Effect Of Pair Coalescence Of Circular Pores On The Overall Elastic Properties

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(2) ARTICLE IN PRESS. JID: SAS. [m5G;May 15, 2019;19:45]. International Journal of Solids and Structures xxx (xxxx) xxx. Contents lists available at ScienceDirect. International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr. Effect of pair coalescence of circular pores on the overall elastic properties L. Lanzoni a, E. Radi b, I. Sevostianov c,∗. Q1. a. Dipartimento di Ingegneria “Enzo Ferrari”, Università di Modena e Reggio Emilia, Via Vivarelli, 10 41125 Modena, Italy Dipartimento di Scienze e Metodi dell’Ingegneria, Università di Modena e Reggio Emilia, Via Amendola, 2 42122 Reggio Emilia, Italy c Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 88001, USA b. a r t i c l e. i n f o. Article history: Received 28 February 2019 Revised 14 May 2019 Accepted 14 May 2019 Available online xxx Keywords: Cylindrical pores Irregular cross-section Bipolar coordinates Compliance contribution tensor Effective elastic. a b s t r a c t The paper focuses on the effect of the pair coalescence of circular pores on the overall elastic properties. An analytic solution for the stress and displacement fields in an infinite elastic medium, containing cylindrical pore with the cross-section formed by two circles, and subjected to remotely applied uniform stresses is obtained. The displacement field on the surface of the pore is then determined as a function of the geometrical parameters. This result is used to calculate compliance contribution tensor for the pore and to evaluate effective elastic properties of a material containing multiple pores of such a shape. © 2019 Elsevier Ltd. All rights reserved.. 1. 1. Introduction. 2. In the present paper we focus on the effect of the pair coalescence of circular pores on the overall elastic properties. The research is motivated mostly be needs to predict properties of porous materials obtained by Gasar technology – process consisting of a melting metal in a gas atmosphere to saturate it with hydrogen and directional solidification (Shapovalov, 1994; Shapovalov and Boyko, 2004). The pores have cylindrical shape and are nucleated heterogeneously. The process is accompanied by pores coalescence. Shapovlov (1998) showed that the pore coalescence becomes prominent for Gasar metals with high porosity. The modeling of the evolution process of pore coalescence has been proposed by Liu et al. (2018). Fig. 1 illustrates the process of the pores coalescence and the resulting shapes of the pores’ cross-sections in Gasar metals. We consider this material in the framework of plane-strain problem and assume that it contains aligned cylindrical inhomogeneities of certain cross-sectional shape. Analytical modeling of materials with inhomogeneities of non-elliptical cross-section is not well developed though many two-dimensional problems have been solved. The main approaches to this problem are:. 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21. ∗. Corresponding author. E-mail address: igor@nmsu.edu (I. Sevostianov).. • Complex variables technique involving conformal mapping of the cross-sectional shape onto a unit circle (Kachanov et al., 1994). For many non-elliptical shapes, the transformation. . z (ζ ) = R. 1. ζ. +. N . . an ζ n. 22 23 24. (1.1). n=1. that maps conformally the exterior of the inhomogeneity in the complex z-plane into the interior of a unit circle in the ζ -plane, is used, with parameters R, N and an corresponding to various shapes; for the elliptical hole, for example, N = 1, R = (a + b)/2 and a1 = (a − b)/(a + b). For “irregular” shapes, a numerical mapping technique can be used (see Tsukrov and Novak, 2004); • Finite element method, that is more universal, applies to inhomogeneities of arbitrary elastic properties, including anisotropic ones, but has lower accuracy than the numerical conformal mapping technique. Comparison of the two methods was given by Tsukrov and Novak (2002). Compressibility of non-elliptical holes has been first analyzed by Zimmerman (1986) on the example of super-circular holes (convex and concave), by Givoli and Elishakoff (1992) and Ekneligoda and Zimmerman (2008a) who considered holes with “corrugated” boundaries and by Ekneligoda and Zimmerman (20 06, 20 08b) who considered shapes having n-fold symmetry axes. Results for the entire compliance contribution tensor of a non-elliptical hole have been obtained by. https://doi.org/10.1016/j.ijsolstr.2019.05.012 0020-7683/© 2019 Elsevier Ltd. All rights reserved.. Please cite this article as: L. Lanzoni, E. Radi and I. Sevostianov, Effect of pair coalescence of circular pores on the overall elastic properties, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019.05.012. 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43.

