A NUMERICAL STUDY OF INDENTATION WITH SMALL SPHERICAL INDENTERS

The finite element method for the conventional theory of mechanism-based strain gradient plasticity is used to study the indentation size effect. For small indenters （e.g., radii on the order of 10μm）, the maximum allowable geometrically necessary dislocation （GND） density is introduced to cap the GND density such that the latter does not become unrealistically high. The numerical results agree well with the indentation hardness data of iridium. The GND density is much larger than the density of statistically stored dislocations （SSD） underneath the indenter, but this trend reverses away from the indenter. As the indentation depth （or equivalently, contact radius） increases, the GND density decreases but the SSD density increases.     

Axisymmetric viscoplastic deformation by the boundary element method

This paper presents a boundary element formulation and numerical implementation of the problem of small axisymmetric deformation of viscoplastic bodies. While the extension from planar to axisymmetric problems can be carried out fairly simply for the finite element method (FEM), this is far from true for the boundary element method (BEM). The primary reason for this fact is that the axisymmetric kernels in the integral equations of the BEM contain elliptic functions which cannot be integrated analytically even over boundary elements and internal cells of simple shape. Thus, special methods have to be developed for the efficient and accurate numerical integration of these singular and sensitive kernels over discrete elements. The accurate determination of stress rates by differentiation of the displacement rates presents another formidable challenge.A successful numerical implementation of the boundary element method with elementwise (called the Mixed approach) or pointwise (called the pure BEM or BEM approach) determination of stress rates has been carried out. A computer program has been developed for the solution of general axisymmetric viscoplasticity problems. Comparisons of numerical results from the BEM and FEM, for several illustrative problems, are presented and discussed in the paper. It is possible to get direct solutions for the simpler class of problems for cylinders of uniform cross-section, and these solutions are also compared with the BEM and FEM results for such cases.     

Shape sensing of 3D frame structures using an inverse Finite Element Method

A robust and efficient computational method for reconstructing the elastodynamic structural response of truss, beam, and frame structures, using measured surface-strain data, is presented. Known as “shape sensing”, this inverse problem has important implications for real-time actuation and control of smart structures, and for monitoring of structural integrity. The present formulation, based on the inverse Finite Element Method (iFEM), uses a least-squares variational principle involving section strains (also known as strain measures) of Timoshenko theory for stretching, torsion, bending, and transverse shear. The present iFEM methodology is based on strain–displacement relations only, without invoking force equilibrium. Consequently, both static and time-varying displacement fields can be reconstructed without the knowledge of material properties, applied loading, or damping characteristics. Two finite elements capable of modeling frame structures are derived using interdependent interpolations, in which interior degrees of freedom are condensed out at the element level. In addition, relationships between the order of kinematic-element interpolations and the number of required strain gauges are established. Several example problems involving cantilevered beams and three-dimensional frame structures undergoing static and dynamic response are discussed. To simulate experimentally measured strains and to establish reference displacements, high-fidelity MSC/NASTRAN finite element analyses are performed. Furthermore, numerically simulated measurement errors, based on Gaussian distribution, are also considered in order to verify the stability and robustness of the methodology. The iFEM solution accuracy is examined with respect to various levels of discretization and the number of strain gauges.     

A simple and accurate mixed FE-DQ formulation for free vibration of rectangular and skew Mindlin plates with general boundary conditions

A simple and accurate mixed finite element-differential quadrature formulation is proposed to study the free vibration of rectangular and skew Mindlin plates with general boundary conditions. In this technique, the original plate problem is reduced to two simple bar (or beam) problems. One bar problem is discretized by the finite element method (FEM) while the other by the differential quadrature method (DQM). The mixed method, in general, combines the geometry flexibility of the FEM and high accuracy and efficiency of the DQM and its implementation is more easier and simpler than the case where the FEM or DQM is fully applied to the problem. Moreover, the proposed formulation is free of the shear locking phenomenon that may be encountered in the conventional shear deformable finite elements. A simple scheme is also presented to exactly implement the mixed natural boundary conditions of the plate problem. The versatility, accuracy and efficiency of the proposed method for free vibration analysis of rectangular and skew Mindlin plates are tested against other solution procedures. It is revealed that the proposed method can produce highly accurate solutions for the natural frequencies of rectangular and skew Mindlin plates with general boundary conditions.     

