A new C0 higher-order layerwise finite element formulation for the analysis of laminated and sandwich plates

doi:10.1016/j.compstruct.2015.04.034

Composite Structures

Volume 131, 1 November 2015, Pages 1-16

https://doi.org/10.1016/j.compstruct.2015.04.034 Get rights and content

Abstract

In this paper, a new layerwise plate formulation based on a C⁰ higher-order finite element model is presented for static and free vibration analyses of laminated composite and sandwich plates. The proposed layerwise theory which is developed for a three layered composite plates, assumes higher-order displacement field for middle layer and first-order displacement field for top and bottom layers. Compatibility conditions are imposed at the layer interface to satisfy the interlaminar displacement continuity. An eight-noded isoparametric element is used to model the plate. The accuracy of the proposed formulation is assessed for linear static and free vibration analyses by comparing the authors’ results with available 3D elasticity, finite element and analytical solutions. It has been shown here that the present finite element formulation is much simpler, straightforward and accurate for static and free vibration analyses of laminated composite and sandwich plates.

Introduction

Laminated composite and sandwich structures are widely used in civil, mechanical, automobile and marine engineering applications due to their high stiffness and strength to weight ratios. These materials can be easily tailored to desired shape, size and weight. Moreover, these specially designed materials have long service life, corrosion and fatigue resistance characteristics, higher energy absorption and self damping capacity. Due to these characteristics, composite and sandwich structures have gained popularity among the designers. A typical sandwich structure consists of one or more layers of high-strength and high-stiffness facesheets (or skin) bonded to one or more flexible core. The facesheets are the primary load carrying members and the core helps transfer of the load between the facesheets. Studying the static and dynamic characteristics of sandwich structures is essential in order to have a deep insight of their behavior for proper analysis and design. Since the elastic properties of the core and the facesheets are significantly different, the accuracy of the results for sandwich structures largely depends on the computational model adopted.

Based on displacement field, two approaches viz., equivalent single layer (ESL) theories and layerwise (LW) theories are mainly considered by researchers. Classical laminate theory (CLT) based on the Kirchhoff’s assumptions and first-order shear deformation theories (FSDT) based on Reissner–Mindlin hypotheses [1], [2] were some of the earliest ESL theories adopted for analysis of laminated composite and sandwich plate and shells. The CLT ignores the shear deformation and hence for layered plate and shell panels, the same is inadequate. Effect of transverse shear deformation can be significant for thick laminates and laminates with high degree of anisotropy. Also, the flexible core used in sandwich plates and shell structures has low shear moduli and undergoes higher shear deformation hence, CLT cannot describe the behavior of sandwich structures accurately. The FSDT considers linear stresses and displacements through the thickness of the plate. This consideration makes the transverse shear stresses, which are parabolic as per the elasticity results, constant along the thickness [3], [4], [5]. To eliminate this discrepancy, the shear correction factors were introduced. Thus, the accuracy of the transverse shear stresses along the thickness of the structure depends on the shear correction factor which is based on type of loading, lamination scheme and boundary conditions [6]. To overcome this drawback, higher-order shear deformation theories (HSDT) were developed where the displacement fields are considered cubic and the transverse shear strains parabolic, Thus eliminating the use of shear correction factor. Lo et al. [7] presented a higher-order theory to study mechanical behavior of laminated plates. Higher-order theories proposed by Reddy [8] and Bhimaradi and Stevens [9] required five degrees of freedom and the computational cost involved is not increased in comparison with the FSDT formulation. However, these theories require shape functions which are C¹ continuous. Pandya and Kant [10], [11] presented a C⁰ continuous higher-order finite element formulation for static analysis of laminated and sandwich plates. Based on a higher-order theory, Ganapathi and Makhecha [12] carried out free vibration analysis of laminated composite plate using a C⁰ eight-noded isoparametric element. An analytical solution based on refined higher-order shear deformation theory for static analysis of laminated and sandwich plates was proposed by Kant and Swaminathan [13]. Some of the other notable contributions made on the higher-order theories are works of Frostig and Thomsen [14], Aagaah et al. [15] and Fiedler et al. [16]. Review articles by Mallikarjuna and Kant [17], Altenbach [18], Zhang and Yang [19], Kreja [20] and Khandan et al. [21] shows the continuous development of shear deformation theories in the last few decades.

Exact solutions for static and dynamic behavior of laminated and sandwich plates were presented by researchers like Pagano [22], [23], Srinivas and Rao [24], Pagano and Hatfield [25], Srinivas [26] and Zenkour [27]. Wang et al. [28] developed a Ritz based solution for free vibration of skew sandwich plates. Higher-order theories based on non-polynomial displacement fields were also proposed by researchers for the analysis of laminated plates [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40].