(3) ARTICLE IN PRESS. JID: SAS 2. [m5G;May 15, 2019;19:45]. L. Lanzoni, E. Radi and I. Sevostianov / International Journal of Solids and Structures xxx (xxxx) xxx. Fig. 1. (a) Pores structure in Gasar Ni-15%Al, intermetallic compound (shape of pores is almost cylindrical, from Drenchev and Sobczak, 2009); (b) and (c) evolution of two pores coalescence in Gasar copper (from Liu et al., 2018).. Fig. 2. (a) two separate circular holes, (b) cross-section formed by two coalesced circular pores of generally different radii.. 44 45 46 47 48 49 50 51 52 53 54 55 56 57. Kachanov et al. (1994) and Jasiuk (1995) for various polygons (convex and concave) and Tsukrov and Novak (20 02, 20 04) for several “irregular” shapes. The present paper continues authors’ work (Lanzoni et al., 2018) on the shapes that may be obtained by union of two circles of generally different diameters (Fig. 2). We consider isotropic elastic plane containing two circular holes of radii r1 and r2 (that may overlap). Thus, the pore shapes may be non-convex and even not simply connected. Instead of the conformal mapping technique (that may be a problem in this case since connectivity of the pore may change) we use an analytic approach based on Fourier series representation or Fourier transform in bipolar coordinates (Jeffery, 1921), (α , β ) (Fig. 3), related to the Cartesian coordinates (x1 ,x2 ) by. . ( x1 + ix2 ) + a α = Re ln ; ( x1 + ix2 ) − a. . ( x1 + ix2 ) + a β = −Im ln ; ( x1 + ix2 ) − a. (1.2) 58. 59 60 61 62 63 64 65 66. a sinh α x1 = ; cosh α − cos β. a sin β x2 = . cosh α − cos β. Fig. 3. Sketch of the bipolar coordinate system.. (1.3). Note, that β -coordinate is multi-valued with a discontinuity of 2π across the segment connecting the foci. Hereinafter, we assume − π < β ≤ π . The two poles of the bipolar coordinates are located on the x1 axis at distance ± a, with a > 0 (the circles in Fig. 2(a) refers to α 1 > 0 and α 2 < 0 whereas Fig. 2(b) shows two overlapping circles with β 1 > 0 and β 2 < 0). First, we consider a single inhomogeneity and solve Neumann boundary value problem in two-steps: (1) assessment of the fun-. damental displacement field related to a remotely applied uniform stress in a homogeneous body and (2) fulfillment of the boundary conditions by adding an extra-term to the fundamental field. This solution is used to construct the compliance contribution tensor of a pore of interest by calculating proper contour integrals. The compliance contribution tensor can be used to calculate overall elastic properties of a material containing parallel cylindrical holes with the cross-sections shown in Fig. 2.. Please cite this article as: L. Lanzoni, E. Radi and I. Sevostianov, Effect of pair coalescence of circular pores on the overall elastic properties, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019.05.012. 67 68 69 70 71 72 73 74.

(4) ARTICLE IN PRESS. JID: SAS. [m5G;May 15, 2019;19:45]. L. Lanzoni, E. Radi and I. Sevostianov / International Journal of Solids and Structures xxx (xxxx) xxx. 3. Fig. 4. Sketch of an infinite plate with (a) two separate holes and (b) two merging holes subjected to remote normal σ ∞ 11 , σ ∞ 22 and shear σ ∞ 12 stress fields along the principal directions x1 , x2 .. Fig. 5. Distribution of the dimensionless stress fields (a) σ αα /σ ∞ 11 ; (b) σ ββ /σ ∞ 11 ; (c) σ αβ /σ ∞ 11 in a plate subjected to a remote stress in the x1 direction for ρ = 3/5, γ = 1.. 75 76 77 78 79 80 81 82 83 84 85 86 87. 2. Two separate circular holes In this Section we briefly summarize the known results about elastic fields in an infinite plate containing two separate circular holes of radii r1 and r2 in an infinite plane separated by the ligament δ between them. For the case of two holes of the same radius the problem was solved by Ling (1947, 1948a) for remotely applied normal loadings and by Karunes (1953) for remotely applied shear loading. Radi (2011) generalized their solutions for two holes of different radii. The geometry of the problem is completely determined by two independent geometrical parameters: for example, ratio of the radii ρ ≡ r1 /r2 and relative length of the ligament γ ≡ δ /r1 (Fig. 4):. xc1 = r1 + δ [1 − (r1 + 0.5δ )/(r1 + r2 + δ )];. α1 = arccos h (xc1 /r1 ); a = r1 sinh (α1 ); α2 = − arcsin h(a/r2 ); xc2 = −r2 cosh α2 . 88 89 90. (2.1). The plane is subjected to the action of remotely applied stresses. ∞ , σ ∞ , and σ ∞ . σ11 22 12. The traction free boundary conditions. σα = ταβ = 0 for α = α1 , α2. (2.2). have to be satisfied at the holes. The stress field, corresponding to the biharmonic Airy stress function χ is given by Jeffery (1921):. 91 92 93. . . ∂ ∂ ∂2 σα = − (cosh α − cos β ) 2 − sinh α − sin β + cosh α hχ ; ∂α ∂β ∂β ∂2 ∂ ∂ σβ = − (cosh α − cos β ) 2 − sinh α − sin β + cos β hχ ; ∂α ∂β ∂α ταβ = (cosh α − cos β ). ∂ 2 hχ , ∂β∂α. (2.4). where. hχ =. 94. χ a. (coshα − cosβ ).. (2.5). The Airy function χ can be represented as the sum of a fundamental stress function χ (0) , which gives the uniform stresses applied at infinity but does not yield vanishing tractions on the circular boundaries, and an auxiliary stress function χ (1) , required to satisfy the boundary conditions (2.2), which gives zero stresses at infinity. Correspondingly,. hχ = hχ( ) + hχ( ) , 0. 1. (2.6). Please cite this article as: L. Lanzoni, E. Radi and I. Sevostianov, Effect of pair coalescence of circular pores on the overall elastic properties, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019.05.012. 95 96 97 98 99 100.

