Summary. Questions of existence and uniqueness for discrete frictional quasi-static incremental problems, rate problems and wedging problems are discussed. Various methods to compute critical bounds for the coefficient of friction which guarantee existence and uniqueness are described, as well as the sharpness of the bounds and their interdependence.
The Encyclopedia of Thermal Stresses is an important interdisciplinary reference work. In addition to topics on thermal stresses, it contains entries on related topics, such as the theory of elasticity, heat conduction, thermodynamics, appropriate topics on applied mathematics, and topics on numerical methods. The Encyclopedia is aimed at undergraduate and graduate students, researchers and engineers. It brings together well established knowledge and recently received results. All entries were prepared by leading experts from all over the world, and are presented in an easily accessible format. The work is lavishly illustrated, examples and applications are given where appropriate, ideas for further development abound, and the work will challenge many students and researchers to pursue new results of their own. This work can also serve as a one-stop resource for all who need succinct, concise, reliable and up to date information in short encyclopedic entries, while the extensive references will be of interest to those who need further information. For the coming decade, this is likely to remain the most extensive and authoritative work on Thermal Stresses
We prove the existence of a solution for an elastic frictional, quasistatic, contact problem with a Signorini non-penetration condition and a local Coulomb friction law. The problem is formulated as a time-dependent variational problem and is solved by the aid of an established shifting technique used to obtain increased regularity at the contact surface. The analysis is carried out by the aid of auxiliary problems involving regularized friction terms and a so-called normal compliance penalization technique.
In the present paper results on existence and uniqueness of solutions to discrete frictional quasi-static unilateral contact problems are given under a condition that the coefficients of friction are smaller than a certain upper bound. This upper bound is defined in terms of an influence matrix for the contact nodes. The results of existence and uniqueness may be ordered into two classes depending on whether regularity conditions for the applied forces are imposed or not. For general loading which has a time derivative almost everywhere it is shown that a solution exists which satisfies governing equations for almost all times. Uniqueness of the solution has been shown only when the problem is restricted to two degrees of freedom. For a loading which is right piecewise analytic, additional results can be obtained. For instance, if each contact node has only two degrees of freedom a unique solution which satisfies governing equeations for all times exists. For the constructed solutions a priori estimates of the displacement field and its time derivate in terms of the applied forces are also given.
This paper explores the effect of initial conditions on the behavior of coupled frictional elastic systems subject to periodic loading. Previously, it has been conjectured that the long term response will be independent of initial conditions if all nodes slip at least once during each loading cycle. Here, this conjecture is disproved in the context of a simple two-node system. Counter examples are presented of “unstable” steady-state orbits that repel orbits starting from initial conditions that are sufficiently close to the steady state. The conditions guaranteeing stability of such steady states are shown to be more restrictive than those required for the rate problem to be uniquely solvable for arbitrary derivative of the external loading. In cases of instability, the transient orbit is eventually limited either by slip occurring at both nodes simultaneously, or by one node separating. In both cases a stable limit cycle is obtained. Depending on the slopes of the constraint lines, the limit cycle can involve two periods of the loading cycle, in which case it appears to be unique, or it may repeat every loading cycle, in which case distinct limit cycles are reached depending on the sign of the initial deviation from the steady state. In the case of instability an example is given of a loading for which a quasi-static evolution problem with multiple solutions exists, whereas all rate problems are uniquely solvable.
