3D finite element analysis of electrostatic deflection and shielding of commercial and FIB-modified cantilevers for electric and Kelvin force microscopy: II. Rectangular shaped cantilevers with asymmetric pyramidal tips

This paper deals with an application of 3D finite element analysis to the electrostatic interaction between (i) a commercial rectangular shaped cantilever (with an integrated anisotropic pyramidal tip) and a conductive sample, when a voltage difference is applied between them, and (ii) a focused ion beam (FIB) modified cantilever in order to realize a probe with reduced parasitic electrostatic force. The 3D modelling of their electrostatic deflection was realized by using multiphysics finite element analysis software and applied to the real geometry of the cantilevers and probes as used in conventional electric and Kelvin force microscopy to evaluate the contribution of the various part of a cantilever to the total force, and derive practical criteria to optimize the probe performances. We report also on the simulation of electrostatic shielding of nanometric features, in order to quantitatively evaluate an alternative way of reducing the systematic error caused by the cantilever-to-sample capacitive coupling. Finally, a quantitative comparison between the performances of rectangular and triangular cantilevers (part I of this work) is reported.     

Mechanical analysis of 3D braided composites: a finite element model

The analysis of 3D braided composites is more difficult due to its complex microstructure. A new type of finite element method is developed to predict the effective moduli and the local stress within 3D braided composites under the 3D mechanical loading. To verify the present method, the material properties of undamaged 3D braided composites predicted in this paper are compared with the previous work. To demonstrate this method, some examples are analyzed.     

Comparison of 2D finite element modeling assumptions with results from 3D analysis for composite skin-stiffener debonding

The influence of two-dimensional finite element modeling assumptions on the debonding prediction for skin-stiffener specimens was investigated. Geometrically nonlinear finite element analyses using two-dimensional plane-stress and plane-strain elements as well as three different generalized plane-strain type approaches were performed. The computed skin and flange strains, transverse tensile stresses and energy release rates were compared to results obtained from three-dimensional simulations. The study showed that for strains and energy release rate computations the generalized plane-strain assumptions yielded results closest to the full three-dimensional analysis. For computed transverse tensile stresses the plane-stress assumption gave the best agreement. Based on this study it is recommended that results from plane-stress and plane-strain models be used as upper and lower bounds. The results from generalized plane-strain models fall between the results obtained from plane-stress and plane-strain models. Two-dimensional models may also be used to qualitatively evaluate the stress distribution in a ply and the variation of energy release rates and mixed mode ratios with delamination length. For more accurate predictions, however, a three-dimensional analysis is required.     

Failure of plain concrete beam at impact load: 3D finite element analysis

In the paper, the results of numerical failure analysis of plain concrete beams loaded by impact three-point bending load are presented and discussed. The theoretical framework for the numerical analysis is continuum mechanics and irreversible thermodynamics. The spatial discretization is performed by the finite element method using update Lagrange formulation. Green–Lagrange stain tensor is used as a strain measure. To account for cracking and damage of concrete, the beam is modeled by the rate sensitive microplane model with the use of the so-called co-rotational stress tensor. Damage and cracking phenomena are modeled within the concept of smeared cracks. To assure objectivity of the analysis with respect to the size of the finite elements, crack band method is used. The contact-impact analysis is based on the mechanical interaction between two bodies—concrete beam (master) and dropping hammer (slave) falling on the mid span of the beam. The contact constrains are satisfied by Lagrange multiplier method, which is adapted for the explicit time integration scheme. To investigate the influence of loading rate on the failure mode of the beam parametric study is carried out. The numerical results are evaluated, discussed and compared with test results known from the literature. It is shown that the beam resistance and failure mode strongly depend on loading rate. For lower loading rates beam fails in bending (mode-I fracture). However, with increasing loading rate there is a transition of the failure mechanism from bending to shear. The results are in good agreement with theoretical and experimental results known from the literature.     

Fixed-point iteration to nonlinear finite element analysis. Part I: Mathematical theory and background

