Tresca Friction Law Research Articles

Asymptotic behavior of a viscoelastic problem with long-term memory and Tresca friction law

Asymptotic behavior of a viscoelastic problem with long-term memory and Tresca friction law

A coupled model of the fluid flow with nonlinear slip Tresca boundary condition

In this article, we consider the finite element discretization of the time-dependent Stokes equations written under the Boussinesq approximation, coupled with the convection-diffusion equation and provided with Tresca's Friction Law. Existence and uniqueness of a solution are established. To solve Stokes variational inequality, we suggest an alternating direction method of multiplier. We derive some a priori estimates and validate the obtained theoretical results by numerical simulations.

Homogenization with the Quasistatic Tresca Friction Law: Qualitative and Quantitative Results

Homogenization with the Quasistatic Tresca Friction Law: Qualitative and Quantitative Results

Non-isothermal non-Newtonian flow problem with heat convection and Tresca's friction law

We consider an incompressible non-isothermal fluid flow with non-linear slip boundary conditions governed by Tresca's friction law. We assume that the stress tensor is given as where θ is the temperature, π is the pressure, u is the velocity and is the strain rate tensor of the fluid while p is a real parameter. The problem is thus given by the p-Laplacian Stokes system with subdifferential type boundary conditions coupled to a elliptic equation describing the heat conduction in the fluid. We establish first an existence result for a family of approximate coupled problems where the coupling term in the heat equation is replaced by a bounded one depending on a parameter , by using a fixed point technique. Then we pass to the limit as δ tends to zero and we prove the existence of a solution to our original coupled problem in Banach spaces depending on p for any .

Asymptotic analysis of quasistatic electro‐viscoelastic problem with Tresca's friction Law

Asymptotic analysis of quasistatic electro‐viscoelastic problem with Tresca's friction Law

Numerical analysis of an evolutionary variational–hemivariational inequality with application to a dynamic contact problem

In this paper, we consider the numerical solution of an evolutionary variational–hemivariational inequality arising in a dynamic contact problem. The material is assumed to be viscoelastic with short memory. The contact is featured by a normal damped response in the normal direction and by the Tresca friction law in the tangential direction. The linear finite elements are used to discretize the spatial variable. Optimal order error estimates are derived for the discrete velocity and discrete displacement under suitable solution regularity assumptions.

Study of generalized Stokes operator in a thin domain with friction law (casep< 2)

In this work, we consider a mathematical model of an incompressible fluid governed by the generalized Stokes system in a stationary regime in a 3‐dimensional thin domain Ωϵwith Tresca friction law. The problem statement and variational formulation of the problem are formulated. Then the estimates on velocity and pressure are proved in a fixed domain. Finally, our main goal concerns the specific Reynolds equation and the uniqueness of the solutionare obtained.

Alternating direction method of multiplier for a unilateral contact problem in electro-elastostatics

We study an alternating direction method of multiplier (ADMM) applied to a unilateral frictional contact problem between an electro-elastic material and an electrically non conductive foundation. The frictional contact is modeled by the Tresca friction law. The resulting coupled problem is non symmetric and non coercive. By eliminating the electric potential, we obtain a symmetric and coercive problem which can be reformulated as a convex minimization problem. We then apply an alternating direction method of multiplier for the numerical approximation. To avoid explicit matrices inverse (due to the elimination of the electric potential) we use a preconditioned conjugate gradient algorithm as an inner solver. Numerical experiments are proposed to illustrate the efficiency of the proposed approach.

The R-linear convergence rate of an algorithm arising from the semi-smooth Newton method applied to 2D contact problems with friction

The goal is to analyze the semi-smooth Newton method applied to the solution of contact problems with friction in two space dimensions. The primal-dual algorithm for problems with the Tresca friction law is reformulated by eliminating primal variables. The resulting dual algorithm uses the conjugate gradient method for inexact solving of inner linear systems. The globally convergent algorithm based on computing a monotonously decreasing sequence is proposed and its R-linear convergence rate is proved. Numerical experiments illustrate the performance of different implementations including the Coulomb friction law.

Evolutional contact with Coulomb's friction on a periodic microstructure

AbstractWe consider the elasticity problem in a heterogeneous domain with an ε‐periodic micro‐structure, ε ≪ 1, including a multiple micro‐contact in a simply connected matrix domain with inclusions completely surrounded by cracks, which do not connect the boundary, or a textile‐like material. The contact is described by the Signorini and Coulomb‐friction contact conditions. In the case of the Coulomb friction, the dissipative functional is state dependent, like in [2]. A time discretization scheme from [2] reduces the contact problem to the Tresca one (with prescribed frictional traction or state independent dissipation) on each time‐increment. We further look for the spatial homogenization. The limiting energy and the dissipation term in the stability condition were obtained for the contact with Tresca's friction law in [4] for closed cracks and can be extended to textile‐like materials. Using these results and the concept of energetic solutions for evolutional quasi‐variational problems from [2], for a uniform time‐step partition, the existence can be proved for the solution of the continuous problem and a subsequence of incremental solutions weakly converging to the continuous one uniformly in time. Furthermore, the irreversible frictional displacements at micro‐level lead to a kind of an evolutional plastic behavior of the homogenized medium. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Development of a Friction Law Respecting Plastic Surface Smoothing

