I. Páczelt

Solution of contact problem using the hp-version of the finite element method☆

Abstract This paper is concerned with the numerical treatment of mechanical contact problems in two dimensions using the p -version of the finite element method coupled with minor iterative modification of the mesh. The method of solution is based on the augmented Lagrangian technique. Examples are presented.

Optimal shape design for contact problems

A new optimization problem of controlling the contact pressurep is presented. We seek the minimum of the maximum contact pressurepmax under the following conditions: (a) the contact pressure should satisfy the inequalityp ≥ 0 as well as the controlling conditionχ = v(x) pmax −p(x) ≥ 0, wherev(x) is the controller function; (b) the gapd between two bodies should be positive or zerod ≥ 0 after deformation; (c) at any possible contact point the conditionpd = 0 should be satisfied and the body that is capable of rigid body motion should be in equilibrium.If the gapd = d(p, Δh) is a linear function of both the pressurep and the contour changeΔh, then the optimization problem is a linear programming problem of restricted basis exchange. Three algorithms are presented for the solution of this problem. Finally, numerical examples will demonstrate the controlling technique for the shape optimization problem of a roller bearing.

Analysis of Thermo-Elastic Wear Problems

In the relative sliding motion of two elastic bodies on contact interfaces both frictional dissipation and wear process occur. The wear process induces shape evolution of contact surface and growth of contact zone. The temperature field is generated by external heat flow and by frictional dissipation on the contact surface. The wear and heat conduction processes are coupled and tend to their steady or quasi-steady states for which the stress and temperature fields are fixed on the translating contact zone and depend only on its size and shape parameters. The steady-state is characterized by the minimum of wear dissipation power for which the stationary conditions generate the contact pressure distribution. The related steady-state temperature field is next specified and the thermal distortion effect is analyzed. The contact shape attained in the steady wear state is optimal as it corresponds to minimal wear rate, and its form depends on both mechanical loading and temeperature field. Several specific examples are presented for translating and rotating punches.

Design curve determination for two-layered wire rope strand using p-version finite element code

Parametric analysis of a two-layered axially loaded strand is performed using the recently developed p-version finite element code, which describes the geometry well and takes into account all possible inter-wire motions and frictional contact between the wires. A special nonlinear contact theory was developed based on the Hertz-theory. It is assumed that the wires have homogenous, isotropic, linear elastic material properties. The developed code is a tool for designing wire rope strands that require low computer resources and short computational time. Case studies are performed to verify and demonstrate the efficiency and applicability of the method. Design curves are presented according to the strand geometry parameters such as helix angle and ratio of the wire radius in the different layers. The optimal geometry parameters for a given strand can be determined using these design curves.

Application of the space enrichment method to problems of mechanical contact

Abstract An extension of the p-version of the finite element method, called the space enrichment method, is described and its application to normal contact problems, using the penalty and augmented Lagrangian formulations, is illustrated by examples. The contacting bodies are represented by the equations of the linear theory of elasticity. The numerical solutions obtained by the space enrichment method and the hp-version of the finite element method are compared. The two methods exhibit very close agreement. The efficiency and accuracy of the space enrichment method are demonstrated.

On Steady Wear States for Monotonic Relative Sliding of Contacting Bodies

The paper presents a synthetic review of recent research carried out by the writers [1-6, 7-9] on contact shape optimization coupled with wear and on the steady wear regimes reached in the transient wear process. It was shown that these regimes can also be specified from the optimality conditions. In the analysis several classes of shape optimization problems were considered, namely minimization of wear volume rate, friction dissipation power or wear dissipation power. It was demonstrated that the contact shape evolution tends to steady or quasi-steady states satisfying the minimum principle of the wear dissipation power, resulting in the coaxiality rule requiring the wear rate vector to be collinear with the rigid body wear velocity vector. The application of steady state wear rules in specification of contact states for selected problems is discussed in the paper. The extension of method is presented for the case of multi-zone contact problems for which both transient and steady states have been analyzed.

Analysis of thermo-mechanical wear problems for reciprocal punch sliding

The relative sliding motion of two elastic bodies in contact induces wear process and contact shape evolution. In the case of a punch sliding on a substrate the transient process tends to a steady state for which the fixed contact stress and strain distribution develops in the contact zone. This state usually corresponds to a minimum of the wear dissipation power. The optimality conditions of the wear dissipation functional provide the contact stress distribution and the wear rate compatible with the rigid body punch motion. The present paper is aimed to extend the previous analyses 1-5 of steady state conditions to cases of periodic sliding of punch, assuming cyclic steady state conditions for both mechanical and thermal fields.

Shape design of rubber part using FEM

Abstract This paper presents a novel solution for shape optimization of compressed rubber parts. The procedure is based on the finite element method (FEM). A special purpose FEM code written in FORTRAN has been developed for the analysis of nearly incompressible axi-symmetric rubber parts. Numerical stability of the code and sensitivity analysis of the FEM input parameters are investigated. The aim of the parameter optimization is to reduce the time consuming FEM computations for the optimization process. The objective of the optimization is to find the optimal shape of the investigated rubber parts with a specified load-displacement curve. A regression model is used to determine the connection between the input and output data calculated by the FEM.

Relative motion of continua with applications to the analysis of nonlinear problems by the finite element method

Abstract The present paper investigates deformations within a solid body assuming that the deformations observed from a fixed coordinate system are composed of two parts, a relative deformation and that of a coordinate system moving arbitrarily, i.e., being deformable with respect to the fixed one. When we apply the finite element method deformations of the solid body (or the finite element mesh moving together with it) are regarded as relative deformations with respect to the coordinate system having a prescribed motion and deformation.

Contact Optimization Problems for Stationary and Sliding Conditions

The contact stress distribution is frequently not regular. It may contain singularities reducing the lifetime of machine elements. In order to eliminate such stress singularities, the application of contact pressure control is recommended in the contact conditions. In the paper, several classes of optimization problems are formulated for stationary and sliding contacts. Further, they are illustrated by specific examples. The relation to wear process is made as a natural way to attain the steady state contact profile satisfying the optimality conditions corresponding to minimization of the wear dissipation rate. It is assumed that the displacements and strains are small and the materials of the contacting bodies are elastic.