An efficient FE model based on combined theory for the analysis of soft core sandwich plate

An efficient C⁰ continuous finite element (FE) model is developed based on combined theory (refine higher order shear deformation theory (RHSDT) and least square error (LSE) method) for the static analysis of soft core sandwich plate. In this (RHSDT) theory, the in-plane displacement field for the face sheets and the core is obtained by superposing a global cubically varying displacement field on a zig-zag linearly varying displacement field with a different slope in each layer. The transverse displacement assumes to have a quadratic variation within the core and it remains constant in the faces beyond the core. The proposed model satisfies the condition of transverse shear stress continuity at the layer interfaces and the zero transverse shear stress condition at the top and bottom of the sandwich plate. The nodal field variables are chosen in an efficient manner to circumvent the problem of C¹ continuity requirement of the transverse displacements. In order to calculate the accurate through thickness transverse stresses variation, LSE method has been used at the post processing stage. The proposed combine model (RHSDT and LSE) is implemented to analyze the laminated composites and sandwich plates. Many new results are also presented which should be useful for future research.     

An efficient FE model and Least Square Error method for accurate calculation of transverse shear stresses in composites and sandwich laminates

Accurate evaluation of transverse stresses in laminated composites and sandwich plates using 2D FE models involves cumbersome post-processing techniques. In this paper a simple and efficient method has been proposed for accurate evaluation of through-the-thickness distribution of transverse stresses in composites and sandwich laminates by using a displacement based C⁰ FE model (2D) derived from Refined Higher Order Shear Deformation Theory (RHSDT) and a Least Square Error (LSE) method. The C⁰ FE model satisfies the inter-laminar shear stress continuity conditions at the layer interfaces and zero transverse shear stress conditions at the top and bottom of the plate. In this model the first derivatives of transverse displacement have been treated as independent variables to circumvent the problem of C¹ continuity associated with the above plate theory (RHSDT). The LSE method is applied to the 3D equilibrium equations of the plate problem at the post-processing stage, after in-plane stresses are calculated by using the above FE model based on RHSDT. Thus the proposed method is quite simple and elegant compared to the usual method of integrating the 3D equilibrium equations at the post-processing stage for calculation of transverse stresses in a composite laminate. In the proposed method, the first two equations of equilibrium are utilized to compute the transverse shear stress variation through the thickness of a laminated plate whereas the third equation of equilibrium gives the normal stress variation. Accuracy of the proposed method is demonstrated in the numerical examples through comparison of the present results with those obtained from different models based on higher order shear deformation theory (HSDT) and 3D elasticity solutions.     

Accurate calculation of transverse shear stresses for soft-core sandwich laminates

Accurate evaluation of transverse stresses in soft-core sandwich laminates using the existing 2D finite element (FE) models involves cumbersome post-processing techniques. In this paper, a simple and robust method is proposed for accurate evaluation of through-the-thickness distribution of transverse stresses in soft-core sandwich laminates by using a displacement-based C⁰ continuous 2D FE model derived from refined higher-order shear deformation theory (RHSDT) and a least square error (LSE) method. In this refined higher-order shear deformation theory (RHSDT), the in-plane displacement field for the face sheets and the core is obtained by superposing a global cubically varying displacement field on a zigzag linearly early varying displacement field. The transverse displacement is assumed to have a quadratic variation within the core, and it remains constant in the faces beyond the core. The proposed C⁰ FE model satisfies the condition of transverse shear stress continuity at the layer interfaces and the zero transverse shear stress condition at the top and bottom of the sandwich plate. The nodal field variables are chosen in an efficient manner to circumvent the problem of C¹ continuity requirement of the transverse displacements associated with the RHSDT. The LSE method is applied to the 3D equilibrium equations of the plate problem at the post-processing stage, after in-plane stresses are calculated by using the above FE model based on RHSDT. Thus, the proposed method is quite simple and elegant compared to the usual method of integrating the 3D equilibrium equations at the post-processing stage for the calculation of transverse stresses in a sandwich laminates. The accuracy of the proposed method is demonstrated in the numerical examples through the comparison of the present results with those obtained from different models based on HSDT and 3D elasticity solutions.     

