Investigation of plane elastic contact allowing for friction

This paper describes an analysis of plane elastic contact using the finite element method. Solids in contact are subject to small strains, and friction is present between them. Triangular elements are used, in plane strain or stress, with, in each node, six parameters which are the displacements and their partial derivatives combined to obtain strains or stresses according to the continuities to be satisfied through the contact. Opposite nodal points are defined on the two surfaces, in the area which is expected to be in contact under load. Stiffness matrices are first calculated for each solid, respectively and, after that, coupled relatively to the degrees of freedom of all the previous common nodal points. Loads are applied progressively and the equilibrium state is sought, for each increment, by means of an iterative procedure: (1) the contact area is defined; if the calculated normal stress is tension at some opposite nodal points, these are uncoupled; (2) according to the friction law used, the coupled nodal points are labelled “adhering” or “sliding” and, in this case, the tangential stress is made equal to the maximum allowable value. The method is checked for the case with zero friction, the well known Hertz problem. Results are in very good agreement with the analytical solution, for both the contact area dimensions and the pressure distribution. The case of a connecting rod with a loose-fitting bush in one joint is investigated. The calculated normal pressure distribution is shown, at two load levels, for several values of the clearance. As verification, the external circumferential stress distribution is compared with strain-gage measurements.     

Finite element analysis of elastodynamic sliding contact problems with friction

This paper presents a general but effective finite element technique to analyze elastodynamic sliding contact problems with friction. The deformed contact area is obtained using the constraint conditions developed by a quadratic mathematic programming technique. Lagrangian multipliers are introduced to evaluate the contact pressures due to friction and determine the adhesion or release of contact surface. Based on these, the sliding process between two contact surfaces is accurately modelled. This work also provides the correction formulae for modelling the transient response of velocities and accelerations on the contact surface when the initial impact or release of the contact surface occurred. Several numerical examples are carried out to demonstrate the validity and accuracy of this work. The complete scheme of the transient sliding response of two contact components subjected to oblique impact loadings is drawn.     

Moving finite element analysis for the elastic beams in contact problems

The contact problem of a linear elastic beam with a rigid barrier is considered, in which the contact surface is assumed to be frictionless. The problem contains the characteristic of moving boundaries as the contact regions expand or reduce in size when the external loads alter. Starting from the variational principles, we derived the interface equations as well as the governing equation. In addition to matching the deflections and the slopes with the rigid barrier at the marginal nodes of the contact regions, the interface equations also have to be satisfied there. In essence, the interface equations are designed to locate the unknown and moving marginal nodes. Although it is confined within the framework of small deformation and linear elastic material behavior, the problem exhibits high nonlinearity due to the moving nature of the marginal nodes. In this paper, we present a moving finite element analysis employing incremental procedures and an iterative numerical scheme to tackle the problem. A fixed number of two-node beam elements are used in the moving finite element analysis and the size of the elements varies as the loads alter. A couple of examples, whose exact solutions are obtainable, are chosen to demonstrate the accuracy and efficiency of the proposed numerical algorithm.     

Boundary element analysis of elastic contact problems using gap finite elements

Gap finite elements provide a practical approach for dealing with elastic contact problems, and it is not possible to derive something similar with boundary elements. This work introduces a simple technique for the analysis of elastic contact problems by coupling a gap finite element subregion with boundary element subregions. The developed algorithm proves to be accurate and reliable, combining the advantages within contact problems of both the gap finite element and the boundary element method.     

Hybrid MoM/FEM modeling of loaded enclosure with aperture in EMC problems

To predict electromagnetic field distribution inside a loaded enclosure with the aperture, we present a hybrid formulation which combines method of moments and finite element method. To predict efficiency of the method, the shielding effectiveness of an empty enclosure is compared with the literature. Then the method is applied to dielectric slab loaded enclosure with the single aperture, and the shielding effectiveness, ratio of stored electrical energy, and dissipated power variations are obtained. © 2008 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2009.     

A solution method for static and dynamic analysis of three-dimensional contact problems with friction

A solution method is presented for the analysis of contact between two (or more) three-dimensional bodies. The surfaces of the contacting bodies are discretized using quadrilateral surface segments. A Lagrange multiplier technique is employed to impose that, in the contact area, the surface displacements of the contacting bodies are compatible with each other. Distributed contact tractions over the surface segments are calculated from the externally applied forces, inertia forces and internal element stresses. Using the segment tractions, Coulomb's law of friction is enforced in a global sense over each surface segment. The time integration of dynamic response is performed using the Newmark method with parameters and . Using these parameters the energy and momentum balance criteria for the contacting bodies are satisfied accurately when a reasonably small time step is used.

The applicability of the algorithm is illustrated by selected sample numerical solutions to static and dynamic contact problems.     

An iterative procedure for finite-element stress analysis of frictional contact problems

A simple, efficient, versatile and easily adaptable, iterative finite-element technique is described for solving frictional contact problems. The method is based on logical steps to establish the contact geometry and regions of slip and nonslip. Unlike previous techniques, the approach can be extended readily to multiple contact surfaces. The scheme is demonstrated by analyzing a mechanical joint in orthotropic wood. In this case, mixed coordinate systems are used to enhance accuracy of the stresses near the pin contact region. The numerically computed values agree with those reported elsewhere.     

