Layerwise Model Research Articles

A time-domain high-order spectral finite element for the simulation of symmetric and anti-symmetric guided waves in laminated composite strips

A time domain spectral finite element is developed for improving the efficiency of numerical simulations of guided waves in laminated composite strips. The finite element relies on a new generalized laminate mechanics model formulated to represent symmetric and anti-symmetric Lamb waves. The laminate mechanics incorporate third-order polynomial terms for the approximation of axial and transverse displacement fields through the thickness and consider the displacements of the upper and lower surfaces as degrees of freedom. The laminate theory formulation is easily expanded to a high-order layerwise model. Based on the resultant governing equations of the laminate section, a new finite element with 8 nodal degrees of freedom is formulated; its nodes are collocated with Gauss–Lobatto–Legendre integration points in order to improve computational efficiency. Stiffness and mass matrices are assembled and the transient response is predicted using the explicit central differences time integration scheme. The transient response of Aluminum, Carbon Fiber Reinforced Polymer laminated and sandwich strips is investigated. Numerical results are validated against a semi-analytical solution. The accuracy and computational efficiency of the introduced element regarding the prediction of symmetric and anti-symmetric wave propagation is also quantified.

Advanced models for free vibration analysis of laminated beams with compact and thin-walled open/closed sections

In this paper, refined one-dimensional beam theories are implemented for the free vibration analysis of laminated beams with compact and thin-walled cross-sections. The proposed models are based on the Carrera Unified Formulation, which was formerly introduced for the analysis of plates and shells and recently extended to beam structures by the first author and his co-workers. Carrera Unified Formulation is a hierarchical modelling technique leading to very accurate and computationally efficient finite element theories. According to the latest developments in the framework of Carrera Unified Formulation, refined beam models are implemented using either Taylor-like or Lagrange-like polynomials in order to expand the unknown kinematic variables on the cross-section of the beam. Equivalent single layer models result from the former approach. On the other hand, if Lagrange polynomials are used, layer-wise models are produced. In this work, a classical one-dimensional finite element formulation along the beam length is used to develop numerical applications. A number of laminated beam structures are analyzed and particular attention is given to laminated box beams with open and closed cross-sections. The frequencies and the mode shapes obtained with the present refined beam elements are compared with solid/shell finite element solutions from the commercial code MSC/Nastran and, when possible, with those found in the literature. The modal assurance criterion is used for model-to-model comparisons so as to demonstrate the enhanced capabilities of the proposed formulation in investigating the free vibration characteristics of both compact and thin-walled box laminated beams.

Variable Kinematics Layerwise Model for Analysis of Bonded Joints

Variable Kinematics Layerwise Model for Analysis of Bonded Joints

Analysis of bonded joints with laminated adherends by a variable kinematics layerwise model

A displacement-based zig-zag plate model with variable in-plane and through-the-thickness representation and fixed degrees of freedom is developed and applied to the analysis of bonded joints. Just the in-plane displacements and the shear rotations of the middle plane are used as primary variables. The continuity functions enable an a priori fulfilment of the out-of-plane stress contact conditions at the interfaces between adjacent layers. High-order contributions to displacements are included to meet the stress boundary conditions at the upper and lower faces and to allow the in-plane representation to be varied, e.g. from the adherends to the overlap. In this way, it is possible to better simulate the variation of solutions and the stress boundary conditions at the ends of the overlap. Closed form expressions of these quantities are obtained using symbolic calculus, which, once incorporated in the model, simplify the obtainment of governing equations and speed-up computations. As the representation can vary from point to point, the present model permits an accurate analysis of laminates with general boundary conditions and of bonded joints under a unified approach. Applications are presented to sample cases of bonded joints with laminated adherends using appropriate series expansions of displacements. Linear and nonlinear benchmark test cases for single- and double-lap joints taken from the literature are considered. The results show that accurate stress predictions are computed with a low computational effort in all the cases considered. A good accuracy is achieved just using one component in the series expansion, which implies solving a 3×3 system.