Inter-laminar stresses in a laminated shallow shell panel

The evaluation of inter-laminar shear stresses in laminated shells using 2D finite element models involves cumbersome post-processing techniques. In this paper, a simple and efficient method has been proposed for accurate evaluation of transverse shear stresses in laminated composite shells by using a displacement-based C⁰ FE model derived from higher-order shear deformation theory (HSDT) and a least square error (LSE) method. In order to include the effect of three curvature terms in the strain components of composite shells, Sander’s approximations are followed. In this model, the first derivatives of transverse displacement have been treated as independent variables to overcome the problem of C¹ continuity in the FE implementation associated with the present shell theory (HSDT). The LSE method is applied at the post-processing stage, after in-plane stresses are calculated by using the present FE model based on HSDT. Thus, the proposed method is quite simple compared to the usual method of integrating the 3D equilibrium equations for the calculation of transverse stresses in laminated composite shells. The accuracy of the method is demonstrated in the numerical examples by comparison of the present results with those obtained from different models based on HSDT, exact analytical and 3D elasticity solutions.     

Reddy-type zig-zag model for multilayered composite plate based on the HW variational theorem

Reddy's higher-order theory is quite attractive, but it could not describe a zig-zag shape distribution of in-plane displacement through the thickness direction and violates the continuity of transverse shear stresses at interfaces. This is due to neglect of the zig-zag function in the in-plane displacement field. Thus, a Reddy-type higher-order zig-zag theory is developed for analysis of multilayered composite plates. The developed model differs from existing ones by two features. First, a Reddy-type zig-zag function (RZZF) satisfying the bounding surface free traction condition is constructed. By introducing the RZZF into Reddy's model, a Reddy-type higher-order zig-zag model can be obtained. Second, a functional suitable for composite plate has been presented to obtain improved transverse shear stresses by employing the three-field Hu–Washizu (HW) variational principle. It is significant that the higher-order derivatives of displacement parameters in expression of transverse shear stresses have been eliminated, which is convenient for the model's finite element implementation. Equilibrium equations and analytical solution can be also presented by means of the HW variational principle. The performance of the proposed model is tested with different numerical examples, and numerical results show its accuracy and range of applicability.     

A new FE model based on higher order zigzag theory for the analysis of laminated sandwich beam with soft core

A new finite element (FE) model has been developed based on higher order zigzag theory (HOZT) for the static analysis of laminated sandwich beam with soft core. In this theory, the in-plane displacement variation is considered to be cubic for both the face sheets and the core. The transverse displacement is assumed to vary quadratically within the core while it remains constant in the faces beyond the core. The proposed model satisfies the condition of transverse shear stress continuity at the layer interfaces and the zero transverse shear stress condition at the top and bottom of the beam. The nodal field variables are chosen in an efficient manner to overcome the problem of continuity requirement of the derivatives of transverse displacements. A C₀ quadratic beam finite element is implemented to model the HOZT for the present analysis. Numerical examples covering different features of laminated composite and sandwich beams are presented to illustrate the accuracy of the present model. Many new results are also presented which should be useful for future research.     

Stability Analysis of Laminated Soft Core Sandwich Plates Using Higher Order Zig-Zag Plate Theory

C₀ finite element model based on higher order zig-zag plate theory is used to study the stability analysis of laminated sandwich plates. The in-plane displacement field is obtained by superposing a global cubically varying displacement field on a zig-zag linearly varying displacement field with different slope in each layer. The transverse displacement assumes to have a quadratic variation within the core and constant in the faces. The conditions regarding transverse shear stress at layer interfaces and top and bottom are satisfied. Numerical examples covering different features of laminated sandwich plates are presented to illustrate the accuracy of the model.     

A solution of laminated cylindrical shells using an unconstrained third-order theory

An unconstrained third-order shear deformation theory is presented for the analysis of laminated anisotropic cylindrical shells. Based on the realistic through-thickness distribution of the in-plane displacements, a zig-zag function is used to approximate the piece-wise nature of the displacements. The zero-shearing condition on the laminate surfaces and continuous conditions for the transverse shear stresses on the inter-laminar surfaces have been considered for the final stresses calculation, the displacement functions remain to be unconstrained. This theory is very useful for the finite element analysis because it requests only C⁰ continuity for the assumed displacement fields. By comparing with three-dimensional elasticity theory for laminated orthotropic cylindrical shell, the performance of the present theory is verified. The problems solved in this paper illustrate that the present theory is very accurate for the thin and moderately thick shells.     

