

AC

CE

following requirement of the surface parameters:

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