平面广义四节点等参元GQ4及其性能探讨

广义节点有限元是将传统有限元方法中的节点广义化,在不增加节点个数的前提下,仅通过提高广义节点的插值函数的阶次,从而达到提高有限元解精度的目的.与现有的p型和hp型有限元不同,在这种新的有限元中,节点自由度全部定义在节点处,在理论与程序实现上与传统有限元方法具有很好的相容性,传统有限元方法是这种新方法的广义节点退化为0阶时的特殊情形.文中主要讨论了这一新方法的四节点等参元（记为GQ4）的形式.对GQ4进行的各种数值试验表明,所发展的广义四节点等参单元具有精度高且无剪切自锁与体积自锁等的特点.     

两类基于局部标架梁单元的闭锁缓解方法

对于大转动、大变形柔性体的刚柔耦合动力学问题,基于李群SE(3)局部标架(local frame formulation, LFF)的建模方法能够规避刚体运动带来的几何非线性问题,离散数值模型中广义质量矩阵与切线刚度矩阵满足刚体变换的不变性,可明显地提高柔性多体系统动力学问题的计算效率. 有限元方法中,闭锁问题是导致单元收敛性能低下的主要原因, 例如梁单元的剪切以及泊松闭锁.多变量变分原理是缓解梁、板/壳单元闭锁的有效手段. 该方法不仅离散位移场,同时离散应力场或应变场, 可提高应力与应变的计算精度. 本文基于上述局部标架,研究几类梁单元的闭锁处理方法, 包括几何精确梁(geometrically exact beam formulation, GEBF)与绝对节点坐标(absolute nodal coordinate formulation, ANCF)梁单元. 其中, 采用Hu-Washizu三场变分原理缓解几何精确梁单元中的剪切闭锁,采用应变分解法缓解基于局部标架的ANCF全参数梁单元中的泊松闭锁. 数值算例表明,局部标架的梁单元在描述高转速或大变形柔性多体系统时,可消除刚体运动带来的几何非线性, 极大地减少系统质量矩阵和刚度矩阵的更新次数.缓解闭锁后的几类局部标架梁单元收敛性均得到了明显提升.      

Higher-order shear deformable theories for flexure of sandwich plates—Finite element evaluations

A simple isoparametric finite element formulation based on a higher-order displacement model for flexure analysis of multilayer symmetric sandwich plates is presented. The assumed displacement model accounts for non-linear variation of inplane displacements and constant variation of transverse displacement through the plate thickness. Further, the present formulation does not require the fictitious shear correction coefficient(s) generally associated with the first-order shear deformable theories. Two sandwich plate theories are developed: one in which the free shear stress conditions on the top and bottom bounding planes are imposed and another, in which such conditions are not imposed. The validity of the present development(s) is established through, numerical evaluations for deflections/stresses/stress-resultants and their comparisons with the available three-dimensional analyses/closed-form/other finite element solutions. Comparison of results from thin plate. Mindlin and present analyses with the exact three-dimensional analyses yields some important conclusions regarding the effects of the assumptions made in the CPT and Mindlin type theories. The comparative study further establishes the necessity of a higher-order shear deformable theory incorporating warping of the cross-section particularly for sandwich plates.     

Finite element-boundary integral formulation for electromagnetic scattering

A new formulation is described which combines the most robust attributes of the volume finite element and surface integral equation approaches to electromagnetic boundary value solutions. The result is a numerical technique which may be applied to scattering problems involving configurations having metallic surfaces and inhomogeneous penetrable material situated in open spatial regions. This is accomplished by way of coupling internal region finite element modal field solutions to equivalent currents on the surrounding boundary surface through an appropriate surface integral equation. The method is demonstrated for the special case of scattering by axisymmetric inhomogeneous penetrable objects. Example numerical calculations are presented for validation of the procedure and potential problem areas are discussed.     

