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.     

3D‐based hp‐adaptive first‐order shell finite element for modelling and analysis of complex structures—Part 2: Application to structural analysis

The paper presents a 3D‐based adaptive first‐order shell finite element to be applied to hierarchical modelling and adaptive analysis of complex structures. The main feature of the element is that it is equipped with 3D degrees of freedom, while its mechanical model corresponds to classical first‐order shell theory. Other useful features of the element are its modelling and adaptive capabilities. The element is assigned to hierarchical modelling and hpq‐adaptive analysis of shell parts of complex structures consisting of solid, thick‐ and thin‐shell parts, as well as of transition zones, where h, p and q denote the mesh density parameter and the longitudinal and transverse orders of approximation, respectively. The proposed hp‐adaptive first‐order shell element can be joined with 3D‐based hpq‐adaptive hierarchical shell elements or 3D hpp‐adaptive solid elements by means of the family of 3D‐based hpq/hp‐ or hpp/hp‐adaptive transition elements. The main objective of the first part of our research, presented in the first part of the paper, was to provide non‐standard information on the original parts of the element algorithm. Here we describe the second part of the research, devoted to the methodology and results of the application of the element to various plate and shell problems. The main objective of this part is to verify algorithms of the element and to show its usefulness in modelling and adaptive analysis of shell and plate parts of complex structures. In order to do that, there is a presentation of the results of a comparative analysis of model plate and shell problems using the classical and our elements, and equidistributed and integrated Legendre shape functions. For the plate problem a comparison of the results obtained from the adaptive and non‐adaptive analysis is also included. Additionally, some advantages of the application of our element are shown through a comparative analysis of p‐convergence of the thin plate problem and an adaptive analysis of the exemplary complex structure. Copyright © 2006 John Wiley & Sons, Ltd.     

A unified 3D‐based technique for plate bending analysis using scaled boundary finite element method

This paper presents a unified technique for solving the plate bending problems by extending the scaled boundary finite element method. The formulation is based on the three‐dimensional governing equation without enforcing the kinematics of plate theory. Only the in‐plane dimensions are discretised into finite elements. Any two‐dimensional displacement‐based elements can be employed. The solution along the thickness is expressed analytically by using a matrix function. The proposed technique is consistent with the three‐dimensional theory and applicable to both thick and thin plates without exhibiting the numerical locking phenomenon. Moreover, the use of higher order spectral elements allows the proposed technique to better represent curved boundaries and to achieve high accuracy and fast convergence. Numerical examples of various plate structures with different thickness‐to‐length ratios demonstrate the applicability and accuracy of the proposed technique. Copyright © 2012 John Wiley & Sons, Ltd.     

A triangular plate element for thermo‐elastic analysis of sandwich panels with a functionally graded core

A sandwich construction is commonly composed of a single soft isotropic core with relatively stiff orthotropic face sheets. The stiffness of the core may be functionally graded through the thickness in order to reduce the interfacial shear stresses. In analysing sandwich panels with a functionally gradient core, the three‐dimensional conventional finite elements or elements based on the layerwise (zig‐zag) theory can be used. Although these elements accurately model a sandwich panel, they are computationally costly when the core is modelled as composed of several layers due to its grading material properties. An alternative to these elements is an element based on a single‐layer plate theory in which the weighted‐average field variablescapture the panel deformation in the thickness direction. This study presents a new triangular finite element based on {3,2}‐order single‐layer theory for modelling thick sandwich panels with or without a functionally graded core subjected to thermo‐mechanical loading. A hybrid energy functional is employed in the derivation of the element because of a C¹ interelement continuity requirement. The variations of temperature and distributed loading acting on the top and bottom surfaces are non‐uniform. The temperature also varies arbitrarily through the thickness. Copyright © 2006 John Wiley & Sons, Ltd.     

