Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 54, 2, pp. 593-600, Warsaw 2016 DOI: 10.15632/jtam-pl.54.2.593 THE THERMOELASTIC PROBLEM FOR A PENNY-SHAPED ANTICRACK WITH HEAT CONDUCTIVITY IN A TRANSVERSELY ISOTROPIC SPACE Andrzej Kaczyński Warsaw University of Technology, Faculty of Mathematics and Information Science, Warsaw, Poland e-mail: akacz@mini.pw.edu.pl Bohdan Monastyrskyy Protein Structure Prediction Center, University of California Davis, Davis, CA USA; e-mail: bmonast@gmail.com An analytical solution of a 3D transversely isotropic thermoelastic problem of a uniform heat flowdisturbed by a penny-shaped rigid sheet-like inclusion (anticrack)with some small conductivity is obtained by using the potential theory method. The behaviour of thermal stresses near the edge of the disc is analysed fromthe standpoint of themechanics of fracture initiation. Keywords: transversely isotropic space, circular anticrack, heat flow, singular integral equ- ations, thermal stress singularities 1. Introduction The study of thermal stresses in solids containing foreign inhomogeneities has great importan- ce for the evaluation of the strength of materials and structures which operate under thermal actions. The rapid development of high-strength composite materials has driven researches to take into account the influence of anisotropy in thermomechanical fields for fractured bodies. In addition to cracks, rigid lamellate inclusions (also called anticracks, for brevity) are objects aroundwhich stress concentrations occur, whichwill stimulate failure ofmaterials.Most of rese- arch works discuss 2D problems dealing with these defects. Owing tomathematical complexity, only few publications on the subject within 3D statement of thermoelastic anticrack problems can be found in the literature (see Kit and Khay, 1989; Stadnyk, 1994, 2011; Podil’chuk, 2001; Chaudhuri, 2003; Kaczyński and Kozłowski, 2009; Kaczyński andMonastyrskyy, 2013). This work treats a rigid penny-shaped inclusion obstructing a uniform perpendicular heat flow in a transversely isotropic space. It may be regarded as a sequel to our papers (Kaczyński andMonastyrskyy, 2009; Kaczyński, 2014; see also extensive references therein) in which a clas- sical condition of thermal insulation of the inclusion faceswas assumed.Thepresent contribution focuses on the determination of a stationary temperature field with more general thermal con- dition by taking into account certain conductivity of a rigid inclusion. The associated problem of induced thermal stresses is reduced to a two-dimensional singular equation with the unknown normal stress discontinuity across inclusion faces, a closed-form solution to which is found by use ofDyson andGalin theorems. Relations for the evaluation of stresses near the inclusion edge are presented and interpreted from a fracture perspective. Moreover, thermal and mechanical fields for thermally conductive and insulated anticracks are compared. 2. Thermoelastostatics of transversely isotropic materials Let us recall the basic relations of uncoupled thermoelasticity for homogeneous transversely isotropic materials. Referring to a Cartesian coordinate system (X1,X2,X3) and denoting the 594 A. Kaczyński, B.Monastyrskyy temperature, fluxes, displacements and stresses by T , qi, ui, σij, respectively, the governing equations for an infinite transversely isotropic thermoelastic solid whose isotropic plane is per- pendicular to theX3-axis and, in absence of body forces andheat sources, are (Ding et al., 2006; Kaczyński, 2014) T,γγ +k −2 0 T,33 =0 qα =−k1T,α q3 =−k3T,3 1 2 (c11+ c12)uγ,γα+ 1 2 (c11−c12)uα,γγ + c44uα,33+(c13+ c44)u3,3α =β1T,α (c13+ c44)uγ,γ3+ c44u3,γγ + c33u3,33 =β3T,3 σ3α = c44(uα,3+u3,α) σ33 = c13uγ,γ + c33u3,3−β3T σ12 = 1 2 (c11− c12)(u1,2+u2,1) σ11 = c11u1,1+ c12u2,2+c13u3,3−β1T σ22 = c12u1,1+ c11u2,2+c13u3,3−β1T (2.1) Moreover, k0 = √ k1 k3 β1 =(c11+ c12)α1+ c13α3 β3 =2c13α1+ c33α3 (2.2) In the equations given above, k1(α1) and k3(α3) denote the coefficients of conductivity (of thermal expansion) in the plane isotropy and along theX3-axis of rotationalmaterial symmetry, respectively, and c11, c12, c13, c33, c44 are five independent elastic constants. Indices i,j run over 1,2,3 while indicesα,γ run over 1,2. Summation convention holds unless otherwise stated. Subscriptsprecededbyacomma indicatepartial differentiationwith respect to thecorresponding coordinates. 3. Formulation of the anticrack problem Consider a transversely isotropic space weakened by a penny-shaped rigid inclusion (anticrack) subjected to a uniform steady-flow of heat q0 in the direction of the negative X3-axis as shown in Fig. 1. The anticrack region S on the mid-plane of transverse isotropy x3 = 0 is denoted as r≡ √ x21+x 2 2 ¬ a. Weare facedwith theboundary-valuevalueproblem:find the fieldsT andui suitable smooth onR3−S such that Eqs. (2.1) hold, subject to the following boundary conditions: — thermal conditions taking into account the thermal conductivity within the anticrack S — mechanical conditions for (x1,x2,x3 =0 ±)∈S with a small constant ε characterizing the rigid vertical translation u1 =u2 =0 u3 = ε (3.1) — thermal andmechanical conditions at infinity q1 = q2 =0 q3 =−q0 (q0 > 0) σij =0 (3.2) The thermoelastic problem for a penny-shaped anticrack... 595 Fig. 1. A transversely isotropic space with a penny-shaped conductive anticrack subjected to a perpendicular uniform flow 4. Solution Bymeans of the superpositionprinciple, it is convenient to represent the solution to theanticrack boundary-value problem as a sum of two components, namely T =T(0)+ T̃ ui =u (0) i + ũi σij =σ (0) ij + σ̃ij (4.1) where the components attached by 0 describe the basic state of the defect-free solid, and the components with the tilde represent perturbations due to the anticrack. The results for the first 0-problem are found to be given by Kaczyński (2014) T(0) = q0 k3 x3 u(0)α = β3q0 k3(2c13+ c33) xαx3 u (0) 3 = β3q0 2k3(2c13+ c33) (x23−x21−x22) σ (0) ij =0 (4.2) Attentionwill benext drawn to the corrective solution of the perturbedproblem.Thedistur- bing thermal field T̃ , decaying at infinity, is determined by solving quasi-Laplace equation (2.1)1 with applying the followingmodel expressions related to the rigid disc S, given fromKaczyński andMonastyrskyy (2009) T̃,3(x1,x2,x3 =0 +)− T̃,3(x1,x2,x3 =0−)= 0 T̃(x1,x2,0 +)− T̃(x1,x2,0−)−k3R(x1,x2)T̃,3(x1,x2,0+)= q0R(x1,x2) (4.3) whereR(x1,x2) is interpreted as the thermal anticrack resistance. From the potential theory (Kellogg, 1953), the solution is expressed as follows T̃(x1,x2,x3)= ∂ω̃(x1,x2,z0) ∂z0 ∣∣∣∣∣ z0=k0x3 (4.4) with ω̃(x1,x2,z0)=− 1 2π ∫∫ S ω(ξ1,ξ2) dξ1dξ2√ (x1−ξ1)2+(x2−ξ2)2+z20 (4.5) 596 A. Kaczyński, B.Monastyrskyy Using thewell-known property of a simple layer potential, the satisfaction of Eq. (4.3)2 leads to an integro-differential singular equation of the Newton type for the unknown density of the potential of the single layer ω(ξ1,ξ2) 2ω(x1,x2)− √ k1k3R(x1,x2) 2π ∆ ∫∫ S ω(ξ1,ξ2) dξ1dξ2√ (x1− ξ1)2+(x2− ξ2)2 = q0R(x1,x2) (4.6) in which∆≡ ∂2 ∂x2 1 + ∂ 2 ∂x2 2 stands for the two-dimensional Laplace operator. Assuming next that R(x1,x2)= R̃(r)=R0 √ a2−r2 R0 > 0 (4.7) an analytical solution to Eq. (4.6) is achieved in the form ω(x1,x2)= ω̃(r)= 2q̃ π √ k1k3 √ a2−r2 (4.8) with q̃= q0 ( 1+ 4 π √ k1k2R0 )−1 ¬ q0 (4.9) Inserting Eq. (4.8) into (4.5) and after integration we arrive at the following elementary formulas for the main thermal potential ω̃ for x3 ­ 0 (see Fabrikant, 1989) ω̃(x1,x2,z0)=− q̃ 2π √ k1k3 [ (2a2+2z20 −r2)sin−1 a l20 − 2a2−3l210 a √ l220−a2 ] (4.10) and, in view of Eqs. (4.4) and (2.1)2, for the temperature T̃ and heat fluxes q̃i T̃(x1,x2,x3)=− 2q̃ π √ k1k3 ( k0x3 sin −1 a l20 − √ a2− l210 ) q̃α = 2q̃a2 π √ k1 k3 xα √ a2− l210 l220(l 2 20− l210) q̃3 = 2q̃ π ( sin−1 a l20 − a √ l220−a2 l220− l210 ) (4.11) Here l1 = l1(x3)= 1 2 [√ (r+a)2+x23− √ (r−a)2+x23 ] l10 = l1(z0) l2 = l2(x3)= 1 2 [√ (r+a)2+x23+ √ (r−a)2+x23 ] l20 = l2(z0) (4.12) In the inclusion plane x3 =0 ± (making use of the relations l10|x3=0 =min(a,r), l20|x3=0 = max(a,r)), we obtain T(r,0±)=    ± 2q̃ π √ k1k3 √ a2−r2 0¬ r¬ a 0 r >a qr(r,0 ±)=−k1 ∂T(r,0±) ∂r =    ±2q̃ π √ k1 k3 r√ a2−r2 0¬ r¬ a 0 r >a q3(r,0 ±)=−k3T,3(r,0±)=    q̃−q0 0¬ r a (4.13) The thermoelastic problem for a penny-shaped anticrack... 597 It follows from these formulas that the rigid inclusion acts as an obstruction to the given heat flow, producing thermal local disturbances such as the jump of temperature T across the inclusion plane and the infinite increase of the heat fluxes in the interior vicinity of the inclusion edge. Nowwe pass to the non-trivial perturbed elastic problem, affixed by the tilde and connected with the determination of the induced state of stress and deformation resulting from the known disturbed temperature T̃ =(∂ω̃/∂z0)|z0=k0x3. Because of the anti-symmetry of the temperature and stress system, and bearing inmind Eqs. (3.1), (4.1) and (4.2), it reduces to that of the half space x3 ­ 0 subjected to the following mixed boundary conditions ũα(x1,x2,x3 =0 +)= 0 (x1,x2)∈R2 ũ3(x1,x2,x3 =0 +)= β3q0 2k3(2c13+ c33) (x21+x 2 2)+ε (x1,x2)∈S (4.14) and σ̃33(x1,x2,x3 =0 +)= 0 (x1,x2)∈R2−S ũi =O(|x|−1) |x|= √ x21+x 2 2+x 2 3 →∞ (4.15) Moreover, having found the distribution of the normal stress σ̃33|S+ ≡ q(x1,x2) in the region S, the unknown rigid translation ε can be calculated from the equilibrium condition ∫∫ S q(x1,x2) dx1dx2 =0 (4.16) A solution to this problemwas given byKaczyński (2014). Here only themain idea and final results with somemodifications will be presented. An efficient approach is based on the construction of harmonic potentials that satisfy gover- ningequations (2.1)3,4 andarewell suited to theabove-mentioned anticrackboundaryconditions. We take the following displacement representation expressed by potentials φ̃α ≡ φ̃α(x1,x2,zα), zα = tαx3, α=1 or α=2 ũα = ( φ̃1+ φ̃2+ c1 ∞∫ z0 ω̃(x1,x2,z0) dz0 ) ,α ũ3 =mαtα ∂φ̃α ∂zα + c2k0ω̃ (4.17) with the potentials satisfying the harmonic equations ( ∆+ ∂2 ∂z2α ) φ̃α =0 α=1,2 (no sum on α) (4.18) Here the constants mα, cα, tα are given in Appendix A of Kaczyński (2014). Note that the general case t1 6= t2, tα 6= k0 is considered. Next we put φ̃α =(−1)αf̃(x1,x2,zα)+aα ∞∫ zα ω̃(x1,x2,zα) dzα α=1,2 (no sum on α) (4.19) where ( ∆+ ∂ ∂x23 ) f̃(x1,x2,x3)= 0 (4.20) 598 A. Kaczyński, B.Monastyrskyy and choose the constants aα in order to satisfy a part of boundary conditions (3.1). In this way, the perturbed anticrack problem reduces to the determination of a potential function f̃ in the upper half-space, decaying at infinity and satisfying the following mixed conditions on the X1X2-plane ∂f̃(x1,x2,x3) ∂x3 ∣∣∣∣∣ x3=0+ = 1 m2t2−m1t1 f0(x1,x2) (x1,x2)∈S ∂2f̃(x1,x2,x3) ∂x23 ∣∣∣∣∣ x3=0+ =0 (x1,x2)∈R2−S (4.21) where f0(x1,x2)= f̃0(r)=−β∗ω̃(r,0)+Ar2+ε= β∗q̃a2 2 √ k1k3 +ε+ ( A− β∗q̃ 4 √ k1k3 ) r2 (4.22) with the following constants β∗ = c2k0−aαmαtα a1 = c1(1+m2)− δ3c44 m1−m2 a2 = −c1(1+m1)+ δ3c44 m1−m2 δ3 =β3− c1c13−c2c33k20 A= β3q0 2k3(2c13+ c33) (4.23) It