On the elastoplastic cyclic analysis of plane beam structures using a flexibility-based finite element approach

This paper presents the extension of a flexibility-based large increment method (LIM) for the case of cyclic loading. In the last few years, LIM has been successfully tested for solving a range of non-linear structural problems involving elastoplastic material models under monotonic loading. In these analyses, the force-based LIM algorithm provided robust solutions and significant computational savings compared to the displacement-based finite element approach by using fewer elements and integration points. Although in cyclic analysis a step-by-step solution procedure has to be adopted to account for the plastic history, LIM will still have many advantages over the traditional finite element method. Before going into the basic idea of this extension, a brief discussion regarding LIM governing equations is presented followed by the proposed solution procedure. Next, the formulation is specified for the treatment of the elastic perfectly plastic beam element. The local stage for the beam behavior is discussed in detail and the required improvement for the LIM methodology is described. Illustrative truss and beam examples are presented for different non-linear material models. The results are compared with those obtained from a standard displacement method and again highlight the potential benefits of the proposed flexibility-based approach.     

A stabilized formulation for incompressible plasticity using linear triangles and tetrahedra

In this paper, a stabilized finite element method to deal with incompressibility in solid mechanics is presented. Both elastic and J2-plastic constitutive behavior have been considered. A mixed formulation involving pressure and displacement fields is used and a continuous linear interpolation is considered for both fields. To circumvent the Babuška–Brezzi condition a stabilization technique based on the orthogonal sub-scale method is introduced. The main advantage of the method is the possibility of using linear triangular or tetrahedral finite elements, which are easy to generate for real industrial applications. Results are compared with standard Galerkin and Q1P0 mixed formulations in either elastic or elasto-plastic incompressible problems.     

A polygonal finite element formulation for modeling nearly incompressible materials

The objective of the present paper is to develop a finite element formulation for modeling nearly incompressible materials at large strains using polygonal elements. The present finite element formulation is a simplified version of the three-field mixed formulation and, in particular, it reduces the functional of the internal potential energy by expressing the field of the average volume-change in terms of the displacement field, where the latter is discretized using the Wachspress shape functions. The reduced mixed formulation eliminates the volumetric locking in nearly incompressible materials and enhances the computational efficiency as the static condensation is circumvented. A detailed implementation of the finite element formulation is presented in this study. Also, different example problems, including eigenvalue analysis, nonlinear patch test and other benchmark problems are presented for demonstrating the accuracy and the reliability of the developed formulation for polygonal elements.

     

A NEW SPATIAL THIN-WALLED BEAM ELEMENT INCLUDING TRANSVERSE AND TORSIONAL SHEAR DEFORMATION

基于Timoshenko梁及Benscoter薄壁杆件理论,建立了考虑剪切变形、弯扭耦合以及翘曲剪应力影响的空间任意开闭口薄壁截面梁单元. 通过引入单元内部结点,对弯曲转角和翘曲角采用三节点Lagrange独立插值的方法,考虑了剪切变形和翘曲剪应力的影响并避免了横向剪切锁死问题;借助载荷作用下薄壁梁的截面运动分析,在位移和应变方程中考虑了弯扭耦合的影响. 通过数值算例将该单元的计算结果与理论解以及商用有限元软件和其他文献中的数值解进行对比和验证,结果对比表明该薄壁梁单元具有良好的精度和收敛性.     

Finite versus small strain discrete dislocation analysis of cantilever bending of single crystals

Plastic size effects in single crystals are investi-gated by using finite strain and small strain discrete dislo-cation plasticity to analyse the response of cantilever beam specimens. Crystals with both one and two active slip sys-tems are analysed, as well as specimens with different beam aspect ratios. Over the range of specimen sizes analysed here, the bending stress versus applied tip displacement response has a strong hardening plastic component. This hardening rate increases with decreasing specimen size. The hardening rates are slightly lower when the finite strain discrete disloca-tion plasticity (DDP) formulation is employed as curving of the slip planes is accounted for in the finite strain formulation. This relaxes the back-stresses in the dislocation pile-ups and thereby reduces the hardening rate. Our calculations show that in line with the pure bending case, the bending stress in cantilever bending displays a plastic size dependence. How-ever, unlike pure bending, the bending flow strength of the larger aspect ratio cantilever beams is appreciably smaller. This is attributed to the fact that for the same applied bend-ing stress, longer beams have lower shear forces acting upon them and this results in a lower density of statistically stored dislocations.     

