scholarly journals Tykhonov Well-Posedness and Convergence Results for Contact Problems with Unilateral Constraints

Technologies ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 1
Author(s):  
Mircea Sofonea ◽  
Meir Shillor
Keyword(s):  
Frictional Contact ◽  
Deformable Body ◽  
Contact Problems ◽  
Well Posedness ◽  
Unilateral Constraints ◽  
Unified Approach ◽  
Weak Formulation ◽  
Abstract Result ◽  
Convergence Results ◽  
Static Contact

This work presents a unified approach to the analysis of contact problems with various interface laws that model the processes involved in contact between a deformable body and a rigid or reactive foundation. These laws are then used in the formulation of a general static frictional contact problem with unilateral constraints for elastic materials, which is governed by three parameters. A weak formulation of the problem is derived, which is in the form of an elliptic variational inequality, and the Tykhonov well-posedness of the problem is established, under appropriate assumptions on the data and parameters, with respect to a special Tykhonov triple. The proof is based on arguments on coercivity, compactness, and lower-semicontinuity. This abstract result leads to different convergence results, which establish the continuous dependence of the weak solution on the data and the parameters. Moreover, these results elucidate the links among the weak solutions of the different models. Finally, the corresponding mechanical interpretations of the conditions and the results are provided. The novelty in this work is the application of the Tykhonov well-posedness concept, which allows a unified and elegant framework for this class of static contact problems.

Optimal control for antiplane frictional contact problems involving nonlinearly elastic materials of Hencky type

2017 ◽  
Vol 23 (3) ◽  
pp. 308-328 ◽  
Author(s):  
Andaluzia Matei ◽  
Sorin Micu ◽  
Constantin Niţǎ
Keyword(s):  
Optimal Control ◽  
Frictional Contact ◽  
Variational Equation ◽  
Optimal Solution ◽  
Contact Problems ◽  
Point Of View ◽  
Friction Laws ◽  
Convergence Results ◽  
Positive Exponent ◽  
Nonlinearly Elastic

We consider an antiplane contact problem modeling the friction between a nonlinearly elastic body of Hencky type and a rigid foundation. We discuss the well-posedness of the model by considering two friction laws. Firstly, Tresca’s law is used to describe the friction force and leads to a variational inequality. Alternatively, a regularizing power law with a positive exponent r is considered and gives, from the mathematical point of view, a variational equation. In both contexts, we address a boundary optimal control problem by minimizing, on a nonconvex set, a cost functional with two arguments. We show the existence of at least one optimal pair for each problem. Finally, we deliver some convergence results proving that the optimal solution of the regular problem tends, when r goes to zero, to an optimal solution of the first one.

History-dependent quasi-variational inequalities arising in contact mechanics

2011 ◽  
Vol 22 (5) ◽  
pp. 471-491 ◽  
Author(s):  
MIRCEA SOFONEA ◽  
ANDALUZIA MATEI
Keyword(s):  
Variational Inequalities ◽  
Frictional Contact ◽  
Deformable Body ◽  
Regularity Result ◽  
Contact Problems ◽  
Uniqueness Result ◽  
Constitutive Law ◽  
Signorini Condition ◽  
Normal Damped Response ◽  
Weak Solvability

We consider a class of quasi-variational inequalities arising in a large number of mathematical models, which describe quasi-static processes of contact between a deformable body and an obstacle, the so-called foundation. The novelty lies in the special structure of these inequalities that involve a history-dependent term as well as in the fact that the inequalities are formulated on the unbounded interval of time [0, +∞). We prove an existence and uniqueness result of the solution, then we complete it with a regularity result. The proofs are based on arguments of monotonicity and convexity, combined with a fixed point result obtained in [22]. We also describe a number of quasi-static frictional contact problems in which we model the material's behaviour with an elastic or viscoelastic constitutive law. The contact is modelled with normal compliance, with normal damped response or with the Signorini condition, as well, associated to versions of Coulomb's law of dry friction or to the frictionless condition. We prove that all these models cast in the abstract setting of history-dependent quasi-variational inequalities, with a convenient choice of spaces and operators. Then, we apply the abstract results in order to prove the unique weak solvability of each contact problem.

On the solvability of mixed variational problems with solution-dependent sets of Lagrange multipliers

2013 ◽  
Vol 143 (5) ◽  
pp. 1047-1059 ◽  
Author(s):  
Andaluzia Matei
Keyword(s):  
Lagrange Multipliers ◽  
Frictional Contact ◽  
Variational Equation ◽  
Deformable Body ◽  
Variational Problems ◽  
Abstract Result ◽  
Continuous Maps ◽  
Mixed Variational Problem ◽  
Fixed Point Technique ◽  
Weak Solvability

We study an abstract mixed variational problem, the set of the Lagrange multipliers being dependent on the solution. The problem consists of a system of a variational equation and a variational inequality. We prove the existence of the solution based on a fixed-point technique for weakly sequentially continuous maps. We then apply the abstract result to the weak solvability of a boundary-value problem that models the frictional contact between a cylindrical deformable body and a rigid foundation.