(5) ARTICLE IN PRESS. JID: SAS 4. [m5G;May 15, 2019;19:45]. L. Lanzoni, E. Radi and I. Sevostianov / International Journal of Solids and Structures xxx (xxxx) xxx. Fig. 6. Distribution of the dimensionless stress fields (a) σ αα /σ ∞ 22 ; (b) σ ββ /σ ∞ 22 ; (c) σ αβ /σ ∞ 22 in a plate subjected to a remote stress in the x2 direction for ρ = 3/5, γ = 1.. Fig. 7. Distribution of the dimensionless stress fields (a) σ αα /σ ∞ 12 ; (b) σ ββ /σ ∞ 12 ; (c) σ αβ /σ ∞ 12 in a plate subjected to a remote shear stress in the x1 ×2 plane for ρ = 3/5, γ = 1.. 101. where. hχ(0). 2 2 ∞ ∞ ∞ (σ11 sin β + σ22 sinh α − 2σ12 sin β sinh α ) = . 2(cosh α − cos β ). The components of the corresponding displacement vector are given by Jeffery (1921). (2.7). 102. hχ(1) = [B α + K ln(cosh α − cos β )](cosh α − cos β ) +. ∞ . φn (α ) cos nβ + ψn (α ) sin nβ .. (2.8). n=1. 103. Functions ϕ n (α ) and ψ n (α ) are given by. φn (α ) = An cosh(n + 1 )α + Bn cosh(n − 1 )α + Cn sinh(n + 1 )α + Dn sinh(n − 1 )α ; ψn (α ) = an cosh(n + 1 )α + bn cosh(n − 1 )α + cn sinh(n + 1 )α + dn sinh(n − 1 )α , 104 105. . 2μuα = (cosh α − cos β ). . hχ κ −1 ∂ 2 ∂α cosh α − cos β . . hQ κ +1 ∂ − , 4 ∂β cosh α − cos β hχ κ −1 ∂ 2μuβ = (cosh α − cos β ) 2 ∂β cosh α − cos β κ +1 ∂ hQ + , 4 ∂α cosh α − cos β. . ∞ α − cos β − σ11∞ − σ22 sin β sinh α hQ = cosh α − cos β α β + 2B β − 4K tan−1 tanh cot (cosh α − cos β ) ∞ 2τ12 cosh α + sinh. (2.9). The integration constants B, K, An , Bn , Cn , Dn , an , bn , cn , dn are given in the Appendix A1.. 2. 2. 107. (2.10). where κ = 3–4 ν or κ = (3 − ν ) / (1 + ν ) for plane strain or plane stress state, respectively, and. 106. 2. Please cite this article as: L. Lanzoni, E. Radi and I. Sevostianov, Effect of pair coalescence of circular pores on the overall elastic properties, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019.05.012. 108 109.

(6) ARTICLE IN PRESS. JID: SAS. [m5G;May 15, 2019;19:45]. L. Lanzoni, E. Radi and I. Sevostianov / International Journal of Solids and Structures xxx (xxxx) xxx. 5. Fig. 8. Dimensionless hoop stress σ ββ along the contour of the hole (a) with α = α 1 and (b) with α = α 2 for some values of ρ and γ = 1.. + 2(A1 sinh 2α + C1 cosh 2α ) sin β. stress along the contours of the pores for some values of ρ ≡ r2 /r1 , γ ≡ δ /r1 = 1. Fig. 9 provides the same information for different values of γ and ρ = 2 .. − 2(a1 sinh 2α + c1 cosh 2α ) cos β ∞ +2 {[An sinh (n + 1)α + Bn sinh(n − 1 )α. 3. Two overlapped circular holes. n=2. + Cn cosh (n + 1 )α + Dn cosh (n − 1 )α ] sin nβ + − [an sinh (n + 1 )α + bn sinh (n − 1 )α + cn cosh (n + 1 )α + dn cosh (n − 1 )α ] cos nβ}. 110 111 112. (2.11). Figs. 5–7 show distribution of the dimensionless stress fields ∞ , σ ∞ and σ ∞ , rein a plate subjected to a remote stresses σ11 22 12 spectively. Fig. 8 illustrates distribution of the dimensionless hoop. The modeling of two overlapping circles differs considerably from the case discussed in Section 2: the circular contours represent two curves of constant β (0 ≤ β 1 < π ,−π ≤ β 2 < 0) for α ∈ ( − ∞, ∞) (Fig. 8). In this case, Fourier transforms have to be applied instead of the Fourier series (see, for example, Ling, 1947; 1948b; Dutt, 1960). The geometry of the problem is completely defined by three independent parameters, e.g. coordinates. Please cite this article as: L. Lanzoni, E. Radi and I. Sevostianov, Effect of pair coalescence of circular pores on the overall elastic properties, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019.05.012. 113 114 115 116 117 118 119 120 121 122 123.

(7) ARTICLE IN PRESS. JID: SAS 6. [m5G;May 15, 2019;19:45]. L. Lanzoni, E. Radi and I. Sevostianov / International Journal of Solids and Structures xxx (xxxx) xxx. Fig. 9. Dimensionless hoop stress σ ββ along the contour of the hole (a) with α = α 1 and (b) with α = α 2 for some values of γ and ρ = 2.. 124 125 126 127 128 129 130 131. yc 1 , yc 2 of the centers of two circles and the focal distance a. Then, β 1 = arctan a/yc 1 , β 2 = arctan a/yc 2 , r1 = a/sin |β 1 |, r2 = a/sin |β 2 |, and the area included in the contour reads A = r1 2 (π − β 1 ) + r2 2 (π + β 2 ) + a2 (cot β 1 − cot β 2 ). In contrast to the case of 2 separate holes, here the ligament δ turns out to be a negative quantity defined as δ = yc 1 − r1 − (yc 2 + r2 ). The form of the fundamental stress function is the same as in (2.7), whereas the auxiliary stress functions are taken as follows:. hχ(1) =. 0. ∞. F (s, β ) cos sα + G(s, β ) sin sα ds,.

(8). hχ(2) = (cosh α − cos β ) K log. cosh α − cos β , cosh α + cos β. (3.1). where. 132. F (s, β ) = fF (s ) sin β sinh sβ + kF (s ) cos β cosh sβ + gF (s ) sin β cosh sβ + hF (s ) cos β sinh sβ ; G(s, β ) = fG (s ) sin β cosh sβ + kG (s ) cos β sinh sβ + gG (s ) sin β sinh sβ + hG (s ) cos β cosh sβ .. (3.2). Please cite this article as: L. Lanzoni, E. Radi and I. Sevostianov, Effect of pair coalescence of circular pores on the overall elastic properties, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019.05.012.

