A formulation of arbitrarily shaped surface elements for three-dimensional large deformation contact with friction

The paper introduces a general theory for the numerical simulation of large deformation contact problems. The contacting bodies under consideration may be of two- or three-dimensional shape modelled by finite elements. A contact finite element which can be applied to handle multi-body contact as well as contact with rigid bodies is developed. The element is universal in the sense that it can be used as a surface element for any known finite element model and includes friction. The frictional behaviour of the model obeys Coulomb's law of friction distinguishing between sticking and sliding contact. The algorithmic treatment is based on a penalty formulation for the normal and sticking contact. The corresponding consistent tangential stiffness matrices are derived, leading to an overall quadratic convergence behaviour for the method. This feature is demonstrated in a number of representative examples. © 1997 by John Wiley & Sons, Ltd.     

Frictional contact of flexible and rigid bodies

 This paper is about planar frictional contact problems of both flexible and rigid bodies. For the flexible case a nonlinear finite element formulation is presented, which is based on a modified Coulomb friction law. Stick-slip motion is incorporated into the formulation through a radial return mapping scheme. Linearly interpolating four node elements and three node contact elements are utilized for the finite element discretization. The corresponding tangent stiffness matrices and residual vectors of the equations of motion are presented. In the rigid body case the contact problem is divided into impact and continual contact, which are mathematically described by linear complementarity problems. The impact in normal direction is modeled by a modified Poisson hypothesis, which is adapted to allow multiple impacts. The formulation of the tangential impact is grounded on Coulombs law of friction. The normal contact forces of the continual contact are such that colliding bodies are prevented from penetration and the corresponding tangential forces are expressed by Coulombs law of friction. Examples and comparisions between the different methods are presented. Received: 10 January 2001     

Finite element method used in contact problems with dry friction

In this paper, we present a formulation of numerical approximations of the frictional quasi-contact problem with dry friction between a deformable body and a foundation with possibility to consider the case of two deformable bodies. We consider numerical approximations of 3D static contact problem with dry friction, using finite contact elements. Saddle-point algorithm, Lagrange incremental multipliers method and penalty functions are used to enforce finite element surface contact constrains for incremental formulation of the frictional quasi-static problem. Some typical examples in the elastic contact problems with dry friction are presented.     

Finite element hydrodynamic friction model for metal forming

The complex contact conditions on the three-dimensional (3-D) tooling-workpiece interface, such as non-penetrations, slip–stick phenomena and friction forces due to the relative motion of contacting surfaces, are of vital importance in metal forming operations. Usually, a lubricant is provided as an interface medium between the tool and the workpiece to avoid strain localization, wear and surface damage. Hence, a simple friction law such as Coulomb friction, involving only a constant friction coefficient, cannot model the contact phenomena accurately. In this research, a realistic friction model, which accounts for the tribological behaviour, and most importantly, the effect of surface roughness on the lubricated contact, is developed. This model has been implemented in a 3-D arbitrary Lagrangian Fulerian finite element code for metal forming analysis. The applicability of the proposed model is demonstrated by the simulation of fluid-lubricated thrust bearing and sheet metal stretch forming.     

A parallel finite element contact/impact algorithm for non-linear explicit transient analysis: Part I—The search algorithm and contact mechanics

A contact algorithm has been developed and implemented in a non-linear dynamic explicit finite element program to analyse the response of three-dimensional shell structures. The contact search algorithm accounts for initial contact, sliding, and release through the use of a parametric representation of the motion of points located on the surface of the structure combined with a contact surface representation which approximates the actual surface by means of triangular search planes. The mechanics of contact is handled by taking advantage of the fact that an explicit time integration scheme results in very small displacements during a time step. The amount of overlap of the discrete representation of the surfaces which occurs at contact is taken as a measure of the approach of the surfaces. Hence, experimental data which relates approach to normal contact pressure can be used to determine the contact pressure applied to the finite element model of the surface as contact evolves. The friction model also incorporates experimental data on the dependence of the coefficient of friction on both the relative sliding velocity and on the relative tangential displacement between surfaces in contact observed in friction tests. The parallel implementation of this contact algorithm and its performance on a 128-processor distributed-memory multiprocessor computer is discussed in Part II of this paper.     

