Nonlinear behaviour of piezoceramics under weak electric fields. Part-II: Numerical results and validation with experiments

When excited near resonance in the presence of weak electric fields, piezoceramic materials exhibit typical nonlinearities similar to a Duffing oscillator such as jump phenomena and presence of superharmonics in the response spectra. In an accompanying paper, a generalized nonlinear 3D finite element formulation has been developed incorporating quadratic and cubic terms in the electric enthalpy density function and the virtual work done by damping forces. In this paper, the formulation has been validated by conducting experiments on test pieces of various geometries and of three different materials (in all, four case studies). Both proportional damping and nonlinear damping formulations have been used to predict the frequency response of these systems. Newmark-β method has been used to obtain the dynamic response of the systems using FE analysis. It is demonstrated that the nonlinear finite element model is able to predict the responses of the various test cases studied and the results match very well with those of experimental observations.     

Non-linear longitudinal vibrations of piezoceramics excited by weak electric fields

Typical non-linear effects, e.g. dependence of the resonance frequency on the amplitude, superharmonics in spectra and a non-linear relationship between excitation voltage and vibration amplitude as well as jump phenomena are observed in experiments with piezoceramics excited at resonance by weak electric fields. These non-linear effects can be observed for both the piezoelectric 31- and the 33-effect. In contrast to the well-known non-linear effects exhibited by piezoceramics in the presence of strong electric fields, these effects are not described in detail in the literature.In this paper, we attempt to model these phenomena using an electric enthalpy density to capture the cubic-like effects observed in the experiments. The equations of motion for the system under consideration are derived via the Ritz method using Hamilton's principle. The ‘non-linear’ parameters are identified and the numerical results are compared to those obtained experimentally. The effects described herein may have a significant influence in structures excited close to resonance frequencies via piezoelectric elements.     

An analytical formulation in 3D domain for the nonlinear response of piezoelectric slabs under weak electric fields

Piezoceramic materials exhibit different types of nonlinearities depending upon the magnitude of the mechanical and electric field strength within the body. Some of the nonlinear phenomena observed under weak electric fields near resonance frequency excitation are the presence of superharmonics in the response spectra and the jump phenomena etc. In this work, an analytical solution for the nonlinear response of rectangular piezoceramic slabs have been obtained by Rayleigh–Ritz method and perturbation technique in the 3-D domain using a generalized nonlinear electric enthalpy density function. Forced vibration experiments (excitation with electric field) have been conducted on a rectangular piezoceramic slab at varying electric field amplitudes and the analytical solutions have been shown to compare very well with the experimental results.     

Static wetting on flexible substrates: a finite element formulation

In static wetting on an elastic substrate, force exerted by the liquid–vapour surface tension on a solid surface deforms the substrate, producing a capillary ridge along the contact line. This paper presents a finite element formulation for predicting elastic deformation, close to the static wetting line (with angle of contact=90o and σSV=σSL).The substrate deformation is modelled with the Mooney–Rivlin constitutive law for incompressible rubber‐like solids. At the contact line, a stress singularity is known to arise, due to the surface tension acting on a line of infinitesimal thickness. To relive the stress singularity, either (i) the surface tension is applied over a finite contact region (of macroscopic thickness), or (ii) the solid crease angle is fixed. These two options suggest that normal component of Neumann's triangle law of forces, for the three surface tensions, is not applicable for elastic substrates (as for rigid ones). The vertical displacement of the contact line is a strong function of liquid/vapour surface tension and shear modulus of the solid. Copyright 2004 John Wiley & Sons, Ltd.     

