Quasistatic Viscoelastic Contact with Normal Compliance and Friction

We prove the existence of a unique weak solution to the quasistatic problem of frictional contact between a deformable body and a rigid foundation. The material is assumed to have nonlinear viscoelastic behavior. The contact is modeled with normal compliance and the associated version of Coulomb's law of dry friction. We establish the continuous dependence of the solution on the normal compliance function. Moreover, we prove the existence of a unique solution to the problem of sliding contact with wear. This revised version was published online in August 2006 with corrections to the Cover Date.     

Dynamic contact problem with adhesion and damage between thermo–electro–elasto-viscoplastic bodies

We study of a dynamic contact problem between two thermo–electro–elasto-viscoplastic bodies with damage and adhesion. The contact is frictionless and is modeled with normal compliance condition. We derive variational formulation for the model and prove an existence and uniqueness result of the weak solution. The proof is based on arguments of evolutionary variational inequalities, parabolic inequalities, differential equations, and fixed point theorem.     

Analysis of viscoelastic contact with normal compliance,friction and wear diffusionAnalyse d'un problème viscoélastique avec compliance normale,frottement et diffusion d'usure

We consider a quasistatic problem of frictional contact between a viscoelastic body and a moving foundation. The contact is with wear and is modeled by normal compliance and a law of dry friction. The novelty in the model is that it allows for the diffusion of the wear debris over the potential contact surface. Such kind of phenomena arise in orthopaedic biomechanics and influence the properties of joint prosthesis. We derive a weak formulation of the problem and state that, under a smallness assumption on the problem data, there exists a unique weak solution for the model. To cite this article: M. Shillor et al., C. R. Mecanique 331 (2003).     

Numerical approximation of a viscoelastic frictional contact problemApproximation numérique d'un problème viscoélastique de contact avec frottement

We consider a fully discrete scheme for a quasistatic frictional contact problem between a viscoelastic body and an obstacle. The contact is bilateral, the friction is modeled with Tresca's law and the behavior of the material is described with a viscoelastic constitutive law with long memory. We state an existence and uniqueness result for the discrete solution, followed by error estimate results. Then, we present numerical simulations in the study of a two-dimensional test example. To cite this article: Á. Rodríguez-Arós et al., C. R. Mecanique 334 (2006).     

Normal Compliance Contact Models with Finite Interpenetration

We study contact problems with contact models of normal compliance type, where the compliance function tends to infinity for a given finite interpenetration. Such models are physically more realistic than standard normal compliance models, where the interpenetration is not restricted, because the interpenetration is typically justified by deformations of microscopic asperities of the surface; therefore it should not exceed a certain value that corresponds to a complete flattening of the asperities. The model can be interpreted as intermediate between the usual normal compliance models and the unilateral contact condition of Signorini type. For the static problem without friction, we prove the existence and uniqueness of solutions and establish the equivalence to an optimization problem. For the static problem with Coulomb friction, we show the existence of a solution. The analysis is based on an approximation of the problems by standard normal compliance models, the derivation of regularity results for these auxiliary problems in Sobolev spaces of fractional order by a special translation technique, and suitable limit procedures.     

Numerical solutions to regularized long wave equation based on mixed covolume method

The mixed covolume method for the regularized long wave equation is developed and studied. By introducing a transfer operator γh , which maps the trial function space into the test function space, and combining the mixed finite element with the finite volume method, the nonlinear and linear Euler fully discrete mixed covolume schemes are constructed, and the existence and uniqueness of the solutions are proved. The optimal error estimates for these schemes are obtained. Finally, a numerical example is provided to examine the efficiency of the proposed schemes.     