Relative Slip on Contact Surface under Partial Slip Fretting Fatigue Condition

Abstract:  A combined experimental–numerical approach was utilized to characterize the relative slip along the contact surface and its features under the partial slip fretting fatigue condition. Relative displacements at two locations on the substrate (specimen) and fretting pad were measured in fretting fatigue tests. These measurements were then utilized to validate finite element analysis. Effects of the coefficient of friction on the relative slip and contact condition were investigated. The stress state along the contact surface was also investigated. Two contact geometries were analysed: cylinder-on-flat and flat-on-flat. There was no change in relative displacement between locations away from the contact surface because of the change in the coefficient of friction, while relative slip on the contact surface was affected by coefficient of friction. In addition, stick/slip sizes were affected by the change in coefficient of friction. Comparison between present and previous finite element models showed that stress state, as well as a critical plane-based crack-initiation parameter, was not much different between these approaches, while relative slip on the contact surface changed considerably.     

Simulation of surface pitting due to contact loading

A computational model for simulation of surface pitting of mechanical elements subjected to rolling and sliding contact conditions is presented. The two-dimensional computational model is restricted to modelling of high-precision mechanical components with fine surface finishing and good lubrication, where the cracks leading to pitting are initiated in the area of largest contact stresses at certain depth under the contacting surface. Hertz contact conditions with addition of friction forces are assumed and the position and magnitude of the maximum equivalent stress is determined by the finite element method. When the maximum equivalent stress exceeds the local material strength, it is assumed that the initial crack develops along the slip line in a single-crystal grain. The Virtual Crack Extension method in the framework of finite element analysis is then used for two-dimensional simulation of the fatigue crack propagation under contact loading from the initial crack up to the formation of the surface pit. The pit shapes and relationships between the stress intensity factor and crack length are determined for various combinations of contacting surface curvatures and loadings. The model is applied to simulation of surface pitting of two meshing gear teeth. Numerically predicted pit shapes in the face of gear teeth show a good agreement with the experimental observations. © 1998 John Wiley & Sons, Ltd.     

Prevention of RCF damage in curved track through development of the INFRA‐STAR two‐material rail

Results from the European 5th frame research project ‘INFRA‐STAR’ are presented. The goal of the project is to prevent rolling contact fatigue (RCF) and to reduce squeal noise in curves by applying an additional surface layer material on the top of the railhead, resulting in a two‐material rail. In INFRA‐STAR, a dynamic train–track interaction model is used to provide the contact forces. Wheel‐rail profiles, wheel‐rail friction, vehicle data, track data and operating conditions are included to calculate the wheel‐rail contact forces and spin moments, contact positions and load distributions in the contact patch. The contact pressure, friction coefficient, coating thickness, material properties of the coating and the rail material are used in finite element calculations and shakedown theory to calculate shakedown limits, which are then used to predict the RCF performance of the system. The paper details the work on theoretical modelling, twin disc testing, metallurgical research and field testing completed to date (August 2002, the project just passed midterm). The development of the surface layer application methods that are used, and the further objectives of the INFRA‐STAR project, are discussed.     

Characterization of Probe Dynamic Behaviors in Critical Dimension Atomic Force Microscopy

This paper describes a detailed computational model of the interaction between an atomic force microscope probe tip and a sample surface. The model provides analyses of dynamic behaviors of the tip to estimate the probe deflections due to surface intermittent contact and the resulting dimensional biases and uncertainties. Probe tip and cantilever beam responses to intermittent contact between the probe tip and sample surface are computed using the finite element method. Intermittent contacts with a wall and a horizontal surface are computed and modeled, respectively. Using a 75 nm Critical Dimension (CD) tip as an example, the responses of the probe to interaction forces between the sample surface and the probe tip are shown in both time and frequency domains. In particular, interaction forces between the tip and both a vertical wall and a horizontal surface of a silicon sample are modeled using Lennard-Jones theory. The Snap-in and Snap-out of the probe tip in surface scanning are calculated and shown in the time domain. Based on the given tip-sample interaction force model, the calculation includes the compliance of the probe and dynamic forces generated by an excitation. Cantilever and probe tip deflections versus interaction forces in the time domain can be derived for both vertical contact with a plateau and horizontal contact with a side wall. Dynamic analysis using the finite element method and Lennard-Jones model provide a unique means to analyze the interaction of the probe and sample, including calculation of the deflection and the gap between the probe tip and the measured sample surface.     

