Analysis Of Multilayered Composite Plates Research Articles

A discontinuous Galerkin formulation for nonlinear analysis of multilayered shells refined theories

A novel pure penalty discontinuous Galerkin method is proposed for the geometrically nonlinear analysis of multilayered composite plates and shells, modelled via high-order refined theories. The approach allows to build different two-dimensional equivalent single layer structural models, which are obtained by expressing the covariant components of the displacement field through-the-thickness via Taylor’s polynomial expansion of different order. The problem governing equations are deduced starting from the geometrically nonlinear principle of virtual displacements in a total Lagrangian formulation. They are addressed with a pure penalty discontinuous Galerkin method using Legendre polynomials trial functions. The resulting nonlinear algebraic system is solved by a Newton–Raphson arc-length linearization scheme. Numerical tests involving plates and shells are proposed to validate the method, by comparison with literature benchmark problems and finite element solutions, and to assess its features. The obtained results demonstrate the accuracy of the method as well as the effectiveness of high-order elements.

Static and dynamic NURBS-based isogeometric analysis of composite plates under hygrothermal environment

In this paper, a computationally efficient isogeometric plate model, employing nonpolynomial shear deformation theory (NPSDT), for static and dynamic analysis of laminated and sandwich composite plates under hygrothermal environment is presented. Transient and steady-state response using Rayleigh damping model and fast Fourier transformation of transient response have also been obtained for the hygrothermal environment. The non-uniform rational B-splines (NURBS) based formulation has been considered, which attribute only five-degree of freedom and satisfies the stringent continuity requirement of the NPSDT model (C1-continuity) without any additional variables. A total Lagrangian approach in conjunction with Hamilton’s principle is utilized to formulate the governing equations for thermal bending and subsequent dynamic analysis of multilayered composite plates. To model stress stiffening effect due to hygrothermal load, both von Kármán and Green–Lagrange strain displacement relations are incorporated and obtained solutions are compared. It has been shown that at higher temperature (around one fourth of the critical buckling temperature), the consideration of Green–Lagrange strain relationship for modeling stiffening effect due to hygrothermal load is important for better accuracy. Further, it has been shown that bending strip is essential in case of higher-order shear deformation theory (HSDT) to get more accurate solution. In addition, the obtained transient damped/undamped solutions may be used to get the natural frequency from their FFT solution.

An Enhancement of the Warping Shear Functions of Refined Zigzag Theory

Abstract The paper presents an enhancement in Refined Zigzag Theory (RZT) for the analysis of multilayered composite plates. In standard RZT, the zigzag functions cannot predict the coupling effect of in-plane displacements for anisotropic multilayered plates, such as angle-ply laminates. From a computational point of view, this undesirable effect leads to a singular stiffness matrix. In this work, the local kinematic field of RZT is enhanced with the other two zigzag functions that allow the coupling effect. In order to assess the accuracy of these new zigzag functions for RZT, results obtained from bending of angle-ply laminated plates are compared with the three-dimensional exact elasticity solutions and other plate models used in the open literature. The numerical results highlight that the enhanced zigzag functions extend the range of applicability of RZT to the study of general angle-ply multilayered structures, maintaining the same seven kinematic unknowns of standard RZT.

Geometrically nonlinear flexural analysis of multilayered composite plate using polynomial and non-polynomial shear deformation theories

In this paper, the geometrically nonlinear bending analysis of multilayered composite plates is carried out through a rigorous comparative study employing Green-Lagrange and von Kármán nonlinearity. The responses are analyzed for various transverse loads and boundary conditions, and emphasized the impending structural condition, which requires the consideration of full geometric nonlinearity. The study is conducted using a C0 finite element plate model using third-order and non-polynomial shear deformation theories (TSDT and NPSDT). The principle of virtual work is utilized to formulate the weak-form of governing equations, and discretized using nine-noded Lagrange elements with seven degrees-of-freedom per node. The performance of the present model has been validated by comparing present results with those available in the literature and obtained ANSYS results. The results reveal that the consideration of Green-Lagrange nonlinearity is essential for plates with all sides subjected to movable simply supported boundary conditions or with one free edge. The present study also provides a clear idea about the utilization of TSDT and NPSDT for the anti-symmetric and symmetric cross-ply plate, respectively, to get more accurate solution. Further, the effect of penalty for various theories and problems is also highlighted, and its prominent impact is asserted for some cases.