LEFM Analysis of V-Notched Aluminum Plates Using Higher-Order Layerwise Model

Higher-order layerwise model is proposed to determine stress intensity factors using virtual crack closure technique for V-notched plates. Present method is based on p-convergent approach and adopts the concept of subparametric element. In assumed displacement field, strain-displacement relations and 3-D constitutive equations of a layer are obtained by combination of 2-D and 1-D higher-order shape functions. Thus, it allows independent implementation of p-refinement for in-plane and transversal displacements. In the proposed elements, the integrals of Legendre polynomials and Gauss-Lobatto technique are employed to interpolate displacement fields and to implement numerical quadrature, respectively.

Shell finite element based on the Proper Generalized Decomposition for the modeling of cylindrical composite structures

The introduction of the Proper Generalized Decomposition (PGD) is presented for the layer-wise modeling of heterogeneous cylindrical shells. The displacement field is approximated as a sum of separated functions of the in-plane coordinates and the transverse coordinate. This choice yields to an iterative process that consists of solving a 2D and 1D problem successively at each iteration. In the thickness direction, a fourth-order expansion in each layer is considered. For the in-plane description, classical Finite Element method is used. The approach is assessed through mechanical tests for thin/thick and deep/shallow laminated cylindrical shells. Both convergence rate and accuracy are discussed.

A p-version layerwise model for large deflection of composite plates with curvilinear fibres

A new p-version finite element based on a zig-zag layerwise theory is developed. With the proposed model, one can accurately determine the behaviour of laminated composites with different material properties, straight or curvilinear fibres. Comparative studies with published results and with results from finite element software Abaqus are performed to verify the new model. Deflections in the linear and non-linear regimes are then calculated for several constant and variable stiffness composite laminates (VSCL), in which the fibre orientation angle varies linearly in each layer. In a few cases, VSCL plates show better performance in comparison to CSCL plates both in the linear and non-linear regimes. It is also found that it is possible to take advantage of VSCL by varying fibre orientations only in some plies of a laminate, hence reducing manufacturing costs, as well as the amount of gaps and overlaps.

A layerwise theory for laminated composites in the framework of isogeometric analysis

Through-thickness modeling of laminated composites using a displacement-based isogeometric layerwise theory is presented. Layerwise theories provide accurate predictions of the three-dimensional stress states that are of prime importance in structural design. This is in sharp contrast to the class of equivalent-single-layer theories that yield no or limited information of three-dimensional stress states. The key idea of layerwise theories is to distinguish and separate the functions of approximation employed in the in-plane and out-of-plane directions. The rationale behind this choice emanates from the underlying physics, due to the balance of linear momentum and continuity of traction, the function describing the transverse displacement field should be C0-continuous at the interface between plies of different fiber angle orientation. The latter condition can be naturally facilitated through conscious use of isogeometric refinement schemes. Finally, a multiple model analysis is introduced. The aim is to demonstrate the use of the different models within predefined regions of a single laminate. The multiple model analysis concept is employed to simulate laminates with existing delaminations. The proposed models are verified considering laminated composite plates. The numerical results confirm the accuracy of the proposed models and shown that the isogeometric layerwise model outperforms its Lagrange polynomial-based counterpart.

