Signorini Condition Research Articles

Quadratic finite element approximation of the Signorini problem

Applying high order finite elements to unilateral contact variational inequalities may provide more accurate computed solutions, compared with linear finite elements. Up to now, there was no significant progress in the mathematical study of their performances. The main question is involved With the modeling of the nonpenetration Signorini condition on the discrete solution along the contact region. In this work we describe two nonconforming quadratic finite element approximations of the Poisson-Signorini problem. responding to the crucial practical concern of easy implementation, and we present the numerical analysis of their efficiency. By means of Falk's Lemma we prove optimal and quasi-optimal convergence rates according to the regularity of the exact solution.

On the signorini frictionless contact problem for linear viscoelastic materials

We consider a mathematical model which describes the contact between a deformable body and an obstacle, the so-called foundation. The body is assumed to have a linear viscoelastic behavior that we model with the Kelvin-Voigt constitutive law. The contact is frictionless and is modeled with the well-known Signorini condition in the form with a zero gap function. We derive a weak formulation of the model and prove an existence and uniqueness result of the solution. The proof is based on a regularization method involving normal compliance frictionless contact conditions followed by compactness and lower semicontinuity arguments. We also prove that the solution of the problem converges to the solution of the corresponding elastic problem, as the viscosity tensor converges to zero.

Elastic beam in adhesive contact

Abstract Dynamic and quasistatic processes of contact with adhesion between an elastic or viscoelastic beam and a foundation are considered. The contact is modeled with the Signorini condition when the foundation is rigid, and with normal compliance when it is deformable. The adhesion is modeled by introducing the bonding function β, the evolution of which is described by an ordinary differential equation. The existence and uniqueness of the weak solution for each of the problems is established using the theory of variational inequalities, fixed point arguments and the existence and uniqueness result in Commun. Contemp. Math. 1(1) (1999) 87–123. The numerical approximations of the quasistatic problem with normal compliance are considered, based on semi-discrete and fully discrete schemes. The convergence of the solutions of the discretized schemes is proved and error estimates for these approximate solutions are derived.

Equilibrium problem for thermoelectroconductive body with the Signorini condition on the boundary

We investigate a boundary value problem for a thermoelectroconductive body with the Signorini condition on the boundary, related to resistance welding. The mathematical model consists of an energy-balance equation coupled with an elliptic equation for the electric potential and a quasistatic momentum balance with a viscoelastic material law. We prove the existence of a weak solution to the model by using the Schauder fixed point theorem and classical results on pseudomonotone operators. Copyright © 2001 John Wiley & Sons, Ltd.

Homogenization of a mixed problem with Signorini condition for an elliptic operator in a perforated domain

For the case of a 2-connected -periodic perforated space with a bounded domain selected in it the homogenization property as is proved for the boundary-value problem for a second-order elliptic operator in the domain with one-sided condition of Signorini type on the boundaries of “cavities” and with Dirichlet condition on the outer boundary.

Equilibrium problem for thermoelectroconductive body with the Signorini condition on the boundary

AbstractWe investigate a boundary value problem for a thermoelectroconductive body with the Signorini condition on the boundary, related to resistance welding. The mathematical model consists of an energy‐balance equation coupled with an elliptic equation for the electric potential and a quasistatic momentum balance with a viscoelastic material law. We prove the existence of a weak solution to the model by using the Schauder fixed point theorem and classical results on pseudomonotone operators. Copyright © 2001 John Wiley & Sons, Ltd.

Rapid sliding contact on a highly elastic pre-stressed material

A rigid smooth indentor slides at a constant sub-sonic speed on a half-space of an isotropic compressible neo-Hookean material that is initially subjected to a pre-stress aligned with the surface. A dynamic steady-state situation of plane strain is considered and, following Beatty and Usmani and Green and Zerna, is viewed as contact-triggered infinitesimal deformations superposed upon finite deformations due to pre-stress. The neo-Hookean material treated behaves for small strains as a linear elastic solid with Poisson's ratio 1/4. Exact solutions for both deformations are presented and, for a range of acceptable pre-stress values, these illustrate the anisotropy induced by pre-stress and a critical sliding speed corresponding to the Rayleigh speed. Imposition of the unilateral Signorini conditions of contact show that the Rayleigh speed is the upper bound for sub-sonic sliding. For pre-stress levels that fall outside this range, however, either a negative Poisson effect occurs, or a Rayleigh wave does not exist and the Signorini conditions cannot be satisfied for any sub-sonic speed.

Derivatives of the energy functional for 2D-problems with a crack under Signorini and friction conditions

We consider the two-dimensional elasticity problem for an elastic body with a crack under unilateral constraints imposed at the crack. We assume that both the Signorini condition for non-penetration of the crack faces and the condition of given friction between them are fulfilled. The problem is non-linear and can be described by a variational inequality. Varying the shape of the crack by a local coordinate transformation of the domain, the first derivative of the energy functional to the problem with respect to the crack length is obtained, which gives the criterion for the crack growing. The regularity of the solution is discussed and the singular solution is performed. Copyright © 2000 John Wiley & Sons, Ltd.