We consider the class of two or three-dimensional discrete contact problems in which a set of contact nodes can make frictional contact with a corresponding set of rigid obstacles. Such a system might result from a finite element discretization of an elastic contact problem after the application of standard static reduction operations. The Coulomb friction law requires that the tractions at any point on the contact boundary must lie within or on the surface of a friction cone, but the exact position of any stuck node (i.e., a node where the tractions are strictly within the cone) depends on the initial conditions and/or the previous history of loading. If the long-term loading is periodic in time, we anticipate that the system will eventually approach a steady periodic cycle. Here we prove that if the elastic system is uncoupled, meaning that changes in slip displacements alone have no effect on the instantaneous normal contact reactions, the time-varying terms in this steady cycle are independent of initial conditions. In particular, we establish the existence of a unique permanent stick zone T comprising the set of all nodes that do not slip after some finite number of cycles. We also prove that the tractions and slip velocities at all nodes not contained in T approach unique periodic functions of time, whereas the (time-invariant) slip displacements in T may depend on initial conditions. Typical examples of uncoupled systems include those where the contact surface is a plane of symmetry, or where the contacting bodies can be approximated locally as half spaces and Dundurs mismatch parameter beta = 0. An important consequence of these results is that systems of this kind will exhibit damping characteristics that are independent of initial conditions. Also, the energy dissipated at each slipping node in the steady state is independent of initial conditions, so wear patterns and the incidence of fretting fatigue failure should also be so independent.
Contains a collection of invited papers dedicated to the memory of two great mathematicians, Gaetano Fichera and Panagis Panagiotopoulos. The book is centered around the seminal research of G Fichera on the Signorini problem, hemivariational inequalities, nonsmooth global optimization, and regularity results for variational inequatities.
It is well known that contact and friction in thermoelasticity result in mathematical problems which may lack solutions or have multiple solutions. Previously, issues related to thermal contact and issues related to frictional heating have been discussed separately. In this work, the two effects are coupled. Theorems of existence and uniqueness of solutions in two or three space dimensions are obtained - essentially extending, to frictional heating, results due to Duvaut, which were built on Barber's heat exchange conditions. Two qualitatively different existence results are given. The first one requires that the contact thermal resistance goes to zero at least as fast as the inverse of the contact pressure. The second existence theorem requires no such growth condition, but requires instead that the frictional heating, i.e. the sliding velocity times the friction coefficient, is small enough. Finally, it is shown that a solution is unique if the inverse of the contact thermal resistance is Lipschitz continuous and the Lipschitz constant, as well as the frictional heating, is small enough.
Given a curvilinear geometric object in R3, made up of properly-joined parametric patches defined in terms of control points, it is of interest to know under what conditions the object will retain its original topological form when the control points are perturbed. For example, the patches might be triangular BΘzier surface patches, and the geometric object may represent the boundary of a solid in a solid-modeling application. In this paper we give sufficient conditions guaranteeing that topological form is preserved by an ambient isotopy. The main conditions to be satisfied are that the original object should be continuously perturbed in a way that introduces no self-intersections of any patch, and such that the patches remain properly joined. The patches need only have C0 continuity along the boundaries joining adjacent patches. The results apply directly to most surface modeling schemes, and they are of interest in several areas of application.
We investigate frictional contact problems for discrete linear elastic structures, in particular the quasistatic incremental problem and the rate problem. It is shown that sharp conditions on the coefficients of friction for unique solvability of these problems are the same. We also give explicit expressions of these critical bounds by using a method of optimization. For the case of two spatial dimensions the conditions are formulated as a huge set of non symmetric eigenvalue problem. A computer program for solving these problems was designed and used to compute the critical bounds for some structures of relative small size, some of which appeared in the literature. The results of a variety of numerical experiments with uniform and non uniform distributions of the frictional properties are presented. (C) 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
For static or incremental contact problems with Coulomb friction there are satisfactory and well known existence results for the coercive case, i.e., when the elastic body is anchored so that rigid body motions are not possible, see [3, 1, 6, 7, 2]. The articles by Jaruusek and Cocu, [7, 2] indeed contain results for the noncoercive case, i.e., when rigid body motions are possible. However, the compatibility conditions which are used to ensure the existence of a solution, are the same that guarantee that the corresponding contact problem without friction has a solution. The condition is essentially that the applied force field should push the elastic body towards the obstacle. One of few previous articles containing friction-dependent compatibility conditions is.
Subdivision surfaces permit a designer to specify the approximate form of a surface defining an object and to refine and smooth the form to obtain a more useful or attractive version of the surface.
