| Thermoelastic material with voids is one of practical utility materials in both structural and functional forms, which is common in various types of geological, biological and synthetic materials. Porous material is extensively applied in many engineering fields due to their advantageous properties, such as low relative density, high specific strength, light weight and thermal insulation. During the past half century, the research on the theories of thermoelastic materials with voids has attracted interests of many scholars, but most results were obtained based on the linear hypothesis and limited to the frequency domain. However, solid mechanics problems in engineering and science are essentially nonlinear. Therefore, the study on the nonlinear static and dynamical characteristics of thermoelastic materials and structures with voids not only has the academic significance, but also has the reference value in engineering departments.In this dissertation, the first part is the research background, including the research purpose and significance, the progress, difficulty and defect of the research for thermoelastic materials and structures with voids. Based on that, theoretical analyses and numerical simulations of static and dynamical problems for thermoelastic materials and structures with voids under the finite deformation are studied systematically according to the theory of thermoelastic materials with voids. New theoretical and numerical results are obtained. The main contents are summarized as follows:1. The generalized variational principles and relevant mathematical models of thermoelastic solids with voids under the finite deformation are presented. Based on continuous mechanics, fundamental equations and functional for thermoelastic solids with voids under the finite deformation are given. Then the relevant generalized Hamilton variational principles and mathematical models of thermoelastic solids with voids are derived. Furthermore, based on the Kirchhoff-Love hypothesis and introducing the moments caused by the change of void volume and temperature, the Hamilton variational principle is extended to the thermoelastic structures with voids under the finite deformation. A complete nonlinear model of thermoelastic Karman-type plates with voids is presented, in which the influences of the middle plane force and inertia, as well as rotation inertia have been considered. At the same time, a generalized nonlinear theory of thermoelastic beams with voids is presented, in which the influences of the axial force, neutral layer and rotation inertia are all considered. These models are very important for solving nonlinear problems of thermoelastic beams and plates with voids.2. Numerical examples of the nonlinear mathematical model of thermoelastic beams with voids are presented, and their nonlinear mechanical characteristics are studied. (1) For thermoelastic beams with voids with simply supported edges, a truncated nonlinear system is obtained by adopting the Galerkin averaging method to simplify the nonlinear mathematical model. Then the Runge-Kutta method with varying time-step is adopted to calculate the truncated system numerically. The dynamical characteristics of the system are investigated and compared for four kinds of beams, namely, thermoelastic beams with voids (TEVB), thermoelastic beams (TEB), elastic beams with voids (EVB) and elastic beams (EB). The influences of parameters are all investigated, and some valuable conclusions are obtained. (2) For thermoelastic beams with voids with clamped edges, the differential quadrature method (DQM) is used to discrete the nonlinear system on the spatial domain. Then, the Newton-Raphson method and Runge-Kutta method are adopted to calculate the static and dynamical systems, respectively. The mechanical characteristics of the system are investigated for aforesaid four kinds of beams, and the influences of parameters are all investigated. (3) The dynamical characteristics of thermoelastic beams with voids subjected to aerodynamic load are studied. It shows that it is easier to occur unbounded instability for a thermoelatic beam with voids subjected to the aerodynamic load. 3. From the complete nonlinear model of thermoelastic Karman-type plates with voids presented in this paper, nonlinear mechanical characteristics of plates are studied. For rectangular plates with simply supported edges, the Galerkin averaging method is adopted to simplify the nonlinear mathematical model. Then the Runge-Kutta method with varying time-step is adopted to calculate the truncated system numerically. The dynamical characteristics of the system, such as time history curves, are obtained. The numerical results of truncated systems with 1-order and 2-order are compared. The dynamical characteristics of the system for four kinds of plates, namely, thermoelastic plates with voids (TEVP), thermoelastic plates (TEP), elastic plates with voids (EVP) and elastic plates (EP), are investigated in