This paper sets the stage for the implementation of the Fixed-Point Iteration (FPI) to nonlinear finite element analysis as an alternative to the existing Newton-Raphson Method (NRM) or its derivatives. The superiority of the former method over the latter one is such that it enables one to obtain nonlinear structural static or dynamic responses without inverting the structural stiffness matrix. In the first part of the paper, a new convergence correction/acceleration factor has been developed for the FPI when applied to a single nonlinear algebraic equation. This new factor causes the iteration function of the equation under consideration to rotate about an axis that passes through one of its fixed points or roots. Using this observation, the slope of the iteration function can be adjusted in the neighbourhood of a specific root to ensure the convergence of the FPI. It is found that the optimum choice of the new factor corresponds to a zero slope, evaluated at the root, of the iteration function. The rate of convergence and the error estimate of this form of the FPI is developed and compared with the NRM. The equilibrium positions of a nonlinear loaded softening spring have been obtained by both methods as an illustrative numerical example to measure the effect of the new factor on the convergence rate. The second part of the paper extends the above concept to find the solution of a linear system of algebraic equations using the FPI. This leads to a better diagonal approximate inverse for the Jacobi iteration, or method of simultaneous displacements. If the elements of the solution vector of a specific system are all equal, the new Jacobi iteration becomes an exact method and the solution is obtained in one iteration. The concept is also extended to the Gauss-Seidel iteration, or method of successive displacements. Systems involving symmetric as well as nonsymmetric coefficient matrices have been used as numerical examples and are presented. For future implementation to nonlinear finite element analysis, the active column or the skyline (or the non-zero profile) FPI algorithms are developed for programming considerations.     

A finite element model for the analysis of 3D axisymmetric laminated shells with piezoelectric sensors and actuators

This paper presents the development of a semi-analytical axisymmetric shell finite element model with piezoelectric layers using the 3D linear elasticity theory. The piezoelectric effect of the material could be used as sensors and/or actuators in way to control shell deformation. In the present 3D axisymmetric model, the equations of motion are expressed by expanding the displacement field using Fourier series in the circumferential direction. Thus, the 3D elasticity equations of motion are reduced to 2D equations involving circumferential harmonics. In the finite element formulation the dependent variables, electric potential and loading are expanded in truncated Fourier series. Special emphasis is given to the coupling between symmetric and anti-symmetric terms for laminated materials with piezoelectric rings. Numerical results obtained with the present model are found to be in good agreement with other finite element solutions.     

Modeling microarchitecture and mechanical behavior of nacre using 3D finite element techniques Part I Elastic properties

Three dimensional finite element models of nacre were constructed based on reported microstructural studies on the 'brick and mortar' micro-architecture of nacre. 3D eight noded isoparametric brick elements were used to design the microarchitecture of nacre. Tensile tests were simulated using this model. The tests were conducted at low stresses of 2 MPa which occur well within the elastic regime of nacre and thus effects related to locus and extent of damage were ignored. Our simulations show that using the reported values of elastic moduli of organic (0.005 GPa) and aragonitic platelets (205 GPa), the displacements observed in nacre are extremely large and correspond to a very low modulus of 0.011 GPa. The reported elastic modulus of nacre is of the order of 50 GPa. The reason for this inconsistency may arise from two possibilities. Firstly, the organic layer due to its multilayered structure is possibly composed of distinct layers of different elastic moduli. The continuously changing elastic modulus within the organic layer may approach modulus of aragonite near the organic-inorganic interface. Simulations using variable elastic moduli for the organic phase suggest that an elastic modulus of 15 GPa is consistent with the observed elastic behavior of nacre. Another explanation for the observed higher elastic modulus may arise from localized platelet-platelet contact. Since the observed modulus of nacre lies within the above two extremes (i.e. 15 GPa and 205 GPa) it is suggested that a combination of the two possibilities, i.e. a higher modulus of the organic phase near the organic-inorganic interface and localized platelet-platelet contact can result in the observed elastic properties of nacre.     

考虑界面结合的SiCP/6061Al 复合材料时间相关棘轮行为的三维有限元分析

采用多颗粒三维单胞模型和复合材料细观有限元分析方法,借助先进循环黏塑性本构模型的有限元实现,对SiC颗粒增强6061Al复合材料的室温、高温时间相关单轴棘轮行为进行数值模拟。讨论了颗粒排列方式和界面结合状态的变化对复合材料棘轮行为的影响;同时,分析了复合材料中基体和界面的微观变形特征及其演变规律;最后,选取一组合理的微结构参数,对复合材料的时间相关棘轮行为进行了数值模拟,并通过与已有实验结果的比较,检验了有限元分析的合理性。结果表明：多颗粒代表性体积单元能够反映复合材料更多的微观细节;颗粒排列方式的变化显著影响复合材料的整体棘轮行为;界面结合状态越好,产生的棘轮变形越小;具有合理参数值的弱界面模型给出的时间相关棘轮变形预测结果比完好界面模型的结果更接近实验值。     

Efficient finite element with physical and electric nodes for transient analysis of smart piezoelectric sandwich plates

This paper presents an assessment of a recently developed quadrilateral element with four physical nodes and one electrical node, based on a coupled improved zigzag theory, for the transient response of smart sandwich plates with electroded piezoelectric sensors and actuators. The novel features of the element include the use of electric nodes to model the equipotential condition of the electroded surface of sensors and actuators, and the inclusion of the d ₃₃-effect on transverse deflection. The assessment is done for a skew plate in comparison with a converged three-dimensional finite element solution obtained using ABAQUS. The accuracy as well as the computational efficiency of the present element is highlighted.     