On the dynamic vibrations of an elastic beam in frictional contact with a rigid obstacle

Existence and uniqueness results are established for weak formulations of initial-boundary value problems which model the dynamic behavior of an Euler-Bernoulli beam that may come into frictional contact with a stationary obstacle. The beam is assumed to be situated horizontally and may move both horizontally and vertically, as a result of applied loads. One end of the beam is clamped, while the other end is free. However, the horizontal motion of the free end is restricted by the presence of a stationary obstacle and when this end contacts the obstacle, the vertical motion of the end is assumed to be affected by friction. The contact and friction at this end is modelled in two different ways. The first involves the classic Signorini unilateral or nonpenetration conditions and Coulomb's law of dry friction; the second uses a normal compliance contact condition and a corresponding generalization of Coulomb's law. In both cases existence and uniqueness are established when the beam is subject to Kelvin-Voigt damping. In the absence of damping, existence of a solution is established for a problem in which the normal contact stress is regularized.The work of the last two authors was supported in part by Oakland University Research Fellowships.     

Examples of non-uniqueness and non-existence of solutions to quasistatic contact problems with friction

Summary This work considers a contact problem with friction involving one contact point and two degrees-of-freedom. The contacting structure is linear elastic. Two different models of contact interaction are considered, the classical Signorini unilateral contact law and a normal compliance law. Coulomb's law of friction is used. All possible so-called rate problems are solved, from which one concludes that the quasistatic problem may possess non-uniqueness and non-existence of solutions. In the case of the normal compliance law this can be explained by a softening structural response. For Signorini's law softening explains only some of the possible situations where non-uniqueness can occur.

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Beispiele der Nichteindeutigkeit und Nichtexistenz der Lösung quasistatischer Kontaktprobleme mit Reibung

Übersicht In dieser Arbeit wird ein Kontaktproblem mit Reibung behandelt, das einen Kontaktpunkt und zwei Freiheitsgrade einschließt. Die kontaktgebende Struktur ist linearelastisch. Zwei verschiedene Modelle der Kontaktwirkung sind berücksichtigt: Erstens das klassische einseitige Signorini-Kontaktgesetz und zweitens ein Gesetz für die Nachgiebigkeit in Normalenrichtung. Das Coulombsche Reibungsgesetz wird verwendet. Alle möglichen sogenannten Geschwindigkeitsprobleme werden gelöst, woraus geschlossen wird, daß das quasistatische Problem Nichteindeutigkeit und Nichtexistenz der Lösung besitzen kann. Im Fall des Nachgiebigkeitsgesetzes kann dieses als abfallende Struktursteifigkeit erklärt werden. Im Fall eines Signorini-Gesetzes erklärt dieses nur einige der möglichen Situationen, wo Nichteindeutigkeit auftreten kann.

     

L2 Well-Posedness of Planar Div-Curl Systems

Criteria for the existence and uniqueness of solutions of div-curl boundary value problems on bounded planar regions with nice boundaries are developed. The boundary conditions to be treated include prescribed normal component of the field, tangential component of the field and disjoint combinations of these conditions. Under natural assumptions on the data, when either tangential or normal components are given on the whole boundary, weak (finite-energy) solutions exist if and only if a compatibility condition holds. If the region is simply connected this solution is unique. When the region is multiply connected, there is a finite-dimensional family of solutions. The dimension of the solution space is the Betti number of the region. The problem is well-posed with a unique solution when certain line integrals are further prescribed. L ² estimates of the solutions are given. If mixed tangential, and normal, components of the field are specified on different parts of the boundary, no compatibility condition is required for solvability. In general, though, there is considerable non-uniqueness of solutions. Well-posedness is recovered by specifying certain line integrals. L ² estimates of the solutions are given. The dimensionality of the solution space depends on the topology of the boundary data. These results depend on certain weighted orthogonal decompositions of L ² vector fields on the region which are related to classical Hodge-Weyl decomposition results.     

Relative Entropy and the Stability of Shocks and Contact Discontinuities for Systems of Conservation Laws with non-BV Perturbations

We develop a theory based on relative entropy to show the uniqueness and L ² stability (up to a translation) of extremal entropic Rankine?CHugoniot discontinuities for systems of conservation laws (typically 1-shocks, n-shocks, 1-contact discontinuities and n-contact discontinuities of large amplitude) among bounded entropic weak solutions having an additional trace property. The existence of a convex entropy is needed. No BV estimate is needed on the weak solutions considered. The theory holds without smallness conditions. The assumptions are quite general. For instance, strict hyperbolicity is not needed globally. For fluid mechanics, the theory handles solutions with vacuums.     

