An Approximation of Solutions for the Problem with Quasistatic Contact in the Case of Dry Friction

Nicolae Pop,Adrian I Pop,Miorita Ungureanu

doi:10.3390/math9080904

Nicolae Pop, Adrian I Pop + Show 1 more

Open Access

https://doi.org/10.3390/math9080904

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Abstract

In this paper, we discuss the question of finding an optimal control for the solutions of the problem with dry friction quasistatic contact, in the case that the friction law is modeled by a nonlocal version of Coulomb’s law. In order to get the necessary optimality conditions, we use some regularization techniques, and this leads us to a problem of control for an inequality of the variational type. The optimal control problem consists, in our case, of minimizing a sequence of optimal control problems, where the control variable is given by a Neumann-type boundary condition. The state system is represented by a limit of a sequence, whose terms are obtained from the discretization, in time with finite difference and space with the finite element method of a regularized quasistatic contact problem with Coulomb friction. The purpose of this optimal control problem is that the traction force (the control variable) acting on one side of the boundary (the Neumann boundary condition) of the elastic body produces a displacement field (the state system solution) close enough to the imposed displacement field, and the traction force from the boundary remains small enough.

Highlights

The quasistatic model of the contact problems implies ignoring the inertial effects.In addition, we specify that the modeling of the boundary conditions must be given by time-dependent functions
We obtain an incremental issue that is equivalent to a sequence of static contact problems, and the sequence of solutions of these static contact problems converges to the solution of the quasistatic problem. Another important feature of the quasistatic model is that it allows the size of the contact area, generally unknown, to be approximated much better than with other methods, and this is because after each incremental step, it is possible to check the actual size of the contact area
Considering a division of the interval [0, T ] by the points (t0, t1, . . . , t N ), it results in a new formulation of Problem (P1 ), one of an incremental type, in which we use an approximation for the derivative of the function u with respect to time, with the help of the backward finite difference

Summary

Introduction

The quasistatic model of the contact problems implies ignoring the inertial effects. In addition, we specify that the modeling of the boundary conditions must be given by time-dependent functions. We obtain an incremental issue that is equivalent to a sequence of static contact problems, and the sequence of solutions of these static contact problems converges to the solution of the quasistatic problem Another important feature of the quasistatic model is that it allows the size of the contact area, generally unknown, to be approximated much better than with other methods, and this is because after each incremental step, it is possible to check the actual size of the contact area. The authors of the article aim to: provide a correct (well-posed) formulation of quasistatic contact problems with friction in elasticity, present the necessary and sufficient hypotheses for the existence and uniqueness of the solution, perform the discretization in space and time of equilibrium equations, and use regularization methods to avoid non-differentiable terms and convergent and numerically stable minimization algorithms for solving boundary optimal control problems.

Presentation of Equilibrium Equations in Classical and Variational Form

A Discretization of the Problems of Contact with Coulomb-Type Friction

A General Formulation for the Problem of Boundary Optimal Control

The Optimal Control Problem for the Regularized Contact State

Conclusions

Methods