scholarly journals Numerical analysis and simulations of a frictional contact problem with damage and memory

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Hailing Xuan ◽  
Xiaoliang Cheng
Keyword(s):  
Contact Problem ◽  
Frictional Contact ◽  
Numerical Solutions ◽  
Deformable Body ◽  
Mechanical Damage ◽  
Coupled System ◽  
Constitutive Law ◽  
Optimal Order ◽  
Order Error ◽  
Nonlinear Parabolic

<p style='text-indent:20px;'>In this paper, we study a frictional contact model which takes into account the damage and the memory. The deformable body consists of a viscoelastic material and the process is assumed to be quasistatic. The mechanical damage of the material which caused by the tension or the compression is included in the constitutive law and the damage function is modelled by a nonlinear parabolic inclusion. Then the variational formulation of the model is governed by a coupled system consisting of a history-dependent hemivariational inequality and a nonlinear parabolic variational inequality. We introduce and study a fully discrete scheme of the problem and derive error estimates for numerical solutions. Under appropriate solution regularity assumptions, an optimal order error estimate is derived for the linear finite element method. Several numerical experiments for the contact problem are given for providing numerical evidence of the theoretical results.</p>

scholarly journals Variational and numerical analysis of the Signorini′s contact problem in viscoplasticity with damage

2003 ◽  
Vol 2003 (2) ◽  
pp. 87-114 ◽  
Author(s):  
J. R. Fernández ◽  
M. Sofonea
Keyword(s):  
Contact Problem ◽  
Numerical Solutions ◽  
Mechanical Damage ◽  
Parabolic Type ◽  
Approximate Solutions ◽  
Damage Function ◽  
Optimal Order ◽  
Order Error ◽  
Fully Discrete ◽  
Solution Regularity

We consider the quasistatic Signorini′s contact problem with damage for elastic-viscoplastic bodies. The mechanical damage of the material, caused by excessive stress or strain, is described by a damage function whose evolution is modeled by an inclusion of parabolic type. We provide a variational formulation for the mechanical problem and sketch a proof of the existence of a unique weak solution of the model. We then introduce and study a fully discrete scheme for the numerical solutions of the problem. An optimal order error estimate is derived for the approximate solutions under suitable solution regularity. Numerical examples are presented to show the performance of the method.

scholarly journals Numerical Analysis of a Sliding frictional contact problem with Normal Compliance and Unilateral Contact

2020 ◽  
Author(s):  
Yahyeh Souleiman ◽  
Mikael Barboteu
Keyword(s):  
Numerical Analysis ◽  
Contact Problem ◽  
Frictional Contact ◽  
Nonlinear Partial Differential Equations ◽  
Viscoelastic Body ◽  
Sliding Contact ◽  
Memory Term ◽  
Optimal Order ◽  
Normal Compliance ◽  
Order Error

Abstract This paper represents a continuation of [15] and [18]. Here, we consider the numerical analysis of a non trivial frictional contact problen in a form of a system of evolution nonlinear partial differential equations. The model describes the equilibrium of a viscoelastic body in sliding contact with a moving foundation. The contact is modeled with a multivalued normal compliance condition with memory term restricted by a unilateral constraint, and is associated to a sliding version of Coulomb's law of dry friction. After a description of the model and some assumptions, we derive a variational formulation of the problem, which consists of a system coupling a variational inequality for the displacement field and a nonlinear equation for the stress field. Then, we introduce a fully discrete scheme for the numerical approximation of the sliding contact problem. Under certain solution regularity assumptions, we derive an optimal order error estimate and we provide numerical validation of this result by considering some numerical simulations in the study of a two-dimensional problem.

scholarly journals A frictionless contact problem for viscoelastic materials

2002 ◽  
Vol 2 (1) ◽  
pp. 1-21 ◽  
Author(s):  
Mikäel Barboteu ◽  
Weimin Han ◽  
Mircea Sofonea
Keyword(s):  
Numerical Scheme ◽  
Evolution Equations ◽  
Deformable Body ◽  
Maximal Monotone ◽  
The Body ◽  
Constitutive Law ◽  
Optimal Order ◽  
Basic Solution ◽  
Order Error ◽  
Uniqueness Results

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 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 a form with a zero gap function. We present two alternative yet equivalent weak formulations of the problem and establish existence and uniqueness results for both formulations. The proofs are based on a general result on evolution equations with maximal monotone operators. We then study a semi-discrete numerical scheme for the problem, in terms of displacements. The numerical scheme has a unique solution. We show the convergence of the scheme under the basic solution regularity. Under appropriate regularity assumptions on the solution, we also provide optimal order error estimates.

scholarly journals Theory of Second Order Numerical Simulation Method of Enhanced Oil Production

2019 ◽  
Vol 11 (2) ◽  
pp. 73
Author(s):  
Yirang Yuan ◽  
Aijie Cheng ◽  
Danping Yang ◽  
Changfeng Li
Keyword(s):  
A Priori Estimates ◽  
A Priori ◽  
Oil Production ◽  
Coupled System ◽  
Simulation Method ◽  
Second Order ◽  
Optimal Order ◽  
Difference Operators ◽  
Order Error ◽  
L2 Norm

A kind of second-order implicit upwind fractional steps finite difference method is presented in this paper to numerically simulate the coupled system of enhanced (chemical) oil production in porous media. Some techniques, such as the calculus of variations, energy analysis method, commutativity of the products of difference operators, decomposition of high-order difference operators and the theory of a priori estimates are introduced and optimal order error estimates in l2 norm are derived.