A considerable amount of mathematical theory is required to understand the characteristics of the resulting surfaces, and this book
• provides a careful and rigorous presentation of the mathematics underlying subdivision surfaces as used in computer graphics and animation, explaining the concepts necessary to easily read the subdivision literature;
• organizes subdivision methods into a unique and unambiguous hierarchy to facilitate insight and understanding;
• gives a broad discussion of the various methods and is not restricted to questions related to regularity of subdivision surfaces at so-called extraordinary points.
Introduction to the Mathematics of Subdivision Surfaces is excellent preparation for reading more advanced texts that delve more deeply into special questions of regularity. The authors provide exercises and projects at the end of each chapter. Course material, including solutions to the exercises, is available on an associated Web page.
Volino and Thalmann have published a conjecture proposing sufficient conditions for non-selfintersection of surfaces. Such conditions may be used in solid modeling, computer graphics, and other application areas, as a basis for collision-detection algorithms. In this paper we clarify certain of the hypotheses of the proposed theorem, and give a proof. A brief summary of possible pitfalls related to using the conditions, when the hypotheses of the formal theorem given here are not satisfied, is also given. We also give examples, and show that the theorem can be extended to domains that are not simply connected. © 2006 Elsevier B.V. All rights reserved.
The problem of maintaining consistent representations of solids in computer-aided design and of giving rigorous proofs of error bounds for operations such as regularized Boolean intersection has been widely studied for at least two decades. One of the major difficulties is that the representations used in practice not only are in error but are fundamentally inconsistent. Such inconsistency is one of the main bottlenecks in downstream applications. This paper provides a framework for error analysis in the context of solid modeling, in the case where the data is represented using the standard representational method, and where the data may be uncertain. Included are discussions of ill-condition, error measurement, stability of algorithms, inconsistency of defining data, and the question of when we should invoke methods outside the scope of numerical analysis. A solution to the inconsistency problem is proposed and supported by theorems: it is based on the use of Whitney extension to define sets, called Quasi-NURBS sets, which are viewed as realizations of the inconsistent data provided to the numerical method. A detailed example illustrating the problem of regularized Boolean intersection is also given.
We consider first quasistatic evolution problems with Coulomb friction in elasticity and then so called wedging problems and their relation to the evolution problems.
Reflection, transmission and dissipation of the energy of an incident extensional wave at a linearly viscoelastic junction between two uniform and collinear linearly elastic bars are considered. The junction consists of a finite number of uniform segments of the same material and length. The optimum shape of a junction with given material, length and number of segments which maximizes the energy transmission for given input and output bars and a given incident wave of finite duration is determined numerically with the use of a quasi-Newton method. Results are presented for rectangular incident waves of different durations and 40-segment junctions of standard linear solid material. In the special case of linearly elastic material, the optimum junctions have piece-wise constant characteristic impedances with a certain number of plateaux of equal lengths. These plateaux are independent of the number of segments provided that this number is an integral multiple of the number of plateaux. The optimum viscoelastic junctions have the appearance of deformed and displaced versions of their elastic counterparts. Thus, the plateaux of the elastic junctions are increasingly deformed and displaced with increased damping and, less markedly, with decreased response time of the material. The transitions between these plateaux of a junction appear to be discontinuous, similarly as in the case of elastic material. The apparent discontinuities become less notable with increased damping of the material.
We present a sufficient condition for the existence of solutions to noncoercive incremental friction problems for discrete systems in contact with obstacles. By discrete system we mean that the displacement of the object can be discribed by a finite number of displacement variables, and by noncoercive we mean that the stiffness matrix of the object is semidefinite. We have a noncoercive friction problem when the object is not fixed to a support. This means that the friction forces need to balance the applied forces if the object is to remain stationary. This is manifested in our condition for existence of solutions. This condition is a compatibility condition on the applied force field, and if it is violated there exists a nontrivial solution to a corresponding dynamical problem. © 2006 Wiley-VCH Verlag GmbH & Co. KGaA.