复合材料夹层板振动分析的精化锯齿理论和三角形板单元

基于精化锯齿理论,构造了六节点三角形协调板单元并推导了夹层板自由振动问题有限元列式。不同于已有锯齿理论,精化锯齿理论特点是面内位移不含有横向位移一阶导数,构造有限元时仅需要C0 插值函数。为验证单元性能,分析了软核夹层板自由振动问题。结果表明,该文构造的单元能准确计算软核夹层板固有频率,然而基于已有锯齿理论建立的不协调元计算结果精度较低。     

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

The available plate theories have been generally calibrated using linear strain–displacement expressions. Furthermore, many of them do not consider the transverse normal stress continuity and the transverse flexibility of the sandwich plates. Majority of the investigations performed so far in the buckling analysis of the sandwich plates, have been restricted to linear buckling analysis of the perfect sandwich plates based on theories that either violate the continuity condition of the transverse stresses at the layer interfaces or do not satisfy the mentioned condition when nonlinear strain–displacement expressions are used. Therefore, their results may be unreliable for nonlinear stress and buckling (especially in the postbuckling region) analyses. In the present paper, nonlinear strain–displacement expressions are employed for imperfect sandwich plates subjected to thermo-mechanical loads to propose an accurate global–local theory that satisfies the continuity of all of the transverse stress components. The theory is presented in a compact matrix form. Compatible Hermitian elements with C¹ continuity are employed to enhance the results. Buckling and wrinkling loads are detected employing a criterion previously published by the author. Comparisons made in the paper with results reported by well-known references, confirm the accuracy and the efficiency of the proposed theory and the relevant solution algorithm.     

Behavior of laminated sandwich plates having interfacial imperfections by a new refined element

The static response of laminated sandwich plates having imperfections at the layer interfaces is investigated by a refined plate theory. The plate theory represents parabolic through thickness variation of transverse shear stresses, which are continuous at the layer interfaces and become zero at the plate top and bottom surfaces. In this plate model the interfacial imperfection is represented by a linear spring-layer. Moreover, with all these features of an accurate modeling, it involves unknowns only at the reference plane of the plate. To have generality in the analysis, finite element method is adopted. But any existing plate element cannot be used, as the plate theory demands certain inter-elemental continuity. Thus an attempt has also been made to develop a new triangular element. As there is no published result on imperfect sandwich plates, the problems of perfect sandwich plates and ordinary laminated plate with inter-laminar imperfection are used for validation.     

Higher order zig-zag plate theory under thermo-electric-mechanical loads combined

A higher order zig-zag plate theory is developed to refine the predictions of the mechanical, thermal, and electric behaviors partially coupled. The in-plane displacement fields are constructed by superimposing linear zig-zag field to the smooth globally cubic varying field through the thickness. Smooth parabolic distribution through the thickness is assumed in the out-of-plane displacement in order to consider transverse normal deformation and stress. The layer-dependent degrees of freedom of displacement fields are expressed in terms of reference primary degrees of freedom by applying interface continuity conditions as well as bounding surface conditions of transverse shear stresses. Artificial shear correction factors are not needed in the present formulation. Thus the proposed theory has only seven primary unknowns and they do not depend upon the number of layers. Through the numerical examples of partially coupled analysis, the accuracy and efficiency of the present theory are demonstrated. The present theory is suitable in the predictions of deformation and stresses of thick smart composite plate under mechanical, thermal, and electric loads combined.     

A global-local higher order theory including interlaminar stress continuity and C0 plate bending element for cross-ply laminated composite plates

A C⁰-type global-local higher order theory including interlaminar stress continuity is proposed for the cross-ply laminated composite and sandwich plates in this paper, which is able to a priori satisfy the continuity conditions of transverse shear stresses at interfaces. Moreover, total number of unknowns involved in the model is independent of number of layers. Compared to other higher-order theories satisfying the continuity conditions of transverse shear stresses at interfaces, merit of the proposed model is that the first derivatives of transverse displacement w have been taken out from the in-plane displacement fields, so that the C⁰ interpolation functions is only required during its finite element implementation. To verify the present model, a C⁰ three-node triangular element is used for bending analysis of laminated composite and sandwich plates. It ought to be shown that all variables involved in present model are discretized by only using linear interpolation functions within an element. Numerical results show that the C⁰ plate element based on the present theory may accurately calculate transverse shear stresses without any postprocessing, and the present results agree well with those obtained from the C¹-type higher order theory. Compared with the C¹ plate bending element, the present finite element is simple, convenient to use and accurate enough.     

