Thermoelastic axisymmetric problem for a two-layer cylinder

Abstract

The majority of devices and units of microelectronics are multilayer structures made of materials with differing coefficients of thermal expansion and elastic constants. Thermal stresses which arise in such systems due to temperature changes when manufactured or in operation may result in a breakdown, or plastic deformation or in a change of the physical properties of materials. At the same time, due to adopted assumptions the existing design models do not describe the stressed states in real systems of finite dimensions. The designs in [1–3] are obtained on the basis of the engineering theory of beams, and in [4, 5] the obtained solution was for the infinite strip in a half-space. In the present article a right circular cylinder of radius R was used as a mathematical model which was cut by the plane z = 0 into two layers of thickness H or H* (Fig. 1). In our considerations the quantities referring to the layer 2 are distinguished by an asterisk. The cylinder deformation problem due to the temperature lowering from t₁ to T₂ was solved within the framework of the linear theory of thermoelasticity. It was assumed that the material of each layer is homogeneous and isotropic, that the temperature is independent of the coordinates, and that the coefficients of thermal expansion α and α* are independent of T. Two formulations are analyzed.