Experimental study on mechanical properties and fracture toughness of structural thick plate and its butt weld along thickness and at low temperatures

The thick plate induces the variation of mechanical properties and fracture toughness, especially in cold regions. At the low temperature, the brittle behaviour of steel becomes worse. A series of tests (such as uniaxial tensile test and three‐point bending test) were carried out at low temperature to investigate the mechanical properties and fracture toughness of structural steel plates of Q345B with thickness of 60 to 150 mm, as well as the fracture toughness of 150 mm thick butt welded plate. The test specimens are all manufactured from plates along thickness with small size, and the tensile test specimens included through‐thickness specimens additionally. The ductility index (percentage reduction of area) and the fracture toughness index (critical CTOD values) all decrease with the temperature decreases and the distance from plate surface increases. The results obtained in this paper provide technical basis for preventing brittle fracture of thick plate steel structures in cold regions.     

A coupled hp‐finite element scheme for the solution of two‐dimensional electrostrictive materials

As part of the ongoing research within the field of computational analysis for the coupled electro‐magneto‐mechanical response of smart materials, the problem of linearised electrostriction is revisited and analysed for the first time using the framework of hp‐finite elements. The governing equations modelling the physics of the dielectric are suitably modified by introducing a new total Cauchy stress tensor (A. Dorfmann and R.W. Ogden. Nonlinear electroelasticity. Acta Mechanica, 174:167–183, 2005), which includes the electrostrictive effect and a staggered partitioned scheme for the numerical solution of the coupling phenomena. With the purpose of benchmarking numerical results, the problem of an infinite electrostrictive plate with a circular/elliptical dielectric insert is revisited. The presented analytical solution is based on the theoretical framework for two‐dimensional electrostriction proposed by Knops (R.J. Knops. Two‐dimensional electrostriction. Quarterly Journal of Mechanics and Applied Mathematics, 16:377–388, 1963) and uses classical techniques of complex variable analysis. Our presentation, to the best of our knowledge, provides the first correct closed form expression for the solution to the infinite electrostrictive plate with a circular/elliptical dielectric insert, correcting the errors made in previous presentations of this problem. We use this analytical solution to assess the accuracy, efficiency and robustness of the hp‐formulation in the case of nearly incompressible electrostrictive materials. Copyright © 2012 John Wiley & Sons, Ltd.     

Vibration analysis of variable stiffness laminated composite sandwich plates

Abstract

This paper presents the free vibration analysis of a variable stiffness laminated composite sandwich plates. The fiber orientation angle of the face sheets (Skin) is assumed to vary linearly with the x-axis. The problem formulation is based on the higher-order shear deformation plate theory HDST C⁰ coupled with p-version of finite element method. The elements of the stiffness and mass matrices are calculated analytically. The sandwich plate is presented with a uniform mesh of four p-elements and the convergence properties are achieved by increasing the degree p of the hierarchical shape functions. A calculation program is developed to determine the fundamental frequencies for different physical and mechanical parameters such as plate thickness, core to face sheets thickness ratio, orientation angle of curvilinear fibers and boundary conditions. The results obtained show a good agreement with the available solutions in the literature. New comparison study of vibration response of laminated sandwich plate between the straight and curvilinear fibers is presented.     

A linearised hp–finite element framework for acousto‐magneto‐mechanical coupling in axisymmetric MRI scanners

We propose a new computational framework for the treatment of acousto‐magneto‐mechanical coupling that arises in low‐frequency electro‐magneto‐mechanical systems such as magnetic resonance imaging scanners. Our transient Newton–Raphson strategy involves the solution of a monolithic system obtained from the linearisation of the coupled system of equations. Moreover, this framework, in the case of excitation from static and harmonic current sources, allows us to propose a simple linearised system and rigorously motivate a single‐step strategy for understanding the response of systems under different frequencies of excitation. Motivated by the need to solve industrial problems rapidly, we restrict ourselves to solving problems consisting of axisymmetric geometries and current sources. Our treatment also discusses in detail the computational requirements for the solution of these coupled problems on unbounded domains and the accurate discretisation of the fields using hp–finite elements. We include a set of academic and industrially relevant examples to benchmark and illustrate our approach. Copyright © 2017 The Authors. International Journal for Numerical Methods in Engineering Published by John Wiley & Sons, Ltd.     