A numerical analysis of contact and limit-point behavior in a class of problems of finite elastic deformation

Finite element methods for the analysis of bifurcations, limit-point behavior, and unilateral frictionless contact of elastic bodies undergoing finite deformation are presented. Particular attention is given to the development and application of Riks-type algorithms for the analysis of limit points and exterior penalty methods for handling the unilateral constraints. Applications focus on the problem of finite axisymmetric deformations, snap-through, and inflation of thick rubber spherical shells.     

Some reflections on elastoplastic stress calculation in finite element analysis

The set of stress invariants (J₁,J'₂,θ) is very frequently employed in finite element programs for elastoplastic analysis. Some precautions are however required when computing the stress states of contact with the yield surface. The numerical problems that may arise and simple measures to avoid them are described in this paper.     

Constitutive error estimator for the control of contact problems involving friction

In this paper, we propose an error estimator for unilateral contact problems involving friction. The use of a bipotential to formulate the constitutive relation on the contact zone leads to a very straightforward construction of an error in constitutive relation estimator for this kind of problem. Its implementation is described in detail for the case of two elastic bodies and two 2D examples are presented. An error indicator which enables one to estimate the quality of the numerical solution of the non-linear problem is proposed.     

Deformation analysis of thin elastic membranes in multiple contact

This article presents the deformation analysis of thin elastic membranes, supported at the boundaries, and lifted by multiple rigid indentors. This study is motivated by a problem that is encountered in the design of flexible elastomeric surface tooling for prototyping free-form shapes made of composite materials. The deformation analysis was carried out using a mathematical programming approach and assuming contacts between the membrane and indentors to be frictionless. The Mooney form of strain energy function is used for potential energy calculations. The conjugate gradient method in the presence of contact constraints is used for energy minimization. The results are presented for three numerical examples describing the realizability of ruled and doubly curved surface shapes.     

A new method of stress calculation in elastic kinetics problems

The conventional difference scheme described by Courant et al. [2] in 1928 is only a finite element method with a special integral approximation. A new method of calculating stress is presented in this paper, according to which the stresses acting on some small regions just satisfy the conservative difference equations for the kinetic equation. The results calculated for an elastic plane problem show that the error of stress obtained by this method is less than that yielded by the finite element method.     

Nonsmooth analysis and quasi-convexification in elastic energy minimization problems

An energy minimization problem for a twocomponent composite with fixed volume fraction is considered. Two questions are studied. The first is the dependence of the minimum energy on the constraints and parameters. The second is the rigorous justification of the method of Lagrange multipliers for this problem. It is possible to treat only cases with periodic or affine boundary condition. It is also found that the constrained energy is a smooth and convex function of the constraints. It is shown that the Lagrange multiplier problem is a convex dual of the problem with constraints. Moreover, it is shown that these two results are closely linked with each other. The main tools are the Hashin-Shtrikman variational principle and some results from nonsmooth analysis.This work was done while the author was a student at the Courant Institute.     

A plane stress to plane strain functional technique to compute stress-intensity factors in three-dimensional problems

A functional technique to calculate stress-intensity factors for cracks intersecting or approaching free surfaces is presented. Results are compared to previous finite elements and alternating solutions. The comparison shows that use of a full plane strain formula could be questionable in three-dimensional problems where cracks approach or intersect free surfaces.     

Theory of finite element method for elastic contact problems of solid bodies

In the paper technique for solving two dimensional (2D) friction/gap closure interface problems is described. These technique can be applied to various problems, with more or less success, depending on how well the real geometry and loading conditions agree with mathematical model.The method given here is an iterative procedure. It is general and could be extended to solve other than elastic problems.     

Transient analysis of sliding contact of layered elastic/plastic solids with rough surfaces

 Based on the boundary element method (BEM) and a variational principle, a numerical model is developed to analyze the time – transient sliding contact of two layered elastic/plastic solids. Two cases are considered: one is the loading/sliding/unloading of a rough surface on a smooth surface, and the other is of two rough surfaces. Contact statistics, contact pressure profile and stress distribution are predicted at each time step with updated surface roughness. The results are used to study the effect of surface roughness, physical properties of the layer and the substrate, and lubricant film thickness on friction, stiction, and wear. Discussion on the integration of this contact model into advanced tribological models, e.g., wear model, is also presented. Received: 28 June 2002/Accepted: 23 October 2002 Currently at: Seagate Technology, Pittsburgh, PA Paper presented at the 13th Annual Symposium on Information Storage and Processing Systems, Santa Clara, CA, USA, 17–18 June, 2002     

Alternative FEM formulations for the inelastic large deflection analysis of plane frames

The present paper is concerened with the analysis of plane frames of arbitrary geometry undergoing large elasto-plastic deformations. On the basis of so-called mixed-hybrid variational theorems wherein displacements and stress resultants represent independent variables, three alternative finite element schemes are proposed. Their theoretical and computational advantages in comparison with the standard assumed displacement formulation are pointed out and discussed in detail.     

A parallel multigrid method for constrained minimization problems and its application to friction,contact, and obstacle problems

The parallel solution of constrained minimization problems requires special care to be taken with respect to the information transfer between the different subproblems. Here, we present a nonlinear decomposition approach which employs an additional nonlinear correction step along the processor interfaces. Our approach is generic in the sense that it can be applied to a wide class of minimization problems with strongly local nonlinearities, including even nonsmooth minimization problems. We also describe the implementation of our nonlinear decomposition method in the object oriented library ObsLib \(++\). The flexibility of our approach and its implementation is presented along different problem classes as obstacle problems, frictional contact problems and biomechanical applications. For the same examples, number of iterations, computation time, and parallelization speedup are measured, and the results demonstrate that the implementation scales reasonably well up to 4096 processors.