Local analysis and global nonlinear behaviour of periodic assemblies of bodies in elastic contact

This work attempts to capture the effects of microstructural changes on the overall response of a composite made of bodies in elastic contact, and to define numerically a homogenised constitutive relationship for the global behaviour. The analysis is restricted to small strains, plane problems and monotonic proportional loads. An important feature of this work is the quasi-static frictional contact analysis of the microstructure composed of deformable elastic bodies by means of parametric quadratic programming principle and its corresponding algorithmic implementation. The generality of the homogenisation algorithm, which, in principle, can be applied to a large variety of non-linear behaviour affecting the representative volume element, is clearly described. Numerical examples are given to demonstrate the efficiency and validity of the algorithm presented in the paper. Received 15 August 1998     

A direct automated procedure for frictionless contact problems

The solution of elastostatic bodies in frictionless contact is obtained by an automated direct method which exploits the theory of linear elasticity and circumvents the need for the inclusion of artificial interface elements, mathematical programming techniques or computation of contact pressure. The method is simple and economical to use and can be easily appended to existing numerical schemes such as the finite element method. The formulation and numerical algorithm are presented for body combinations which are independent of relative tangential displacements along the contact surface. The method is illustrated through an elementary example amenable to hand calculation. Numerical results for more realistic problems are given and compared to known solutions. It is concluded that the method provides a powerful means for both the analysis and design of contacting bodies when used in conjunction with a finite element computer program.     

Application of an improved contact algorithm for penetration analysis in SPH

A numerical algorithm to analyze an elasto-plastic contact problem using the smoothed particle hydrodynamics (SPH) method is proposed. In order to solve the contact problem with SPH when both bodies are deformable, a variational equation based on the virtual work principle is derived and its solution is obtained by the penalty method. Methods to find the boundary particles and to determine the boundary normal vector are reviewed to apply to the contact algorithm. In calculating the actual penetration and penetration rate, a method that can determine the actual normal boundary vector through the contact relationship with a contact surface is proposed. A numerical simulation is conducted to validate the proposed method in cylindrical coordinates. A steel ball 10 mm in diameter impacting a thin steel plate of 1 mm thickness at a high velocity such as 200 m/s is chosen to verify that the contact algorithm can be applied to the penetration problem. The final shape results obtained by the proposed contact algorithm are more similar to the experimental results than the conventional SPH analysis results.     

A finite element model for contact analysis of multiple Cosserat bodies

The objective of this paper is to develop a finite element model for multi-body contact analysis of Cosserat materials. Based on the parametric virtual work principle, a quadratic programming method is developed for finite element analysis of contact problems. The contact problem with friction between two Cosserat bodies is treated in the same way as in plastic analysis. The penalty factors, that are normally introduced into the algorithm for contact analysis, have a direct influence on accuracy of solution. There is no available rule for choosing a reasonable value of these factors for simulation of contact problems of Cosserat materials, and they are therefore cancelled through a special technique so that the numerical results can be of high accuracy. Compared with the conventional work on Cosserat elasticity, the newly developed model is on the contact analysis of the Cosserat materials and is seldom found in the existing literatures. Four examples are computed to illustrate the validity and importance of the model developed.     

弹性接触颗粒状周期性结构材料力学分析的均匀化方法(Ⅰ) ——局部RVE分析

研究工作目的是建立弹性接触颗粒状组成周期性结构材料力学分析的均匀化模型。首先对具有周期性构造的弹性接触颗粒材料力学的微观(小尺度)与宏观两级均匀化方法的研究现状进行了简要回顾,进而发展了问题局部RVE分析的有限元求解技术,该方法考虑了弹性接触体的粘着界面特性,并基于参变量变分原理提出了问题求解的参数二次规划算法,为宏观均匀化分析工作打下基础。      

Solution of large deformation impact problems with friction between Blatz–Ko hyperelastic bodies

Feng et al. [Z.Q. Feng, F. Peyraut, N. Labed, Solution of large deformation contact problems with friction between Blatz–Ko hyperelastic bodies, Int. J. Eng. Sci. 41 (2003) 2213–2225] have proposed a study of contact problems between Blatz–Ko hyperelastic bodies in static cases using the bi-potential method. The extension of this method for dynamic analysis of impact problems is realized in the present work. The total Lagrangian formulation is adopted to describe large strains and large displacements non-linear behavior. A first order algorithm is applied for the numerical integration of the time-discretized equation of motion. Numerical examples are carried out in two cases: rigid–deformable and deformable–deformable–rigid impacts in 2D. Numerical results show that the proposed approach is robust and efficient and the physical energy dissipation phenomena are apparently illustrated.     

