Interface crack between different orthotropic media under uniform heat flow

In this paper, plane thermo-elastic solutions are presented for the problem of a crack in two bonded homogeneous orthotropic media with a graded interfacial zone. The graded interfacial zone is treated as a nonhomogeneous interlayer having spatially varying thermo-elastic moduli between dissimilar, homogeneous orthotropic half-planes, which is assumed to vary exponentially in the direction perpendicular to the crack surface. Using singular integral equation method, the mixed boundary value conditions with respect to the temperature field and those with respect to the stress field are reduced to a system of singular integral equations and solved numerically. Numerical results are obtained to show the influence of non-homogeneity parameters of the material thermo-elastic properties, the orthotropy parameters and the dimensionless thermal resistance on the temperature distribution and the thermal stress intensity factors.

transient-temperature field in a ceramic/metal functionally graded plate. In addition, assuming the surfaces of the crack are insulated, thermal stresses around a crack in the interfacial layer between two dissimilar elastic half-planes are studied by Itou (2004). With the introduction of the thermal resistance concept, the thermal stress intensity factors for the interface crack between functionally graded layered structures under the thermal loading are investigated by . Zhou and Lee (2011) studied the thermal fracture problem of a functionally graded coating-substrate structure of finite thickness with a partially insulated interface crack subjected to thermal-mechanical supply. Chen (2005) obtained the thermal stress intensity factors (TSIFS) of a graded orthotropic coating-substrate structure with an interface crack. Zhou et al. (2010) considered the thermal response of an orthotropic functionally graded coating-substrate structure with a partially insulated interface crack.
Using mesh-free model, Dai et al. (2005) studied the active shape control as well as the dynamic response repression of the functionally graded material (FGM) plate containing distributed piezoelectric sensors and actuators. Natarajan et al. (2011) considered the linear free flexural vibration of cracked functionally graded material plates by using the extended finite element method. Using extended finite element method, fatigue crack growth simulations of bi-material interfacial cracks have been considered under thermoelastic loading (Pathak et al. 2013). Using element free Galerkin method, Pathak et al. (2014) studied quasi-static fatigue crack growth simulations of homogeneous and bimaterial interfacial cracks under mechanical as well as thermo-elastic load.
Layered FGM structure are very import in practical engineering (Sofiyev and Avcar 2010;Sofiyev et al. 2012;. The research of thermal elastic crack problem in layered structure is helpful for the design and application of functionally graded materials. This paper explores the thermal-mechanical response of layered and graded structures using the integral equation approach. The analytical results of the cracked layered material systems with the material properties in the graded coating varying as an exponential function has been obtained by using the integral transform technique. The surface of the crack is assumed to be part of the thermal insulation. The temperature distributions along the crack line are presented. The TSIFS under thermo-mechanical loadings are obtained, which is very important for the designing of layered orthotropic media.

Problem formulation
As shown in Fig. 1, the problem under consideration consists of a functionally graded orthotropic strip (FGOS) of thickness h bonded to two homogeneous semi-infinite orthotropic media with a partially insulated interface crack of length 2c along the x-axis is considered. The subscript j(j = 1, 2, 3) indicates the FGOS and two semi-infinite orthotropic media respectively. The remaining thermo-mechanical properties depend on the y-coordinate only and are modeled by an exponential function y are the thermal conductivities for the homogeneous orthotropic substrate II, and δ is an arbitrary nonzero constant.
The temperature satisfies Substituting Eqs.
(1) and (2) into the Eq. (3), the heat equation can be given by where k xy0 = k (2) x /k (2) y . The heat flux components are written as We define the following dimensionless quantities ij , G ij , G

,C
ij ,G

,C
ij ,G where α 0 and E 0 are the typical values of the coefficient of linear thermal expansion and the Young's modulus of elasticity for the homogeneous orthotropic substrate, respectively. But for simplicity, in what follows, the bar appearing with the dimensionless quantities is omitted. The Duhamel-Neumann constitutive equations for the plane thermo-elastic problem are given by Nowinski (1978) in which The elastic stiffness coefficients and the coefficients of the linear thermal expansion in dimensionless form are modeled to take the following forms where superscripts 1, 2 refer to the FGOS and the homogeneous orthotropic substrate II, respectively, β and γ are graded parameters. The properties of material 3 can be found in Eq. (11) when y is taken as h. In Eq. (11), elastic stiffness coefficients in dimensionless form can be represented by the Young's moduli and the Poisson's ratios as where ν ij are the Poisson's ratios and assumed to be constant. E (2) xx and E (2) yy are Young's moduli for the homogeneous orthotropic substrate II, respectively.
Substituting Eq. (9) into the equations of equilibrium for the forces reduces these equations to the forms (9) ∂T 2 ∂y

Boundary conditions
The temperature filed can be provided using the following boundary condition where Bi = 1/k (1) y (0)/R c is dimensionless thermal resistance through the crack region. R c is the thermal resistance through the crack region.
The boundary conditions of the stress and displacement field can be given by

Heat conduction problem
By using Fourier transform, the solutions of Eqs. (4) and (5) are given by where M k (ω)(k = 1 − 6) can be found in "Appendix 1". s k , p k and o k are the roots of the characteristic polynomials, which can be given by

Introducing the unknown density function
From (17), we obtain where the kernel H (x, u) can be found in "Appendix 1".