In ESL, the displacements and transverse strain components are considered to be continuous functions across the laminate thickness leading to discontinuous interlaminar stresses at layer interface because of different elastic constants in each layer. Therefore, ESL theories fail to capture precise static and free vibration behavior particularly for sandwich structures. This drawback in ESL was circumvented by the layerwise theories in which the displacements are assumed at the mid surface of each laminate and maintaining the continuity of the displacements at the layer interface. Therefore, in layerwise theories, the derivatives of the displacement fields become discontinuous functions through the thickness and transverse stresses can be continuous at the interfaces having different elastic properties. Layerwise displacement fields are capable of capturing correct behavior of both thin and thick laminated as well as sandwich composite plates. Many researchers worked on the development and application of layerwise theories. Di Sciuva [41], [42] proposed a piecewise linear displacement model that satisfied displacement and transverse stress continuity for static and dynamic analysis of orthotropic plates. Toledano and Murakami [43] solved the bending problem for an arbitrary laminate configuration using a layerwise displacement model. Cho et al. [44] presented a higher-order displacement field for each layer to determine natural frequencies of laminated composites under simply supported boundary conditions. Lu and Liu [45] analyzed transverse shear stresses of thick and thin laminated composites using a layerwise theory. A generalized layerwise 2D displacement based approach for static analysis of thick laminated plate was presented by Robbins and Reddy [46]. Ferreira [47], [48] provided meshless solutions for static and free vibration analyses of laminated composite and sandwich plates using a layerwise theory. Fares and Elmarghany [49] presented analytical solutions for static analysis of laminated composite plates using a layerwise displacement model. Plagianakos and Saravanos [50] using a higher-order layerwise theory and a Ritz based solution technique discussed static behavior of laminated and sandwich plates. Rao et al. [51] used a layerwise theory and introduced a semi analytical approach for predicting the natural vibrations of laminated and sandwich plates. Cetkovic and Vuksanovic [52] using generalized plate theory (GLPT) of Reddy [53] carried out static and dynamic analyses of laminated composite and sandwich plates. Recently, Maturi et al. [54] presented meshless solutions for static analysis of laminated and sandwich plates using a new layerwise theory. An automatic approach for generation of displacement fields using ESL and layerwise theories was developed and employed for evaluation of various theories by Carrera and co workers [55], [56], [57], [58], [59], [60]. Carrera [61] reviewed the issues involved in modeling of multilayered flat and curved structures using Reissner-mixed variational theorem (RMVT) and presented the RMVT based results for plate and shell structures. Review articles by Carrera [62], [63], Carrera and Brischetto [64] reflect the continuous development in the field of ESL and layerwise theories. Demasi [65], [66], [67], [68], [69] presented mixed plate theories based on generalized Carrera unified formulation for composite plates. A generalized layerwise higher-order model for static analysis of laminated and sandwich plates was proposed by Mantari et al. [70].

Analytical methods involves cumbersome solution procedure and are applicable only for certain boundary conditions, hence analysis based on finite element method (FEM) became popular [53], [71], [72], [73]. Sheikh and Chakrabarti [74] employed a six-noded non-confirming triangular element along with higher-order theory of Reddy [8] for static analysis of laminated composite plates. Chakrabarti and Sheikh [75], [76] developed a C¹ continuous six-noded triangular plate element for static and dynamic analyses of laminated composite and sandwich plates. Kapuria and Kulkarni [77], Kulkarni and Kapuria [78], [79] carried out static and free vibration analyses of composite plates by using a four-noded quadrilateral and improved discrete Kirchhoff quadrilateral (IDKQ) interpolation functions earlier proposed by Jeyachandrabose et al. [80]. Zhen and Wanji [81] studied free vibration of laminated composite and sandwich plates by using a zig-zag theory and three-noded triangular element satisfying C¹ continuity. Akhras and Li [82] based on higher-order zig-zag theory and a finite strip method carried out static analysis of composite plate. Pandit et al. [83], [84] proposed an improved zig-zag theory in which transverse displacement is assumed to be quadratic over core and constant over facesheet. Chalak et al. [85], [86] carried out static and dynamic analyses of laminated composite and sandwich plates using a higher-order zig-zag plate theory. Khandelwal et al. [87], [88], [89] proposed a refined higher-order shear deformation theory and combined it with least square error method for carrying out static analysis of sandwich plates using a nine-noded quadratic plate element. Static and dynamic analyses of laminated composite and sandwich plates using inverse hyperbolic, trigonometric zig-zag theory was presented by Sahoo and Singh [90], [91], [92], [93].