A coupled hygrothermo-mechanical viscoelastic analysis of multilayered composite plates for long-term creep behaviors

This paper investigates the coupling hygrothermo-mechanical behaviors of viscoelastic composite laminated plates by using a higher-order zigzag theory. All in-plane displacement, temperature, and moisture fields through the thickness are superimposed by the local linear zigzag terms on a global cubic polynomial distribution. The Laplace transform is employed to avoid suche time-dependent integration of a viscoelastic constitutive equation as well as to improve accuracy and efficiency. Thereby, all formulations are simplified in the Laplace domain. The hygrothermal variational principle is employed to determine the temperature and moisture distributions through the thickness. Then, viscoelastic relaxation moduli with various hygrothermal shift factors in each layer are calculated in the general form of Prony series. The final numerical results of stresses and deformations are obtained by using inverse Laplace techniques. To demonstrate the efficiency and accuracy of the proposed theory, some numerical examples for long-term creep process are performed. The proposed theory shows powerfully application for the prediction of fully coupled hygro-thermo-mechanical behaviors for thick viscoelastic composite laminates. Moreover, it can handle partial problems such as thermal conductivity, hygral absorption, thermo-elastic, thermo-viscoelastic analyses.

Layerwise mixed elements with node-dependent kinematics for global–local stress analysis of multilayered plates using high-order Legendre expansions

Carrera Unified Formulation (CUF) is taken a step further to render node-dependent kinematics (NDK) capabilities to new layerwise finite elements based on Reissner’s Mixed Variational Theorem (RMVT), especially suited for global–local stress analysis of multilayered plates, ensuring high numerical accuracy and computational efficiency, all together. In the framework of CUF, as introduced originally for multilayered structures, any degree of kinematic refinement can be considered in agreement with Equivalent Single-Layer (ESL) or Layer-Wise (LW) theories to develop advanced finite element models, whether based on the Principle of Virtual Displacements (PVD) or RMVT. The degree of kinematic refinement, which usually holds equally for the entire element, can be taken a step further, by being assigned locally to each of its nodes, making full use of CUF to render NDK capabilities to the elements. Besides, even though the elements can adopt any type of nodal shape functions, high p-order hierarchical Legendre expansions (HLE) can also be combined with NDK, achieving excellent convergence rates. These capabilities combined, explored first under the PVD, are for once integrated in the proposed elements under RMVT to further benefit accurate stress analysis. These elements can be applied throughout the entire mesh, adapting to local, transitional and global regions straightforwardly, providing high numerical accuracy, locally, with minimal computational efforts, globally. The numerical results focus on stress analysis of multilayered composite plates, including local effects, to demonstrate the predictive capabilities of the proposed RMVT-based LW elements with NDK and HLE combined, considering well-known benchmark three-dimensional exact solutions for assessment.

Mesh-free thermal buckling analysis of multilayered composite plates based on an nth-order shear deformation theory

This study aims at exploring buckling behavior of multilayered composite plates including functionally graded material (FGM) layer in thermal environments by using a mesh-free method. The thermal buckling of the composite plate laminated with FGM layer is formulated by an improved Moving Kriging (MK) meshless method based on an nth-order shear deformation theory which enables the optimal order number to provide the best prediction to be chosen. In the improved MK mesh-free method, the covariance basis function is presented in a compactly supported form to build the shape functions with no fitting parameters. Four types of multilayered composite plates with FGM layer subjected to in-plane and through-thickness temperature changes are considered, and the material properties are both position and temperature dependent. Performance accuracy of the proposed mesh-free method is first confirmed by comparing the computed results with the reference solutions available in the literature, followed by the detailed parametric studies in which the effects of the ingredient fraction, FGM constituents, composition scheme, plate geometric parameter and boundary condition are scrutinized focusing on temperature dependency of the material properties.

