Appl. Math. Mech. -Engl. Ed. 30(10), 1221 1232 (2009) DOI: 10.1007/s10483-009-1002-x c Shanghai University and Springer-Verlag 2009 Applied Mathematics and Mechanics (English Edition) Interaction etween heat dipole and circular interfacial crack Wan-shen XIAO ( ), Chao XIE ( ), You-wen LIU ( ) (College of Mechanics and Aerospace, Hunan University, Changsha 410082, P. R. China) (Communicated y Xing-ming GUO) Astract The heat dipole consists of a heat source and a heat sink. The prolem of an interfacial crack of a composite containing a circular inclusion under a heat dipole is investigated y using the analytical extension technique, the generalized Liouville theorem, and the Muskhelishvili oundary value theory. Temperature and stress fields are formulated. The effects of the temperature field and the inhomogeneity on the interfacial fracture are analyzed. As a numerical illustration, the thermal stress intensity factors of the interfacial crack are presented for various material cominations and different positions of the heat dipole. The characteristics of the interfacial crack depend on the elasticity, the thermal property of the composite, and the condition of the dipole. thermoelasticity, heat dipole, interfacial crack, circular inclusion, inhomo- Key words geneity Chinese Lirary Classification TB381, O343.7 2000 Mathematics Suject Classification 74R10, 74F05 Nomenclature R, inclusion radius; L, interfacial crack arc; L, interface arc; S +,S, inclusion domain and matrix domain; I, II, material numers for inclusion and matrix; μ j,κ j, elasticity constants of the jth material; ρ, polar radius of heat dipole center location; θ, polar angle of heat dipole center location; r, half span length of heat dipole; z 1,z 2, coordinates of heat source and heat sink; ϕ, angle etween the span line and the x- axis; α, half of the central angle of the crack arc; T ( ), Re( ), temperature field function; real part of a complex variale or a complex function; i, imaginary unit; ( ), conjugate value of a complex parameter; g( ), the real part of its differential is the complex function for temperature; q, heat flux; Q, total heat transfer rate; k t, the coefficient of heat conductivity; σ, normal stress; τ, shear stress; ϕ( ),ψ( ), complex potential functions for stress; Φ( ), Ψ( ), differential function of complex potential for stress; β, heat expansion coefficient; u, v, displacements; Received Sept. 25, 2008 / Revised Sept. 1, 2009 Corresponding author Wan-shen XIAO, Prof., Ph. D., E-mail: xwshndc@126.com, xwshndc@hnu.cn
1222 Wan-shen XIAO, Chao XIE, and You-wen LIU e, f, K, W, δ, S, J, N, O, constants; K j,k jl, stress intensity factors of crack tip; Y ( ), ω( ), Plemelj function; transform function. Introduction The interaction etween inhomogeneity and thermoelasticity has een the topic of considerale research during the past decades [1-2]. Mindlin and Cheng [1] analyzed thermoelastic stress in the semi-infinite solid emedded with an inclusion. Zhu and Muquid [2] presented an analytical solution for the two-dimensional thermoelastic case of multiple interacting inhomogeneities y employing the complex potentials of Muskhelishvili [3]. Recently, more and more researchers focus on the investigation of the thermoelastic prolems with various defects occurring frequently in most engineering structures. The analysis for thermal interfacial crack prolems in dissimilar anisotropic media under the uniform heat flux was provided y Chao and Chang [4] asedonthe Hilert theory and the analytical