Analytical formulation of 2-D aeroelastic model in weak ground effect

This paper deals with the aeroelastic modeling and analysis of a 2-D oscillating airfoil in ground effect, elastically constrained by linear and torsional springs and immersed in an incompressible potential flow (typical section) at a finite distance from the ground. This work aims to extend Theodorsen theory, valid in an unbounded flow domain, to the case of weak ground effect, i.e., for clearances above half the airfoil chord. The key point is the determination of the aerodynamic loads, first in the frequency domain and then in the time domain, accounting for their dependence on the ground distance. The method of images is exploited in order to comply with the impermeability condition on the ground. The new integral equation in the unknown vortex distribution along the chord and the wake is solved using asymptotic expansions in the perturbation parameter defined as the inverse of the non-dimensional ground clearance of the airfoil. The mathematical model describing the aeroelastic system is transformed from the frequency domain into the time domain and then in a pure differential form using a finite-state aerodynamic approximation (augmented states). The typical section, which the developed theory is applied to, is obtained as a reduced model of a wing box finite element representation, thus allowing comparison with the corresponding aeroelastic analysis carried out by a commercial solver based on a 3-D lifting surface aerodynamic model. Stability (flutter margins) and response of the airfoil both in frequency and time domains are then investigated. In particular, within the developed theory, the solution of the Wagner problem can be directly achieved confirming an asymptotic trend of the aerodynamic coefficients toward the steady-state conditions different from that relative to the unbounded domain case. The dependence of flutter speed and the frequency response functions on ground clearance is highlighted, showing the usefulness of this approach in efficiently and robustly accounting for the presence of the ground when unsteady analysis of elastic lifting surfaces in weak ground effect is required.     

A non-linear 3-D finite element analysis of soil failure with tillage tools

A non-linear 3-D finite element model was developed to study the soil failure under a narrow tillage blade. The weighted residual method was applied to formulate the finite element model. The Duncan and Chang hyperbolic stress-strain model was used in the analysis. This finite element model also takes into account friction at the soil-tool interface, and progressive and continuous cutting. A FORTRAN program was written to carry out the finite element analysis. The results provided soil forces, a progressive developed failure zone, displacement field and stress distribution along the tool surface. Tillage were conducted in the laboratory soil bin to verify soil forces from the finite element analysis. The comparison between the results from the finite element model and those from the soil bin tests was reasonably good.     

Variational formulation of a three-dimensional surface-related solid-shell finite element

The solution of structural analysis problems, especially of shell structures, demands an efficient numerical solution strategy. Since unilateral contact problems are investigated, the shell model is formulated with respect to one of the outer surfaces, i.e., the shell formulation is surface-related. In particular, the investigation of textile reinforced strengthening layers (Brameshuber (ed.) in State-of-the-Art Report of RILEM Technical Commitee 201—TRC, 2006) will be carried out by this approach. Since shells are three-dimensional structures, i.e., bodies, the field equations of continuum mechanics are the starting point. This set of partial differential equations with pertinent boundary conditions has to be solved. An efficient numerical solution of this problem becomes easier, if the problem is reformulated using variational formalism. A corresponding mathematically abstract formulation of the underlying variational principle of the three-dimensional surface-related solid-shell finite element is stated. The discretization of the mathematically abstract principle is, among others, the source of several locking phenomena. The presented shell formulation assumes linear shell kinematics with six displacement parameters, circumventing a rotation formulation. This low-order shell kinematics produces parasitical strains and stresses, leading to poor approximations of the solution or even useless results. Therewith, extensions and/or adjustments of well-known techniques to prevent or at least reduce locking like the assumed natural strain method (Simo and Hughes in J Appl Mech 53:52–54, 1986) and the enhanced assumed strain method (Simo and Rifai in Int J Numer Methods Eng 29:1595–1638, 1990) have to be carried out. Using these adapted methods, a reliable and efficient solid-shell element with tremendously reduced locking properties is obtained. This concept comprises the utilization of unmodified three-dimensional constitutive relations by a minimal number of kinematical parameters. Finally, two nonlinear examples illustrate the reliability and the efficiency of the new solid-shell element.     

Nonlinear finite element analysis of no-tension masonry structures

A numerical approach for structural analysis of masonry walls in plane stress conditions is presented. The assumption of a perfectly no-tension material (NTM) constitutive model, whose relevant equations are in the form of classical rate-independent associated flow laws of elastoplastic material, allows one to adopt numerical procedures commonly used in computational plasticity. An accuracy analysis on the integration algorithm employed in the solution of constitutive relations has been carried out. The results obtained for some relevant case-studies and their comparison with data, available in the literature show the effectiveness of the proposed method.