Efficient three-node triangular element based on a new mixed global-local higher-order theory for multilayered composite plates

An accurate and computationally attractive global-local higher-order theory (GLHT) is developed for the linearly elastic analysis of cross-ply multilayered composite plates. The theory is derived using the kinematic assumptions of GLHT in conjunction with the Reissner mixed variational principle. For a low-order linear element, it is difficult to accurately compute the transverse shear stresses even applying the three-dimensional equilibrium equation post-processing technique. The reason for this difficulty is that the higher-order derivatives of displacement variables are included in the transverse shear stress fields after using the post-processing technique. Thus, by employing the Reissner mixed variational principle, the higher-order derivatives of displacement variables have been removed from the transverse shear stress components before the finite element procedure is implemented. Based on the mixed GLHT, a computationally efficient C0-type three-node triangular plate element with linear interpolation function is proposed for the analysis of multilayered composite plates. The advantage of the present formulation is that no post-processing approach is needed to calculate the transverse shear stresses while maintaining the computational accuracy of a linear plate element. Performance of the proposed element is assessed by comparing with several benchmark solutions. Numerical results show that the present elements can robustly and accurately predict the displacements and stresses of multilayered composite plates.

A new mixed-field theory for bending and vibration analysis of multi-layered composite plate

A novel mixed-field theory with relatively low number of unknown variables is introduced for bending and vibration analysis of multi-layered composite plates. The presented plate theory is derived from a parametrized mixed variational principle which is introduced recently by the first author. A global-local kinematic with a layer-independent number of variables is assumed for the description of the displacements of the plate. The considered kinematic stratifies the displacement and transverse stress continuity conditions at the mutual interfaces of the layers. It also fulfill the homogenous boundary conditions of the shear stresses on the upper/lower surfaces of the plates without using the shear correction factor. One-unknown variable fields which satisfy a priori the continuity conditions at the adjacent interfaces between layers and the zero boundary conditions on the bounding surfaces are considered for the approximation of the transverse shear stresses. The transverse normal stress along the total thickness of the multi-layered plate is approximated via a quadratic polynomial. The presented mixed-field plate theory has been validated through comparison of the bending and vibration analysis results with those obtained from the three-dimensional (3D) theory of elasticity and the results of the other classical and high-order plate theories.

Hygrothermal analysis of multilayered composite plates by variable kinematic finite elements

ABSTRACTAdvanced plate models with variable kinematics for steady state hygrothermal analysis of composite laminates are proposed. The refined models discussed include both layer-wise (LW) and equivalent single layer (ESL) models, and the Carrera Unified Formulation (CUF) is used. The mixed interpolation of tensorial component (MITC) method is applied to a nine-node element to contrast the shear locking phenomena. The governing equations are derived from the principle of virtual displacement (PVD) taking into account elastic mechanical, thermal and hygroscopic effects. Through-the-thickness variations of temperature and moisture concentration are calculated by solving the Fourier equation and the Fick law, respectively. Cross-ply plates with symmetrical lamination and simply supported edges subjected to bisinusoidal thermal/hygroscopic loads are analyzed considering various thickness ratios. Results obtained with assumed linear and calculated temperature/hygroscopic profiles are compared. Variable kinematics with a variety of thickness functions are compared regarding both accuracy and computational costs. The results show that all the kinematics proposed can approximate the transverse shear stress distribution through the thickness with satisfactory accuracy when sufficient expansion terms are adopted. In some cases, miscellaneous expansions can lead to significant reductions in computational costs. The results presented here can be used as benchmark solutions for future works.

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

ABSTRACTReddy'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.

