A generalized global–local high-order theory for bending and vibration analyses of sandwich plates subjected to thermo-mechanical loads

Abstract

Although the global higher-order shear deformation theories may predict the gross responses of the sandwich plates sufficiently accurate, their results may show considerable errors in predicting the local effects. Layerwise and mixed layerwise theories are computationally expensive and generally, the interlaminar transverse stresses continuity conditions are not enforced in the former category of theories. Majority of the available zigzag and global–local theories suffer from the point that the transverse normal stress continuity that influences the transverse deformation significantly, especially in sandwich plates with soft-cores, is not satisfied at the layer interfaces.

In the present paper, a generalized global–local theory that guarantees the continuity condition of all of the displacement and transverse stress components and considers the transverse flexibility under thermo-mechanical loads is introduced. One of the advantages of the present theory is that the number of unknown parameters is independent of the number of the layers. Furthermore, all stress components are considered in the formulations. Therefore, in contrast to the available works, the theory may be used for sandwich plates with stiff or soft cores. In contrast to the available global–local formulations, the present formulation is developed in a compact matrix form that makes it more desirable for computerized solutions. The present theory may be considered as a generalized layerwise theory with an optimized computational time. Compatible quadrilateral Hermitian elements are employed to further enhance the accuracy of the results. Validity, advantages, and efficiency of the present theory are investigated for different local and global behaviors of the layered composite and sandwich plates. Comparison of the present results with those of the three-dimensional theory of elasticity and the available plate theories confirms the efficiency and accuracy of the proposed theory. Results reveal that the global theories (e.g. the higher-order shear deformation theories) may encounter serious accuracy problems even in predicting the gross responses of the sandwich plates.

Introduction

Sandwich structures have been extensively used in aerospace, aeronautic, automotive, naval, underwater, and building structures. Sandwich plates may be used for constructing light-weight structures with high strength or stiffness to weight ratios, noise, vibration, and harshness (NVH) isolation, thermal insulation, or constructing structures with discrete functionally graded layers. In many applications, the sandwich plate is a laminated construction, which consists of two or more stiff sheets (including the face-sheets) and one or multiple thicker low strength flexible cores. Since the variations in the rigidity and the material properties of the layers are much more severe in sandwich plates with soft cores in comparison to the traditional composite plates, influence of the transverse shear and normal strains and stresses are more significant in the mentioned sandwich plates, especially for severe temperature gradients and high mechanical loads. Therefore, continuity conditions of the interlaminar transverse shear and normal stresses should be accurately imposed to accurately model the structural and dynamical behaviors of the mentioned plates.

There are limited exact solutions in literature [1], [2], [3], [4], [5], [6] that have been provided for vibration and interlaminar responses analyses of the laminated composite and sandwich plates. They have been derived based on the classic linear three-dimensional theory of elasticity. Although results of the global (equivalent single layer) plate theories [7], [8], [9], [10], [11], [12], [13], [14], [15] may be sufficiently accurate for predicting global responses of the sandwich structures, their results may exhibit considerable errors in predicting the local behaviors. Layerwise plate theories [16], [17], [18] have been proposed to overcome some of the shortcomings of the equivalent single layer theories. Despite the capability to trace the local variations in each layer more accurately, these theories mainly suffer from two points: (1) number of the nodal unknowns becomes very large as the number of the layers increases, and (2) continuity of the transverse interlaminar stresses is not guaranteed especially when elements with C⁰ continuity are employed.

To enforce the continuity conditions of the transverse shear and normal stresses, mixed layerwise theories [19], [20] that assume the transverse stress and displacement quantities of each layer to be two independent fields, have been developed. Shariyat and Eslami [21] and Eslami et al. [22], [23] proposed the hybrid higher-order and layerwise theories. The governing equations were expressed based on mixed formulations consist of stress and displacement components. Rao and Desai [24] developed a higher order mixed layerwise theory for the sandwich plates. This theory accounted for the continuity of the transverse stresses at the layer interfaces. Extensive studies were presented by Demasi [25] and Carrera and Brischetto [26] for the mixed layerwise plate theories. Recently, Plagianakos and Saravanos, [17], [27] presented a higher-order double superposition layerwise theory for thick composite and sandwich plates. Interlaminar shear stress continuity conditions were imposed.