Analysis of a Signorini problem with friction

This paper deals with the study of a nonlinear problem of frictional contact between an elastic body and a rigid foundation. The elastic constitutive law is assumed to be nonlinear and the contact is modelled with Signorini's conditions and a version of Coulomb's law of dry friction. We present two weak formulations of the problem and establish existence and uniqueness results, using arguments of elliptic variational inequalities and a fixed-pint property. Moreover, we prove some equivalence results and study the behaviour of the solution when the coefficient of friction tends to zero.

The solvability of a coupled thermoviscoelastic contact problem with small Coulomb friction and linearized growth of frictional heat

A coupled thermoviscoelastic frictional contact problem is investigated. The contact is modelled by the Signorini condition for the displacement velocities and the friction by the Coulomb law. The heat generated by friction is described by a non-linear boundary condition with at most linear growth. The weak formulation of the problem consists of a variational inequality for the elasticity part and a variational equation for the heat conduction part. In order to prove the existence of a solution to this problem we first use an approximation of the Signorini condition by the penalty method. The existence of a solution for the approximate problem is shown using the fixed-point theorem of Schauder. This theorem is applied to the composition of the solution operator for the contact problem with given temperature field and the solution operator for the heat equation problem with known displacement field. To obtain this proof, the unique solvability of both problems is necessary. Due to this reason it is necessary to introduce the penalty method. While the penalized contact problem has a unique solution, this is not clear for the original contact problem. The solvability of the original frictional contact problem is verified by an investigation of the limit for vanishing penalty parameter. Copyright © 1999 John Wiley & Sons, Ltd.

Existence of a solution to a coupled elliptic system with a Signorini condition

The existence of the solution to an elliptic system arising in electrochemical modelling is proven here. The elliptic system of interest here is composed of two diffusion equations; one of them is posed with a Dirichlet condition which couples it to the other equation on an interface, and a Signorini condition on one of the boundaries. The other one is posed with Neumann conditions, which are also coupled at an interface. The existence of the solution is proven by using Schauder's fixed point theorem, which requires some previous local regularity properties of the solution to the ``Signorini problem".

Contact between deformable solids: The fully coupled approach

This paper presents a finite element technique to handle frictional contact between bodies submitted to large deformations. Contact finite elements, based on a penalty method, are derived from the virtual work principle. The originality lies in the fact that the contact conditions (Signorini's conditions and Coulomb friction law) are expressed at some integration points and not at the nodal points. The development of contact introduces a coupling between the Degrees Of Freedom (DOF) of the bodies. In certain cases, this coupling can be neglected in the numerical strategy.

Homogenization of an elastic material with inclusions in frictionless contact

The homogenization method is applied to the problem of elastic deformations of an array of holes, or pores, filled with softer inclusions and imbedded in a stiffer matrix. The contact of the inclusions with the matrix is assumed to be frictionless and is modeled by Signorini's condition. A priori estimates are established for solutions to the problem with χ that permit passing to the homogenization limit ϵ → 0. The limit problem is identified, and the two-scale convergence of the solutions to the unique limit solution is proven.

Analysis of a frictionless contact problem for elastic bodies

This paper deals with a nonlinear problem modelling the contact between an elastic body and a rigid foundation. The elastic constitutive law is assumed to be nonlinear and the contact is modelled by the well-known Signorini conditions. Two weak formulatio

Hertz problem for a rigid punch moving across the surface of a semi-infinite elastic solid

The elastodynamic problem of a rigid punch moving at a constant sub-Rayleigh speed across the surface of an elastic half-space is investigated in the present paper. The unknown contact region is determined as part of solution from the unilateral or Signorini conditions. Numerical results are plotted showing how the eccentricity of the contact ellipse changes with the punch speed. Some asymptotic properties of the solution for the case where the punch speed is comparable with the Rayleigh wave speed are explored in details.

Simulation numérique des problèmes de contact avec frottement

The purpose of this paper is to discuss three questions related to numerical simulation of dynamical or quasi-static problems in presence of unilateral contact and dry friction.- Choosing contact and friction laws: among several possible choices, Signorini's conditions and Coulomb's law are presented and commented.- The dynamical equations together with contact and friction relations are written down. For numerical purpose these equations have to be discretized. An explicit and an implicit typical numerical scheme, together with an explicit and implicit scheme for contact and friction relations are proposed, allowing to exhibit four classes of methods using local or general variables.- These methods of resolution will be illustrated in the particular case of a one dimentional dynamical frictional problem.

Duality applied to contact problems with friction

The duality theory of Mosco, Capuzzo-Dolcetta, and Matzeu for variational and quasi-variational inequalities is extended. Then it is applied to two problems of contact with friction of an elastic body with a rigid foundation. The more realistic normal compliance condition is used in place of Signorini's condition on the contact surface.

Numerical Methods For Three DimensionalDynamical Frictional Contact Problems

The dynamical frictional contact problem between deformable or rigid bodies is considered. A discrete model is obtained using such methods as finite element methods. Unilateral contact and dry friction between a node candidate to contact and the antagonist body are expressed through Signorini's conditions Lnd Coulomb's law. The system of equations governing the problem is solved using a special relaxation method. It relies on the property that the dynamical contact friction problem for a single node has a straightforward solution. This method is fully implicit.