Bending of cross-ply laminated composites: An accurate and efficient plate theory based upon models of Lekhnitskii and Ren

Bending laminated composites results in a distinctive zig-zag shaped deformation pattern, accordingly jumping transverse shear strains at layer interfaces, but continuous courses of transverse shear stresses there. An accurate representation of this laminate-specific mechanical behavior in terms of plate theories is challenging, even more if computational efficiency is aimed for. Here, an axiomatic equivalent single layer plate theory for cross-ply laminated composites is presented, which is based on the work of Lekhnitskii and Ren and delivers accurate deformation and stress prognoses at the cost of six solution variables. Fulfilling transverse stress continuity, the infinitesimal equilibrium equations are considered in order to derive an appropriate ansatz for the transverse shear stresses including the influence of all plane stress reduced stiffness components. However, the effect of the normal stress σ_zz is neglected, and deflection w is assumed constant across the plate thickness. The equilibrium equations and corresponding boundary conditions of the plate theory are derived by application of the principle of virtual displacements. Numerical results for symmetrical and non-symmetrical composites as well as for typical sandwich plates obtained by the present theory show good agreement with corresponding exact elasticity solutions given by Pagano, even for thick plates.     

Refined global–local higher-order theory for angle-ply laminated plates under thermo-mechanical loads and finite element model

To analyze angle-ply laminated composite and sandwich plates coupled bending and extension under thermo-mechanical loading, a refined global–local higher-order theory considering transverse normal strain is presented in this work. Hitherto, present theory for angle-ply laminates has never been reported in the literature, and this theory can satisfy continuity of transverse shear stresses at interfaces. In addition, the number of unknowns in present model is independent of layer numbers of the laminate. Based on this theory as well as methodology of the refined triangular discrete Kirchhoff plate element, a triangular laminated plate element satisfying the requirement of C¹ continuity is presented. Numerical results show that the present refined theory can accurately analyze the bending problems of angle-ply composite and sandwich plates as well as thermal expansion problem of cross-ply plates, and the present refined theory is obviously superior to the existing global–local higher-order theory proposed by Li and Liu [Li XY, Liu D. Generalized laminate theories based on double superposition hypothesis. Int J Numer Meth Eng 1997;40:1197–212]. After ascertaining the accuracy of present model, the distributions of displacements and stresses for angle-ply laminated plates under temperature loads are also given in present work. These results can serve as a reference for future investigations.     

A generalized layerwise higher-order shear deformation theory for laminated composite and sandwich plates based on isogeometric analysis

This paper presents a generalized layerwise higher-order shear deformation theory for laminated composite and sandwich plates. We exploit a higher-order shear deformation theory in each layer such that the continuity of the displacement and transverse shear stresses at the layer interfaces is ensured. Thanks for enforcing the continuity of the displacement and transverse shear stresses at an inner-laminar layer, the minimum number of variables is retained from the present theory in comparison with other layerwise theories. The method requires only five variables, the same as what obtained from the first- and higher-order shear deformation theories. In comparison with the shear deformation theories based on the equivalent single layer, the present theory is capable of producing a higher accuracy for inner-laminar layer shear stresses. The free boundary conditions of transverse shear stresses at the top and bottom surfaces of the plate are fulfilled without any shear correction factors. The discrete system equations are derived from the Galerkin weak form, and the solution is obtained by isogeometric analysis (IGA). The discrete form requires the C¹ continuity of the transverse displacement, and hence NURBS basis functions in IGA naturally ensure this condition. The laminated composite and sandwich plates with various geometries, aspect ratios, stiffness ratios and boundary conditions are studied. The obtained results are compared with the 3D elasticity solution, the analytical as well as numerical solutions based on various plate theories.     