A semi-analytical finite element model for the analysis of laminated 3D axisymmetric shells: Bending, free vibration and buckling

A general semi-analytical finite element model is developed for bending, free vibration and buckling analysis of shells of revolution made of laminated orthotropic elastic material. The 3D elasticity theory is used and the equations of motion are obtained by expanding the displacement field and load in the Fourier series in terms of the circumferential coordinate, θ. The coefficients of the expansion are functions of (r, z), and they are approximated using the finite element method. This leads to a semi-analytical finite element in the (r, z) plane. The element is validated by comparing the present results with the analytical and numerical solutions available in the literature.     

Determination of T-stress from finite element analysis for mode I and mixed mode I/II loading

The elastic T-stress has been recognised as a measure of constraint around the tip of a crack in contained yielding problems. A review of the literature indicates that most methods for obtaining T are confined to simple geometry and loading configurations. This paper explores direct use of finite element analysis for calculating T. It is shown that for mode I more reliable results with less mesh refinement can be achieved if crack flank nodal displacements are used. Methods are also suggested for calculating T for any mixed mode I/II loading without having to calculate stress intensity factors. There is good agreement between the results from the proposed methods and analytical results. T-stress is determined for a test configuration designed to investigate brittle and ductile fracture in mixed mode loading. It is shown that in shear loading of a cracked specimen T vanishes only when a truly antisymmetric field of deformation is provided. However this rarely happens in practice and the presence of T in shear is often inevitable. It is shown that for some cases the magnitude of T in shear is much more than that for tension. The effect of crack length is also investigated. This revised version was published online in August 2006 with corrections to the Cover Date.     

2D finite element analysis of thermally bonded nonwoven materials: Continuous and discontinuous models

Due to a random structure of nonwoven materials, their non-uniform local material properties and nonlinear properties of single fibres, it is difficult to develop a numerical model that adequately accounts for these features and properly describes their performance. Two different finite element (FE) models – continuous and discontinuous – are developed here to describe the tensile behaviour of nonwoven materials. A macro-level continuum finite element model is developed based on the classic composite theory by treating the fibrous network as orthotropic material. This model is used to analyse the effect of thermally bonding points on the deformational behaviour and deformation mechanisms of thermally bonded nonwoven materials at macro-scale. To describe the effects of discontinuous microstructure of the fabric and implement the properties of polypropylene fibres, a micro-level discontinuous finite element model is developed. Applicability of both models to describe various deformational features observed in experiments with a real thermally bonded nonwoven is discussed.     

3D finite element analysis of tensile notched strength of 2/2 twill weave fabric composites with drilled circular hole

This paper presents the micromechanical three-dimensional finite element models of the 2/2 twill weave T300 carbon/epoxy woven fabric composite laminates with drilled circular holes of different sizes. A fiber breakage failure criterion for predicting the ultimate tensile notched strength of fiber dominated composites is also proposed. It is found that the location of failure initiation for laminates with large hole size is different from those for laminates with smaller holes while the stress concentration may not occur at the notch roots for the fiber dominated laminates. Based on the uniaxial, shear and von Mises stress distributions in the yarn and matrix, the influence of hole-size on the stress distributions and stress concentration is discussed. Standard tensile tests with modifications are performed for this particular type of woven fabric composites. The apparent strain concentration factors and notched strengths determined by experiments are presented and the finite element models are verified by satisfactory correlation between prediction and experiment.     

A 3D finite element approach for the coupled numerical simulation of electrochemical systems and fluid flow

A comprehensive finite element method for three‐dimensional simulations of stationary and transient electrochemical systems including all multi‐ion transport mechanisms (convection, diffusion and migration) is presented. In addition, non‐linear phenomenological electrode kinetics boundary conditions are accounted for. The governing equations form a set of coupled non‐linear partial differential equations subject to an algebraic constraint due to the electroneutrality condition. The advantage of a convective formulation of the ion‐transport equations with respect to a natural application of homogeneous flux boundary conditions is emphasized. For one of the numerical examples, an analytical solution for the coupled problem is provided, and it is demonstrated that the proposed computational approach is robust and provides accurate results. Copyright © 2011 John Wiley & Sons, Ltd.     