Variational base and solution strategies for non-linear force-based beam finite elements

This paper presents the variational bases for the non-linear force-based beam elements. The element state determination of these elements is obtained exactly from a two-field functional with independent stress and strain fields. The variational base of the non-linear force-based beam elements implemented in a general purpose displacement-based finite element program requires the inclusion of independent displacement field in the formulation. For this purpose, a three-field functional is considered with independent displacement, stress, and strain fields. Various local and global solution strategies come out from the mixed formulation of the beam element, and these are shown to yield the algorithms presented for non-linear force formulation beam elements in literature; thus removing any doubts on their variational bases. The presented numerical examples demonstrate the accuracy and robustness of the solution algorithms adapted for mixed formulation elements over popularly used displacement-based beam finite elements even for large structural systems.     

The application of 2D base force element method (BFEM) to geometrically non-linear analysis

Based on the concept of the base forces by Gao, a new finite element method – the base force element method (BFEM) on complementary energy principle for two-dimensional geometrically non-linear problems is presented. A 4-mid-node plane element model of the BFEM for geometrically non-linear problem is derived by assuming that the stress is uniformly distributed on each sides of a plane element. The explicit formulations of the control equations for the BFEM are derived using the modified complementary energy principle. The BFEM is naturally universal for small displacement and large displacement problems. A number of example problems are solved using the BFEM and the results are compared with corresponding analytical solutions and those obtained from the standard displacement finite element method. A good agreement of the results, and better performance of the BFEM, compared to the displacement model, in the large displacement and large rotation calculations, is observed.     

Hybrid-Trefftz displacement element for spectral analysis of bounded and unbounded media

The hybrid-Trefftz displacement element is applied to the elastodynamic analysis of bounded and unbounded media in the frequency domain. The displacements are approximated in the domain of the element using local solutions of the wave equation, the Neumann conditions are enforced directly and the surface forces are approximated on the Dirichlet and inter-element boundaries of the finite element mesh. Two alternative elements are developed to model unbounded media, namely a finite element with absorbing boundaries and an unbounded element that satisfies explicitly the Sommerfeld condition. The finite element equations are derived from the fundamental relations of elastodynamics written in the frequency domain. The numerical implementation of these equations is discussed and numerical tests are presented to assess the performance of the formulation.     

A finite deformation theory of higher-order gradient crystal plasticity

For higher-order gradient crystal plasticity, a finite deformation formulation is presented. The theory does not deviate much from the conventional crystal plasticity theory. Only a back stress effect and additional differential equations for evolution of the geometrically necessary dislocation (GND) densities supplement the conventional theory within a non-work-conjugate framework in which there is no need to introduce higher-order microscopic stresses that would be work-conjugate to slip rate gradients. We discuss its connection to a work-conjugate type of finite deformation gradient crystal plasticity that is based on an assumption of the existence of higher-order stresses. Furthermore, a boundary-value problem for simple shear of a constrained thin strip is studied numerically, and some characteristic features of finite deformation are demonstrated through a comparison to a solution for the small deformation theory. As in a previous formulation for small deformation, the present formulation applies to the context of multiple and three-dimensional slip deformations.     

Boundary integral equations for 2D elasticity and its application in discrete dislocation simulation in finite body: 2. Numerical implementation

In the previous paper by Yu and Diab (2013), several sets of boundary integral equations are derived for general anisotropic materials and corresponding equations for materials with different classes of symmetry are deduced. The work presented herein implements two sets of boundary element schemes to numerically solve the stress field. The integration on the element that has the singular point of the kernel is bounded and can be evaluated analytically. Four benchmark elastic problems are solved numerically to show the advantage of the two schemes over the conventional boundary element formulation in eliminating the boundary layer effect. The one with the weaker singularity has better convergence and gives more accurate results. The presented formulation also provides a direct approach to solve for stress field in a finite solid body in the presence of dislocations. Combined with discrete dislocations dynamics, boundary value problems with dislocations in finite bodies can be solved. Two examples, bending of a single crystal beam and pure shearing of a polycrystalline solid, are simulated by discrete dislocation dynamics using the scheme that has the weaker singularity. The comparisons with the published results using the well-established superposition technique validate the proposed formulation and show its quick convergence.     

应力函数及其对偶理论在有限元中的应用

借助于Cosserat连续介质模型，探讨了应力函数和位移对避免有限元C$^{1}$ 连续性困难的互补性作用. 通过对应力函数对偶理论的深入分析，为将应力函数列式得到的 余能单元转化为具有一般位移自由度的势能单元提供了严格的理论基础，在此基础上， 给出应用应力函数构造有限元的一般方法.