continuation. With the comination of Stroh s formulation, the conformal mapping, the perturation technique, and the analytical continuation, a general analytical thermoelectroelastic solution for an elliptic piezoelectric inclusion emedded in an infinite piezoelectric matrix was otained y Qin [5]. Xiao and Wei [6] explored the interaction etween a screw dislocation and a circle interfacial crack in a piezoelectric composite under a heat flow y comining the theory of sectional homorphic functions and the Riemann oundary prolem technique. Pham et al. [7] studied the interaction etween a cracked hole and a line crack y using Green s function, the mapping function, and the complex variale method. Hasee et al. [8] solved the prolem of a rigid inclusion interacting with a line crack in an infinite plane sujected to a uniform heat flux y employing the complex variale method along with the rational mapping technique. The temperature field distriution or the heat flux is assumed to e known in advance in all the aforementioned articles. However, in most of the practical cases, only the information aout the heat source is known. Therefore, it is significant to examine the heat source in thermal structure prolems. Chao and Shen [9] solved the prolem of a circular inhomogeneity with a point heat source ased upon the use of the Laurent series expansion of the corresponding complex potentials and the analytical continuation method. Applying the analytical continuation method and the Fourier expansion technique, Chao and Tan [10] otained a solution in a sequential form for the annular prolem of a point heat source. Rahman [11] reduced an axisymmetric contact prolem of a point heat source to an integral equation y means of potential methods. Chao and Chen [12] provided a general analytical solution for an isotropic trimaterial interacted with a point heat source ased on the analytical continuation method. Hasee and Wang [13] formulated the prolems of various holes or a rigid inclusion with deondings emedded in an elastic material sujected to a heat source and a sink y applying the complex theory. However, to the est of our knowledge, the prolem of deonded thermoelastic inclusion interacting with a heat dipole has not een considered. The aim of this paper is to study the interaction etween a circular elastic inclusion and an interfacial crack under a heat dipole. The closed forms of the solutions of the temperature and stress fields are otained with the analytical continuation technique of the complex potential. Numerical curves show that the thermal dipole and the mismatch etween the inclusion and the matrix material convert the shielding and anti-shielding effects on the interfacial crack, and the temperature and the inclusion affect the interfacial crack characteristics. 1 Prolem description As shown in Fig. 1, the medium I with the shear modulus μ 1 occupies the region S + that is the interior of the circle of the radius R, while the medium II with the shear modulus μ 2