Thermal buckling of cross-ply laminated composite and sandwich plates according to a global higher-order deformation theory

A two-dimensional global higher-order deformation theory is presented for thermal buckling of cross-ply laminated composite and sandwich plates. By using the method of power series expansion of continuous displacement components, a set of fundamental governing equations which can take into account the effects of both transverse shear and normal stresses is derived through the principle of virtual work. Several sets of truncated Mth-order approximate theories are applied to solve the eigenvalue problems of a simply supported multilayered plate. Modal transverse shear and normal stresses can be calculated by integrating the three-dimensional equations of equilibrium in the thickness direction, and satisfying the continuity conditions at the interface between layers and stress boundary conditions at the external surfaces. Numerical results are compared with those of the published three-dimensional layerwise theory in which both in-plane and normal displacements are assumed to be C⁰ continuous in the continuity conditions at the interface between layers. Effects of the difference of displacement continuity conditions between the three-dimensional layerwise theory and the global higher-order theory are clarified in thermal buckling problems of multilayered composite plates.     

A global/local third-order Hermitian displacement field with damaged interfaces and transverse extensibility: analytical formulation

A third-order Hermitian zig-zag plate theory is presented as development of the classical cubic zig-zag displacement field. In addition to the capabilities of the previous model ((i) transverse shear flexibility, (ii) through-the-thickness continuity of the transverse shear stresses, (iii) traction-free condition on the two external surfaces of the laminate and (iv) possibility to study damaged interfaces), the Hermitian model offers some interesting improvements ((i) through-the-thickness linear varying transverse displacement, (ii) evaluation of the normal transverse deformability in general and of the corresponding normal stress in particular, (iii) traction equilibrium condition on the external surfaces and (iv) use of the displacements and transverse shear stresses of the external surfaces as degrees of freedom of the plate model). By means of the virtual work principle of the three-dimensional linear elasticity theory, the two-dimensional equations of motion and boundary conditions are obtained. Some numerical results are finally presented to show the particular nature of the through-the-thickness Hermitian shape functions and to test the model performances in evaluating the transverse normal stress.     

A C-type higher-order theory for bending analysis of laminated composite and sandwich plates

In this paper, a C⁰-type higher-order theory is developed for bending analysis of laminated composite and sandwich plates subjected to thermal/mechanical loads. The total number of unknowns in the present theory is independent of number of layers. The continuity conditions of transverse shear stresses at interfaces are a priori enforced. Moreover, the conditions of zero transverse shear stresses on the upper and lower surfaces are also considered. Based on the developed higher order theory, the typical solutions are presented for comparison. It is very important that the first derivatives of transverse displacement w have been taken out from the in-plane displacement fields of the proposed model, so that its finite element counterparts may avoid using the C¹ interpolation functions. To assess the developed theory, the C¹-type higher-order theory is chosen for comparison. Numerical results show that the present model can accurately predict the thermal/mechanical response of laminated composite and sandwich plates. Moreover, the present model is able to accurately calculated transverse shear stresses directly from constitutive equations without any postprocessing methods.     

Higher-order global-local theory with novel 3D-equilibrium-based corrections for static,frequency, and dynamic analysis of sandwich plates with flexible auxetic cores

In the present article, a high-order global-local theory with three-dimensional elasticity corrections is employed to trace the local and instantaneous variations of lateral deflections and stress components of sandwich plates with auxetic (negative Poisson ratio) cores under static and dynamic loads. Effects of the auxetecity of the core material on the natural frequencies are evaluated as well. The governing equations are extracted based on Hamilton's principle. The main novelties of the present research in comparison to the available literature and previous researches of the second author of the present paper are: (i) Presenting a higher-order global-local plate theory with a novel equilibrium-based three-dimensional elasticity corrections, (ii) Incorporation of the transverse flexibility of the core; a fact that is crucial when studying behaviors of thick or soft core sandwich plates, (iii) Frequency and dynamic behavior analyses (in addition to the traditional static analysis) of sandwich plates with soft cores by means of the presented accurate global-local theory, and (iv) Investigation of the negative Poisson ratio (auxeticity) effects of the core material on the static (stress) and dynamic responses and natural frequencies. All these items are accomplished here, for the first time. Since the transverse shear stresses are extracted based on the three-dimensional elasticity theory, in contract the traditional constitutive-based theories, the inter-laminar continuity condition of the transverse shear stresses is met. The verification results show that the presented finite element formulation leads to highly accurate results, even for thick or soft core sandwich plates. A comprehensive parametric study is accomplished to evaluate effects of the auxeticity of the core material and transverse compliance of the core on the resulting displacement and stress distributions, natural frequencies, and dynamic responses. Results reveal that auxeticity of the core material decreases the global and relative stresses and lateral deflections of the face sheets and the core compliance may lead to asynchronized movements of the face sheets and strengthen the local bending and extensions.