A Geometrical Approach to the Study¶of Unbounded Solutions¶of Quasilinear Parabolic Equations

In this article, we are interested in the existence and uniqueness of solutions for quasilinear parabolic equations set in the whole space ? ^N . We consider, in particular, cases when there is no restriction on the growth or the behavior of these solutions at infinity. Our model equation is the mean-curvature equation for graphs for which Ecker and Huisken have shown the existence of smooth solutions for any locally Lipschitz continuous initial data. We use a geometrical approach which consists in seeing the evolution of the graph of a solution as a geometric motion which is then studied by the so-called “level-set approach”. After determining the right class of quasilinear parabolic PDEs which can be taken into account by this approach, we show how the uniqueness for the original PDE is related to “fattening phenomena” in the level-set approach. Existence of solutions is proved using a local L ^∞ bound obtained by using in an essential way the level-set approach. Finally we apply these results to convex initial data and prove existence and comparison results in full generality, i.e., without restriction on their growth at infinity.     

Existence and Stability of Solutions for Steady Flows of Fibre Suspension Flows

We establish existence, uniqueness, convergence and stability of solutions to the equations of steady flows of fibre suspension flows. The existence of a unique steady solution is proven by using an iterative scheme. One of the restrictions imposed on the data confirms a well known fact proven in Galdi and Reddy (J Non-Newtonian Fluid Mech 83:205–230, 1999), Munganga and Reddy (Math Models Methods Appl Sci 12:1177–1203, 2002) and Munganga et al. (J Non-Newtonian fluid Mech 92:135–150, 2000) that the particle number N _p must be less than 35/2. Exact solutions are calculated for Couette and Poiseuille flows. Solutions of Poiseuille flows are shown to be more accurate than those of Couette flow when a time perturbation is considered.     

Painlevé paradox during oblique impact with friction

In analyses using non-smooth dynamics, oblique impact of rough bodies in an unsymmetrical configuration can result in self-locking or “jam” at the sliding contact if the coefficient of friction is sufficiently large; this has been termed, Painlevé’s paradox. In the range of configurations and coefficients of friction where Painlevé’s paradox occurs, analyses based on rigid body dynamics give results indicating that either there are multiple solutions or the solution is nonexistent. This conundrum has been resolved by considering that the contact has small normal and tangential compliance which is representative of deformability in a local region around the contact point. An analysis using a hybrid model which includes local compliance of the contact region has calculated the time-dependent changes in relative motion of colliding bodies for a range of incident angles of obliquity, tan^?1[?V₁(0)/V₃(0)] where V₁(0)and V₃(0) are the incident tangential and normal relative velocities at the contact point, respectively. The paradox is shown to result from a negative relative acceleration of the contact points during an initial period of sliding – a negative acceleration that is inconsistent with the assumption of rigid-body contact.     

The Bending-Gradient Theory for Thick Plates: Existence and Uniqueness Results

This paper is devoted to the mathematical justification of the Bending-Gradient theory which is considered as the extension of the Reissner-Mindlin theory (or the First Order Shear Deformation Theory) to heterogeneous plates. In order to rigorously assess the well-posedness of the Bending-Gradient problems, we first assume that the compliance tensor related to the generalized shear force is positive definite. We define the functional spaces to which the variables of the theory belong, then state and prove the existence and uniqueness theorems of solutions of the Bending-Gradient problems for clamped and free plates, as well as for simply supported plates. The obtained results are afterward extended to the general case, i.e., when the compliance tensor related to generalized shear forces is not definite.     

