Eigenvalue buckling analysis of cracked functionally graded cylindrical shells in the framework of the extended finite element method

Abstract

In this study, eigenvalue buckling analysis for cracked functionally graded cylindrical shells is performed using eight noded degenerated shell elements in the framework of the extended finite element method. First, validity and efficiency of the proposed method in comparison with available results are examined and then the approach is utilized for examining cracked FGM cylindrical shells subjected to different loading conditions, including axial compression, axial tension and combined internal pressure and axial compression. Also, the effects of various parameters such as crack length and angle, gradient index of the material, aspect ratio of the cylinder and internal pressure on the buckling behavior are extensively investigated.

Introduction

Composite shell structures have been extensively utilized for a broad range of applications including space crafts, airplane fuselage, thermal coating barriers, defense systems and many others due to their considerably high strength to weight ratio. Nevertheless, fiber reinforced laminate shells suffer from many disadvantages such as delamination vulnerability to impact loading, low resistance to sustain in thermal environments. Functionally graded materials are a new class of composite materials which have obviated the interface-related problems in traditionally laminate composites due to their continuous and smooth variation of materials across the thickness. Usually, these materials consist of a ceramic surface, which can be imposed to a high gradient of thermal loading, gradually changed into a metallic surface on the other side to withstand in mechanical loadings. As a result, functionally graded cylindrical shells have drawn special attention from the application and theoretical points of view. Despite their superior characteristics, there are several failure modes which may endanger the overall safety of these structures. Among them, buckling can be one of the dominant failure modes. On the other hand, because functionally graded materials are often used in extreme environments, they are highly prone to imperfections such as voids and cracks in their structure during the production and life service. Hence, it is vital to perform accurate buckling analyses for cracked functionally graded cylindrical shells to allow for better and more reliable designs.

Many comprehensive studies have been performed on the buckling behavior of functionally graded cylindrical shells in the past decade. Ng et al. [1] presented an analytical formulation for dynamic buckling behavior of FGM cylindrical shells subjected to periodic loadings. They found out that the gradient index of the material could crucially affect the buckling behavior of the problem. Another analytical solution was presented by Sofiyev [2] concerning the buckling analysis of FGM cylindrical shells subjected to axial dynamic loadings. Further analytical studies for buckling and postbuckling of functionally graded cylindrical shells can be found in [3], [4], [5], [6], [7], [8], [9], [10].

On the other hand, there are only limited works on the buckling of cracked cylindrical shells. Esteknachi and Vafai [11] studied the buckling behavior of isotropic cracked cylindrical shells subjected to axial loading using the classical finite element method. They used a mesh zooming scheme for adaptive generation of the mesh of the cylindrical shell so that with approaching to the crack tip the size of the elements would decrease from the standard size of the uncracked regions to a very finer size to better capture the crack tip stress singularity. They also performed a similar study for cracked plates [12]. The effect of internal pressure on the buckling behavior of cracked cylindrical shells subjected to combined internal pressure and axial compression was investigated by Vaziri and Estekanchi [13] using the commercial FEM package ANSYS. They concluded that the effect of the internal pressure on the buckling stresses became completely different depending on the crack being axial or circumferential. For instance, when the crack was in the axial direction of the cylinder, the internal pressure had a detrimental effect on the buckling stresses, whereas, for the circumferential crack, the internal pressure had a stabilizing effect on the buckling behavior [13]. Also, Vaziri [14] carried out a linear eigenvalue buckling analysis using the finite element method to study the effect of crack length, crack orientation and the sequence of the lamina on the buckling behavior of composite cylindrical shells under axial compression. Dynamic stability and vibration of cracked cylindrical shells under compressive and tensile periodic loadings were investigated using the finite element method by Javidruzi et al. [15]. They showed that the existence of crack could considerably decrease the natural frequency of the shell. Also, Tafreshi [16], [17], [18] performed a series of delamination buckling and postbuckling analysis for laminate composite cylindrical shells under various loading conditions by the finite element method.

In addition, there are a few works which have addressed the buckling analysis of cracked plates in the framework of the extended finite element method. Recently, Nasirmanesh and Mohammadi [19] performed an eigenvalue buckling analysis for cracked composite plates using the extended finite element method. They examined several problems and thoroughly investigated the effects of different parameters such as crack lengths, crack angles and direction of fibers on the buckling behavior of composite plates. They also concluded that even for the tensile loading, changes in the fiber direction can alter the local instability around the crack faces to a global buckling mode.

Natarajan et al. [20] carried out thermo-mechanical buckling analysis of cracked functionally graded plates in the framework of the partition of unity method and examined the effects of gradient index of the material and crack lengths on the critical temperature and critical buckling stresses. Another XFEM buckling analysis for cracked FGM rectangular plates subjected to compressive loading was reported by Liu et al. [21]. Baiz et al. [22] used the smooth curvature method to study the effects of crack lengths and locations on the critical buckling stresses for isotropic plates. None of the existing works have studied the buckling behavior of cracked functionally graded shells. The novelty of the present study is, for the first time, to propose an XFEM shell formulation to carry out an eigenvalue buckling analysis for cracked FGM cylindrical shells.

XFEM was motivated by disadvantages of the classical finite element method for fracture analysis; including the need for mesh conformity to crack path and incapability to capture the exact stress field near a crack tip. In XFEM, while cracks are represented independent of the mesh, the exact analytical stress field around the crack tip is achieved. The method has been extended to static and dynamic orthotropic problems for both fixed and propagating cracks [23], [24], [25], [26], [27], [28], [29], [30], bi-materials [31], [32] and FGMs [33]. Recently, Rashetnia and Mohammadi [34] proposed a new set of tip enrichment functions for studying the fracture behavior of rubber-like materials which experience large deformations. They concluded that the logarithmic set of enrichment functions are more accurate and efficient.

There are other approaches that are capable of handling complex problems including nonlinear dynamic fracture and fluid driven fracture of plates and shells. For instance, Nguyen-Thanh et al. [35] proposed a model based upon the extended isogeometric method in accordance with the Kirchhoff-Love theory to analyze thin shells. In addition, a meshfree method was proposed for nonlinear dynamic fracture analysis of thin shells, which allowed to predict crack propagation across the thickness of shell [36], [37]. Recently, the phase-field method has been utilized to study the fracturing behavior of plates and shells [38], [39] with the goal of avoiding explicit track of cracks in special problems.

The present paper is organized as follows: the basic formulation of the functionally graded problem is presented. Definition of degenerated shell elements is followed by deriving the stability equations in the framework of the extended finite element method. Numerical simulations are presented and discussed for verifying the proposed method and then extended to cracked FGM problems in Section 3. Finally, a brief review of the concluding remarks is presented.