Discrete element simulation of railway ballast: modelling cell pressure effects in triaxial tests

The paper investigates reproducing the effects of confining pressure on the behaviour of scaled railway ballast in triaxial tests in discrete element models (DEM). Previous DEM work, using a standard Hertzian elastic contact law with an elastic–perfectly plastic tangential slip model, has been unable to replicate the behaviour observed in laboratory tests across a range of confining pressures without altering both the material stiffness and the inter-particle friction. A new contact law modelling damage at the contacts between particles is introduced. Particle contact is via spherically-capped conical asperities, which reduce in height if over-stressed. This introduces plasticity to the behaviour normal to the contact surface. In addition, the inter-particle friction angle is varied as a function of normalized contact normal force. At relatively low normal forces the friction angle must be increased for peak mobilized friction angles to match the laboratory data, an effect that is attributed to interlocking at the scale of surface roughness. Simulation results show close agreement with laboratory data.     

Z-pins对层合复合材料非对称分层增韧作用参数分析——Ⅱ型层间韧性

利用纤维Z-pins的细观力学模型构造了相应的Z-pin单元,结合考虑一阶剪切变形的梁单元,建立了用于分析含非对称分层采用Z-pins增韧的端部开口弯曲试件(End notched flexure,ENF)的有限元模型,并在分层裂纹面上引入接触单元以防止分析过程中2个分层子梁在端部开口处的相互嵌入。通过数值算例分析了Z-pins对含非对称分层的ENF试件Ⅱ型层间韧性的增强作用。参数分析表明,当分层位置靠近层合板的表面时,Z-pins的增韧作用明显下降,Z-pins对Ⅱ型层间韧性的增强作用主要由2个分层子层中较薄子层决定,但另一分层的厚度也对Ⅱ型层间韧性有一定的影响。     

高速切削时摩擦系数对切削影响的数值模拟

高速切削加工中,刀屑间的摩擦系数对切削产生重要影响．由于刀屑接触的复杂性,它们之间的摩擦系数很难确定．为了探究高速切削时,刀具与切屑间摩擦系数对切削的影响,采用有限元通用程序ABAQUS／STANDARD对不同摩擦系数下切削过程进行数值模拟．通过对Mises应力和加工表面节点垂直方向位移的比较,得出刀屑间的摩擦系数对剪切角有较大影响,摩擦系数增大,剪切角随之减小．高速切削既能够提高切削效率,又能提高加工精度．     

A two-dimensional elastic–plastic finite element analysis of friction effects on sliding crack surfaces in full or partial contact

Sliding crack surfaces are analysed, that are completely or partially in contact, using a two-dimensional plane-stress elastic–plastic finite element technique. Our in-house program was modified to account for the friction which acts between two rough mating surfaces. The analysis is applied to a cantilever beam cracked along its span through its centroidal plane. Twelve cracks with length-to-span ratios ranging from zero to 0.5 were analysed. The effect of friction was investigated by considering 0, 0.4, 0.8, 1.2 and 1.6 as values for the coefficient of friction with each crack length. The results show the influence of friction on the beam stiffness, strain energy release rate, modes of crack tip and surface displacements, and the development of plastic deformation. The present finite element outputs assist in the explanation of experimental events associated with mode II crack tip displacement data found in the literature.     

弹性压头与复合材料层合梁间的弹性接触分析

冲击器与复合材料接触性态的研究是复合材料低速冲击响应研究的关键.本文采用处理弹性接触问题的有限元混合法研究了弹性压头与复合材料层合梁间的弹性接触问题.用数值算例讨论了静力接触定律和压头弹性、层合梁铺设方式及摩擦等因素对接触内力和层合梁内部层间应力大小和分布的影响,得到一些有益的结论.     

Friction coefficient estimation in shaft/bush interference using finite element model updating

Reliable finite element modeling has a great degree of importance for studies related to the mechanical design in industry. Finite element modeling is often associated with some uncertainties. Model updating is a tool which can be used to overcoming these uncertainties. This method uses experimental data of the structural components to modify the initial FE model. A typical use of model updating is for interference joint of a shaft and bush. In the modeling of the interference joint for known materials, an approximate value is assumed for friction coefficient of contact areas. Based on intuition, engineering judgments or previous tests, it is difficult to accurately estimate this coefficient. In this paper, model updating is used to estimate and update the friction coefficient in the contact surface of interference shaft and bush joints. Suitable experiments were conducted and the initial finite element model was modified based on the experimental extraction strength of interference fit joints. Finally, typical values of stress fields were extracted from the new model with an updated value of the friction coefficient.     

