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.

In this study, the variational method and numerical simulation technique were used to solve the problem of bimodular functionally graded thin plates under large deformation. During the application of the variational method, the functional was established on the elastic strain energy of the plate while the variation in the functional was realized by changing undetermined coefficients in the functional. As a result, the classical Ritz method was adopted to obtain the important relationship between load and maximum deflection that is of great concern in engineering design. At the same time, the numerical simulation technique was also utilized by applying the software ABAQUS6.14.4, in which the bimodular effect and functionally graded properties of the materials were simulated by subareas in tension and compression, as well as the layering along the direction of plate thickness, respectively. This study indicates that the numerical simulation results agree with those from the variational solution, by comparing the maximum deflection of the plate, which verifies the validity of the variational solution obtained. The results presented in this study are helpful for the refined analysis and optimization design of flexible structures, which are composed of bimodular functionally graded materials, while the structure is under large deformation.

The biparametric perturbation method is applied to solve the improved Föppl–von Kármán equation, in which the improvements of equations come from two different aspects: the first aspect concerns materials, and the other is from deformation. The material considered in this study has bimodular functionally graded properties in comparison with the traditional materials commonly used in classical Föppl–von Kármán equations. At the same time, the consideration for deformation deals with not only the large deflection as indicated in classical Föppl–von Kármán equations, but also the larger rotation angle, which is incorporated by adopting the precise curvature formulas but not the simple second-order derivative term of the deflection. To fully demonstrate the effectiveness of the biparametric perturbation method proposed, two sets of parameter combinations, one being a material parameter with central defection and the other being a material parameter with load, are used for the solution of the improved Föppl–von Kármán equations. Results indicate that not only the two sets of solutions from different parameter combinations are consistent, but also they may be reduced to the single-parameter perturbation solution obtained in our previous study. The successful application of the biparametric perturbation method provides new ideas for solving similar nonlinear differential equations.