Interaction etween heat dipole and circular interfacial crack 1223 occupies the region S that is the exterior of the circle. The interfacial crack L with the end points a and lies along a part of the interface etween two materials. L is the remainder of the interface. The heat dipole consists of a heat source with the thermal field strength q 0 at the point z 1 (z 1 = ρe iθ + re iϕ )andaheatsinkwiththethermal field strength q 0 at the point z 2 (z 2 = ρe iθ re iϕ ). Oviously, the length of the heat dipole arm is 2r. Both the heat source and the heat sink are located in the domain S, namely, outside the interface circle. Let the circle centre e placed at the origin of the complex plane. The circle can e expressed as z = t = Re iθ, where R is the circle radius. For the plane strain prolem, κ =3 4ν, where ν is the Poisson ratio. y z 1 2r z 2 φ II(μ 2, κ 2 ) S _ R ρ I(μ 1, κ 1 ) S + O θ α a L' x L Fig. 1 The deonded circular inclusion composite with a heat dipole 2 Solutions of temperature field Consider a two-dimensional prolem. When the heat exchange reaches a steady condition, the temperature field will not vary with the time t. Thus, the temperature field function T (x, y) is governed y Laplace equation 2 T =0. Therefore, T (x, y) can e expressed y the real part of an analytical complex function as T =Re[g (z)]. Since the Cartesian rectangular coordinate T components of the heat flux vector q x and q y caneexpressedasq x = k t x and q T y = k t y, where k t is the coefficient of the heat conductivity, we have the equations governing the twodimensional heat exchange: q x +iq y = k t g (z), Q = (q x dy q y dx) = k t Im[g (z)]. (1) Here, Q is the total heat transfer rate, and i is the imaginary unit. The ar over an entity represents the conjugate value. Suscripts 1 and 2 are used to denote the quantities defined in S + and S. For the steady state heat conduction prolem, the temperature potential g (z) can e written as g 2 (z) =Q 0ln z z 1 + g 20 z z (z), z S, 2 where Q 0 = q 0 /(2πk 2t ). The function g 20 (z) is holomorphic in S, namely, g 20 (z) =O(1/z) ( z ). While the function g 1 (z) is holomorphic in S+, and it can e expanded into the Taylor series g 1 (z) =a 0 + a k z k, z S +. k=1
1224 Wan-shen XIAO, Chao XIE, and You-wen LIU The connective conditions of the heat transfer rate and the temperature for the present prolem can e expressed as Q + 1 (t) =Q 2 (t), T+ 1 (t) =T 2 Q + 1 (t) =Q 2 (t) =0, t L. (t), t L; Here, the superscripts + and are used to denote the oundary values of the physical quantities as they approach the interface from S + and S, respectively. Using the analytical continuation method of the complex potential, we have Here, g 1 (z) = g 2 (z) = k [ ( 2t X 0 (z)g 1 (z)+q 0 ln z z 1 k 1t + k 2t +ln z z 1 z z 2 z z2 k 1t X 0 (z)g 1 (z)+ k ( 2tQ 0 ln z z 1 +ln z z 1 k 1t + k 2t k 1t + k 2t z z 2 z z2 G 1 (z) = 2 j=1 ( ) G zj (z)+g z j (z) + G (z), G zj (z) =( 1) j+1 Q 0ln(z z j ), X 0 (z j ) G z j (z) =( 1) j Q 0ln(z zj ) X 0 (zj ), G (z) =(z R cos α)d 0 +(z 2 z 1 + z1 z2)q 0, P (z) =c n z n + c n 1 z n 1 + + c 0, )] k 1tD 0 X 0 (z) =(z a) 1/2 (z ) 1/2, D 0 = ā 0, c 1 =0, c 0 =0. Putting z = 0 in Eq. (2) generates the following equation for a 0 : a 0 = 3 Solutions of stress fields, (2) k 1t + k 2t ) k 1tD 0. (3) k 1t + k 2t k [ ( 2t X 0 (0)G 1 (0) + Q 0 ln z )] 1 +ln z 1 k 1t + k 2t z 2 z2 + k 1tā 0. (4) k 1t + k 2t According to Ref. [3], we can express the stress and displacement of thermoelasticity as σ x + σ y =2(Φ(z)+Φ(z)), σ y σ x +2iτ xy =2( zφ (z)+ψ(z)), (5) 2μ(u +iv) =κϕ(z) zϕ (z) ψ(z)+2μβ g (z)dz. (6) Here, Φ(z) =ϕ (z), Ψ(z) =ψ (z), ϕ 2 (z) =A 1 zln(z z 1 )+B 1 ln(z z 1 )+A 2 z ln(z z 2 )+B 2 ln(z z 2 )+ϕ 20 (z), z S, (7) ψ 2 (z) =C 1 ln(z z 1 )+C 2 ln(z z 2 )+ψ 20 (z), z S, (8) where ϕ 20 (z) andψ 20 (z) are holomorphic in S.