基于接触分析的转定子系统整周碰摩故障模拟

以一个单跨双盘柔性转子系统为研究对象,建立转子系统的有限元模型,基于接触动力学理论,将转子和定子简化为一个点-点接触单元,通过转定子间的圆形间隙变化来模拟转定子的分离及整周接触,并通过碰摩力耦合转定子模型,采用增广的拉格朗日方法处理接触约束条件,用库仑摩擦模型模拟转定子之间摩擦,考虑不同转速、转定子间隙、转定子法向接触刚度、阻尼和摩擦系数对转子系统动力学特性的影响。研究结果表明:转速和转定子法向接触刚度对系统响应影响最大,转定子法向接触阻尼和摩擦系数次之,转定子间隙影响最小。     

An insight into mode II fracture toughness testing using SCB specimen

The effect of friction forces between the test specimen and its bottom supports on the mode II fracture toughness values obtained using the semicircular bend (SCB) specimen is investigated. First, a number of experiments were conducted on SCB specimen in order to determine the mode II fracture toughness of polymethyl methacrylate (PMMA) according to the conventional approaches available in the literature. Three different types of supports that have been frequently employed by researchers in recent years were used to evaluate the effect of support type on the fracture loads. It was found that the friction forces between the supports and the SCB specimen have a significant effect on the value of mode II fracture toughness measured using the SCB samples. Then, the specimen was simulated using finite element method for more detailed investigation on the near crack tip stress field evolution when friction forces increase between the supports and the SCB specimen. The finite element results confirmed that the type of support affects not only the stress intensity factors K_I and K_II but also the T‐stress. The experimental and numerical results showed that the use of the crack tip parameters available in literature for frictionless contact between the supports and the SCB specimen can result in significant errors when the mode II experiments are performed by using the fixed or roller‐in‐grove types of supports.     

Lagrange constraints for transient finite element surface contact

A new approach to enforce surface contact conditions in transient non-linear finite element problems is developed in this paper. The method is based on the Lagrange multiplier concept and is compatible with explicit time integration operators. Compatibility with explicit operators is established by referencing Lagrange multipliers one time increment ahead of associated surface contact displacement constraints. However, the method is not purely explicit because a coupled system of equations must be solved to obtain the Lagrange multipliers. An important development herein is the formulation of a highly efficient method to solve the Lagrange multiplier equations. The equation solving strategy is a modified Gauss-Seidel method in which non-linear surface contact force conditions are enforced during iteration. The new surface contact method presented has two significant advantages over the widely accepted penalty function method: surface contact conditions are satisfied more precisely, and the method does not adversely affect the numerical stability of explicit integration. Transient finite element analysis results are presented for problems involving impact and sliding with friction. A brief review of the classical Lagrange multiplier method with implicit integration is also included.     

Numerical modelling of micro-pitting of gear teeth flanks

A two‐dimensional computational model for simulation of contact fatigue of gear teeth flanks is presented. In the model, it is assumed that the initial crack of length 0.015 mm is initiated at the surface due to previous mechanical or heat treatment of the material as well as a consequence of the running in process. The discretized model with the initial crack is then subjected to normal contact pressure, which takes into account the elasto‐hydro‐dynamic (EHD) lubrication conditions, and tangential loading due to friction between contacting surfaces. The model also considers the moving contact of gear flanks, fluid trapped in the crack and residual stresses due to heat treatment of the material on crack propagation. The virtual crack extension (VCE) method, implemented in the finite element method, is then used for simulating the fatigue crack growth from the initial crack up to the formation of the surface pit. The computational results show that the initial surface crack of length 15 μm and the considered boundary conditions lead to the appearance of very small surface pits, which can be termed as micro‐pitting on gear teeth flanks. The numerical results correspond well with available experimental data.     

Space–time approach to numerical analysis of a string with a moving mass

Inertial loading of strings, beams and plates by mass travelling with near‐critical velocity has been a long debate. Typically, a moving mass is replaced by an equivalent force or an oscillator (with ‘rigid’ spring) that is in permanent contact with the structure. Such an approach leads to iterative solutions or imposition of artificial constraints. In both cases, rigid constraints result in serious computational problems. A direct mass matrix modification method frequently implemented in the finite element approach gave reasonable results only in the range of relatively low velocities. In this paper we present the space–time approach to the problem. The interaction of the moving mass/supporting structure is described in a local coordinate system of the space–time finite element domain. The resulting characteristic matrices include inertia, Coriolis and centrifugal forces. A simple modification of matrices in the discrete equations of motion allows us to gain accurate analysis of a wide range of velocities, up to the velocity of the wave speed. Numerical examples prove the simplicity and efficiency of the method. The presented approach can be easily implemented in the classic finite element algorithms. Copyright © 2008 John Wiley & Sons, Ltd.