Variable kinematic finite element models of multilayered composite plates coupled with acoustic fluid

ABSTRACT

This article presents a novel and advanced finite element formulation of the structural-acoustic problem involving thin and thick multilayered composite plates coupled with a cavity. Exploiting the Carrera's unified formulation, many plate and fluid-structure interface elements based on different kinematic models, including higher-order equivalent single-layer and layerwise theories, are developed within a single mathematical framework. Accordingly, a large number of vibro-acoustic models can be easily obtained and selected according to the accuracy requirements of the application. In particular, it is shown that refined models can be adopted in those cases where models relying on traditional or low-order plate theories fail in providing the correct estimation of the fluid-structure coupling. The proposed formulation is also validated with respect to some reference cases available in the literature.     

Development of semianalytical axisymmetric shell models with embedded sensors and actuators

This paper presents the development of two semianalytical axisymmetric shell finite element models, which have the possibility of having embedded and/or surface-bonded piezoelectric ring actuators and/or sensors. A mixed finite element approach is used, which combines the equivalent single-layer higher-order shear deformation theory, to represent the mechanical behavior with a layerwise discretization in the thickness direction to represent the distribution of the electrical potential of each piezoelectric layer of the frusta conical finite element. The electrical potential function is represented through a layerwise discretization in the thickness direction and can be assumed linear or quadratic with two or six electrical potential ring nodes per piezoelectric layer. The displacement field and the electrical potential are expanded by Fourier series in the circumferential direction, considering symmetric and anti-symmetric terms. Several examples are presented and discussed to illustrate the accuracy and capabilities of both models.     

Optimal design and parameter estimation of frequency dependent viscoelastic laminated sandwich composite plates

Recent developments in optimization and parameter estimation of frequency dependent passive damping of sandwich structures with viscoelastic core are presented in this paper. A finite element model for anisotropic laminated plate structures with viscoelastic frequency dependent core and laminated anisotropic face layers has been formulated, using a mixed layerwise approach, by considering a higher order shear deformation theory (HSDT) to represent the displacement field of the viscoelastic core, and a first order shear deformation theory (FSDT) for the displacement fields of adjacent laminated face layers. The complex modulus approach is used for the viscoelastic material behaviour, and the dynamic problem is solved in the frequency domain, using viscoelastic material data for the core, assuming fractional derivative constitutive models. Constrained optimization of passive damping is conducted for the maximisation of modal loss factors, using the Feasible Arc Interior Point Algorithm (FAIPA). Identification of the frequency dependent material properties of the sandwich core is conducted by estimating the parameters that define the fractional derivative constitutive model. Optimal design and parameter estimation applications in sandwich structures are presented and discussed.     

Dynamic characteristics of cylindrical hybrid panels containing viscoelastic layer based on layerwise mechanics

The viscoelastic damping model of the cylindrical hybrid panels with co-cured, free and constrained layers has been developed and investigated by using the refined finite element method based on the layerwise shell theory. The transverse shear and normal strains and the curved geometry are exactly taken into account in the present layerwise shell model, which can depict the zig-zag in-plane and out-of-plane displacements. The damped natural frequencies, modal loss factors and frequency response functions of cylindrical viscoelastic hybrid panels are compared with those of the base composite panel without a viscoelastic layer. The difference in the free vibration and damping of the thin and thick composite laminates and the viscoelastic sandwiched beam between full and partial layerwise theories is verified by comparison with the published results. Various damping characteristics of cylindrical hybrid panels with free viscoelastic layer, constrained layer damping, and co-cured sandwich laminates are investigated. Present results show that the full layerwise damping model accurately predicted the vibration and damping of the cylindrical hybrid panels with viscoelastic layers.     

Vibration Modeling of Multilayer Composite Structures with Viscoelastic Layers

This work deals with the vibration of orthotropic multilayer sandwich structures with viscoelastic core. A finite element model is derived from a classical zigzag model with shear deformation in the viscoelastic layer. The aim of the present work is to establish numerical models and develop numerical tools to design multilayer composites structures with high damping properties. To fulfill this purpose, a finite element model has been developed for vibration analysis of a sandwich plate (elastic orthotropic)/(viscoelastic orthotropic)/(elastic orthotropic). A numerical study from the variation of the damping properties of the structures was performed according to the faces materials fibers orientation.     

