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This paper presents a theoretical study of the thermally driven behavior of a shape memory alloy (SMA)/FGM actuator under arbitrary loading and boundary conditions by developing an integrated mathematical model. The model studied is established on the geometric parameters of the three-dimensional laminated composite box beam as an actuator that consists of a functionally graded core integrated with SMA actuator layers with a uniform rectangular cross section. The constitutive equation and linear phase transformation kinetics relations of SMA layers based on Tanaka and Nagaki model are coupled with the governing equation of the actuator to predict the stress history and to model the thermo-mechanical behavior of the smart shape memory alloy/FGM beam. Based on the classical laminated beam theory, the explicit solution to the structural response of the structure, including axial and lateral deflections of the structure, is investigated. As an example, a cantilever box beam subjected to a transverse concentrated load is solved numerically. It is found that the changes in the actuator's responses during the phase transformation due to the strain recovery are significant.

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An initial-boundary value problem that is Duffing's oscillator with time varying coefficients will be studied. Using Banach's fixed-point theorem, the existence of periodic solution of the equation will be predicted. The method applied in this paper is the Schauder second fixed point theorem, which includes the response of structures under vibratory force systems. As an example, the dynamics of nonlinear simply supported rectangular thin plate under influence of a relatively moving mass is studied. By expansion of the solution as a series of mode functions, the governing equations of motion are reduced to an ordinary differential equation for time development vibration amplitude, which is Duffing's oscillator. Finally, a parametric study is developed, after that some numerical examples are solved, and the validity of the present analysis is clearly shown.

This work investigates the free vibration and buckling of a two-layered cylindrical shell structure made of an elastic embedded functionally graded (FG) shell subjected to combined static and periodic axial forces. Such structures are widely used in chemical and nuclear reactors, space and aerial industries, and so on. Material properties of an FG cylindrical shell are considered to be temperature dependent and graded in the thickness direction according to a power-law distribution in terms of the volume fractions of the constituents. Theoretical formulations are presented based on two different methods of the first-order shear deformation theory considering the transverse shear strains and the rotary inertias and the classical shell theory. The results obtained show that the effect of transverse shear and rotary inertias on free vibration of an FG cylindrical shell is dependent on the material composition, deformation mode, and geometry parameters of the shells. It is concluded that the application of an outer elastic layer increases elastic stability.