Solution of large deformation contact problems with friction between Blatz-Ko hyperelastic bodies

The present paper is devoted to the analysis of the contact problems with Coulomb friction and large deformation between two hyperelastic bodies. One approach to separate the material nonlinearity and contact nonlinearity is presented. The total Lagrangian formulation is adopted to describe the geometrically nonlinear behavior. Nondifferentiable contact potentials are regularized by means of the augmented Lagrangian method. Numerical examples are carried out in two cases: rigid-deformable contact and deformable-deformable contact with large slips. The numerical results prove that the proposed approach is robust and efficient concerning numerical stability.     

Fatigue crack propagation in rocker and roller-rocker bearings of railway steel bridges

In this work, a method is proposed for rolling contact fatigue crack propagation analysis using contact and fracture theories in conjunction with fatigue laws. The proposed method is used in the fatigue analysis of rocker and roller-rocker bearings of a railway open web girder bridge which is instrumented with strain gages. Using a contact algorithm based on the minimum energy principle for bodies in rolling contact with dry friction, the normal and tangential pressure distribution are computed. It is seen that the most critical location of a crack in bearings is at a point very close to the contact region, as expected.     

Semi-Lagrangian reproducing kernel particle method for fragment-impact problems

Fragment-impact problems exhibit excessive material distortion and complex contact conditions that pose considerable challenges in mesh based numerical methods such as the finite element method (FEM). A semi-Lagrangian reproducing kernel particle method (RKPM) is proposed for fragment-impact modeling to alleviate mesh distortion difficulties associated with the Lagrangian FEM and to minimize the convective transport effect in the Eulerian or Arbitrary Lagrangian Eulerian FEM. A stabilized non-conforming nodal integration with boundary correction for the semi-Lagrangian RKPM is also proposed. Under the framework of semi-Lagrangian RKPM, a kernel contact algorithm is introduced to address multi-body contact. Stability analysis shows that temporal stability of the kernel contact algorithm is related to the velocity gradient between two contacting bodies. The performance of the proposed methods is examined by numerical simulation of penetration processes.     

Boundary element analysis of two-dimensional elastoplastic contact problems

In this paper a boundary element method for two-dimensional elastoplastic stress analysis of frictional contact problems is presented. The bodies in contact are treated as separate regions. The contact conditions are used to join the different system of equations for different regions of contacted bodies and, hence, an overall system of equation is obtained. An incremental and iterative procedure can be used to find the contact load, or the contact extent and the proper contact conditions. To include the plastic deformation in the analysis, the initial strain algorithm is employed. Elastic-perfectly plastic or work-hardening material behaviour can be assumed. For the numerical analysis, an isoparametric three-noded line elements are used to represent the boundary and eight-noded quadrilateral or six-noded triangular elements are used for the interior of the domain. The displacement rates and traction rates are assumed to vary quadratically and the shape functions for the interior strain rates are also of quadratic type. As an example, the behaviour of an elastic and elastoplastic body with a smooth, circular inclusion under the increasing load is presented.     

Use of the Constrained Optimization Algorithms in Some Problems of Fracture Mechanics

In this paper we consider an algorithm of constrained optimization which arises from boundary variational principles of elastodynamics for bodies with cracks and unilateral constraints on the cracks edges. Variational formulation of unilateral contact problems with friction was considered, boundary variational functionals used with boundary integral equations were obtained and algorithm for solution of the unilateral contact problem with friction was developed. Some numerical results for 3-D elastodynamic unilateral contact problem for bodies with cracks are presented.     

Tangent operators,sensitivity expressions,and optimal design of non-linear elastica in contact with applications to beams

The tangent operator and design sensitivity expressions for non-linear elastica subject to frictionless unilateral contact are derived and computed via the finite element method. The sensitivity computations are then combined with a numerical programming package to create an optimal design environment. To exemplify the optimal design environment, a beam's contact surface is contoured to minimize bending stress. This work combines the research on the sensitivity analysis of non-linear elastic bodies subject to constraints, finite strain non-linear elastic beam analysis and contact analyses. The analysis is valid for any smooth contact surface; and specialized for the case in which the surface is represented by a cubic spline. The direct differentiation method is utilized to perform the sensitivity analysis. In an example problem, the sensitivity analysis is verified by finite difference computations and then combined with a numerical optimization program to design the stop profile of a valve.     

NBS contact detection algorithm for bodies of similar size

Large-scale discrete element simulations, as well as a whole range of related problems, involve contact of a large number of separate bodies. In this context an efficient and robust contact detection algorithm is necessary. There has been a number of contact detection algorithms with total detection time (CPU time needed to detect all couples close to each other) proportional to Nln(N) (where N is the total number of separate bodies) reported in recent years. In this work a contact detection algorithm with total detection time proportional to N is reported. The algorithm is termed NBS, which stands for no binary search. In other words, the proposed algorithm involves no binary search at any stage. In addition the performance of the algorithm in terms of total detection time is not influenced by packing density, while memory requirements are insignificant. The only limitation of the algorithm is its applicability to the systems comprising bodies of similar size. © 1998 John Wiley & Sons, Ltd.     