Interaction etween heat dipole and circular interfacial crack 1225 The stress field of thermoelasticity must satisfy the single-valued conditions of the force and displacement. Therefore, B 1 C 2μ 2 β 2 k 2t Q 0 1 =0, A 1 = (k 1t + k 2t )(κ 2 +1), (9) B 2 C 2μ 2 β 2 k 2t Q 0 2 =0, A 2 = (k 1t + k 2t )(κ 2 +1). Inserting Eq. (9) into the differentiation of Eqs. (7) and (8), we otain the differential function of stresses: Φ 2 (z) =A 1 ln z z 1 z z 2 +Φ 20 (z), z S, (10) Ψ 2 (z) =Ψ 20 (z), z S, (11) where Φ 20 (z) andψ 20 (z) are the holomorphic functions, and they vanish at infinity. The stress interfacial connecting condition can e expressed as We have [Φ 1 (t)+φ 2 (t)] + =[Φ 1 (t)+φ 2 (t)], t L + L. (12) Φ 1 (z)+φ 2 (z) =G(z)+F, (13) in which F = Φ 1 (0), G(z) =A 1 ln z z 1 + A 1 ln z ( z 2 z z z 2 z z1 + A 2 1 z z2 z 1 z z1 ). (14) The interfacial connecting condition for the displacements can e expressed as κ 1 Φ + 1 μ (t)+ 1 Φ 1 1 μ (t)+2β 1g 1(t) = κ 2 Φ + 2 1 μ (t)+ 1 Φ 2 2 μ (t)+2β 2g 2(t), 2 t L. (15) We otain where Φ + 20 (t) eφ 20 (t) =fg(t)+wi(t)+jf, (16) e = κ 2μ 1 + μ 2, f = (1 + κ 2)μ 1, J = (1 + κ 1)μ 2, μ 1 + κ 1 μ 2 μ 1 + κ 1 μ 2 μ 1 + κ 1 μ 2 I(t) =2β 1 g 1 (t) 2β 2g 2 (t), W = μ 1μ 2. μ 1 + κ 1 μ 2 The general solution of Eq. (16) can e written as [3] It follows that Φ 20 (z) = Y 0(z) 2πi L Φ 20 (z) = fg(z)+wi(z)+jf 1 e fg(t)+wi(t)+jf Y + 0 (t)(t z) dt + R(z)Y 0 (z). (17) ( H1 (z) ) Y 0 (z) 1 e + R(z), (18)
1226 Wan-shen XIAO, Chao XIE, and You-wen LIU where Y 0 (z) =(z a) 1 2 iδ (z ) 1 2 +iδ, δ = ln e 2π, 2 H 1 (z) = (H zj (z)+h z j (z)) + H (z), j=1 H zj (z) = 1 { ( 1) j fa 1 ln(z z j )+ 2W [(β 2 k 1t β 1 k 2t )G zj X 0 (z j ) Y 0 (z j ) k 1t + k 2t } +( 1) j (β 1 β 2 )k 2t Q 0 ln(z z j )], H z j (z) = 1 { [ Y 0 (zj ) ( 1) j fa 1 ln(z zj )+ z ] j z zj + 2W [(β 1 k 2t β 2 k 1t )G z k 1t + k j X 0 (zj ) 2t ( 1) j (β 1 β 2 )k 2t Q 0 ln(z zj }, )] [ H (z) = z R ] 2 (eiα1 +e iα2 ) irδ(e iα1 e iα2 ) [JF +2W ( k 1t D 0 )(β 1 β 2 )] + G f1. Here, G f1 = fa 1 (z 2 z 1 )+ 2W k 1t + k 2t [(β 1 k 2t β 2 k 1t )G +(β 1 β 2 )k 2t Q 0 (z 2 z 1 + z 2 z 1 )]. Sustituting Eq. (18) into Eq. (10) yields the following equation: Φ 2 (z) = JG(z)+WI(z)+JF 1 e Y 0(z) 1 e H 1 (z). (19) Then inserting Eq. (19) into Eq. (13) generates the following equation: Φ 1 (z) = fg(z) WI(z) ff 1 e + Y 0(z) 1 e H 1 (z). (20) Putting z = 0 in Eq. (20) gives Φ 1 (0) = fg(0) WI(0) + fφ 1(0) 1 e 4 Stress intensity factors of the crack tips Take a coordinate transformation + Y 0(0) 1 e H 1 (0). (21) z = ω(z) =iexp(iα)[z ir (R/2) sin(2α)]. (22) The Z-plane is shown in Fig. 2. The stress intensity factors of the interfacial crack tips can e written as [14] K 1 ik 2 =2 2e πδ lim (Z 1 ) 0.5 iδ Φ 2 [ω(z)] on 1. (23) Z 1 Sustituting Eq. (22) into Eq. (23) yields K 1 ik 2 = i2 2exp[(π α)δ iα/2] (1 e)(2r sin α) 1 2 +iδ H 1 (). (24)
Interaction etween heat dipole and circular interfacial crack 1227 Y S + S _ 2α R O 1 X Fig. 2 The new coordinate plane The intensity factor expression of the interfacial crack tip and all the solutions prior to it cannot hold true unless there is no contact stress on the crack surfaces. The non-contacting characteristic etween the crack surfaces means the inequality of the radial displacement difference [ Re e iθ( κ 2 ϕ 2 (z) zϕ 2 (z) ψ 2(z)+2μ 2 β 2 g 2 (z)dz μ 2 κ 1ϕ 1 (z) zϕ 1 (z) ψ 1(z)+2μ 1 β 1 g 1 (z)dz )] 0, θ <α. (25) μ 1 5 Results and discussions The stress intensity factor of the crack tip is normalized y the parameters K10 ik 20 = K 1 ik 2. (26) μ 2 β 2 Q 0 1 2 iδ Consider the special case where the crack is symmetric aout the x-axis. All the discussions elow are ased on the prerequisites α 1 = α 2 = α = π/12, θ =0, ρ/r =2. (I) The variation law of the type I crack intensity factors (a) ϕ =0 For a given ratio ρ/r (ρ >R), as θ =0andϕ =0, the heat source is located closer to the crack than the heat sink is. Assume that κ 1 = κ 2 = 1.8, and the two material components have the same thermal parameter values, i.e., k 1t = k 2t and β 1 = β 2, ut different shear modulus values. For various shear modulus ratios μ 2 /μ 1, the curves of the crack stress intensity factor versus the heat dipole semi-span are depicted in Fig. 3. With the growth of r, the heat source gradually approaches the crack, while the heat sink departs further and further away from the crack. As a result, the anti-shielding effect on the crack tip ecomes strong, i.e., the crack is much easier to develop. This trend is reasonale ecause the heat source heats the materials, and consequently the materials expand and the hole also expands, namely, the interfacial crack enlarges. The closer
1228 Wan-shen XIAO, Chao XIE, and You-wen LIU the heat source is to or the farther the heat sink is from the interfacial crack, the easier the crack grows. Moreover, the smaller the shear modulus ratio μ 2 /μ 1 is, i.e., the softer the matrix is, the stronger the anti-shielding effect ecomes. Suppose that κ 1 = κ 2 =1.8 andμ 2 =2μ 1 (the inclusion is softer than the matrix), and the two materials are identical in the thermal expansion, i.e., β 1 = β 2. The curves of the crack stress intensity factor versus the heat dipole semi-span r are shown in Fig. 4 for various ratios of the thermal conductivity coefficients k 2t /k 1t. When k 2t /k 1t is relatively high (i.e., k 2t /k 1t 1 in this figure), the greater the semi-span r is, in other words, the closer the heat source is to and the farther the heat sink is from the interfacial crack, the stronger the anti-shielding effect on the crack tip ecomes. Therefore, the crack grows easier. Additionally, the greater the thermal conductivity coefficient ratio k 2t /k 1t is (namely, compared with the inclusion, the etter the matrix in the heat conduction is), the stronger the anti-shielding effect on the crack tip ecomes. K 10 1.8 1.6 1.4 μ 2 /μ 1 = 2 μ 2 /μ 1 = 5 μ 2 /μ 1 = 8 μ 2 /μ 1 = 1/5 1.2 φ=0, κ 1 =κ 2 =1.8 1.0 β 1 =β 2, k 1t =k 0.8 2t 0.6 0.4 0.2 10 K 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 k 2t /k 1t = 1 k 2t /k 1t = 2 k 2t /k 1t = 5 k 2t /k 1t = 1/5 φ=0, κ 1 =κ 2 =1.8 μ 2 =2μ 1, β 1 =β 2 Fig. 3 factor K10 semi-span r for various modulus ratios μ 2/μ 1 Fig. 4 factor K10 semi-span r for various thermal conductivity coefficient ratios k 2t/k 1t Under the premise of that κ 1 = κ 2 =1.8, μ 2 =2μ 1 (the inclusion is softer than the matrix), and the two material components have the same thermal conductivity, i.e., k 2t /k 1t =1, Fig. 5 shows the curves of the crack stress intensity factor versus the heat dipole semi-span r for various ratios of the thermal expansion coefficients β 2 /β 1. In this figure, if the ratios of the thermal expansion coefficients are relatively small, e.g., β 2 /β 1 =0.5 orβ 2 /β 1 =1, with the increase of the semi-span r (i.e., the heat source approaches and the heat sink deviates far from the interfacial crack gradually), the anti-shielding effect on the crack tip ecomes stronger (i.e., the crack grows easier). As β 2 /β 1 is relatively great, for instance, β 2 /β 1 =2, the crack tip stress intensity factor increases efore the curve peak point. Then, it drops corresponding to the increase of r. Nevertheless, when β 2 /β 1 = 5, the curve ecomes monotonously decreasing. This means that, when β 2 /β 1 =5, with the growth of the semi-span