To retain the benefits of less computational times of the equivalent single-layer theories and the accuracy of the layerwise theories, zigzag theories with linear [28], [29], [30], [31], [32] or high-order [33], [34], [35], [36], [37], [38], [39] local layerwise zigzag functions were proposed. Due to imposing the continuity conditions of the transverse shear stresses, the number of unknowns in these models is independent of the number of the layers. Kapuria and Achary [40] presented a third-order zig-zag theory for the sandwich plates. Ganapathi et al. [41] investigated the nonlinear dynamics behavior of the sandwich plates based on a zigzag theory. Kim [42] developed two theories for the sandwich plates via the mixed variational formulation for free vibration studies. Recently, Icardi [43] developed a mixed, sublaminate element in which displacements and stresses were approximated by two independent zigzag models. In addition to the contact conditions on the interlaminar shears, those on the transverse normal stress and stress gradient were also fulfilled. Pandit et al. [44] also proposed an improved cubic zigzag theory for static analysis of laminated sandwich plates with transversely compressible cores. The transverse displacement was assumed to vary quadratically within the core while it remained constant through the faces. A refined first-order Reissner–Mindlin-type nonlinear zigzag shear deformation theory was presented by Fares and Elmarghany [45]. Continuity of the transverse shear and normal stresses was taken into account. Kulkarni and Kapuria [46] studied the free vibration responses of the sandwich plates employing a third-order zigzag theory. Recently, Brischetto et al. [47], added the zigzag function to the known higher-order theories for bending analysis of sandwich flat panels.

In addition to the transverse shear strains and stresses, effects of the transverse normal strain and stress within the core are significant for thick soft cores [26], [48], [49], [50], [51], [52], [53]. Transverse flexibility of the flexible core considerably affects the overall behavior of the sandwich plate and it is crucial for investigating the local effects and failures. Pai and Palazotto [54] presented a layerwise high-order theory to satisfy continuity conditions of the interlaminar shear and normal stresses in the sandwich plates. Matsunaga [55], [56] presented a global higher-order deformation theory based on a truncated mth-order power-series for the evaluation of interlaminar stresses and displacements in sandwich plates subjected to thermal loadings. Although the transverse stresses continuity was not enforced in the governing equations, transverse shear and normal stresses were calculated by integrating the three-dimensional equations of equilibrium in the thickness direction and satisfying the continuity conditions at the layer interfaces.

Although the idea of superimposing local variations on the global ones in the interpolating equations to enhance their capability to trace the local variations was already proposed, e.g. by Pagano and Soni [3] and Robbins and Reddy [57], the first zigzag global–local plate theory based on the double superposition of the local interpolating expressions was proposed by Li and Liu [58], [59]. The number of the unknowns was independent of the layer number of the laminates. Using this theory, the in-plane stresses and transverse shear stresses can be predicted accurately by constitutive relations. Based on the global–local higher-order theory of Li and Liu [59], Zhen and Wanji [60] studied the free vibration of the laminated composite and sandwich plates, employing a refined element satisfying C¹ weak-continuity conditions. They concluded that the global theories overestimate the natural frequencies for these special structures. Zhen and Wanji [61], [62] studied the effect of the order of the global expression in the mth-order global–local theory on the accuracy of solutions of bending, vibration, and linear static buckling of plates subjected to thermal/mechanical loading. They concluded that the seventh-order global expressions seem to be the upper limit in the global–local theory. All of the mentioned double superposition global–local references have discarded effects of the transverse normal deformation and stress.

Carrera [26], [63] has claimed that transverse normal strain cannot be neglected for thermal bending and expansion problems. Based on these observations, a refined global–local higher-order theory was proposed by Zhen and Wanji [64], [65], [66] for laminated plates subjected to coupled bending and extension under thermo-mechanical loading. Transverse normal deformations were approximated by a second-order polynomial in terms of the global transverse coordinate.

In the present paper, the double superposition idea proposed by Li and Liu is extended to propose a generalized higher-order global–local theory that takes into account the transverse normal stress continuity at the layer interfaces as well as the transverse compressibility of the layers. Therefore, in contrast to theories proposed so far in this field, the transverse displacement is also assumed to have higher-order global and local components. Another advantage of the present formulation is its compact matrix form that makes it more adequate and pleasant for computerized solutions. Results of the present theory are compared with results of the available theories to examine the accuracy and the efficiency of the theory. To further enhance the accuracy of the finite element model, a compatible C¹ Hermetian element is employed to obtain stress distributions and modal responses of sandwich plates subjected to thermo-mechanical loads.