Application of thermo-elastic dislocation on a cracked layer under temperature field

The effect of steady-state thermal loading on a cracked layer is investigated. A Volterra type thermo-elastic dislocation is introduced in a layer which is free of traction on the boundaries. The assumptions of quasi-static, steady-state condition are employing and the uncoupled theory of thermo-elasticity is considered. The Fourier transformation is utilized to obtain temperature distribution and stress fields in the layer containing dislocation. The temperature field is also derived in the layer with specified temperature at the boundaries and in the absence of any defects. By means of the distributed dislocation technique, the dislocation solution is introduced into the layer to derive integral equations for dislocation density functions on the surfaces of cracks. These equations are Cauchy singular and are solved numerically. The solutions are employed to determine stress intensity factors (SIFs) for cracks in both cases of the impermeable and partially permeable heat flux. Introduction Structures containing defects are vulnerable to thermal loading. The mutual effects of heat flux and thermal stress on interacting cracks may induce excessive SIFs resulting in the instability of cracks and the failure of structures. The stress analysis of cracked elastic bodies subjected to thermal loads was carried out by several researchers. Some investigations regarding uncoupled thermal analysis of half-planes and layers containing cracks relevant to the present study, are enumerated here. A halfplane weakened by an insulated crack under uniform heat flow was solved by Sekine [1] and modes I and II SIFs were determined for a crack with arbitrary orientation. Nied [2] studied the effects of thermal shocks in a strip weakened by an edge crack. The strip was insulated at a boundary and cooled by surface convection at the other boundary. Lam et al. [3] analyzed mixed mode fracture of cracked strips with varying crack surface heat conductivity under uniform heat flow. Transient thermal stress in a strip with an edge crack was the subject of investigation by Rizk and Radwan [4].The elastic strip was insulated at one face and cooled suddenly on the other boundary. A strip containing a crack perpendicular to the boundary under sudden surface heating was solved by Rizk [5]. Jin and Noda [6] considered a graded half-plane with exponentially varying material properties having an edge crack. The medium was under steady heat flux and thermal SIF for various material constants was determined. Liu and Kardomateas [7] modeled an insulated crack in an anisotropic half-plane under a uniform heat flux by a continuous distribution of dislocations to determine thermal SIFs. Their solution was based on the formulations derived by Clements [8] and Sturla and Barber [9] in conjunction with the image method. The problem of two periodic edge cracks in an isotropic elastic strip located symmetrically along the free boundaries and quenched by a ramp function temperature change was investigated by Rizk [10]. In the present article, a layer with free boundaries is considered. A Volterra type thermo-elastic dislocation is introduced and the temperature distribution and stress fields are derived in the layer under the assumptions of quasi-static, steady-state employing the uncoupled theory of thermoelasticity. The stress components and heat flux are Cauchy singular at dislocation location. The distributed dislocation technique is utilized to perform a set of integral equations for the layer weakened by multiple cracks subjected to general temperature distribution at the boundaries. These equations are Cauchy singular. Only the case of the complete opening of cracks is studied. A crack surface may be partially heat permeable. The integral equations are solved numerically and SIFs are determined. Several examples are solved for layers with cracks having different geometries and the effect of the ratio of heat permeability on SIFs is studied. Solution of thermo-elastic dislocations. An elastic isotropic layer which is free at the boundaries is considered, Fig.1. The layer is stress free at the ambient temperature . A thermo-elastic dislocation is located at a point with coordinates ( ).The dislocation line is extended in the positive direction of the x-axis. In the uncoupled theory of thermo-elasticity, utilizing Fourier’s law of heat conduction and under steady-state situation, the temperature field is governed by the Laplace equation