Analysis of multilayered composite plates resting on elastic foundations in thermal environment using a hyperbolic model

This paper is devoted to investigate the effect of hyperbolic thermal condition on the bending response of simply-supported shear-deformable multilayered composite plates resting on elastic foundations using a new hyperbolic model. Through this model, the shear stresses are devolved to zero at the top and bottom surfaces of the plate. The governing equations are based on the present hyperbolic model and include the plate–foundation interaction and thermal effect. The effects of temperature concentration, foundation parameters, transverse shear deformation, aspect ratio, side-to-thickness ratio on deflections and stresses are investigated. Some validation examples are presented to show the accracy of the present model comparing with the corresponding models in the literature.

The Certain Generalized Stresses Method for static analysis of multilayered composite plates with variability of material and physical properties

In this paper, the Certain Generalized Stresses Method (CGSM) is developed for static probabilistic analysis of multilayered composite plates modeled by finite elements. The material and the physical properties are considered as random parameters. The variability of a displacement can be evaluated by a Monte Carlo Simulation (MCS) using an explicit expression. For calculating the variability of a displacement, only one nominal finite element analysis with two load cases is required. The variability of strains, stresses and failure criteria can also be evaluated by using the CGSM. Two examples, a two-layer bending plate and an eight-layer bending plate, are studied. The results are compared with those obtained by the direct MCS, considered as a reference, and those presented in the literature. The comparison shows that the CGSM+MCS approach provides quite accurate results and highlights the high computational efficiency of the proposed method.

Three-dimensional analysis of multi-layer composite plates of arbitrary shape and boundary conditions with shear slip interfaces

ABSTRACTAn accurate semi-analytical method for three-dimensional static analysis of composite plates consisting of several anisotropic layers with shear slip imperfection is presented. The geometry and boundary conditions of the composite plate are arbitrary. At given interfaces, the imperfections are modeled via linear springs. The in-plane displacements are represented by appropriate expressions such that their discontinuity across the interfaces is incorporated accordingly. Also, the out-of-plane displacement normal to the interface must be continuous while the out-of-plane normal strain is discontinuous across the interface. In this approach, the essential boundary conditions are satisfied exactly, and the continuity of the inter-layer traction is captured with high accuracy.

Thermo-elastic non-linear analysis of multilayered plates and shells

Geometrically nonlinear FEM analysis of multilayered composite plates and shells is performed in order to resolve the stability problem of the structures being under the influence of temperature field. The Riks–Wempner–Ramm algorithm with a specially modified multi-choice unloading condition has been implemented in authors’ numerical code. As the representation of multilayered medium the Equivalent Single Layer approach with the First Order Shear Deformation kinematics is employed. The effectiveness of the proposed model is examined in numerical examples with reference solutions available in the literature. Presented study proves that the proposed approach can be very effective in the analysis of stability of thin-walled thermally loaded panels.

A layerwise mixed least-squares finite element model for static analysis of multilayered composite plates

A layerwise finite element model is developed in a mixed least-squares formulation for static analysis of multilayered composite plates. The model assumes a layerwise variable description of displacements, transverse stresses and in-plane strains, taken as independent variables. The mixed formulation allows to completely and a priori fulfil the known C z 0 requirements, which refer to the zig-zag form of displacements in the thickness direction and the interlaminar continuity of transverse stresses, due to compatibility and equilibrium reasons, respectively. This contrasts with layerwise displacement-based models that usually cannot a priori account for the interlaminar continuity of transverse stresses. In addition, the benefit of mixed least-squares formulation, as opposed to mixed weak form models, is that it leads to a variational unconstrained minimization problem, where the finite element approximating spaces can be chosen independently. Numerical examples are shown to assess the layerwise mixed least-squares model predictive capabilities compared to three-dimensional elasticity solutions and also other finite element results available in literature. Most notably, the present model is able to achieve accurate results in very good agreement with three-dimensional solutions and is shown to be insensitive to shear-locking.