Classical and advanced multilayered plate elements based upon PVD and RMVT. Part 2: Numerical implementations

This paper presents numerical evaluations related to the multilayered plate elements which were proposed in the companion paper (Part 1). Two‐dimensional modellings with linear and higher‐order (up to fourth order) expansion in the z‐plate/layer thickness direction have been implemented for both displacements and transverse stresses. Layer‐wise as well as equivalent single‐layer modellings are considered on both frameworks of the principle of virtual displacements and Reissner mixed variational theorem. Such a variety has led to the implementation of 22 plate theories. As far as finite element approximation is concerned, three quadrilaters have been considered (four‐, eight‐ and nine‐noded plate elements). As a result, 22×3 different finite plate elements have been compared in the present analysis. The automatic procedure described in Part 1, which made extensive use of indicial notations, has herein been referred to in the considered computer implementations. An assessment has been made as far as convergence rates, numerical integrations and comparison to correspondent closed‐form solutions are concerned. Extensive comparison to early and recently available results has been made for sample problems related to laminated and sandwich structures. Classical formulations, full mixed, hybrid, as well as three‐dimensional solutions have been considered in such a comparison. Numerical substantiation of the importance of the fulfilment of zig‐zag effects and interlaminar equilibria is given. The superiority of RMVT formulated finite elements over those related to PVD has been concluded. Two test cases are proposed as ‘desk‐beds’ to establish the accuracy of the several theories. Results related to all the developed theories are presented for the first test case. The second test case, which is related to sandwich plates, restricts the comparison to the most significant implemented finite elements. It is proposed to refer to these test cases to establish the accuracy of existing or new higher‐order, refined or improved finite elements for multilayered plate analyses. Copyright © 2002 John Wiley & Sons, Ltd.     

A new triangular element to model inter‐laminar shear stress continuous plate theory

An efficient triangular element based on an inter‐laminar shear stress continuous plate theory is developed and applied to the analysis of composite and sandwich plates under different situations to study the performance of the element. The plate theory represents parabolic through thickness variation of transverse shear stresses where the continuity condition of these stresses are satisfied at the layer interfaces. It also satisfies transverse shear stress free condition at the top and bottom surfaces of the plate. The most attractive feature of the plate theory is that the basic unknowns are same as those used in first‐order shear deformation theory. The only problem lies with this elegant plate theory is found in its finite element implementation, as it requires C¹ continuity of transverse displacement at the element interfaces. This is a well‐known problem of thin plate elements, which is also found in some other refined plate theories. Although there are some elements based on these refined plate theories but the number of such elements is very few and they possess certain drawbacks in general. Keeping these aspects in view, an attempt has been made in this study to develop a six‐noded triangular element having equal degrees of freedom at each node. Copyright © 2004 John Wiley & Sons, Ltd.     

纯钛燃料电池双极板软模成形工艺研究

目的 研究软模成形过程中塑性应变比r值对双极板成形深度及壁厚的影响,探究不同工艺参数对双极板尺寸的影响规律。方法 通过单向拉伸实验得到纯钛极薄带的力学性能参数,然后采用橡胶软模成形方法制备纯钛燃料电池双极板,利用光学显微镜对制备的双极板尺寸及壁厚进行测量并深入分析。结果 TD取向的r值最大为2.56,沿该方向成形时,纯钛极薄带在载荷为300 kN、软模硬度为77HA条件下得到的双极板深度最大,为0.293 mm;同时,其壁厚减薄较小,在减薄最严重的位置壁厚减薄率仅为13.52%。结论 较大的载荷与适宜的软模硬度能得到较好的双极板深度,对双极板周期无影响;双极板深度、壁厚与r值有关,r值越大,纯钛极薄带抵抗壁厚减薄的能力越强,成形深度越大。     

Development of shear locking‐free shell elements using an enhanced assumed strain formulation