r, the shielding effect on the crack strengthens. Therefore, it is more difficult for the crack to extend. In addition, the greater the thermal expansion coefficient ratio β 2 /β 1 is, the stronger the shielding effect on the crack is, and the greater the resistance to the fracture extending is. Figure 6 displays the curves of the crack stress intensity factor versus the heat dipole semispan r for various Poisson ratios κ 1 and κ 2 with μ 2 =2μ 1,k 2t /k 1t =1, and β 1 = β 2. Corresponding to the growth of the heat dipole semi-span r, the anti-shielding effect on the crack
Interaction etween heat dipole and circular interfacial crack 1229 K 10 1.6 β 2 /β 1 =1 β 2 /β 1 =2 1.4 β 2 /β 1 =5 1.2 β 2 /β 1 =0.5 1.0 0.8 φ=0, κ 1 =κ 2 =1.8 μ 2 =2μ 1, k 1t =k 2t 0.6 0.4 0.2 K 10 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 κ 2 =1.0, κ 1 =2.2 κ 2 =1.8, κ 1 =1.8 κ 2 =2.0, κ 1 =1.4 κ 2 =2.2, κ 1 =1.0 φ=0, μ 2 =2μ 1 β 1 =β 2, k 1t =k 2t Fig. 5 factor K10 semi-span r for various thermal expansion coefficient ratios β 2/β 1 Fig. 6 factor K10 semi-span r for various Poisson ratios κ 1 and κ 2 tip ecomes strong. Therefore, the crack extends more easily. Besides, the greater the Poisson ratio κ 2 /κ 1 is (namely, the matrix possesses a greater coefficient of the lateral deformation than the inclusion), the stronger the anti-shielding effect on the crack is. () ϕ = π For a given ρ/r (ρ >R), when θ =0andϕ = π, the heat sink is closer to the crack than the heat source is. The curves of the crack stress intensity factor versus the heat dipole semi-span r for various ratios of the thermal conductivity coefficients k 2t /k 1t with κ 1 = κ 2 =1.8, μ 2 =2μ 1,and β 1 = β 2 are plotted in Fig. 7. The growth of the heat dipole semi-span r promotes the shielding effect on the crack tip. Therefore, it deters the crack development. In addition, the greater the thermal conductivity coefficient ratio k 2t /k 1t is (namely, the etter the matrix is than the inclusion in the heat conduction), the stronger the shielding effect of the heat sink on the crack is. Consequently, it is more difficult for the crack to extend. Under the condition that κ 1 = κ 2 =1.8 andμ 2 =2μ 1,k 2t /k 1t =1. The curves of the crack stress intensity factor versus the heat dipole semi-span r for various ratios of the thermal expansion coefficients β 2 /β 1 are illustrated in Fig. 8. As β 2 /β 1 1, corresponding to the growth of the heat dipole semi-span r, the heat sink is closer to ut the heat source is farther from the crack. The shielding effect on the crack tip ecomes strong. Hence, the hindrance to the crack extension ecomes greater. Contrarily, if β 2 /β 1 is relatively great, e.g., β 2 /β 1 5asshownin Fig. 8, the dipole exerts an anti-shielding effect on the crack, and with the increase of r, the anti-shielding effect also increases. Therefore, the crack develops easier. (II) The variation law of the type II crack stress intensity factors when ϕ =0 For a given ratio ρ/r (ρ >R), when θ =0andϕ =0, the heat source is closer to the crack than the heat sinks. Assume that the two material components have the same thermal property (that is, k 1t = k 2t and β 1 = β 2 ), ut different shear moduli. Set κ 1 = κ 2 =1.8. For different definite values of the shear modulus ratios μ 2 /μ 1, the curves of the crack stress intensity factors versus the heat dipole semi-span r are plotted in Fig. 9. With the growth of r, the crack is close to the heat source and leaves the heat sinks. The type II shielding effect on the crack tip then gradually
