۱- Introduction

۲- Material properties distribution

۳- Kinematical description

۴- Internal and external virtual work

۵- Plane-stress state

۶- Isotropic elastic material

۷- Finite element discretization

۸- Numerical studies

۹- Conclusions

References

Abstract

In this paper, the thermo-elastic nonlinear analysis of various Functionally Graded (FG) shells under different loading conditions is studied. A second-order isoparametric triangular shell element is presented for this purpose. The element is six-noded, and each node has all six independent degrees of freedom in space. It should be added, the first-order shear deformation theory is induced. Furthermore, Voigt’s model is adopted to define the FG material properties, which are considered to change gradually from one surface to another. The critical temperature is predicted. Both the pre-buckling and post-buckling equilibrium paths are traced. Since the linear eigenvalue analysis leads to wrong responses in the problems with strong nonlinearity, the suggested procedure is performed based on the FEM and more exact estimations are achieved using equilibrium path.

Introduction

Application of composite materials in engineering constructions has a long historical background. From the early usage of straw in mud bricks in masonry structures to the new fiber-matrix laminates applied in aerospace vehicles, all are categorized in the family of composites. Today, nobody has doubts about the advantages of advanced composite materials. Along their widespread usage in industries, the demands for new theories and mathematical modeling capable of predicting their behaviors are increasing rapidly. It is obvious that applying these materials brings some fresh problems that should be considered, as well. For example, laminates show severe stress concentration at the layer interfaces which leads to delamination. Repeated cyclic stresses or impact may cause layers to separate and forming a mica-like configuration of separate layers. As a result, structure can lose significant mechanical toughness. To alleviate this phenomenon, Japanese scientists manufactured a new kind of material, which exhibits a smooth and continues change of material properties through the thickness. This kind of composite was named Functionally Graded (FG) Material. Until now, many efforts have been made to study the behavior of FG materials [1–۳]. Reddy and Chin developed a finite element procedure for FG cylinders and plates, including the thermo-mechanical coupling. They demonstrated the effects of coupling on the temperature distribution, displacements and stresses [4]. Woo and Meguid presented a closed-form solution for large deflection analysis of FG plates and shells. They applied a power law model for material properties’ distribution through the thickness. Their solutions were given in Fourier series format [5]. Patel et al. studied geometrically nonlinear responses and thermo-elastic stability characteristics of the cross-ply laminated cylindrical/conical shells with non-circularity/ovality under uniform temperature rise. It should be mentioned, load-displacement curves were obtained with the aid of FEM. They found that the shells with circular cross-section have a distinct bifurcation point, while noncircular ones show a smooth equilibrium path. Furthermore, the effect of initial perturbation/disturbance/imperfection was discussed.