Study of Dynamics of Functionally Graded Structural Elements Considering Geometric Nonlinearity

Abstract

The primary aim in the field of applied solid mechanics is to cater safe design for engineering components that enables them to withstand static or dynamic service loads for a certain period of time. This can be achieved by improving the stiffness and strength of the materials used in the structure. One such material which fits the above criteria is functionally graded material (FGM). The FGMs are treated as new generation materials in which the material properties vary continuously along spatial directions and claim many advantages over traditional materials. Due to which they may be employed in various advanced structures especially in large space structure and aerospace applications. Due to the severe working conditions in some cases these structures have to bear acute dynamic loading which may lead to failure. Hence it is essential to conduct thorough investigation on the dynamic behaviour of FGMs.

A study of nonlinear dynamics of functionally graded beams and plates subjected to transverse large amplitude is presented. The study is carried out on both uniform and non-uniform FG structures using different material models and taper profiles. Both free and forced vibration analyses are carried out using two distinct methodologies namely whole domain method and finite element method. In case of beam, Euler-Bernoulli beam theory is utilized in both the methods but in the analysis of plate, whole domain method is based on classical plate theory and finite element method is based on Mindlin plate theory. Von Karman’s strain-displacement relations are incorporated into the methodology to account for the geometric nonlinearity present in the system.

The free and forced vibration analyses are carried out separately with different basic assumptions. The nonlinear free vibration problem is solved in two steps where the objective in the first part is to compute the stiffness matrix in deflected configuration through a static analysis. This equivalent stiffness matrix is directly used in dynamic analysis for obtaining eigenvalues and eigenvectors which form the natural frequency and mode shape of the system, respectively. The assumption in the nonlinear forced vibration analysis is that all the forces acting on the system attains equilibrium at the peak amplitude. This unique assumption enables the dynamic problem to be solved as an equivalent static problem. The static analysis in the first part of free vibration is based on principle of minimum total potential energy whereas Hamilton’s principle is used in dynamic analysis of both free and forced vibration.

The two methods used in the present thesis are different in terms of basic philosophy. In whole domain method, the whole system is under consideration and the displacement fields are calculated at a number of computational points within the system. The displacement fields are expressed as linear combinations of orthogonal kinematically admissible functions and unknown parameters. These orthogonal functions are generated at the computational points through Gram Schmidt orthogonalisation procedure from the initial or start functions dependent on flexural and membrane boundary conditions. These functions remain constant for a particular boundary condition so the only variables are the unknown parameters and computing these unknown parameters lead to the displacement fields of the system. In finite element method the domain is discretised into some finite number of elements and the analysis is concentrated in these elements. The displacement field within the element is approximated in terms of nodal displacements and shape functions.

Three models are considered for material variation and the same number of taper profiles is taken for geometry variations. The material models are TFG (Thickness-wise functional gradation), AFG (Length-wise functional gradation) and BFG (Bidirectional functional gradation). On the other hand the taper profiles considered are as follows - linear taper profile, parabolic taper profile and exponential taper profile. The effect of these material models and taper profiles on the dynamics of a system is studied through different values of material gradient parameters and taper parameters.

Validation of current study is done by comparing the results with those already available in the literature. The results are presented in terms of backbone curves and mode shapes in free vibration analysis and frequency-response curves and operational deflection shapes in forced vibration scenario. A general conclusion drawn from the results is that the system exhibits hard spring behaviour i.e. its stiffness increases with increasing response amplitude. Also, the parameters mentioned above significantly affect the dynamic behaviour of the system. The author believes that the results presented in the thesis will serve as benchmark for further research works in this area.