Numerical investigation of functionally graded cylindrical shells and panels using the generalized unconstrained third order theory coupled with the stress recovery
Introduction
Composite circular cylindrical shells are extensively used in many engineering applications. As far as the behavior of cylindrical shells is concerned, by acting on material type, fiber orientation and thickness, a designer can tailor different properties of a laminate to suit a particular application. However, serious shortcomings due to stress concentrations between layers could lead to delamination failures. In order to overcome the variation of the material properties, the functionally graded material (FGM) was proposed by Koizumi and Yamanouchi [1] and Koizumi [2], characterized by a smooth and continuous variation from the core to the external surfaces. The possibility to graduate the material properties through the thickness avoids abrupt changes in the stress and displacement distributions.
Many researchers have furnished several results in the study of the FGM cylindrical shell [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32].
Basset [3] presented an overview on the extension and flexure of cylindrical and spherical thin shells. Bhimaraddi [4] developed a higher order theory for free vibration analysis of circular cylindrical shells. Obata and Noda [5] studied circular hollow cylinders structured from FGM material to analyze steady thermal stress at high temperature. Loy et al. [6] reached frequency spectra of FGM cylindrical shells for simply supported boundary conditions. Hua and Lam [7] calculated the frequency characteristics of a thin rotating cylindrical shell using the generalized differential quadrature method. Horgan and Chan [8] analyzed the deformations of a FG cylinder composed of a compressible isotropic linear elastic material, where the elastic modulus was a power law function of the radius and the Poisson’s ratio was constant. Pradhan et al. [9] investigated the vibration characteristics of a FGM shell made up of stainless steel and zirconia, for various boundary conditions. Liew et al. [10] gave a three dimensional elasticity solution to the free vibration problem of thick cylindrical shell panels of rectangular platform. Wu et al. [11] formulated a high order theory to examine the electromechanical behavior of piezoelectric generic shells with graded material properties in the thickness direction. Zhu et al. [12] discussed the dynamic stability of functionally graded piezoelectric circular cylindrical shells. Shen and Noda [13] characterized post buckling phenomena of FGM under combined axial and radial mechanical loads in high-temperature state. Patel et al. [14] carried out the vibration analysis of a functionally graded shell using a higher order theory. Najafizadeh and Isvandzibaei [15] used a higher order shear deformation plate theory to study the vibration of simply supported FG cylindrical shells with ring supports. Wu and Syu [16] found exact solutions of functionally graded piezoelectric shells under cylindrical bending. Haddadpour et al. [17] conducted the free vibration analysis of functionally graded cylindrical shells including thermal effects. Arshad et al. [18] reported the frequency analysis of functionally graded material cylindrical shells with various volume fraction laws. Iqbal et al. [19] examined the vibration characteristics of FGM circular cylindrical shells filled with fluid using wave propagation approach. Matsunaga [20] treated a higher order shear deformation theory in order to assess the natural frequencies and buckling stresses of functionally graded circular cylindrical shells. Tornabene and Viola [21] and Tornabene [22] dealt with the dynamic behavior of moderately thick FG cylindrical shells, by using the four parameter power law distribution. Zhao et al. [23] calculated the static response and free vibration of FGM cylindrical shells subjected to mechanical or thermomechanical loading using the element-free kp-Ritz Method. Sobhany Aragh and Yas [24], [25] considered the three dimensional analysis of thermal stresses, static and free vibration analysis of continuously graded fiber reinforced cylindrical shells by using the generalized power law distribution. In the studies under consideration, the influence of the power-law exponent and the power-law distribution were investigated. Several symmetric, asymmetric, and classic profiles were considered. A recent work by Arshad et al. [26] furnished a detailed analysis of the effects of the exponential volume fraction law on the natural frequencies of FGM cylindrical shells under various boundary conditions. Alibeigloo [27] estimated the thermoelastic solution to static deformations of functionally graded cylindrical shell bonded to thin piezoelectric layers. Sepiani et al. [28] focused on the vibration and buckling analysis of two layered functionally graded cylindrical shell, considering the effects of transverse shear and rotary inertia. Nie and Batra [29] evaluated exact solutions and material tailoring for functionally graded hollow circular cylinders. Alibeigloo and Nouri [30] developed the static analysis of functionally graded cylindrical shells with piezoelectric layers using differential quadrature method.
Sofiyev [31] presented an analytical study on the dynamic behavior of the infinitely long FGM cylindrical shell subjected to the combined action of axial tension, internal compressive load and ring shaped compressive pressure with constant velocity. Sobhani Aragh and Yas [32] studied the dynamic behavior of four parameter continuous grading fiber reinforced cylindrical panels resting on Pasternak foundation.
In the last decades, numerous studies have been also conducted on FGM cylindrical shells and plates, dealing with a variety of subjects such as thermal elasticity [33], [34], [35], static bending [36], free vibration and dynamic response [37], [38], buckling and post buckling [39], among others.
Literature review shows that there are quite a few numerical works presenting static analysis of FGM cylindrical shells. Moreover, the models proposed by different authors in literature are based on the classical theory, the first order shear deformation theory [33], [40], [41], [42], [43], [44] and the third order shear deformation theory by Reddy [45], [46].
To the best knowledge of the authors, the literature background on the static analysis of FGM cylindrical shells by using the unconstrained shear deformation theory of Leung [47] is quite poor. It should be noticed that in Leung’s theory the additional constraint typical of Reddy’s third order shear deformation theory vanishes [45], [46]. In addition, the use of four parameter power law distributions seems to be absent in the investigation of cylindrical shells, when the initial curvature effect is included in the model and a GDQ solution [21], [22], [48], [49], [50] to the problem is given.