is known from the potential theory (Kellogg, 1953) that the solution to this problem is represented by the Newton potential of a simple layer distributed over the region S as f̃(x1,x2,x3)= 1 2πc44(m1−m2) ∫∫ S q(ξ1,ξ2)ln (√ (x1− ξ1)2+(x2− ξ2)2+x23+x3 ) dξ1dξ2 (4.24) where the unknown layer density q can be identified as the normal stress σ̃33|S+. Taking con- sideration of the first condition in Eq. (4.21), the following governing two-dimensional singular integral equation (similar to that arising in classical contact mechanics) is obtained H̃ ∫∫ S q(ξ1,ξ2) dξ1dξ2√ (x1− ξ1)2+(x2− ξ2)2 =−f0(x1,x2) (x1,x2)∈S (4.25) with f0 given by Eq. (4.22) and H̃ defined by H̃ = m2t2−m1t1 2πc44(m2−m1) = √ c11c33+ c44 2π √ c44c33 √ ( √ c11c33− c13)( √ c11c33+ c13+2c44) (4.26) Taking a solution to this equation in the form (using Dyson’s and Galin’s theorems) q(x1,x2)= q̃(r)= p̃0a 2− p̃2r2 H̃π2 √ a2−r2 0¬ r a σ33(r,0 ±)=    ± p̃2 3H̃π2 2a2−3r2√ a2−r2 0¬ ra σ3r(r,0 ±)=    β̃r 0¬ r a (4.30) where β̃0 = p̃2 c44( √ c11c33− c13) 3( √ c11c33+ c44) β̃= c44 2 [ 3β̃0− q0 δ̃+(c1− c2)k0√ k1k2 ] δ̃= ( √ c11c33− c13)(2c1c44− δ3) c33c44(t1+ t2) − c1(t1+ t2) (4.31) 5. Analysis of the results and conclusions The analytical results obtained in the previous Section are useful in interpreting the mechanics of fracture initiation at the rim of the rigid inclusion. In view of linear fracture mechanics, two failuremechanisms are possible: mode II (edge-sliding) of fracture deformation characterized by the stress intensity factor KII = lim r→a+ √ 2π(r−a)σ3r(r,0)=− 2β̃0a √ a√ π (5.1) and the possible detachment of the material from the inclusion surface described by the stress intensity coefficients S±I = lim r→a− √ 2π(a−r)σ33(r,0±)=∓ p̃2a √ a 3π √ πH̃ (5.2) These parameters can be used in conjunction with a suitable failure criterion. In conclusion, by taking into account some interior conductivity of the anticrack, we ha- ve pointed out that by letting R0 → ∞ (see Eqs. (4.6) and (4.7)) the present solution with 600 A. Kaczyński, B.Monastyrskyy q̃= q0 (cf. Eq. (4.9)) reduces to that dealingwith the case of a thermally insulated rigid circular inclusion obtained in Kaczyński (2014). Moreover, comparison between thermally conductive and insulated anticracks in a transversely isotropic (in particular, isotropic) space has shown only quantitative changes in the temperature and stress distributions. References 1. Chaudhuri R.A., 2003, Three-dimensional asymptotic stress field in the vicinity of the circumfe- renceof apenny-shapeddiscontinuity, International Journal of Solids and Structures,40, 3787-3805 2. DingH., ChenW., ZhangL., 2006,Elasticity of Transversely IsotropicMaterials, SolidMecha- nics and its Applications, 126, Springer, The Netherlands 3. FabrikantV.I., 1989,Applications of Potential Theory inMechanics: A Selection of NewResults, Kluver Academic Publishers, Dordrecht 4. FabrikantV.I., 1991,MixedBoundaryValue problems ofPotential Theory and theirApplications, Kluver Academic Publishers, Dordrecht 5. 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Podil’chuk Yu.,N., 2001, Exact analytical solutions of three-dimensional static thermoelastic problems for a transversally isotropic body in curvilinear coordinate systems, International Applied Mechanics, 37, 6, 728-761 12. StadnykM.M., 1994,Method for the solution of the three-dimensional thermoelastcity for bodies with thin inclusions,Materials Science, 30, 6, 643-653 13. Stadnyk M.M., 2011, Elastic-ellipsoidal heat-conducting inclusion in a body under the action of a heat flux applied at infinity,Materials Science, 47, 3, 284-293 Manuscript received February 22, 2015; accepted for print October 9, 2015