An evaluation of equivalent-single-layer and layerwise theories of composite laminates

A review of equivalent-single-layer and layerwise laminate theories is presented and their computational models are discussed. The layerwise theory advanced by the author is reviewed and a variable displacement finite element model and the mesh superposition techniques are described. The variable displacement finite elements contain several different types of assumed displacement fields. By choosing appropriate terms from the multiple displacement field, an entire array of elements with different orders of kinematic refinement can be formed. The variable kinematic finite elements can be conveniently connected together in a single domain for global-local analyses, where the local regions are modeled with refined kinematic elements. In the finite element mesh superposition technique an independent overlay mesh is superimposed on a global mesh to provide localized refinement for regions of interest regardless of the original global mesh topology. Integration of these two ideas yields a very robust and economical computational tool for global-local analysis to determine three-dimensional effects (e.g. stresses) within localized regions of interest in practical laminated composite structures.     

Quasi-static analysis of multilayered domains with viscoelastic layer using incremental-layerwise finite element method

This paper presents a layerwise finite element formulation for quasi-static analysis of laminated structures with embedded viscoelastic material whose constitutive behavior is represented by the Prony series. To account the time dependence of the constitutive relations of linear viscoelastic materials, the incremental formulation in the temporal domain is used. This approach avoids the use of relaxation functions and mathematical transformations. A computer code based on the presented formulation has been developed to provide the numerical results. The high accuracy of the method is exhibited by comparing the results with existing solutions in the literature and also with those obtained using the ABAQUS software. Finally, and as an application of the presented formulation, the effects of time and load rate on the quasi-static structural response of asphalt concrete (AC) pavements are studied.     

An efficient finite element with layerwise mechanics for smart piezoelectric composite and sandwich shallow shells

In this work, we present a new efficient four-node finite element for shallow multilayered piezoelectric shells, considering layerwise mechanics and electromechanical coupling. The laminate mechanics is based on the zigzag theory that has only seven kinematic degrees of freedom per node. The normal deformation of the piezoelectric layers under the electric field is accounted for without introducing any additional deflection variables. A consistent quadratic variation of the electric potential across the piezoelectric layers with the provision of satisfying the equipotential condition of electroded surfaces is adopted. The performance of the new element is demonstrated for the static response under mechanical and electric potential loads, and for free vibration response of smart shells under different boundary conditions. The predictions are found to be very close to the three dimensional piezoelasticity solutions for hybrid shells made of not only single-material composite substrates, but also sandwich substrates with a soft core for which the equivalent single layer (ESL) theories perform very badly.     

Efficient modeling of smart piezoelectric composite laminates: a review

Current research issues in the development of efficient analysis models and their efficient numerical implementation for smart piezoelectric laminated structures are discussed in this paper. The improved zigzag theories with a layerwise quadratic variation of electric potential have emerged as the best compromise between accuracy and cost for hybrid composite, sandwich and FGM beams and plates. The concept of associating surface potentials to electric nodes and internal potentials to physical nodes is very effective in modeling the equipotential electroded surfaces. Unified formulations for shear and extension mode actuation, and modeling of piezoelectric composite actuators and sensors are discussed. Future challenge lies in developing efficient theories capable of predicting the interlaminar transverse shear stresses in hybrid laminates directly from the constitutive equations.     

A finite element evaluation of single-layer and multi-layer theories for the analysis of laminated plates

A layerwise polynomial expansion along the thickness direction for displacements is assumed to analyse the behaviour of an arbitrary laminated composite plate. In contrast with other proposed approaches and in order to take into account the transverse normal stress distribution, out-of-plane displacements are not assumed to be constant along the thickness. Based on the proposed kinematic assumptions the continuity of the interlaminar stress components at the interface can be also achieved. A finite element procedure is established and plate models are derived in which the stress field is obtained directly from the constitutive relations and not by the integration of the three-dimensional equilibrium equations. Comparisons among the numerical results obtained with the proposed layerwise models, single-layer models, the classical laminate theory and exact three-dimensional elasticity solutions are presented and briefly discussed.     

