Dynamic buckling of FGM truncated conical shells subjected to non-uniform normal impact load
Introduction
Functionally graded materials (FGMs) have been regarded as one of the advanced inhomogeneous composite materials, usually made from metal and ceramic. By gradually and continuously varying the volume fraction of the constituent materials, FGMs possess the gradient in properties. Comprehensive works on the static buckling of structures of purely FGM, or FGM laminated structures have been reported in the literatures. Feldman and Aboudi [1] studied the elastic bifurcation buckling of functionally graded plate under in-plane compressive loadings, the buckling loads of rectangular plates with both simply supported and clamped edges were presented. Javaheri and Eslami [2], [3] investigated the thermal buckling of rectangular FGM plate based on classical as well as higher-order shear deformation theories and obtained the closed form solutions of the problem under several types of thermal loadings. Liew et al. [4] examined the buckling and post-buckling of piezoelectric FGM plates subjected to thermo-electro-mechanical loading. Based on the first-order and the third-order shear deformation theories, investigation of nonlinear thermal bending and post-buckling of circular functionally graded plates were performed by Ma and Wang [5], [6]. Further more, Yang et al. [7] extended their work to the buckling of FGM plates with randomness in material properties and presented second-order statistics for the critical buckling load of rectangular FGM plates. Sofiyev et al. [8] study the stability of a three-layered conical shell containing a functionally graded material layer subjected to axial compressive load. Applying Galerkin’s method, critical axial loads were obtained. Sofiyev [9] also studied the vibration and stability of freely supported FGM truncated and complete conical shells subjected to uniform lateral and hydrostatic pressures. Based on first-order shear deformation theory, Bhangale et al. [10]studied the thermal buckling and vibration behavior of clamped–clamped FGM conical shells in a high-temperature environment by using finite element formulation. Many other works on buckling and post-buckling for FGM structures are also available in the literatures, such as works by Wu [11], Woo et al. [12], Na and Kim [13] and so on.
In all the aforementioned contributions on static buckling of FGM structures, few investigations on the dynamic buckling of FGM structures have been reported, such as in papers by Huang and Han [14] studied nonlinear dynamic buckling of functionally graded cylindrical shells subjected to time-dependent axial load. Static and dynamic stabilities of functionally graded panels subjected to combined thermal and aerodynamic loads are investigated by Sohn and Kim [15]. Based on first-order shear deformation theory, the dynamic thermal buckling behavior of functionally graded spherical caps is studied by Prakash et al. [16]. Ganapathi [17] studied the dynamic stability behavior of a clamped functionally graded materials spherical shell structural element subjected to external pressure load. Shariyat [18] studied the dynamic buckling of a pre-stressed, suddenly heated imperfect FGM cylindrical shell and dynamic buckling of a mechanically loaded imperfect FGM cylindrical shell in thermal environment. For FGM truncated conical shells, Sofiyev [19], [20] studied the buckling of the shells under dynamic axial loading. However, there have been few researches dealing with characteristics for the normal impact buckling of the FGM truncated conical shells. Therefore, in this study, dynamic buckling of imperfect FGM conical shell subjected to normal impact load is investigated by using Galerkin procedure and Runge–Kutta integration scheme. From the characteristics of dynamic response of the shells obtain critical loads according to B-R criterion [21], [22]. The effects of both the physical and geometrical parameters on the impact buckling critical loads are analyzed and discussed.
Section snippets
Problem formulation
The geometry of an FGM truncated conical shell is defined in Fig. 1. R1 and R2 are the average radii of the cone at its small and large edges, h is the cone thickness, φ is the semivertex angle of the cone and L is the cone length along its generator. The conical shell is referred to an orthogonal curvilinear coordinates (s, θ, z) as shown in Fig. 1, in which s is in the generatrix direction measured from the vertex of the shell, θ is in the circumferential direction, and z is in the normal
Numerical results and discussion
In the computation, a functionally graded conical shell made of the constituents of aluminum (material 1) and zirconia (material 2) is considered. The material properties are the same as in Ref. [5]. The displacement fields of a thin FGM conical shell with clamped boundary conditions can be expanded aswhere m is circumferential number. We consider that the conical shell is subjected to
Conclusions
Dynamic buckling of FGM conical shells subjected to non-uniform normal step loads and sinusoidal impulse is discussed. In the analysis, the material properties of FGM shells are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents, geometrically nonlinear large deformation and the initial imperfections are taken into account. According to B-R criterion, the critical loads are obtained. From the research
Acknowledgements
This work is supported by the National Natural Science Foundation of China (No. 10872083), the Scientific Research Development Foundation of Lanzhou University of Technology (No. BS10200902). The authors gratefully acknowledge all the supports.
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