(9) JID: SAS. ARTICLE IN PRESS. [m5G;May 15, 2019;19:45]. L. Lanzoni, E. Radi and I. Sevostianov / International Journal of Solids and Structures xxx (xxxx) xxx. 7. Fig. 10. Distribution of the dimensionless stress fields (a) σ αα /σ ∞ 11 ; (b) σ ββ /σ ∞ 11 ; (c) σ αβ /σ ∞ 11 in a plate subjected to a remote stress in the x1 direction for κ 1 = 1/2 and κ 2 = −1.. Fig. 11. Distribution of the dimensionless stress fields (a) σ αα /σ ∞ 22 ; (b) σ ββ /σ ∞ 22 ; (c) σ αβ /σ ∞ 22 in a plate subjected to a remote stress in the x2 direction for κ 1 = 1/2 and κ 2 = −1.. 133 134 135 136 137. Note also that a symmetric layout is retrieved if β 2 = −β 1 : In such a case one has gF (s) = gG (s) = hF (s) = hG (s) = 0 (see Ling (1948b) for a plate with symmetric overlapped holes subjected to normal loadings and Karunes (1953) for the shear loading). For remotely applied shear loading it is hχ (2 ) = 0.. The fundamental stress function (2.7) can be rewritten, after some algebra, in the following form:. hχ(0) =. ∞ (σ ∞ sin β − 2σ12 sin β sinh α ) ∞ + σ22 cos β , 2(cosh α − cos β ). 138 139. 2. (3.3). where σ ∞ = (σ ∞ 11 − σ ∞ 22 ). Please cite this article as: L. Lanzoni, E. Radi and I. Sevostianov, Effect of pair coalescence of circular pores on the overall elastic properties, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019.05.012. 140.

(10) ARTICLE IN PRESS. JID: SAS 8. [m5G;May 15, 2019;19:45]. L. Lanzoni, E. Radi and I. Sevostianov / International Journal of Solids and Structures xxx (xxxx) xxx. Fig. 12. Distribution of the dimensionless stress fields (a) σ αα /σ ∞ 12 ; (b) σ ββ /σ ∞ 12 ; (c) σ αβ /σ ∞ 12 in a plate subjected to a remote shear stress in the x1 ×2 plane for κ 1 = 1/2 and κ 2 = −1.. The traction-free boundary conditions at the hole are. 141. σβ = 0, ταβ = 0, for β = β1 , β2 ; 1 2 hχ ( ) ( α , β ) + hχ ( ) ( α , β ) = 0. 142 143. for α , β → 0.. (3.4). The first two conditions can be reformulated for the auxiliary stress functions hχ (1 )(α , β ) and hχ (2 )(α , β ) as. ∂ 2 hχ = 0, for β = β1 , β2 . ∂β ∂α 144. and. . ∂2 ∂ ∂ (cosh α − cos β ) 2 − sinh α − sin β + cos β ∂α ∂β ∂α hχ = 0, for β = β1 , β2 . 145. 147. ∞. 149 150 151 152. . where the apex denotes derivative with respect to coordinate β whereas from condition (3.9) one has. ∞ 0. (3.6). Taking the derivative of expression (3.6) with respect to α and using (3.5) one can write. −. 2. . ∞ 0. F (s, β ) sin sα ds = σ ∞. 2. . sin β 2. 1 cosh α + + cosh α − cos β (cosh α − cos β )2. . 2sinh α. −. (cosh α − cos β ). 3. 5 cosh α sinh α. (cosh α − cos β )2 ∞. 0. (3.9). Expressions (3.5) and (3.9) can now be used to find unknown functions fF (s), fG (s), kF (s), kG (s), gF (s) and hF (s). Constant K follows from condition (3.7) for normal loading. Condition (3.6) gives (for β = β 1 , β 2 ).. +. . ∂2 1− h + Ci = 0, for β = βi (i = 1, 2). ∂ α2 χ. σ11∞. ∞ σ22. (1 − cos β cosh α ) sin β sinh α + B0 sinh α (cosh α − cos β )3. 2. . −. 2cosh α + sinh α (cosh α − cos β ) 2. 4. −. (cosh α − cos β )3. ∞ (1 + s2 )G(s, β ) sin sαds = +σ12. ∞ − σ12. 2 sin β sinh α. 2. . 2sinh α. 2. and in turn, integration of (3.8) with respect to α gives. cosh α − cos β + 2K cos β cosh α + cos β. cos2 β sin 2β −K cosh α + cos β (cosh α − cos β )(cos α + cos β )2. 2. . (3.8). 153 154. 2. −. . (1 + s2 )F (s, β ) cos sα ds = K cos β ln. +2K. (3.7). kF (s )ds = 0.. ∂ ∂ 1− h = 0, for β = β1 , β2 , ∂α ∂ α2 χ 148. (3.5). The last of the conditions (3.4) yields. 0. 146. sin 2β sinh 2α ; (cosh α + cos β )2 (cosh α − cos β ) ∞ ∞ (3 − 4 cos β cosh α + cos 2β cosh 2α ) sG (s, β ) cos sα ds = −σ12 , 3 0 2(cosh α − cos β ) (3.10) +K. − Ci ;. 3 cosh α sinh α sin β. (cosh α − cos β )2. 3. (cosh α − cos β )3. − C˜i ;. (3.11). Eq. (3.11)1,2 for α → ∞ yield. . . ∞ Ci = 2K − σ22 + B0 β cos β ,. 155. ∞ C˜i = σ12 sin β ,. for β = βi (i = 1, 2) .. (3.12). Please cite this article as: L. Lanzoni, E. Radi and I. Sevostianov, Effect of pair coalescence of circular pores on the overall elastic properties, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019.05.012.

