J.T. Oden

Models and computational methods for dynamic friction phenomena

Abstract This paper addresses the general problem of formulating continuum models of a large class of dynamic frictional phenomena and of developing computation methods for analyzing these phenomena. Of particular interest are theories which can adequately predict stick-slip motion, frictional damping in structural dynamics, and sliding resistance. This work is divided into three principal parts. In Part I, a large body of experimental and theoretical literature on friction is critically reviewed and interpreted as a basis for models of dynamic friction phenomena. In Part II, continuum models of interfaces are developed which simulate key interface properties identified in Part I. Variational principles for a class of dynamic friction problems are also established. In Part III, finite element models and numerical algorithms for analyzing dynamic friction are presented. Also, a dynamic stability analysis is presented in which it is established that stick-slip motion can be associated with dynamic instability of the governing nonlinear system for certain ranges of slip velocity and coefficient of friction. Numerical results suggest that the new models derived here can satisfactorily depict a large and important class of dynamic friction effects.

Generalized finite element methods for three-dimensional structural mechanics problems

Abstract The present paper summarizes the generalized finite element method formulation and demonstrates some of its advantages over traditional finite element methods to solve complex, three-dimensional (3D) structural mechanics problems. The structure of the stiffness matrix in the GFEM is compared to the corresponding FEM matrix. The performance of the GFEM and FEM in the solution of a 3D elasticity problem is also compared. The construction of p-orthotropic approximations on tetrahedral meshes and the use of a-priori knowledge about the solution of elasticity equations in three-dimensions are also presented.

Toward a universal h-p adaptive finite element strategy, part 1. Constrained approximation and data structure

A portable, power operated, hand cultivator comprising a frame having a motor supported thereon which, through a transmission, oscillates two or more generally vertically disposed cultivator tines extending downwardly from the frame. A handle is provided for easy control and manipulation of the device.

Toward a universal h-p adaptive finite element strategy, part 2. A posteriori error estimation

A harrow having two horizontal elongated tined members the ends of which are driven around substantially vertical axes and including a soil contacting elongated element mounted to each of the tined members.

A New Cloud-Based hp-Finite Element Method

A hybrid computational method for solving boundary-value problems is introduced which combines features of the meshless hp-cloud methods with features of conventional finite elements. The method admits straightforward nonuniform hp-type approximations, easy implementation of essential boundary conditions, is robust under severe distortions of the mesh, and can deliver exponential rates of convergence. Results of numerical experiments are presented.

Nonlocal and nonlinear friction laws and variational principles for contact problems in elasticity.

The use of the classical Coulomb law of friction in the formulation of contact problems in elasticity leads to both physical and mathematical difficulties; the former arises from the fact that this law provides a poor model of frictional stresses at points on metallic surfaces in contact, and the latter is due to the fact that the existence of solutions of the governing equations can be proved only for very special situations. In the present paper, nonclassical friction laws are proposed in an attempt to overcome both of these difficulties. We consider a class of contact problems involving the equilibrium of linearly elastic bodies in contact on surfaces on which nonlocal and nonlinear friction laws are assumed to hold. The physics of friction between metallic bodies in contact is discussed and arguments in support of the theory are presented. Variational principles for boundary-value problems in elasticity in which such nonlinear nonlocal laws hold are then developed. A brief discussion of the questions of existence and uniqueness of solutions to the nonlocal and nonlinear problems is given.

On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors

A method to estimate and control pointwise errors in finite element approximations of elliptic problems is presented as an application of the more general theory of goal-oriented error estimation. In the latter, the accuracy of numerical approximations is assessed in terms of measures that are of practical interest to engineers rather than the classical global energy norm. Goal-oriented error estimation requires the solution of a problem dual to the original problem and the computation of several global error estimates. The performance of the method is carefully tested in the particular case of estimation and control of pointwise errors on a one-dimensional problem.

Goal-oriented error estimation and adaptivity for the finite element method

Abstract In this paper, we study a new approach in a posteriori error estimation, in which the numerical error of finite element approximations is estimated in terms of quantities of interest rather than the classical energy norm. These so-called quantities of interest are characterized by linear functionals on the space of functions to where the solution belongs. We present here the theory with respect to a class of elliptic boundary-value problems, and in particular, show how to obtain accurate estimates as well as upper and lower bounds on the error. We also study the new concept of goal-oriented adaptivity, which embodies mesh adaptation procedures designed to control error in specific quantities. Numerical experiments confirm that such procedures greatly accelerate the attainment of local features of the solution to preset accuracies as compared to traditional adaptive schemes based on energy norm error estimates.

Existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws

IT [S well known that no engineering surfaces are perfectly flat. no matter how precise the machining process used to produce an apparently flat finish. Under magnification one observes that all polished surfaces have undulations that form hills and valleys. the dimensions of which are large in comparison with molecular dimensions. Furthermore. the surface layers (contaminants. adsorbed materials. oxides. work-hardened layers) which cover most exposed metallic surfaces and which meet in actual contact processes. do not have the same mechanical properties as the underlying bulk materials. It is. therefore. natural in developing continuum mechanics models for contact problems. to assign to the interface a separate structure characterized by phenomenological laws independent of the constitutive equations that characterize the parent bulk materials. However. the use of a separate characterization of the interface has not been frequent in continuum mechanics formulations of problems involving the dry contact between solid bodies. Usually. unilateral contact conditions are adopted which simply assert that. when two deformable bodies are pressed together, no mutual penetration of the bodies occurs. In other words. the compressed interface is assumed to have no normal compliance. This approach has led to serious mathematical difficulties, particularly in the formulation of dynamic contact problems. To date. no general theory of existence is available for these problems even in the frictionless case. Although. for finite dimensional problems. conditions for existence and uniqueness have been proved for frictionless [1-8] or frictional [9,10] contact situations. no easy extensions to infinite dimensions have been possible. However. several works have been published that provide important results for particular cases or related problems: the unilateral contact of strings with continuous or discrete obstacles [11-20]; a wave problem in a half-space with a unilateral constraint at the boundary [21] and, recently, the unilateral contact of an axially deforming rod with an obstacle at one of its ends (Schatzman and Bercovier [22]): and the important work ofDuvaut and Lions [23]on dynamic or quasistatic evolution problems involving linearly elastic or viscoelastic bodies subjected to Coulombs friction on a part of the boundary where the normal stresses are prescribed. Many of the unresolved mathematical difficulties can be traced to the requirement of an unilateral (noncompliant) contact constraint. In fact. these unilateral dynamic contact problems are a particular case of general classes of evolution problems governed by second-order (in time) partial differential equations and subjected to unilateral constraints on the unknown

A study of static and kinetic friction

It is the objective of this work to present and discuss various numerical studies on low speed frictional sliding phenomena which strongly suggest that classical interpretations for these phenomena should be critically re-examined and the role played by the dynamic properties of the experimental apparati should be carefully taken into account. In most of the discussions we shall restrict ourselves to dry frictional sliding of metallic bodies and particular emphasis will be given to the importance of norma (and rotational) degrees-of-freedom in dynamic friction phenomena