Friction has an essential influence on metal forming processes and affects the mould filling strongly. Numerical simulation is widely used because they allow for a efficient product design without the time and cost intensive production of prototype moulds. The quality of the simulation results and thus their reliability is determined by the accuracy of the modelling. For this purpose the applied friction law is of great importance. Characteristic of sheet-bulk metal forming is the coexistence of moderate contact pressures like in sheet metal forming and high contact pressures like in bulk metal forming. The Coulomb friction law is suitable for the sheet metal forming process but it predicts too high friction forces for high contact pressures. On the other hand the Tresca friction law is suitable for bulk metal forming but overestimates the friction for low contact pressures. A smooth transition between the Coulomb and Tresca friction law is described by the Shaw friction law and the Wanheim-Bay friction law. An unresolved problem remains the influence of plastic surface smoothing of structured workpiece surfaces. The tribological properties of the surface are altered by the plastic deformation of the surface roughness. As a consequence the real area of contact and thus the friction are larger in unloading and reloading than in the first loading at the same surface pressure. This plays a role in forming processes with multiple stages, where the surface is smoothed by prior forming operations like for example the forming of tailored blanks. Therefore efforts have been made in the numerical modelling of elasto-plastic surface deformation with a halfspace model. This model allows for the efficient modelling of large rough surfaces because it uses only a surface mesh and not an numerically expensive volume mesh like a Finite-Element model. This halfspace model is calibrated and verified with experimental investigations. A friction law taking into account the plastic surface deformation has been developed based on the halfspace simulations. It distinguishes between first loading, where the current surface pressure is higher than all surface pressures which occurred previously, and unloading or reloading, where the friction is higher because the surface is smoothed plastically in a previous load step, where the surface pressure was higher than currently.

A new material model for 2D numerical simulation of serrated chip formation when machining titanium alloy Ti–6Al–4V

A new material constitutive law is implemented in a 2D finite element model to analyse the chip formation and shear localisation when machining titanium alloys. The numerical simulations use a commercial finite element software (FORGE 2005 ®) able to solve complex thermo-mechanical problems. One of the main machining characteristics of titanium alloys is to produce segmented chips for a wide range of cutting speeds and feeds. The present study assumes that the chip segmentation is only induced by adiabatic shear banding, without material failure in the primary shear zone. The new developed model takes into account the influence of strain, strain rate and temperature on the flow stress and also introduces a strain softening effect. The tool chip friction is managed by a combined Coulomb–Tresca friction law. The influence of two different strain softening levels and machining parameters on the cutting forces and chip morphology has been studied. Chip morphology, cutting and feed forces predicted by numerical simulations are compared with experimental results.

Existence for viscoplastic contact with Coulomb friction problems

We present existence results in the study of nonlinear problem of frictional contact between an elastic‐viscoplastic body and a rigid obstacle. We model the frictional contact both by a Tresca′s friction law and a regularized Coulomb′s law. We assume, in a first part, that the contact is bilateral and that no separation takes place. In a second part, we consider the Signorini unilateral contact conditions. Proofs are based on a time‐discretization method, Banach and Schauder fixed point theorems.

Optimal control of an elastic contact problem involving Tresca friction law

In this paper we establish existence results and necessary optimality conditions for a class of optimal control problems where the state is described by elliptic variational inequalities modelling the frictional contact between an elastic body and a rigid foundation. We use the Tresca’s law of dry friction, in which the friction bound is prescribed for the contact condition. We assume that slowly varying time-dependent volume forces and surface tractions act on the body so that the acceleration in the system will be neglected. Existence and uniqueness results for initial and boundary value problems involving Tresca friction law were obtained in [2,6,10] in the case of an elastic body, in [4] in the case of elasto-viscoplastic models and in [3] in the case of perfectly plastic solids. The aim of this paper is to study optimal control problems for mechanical models describing the quasistatic process of bilateral frictional contact between an elastic body and a rigid foundation. The state of this system is governed by a variational inequality

Dual formulation of a quasistatic viscoelastic contact problem with tresca's friction law

We consider quasistatic evolution of a viscoelastic boby which is in bilateral frictional contact with a rigid foundation. We derive two variational formulations for the problem:the primal formulation in terms of the displacements and the dual formulation in terms of the stress field. We prove the existence of a unique solution to each one and establish the equivalence between the two variational formulations. We also prove the continuous dependence of the solution on the friction yield limit.

Evolutionary variational inequalities arising in quasistatic frictional contact problems for elastic materials

We consider a class of evolutionary variational inequalities arising in quasistatic frictional contact problems for linear elastic materials. We indicate sufficient conditions in order to have the existence, the uniqueness and the Lipschitz continuous dependence of the solution with respect to the data, respectively. The existence of the solution is obtained using a time‐discretization method, compactness and lower semicontinuity arguments. In the study of the discrete problems we use a recent result obtained by the authors (2000). Further, we apply the abstract results in the study of a number of mechanical problems modeling the frictional contact between a deformable body and a foundation. The material is assumed to have linear elastic behavior and the processes are quasistatic. The first problem concerns a model with normal compliance and a version of Coulomb′s law of dry friction, for which we prove the existence of a weak solution. We then consider a problem of bilateral contact with Tresca′s friction law and a problem involving a simplified version of Coulomb′s friction law. For these two problems we prove the existence, the uniqueness and the Lipschitz continuous dependence of the weak solution with respect to the data.

Analysis of a quasistatic viscoplastic problem involving tresca friction law

The quasistatic evolution of an elastic-viscoplastic body in bilateral contact with a rigid foundation is considered. The contact involves viscous friction of Tresca type. Two variational formulations of the problem are proposed, followed by existence and uniqueness results. Some properties involving the equivalence between the previous variational formulations, the continuous dependence of the solution with respect to the data as well as a convergence result with respect to the friction yield limit are also obtained.