1230 Wan-shen XIAO, Chao XIE, and You-wen LIU increases. That is, it is more difficult for the crack to extend. Moreover, the smaller the value of the shear modulus ratio is (that is, the softer the matrix material is than the inclusion), the stronger the shielding effect of the type II on the crack is. When κ 1 = κ 2 =1.8, μ 2 =2μ 1 (the inclusion is softer than the matrix material) and the thermal expansion performances of the two material components are the same, i.e., β 1 = β 2. Figure 10 draws the numerical curves of the crack stress intensity factor varying with the heat dipole semi-span r as the thermal conductivity coefficient ratios k 2t /k 1t have various values. When k 2t /k 1t 1, with the increase of r (that is, the heat source is close to the crack and the heat sinks leave the crack), the type II shielding effect of the thermal dipole on the crack tip gradually increases. That is, the crack extension ecomes more difficult. In addition, the greater the thermal conductivity ratio k 2t /k 1t is (i.e., compared with the inclusions, the etter thermal conductivity the matrix material has), the stronger the type II shielding effect on the crack tip is. When k 2t /k 1t =0.2, the thermal dipole with shorter arm generates a shielding effect on the crack tip, while that with a longer arm has an anti-shielding effect on the crack. For the case where κ 1 = κ 2 =1.8,μ 2 =2μ 1 (the inclusion is softer than the matrix material), K 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 k 2t =k 1t =0.5 k 2t =k 1t =1 k 2t =k 1t =5 φ=π, κ 1 =κ 2 =1.8 μ 2 =2μ 1, β 1 =β 2 1.1 K 10 0.2 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 β 2 /β 1 =0.5 β 2 /β 1 =1 β 2 /β 1 =5 β 2 /β 1 =8 φ=π, κ 1 =κ 2 =1.8 μ 2 =2μ 1, k 1t =k 2t Fig. 7 factor K10 semi-span r for various thermal conductivity coefficient ratios k 2t/k 1t Fig. 8 factor K10 semi-span r for various thermal expansion coefficient ratios β 2/β 1 0 1 K 20 1 2 3 4 5 μ 2 /μ 1 = 1/5 μ 2 /μ 1 = 2 μ 2 /μ 1 = 5 μ 2 /μ 1 = 8 φ=0, κ 1 =κ 2 =1.8 β 1 =β 2, k 1t =k 2t K 20 0 1 2 k 2t /k 1t = 1/5 k 2t /k 1t = 1 k 2t /k 1t = 2 3 4 k 2t /k 1t = 5 φ=0, κ 1 =κ 2 =1.8 μ 2 =2μ 1, β 1 =β 2 Fig. 9 factor K20 semi-span r forvariousshearmod- ulus ratios μ 2/μ 1 Fig. 10 factor K20 semi-span r for various thermal conductivity coefficient ratios k 2t/k 1t
Interaction etween heat dipole and circular interfacial crack 1231 and the two material components are identical in the thermal conductivity, i.e., k 2t /k 1t =1, Fig. 11 illustrates the numerical curves of the crack stress intensity factor varying with the heat dipole semi-span r as the thermal expansion coefficient ratios β 2 /β 1 have various values. When the thermal expansion coefficient ratio is relatively small, for example, β 2 /β 1 =0.5 or β 2 /β 1 = 1 in the figure, with the increase of r, the type II shielding effect on the crack tip gradually increases. Namely, the crack development is more difficult when the crack is located nearer the heat source and farther from the heat sink. When the thermal expansion coefficient ratio is relatively large, e.g., β 2 /β 1 = 5 in the figure, with the increase of the thermal dipole semi-span r, the shielding effect on the crack enhances initially, and then weakens. K 20 0.5 0.5 1.0 1.5 2.0 2.5 3.0 3.5 β 2 /β 1 =0.5 β 2 /β 1 =1 β 2 /β 1 =2 β 2 /β 1 =5 φ=0, κ 1 =κ 2 =1.8 μ 2 =2μ 1, k 1t =k 2t Fig. 11 factor K 20 versustheheatdipolesemi-spanr for various thermal expansion coefficient ratios β 2/β 1 6 Conclusions For the case where the