Abstract

The problem of two collinear cracks in an orthotropic solid under antisymmetrical linear heat flow is investigated. It is assumed that there exists thermal resistance to heat conduction through the crack region. Applying the Fourier transform, the thermal coupling partial differential equations are transformed to dual integral equations and then to singular integral equations. The crack-tip thermoelastic fields including the jumps of temperature and elastic displacements on the cracks and the mode II stress intensity factors are obtained explicitly. Numerical results show the effects of the geometries of the cracks and the dimensionless thermal resistance on the temperature change and the mode II stress intensity factors. Also, FEM solutions for the stress intensity factor are used to compare with the solutions obtained using the method. It is revealed that the friction in closed crack surface region should be considered in analyzing the stress intensity factor .

1. Introduction

In engineering problems, the thermoelastic analysis for a cracked material has attracted much interest [1–5]. The thermal stress analysis for a cracked isotropic or anisotropic material has attracted much attention. A large number of related papers have been published to address fracture behaviors of isotropic and anisotropic solid. For example, the singularity of the stress fields near the crack-tip with a specified temperature or heat flux loading is studied by Sih [6]. The stress intensity factors of a central crack in an orthotropic material under a uniform heat flow are given by Tsai [7]. The thermoelasticity problem of two collinear cracks embedded in an orthotropic solid has been considered by Chen and Zhang [8]. By using the -integral obtained from the finite element solutions, the stress intensity factor has been computed by Wilson and Yu [9]. Two alternative approaches for analyzing the nonlinear interaction between two equal-length collinear cracks subjected to remote tensile stress on infinity are developed by Chang and Kotousov [10]. By using a two-dimensional dual boundary element method, the stress intensity factors of a cracked isotropic material under the transient thermoelastic loadings have been calculated by Prasad et al. [11]. The diffraction of plane temperature-step waves by a crack in an orthotropic thermoelastic solid has been investigated by Brock [12]. The steady-state thermoelasticity problem of a cracked fiber-reinforced slab under a state of generalized plane deformation is studied [13]. Using the hyperbolic heat conduction theory and the dual-phase-lag heat conduction model, the transient temperature and thermal stresses around a partially insulated crack in a thermoelastic strip under a temperature impact and the transient temperature field around a partially insulated crack in a half plane are obtained by Hu and Chen [14, 15]. The thermal-medium crack model proposed by Zhong and Lee [16] is applied to investigate the problem of a penny-shaped crack in an infinite isotropic material [17]. Development of a unified model for the steady-state operation of single-phase natural circulation loops is made by Basu et al. [18]. An unsymmetrical end-notched flexure test is described and its suitability for interfacial fracture toughness testing is evaluated [19]. An infinite functionally graded medium with a partially insulated crack subjected to a steady-state heat flux away from the crack region as well as mechanical crack surface stresses is considered [20]. The influence of an axisymmetric partially insulated mixed-mode crack on the coupled response of a functionally graded magnetoelectroelastic material subjected to thermal loading is investigated [21]. The elastic-static problem of a partially insulated axisymmetric crack embedded in a graded coating bonded to a homogeneous substrate subjected to thermal loading is considered [22]. In order to investigate the crack problems in elastic material, many mathematical methods have been widely used such as the integral equation method [23], the linear sampling method [24], and the Trefftz method [25].

In the abovementioned works and other works [26–32], the partially insulated boundary condition is mainly used and the closed-form solutions have been obtained for the thermoelastic field around the penny-shaped crack under the loading of uniform heat flow.

Under the consideration of the antisymmetrical linear heat flow, the partially insulated boundary condition is applied to address the problem of two collinear cracks in an orthotropic solid and FEM solutions for the stress intensity factor are used to investigate the accuracy of this methodology in this paper. Applying the Fourier transform technique and integral equation methods, the crack-tip thermoelastic fields involving the jumps of temperature, the elastic displacements across the cracks, and the mode II stress intensity factors are given in explicit forms. Numerical results show the heat flux to the crack surfaces and the mode II stress intensity factors are dependent on the geometries of cracks and the dimensionless thermal resistance. The thermal resistance in the heat conduction through the crack region is of much importance in analyzing the thermoelastic problem of a cracked material with a thermal loading. Furthermore, FEM solutions for the stress intensity factor are used to compare with the solutions obtained using the method. It is revealed the friction in closed crack surface region has an effect on analyzing the stress intensity factor and should be considered.