A solid-shell layerwise finite element for non-linear geometric and material analysis

The application of layerwise theories to correctly model the displacement field of sandwich structures or laminates with high modulus ratios usually employs plate or facet-shell finite element formulations to compute the element stiffness and mass matrices for each layer. In this work an alternative approach is proposed, using a high performance hexahedral finite element to represent the individual layer mass and stiffness. This eight-node hexahedral finite element is formulated based on the application of the enhanced assumed strain method (EAS) to solve several locking pathologies coming from the high aspect ratio of the finite element and the usual incompressibility condition of the core materials. The solid-shell finite element formulation is introduced in the layerwise theory through the definition of a projection operator, based on the finite element variables transformation matrix. The non-linear geometric and material capabilities are introduced into the finite element formulation, allowing for the representation of large displacements, large deformation and material non-linear behaviors. The developed formulation is numerically tested and benchmarked, being validated by using published experimental results obtained from sandwich specimens.     

Modal validation of sandwich shell finite elements based on a p‐order shear deformation theory including zigzag terms

This paper describes a set of improved C⁰‐compatible composite shell finite elements for evaluating the global dynamic response (natural frequencies and mode shapes) of sandwich structures. Combining a through‐the‐thickness displacement approximation of variable high order with a first‐order zigzag function, the proposed finite elements are suited for modelling sandwich plates and doubly curved shells with a non‐uniform thickness and are more accurate than conventional models based on the first‐ and third‐order shear deformation theories, especially in sandwich panels with highly heterogeneous properties. The new finite element model is then validated by a comparison with the standard shell and 3D solid models. From these investigations, it can be concluded that adding a zigzag function even to high‐order polynomial approximations of the through‐the‐thickness displacement is a useful tool for refining the modelling of sandwich structures. In addition, the proposed formulation is sufficiently versatile to represent with the same level of accuracy the behaviour of thin‐to‐thick laminated shells as well as of strongly heterogeneous sandwich structures. Copyright © 2008 John Wiley & Sons, Ltd.     

基于Layerwise理论的共固化粘弹阻尼复合材料动特性分析

共固化粘弹性复合材料兼具结构承载和阻尼减振功能。针对传统的混合单元法在应用于粘弹性夹层复合材料结构阻尼性能分析时存在着前处理困难、计算规模大、精度低以及难以考虑正交各向异性铺层自身损耗能力的缺点,推导了一种基于Layerwise离散层理论的四节点四边形复合材料层合板单元,并利用直接复特征值解法建立了共固化粘弹性复合材料结构的阻尼性能分析方法。将该方法应用于不同的阻尼结构,分析结果与文献中已公开结果和混合单元法的计算结果进行了对比验证。结果表明,基于离散层理论的层合板单元具有计算精度高、前处理建模简单和计算规模小的优点,可有效应用于复杂共固化粘弹性复合材料结构的阻尼性能分析和设计。     

A generalized layerwise finite element for multi-layer damping treatments

This paper presents a 4-node facet type quadrangular shell finite element, based on a layerwise theory, developed for dynamic modelling of laminated structures with viscoelastic damping layers. The bending stiffness of the facet shell element is based on the Reissner–Mindlin assumptions and the plate theory is enriched with a shear locking protection adopting the MITC approach. The membrane component is corrected by using incompatible quadratic modes and the drilling degrees of freedom are introduced through a fictitious stiffness stabilization matrix. Linear static tests, using several pathological tests, showed good and convergent results. Dynamic analysis evaluation is provided by using two eigenproblems with exact analytical solution, as well as a conical sandwich shell with a closed-form analytical solution and a semi-analytical ring finite element solution. The applicability of the proposed finite element to viscoelastic core sandwich plates is assessed through experimental validation.     

An enhanced layerwise finite element using a robust solid-shell formulation approach

The application of layerwise theories to correctly model the displacement field of sandwich structures or laminates with high modulus ratios, usually employs plate or facet shell finite element formulations to compute the element stiffness and mass matrices for each layer. In this work, a different approach is proposed, using a high performance hexahedral finite element to represent the individual layer mass and stiffness. This 8-node hexahedral finite element is formulated based on the application of the enhanced assumed strain method (EAS) to resolve several locking pathologies coming from the high aspect ratios of the finite element and the usual incompressibility condition of the core materials. The solid-shell finite element formulation is introduced in the layerwise theory through the definition of a projection operator, which is based on the finite element variables transformation matrix. The new finite element is tested and the implemented numerical remedies are verified. The results for a soft core sandwich plate are hereby presented to demonstrate the proposed finite element applicability and robustness.     