heat source end of the heat dipole in the matrix is located near the crack, the softer the matrix compared with the inclusion is, the greater the thermal conductivity coefficient of the matrix is, and the smaller the thermal expansion coefficient of the matrix is or the greater the Poisson ratio of the matrix is, the stronger the anti-shielding effect on the crack is. Hence, it is the easier for the type I interfacial crack to develop. Contrarily, for the case where the heat sink end of the heat dipole is located near the crack, all these trends are reverse. In addition, oth the anti-shielding effects and the shielding effects on the type I interfacial crack ecome more and more ovious with the increase of the heat dipole span. However, compared with the inclusion, the softer the matrix is, the etter the thermal conductivity is, and the smaller the thermal expansion coefficient is, the stronger the type II shielding effect of the thermal dipole on the interfacial crack is. That is, it is more difficult for the interfacial crack to expand. Acknowledgements The support of the Natural Science Foundation of Hunan Province of China (No. 05JJ30140) is gratefully acknowledged. References [1] Mindlin, R. D. and Cheng, D. H. Thermoelastic stress in the semi-infinite solid. Journal of Applied Physics 21(9), 931 933 (1950) [2] Zhu, Z. H. and Muguid, S. A. On the thermoelastic stresses of multiple interacting inhomogeneities. International Journal of Solids and Structures 37(16), 2313 2330 (2000)
1232 Wan-shen XIAO, Chao XIE, and You-wen LIU [3] Muskhelishvili, N. I. Some Basic Prolems of the Mathematical Theory of Elasticity, Noordhoff, Leyden (1975) [4] Chao, C. K. and Chang, R. C. Thermal interface crack prolems in dissimilar anisotropic media. Journal of Applied Physics 72(7), 2598 2604 (1992) [5] Qin, Q. H. Thermoelectroelastic solution for elliptic inclusions and application to crack-inclusion prolems. Applied Mathematical Modelling 25(1), 1 23 (2000) [6] Xiao, W. S. and Wei, G. Interaction etween screw dislocation and circular crack under uniform heat flux (in Chinese). Journal of Mechanical Strength 29(5), 779 783 (2007) [7] Pham, C. V., Hasee, N., Wang, X. F., and Saito, T. Interaction etween a cracked hole and a line crack under uniform heat flux. International Journal of Fracture 13(4), 367 384 (2005) [8] Hasee, N., Wang, X. F., Saito, T., and Sheng, W. Interaction etween a rigid inclusion and a line crack under uniform heat flux. International Journal of Solids and Structures 44(7-8), 2426 2441 (2007) [9] Chao, C. K. and Shen, M. H. On onded circular inclusions in plane thermoelasticity. Journal of Applied Mechanics 64(4), 1000 1004 (1997) [10] Chao, C. K. and Tan, C. J. On the general solutions for annular prolems with a point heat source. Journal of Applied Mechanics 67(3), 511 518 (2000) [11] Rahman, M. The axisymmetric contact prolem of thermoelasticity in the presence of an internal heat source. International Journal of Engineering Science 41(16), 1899 1911 (2003) [12] Chao, C. K. and Chen, F. M. Thermal stresses in an isotropic trimaterial interacted with a pair of point heat source and heat sink. International Journal of Solids and Structures 41(22-23), 6233 6247 (2004) [13] Hasee, N. and Wang, X. F. Complex variale method for thermal stress prolem. Journal of Thermal Stresses 28(6-7), 595 648 (2005) [14] Sih, G. C., Raris, P. C., and Erdogan, F. Crack-tip stress factors for plane extension and plane ending prolem. Journal of Applied Mechanics 29(1), 306 312 (1962)