(11) ARTICLE IN PRESS. JID: SAS. [m5G;May 15, 2019;19:45]. L. Lanzoni, E. Radi and I. Sevostianov / International Journal of Solids and Structures xxx (xxxx) xxx. 156 157. Thus, from Eqs. (3.10) and (3.11), taking into account results (3.12) one has for β = β i (i = 1, 2). 2 a σ ∞ sπ. (1 − cos β cosh α ) sin β sinh α dα 0 (cosh α − cos β )3. 2K ∞ sin 2β sinh 2α + sin sα dα ; sπ 0 (cosh α + cos β )2 (cosh α − cos β ) F (s, βi ) =. ∞. sin sα. ∞ 2 G (s, βi ) = −σ12 sπ ∞ (3 − 4 cos β cosh α + cos 2β cosh 2α ). 2(cosh α − cos β ). 3. 0. 158. and. cos sα dα. (3.13). 2K cosh α − cos β cos β cos sα ln dα cosh α + cos β ( 1 + s2 )π 0 ∞ 4K cos sα + cos2 β dα + cosh α + cos β ( 1 + s2 )π 0 ∞ 2K cos sα 2 − sin 2β dα + ( 1 + s2 )π 0 (cosh α − cos β )(cos α + cos β )2 ∞ σ ∞ 3 cos β 2 +a sin β cos sα 2 ( 1 + s )π 0 (cos α − cos β )2 F (s, βi ) =. 2sin β. 159 160 161 162. ∞ 4σ12 sin β ( 1 + s2 )π. ∞ 0. sinh α sin sα 3. (cosh α − cos β ). 3. dα .. (3.14). 1 ε = (un + nu )dS 2V ∂ V. 176 177 178 179 180 181 182 183 184 185 186 187. (4.1) 188 189 190 191 192 193. (4.2). where H is a fourth-rank compliance contribution tensor of the inhomogeneity. If the inhomogeneity is a pore, the extra overall strain due its presence is given by the well-known expression in terms of an integral over the pore boundary (Hill, 1963):. Note that, through the results reported in Appendix A2, all the Fourier transforms involved in the Eqs. (3.13) and (3.14) can be evaluated in closed form, thus allowing to find the analytic expressions of functions F(s, β ), G(s, β ) and their derivatives:. F (s, β ). Compliance contribution tensors have been first introduced by Horii and Nemat-Nasser (1983) for pores of ellipsoidal shape (explicit formulas connecting compliance contribution tensor and Eshelby tensor for an ellipsoidal pore are given in the appendix of the mentioned paper). Components of this tensor for various twodimensional pores were given by Kachanov et al. (1994) and for ellipsoidal inhomogeneities – by Sevostianov and Kachanov (1999). This tensor connects the extra strain due to the presence of the inhomogeneity under given remotely applied stresses. Indeed, if we consider a representative volume element V containing an isolated inhomogeneity of volume V1 , the average, over representative volume V strain can be represented as a sum. V ε = 1 H : σ 0 V. (cos α − cos β ) ∞ ∞ 2σ12 sin β sin sα dα G(s, βi ) = − ( 1 + s2 )π 0. ∞ ∞ 6σ12 cosh α sinh α sin sα + sin β dα + ( 1 + s2 )π 0 (cosh α − cos β )2 −. 175. where S0 is the compliance tensor of the matrix and σ 0 represents the uniform boundary conditions (tractions on ∂ V have the form t|∂ V = σ 0 • n where σ 0 is a constant tensor); σ 0 can be viewed as far-field (“remotely applied”) stress. The material is assumed to be linear elastic; hence the extra strain due to the inhomogeneity ε is a linear function of σ 0 :. dα ;. 3. 4. Evaluation of the compliance contribution tensor. ε = S 0 : σ 0 + ε. . 2. −. ∞. 9. 194 195 196 197. (4.3). Thus, Neumann boundary value problem has to be solved in order to find the compliance contribution tensor of a pore.. 198. 4.1. Two separate circular inhomogeneities (symmetric with respect to x1 axis). 200. 199. 201. sinh s( π2 − |β| ) sinh sβ = −2K cos β + 4K cos2 β s(1 + s2 ) cosh s2π (1 + s2 ) sinh sπ sin β. − aσ ∞ sin. 2. β csc |β|cschsπ sinh s(π − |β| ) +. sec β sinh s(π − |β| ) − 2s cosh sβ csc |β| − (−2 cot β csc β + sec |β| ) sinh s|β| 2 − K sin 2β ; 2(1 + s2 ) sin |β| cos β sinh sπ. π s(1 + s2 ) cosh s(π − |β| )cschs(π s ) − 1 sin |β|; sπ ( 1 + s 2 ) s cos β sinh s|β| + sinh s2π sinh s( π2 − |β| ) sin |β| F (s, β ) = 4K sin 2β s sin |2β| sinh sπ s ( s cosh s ( π − | β| ) − sinh s(π − |β| ) cot |β| +aσ ∞ sin β ; s sinh sπ ∞ G (s, β ) = 2σ12 cschsπ [cos β cosh s(π − |β| ) − s sin |β| sinh s(π − |β| )]. ∞ G(s, β ) = 2σ12. 163 164 165 166 167 168 169 170 171 172 173 174. System (3.15) imposed for β = β i (i = 1, 2) allows assessing functions fF (s), fG (s), kF (s), kG (s), gF (s), gG (s), hF (s) and hG (s) and, in turn, the stress and displacement fields according to Eqs. (2.4) and (2.10), respectively. For the case of two equal overlapping holes β 1 = − β 2 the expressions of fF (s), kF (s) reported in Ling (1948b) for normal loadings and fG (s), kG (s) reported in Karunes (1953) for shear loadings are exactly retrieved (actually, a misprint occurred in expression (16)1 of Fn reported in Karunes (1954), in which the square in “n2 ” must be removed). Figs. 10–12 illustrate distribution of the dimensionless stress fields in a plate subjected to a remote ∞ , σ ∞ and σ ∞ , respectively. stresses σ11 22 12. (3.15). The components of the unit vector and the infinitesimal arc length on the contour of the two circles with α = const are:. cosh αi cos β − 1 sinh |αi | sin β sign(αi ), n2 = − cosh αi − cos β cosh αi − cos β a sign(αi ) ds = ri dθ = d β , i = 1, 2 cosh αi − cos β. 202 203. n1 = −. (4.4). where θ is the polar angle measured from x1 axis as shown in Fig. 13(a). In the Cartesian coordinate system (x1 , x2 ), the components of the unit vector, the displacement field and the infinitesimal arc length on the contour of the two separate circles with. Please cite this article as: L. Lanzoni, E. Radi and I. Sevostianov, Effect of pair coalescence of circular pores on the overall elastic properties, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019.05.012. 204 205 206 207.