Analysis of a quasistatic contact problem with adhesion and nonlocal friction for viscoelastic materials

A mathematical model is established to describe a contact problem between a deformable body and a foundation. The contact is bilateral and modelled with a nonlocal friction law, in which adhesion is taken into account. Evolution of the bonding field is described by a first-order differential equation. The materials behavior is modelled with a nonlinear viscoelastic constitutive law. A variational formulation of the mechanical problem is derived, and the existence and uniqueness of the weak solution can be proven if the coefficient of friction is sufficiently small. The proof is based on arguments of time-dependent variational inequalities, differential equations, and the Banach fixed-point theorem.     

Quasistatic Evolution of Sessile Drops and Contact Angle Hysteresis

We consider the classical model of capillarity coupled with a rate-independent dissipation mechanism due to frictional forces acting on the contact line, and prove the existence of solutions with prescribed initial configuration for the corresponding quasistatic evolution.We also discuss in detail some explicit solutions to show that the model does account for contact angle hysteresis, and to compare its predictions with experimental observations.     

Sliding of a spherical indenter on a viscoelastic foundation with the forces of molecular attraction taken into account

The problem of sliding of a spherical indenter on a viscoelastic foundation is solved in a quasistatic formulation taking account the forces of adhesive attraction which are considered different at the entrance to and exit from the contact region due to changes in the surface properties during the interaction. It is found that the contact characteristics and the frictional force due to the imperfect elasticity of the foundation depend on the surface and bulk properties of the materials of the interacting bodies and the interaction conditions (load, velocity, etc.).     

Limit models in the analysis of three different layered elastic strips

The mechanical behavior of three types of laminated strips is investigated. They are made of three layers filled with homogeneous, isotropic and elastic materials; the upper and lower layer are called adherents, the middle layer is called adhesive. The first model studies a strip consisting of three layers made of materials with similar stiffness; the second one concerns with a strip in which the adhesive is soft; in particular, we suppose that the elastic stiffness of the middle layer is two orders of magnitude smaller than that of the upper and lower layers; the third case is a strip in which the core is thinner and stiffer than the two adherents: the elastic modula of the adherents are one order of magnitude bigger that those of the adhesive. After identifying a parameter of smallness ε (which measures the thickness and the stiffness of each layer), the limit of the solution when ε tends to zero has been considered. Afterwards, it has been shown that each solution of the simplified models verifies the so-called limit problems, written using a “weak” and a “strong” formulation. The existence and uniqueness of the solutions of each limit problem have been established. The strong convergence of the exact solutions towards the solution of the limit problem of the first model has been established, too.     

Global Well-Posedness for the Generalized Large-Scale Semigeostrophic Equations

We prove existence and uniqueness of global classical solutions to the generalized large-scale semigeostrophic equations with periodic boundary conditions. This family of Hamiltonian balance models for rapidly rotating shallow water includes the L ₁ model derived by R. Salmon in 1985 and its 2006 generalization by the second author. The results are, under the physical restriction that the initial potential vorticity is positive, as strong as those available for the Euler equations of ideal fluid flow in two dimensions. Moreover, we identify a special case in which the velocity field is two derivatives smoother in Sobolev space as compared to the general case. Our results are based on careful estimates which show that, although the potential vorticity inversion is nonlinear, bounds on the potential vorticity inversion operator remain linear in derivatives of the potential vorticity. This permits the adaptation of an argument based on elliptic L ^p theory, proposed by Yudovich in 1963 for proving existence and uniqueness of weak solutions for the two-dimensional Euler equations, to our particular nonlinear situation.     

Bingham Viscoplastic as a Limit of Non-Newtonian Fluids

A new formulation is proposed for the equations of the Bingham viscoplastic. Global existence of x--periodic solutions is proved. A uniqueness theorem is established in the two-dimensional case. A relation with the G. Duvaut--J. L. Lions variational inequality is discussed, and a result on equivalence is obtained. The question of interaction between fluid-rigid phases is studied when the initial state is rigid. A free-boundary problem that describes two-phase one-dimensional flows is considered.