Hemivariational inverse problem for contact problem with locking materials

2021 ◽  
Vol 8 (4) ◽  
pp. 665-677
Author(s):  
Z. Faiz ◽  
◽  
O. Baiz ◽  
H. Benaissa ◽  
D. El Moutawakil ◽  
...  
Keyword(s):  
Inverse Problem ◽  
Contact Problem ◽  
Frictional Contact ◽  
Deformable Body ◽  
Lipschitz Mapping ◽  
Hemivariational Inequalities ◽  
Existence Of A Solution ◽  
Locally Lipschitz ◽  
Uniqueness Of The Solution ◽  
Dependence Result

The aim of this work is to study an inverse problem for a frictional contact model for locking material. The deformable body consists of electro-elastic-locking materials. Here, the locking character makes the solution belong to a convex set, the contact is presented in the form of multivalued normal compliance, and frictions are described with a sub-gradient of a locally Lipschitz mapping. We develop the variational formulation of the model by combining two hemivariational inequalities in a linked system. The existence and uniqueness of the solution are demonstrated utilizing recent conclusions from hemivariational inequalities theory and a fixed point argument. Finally, we provided a continuous dependence result and then we established the existence of a solution to an inverse problem for piezoelectric-locking material frictional contact problem.

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.

INTEGRODIFFERENTIAL HEMIVARIATIONAL INEQUALITIES WITH APPLICATIONS TO VISCOELASTIC FRICTIONAL CONTACT

2008 ◽  
Vol 18 (02) ◽  
pp. 271-290 ◽  
Author(s):  
STANISŁAW MIGÓRSKI ◽  
ANNA OCHAL ◽  
MIRCEA SOFONEA
Keyword(s):  
Frictional Contact ◽  
Deformable Body ◽  
Monotone Operators ◽  
Uniqueness Result ◽  
Second Order ◽  
Material Behavior ◽  
Constitutive Law ◽  
Memory Term ◽  
Hemivariational Inequalities ◽  
Banach Fixed Point Theorem

We consider a class of abstract second-order evolutionary inclusions involving a Volterra-type integral term, for which we prove an existence and uniqueness result. The proof is based on arguments of evolutionary inclusions with monotone operators and the Banach fixed point theorem. Next, we apply this result to prove the solvability of a class of second-order integrodifferential hemivariational inequalities and, under an additional assumption, its unique solvability. Then we consider a mathematical model which describes the frictional contact between a deformable body and a foundation. The process is dynamic, the material behavior is described with a viscoelastic constitutive law involving a long memory term and the contact is modelled with subdifferential boundary conditions. We derive the variational formulation of the problem which is of the form of an integrodifferential hemivariational inequality for the displacement field. Then we use our abstract results to prove the existence of a unique weak solution to the frictional contact model.

Hemivariational inequalities modeling electro-elastic unilateral frictional contact problem

2017 ◽  
Vol 23 (3) ◽  
pp. 329-347 ◽  
Author(s):  
Piotr Gamorski ◽  
Stanisław Migórski
Keyword(s):  
Contact Problem ◽  
Frictional Contact ◽  
Coupled System ◽  
Contact Friction ◽  
Hemivariational Inequalities ◽  
Unilateral Constraints ◽  
Frictional Contact Problem ◽  
Uniqueness Of The Solution ◽  
Pseudomonotone Mappings ◽  
Weak Solvability

We study a class of abstract hemivariational inequalities in a reflexive Banach space. For this class, using the theory of multivalued pseudomonotone mappings and a fixed-point argument, we provide a result on the existence and uniqueness of the solution. Next, we investigate a static frictional contact problem with unilateral constraints between a piezoelastic body and a conductive foundation. The contact, friction and electrical conductivity condition on the contact surface are described with the Clarke generalized subgradient multivalued boundary relations. We derive the variational formulation of the contact problem which is a coupled system of two hemivariational inequalities. Finally, for such system we apply our abstract result and prove its unique weak solvability.

scholarly journals A Quasistatic Contact Problem for Viscoelastic Materials with Slip-Dependent Friction and Time Delay

2012 ◽  
Vol 2012 ◽  
pp. 1-22
Author(s):  
Si-sheng Yao ◽  
Nan-jing Huang
Keyword(s):  
Mathematical Model ◽  
Variational Inequality ◽  
Time Delay ◽  
Contact Problem ◽  
Deformable Body ◽  
Constitutive Law ◽  
Time Dependent ◽  
Delay Term ◽  
Inequality System ◽  
Differential Variational Inequality

A mathematical model which describes an explicit time-dependent quasistatic frictional contact problem between a deformable body and a foundation is introduced and studied, in which the contact is bilateral, the friction is modeled with Tresca’s friction law with the friction bound depending on the total slip, and the behavior of the material is described with a viscoelastic constitutive law with time delay. The variational formulation of the mathematical model is given as a quasistatic integro-differential variational inequality system. Based on arguments of the time-dependent variational inequality and Banach's fixed point theorem, an existence and uniqueness of the solution for the quasistatic integro-differential variational inequality system is proved under some suitable conditions. Furthermore, the behavior of the solution with respect to perturbations of time-delay term is considered and a convergence result is also given.

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