(12) ARTICLE IN PRESS. JID: SAS 10. [m5G;May 15, 2019;19:45]. L. Lanzoni, E. Radi and I. Sevostianov / International Journal of Solids and Structures xxx (xxxx) xxx. Fig. 13. Sketch of the polar coordinate systems used to perform the circular integrals involved in expression (4.2) for the components of the extra overall strain.. Fig. 14. Normalized components of the cavity compliance tensor (a) H1111 μ; (b) H1122 μ; (c) H2222 μ; (d) H1212 μ for some values of ρ . Reference is made to plane strain condition.. 208. α = const are u1 = −uα cos. 209. θ − uβ sin θ ;. u2 = −uα sin. θ + uβ cos θ ; (4.5). cosh αi cos β − 1 sinh |αi | sin β sign(αi ); sin θ = ; cosh αi − cos β cosh αi − cos β. a sign(αi ) dβ , ds = ri dθ = cosh αi − cos β 210. 214. The component of the unit vector and the infinitesimal arc length on the contour at β = const are. 215 216. sinh α sin βi 1 − cosh α cos βi n1 = − sign(βi ); n2 = − sign(βi ) cosh α − cos βi cosh α − cos βi. with. cos θ =. 4.2. Overlapped circles symmetric with respect to x2 axis. ds = ri dθ = − (4.6). βi. dα ,. i = 1, 2. (4.8). For the overlapping holes (Fig. 13(b)), one finds. u1 = uα sin. and. a sign(βi ) cosh α − cos. θ − uβ cos θ ; u2 = −uα cos θ − uβ sin θ ;. 217. (4.9) 218. for α =. α1 > 0 : for α = α1 < 0 :. 211 212 213. n1 = − cos θ ; n1 = cos θ ;. n2 = − sin θ ; n2 = − sin θ .. for β = (4.7). Now, using results of the Section 2 and formulas (4.3) compliance contribution tensor can be calculated for two separate pores (the integral has to be evaluated numerically).. for β =. β1 > 0 : β2 < 0 :. n1 = − cos θ ;. n2 = − sin θ ;. n1 = − cos θ ;. n2 = sin θ ;. (4.10) 219. cos θ =. sinh α sin βi ; cosh α − cos βi. sin θ =. 1 − cosh α cos βi , cosh α − cos βi. Please cite this article as: L. Lanzoni, E. Radi and I. Sevostianov, Effect of pair coalescence of circular pores on the overall elastic properties, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019.05.012.