Effect of numerical artificial corners appearing when using BEM on contact stresses

It has long been established that when a small-displacement problem involving contact between two or more bodies is solved, conforming algorithm approaches based on BEM are reliable and robust. Nevertheless, the authors have shown in a previous paper that a relative sliding between the bodies in contact, even if small, can originate a fictitious corner in the numerical model, which may alter the stress and the displacement predictions. This problem disappears if a non-conforming algorithm is employed. In this paper a new problem with different features but also presenting artificial alteration of the results is addressed. The compression of a rounded punch on a foundation is analysed for many fillet radii. Conforming and non-conforming algorithms are used. Comparison of results shows that effects of an artificial ‘numerical corner’ may alter even the traction distribution relatively far from it. Consequently, the assumed connection between small displacements and initially conforming discretizations with the use of a conforming approach disappears. An adequate knowledge of the contact problem under consideration is then essential to determine whether the singularity is inherent to the problem or is artificial, a non-conforming algorithm then being recommended in this second case.     

The bi-potential method applied to the modeling of dynamic problems with friction

The bi-potential method has been successfully applied to the modeling of frictional contact problems in static cases. This paper presents an extension of this method for dynamic analysis of impact problems with deformable bodies. A first order algorithm is applied to the numerical integration of the time-discretized equation of motion. Using the Object-Oriented Programming (OOP) techniques in C++ and OpenGL graphical support, a finite element code including pre/postprocessor FER/Impact is developed. The numerical results show that, at the present stage of development, this approach is robust and efficient in terms of numerical stability and precision compared with the penalty method.     

MR linear contact detection algorithm

Large‐scale discrete element simulations, as well as a whole range of related problems, involve contact of a large number of separate bodies and an efficient and robust contact detection algorithm is necessary. There has been a number of contact detection algorithms with total detection time proportional to N ln(N) (where N is the total number of separate bodies) reported in the past. In more recent years algorithms with total CPU time proportional to N have been developed. In this work, a novel contact detection algorithm with total detection time proportional to N is proposed. The performance of the algorithm is not influenced by packing density, while memory requirements are insignificant. The algorithm is applicable to systems comprising bodies of a similar size. The algorithm is named MR (Munjiza–Rougier: Munjiza devised the algorithm, Rougier implemented it). In the second part of the paper the algorithm is extended to particles of different sizes. The new algorithm is called MMR (multi‐step MR) algorithm. Copyright © 2005 John Wiley & Sons, Ltd.     

Smooth contact strategies with emphasis on the modeling of balloon angioplasty with stenting

Critical to the simulation of balloon angioplasty is the modeling of the contact between the artery wall and the medical devices. In standard approaches, the 3D contact surfaces are described by means of C⁰‐continuous facet‐based techniques, which may lead to numerical problems. This work introduces a novel contact algorithm where the target surfaces are described by polynomial expressions with C²‐continuity. On the basis of uniform cubic B‐splines, two different parametrization techniques are presented and compared, while the related implementation of the algorithm into a finite element analysis program is described. Two numerical examples are selected to demonstrate the special merits of the proposed contact formulation. The first example is a benchmark contact problem selected to point out the special features of the proposed strategies. The second example is concerned with the simulation of balloon angioplasty and stenting, where the contact between the balloon, the stent and the artery wall is numerically modeled. A patient‐specific 3D model of a stenotic femoral artery serves as a basis. The study concludes by identifying the changes in the mechanical environment of the artery in terms of contact forces and strains by considering two different stent designs. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

Penalty function method for combined finite–discrete element systems comprising large number of separate bodies

Large‐scale discrete element simulations, the combined finite–discrete element method, DDA as well as a whole range of related methods, involve contact of a large number of separate bodies. In the context of the combined finite–discrete element method, each of these bodies is represented by a single discrete element which is then discretized into finite elements. The combined finite–discrete element method thus also involves algorithms dealing with fracture and fragmentation of individual discrete elements which result in ever changing topology and size of the problem. All these require complex algorithmic procedures and significant computational resources, especially in terms of CPU time. In this context, it is also necessary to have an efficient and robust algorithm for handling mechanical contact. In this work, a contact algorithm based on the penalty function method and incorporating contact kinematics preserving energy balance, is proposed and implemented into the combined finite element code. Copyright © 2000 John Wiley & Sons, Ltd.