Multi-scale finite element analysis for tension and ballistic penetration damage characterizations of 2D triaxially braided composite

The multi-scale finite element model is presented to analyze tension and ballistic penetration damage characterizations of 2D triaxially braided composite (2DTBC). At the mesoscopic level, the damage of fiber tows is initiated with 3D Hashin criteria, and the damage initiation of pure matrix is predicted by the modified von Mises. The progressive degradation scheme and energy dissipation method are adopted to capture softening behaviors of tow and matrix. The macro-scale damage model is established by maximum-stress criteria and exponential damage evolution. To simulate interface debonding and inter-ply delamination, a triangle traction–separation law is adopted in each scale. Both scale damage models are verified with available experimental results. Based on numerical predictions, the stress–strain responses and damage developments of 2DTBC under axial and transverse tension loading are studied. For ballistic penetration loading, the meso-scale damage mechanisms of 2DTBC are predicted using 1/4 model, 1/2 model, 1-layer model, 2-layer model and 3-layer model. Then, effects of different model and impactor radius on damage modes are analyzed. Additionally, the macro-scale ballistic penetration behaviors of 2DTBC are simulated and compared with experiment. The prediction results for tension and penetration correlate well with experiment results. Both tension and penetration damage characterizations for tow, matrix within tow, pure matrix, interface and inter-ply delamination are revealed. A comparison of penetration damage between meso- and macro-scale presents a similar crack mechanism between two scales.     

A reduced-order model based on the coupled 1D-3D finite element simulations for an efficient analysis of hemodynamics problems

A reduced-order model for an efficient analysis of cardiovascular hemodynamics problems using multiscale approach is presented in this work. Starting from a patient-specific computational mesh obtained by medical imaging techniques, an analysis methodology based on a two-step automatic procedure is proposed. First a coupled 1D-3D Finite Element Simulation is performed and the results are used to adjust a reduced-order model of the 3D patient-specific area of interest. Then, this reduced-order model is coupled with the 1D model. In this way, three-dimensional effects are accounted for in the 1D model in a cost effective manner, allowing fast computation under different scenarios. The methodology proposed is validated using a patient-specific aortic coarctation model under rest and non-rest conditions.     

A 3D fibre beam-column element with shear modelling for the inelastic analysis of steel structures

A force-based fiber beam-column element is proposed for the capacity assessment of frame structures under high shear. The proposed element is suitable for the performance assessment of large scale steel structures, which are not flexure-dominated. The element formulation follows the assumptions of the Timoshenko beam theory, while its kinematics are obtained through the natural-mode method. The element state-determination phase, instead of uniaxial material laws, typically associated with fiber elements, is based on a three-dimensional law taking into consideration the interaction between axial, bending, shear and torsion. Numerical examples are presented confirming the accuracy and the computational efficiency of the proposed formulation under monotonic, cyclic and dynamic/seismic loading. Compared to experimental results and the results of detailed finite element models, excellent agreement is achieved.     

A variational technique for boundary element analysis of 3D fracture mechanics weight functions: static

The dual boundary element method coupled with the weight function technique is developed for the analysis of three-dimensional elastostatic fracture mechanics mixed-mode problems. The weight functions used to calculate the stress intensity factors are defined by the derivatives of traction and displacement for a reference problem. A knowledge of the weight functions allows the stress intensity factors for any loading on the boundary to be calculated by means of a simple boundary integration without singularities. Values of mixed-mode stress intensity factors are presented for an edge crack in a rectangular bar and a slant circular crack embedded in a cylindrical bar, for both uniform tensile and pure bending loads applied to the ends of the bars. © 1998 John Wiley & Sons, Ltd.     

Global 2D digital image correlation for motion estimation in a finite element framework: a variational formulation and a regularized,pyramidal, multi‐grid implementation

In this study, the inverse problem of reconstructing the in‐plane (2D) displacements of a monitored surface through a sequence of two‐dimensional digital images, severely ill‐posed in Hadamard's sense, is deeply investigated. A novel variational formulation is presented for the continuum 2D digital image correlation problem, and critical issues such as semi‐coercivity and solution multiplicity are discussed by functional analysis tools. In the framework of a Galerkin, finite element discretization of the displacement field, a robust implementation for 2D digital image correlation is outlined, aiming to attenuate the spurious oscillations which corrupt the deformation scenario, especially when very fine meshes are adopted. Recourse is made to a hierarchical family of grids linked by suitable restriction and prolongation operators and defined over an image pyramid. Multi‐grid cycles are performed ascending and descending along the pyramid, with only one Newton iteration per level irrespective of the tolerance satisfaction, as if the problem were linear. At each level, the conventional least‐square matching functional is herein enriched by a Tychonoff regularization provision, preserving the solution against an unstable response. The algorithm is assessed on the basis of both synthetic and truly experimental image pairs. Copyright © 2013 John Wiley & Sons, Ltd.