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Sommario Si presenta un approccio numerico per l'analisi strutturale di pareti in muratura in stato piano di tensione. L'assunzione di un modello costitutivo per materiale perfettamente non resistente a trazione (NTM), le cui equazioni sono esprimibili nella classica forma incrementale delle leggi dello scorrimento plastico di tipo associato per materiali elastoplastici, consente di adottare procedure numeriche comunemente utilizzate in plasticità computazionale. Si conduce un'analisi di accuratezza dell'algoritmo utilizzato nell'integrazione delle equazioni costitutive del modello esaminato. I risultati ottenuti per alcuni casi analizzati ed il confronto effettuato con esempi riportati in letteratura mostrano l'efficienza dell'approccio proposto.

     

钢－混凝土组合梁非线性有限元分析

提出了部分剪力连接钢-混凝土组合梁非线性有限元分析的简化模型，该模型同时考虑了钢、混凝土材料的非线性本构关系和剪力连接件的滑移非线性对组合梁刚度和强度的影响，推导了考虑滑移的剪力连接件单元刚度矩阵。利用该模型计算了连续组合梁的极限荷载、挠度、应力以及滑移，与已有的理论计算结果和实验结果吻合，证明本方法分析钢-混凝土组合梁非线性的可靠性和实用性。     

Nonlinear finite element analysis of laminated composite spherical shell vibration under uniform thermal loading

In this article, nonlinear free vibration behavior of laminated composite shallow shell under uniform temperature load is investigated. The mid-plane kinematics of the laminated shell is evaluated based on higher order shear deformation theory to count the out of plane shear stresses and strains accurately. The nonlinearity in geometry is taken in Green-Lagrange sense due to the thermal load. In addition to that, all the nonlinear higher order terms are taken in the mathematical model to capture the original flexure of laminated panel. A nonlinear finite element model is proposed to discretise the developed model and the governing equations are derived using Hamilton’s principle. The sets of governing equations are solved using a direct iterative method. In order to validate the model, the results are compared with the available published literature and the limitations of the existing models have been discussed. Finally, some numerical experimentation has been done using the developed nonlinear model for different parameters (thickness ratio, curvature ratio, modular ratio, support condition, lamination scheme, amplitude ratio and thermal expansion coefficient) and their effects on the responses are discussed in detail.     

Refined geometrically nonlinear formulation of a thin-shell triangular finite element

A refined geometrically nonlinear formulation of a thin-shell finite element based on the Kirchhoff-Love hypotheses is considered. Strain relations, which adequately describe the deformation of the element with finite bending of its middle surface, are obtained by integrating the differential equation of a planar curve. For a triangular element with 15 degrees of freedom, a cost-effective algorithm is developed for calculating the coefficients of the first and second variations of the strain energy, which are used to formulate the conditions of equilibrium and stability of the discrete model of the shell. Accuracy and convergence of the finite-element solutions are studied using test problems of nonlinear deformation of elastic plates and shells. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 5, pp. 160–172, September–October, 2007.     

Stochastic finite element analysis of local average random fields

This paper discussed the concepts underlying a new approach to representation and analysis of homogeneous random Fields. An orthogonalized isoparametric local average model of random fields is proposed, and the optimum selection of random variables is determined based on the theory of nonlinear programming. The finite element method is used to develop a methodology for random field analysis. Numerical examples showing the accuracy and efficiency of the method are studied.     

有初应力钢管混凝土压弯扭构件非线性有限元分析

钢管初应力是钢管混凝土在大跨度桥梁,高层超高层建筑应用中广大工程技术人员关心的问题.本文在钢材随动强化模型和混凝土边界面模型的基础上编制了计算有初应力的钢管混凝土压弯扭构件全过程曲线的非线性有限元程序,无论在本构关系的选取还是有限元模式的确定,非线性问题的求解都有自己的特色,不仅大大节省了计算时间,也较好地解决了这一工程实际问题.     