Assessment of the Refined Zigzag Theory for bending, vibration, and buckling of sandwich plates: a comparative study of different theories

The Refined Zigzag Theory (RZT) belongs to the zigzag class of approximations for the analysis of laminated composite and sandwich structures. This paper presents the derivation of the non-linear equations of motion and consistent boundary conditions of RZT for multilayered plates. Subsequently, the equations are specialized to the linear boundary value problem of bending and the linear eigenvalue problems of free vibrations and buckling. In order to assess the accuracy of RZT, results concerning the static response, the free vibration frequencies and modal shapes, and the buckling loads of symmetric and un-symmetric sandwich plates, both simply supported and clamped and subjected to several loading conditions, are compared to the three-dimensional exact elasticity solution, high-fidelity FEM solutions, classical and zigzag theories, and accurate layer-wise models or solutions obtained in the open literature by means of other methods. The numerical investigation shows that RZT is highly accurate in predicting the static response, the natural frequencies and the buckling loads of sandwich plates without requiring any shear correction factors. In virtue of its accuracy and of the C0-continuity requirement for shape functions, RZT can be adopted to derive reliable and computationally efficient finite elements suited for large-scale analyses of sandwich structures.

Some Results on Thermal Stress of Layered Plates and Shells by Using Unified Formulation

This work presents some results on two-dimensional modelling of thermal stress problems in multilayered structures. The governing equations are written by referring to the Unified Formulation (UF) introduced by the first author. These equations are obtained in a compact form, that doesn't depend on the order of expansion of variables in the thickness direction or the variable description (layer-wise models and equivalent single layers models). Classical and refined theories based on the Principle of Virtual Displacements (PVD) and advanced mixed theories based on the Reissner Mixed Variational Theorem (RMVT) are both considered. As a result, a large variety of theories are derived and compared. The temperature profile along the thickness of the plate/shell is calculated by solving the Fourier's heat conduction equation. Alternatively, thermo-mechanical coupling problems can be considered, in which the thermal variation is influenced by mechanical loading. Exact closed-form solutions are provided for plates and shells, but also the applications of the Ritz method and the Finite Element Method (FEM) are presented.

Proper Generalized Decomposition and layer-wise approach for the modeling of composite plate structures

In the framework of the design of laminated and sandwich structures, the computation of local quantities needs a layer-wise approach. But, the computational cost of such approach increases with the number of layers. In this work, the introduction of the Proper Generalized Decomposition (PGD) is presented for the layer-wise modeling of heterogeneous structures in order to reduce the number of unknowns. The displacement field is approximated as a sum of separated functions of the in-plane coordinates x,y and the transverse coordinate z. This choice yields to an iterative process that consists of solving a 2D and 1D problem successively at each iteration. In the thickness direction, a fourth-order expansion in each layer is considered. For the in-plane description, classical Finite Element method is used.After a preliminary study to show the relevance of the present approach, mechanical tests for thin to thick laminated and sandwich plates with various boundary conditions are presented. The results are compared with elasticity reference solutions.

Assessment of plate theories for free-edge effects

This paper deals with a comparative study of several laminated plate theories with respect to their capability to capture the steep transverse stress gradients occurring in vicinity of free edges. The considered laminated plate theories pertain to the family of Equivalent Single Layer (ESL) as well as Layer-Wise (LW) descriptions. Reference is made to the classical displacement-based approach as well as to a partially mixed variational formulation, which allows to introduce independent assumptions for the transverse stresses and the displacements. Finite element solutions are obtained for free-edge effects that arise in several representative laminates subjected to uniaxial tension. An equivalent stress measure is proposed for assessing the three-dimensional (3D) stress fields predicted by the various theories. It is shown that refined LW models can provide quasi-3D results that compare well with full 3D FEM computations, whereas ESL models fail to capture the free-edge effects. Present results indicate that free-edge effects induced by a ±45° interface are most critical for the accuracy of laminated plate models.