This paper is motivated by this lack of studies and presents a static analysis of thick FGM cylindrical shells by using an unconstrained third order shear deformation theory. The initial curvature effect is involved in the analytical formulation as it was included in the first order shear deformation theory (FSDT) by Toorani and Lakis in the past decade [51] and recently improved by Tornabene et al. [52]. Furthermore, the stress recovery is worked out.
Firstly, a basic scheme is followed to write the fundamental equilibrium equations. It starts with the definition of the displacement field which includes higher order terms, the strain components, the FGM material by means of a four parameter power law distribution, and the elastic engineering stiffness constants, the stress - strain relations, as well as the relations between the internal actions and the generalized components of displacement and the definition of external applied loads.
Secondly, seven indefinite equilibrium equations are determined by applying the principle of virtual displacements. The fundamental equations are obtained by substituting in them the constitutive equations expressed in terms of generalized components of displacement.
Thirdly, the fundamental equations are discretized via GDQ [53], [54], [55], [56], [57], [58] and the differential equilibrium equations appear in the form of algebraic equations. The boundary conditions also take the analogous algebraic form. The solution is given in terms of generalized components of displacement of nodal points on the middle surface domain.
Fourthly, the through-thickness distribution of in plane stress are given.
Fifthly, the in plane stress components calculated from the constitutive relations by using the third order unconstrained theory are compared with those determined via the first order shear deformation theory, for several types of functionally graded cylindrical shells. Both the transverse shear stress components (τxn, τsn) along the thickness direction are determined from the constitutive equations using the unconstrained first and third order theories, respectively. In order to satisfy the zero shear conditions on the lateral surfaces which is not imposed a priori in the unconstrained theory, the transverse shear stress components are calculated by integrating the 3D differential equilibrium equations in the thickness direction [20], using the in plane stress components (σx, σs, τxs) determined via the constitutive relations. The effects of the material power law function and the initial curvature are discussed and graphically shown in all the numerical results.
Sixthly, the transverse normal stress component is carried out by using the recovery technique, as for the transverse shear stress components. All the recovered transverse stress components are improved as reported in [59].
Finally, in order to prove the validity of the present formulation, the numerical examples proposed by Aghdam et al. [60], Zhao et al. [61], Fereidoon et al. [62] and Ferreira et al. [63], [64] are also considered. The center deflections of isotropic and functionally graded cylindrical panels were obtained in the present study and compared with the ones reported in [60], [61]. The vertical displacements and membrane normal stresses in the central node of functionally graded rectangular plates were carried out and compared with those reported in [62], [63], [64]. The transverse displacement component, the membrane normal and transverse shear stresses calculated at an arbitrary point of functionally graded rectangular plates were compared with the ones derived from Zenkour [65].
Further publications related with the present paper are reported in [66], [67], [68], [69], [70], [71], [72], [73].
Section snippets
Fundamental hypotheses
In this paper, a graded composite circular cylindrical shell is considered. L0, R0, h denote the length, the mean radius and the total thickness of the shell, respectively. The position of an arbitrary point P within the shell is located by the coordinates x(0 ⩽ x ⩽ L0), s(0 ⩽ s ⩽ s0 = ϑR0) upon the middle surface, and ζ directed along the outward normal n, and measured from the reference surface (−h/2 ⩽ ζ ⩽ h/2), as shown in Fig. 1.
When the general case of shell of revolution changes into the case under
Discretized equations and stress recovery
The generalized differential quadrature method (GDQ) [53], [54], [55], [56], [57], [58] is used to discretize the derivatives in the governing Eq. (42), as well as the external boundary conditions and the compatibility conditions. In this paper, the Chebyshev–Gauss–Lobatto (C–G–L) grid distribution is adopted, where the coordinates of grid points along the reference surface are identified by the following relations:
Classes of graded materials
In this numerical study, the static analysis of FGM cylindrical shells is conducted and the through the thickness stress distributions are furnished. The theoretical formulations are based on two shear deformation models: the generalized unconstrained third (GUTSDT) and first order (GFSDT) shear deformation theories. They are labeled as generalized because they are enriched by the initial curvature effect. The stress recovery is also proposed in order to define the correct profile of the
Literature numerical examples worked out for comparison
In this section several numerical examples are considered in order to compare the present results with the existing ones in literature. Aghdam et al. [60] investigated the bending of moderately thick clamped functionally graded (FG) conical panels subjected to uniform and non-uniform distributed loadings. They used the first order shear deformation theory by taking into account the initial curvature effect in the formulation. In the present work, the numerical results reported in [60] for a
Final remarks and conclusion
The cylindrical shell problem described in terms of seven differential Eq. (42) has been solved by using the GDQ method. Among the methods of approximation, the GDQ procedure starts directly from the strong statement of the problem under consideration. It should be noted that the GDQ technique of obtaining algebraic equations does not require the construction of any variational formulation of the problem. As it is well known, the GDQ method is based on the idea that the partial derivative of a
Acknowledgements
This research was supported by the Italian Ministry for University and Scientific, Technological Research MIUR (40% and 60%). The research topic is one of the Centre of Study and Research for the Identification of Materials and Structures (CIMEST) – “M. Capurso” of the Alma Mater Studiorum University of Bologna (Italy).
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