2. Statement of the Problem

As shown in Figure 1, two collinear cracks for an infinite transversely orthotropic material are embedded in an orthotropic solid. We use Cartesian coordinates system and suppose that the two collinear cracks are situated at the segment of . With respect to the state of plane stress, the following constitute equations are where and denote the components of elastic displacement, stands for the temperature change, and are the Poisson ratios, and are Young’s moduli, is the shear modulus, and and are the coefficients of linear expansion. Applying the following equations:we haveFurthermore, applying thermal equilibrium equation leads to where and are the thermal conductivities of a cracked orthotropic material. From the theory of Fourier heat conduction, one has Due to the antisymmetry of the heat flux acting on two collinear cracks, the thermal field in the region is only studied. With application of the crack-face boundary conditions, one hasFor the thermal loading, the antisymmetrical linear heat flow is acting on the crack surfaces; namely, where is the prescribed constant, whereConsidering the thermal resistance in the heat conduction through the crack region, the thermoelastic boundary conditions on the crack surfaces are written aswhere denotes the heat flux to the crack surfaces. The limiting value or represents perfect conduction or perfect insulation on the crack surfaces. In addition, the continuity of elastic displacement and temperature on the crack-free parts of -axis yields

Figure 1 

Two collinear cracks in an orthotropic material under antisymmetrical linear heat flow.

3. Solution Procedure

3.1. Temperature Field

The temperature field is not dependent on the elastic strain; the temperature field can be solved firstly. By applying the Fourier transform technique, the solution of (4) can be defined as is unknown and will be solved. for and for . Hereafter the superscripts and − denote the physical quantities of the upper (i.e., ) and lower (i.e., ) region, respectively. From (5), we obtainUsing (8) and the second relation in (13), we obtainUsing the boundary conditions (7), one hasApplication of the first relation in (13) leads toIn order to solve (17) and (18), we give a definition of the auxiliary function Applying the inverse Fourier transform, one obtainsSubstituting (20) into (17) leads to From the known result [33] Equation (21) can be rewritten as Then, introducing , , , , and , (23) can be written as By virtue of the singular integral equation theory, we can obtain the solution of (21); namely, where is the constant determined by By some computations, we can get . Furthermore, applying (19) and (25) yields Substituting (25) into (27), we get the temperature change on the cracks as When , the temperature change on the cracks of a single crack of length is given as

3.2. Elastic Field

In what follows, we pay an attention to the solutions of the partial differential equations (3). and are written as two parts and correspond to the general solutions under . and are the particular solutions under a thermal flux loading. With application of the Fourier transform, and can be expressed as are unknown to be solved. are the roots of equation as follows: where can be calculated as follows: In addition, and are chosen as Substituting (34) into (3) leads toFrom (31) and (34), the components of stress can be written as It is seen that the elastic field was only induced by the loading . We only consider the antisymmetrical linear heat flow in this paper, and it follows thatFrom (37) and (39), one arrives at Utilizing the second relation of (6) and (12) yields To solve the dual integral equations, we introduce a new auxiliary function . By virtue of theory of the inverse Fourier transform, one gets Making use of (40) and (44) yields Application of (42) and (45) leads to with With application of the inverse Fourier transform and (28), one obtains Recalling the known result [34] Inserting (28) into (48), one gets where From (46) and (50), we can get Considering the following condition: By further calculations, one obtains With application of (43), one has Substituting (52) into (55), we have where and and are the complete elliptic integrals of the first and second kind, respectively. and are the incomplete elliptic integrals of the first and the second kinds with Applying (38) and (45), the shearing stresses near the crack-tips are expressed as Substituting (50) and (52) into (58), one has