A refined sinus plate finite element for laminated and sandwich structures under mechanical and thermomechanical loads

This paper presents a C1 6-node triangular finite element for multilayered composite plates submitted to mechanical and thermomechanical loads. It is based on the sinus model with layer refinement and includes the transverse normal effect. This kinematics allows to exactly ensure both (i) the continuity conditions for displacements and transverse shear stresses at the interfaces between layers of a laminated structure, and (ii) the boundary conditions at the upper and lower surfaces of the plates. It is important to notice that the number of unknowns is independent of the number of layers. This finite element is able to model both thin and very thick plates without any pathologies of the classic plate finite elements (shear locking, spurious modes, etc.). It is built on the Argyris interpolation for bending and the Ganev interpolation for membrane displacements and transverse shear rotations. The representation of the transverse shear strains by cosine functions allows avoiding shear correction factors. In addition, the transverse normal stress can be accurately recovered from equilibrium equations at the post-processing level, using the high degree of interpolation polynomia.Both mechanical and thermomechanical tests for thin and very thick plates are presented in order to evaluate the capability of this new finite element to give accurate results with respect to exact three-dimensional solutions. Both convergence velocity and accuracy are discussed and this new finite element yields very satisfactory results at a low computational cost. In particular, the transverse shear stresses computed from the constitutive relation are well estimated with regards to other equivalent single layer or even with respect to layerwise models.

Bending analysis of laminated and sandwich plates using a layer-wise stress model

In this paper, the behavior of laminated composites is investigated using several high order or layer-wise finite element calculations. A layer-wise model and its dedicated C0 eight-node finite element have been originally specifically developed for interlaminar stresses analysis in free edge problem [1,2] or recently for bonding study [3]. This model is the core of the present comparisons. It is based on a typical layer-wise description which represents the laminate as a superposition of Reissner plates coupled by interfacial stresses. This element has 5n degrees of freedom per node (n is the layers number) and is able to predict interlaminar stresses. These out-of-plane stresses are deduced directly from constitutive equations without post-processing steps. While previous papers dealt with the accuracy of these interface stresses, in the present paper the accuracy and validity of displacements and stresses are established on classical benchmark bending examples for composites and sandwich plates. The aim is also a better positioning and promoting of this stress approach deriving from the work of Pagano [4], which is not usually studied in this way.

Cylindrical bending of multilayered plates with multi-delamination via a layerwise stress approach

Three-dimensional finite element simulations of delamination propagation in multilayered plates are usually expensive in computational time and memory. In this paper, a two-dimensional layerwise model, called the LS1 model, is introduced as an accurate and effective alternative to the finite element method. The LS1 model is extended to the analysis of delamination growth in multilayered plates subjected to cylindrical bending loading. To this end, a general multilayered plate with arbitrary boundary conditions is investigated. By imposing the unilateral contact conditions at delaminated interfaces, analytical LS1 solutions to a general multi-delamination problem are obtained. As application examples, two classical mode I (Double Cantilever) and mode II (End Notched Flexure) delamination tests are investigated. By considering two types of crack resistance curve, the delamination propagation in composite laminates is studied. The LS1 results are found and compared to those of a finite element model. Very good agreements between the results of the two models prove the efficiency and accuracy of the LS1 model for the simulation of delamination propagation in laminates subjected to cylindrical bending.

Accurate free vibration analysis of thermo-mechanically pre/post-buckled anisotropic multilayered plates based on a refined hierarchical trigonometric Ritz formulation

An advanced variable-kinematics trigonometric Ritz formulation built in the extended framework of the Carrera Unified Formulation (CUF) is developed to cope with free vibration response of anisotropic composite plates in a thermo-mechanical pre/post-buckled state. By using the CUF it is not only possible to introduce different expansion orders for each displacement unknown, but also to retain or discard a particular term with respect to its effectiveness and independently form its location in the displacement field generating the so-called mixed axiomatic/asymptotic theories. The proposed formulation is based on Ritz fundamental primary and secondary stiffness, mass and initial stress nuclei which are built in a unified framework that allows them to be invariant regarding to the used kinematic description, the expansion order adopted for each displacement component and the chosen trial functions. Refined equivalent single layer, zig-zag and layer-wise models are assessed by comparison with the 3D elasticity solution. Convergence and accuracy of the presented formulation are critically examined. Several in-plane mechanical load conditions combined with temperature variations are applied and their influence upon the free vibration response is studied. The effects of significant parameters such as stacking sequence, orthotropic ratio and length-to-thickness ratio on the circular frequency parameters are deeply investigated.

