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

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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A CLASS OF QUASISTATIC CONTACT PROBLEMS FOR VISCOELASTIC MATERIALS WITH NONLOCAL COULOMB FRICTION AND TIME-DELAY

Mathematical Modelling and Analysis ◽

10.3846/13926292.2014.956354 ◽

2014 ◽

Vol 19 (4) ◽

pp. 491-508 ◽

Cited By ~ 2

Author(s):

Si-sheng Yao ◽

Nan-jing Huang

Keyword(s):

Mathematical Model ◽

Variational Inequality ◽

Time Delay ◽

Uniqueness Theorem ◽

Existence And Uniqueness ◽

Variational Formulation ◽

Frictional Contact ◽

Contact Problems ◽

Existence And Uniqueness Theorem ◽

Differential Variational Inequality

In this paper, a mathematical model which describes the explicit time dependent quasistatic frictional contact problems is introduced and studied. The material behavior is described with a nonlinear viscoelastic constitutive law with time-delay and the frictional contact is modeled with nonlocal Coulomb boundary conditions. A variational formulation of the mathematical model is given, which is called a quasistatic integro-differential variational inequality. Using the Banach's fixed point theorem, an existence and uniqueness theorem of the solution for the quasistatic integro-differential variational inequality is proved under some suitable assumptions. As an application, an existence and uniqueness theorem of the solution for the dual variational formulation is also given.

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Using Nonlinear Contact Mechanics in Process of Tool Edge Movement on Deformable Body to Analysis of Cutting and Sliding Burnishing Processes

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.474.339 ◽

2014 ◽

Vol 474 ◽

pp. 339-344 ◽

Cited By ~ 4

Author(s):

Jaroslaw Chodor ◽

Leon Kukielka

Keyword(s):

Numerical Simulation ◽

Mathematical Model ◽

Surface Layer ◽

Numerical Methods ◽

Contact Problem ◽

Deformable Body ◽

Comprehensive Analysis ◽

Tool Edge ◽

Nonlinear Contact ◽

Performance Conditions

Properties of the surface layer after cutting or sliding burnishing depend mainly on type of process and its performance conditions. For its comprehensive analysis is necessary to develop an adequate mathematical model and numerical methods of solving it. A common feature of both processes is moving the tool edge on elastic/visco-plastic workpiece. However, these processes are different i.e. the chip formation or chipless forming, therefore, different properties of surface layer depend mainly on: the geometry of the tool edge and its workpiece relative and depth of process. Therefore, this article is about the application of an incremental modelling and numerical solution of the contact problem between movable rigid and elastic/visco-plastic bodies developed in [ to the numerical simulation of physical process of moving a rigid tool on the workpiece.

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Tykhonov triples and convergence results for history-dependent variational inequalities

ITM Web of Conferences ◽

10.1051/itmconf/20203401006 ◽

2020 ◽

Vol 34 ◽

pp. 01006

Author(s):

Mircea Sofonea

Keyword(s):

Mathematical Model ◽

Variational Inequality ◽

Continuous Dependence ◽

Viscoelastic Body ◽

Time Dependent ◽

Well Posedness ◽

Rigid Foundation ◽

Frictionless Contact ◽

Convergence Results ◽

Engineering Sciences

We deal with the Tykhonov well-posedness of a time-dependent variational inequality deﬁned on the unbounded interval of time ℝ+ = [0, +∞ ), governed by a history-dependent operator. To this end we introduce the concept of Tykhonov triple, provide three relevant examples, then we state and prove the corresponding well-posedness results. This allows us to deduce various corollaries which illustrate the continuous dependence of the solution with respect to the data. Our results provide mathematical tools in the analysis of a large number of history-dependent problems which arise in Mechanics, Physics and Engineering Sciences. To give an example, we consider a mathematical model which describes the equilibrium of a viscoelastic body in frictionless contact with a rigid foundation.