Elastoplastic Frictional Contact Study on Interference Fits of Compressor

2011 ◽  
Vol 211-212 ◽  
pp. 535-539
Author(s):  
Ai Hua Liao
Keyword(s):  
Stress Distribution ◽  
Contact Stress ◽  
Frictional Contact ◽  
Contact Problems ◽  
Boundary Problems ◽  
Interference Fit ◽  
Parametric Variational Principle ◽  
Design And Manufacturing ◽  
Contact Stress Distribution ◽  
Fit Tolerance

The impeller mounted onto the compressor shaft assembly via interference fit is one of the key components of a centrifugal compressor stage. A suitable fit tolerance needs to be considered in the structural design. A locomotive-type turbocharger compressor with 24 blades under combined centrifugal and interference-fit loading was considered in the numerical analysis. The FE parametric quadratic programming (PQP) method which was developed based on the parametric variational principle (PVP) was used for the analysis of stress distribution of 3D elastoplastic frictional contact of impeller-shaft sleeve-shaft. The solution of elastoplastic frictional contact problems belongs to the unspecified boundary problems where the interaction between two kinds of nonlinearities should occur. The effect of fit tolerance, rotational speed and the contact stress distribution on the contact stress was discussed in detail in the numerical computation. The study play a referenced role in deciding the proper fit tolerance and improving design and manufacturing technology of compressor impellers.

scholarly journals CONTACT PROBLEMS FOR NONLINEARLY ELASTIC MATERIALS: WEAK SOLVABILITY INVOLVING DUAL LAGRANGE MULTIPLIERS

2010 ◽  
Vol 52 (2) ◽  
pp. 160-178 ◽  
Author(s):  
A. MATEI ◽  
R. CIURCEA
Keyword(s):  
Lagrange Multipliers ◽  
Variational Equation ◽  
Small Deformation ◽  
Contact Problems ◽  
Point Theory ◽  
The Body ◽  
Elastic Materials ◽  
Weak Formulation ◽  
Nonlinearly Elastic ◽  
Weak Solvability

AbstractA class of problems modelling the contact between nonlinearly elastic materials and rigid foundations is analysed for static processes under the small deformation hypothesis. In the present paper, the contact between the body and the foundation can be frictional bilateral or frictionless unilateral. For every mechanical problem in the class considered, we derive a weak formulation consisting of a nonlinear variational equation and a variational inequality involving dual Lagrange multipliers. The weak solvability of the models is established by using saddle-point theory and a fixed-point technique. This approach is useful for the development of efficient algorithms for approximating weak solutions.

scholarly journals Sensitivity Analysis for Frictional Contact Problems with Large Deformation.

1999 ◽  
Vol 65 (637) ◽  
pp. 1859-1866
Author(s):  
Xian CHEN ◽  
Kazuhiro NAKAMURA ◽  
Masahiko MORI ◽  
Toshiaki HISADA
Keyword(s):  
Sensitivity Analysis ◽  
Large Deformation ◽  
Frictional Contact ◽  
Contact Problems

PREDICATION FOR RELATIVE MOTION OF THE COLONOSCOPE IN COLONOSCOPY

2013 ◽  
Vol 13 (03) ◽  
pp. 1350023 ◽  
Author(s):  
WU BIN CHENG ◽  
MICHAEL A. J. MOSER ◽  
SIVARUBAN KANAGARATNAM ◽  
WEN JUN ZHANG
Keyword(s):  
Frictional Force ◽  
Frictional Contact ◽  
Linear Complementarity ◽  
Intestinal Wall ◽  
Deformable Objects ◽  
Dynamic Friction ◽  
Unilateral Constraints ◽  
Common Procedure ◽  
Mechanical Compliance ◽  
Signorini's Problem

Colonoscopy is common procedure frequently carried out. It is not without its problems, which include looping formation. Looping formation prevents the tip of the colonoscope itself from advancing, thus further probing induces a risk of perforation, significant patient discomfort, and failure of colonoscopy. During colonoscopy, the manipulated colonoscope for intubation in the colon goes through the friction between the colonoscope and the colon. Due to major frictional force, the sigmoidal colon forms looping with the scope during intubation. The interactive frictional force between the colon and the colonoscope is highly complex because of frictional contact between two deformable objects. In this paper, contact force computation was formulated into a linear complementarity problem (LCP) by linearizing Signorini's problem, which was adapted into non-interpenetration with unilateral constraints. Frictional force was computed by the mechanical compliance of finite element method (FEM) models with the consideration of dynamic friction between the colonoscope and the intestinal wall. Furthermore, we presented a mathematical model of the elongation of the colon that predicts the motion of scope relative to the intestinal wall in colonoscopy.

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