It is observed from literature that most of the finite element formulation for static and free vibration analyses of sandwich plates are based on zig-zag theory. However, the zig-zag theory has a problem in its finite element implementation as it requires C¹ continuity of transverse displacement at the nodes which is highly undesirable for plate and shell elements and need to be eliminated by imposing certain constrains giving rise to complexities in formulation [86], [94], [95]. Also, to obtain transverse shear stresses from equilibrium equations, higher-order derivative of displacement are required resulting in difficulties in the finite element implementation [96].

Keeping this in mind the present work aims to propose a simplified approach for static and free vibration analyses of laminated and sandwich plates. A new C⁰ layerwise theory is proposed in which in-plane displacement fields are expanded using Taylor’s series for the core and first-order displacement field is assumed for both upper and lower facesheets maintaining the displacement continuity at the layer interfaces. An eight-noded isoparametric plate element with 13 degrees of freedom per node is employed to model the plate using FEM. The accuracy of the present formulation is validated for a wide range of static and free vibration problems of laminated composite and sandwich plates with soft core with different boundary conditions, loading and geometric parameters. It has been shown here that the present finite element formulation is simple, accurate and straightforward with requirement of few elements only.

Section snippets

Mathematical formulation

Consider a rectangular plate having width a, length b and thicknessh along $x, y$ and z axes, respectively. Fig. 1 shows the one dimensional layerwise kinematics of three layered composite. The displacement components $u^{(2)}, v^{(2)}$ and $w^{(2)}$ along $x, y$ and z directions, respectively for the middle layer (core) are expanded using Taylor’s series in terms of thickness coordinate $z^{(2)}$ as [10]: $\begin{matrix} u^{(2)} (x, y, z) = u_{0} (x, y) + z^{(2)} θ_{x}^{(2)} + {(z^{(2)})}^{2} u^{*} + {(z^{(2)})}^{3} θ_{x}^{*} \\ v^{(2)} (x, y, z) = v_{0} (x, y) + z^{(2)} θ_{y}^{(2)} + {(z^{(2)})}^{2} v^{*} + {(z^{(2)})}^{3} θ_{y}^{*} \\ w^{(2)} (x, y, z) = w_{0} (x, y) \end{matrix}$

Finite element formulation

In the present work, an eight noded C⁰ isoparametric element with 13 degrees of freedom per node ${\{\begin{matrix} u_{0}, & v_{0}, & w_{0}, & θ_{x}^{(1)}, & θ_{y}^{(1)}, & θ_{x}^{(2)}, & θ_{y}^{(2)}, & θ_{x}^{(3)}, & θ_{y}^{(3)}, & u^{*}, & v^{*}, & θ_{x}^{*}, & θ_{y}^{*} \end{matrix}\}}^{T}$ is used to develop the finite element model. The subsequent sections deal with detailed derivation of stiffness matrix, mass matrix, load vector and also governing differential equations for present formulation.

Governing differential equation

Hamilton’s principle is used in order to formulate governing static and free vibration problems considered in this work, which is given as: $\int_{t_{1}}^{t_{2}} δ [(T - U + W_{ext})] d t = 0$ where $t_{1}$ and $t_{2}$ is the time interval during which the variation is taken. Substituting for the energy terms from the Eqs. (23), (29) and (34) in Eq. (35). $\int_{t_{1}}^{t_{2}} δ [\frac{1}{2} {\dot{d}}^{T} M \dot{d} - \frac{1}{2} d^{T} Kd + d^{T} Q] d t = 0$ Alternatively, Eq. (36) is written as $\int_{t_{1}}^{t_{2}} [- δ d^{T} M \ddot{d} - δ d^{T} Kd + δ d^{T} Q] d t = 0$ or $\int_{t_{1}}^{t_{2}} δ d^{T} [M \ddot{d} + Kd - Q] d t = 0$ Eq. (38) can be satisfied if the term in square bracket vanish i.e. $M \ddot{d} +$

Results and discussion

The efficiency of the present higher-order layerwise formulation is established by comparing the present static and free vibration analyses results with those available in the literature. Two different boundary conditions considered in the present investigation are all edges simply supported (SSSS) and clamped (CCCC). In SSSS condition, degrees of freedom $u_{0}, w_{0}, θ_{x}^{(1)}, θ_{x}^{(2)}, θ_{x}^{(3)}, u^{*}$ and $θ_{x}^{*}$ are restrained along edges parallel to x axis and $v_{0}, w_{0}, θ_{y}^{(1)}, θ_{y}^{(2)}, θ_{y}^{(3)}, v^{*}$ and $θ_{y}^{*}$ are zero along edges

Conclusions

In this paper a new higher-order layerwise theory is proposed for static and free vibration analyses of laminated composite and sandwich plates. The theory satisfies interlaminar displacement continuity. The finite element model is formulated adopting an eight-noded C⁰ isoparametric element. Performance of the present finite element model is evaluated by comparing the authors’ static and free vibration results with the results available in the published literature. It is worth to mention here