A finite element model for linear viscoelastic behaviour of pultruded thin-walled beams under general loadings

A finite element model for orthotropic thin-walled beams subject to long-term loadings is presented. The hypothesis, rather usual for thin-walled beams, of cross-sections remaining undistorted in their own planes after deformation is introduced, so reducing the number of d.o.f.’s and, consequently, the computational effort of the analysis. The model is used to perform linear viscoelastic analysis of prismatic beams with general cross-sections, i.e., open, closed or multi-cell. As far as the constitutive viscoelastic law is concerned, a generalized linear Maxwell model is adopted. Making use of the exponential algorithm, differential equations are written in incremental form and integration is performed adopting time intervals of variable length. Numerical examples are finally presented, concerning glass-fibre pultruded shapes under long-term loadings. Displacement evolution with time and stress redistribution adopting different creep laws are presented. Convergence features of the proposed finite element and time integration procedure are also shown.     

三点弯曲试样动态冲击特性的有限元分析

本文使用动态有限元技术，对于两种不同几何尺寸，两种不同材料的三点弯曲试样在三类七种不同冲击载荷作用下的动态响应进行了分析，求得了动态应力强度因子随时间的变化规律。并与准静态应力强度因子进行了比较。计算结果表明：将冲击载荷历史代入静态公式确定动态应力强度因子的做法是不正确的，要求得动态应力强度因子，必须对试样进行完全的动态分析。当材料的Ｅ／ρ值相同时，动态应力强度因子的响应曲线完全相同。而动态应力强度因子分别与加载点的位移及裂纹的张开位移之间存在着与准静态情况下各自相同的线性关系。这与资料［５］［６］中的结论完全相同。     

降落伞膜索非线性有限元模拟

用膜单元和索单元模拟降落伞织物绳索系统,基于完全拉格朗日格式的非线性有限元方法编程计算降落伞的结构动力学特性。采用增量与迭代混合方法改善非线性计算的收敛特性并结合HHT隐式时间推进方法减小整体迭代计算量。使用修正应力应变张量导数的方法模拟膜单元单向应力状态并针对膜单元和索单元分别进行了非线性有限元计算验证。最后针对C-9型降落伞建立三维有限元模型,根据设定流速对伞衣施加均匀压强载荷,将模拟展开的结果与使用相同模型、不同方法商业软件的文献进行对比,显示了隐式非线性有限元方法模拟降落伞膜索系统大变形动力学的能力。     

弹塑性板壳结构非线性有限元分析

本文根据塑性流动理论的基本公式，由隐式积分导出了与路径无关的变量更新算法和一致切线模量。采用单元广义应力应变直接离散塑性流动定律，构造了杂交应力单元一致切线刚度矩阵的显式表达式，编制了结构有限元程序ＳＡＦＥ，数值算例表明：本文的计算方法和计算程序是正确可靠的，可用于弹塑性板壳结构的非线性分析，计算结果屈曲临界载荷和极限承载能力。     

有限元方法中的光滑积分伪弱形式

提出了一种光滑积分伪弱形式,将光滑积分拓展至被积函数非偏导项求解。结合光滑应变技术和伪弱形式,可实现有限元系统方程统一光滑积分求解,即对刚度矩阵和质量矩阵中的应变矩阵和形函数矩阵均可进行光滑积分处理,并转化为光滑子域的边界积分。光滑积分伪弱形式与光滑应变技术比较,增加了形函数矩阵不定积分处理过程,且没有降低有限元求解对形函数连续性的要求。不过,伪弱形式改变了单元积分的求解形式,连续质量矩阵求解也无需坐标映射和雅可比矩阵计算。以轴对称二维问题为研究对象,结果表明极度不规则三角形和四边形单元光滑积分伪弱形式在静态和动态有限元方程求解中也具有很好的精度。     

Dual variational formulation for Trefftz finite element method of elastic materials

A dual variational principle is presented for Trefftz finite element analysis. The proof of the stationary conditions of the variational functional and the theorem on the existence of extremum are provided in this paper. They are boundary displacement condition, surface traction condition and interelement continuity condition. Based on the assumed intraelement and frame fields, element stiffness matrix equation is obtained which can easily be implemented into computer programs for numerical analysis with Trefftz finite element method. Two numerical examples are considered to illustrate the effectiveness and applicability of the proposed element model.