Solution of the Thermoelastic Problem for a Two-Dimensional Curved Beam with Bimodular Effects

Abstract

In classical thermoelasticity, the bimodular effect of materials is rarely considered. However, all materials will present, in essence, different properties in tension and compression, more or less. The bimodular effect is generally ignored only for simple analysis. In this study, we theoretically analyze a two-dimensional curved beam with a bimodular effect and under mechanical and thermal loads. We first establish a simplified model on a subarea in tension and compression. On the basis of this model, we adopt the Duhamel similarity theorem to change the initial thermoelastic problem as an elasticity problem without the thermal effect. The superposition of the special solution and supplement solution of the Lamé displacement equation enables us to satisfy the boundary conditions and stress continuity conditions of the bimodular curved beam, thus obtaining a two-dimensional thermoelastic solution. The results indicate that the solution obtained can reduce to bimodular curved beam problems without thermal loads and to the classical Golovin solution. In addition, the bimodular effect on thermal stresses is discussed under linear and non-linear temperature rise modes. Specially, when the compressive modulus is far greater than the tensile modulus, a large compressive stress will occur at the inner edge of the curved beam, which should be paid with more attention in the design of the curved beams in a thermal environment.

The Thermal Stress Problem of Bimodular Curved Beams under the Action of End-Side Concentrated Shear Force

“…For the realization of the tension-compression subarea, we needed to determine the position of the unknown neutral layer of the beam under pure bending first. The solution was obtained in our previous study [31], which gave:…”

“…Following this demand, some scholars have carried out research on bimodular beams in a thermal environment [28][29][30][31]. Wen et al [28] first obtained a two-dimensional thermoelastic solution of a bimodular beam under thermal and mechanical loads, where the Duhamel similarity theorem was used to transform the thermoelastic problem into a pure elastic problem.…”

“…More recently, He et al [31] first investigated a bimodular curved bar under pure bending in a thermal environment; the application of this problem may be easily found in mechanical engineering. For example, if a certain portion between two adjacent crosssections of a ring is cut out (see Figure 2a), joining the ends of the ring again by welding or another means gives a ring with an initial stress, that is, there is stress in the ring in the absence of an external force.…”

“…The bimodular effect on stress distribution is discussed in Section 5, and Section 6 contains the concluding remarks. In our previous study aiming at the pure bending problem [31], it was found that the thermoelastic plane stress problem with a bimodular effect can also be transformed into a purely elastic problem under known body forces and known surface forces, which is the familiar Duhamel similarity theorem. However, once the beam is no longer under pure bending but under transverse bending, the axisymmetric stress state in pure bending is no longer maintained, as shown in Figure 3.…”

Dynamic Analysis of Sigmoid Bidirectional FG Microbeams under Moving Load and Thermal Load: Analytical Laplace Solution

“…Based on generalized elasticity theory and modified couple stress theory [62], the strain ε, Cauchy stress σ B , symmetric curvature χ, and the couple stress m are given by [63,64]:…”