(13) ARTICLE IN PRESS. JID: SAS. [m5G;May 15, 2019;19:45]. L. Lanzoni, E. Radi and I. Sevostianov / International Journal of Solids and Structures xxx (xxxx) xxx. ds = ri dθ = −. a sign(βi ) dα . cosh α − cos βi. (4.11). 225. Taking into account that the area of the pore cross-section Ai = ri2 (π − |βi + (sin |2β1 | )/2| ), one can use results of Section 3 and formula (4.3) to evaluate the compliance contribution tensor. Fig. 14 illustrates the dimensionless components of the compliance contribution tensor in dependence on δ /r1 for different values of r2 /r1 .. 226. 5. Concluding remarks. 227. In this paper, we calculated compliance contribution tensor of two separate or intersecting circular pores. For this goal, we first considered two holes and solved Neumann boundary value problem in two-steps: (1) assessment of the fundamental displacement field related to a remotely applied uniform stress in a homogeneous body and (2) fulfillment of the boundary conditions in the problem with pores by adding an extra-term to the fundamental field. This solution was used to construct the compliance contribution tensor of the combination of two circular pores by calculating proper contour integrals. Plots for H1111 and H2222 (Fig. 14(a) and (c)) generally reproduce the curves for two corresponding components of the resistivity contribution tensor (Lanzoni et al., 2018). At the same time, components H1122 and H1212 behave differently and may show non-monotonic not only near δ /r1 = 1 (point where the circles touch each other), but also when δ /r1 > 1 or δ /r1 < 1 (see Fig. 14(b) and (d)). This is in agreement with expressions for crossproperty connections for a material with ellipsoidal/elliptical pores and inhomogeneities derived by Sevostianov and Kachanov (2002, 2008). In particular, for a two-dimensional elliptical hole. 220 221 222 223 224. 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 Q2 244 245. . (a1 + a2 )2 2a1 a2. R= 247. . . E0 H1111 = k0 R11 1 + E0 H1212 = 248 249 250 251 252 253 254 255 256 257 258 259. . 1 ; k0 R22. 1 1 + ; k0 R11 k0 R22. (5.2). . E0 H2222 = k0 R22 1 +. E0 H1122 =. (R11 + R22 )2 R11 R22. (5.3). Components R11 and R22 that behave oppositely (one increases when another decreases) enter expressions for H1122 and H1212 in concurrent manner, so that their combined effect may be quite complex. The case δ /r1 = −2, r/r1 = 1 corresponds to an isolated circular inhomogeneity. In this case, the well-known result for the compliance contribution tensor of a circular hole (see Horii and NematNasser, 1983) is recovered. The compliance contribution tensor constitutes the basic building block for calculation of the overall elastic properties of a material containing parallel cylindrical holes with the cross-sections shown in Fig. 2 (see Kachanov and Sevostianov, 2018). 261. Financial support from the Italian Ministry of Education, University and Research (MIUR) in the framework of the Project PRIN 2017 Modelling of constitutive laws for traditional and innovative building materials (code 2017HFPKZY) is gratefully acknowledged. IS and ER acknowledge financial support of Indam-GNFM "Gruppo Nazionale per la Fisica Matematica".. 264 265 Q3 266. A1 =. 1 2 2 cosh (α1 + α2 ){2 f (α1 )sinh α2 − 2 f (α2 )sinh α1 D + [g(α1 ) − g(α2 )] tanh (α1 + α2 )},. (A1.1). 1 2 2 B1 = cosh (α1 − α2 ) 2 f (α2 )sinh α1 − 2 f (α1 )sinh α2 D −[g(α1 ) + g(α2 )]tanh (α1 − α2 ) + g(α2 )sinh2α1 − g(α1 )sinh2α2 }. (A1.2) 270. C1 =. 1 cosh (α1 + α2 ){g(α2 ) − g(α1 ) D + [ f (α1 ) − f (α2 )] tanh (α1 + α2 ) + f (α2 )sinh2α1 − f (α1 )sinh2α2 }. (A1.3) 271. 2 B = B = cosh(α1 − α2 ){[ f (α1 ) + f (α2 )] tanh(α1 − α2 ) D + g(α2 ) − g(α1 )}, (A1.4) where:. 272. f ( α ) = 2K e. −|α|. sinh. −2|α|. α − ( σ − σ )e sign α , ∞ α + σ22 sinh |α| ), ∞ 22. ∞ g(α ) = cosh 2α − e−|α| (σ11 cosh. ∞ 11. (A1.5). D = 2 sinh(α1 − α2 )(sinh. 2. α1 + sinh2 α2 ). (A1.6). a1 = τ. ∞e 12. −2|α1 |. cosh 2α2 − e−2|α2 | cosh 2α1 , sinh 2(α1 − α2 ). −2|α1 | sinh 2α2 + e−2|α2 | sinh 2α1 ∞e c1 τ12 . sinh 2(α1 − α2 ). 274. (A1.7) 275. (A1.8) 276. Acknowledgments. 263. 268. . 1 ; k0 R11. 260. 262. A1. Integration constants used in Section 2. ( e1 e2 + e2 e1 ) ( e1 e2 + e2 e1 ) − ( e1 e1 e2 e2 + e2 e2 e1 e1 ). 1 a1 + a2 a1 + a2 e1 e1 + e2 e2 k0 a1 a2. So that. 267. 273. (5.1). 246. Appendices. 269. 1 ( 2a2 + a1 ) ( 2a1 + a2 ) H = e1 e1 e1 e1 + e2 e2 e2 e2 E0 a1 a2 +. 11. 1 An = {Pn (α1 , α2 )n (α1 ) + Pn (α2 , α1 )n (α2 ) Hn + Qn (α1 , α2 )n ∗ (α1 ) + Qn (α2 , α1 )n ∗ (α2 ). (A1.9) 277. 1 Bn = {P−n (α1 , α2 )n (α1 ) + P−n (α2 , α1 )n (α2 ) Hn +Q−n (α1 , α2 )n ∗ (α1 ) + Q−n (α2 , α1 )n ∗ (α2 )}. (A1.10) 278. 1 {Un (α1 , α2 )n (α1 ) + Un (α2 , α1 )n (α2 ) Hn + [Vn (α1 , α2 ) + cosh(2nα2 − (n − 1 )α1 )]n ∗ (α1 ). Cn = −. + Vn (α2 , α1 )n ∗ (α2 )}. (A1.10A) 279. 1 Dn = {U−n (α1 , α2 )n (α1 ) + U−n (α2 , α1 )n (α2 ) Hn +V−n (α1 , α2 )n ∗ (α1 ) + V−n (α2 , α1 )n ∗ (α2 )}. (A1.11) 280. an =. 1 {Pn (α1 , α2 )n (α1 ) + Pn (α2 , α1 )n (α2 ) Hn + Qn (α1 , α2 )n ∗ (α1 ) + Qn (α2 , α1 )n ∗ (α2 )}. (A1.12). Please cite this article as: L. Lanzoni, E. Radi and I. Sevostianov, Effect of pair coalescence of circular pores on the overall elastic properties, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019.05.012.