Thermal stress analysis
By using the standard Fourier transforms to Eqs. (13)-(15), following results for the displacement fields for the FGOS and two homogeneous orthotropic media are obtained are given in "Appendix 1". m j (j = 1 − 4) and n j (j = 1 − 4) are the roots of the characteristic polynomials, which can be given by where

Substituting Eqs. (24)-(26) into Eqs. (18)-(19), we obtain
where K ij (x, u)(i, j = 1, 2), ω 1 (x) T , ω 2 (x) T are given in "Appendix 2". The singular integral Eq. (31) are solved numerically with the unknown density functions R 1 (u) and R 2 (u) having the following form Once R 1 (u) and R 2 (u) have been determined, the thermal stress intensity factors ahead of the crack tip can be defined and calculated as follows

Numerical results and discussion
In this paper, the orthotropy and non-homogeneity parameters of Tyrannohex can be found in Ootao and Tanigawa (2005). The material properties can be given by In the presented results the values of the thermal stress intensity factors are normalized by k 0 = E 2 Q 0 α 2 √ c/k (2) y . The crack is located along the interval −1 ≤ x ≤ 1. Figure 2a, b show the effects of the thermal conductivity parameter δ on the crack surface temperature when Bi = 0.1 and Bi = 0.5, respectively. From Fig. 2a, b, it can be found that the temperature jump across the crack surfaces increases with an decrease of the absolute values of δ. At the other hand, for smaller value of Bi, the temperature will become more pronounced. As expected, the temperature jump across the crack becomes more pronounced as the crack surfaces become more insulated, that is, as Bi decreases. (31) GPa, E yy = 87 GPa, ν xy = 0.15, ν yx = 0.09667, α xx = 0.32 × 10 −5 / • C, α yy = 0.32 × 10 −5 / • C, k x = 2.81 W/m • C, k y = 3.08 W/m • C Figure 3a, b show the effects of the thermal conductivity parameter δ and k xy0 on the mode I and k xy0 = 0.5II thermal stress intensity factors. It can be found that the mode I thermal stress intensity factors increases with an increase of the thermal conductivity parameter δ for either or k xy0 = 2.0; while increases with an increase of k xy0 for both δ = −1.0 and δ = 1.0. And the values of mode II thermal stress intensity factors decreases with the increasing of the thermal conductivity parameter δ regardless of the value of k xy0 . Meanwhile, the values of mode II thermal stress intensity factors decreases with the increasing of an increase of k xy0 regardless of the value o α (2) xx f δ. Figure 4a, b illustrate the effects of the stiffness parameter β and E (2) xx on the mode I and II thermal stress intensity factors. It can be seen that the mode I thermal stress intensity factors increases with a decrease of the stiffness parameter β for both E (2) xx = 0.5 and a b Fig. 2 Influences of thermal conductivity parameter δ on the normalized crack surfaces and crack extend line y = 0 temperatures T (x, 0 + )/T 0 and T (x, 0 − )/T 0 , T 0 = Q 0 c/k (2) y , h = 1.0, k xy0 = 2.0, a Bi = 0.1, b Bi = 0.5 E (2) xx = 2.0; while increases with an increase of E (2) xx regardless of the value of the stiffness parameter β. For the mode II thermal stress intensity factors, the contrary is the case. Figure 5a, b show the II effects of the thermal expansion parameter γ and on the mode I and II thermal stress intensity factors. It may be obtained that the absolute values of both mode I and mode II thermal stress intensity factors increases with an increase of the thermal expansion parameter γ for either k xy0 = 0.5 or k xy0 = 2.0; and the absolute values of both mode I and mode II thermal stress intensity factors increases with an increase of α (2) xx . a b Fig. 3 Influences of the thermal conductivity parameter δ and k xy0 on the normalized thermal stress intensity factors, h = 1.0. a mode I . b mode II Figure 6a, b illustrate the effects of different thickness of functionally graded orthotropic strip on the mode I and II thermal stress intensity factors when δ = −1.0 and δ = 1.0, respectively. We can see that the mode I and thermal stress intensity factors increase or decrease with the increasing of h, and then reach a steady value.