An extended layerwise/solid-element method of stiffened composite plates with delaminations and transverse crack

The stiffened composite plates with the transverse crack and delamination were studied in this paper, and an extended layerwise/solid-element (XLW/SE) method was developed. In the proposed method, the governing equations of composite plates and stiffeners were established based on the extended layerwise method and 3D solid elements, respectively. The final governing equation of stiffened composite plates is assembled by using the compatibility conditions and internal force equilibrium conditions at the joint interface between the plates and stiffeners. For the stiffened composite plates with damages, the XLW/SE method can obtain the local stress and displacement fields accurately and simulate the in-plane transverse cracks and delaminations simultaneously, considering complicated stiffeners without any assumptions. In the numerical examples, the results obtained by the proposed method are compared with those obtained by the 3D elastic models developed in the general finite element code, and the good agreements were achieved for the stiffened composite plates with/without delaminations and/or transverse crack.     

A Viscoelastic Sandwich Finite Element Model for the Analysis of Passive,Active and Hybrid Structures

In this paper we present a finite element model for the analysis of active sandwich laminated plates with a viscoelastic core and laminated anisotropic face layers, as well as piezoelectric sensor and actuator layers. The model is formulated using a mixed layerwise approach, by considering a higher order shear deformation theory (HSDT) to represent the displacement field of the viscoelastic core and a first order shear deformation theory (FSDT) for the displacement field of the adjacent laminated anisotropic face layers and exterior piezoelectric layers. The dynamic problem is solved in the frequency domain with viscoelastic frequency dependent material properties for the core. Control laws are also implemented for the piezoelectric sensors and actuators. The model behaviour in dynamics is assessed with the few solutions found in the literature, including experimental data, and a laminated composite active sandwich application is proposed. In this numerical application, velocity feedback control law is implemented for active control, using co-located piezoelectric patch sensors and actuators.     

An improved discrete Kirchhoff quadrilateral element based on third‐order zigzag theory for static analysis of composite and sandwich plates

A new improved discrete Kirchhoff quadrilateral element based on the third‐order zigzag theory is developed for the static analysis of composite and sandwich plates. The element has seven degrees of freedom per node, namely, the three displacements, two rotations and two transverse shear strain components at the mid‐surface. The usual requirement of C¹ continuity of interpolation functions of the deflection in the third‐order zigzag theory is circumvented by employing the improved discrete Kirchhoff constraint technique. The element is free from the shear locking. The finite element formulation and the computer program are validated by comparing the results for simply supported plate with the analytical Navier solution of the zigzag theory. Comparison of the present results with those using other available elements based on zigzag theories for composite and sandwich plates establishes the superiority of the present element in respect of simplicity, accuracy and computational efficiency. The accuracy of the zigzag theory is assessed by comparing the finite element results of the square all‐round clamped composite plates with the converged three‐dimensional finite element solution obtained using ABAQUS. The comparisons also establish the superiority of the zigzag theory over the smeared third‐order theory having the same number of degrees of freedom. Copyright © 2006 John Wiley & Sons, Ltd.     

Unified formulation for a family of advanced finite elements for smart multilayered plates

Abstract

Families of layer-wise and equivalent single-layer advanced finite elements for the analysis of smart multilayered plates are formulated in a unified framework. The proposed modeling strategy reduces the multifield problem to an effective mechanical plate by the condensation of the electromechanical state into the plate kinematics, which is assumed as a variable order expansion along the plate thickness. Carrera Unified Formulation is invoked to derive the elemental stiffness and mass matrices and the mechanical and magneto-electric equivalent forces. The obtained smart plate finite element equations involves kinematical variables only and this extends the tools developed for multilayered composite plates to smart laminates. Results for simply-supported magneto-electro-elastic multilayered square plates are presented to validate the proposed modeling approach and finite elements and to investigate their features.