The degenerated approach for shell elements of Ahmad and co‐workers is revisited in this paper. To avoid transverse shear locking effects in four‐node bilinear elements, an alternative formulation based on the enhanced assumed strain (EAS) method of Simo and Rifai is proposed directed towards the transverse shear terms of the strain field. In the first part of the work the analysis of the null transverse shear strain subspace for the degenerated element and also for the selective reduced integration (SRI) and assumed natural strain (ANS) formulations is carried out. Locking effects are then justified by the inability of the null transverse shear strain subspace, implicitly defined by a given finite element, to properly reproduce the required displacement patterns. Illustrating the proposed approach, a remarkably simple single‐element test is described where ANS formulation fails to converge to the correct results, being characterized by the same performance as the degenerated shell element. The adequate enhancement of the null transverse shear strain subspace is provided by the EAS method, enforcing Kirchhoff hypothesis for low thickness values and leading to a framework for the development of shear‐locking‐free shell elements. Numerical linear elastic tests show improved results obtained with the proposed formulation. Copyright © 2001 John Wiley & Sons, Ltd.     

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.     

3D meso-mechanical analysis of concrete specimens under biaxial loading

In recent years, the mechanics of materials group at ETSECCPB‐UPC has developed an approach for meso‐mechanical analysis of concrete using zero‐thickness interface elements in 2D and more recently in 3D. In this methodology, the meso‐structure is generated with in‐house developed computer programs based on Voronoï/Delaunay theory. In the analysis, continuum elements are assumed linear elastic. Non‐linearity and fracture phenomena are made possible by the systematic use of zero‐thickness interface elements inserted on a priori determined potential fracture planes. In this paper, the results obtained for a 3D specimen under biaxial loading are presented. The results turn out to be very satisfactory and, in particular, it is observed that even the specimens which contain a reduced number of aggregates (14 in the present calculations) lead to a realistic failure envelope under biaxial loading, and they also capture the tendencies of cracking and fracture orientations observed in experiments for different rates of biaxial loading. The special limit case of biaxial loading under restrained out‐of‐plane deformations is also analysed, leading to practically elastic behaviour as shown by available experimental evidence.     

C0 REISSNER–MINDLIN MULTILAYERED PLATE ELEMENTS INCLUDING ZIG-ZAG AND INTERLAMINAR STRESS CONTINUITY

Concerning composites plate theories and FEM (Finite Element Method) applications this paper presents some multilayered plate elements which meet computational requirements and include both the zig-zag distribution along the thickness co-ordinate of the in-plane displacements and the interlaminar continuity (equilibrium) for the transverse shear stresses. This is viewed as the extension to multilayered structures of well-known C⁰ Reissner–Mindlin finite plate elements. Two different fields along the plate thickness co-ordinate are assumed for the transverse shear stresses and for the displacements, respectively. In order to eliminate stress unknowns, reference is made to a Reissner mixed variational theorem. Sample tests have shown that the proposed elements, named RMZC, numerically work as the standard Reissner–Mindlin ones. Furthermore, comparisons with other results related to available higher-order shear deformation theories and to three-dimensional solutions have demonstrated the good performance of the RMZC elements.     

Mixed plate bending elements based on least‐squares formulation

A finite element formulation for the bending of thin and thick plates based on least‐squares variational principles is presented. Finite element models for both the classical plate theory and the first‐order shear deformation plate theory (also known as the Kirchhoff and Mindlin plate theories, respectively) are considered. High‐order nodal expansions are used to construct the discrete finite element model based on the least‐squares formulation. Exponentially fast decay of the least‐squares functional, which is constructed using the L₂ norms of the equations residuals, is verified for increasing order of the nodal expansions. Numerical examples for the bending of circular, rectangular and skew plates with various boundary conditions and plate thickness are presented to demonstrate the predictive capability and robustness of the new plate bending elements. Plate bending elements based on this formulation are shown to be insensitive to both shear‐locking and geometric distortions. Copyright © 2004 John Wiley & Sons, Ltd.