Enhanced layerwise model for laminates with imperfect interfaces – Part 1: Equations and theoretical validation

The aim of this paper is the enhancement and validation of a layerwise model applied to the analysis of laminates with thin layers of an elastic–plastic adhesive. The thin adhesive layers are modelled as imperfect interfaces across which displacement discontinuities exist. In a previous paper, the constitutive equations of the imperfect interfaces were empirically established without following the layerwise logic. The model equations are revisited and a solid theoretical justification of the new enhanced equations is obtained by making use of the Hellinger–Reissner functional. A theoretical validation of the model is performed by comparing its predictions to those of a solid finite element resolution in the case of a T-peel joint. The results of the enhanced version of the model are very accurate whereas those of the previous version are erratic for the considered joint. As compared to the solid finite element method, an important saving in computational cost is achieved.

Enhanced layerwise model for laminates with imperfect interfaces – Part 2: Experimental validation and failure prediction

In this paper, the layerwise model for laminates with imperfect interfaces and enhanced in Part 1 of this work is confronted with experimental results. The model calculations are validated by comparing its sliding predictions in double lap adhesive joints to the sliding measurements performed in a previous paper. The model predictions agree with the experimental results. Finally, the model is applied to the failure analysis in double lap joints and T-peel joints exhibiting adherend, adhesive and cohesive failures. The model calculations and pertinent failure criteria provide accurate predictions and may become a helpful tool suitable to the design of adhesive joints.

Advanced variable kinematics Ritz and Galerkin formulations for accurate buckling and vibration analysis of anisotropic laminated composite plates

Accurate free-vibrations and linearized buckling analysis of anisotropic laminated plates with different lamination schemes and simply supported boundary condition are addressed in this paper. Approximation methods, such as Rayleigh-Ritz, Galerkin and Generalized Galerkin, based on Principle of Virtual Displacement are derived in the framework of Carrera’s Unified Formulation (CUF). CUF widely used in the analysis of composite laminate beams, plates and shells, have been here formulated both for the same and different expansion orders, for the displacement components, in the thickness layer-plate direction. An extensive assessment of advanced and refined plate theories, which include Equivalent single Layer (ESL), Zig-Zag (ZZ) and Layer-wise (LW) models, with increasing number of displacement variables is provided. Accuracy of the results is shown to increase by refining the theories. Convergence studies are made in order to demonstrate that accurate results are obtained examining thin and thick plates using trigonometric approximation functions. The effects of boundary terms, upon frequency parameters and critical loads are evaluated. The effects of the various parameters (material, number of layers, fiber orientation, thickness ratio, orthotropic ratio) upon the frequencies and critical loads are discussed as well. Numerical results are compared with 3D exact solution when available from the open literature.

Axisymmetric bending of functionally graded circular magneto-electro-elastic plates

Axisymmetric bending of functionally graded circular magneto-electro-elastic plates of transversely isotropic materials is analyzed based on linear three-dimensional theory of elasticity coupled with magnetic and electric fields. The transverse loads are expanded in Fourier-Bessel series and therefore can be arbitrarily distributed along the radial direction . The radial distributions of the displacements are assumed in combination of Fourier-Bessel series and polynomials as well as the electric potential and magnetic potential. If the material properties obey the exponential law along the thickness of the plate, two three-dimensional exact solutions for two unusual boundary conditions can be derived since they satisfy the governing equations and specified boundary conditions point by point. For simply supported or clamped boundary, the obtained solutions satisfy the governing equations exactly and the boundary conditions approximately. A layer wise model is also introduced to treat with the plates whose material property components vary independently and arbitrarily along the thickness of the plates. The numerical results are finally tabulated and plotted to demonstrate the presented method and agree well with those from finite element methods .