Layerwise mixed least-squares finite element models for static and free vibration analysis of multilayered composite plates

Abstract Layerwise finite element models are developed based on a mixed least-squares formulation for both static and free vibration analysis of multilayered composite plates. The models assume a layerwise variable description of displacements, transverse stresses and in-plane strains, taken as independent variables. Altogether, the benefits of this finite element formulation are twofold. First, the layerwise mixed formulation enables the fulfilment of the so-called C z 0 requirements, completely and a priori. Specifically, the interlaminar continuity of displacements and transverse stresses, due to compatibility and equilibrium conditions, are both in the form of C0 continuous functions in the thickness z-direction. Second, the least-squares formulation leads to a variational unconstrained minimization problem, where the finite element approximating spaces can be chosen independently. Therefore, contrary to mixed weak form models, the mixed least-squares models are able to by-pass stability conditions, also known as inf-sup conditions. Ultimately, the model for static analysis yields a symmetric positive definite system of linear equations, whereas the model for free vibration analysis yields a symmetric quadratic eigenvalue problem. The numerical examples show that the models predictive capabilities are in excellent agreement with three-dimensional exact solutions, from very thick to very thin plates, and are shown to be insensitive to shear locking.

Thermo-mechanics of undamaged and damaged multilayered composite plates: a sub-laminates finite element approach

In this work a plate mixed finite element based on a refined zig-zag plate model is formulated for the analysis of multilayered composite plates subjected to thermal and mechanical loads. The zig-zag plate model is characterised by: (i) through-the-thickness cubic in-plane displacements and linear transverse displacement, (ii) transverse shear stresses continuity, (iii) satisfaction of traction equilibrium conditions on top and bottom faces of the laminate thus allowing for the presence of tangential distributed loads, (iv) the transverse normal deformability is taken into account by means of a σ33 supposed constant along the thickness and by the use of the complete constitutive equations of orthotropic materials, (v) use, as degrees of freedom of the model, of displacements and transverse shear stresses of external faces of the laminate. In particular, this last property may be extended to the associated plate finite element that, having its nodal degrees of freedom on the external faces, allows a through-the-thickness discretisation in some sub-domains called sub-laminates. Some numerical results are presented in order to show that the use of more sub-laminates improves the capability of the approach to capture with high accuracy the though-the-thickness distributions of transverse displacements, strains and stresses also for very thick plates.

Optimization of Multilayered Composite Structures with Randomly Distributed Mechanical Properties

The variability of mechanical properties commonly encountered in composite structures should be taken into account in the optimal design. The present paper discusses such a possibility by using the fuzzy-set and antioptimization approaches. The main attention is focused on the formulation of optimization problems. The formulation and methods proposed are successfully implemented in the buckling analysis of multilayered composite plates. The problem of optimum stacking sequence for plates is also solved.

CLOSED-FORM SOLUTIONS TO ASSESS MULTILAYERED-PLATE THEORIES FOR VARIOUS THERMAL STRESS PROBLEMS

This paper represents a further development of the first author's works on two-dimensional modeling for thermal stress analysis of multilayered composite plates. The governing equations are written by referring to the unified compact formulation. These equations have been obtained in a form that is not affected by the order of the expansion in the thickness plate direction z or by variable descriptions (layer-wise models and equivalent single layers models). Classical theories based on the principle of virtual displacements and advanced mixed theories based on the Reissner mixed variational theorem are both considered. As a result, a large variety of theories are derived and compared. The temperature profile T P in the direction z is calculated by solving the heat conduction problem and it is compared to the case in which T P is assumed linear in z. Exact closed-form solutions have been derived for the case of the in-plane harmonic distribution of displacements, transverse stress variables, and temperature fields. The Fourier expansion was then used to solve problems related to uniform, triangular, bitriangular (tentlike), and localized in-plane distribution of temperature. In some cases more than 25 theories were compared. The effect of transverse shear deformation, the zig-zag form of displacement fields, and interlaminar continuity of transverse stresses (both shear and normal components) have been evaluated in the framework of both classical and mixed theories.