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

Mathematical Control & Related Fields ◽

10.3934/mcrf.2021037 ◽

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>

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A regularization method for a viscoelastic contact problem

Mathematics and Mechanics of Solids ◽

10.1177/1081286516676869 ◽

2016 ◽

Vol 23 (2) ◽

pp. 181-194

Author(s):

Flavius Pătrulescu

Keyword(s):

Mathematical Model ◽

Contact Problem ◽

Variational Inequalities ◽

Long Memory ◽

Regularization Method ◽

Viscoelastic Body ◽

Constitutive Law ◽

Normal Compliance ◽

Viscoelastic Contact

We consider a mathematical model which describes the quasistatic contact between a viscoelastic body and a deformable obstacle, the so-called foundation. The material’s behaviour is modelled with a viscoelastic constitutive law with long memory. The contact is frictionless and is defined using a multivalued normal compliance condition. We present a regularization method in the study of a class of variational inequalities involving history-dependent operators. Finally, we apply the abstract results to analyse the contact problem.

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Mathematical model to predict the characteristics of polarization in dielectric materials: The concept of piezoelectrcity and electrostriction

10.31221/osf.io/39572 ◽

2019 ◽

Author(s):

Chem Int

Keyword(s):

Mathematical Model ◽

Electric Field ◽

Lead Zirconate Titanate ◽

Dielectric Material ◽

Strain Curve ◽

Dielectric Materials ◽

Lead Zirconate ◽

Simulation Software ◽

Constitutive Law ◽

Basic Material

Model was developed for the prediction of polarization characteristics in a dielectric material exhibiting piezoelectricity and electrostriction based on mathematical equations and MATLAB computer simulation software. The model was developed based on equations of polarization and piezoelectric constitutive law and the functional coefficient of Lead Zirconate Titanate (PZT) crystal material used was 2.3×10-6 m (thickness), the model further allows the input of basic material and calculation of parameters of applied voltage levels, applied stress, pressure, dielectric material properties and so on, to generate the polarization curve, strain curve and the expected deformation change in the material length charts. The mathematical model revealed that an application of 5 volts across the terminals of a 2.3×10-6 m thick dielectric material (PZT) predicted a 1.95×10-9 m change in length of the material, which indicates piezoelectric properties. Both polarization and electric field curve as well as strain and voltage curve were also generated and the result revealed a linear proportionality of the compared parameters, indicating a resultant increase in the electric field yields higher polarization of the dielectric materials atmosphere.

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Modal Analysis of the Axial Compressor Blade: Advanced Time-Dependent Electronic Interferometry and Finite Element Method

International Journal of Turbo and Jet Engines ◽

10.1515/tjj-2020-0014 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Mykhaylo Tkach ◽

Serhii Morhun ◽

Yuri Zolotoy ◽

Irina Zhuk

Keyword(s):

Experimental Data ◽

Mathematical Model ◽

Finite Element ◽

High Reliability ◽

Axial Compressor ◽

Time Dependent ◽

Experimental Setup ◽

Vibration Modes ◽

Compressor Blades ◽

Isoparametric Finite Element

AbstractNatural frequencies and vibration modes of axial compressor blades are investigated. A refined mathematical model based on the usage of an eight-nodal curvilinear isoparametric finite element was applied. The verification of the model is carried out by finding the frequencies and vibration modes of a smooth cylindrical shell and comparing them with experimental data. A high-precision experimental setup based on an advanced method of time-dependent electronic interferometry was developed for this aim. Thus, the objective of the study is to verify the adequacy of the refined mathematical model by means of the advanced time-dependent electronic interferometry experimental method. The divergence of the results of frequency measurements between numerical calculations and experimental data does not exceed 5 % that indicates the adequacy and high reliability of the developed mathematical model. The developed mathematical model and experimental setup can be used later in the study of blades with more complex geometric and strength characteristics or in cases when the real boundary conditions or mechanical characteristics of material are uncertain.

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