The conventional form of heat conduction, Fourier’s law, has broad and successful applications in the thermal systems which have a large spatial dimension with the focus of its long time behavior. However, for problems involving high temperature gradient, materials with porosity or multiple phases, ultrafast heating and/or, micro/nano-scale heat conduction, the Fourier heat conduction is not accurate. This is due to the incorrect assumption of the infinite speed of heat propagation, which in turn, comes from the inability of Fourier heat conduction in considering the microstructural interactions and nonhomogeneity effect of the material.

The field of Thermal Stresses lies at the crossroad of Stress Analysis, Theory of Elasticity, Thermoelasicity, Heat Conduction Theory, and advanced methods of Applied Mathematics. Each of these areas is covered to some extent and explained step by step in this thesis. The heat conduction theory employed here eliminates the paradox of an infinite velocity of heat propagation by employing a more general, functional relation between heat flow and temperature gradient than the existing theory. Also, as a first attempt, the combined application of the differential quadrature method (DQM) and the Newton Raphson method is used to solve the hyperbolic (non-Fourier) and dual-phase-lag (DPL) heat conduction equations to obtain temperature, displacements and nonlinear frequency in the functionally graded (FG) nanocomposite Timoshenko beam and cylinder of different sizes. The hyperbolic heat conduction is solved to obtain temperature in the spatial and temporal domains. Then by implementing the obtained temperature in thermoelastic equations, the displacements and stresses are obtained at each time step. Here, the time domain is divided into a few blocks. In each block, there are several time levels, and the numerical results at these time levels are obtained simultaneously. Through this way, the numerical solution at the (n+1)th time level depends on the solutions at previous levels from the 1st to the nth levels. The results in the temporal domain are obtained using the Newton-Raphson method.

In general, the variation of temperature field within an elastic continuum results in thermal stresses. So, thermally induced vibration is investigated after obtaining the temperature distribution and thermal forces of carbon nanotube (CNT) reinforced nanocomposite beams and shells. The influence of temperature field in the governing equations of thermoelasticity is reflected through the constitutive law. The theory of linear thermoelasticity is based on linear addition of thermal strains to mechanical strains. All material properties such as heat capacity (Cp), thermal relaxation time (τ), density (ρ) and thermal conductivity (K) are considered as a function of both temperature and CNT volume fraction. While the equilibrium and compatibility equations of the nanocomposite remain the same as for elasticity problems, the main difference rests in the constitutive law where the effect of volume fraction and distribution of CNTs is reflected in the thermoelastic response of nanocomposite.

It has been shown that in certain situations, non-Fourier heat conduction models such as the Cattaneo and Vernotte (CV) and DPL show interesting results like temperature overshoot phenomena observed in a slab subjected to sudden temperature rise on its boundaries. As the vast majority of devices with micro- and nano-scale dimensions emerge in various micromechanical and microelectronic systems, it seems crucial to accurately measure the imposed temperature. The overshooting phenomenon, which is investigated in this research may lead to permanent damages on the sensitive electrical devices if not handled properly. Accordingly, the effects of this phenomenon on the deformation, and vibrational behavior of the beams and shells are investigated to recognize the importance of using non-conventional heat conduction methods. We showed that non-Fourier heat conduction models would play important roles in the thermoelastic design of nanocomposite structures.