(14) ARTICLE IN PRESS. JID: SAS 12. (1 − cos β cosh α ) sinh α sin sα ds 0 (cosh α − cos β )3 π (s cosh s(π − |β| ) − sinh s(π − |β| ) cot |β| ) = s ; 2 sinh sπ. 281. 1 bn = {P−n (α1 , α2 )n (α1 ) + P−n (α2 , α1 )n (α2 ) Hn + Q−n (α1 , α2 )n ∗ (α1 ) + Q−n (α2 , α1 )n ∗ (α2 )}. (A1.13). 282. + Vn (α2 , α1 )n ∗ (α2 )} 283. 298. sinh 2α. ∞. sin sα ds (cosh α − cos β )(cosh α + cos β )2. s sinh s|β| cos β + sinh s2π sinh s π2 − |β| sin |β| = 2π ; (A2.2) sinh sπ sin 2|β|. ∞. 0. (A1.15). for n ≥ 2, where. 1 Pn (ξ , η ) = (sinh(ξ + nη ) sinh n(ξ − η ) n+1 + n sinh(ξ + nξ ) sinh(ξ − η ) ),. (A2.1). 0. (A1.14). 1 dn = {U−n (α1 , α2 )n (α1 ) + U−n (α2 , α1 )n (α2 ) Hn + V−n (α1 , α2 )n ∗ (α1 ) + V−n (α2 , α1 )n ∗ (α2 )}. ∞. 1 cn = − {Un (α1 , α2 )n (α1 ) + Un (α2 , α1 )n (α2 ) Hn + [Vn (α1 , α2 ) + cosh(2nα2 − (n − 1 )α1 )]n ∗ (α1 ). 284. [m5G;May 15, 2019;19:45]. L. Lanzoni, E. Radi and I. Sevostianov / International Journal of Solids and Structures xxx (xxxx) xxx. ∞. 0. 299. cos sα sinh s(π − |βi | ) dα = π ; (cosh α − cos βi ) sinh sπ sin |βi |. (A2.3) 300. cos sα sinh s|βi | dα = π ; (cosh α + cos βi ) sinh sπ sin |βi |. (A2.4) 301. (A1.16). (cosh α + cos βi ). 2. 0. 285. cos sα. ∞. dα = π. s cosh βi − cot |βi | sinh s|βi | sinh sπ sin. 2. |βi |. (A2.5). Qn (ξ , η ) = cosh (ξ + nη )sigh n(ξ − η ) − n cosh(η + nξ ) sinh(ξ − η ),. (A1.17) Un (ξ , η ) =. 1 [cosh(ξ + nη ) sinh n(ξ − η ) n+1 + n cosh(η + nξ ) sinh(ξ − η )],. 302. π. ∞ sinh s 2 − |βi | cosh α − cos βi π ; log( ) cos sα dα = −π cosh α + cos βi s cosh s 2 0. 286. ;. (A2.6). (A1.18). 303. 287. Vn (ξ , η ) = sinh(ξ + nη ) sinh n(ξ − η ) − n sinh(η + nξ ) sinh(ξ − η ),. (A1.19) 288. 289. Hn = 2n{sinh [n(α1 − α2 )] − n2 sinh (α1 − α2 )}. 2. 2. 0. (A1.20). 304. cos sα dα (cosh α − cos βi )3 3s cosh s(π − |βi | ) cot |βi | + (s2 − 2 + 3csc2 βi ) sinh s(π − |βi | ) =π ; 3 sin |βi | sinh sπ. ∞ ∞ n ∗ (α ) = 2K e−n|a| sinh α − (σ22 − σ11 )n gn (α )sign α , (A1.21). 291. ∞. 0. 290. n (α ) = −e−n|α| [2 K (cosh α + n sign|α| ) ∞ ∞ ∞ + (σ22 − σ11 ) n (n2 − 1 ) sinh|σ22 |. cos sα s cosh s(π − |βi | ) + cot |βi | sinh s(π − |βi | ) dα = π ; 2 sin βi sinh sπ (cosh α − cos βi )2 (A2.7). ∞. (A2.8). (A1.22). cos sα dα = (cosh α − cos β )(cosh α + cos β )2 sec|β| sinh s(π − |β| ) cos β − 2s cosh sβ csc β − (sec|β| − 2 cot |β| csc β ) sinh s|β| =π ; 4 sin |β| cos β sinh sπ ∞. 0. (A2.9) 305. 292. n ∗ ( α ) = 2 n g n ( α ). (A1.23). ∞ n (α ) = 2 τ12 n (n2 − 1 ) e−n|α| sinh α .. (A1.24). Finally, the constant K follows from the condition. 293. ∞ . 295. 0. cosh α sinh s(π − |βi | ) cos βi cos(sα )dα =π ; (cosh α − cos βi ) sinh sπ sin |βi | (A2.10) 306. ∞. cosh α. ∞. sinh α. cos(sα )dα (cosh α − cos βi )2 s cosh s(π − |βi | ) cot |βi |+csc2 |βi | sinh s(π − |βi | ) =π ; (A2.11) sinh sπ sin |βi | 0. ( An + Bn ) = 0.. (A1.25). n=1. 294. ∞. after the introduction of the constants An and Bn , for n ≥ 1. A2. Useful Fourier transforms. 307. sin(sα )dα (cosh α − cos βi )2 sinh sπ cosh sβi − cosh sπ sinh |sβi | = sπ ; sinh sπ sin |βi | 0. 296 297. The following definite integrals have been used to find expressions (3.13) and (3.14):. (A2.12). Please cite this article as: L. Lanzoni, E. Radi and I. Sevostianov, Effect of pair coalescence of circular pores on the overall elastic properties, International Journal of Solids and Structures, https://doi.org/10.1016/j.ijsolstr.2019.05.012.

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