detail, and the influences of parameters are considered. Besides, the dynamical characteristics of thermoelastic plates with voids subjected to the aerodynamic load are studied. It shows that it is easier to occur unbounded instability for a thermoelatic plate with voids subjected to the aerodynamic load.4. The linear mathematical model of thermoelastic thin plate with voids of arbitrary shape is established. As an application, the analytical solution of the axial symmetric problem of thermoelastic thin circular plate is obtained. The response of deflections and void volumes of TEVP, TEP, EVP and EP are compared, and the effects of thick-diameter ratio and temperature on the deflection and void volume are investigated.5. The mechanical characteristics of thermoelastic half-plane with voids are investigated. Firstly, the steady-state dynamical response model of thermoelastic half-plane with voids subjected to continuous harmonic loads on a finite region of the surface is established. Semi-analytic solution, namely the Fourier transform method-one of integral-transform method (ITM) and its numerical inverse transform, as well as the differential quadrature element method (DQEM) are adopted to solve the problem, respectively. It reveals that, ITM has exceptional advantages in solving the semi-infinite plane problem since the analytic expressions of solutions can be obtained in the transform domain. Then relatively accurate results can be obtained easily via operating the inverse transform or numerical integration of these analytic expressions. Moreover, one can see that the results obtained by ITM and DQEM are coincident with each other. In addition, it can be seen that DQEM has merits in dealing with the problems with discontinuous loads, such as less computation and high accuracy.6. The dynamical response problem of 2-D thermoelastic materials with voids subjected to a periodic loading is studied. Firstly, the mathematical model of a 2-D thermoelastic infinite strip with voids subjected to a periodic loading is presented based on the linear theory of thermoelastic materials with voids, including momentum balance equations, the balance of equilibrated force, energy equation, periodic boundary conditions, and initial conditions. On this basis, the DQM and finite differential method (FDM) are adopted to discrete the governing equations on spatial and temporal domain, respectively. Then the calculation is implemented to the discrete system. As numerical examples, the transient dynamical characteristics of an infinite thermoelastic strip with voids subjected to a plane harmonic loading and limiting car-loading are investigated. The effect of the car speed on the settlement is studied. The DQM has advantages in solving the problems with periodic boundary conditions, such as high accuracy and efficiency, and good convergence.7. The nonlinear mechanical characteristics of thermoelastic half-plane with voids under the finite deformation are investigated. From the nonlinear mathematical model of thermoelastic solids with voids presented in this paper, the DQM is adopted to discrete the nonlinear governing equations and boundary conditions on spatial domain, and the FDM is adopted to discrete the time derivatives on temporal domain. Then the Newton-Raphson method is used to solve the nonlinear algebraic equations iteratively. One can see that the results obtained by the linear model and the nonlinear model are different when the load is relatively large, and the DQM has unique advantages in solving nonlinear problems for the less computation and high accuracy. From the study on the linear and nonlinear mechanical characteristics of thermoelastic materials and structures with voids in this paper, the primary conclusion is that the presence of voids enlarges the deformation in the materials and structures, while the temperature plays an opposite role in the deformation because the external work is partially transformed to the thermal energy and to be dissipated.Thermoelasticity of materials with voids is essentially a multi-physical field coupled problem. It is very difficult to obtain the analytic or semi-analytic solutions when the effect of nonlinear deformation of structures is also considered. Hence, one has to use suitable numerical methods to solve the coupled problem. Taken into consideration the merits of DQM, such as functional-independence, simple formulae, high accuracy, less computation and memory needed, and good convergence, it is adopted to discrete on spatial domain. It is very successful to apply the numerical methods presented in this paper to deal with the linear and nonlinear problems of thermoelastic beams, plates and half-planes with voids and the problems with periodic boundary conditions. Furthermore, the DQM is extended to the DQEM to solve the problems with irregular region and discontinuous conditions, which broadens the application fields of the DQM. |
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