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Contact problems for bodies with complex coatings

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Kirill Kazakov
Kirill Kazakov
Kirill Kazakov
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Svetlana Kurdina
Svetlana Kurdina
Svetlana Kurdina
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The paper deals with various statements and mathematical models of contact and contact‐wear problems for bodies with coatings. It is shown that the mathematical models for a number of such problems can be represented as a mixed integral equation or a system of mixed integral equations with additional conditions. It is also shown that these equations contain rapidly varying or even discontinuous functions in the case of interaction between bodies of complex shape and with some surface properties. Therefore, it is necessary to use a special approach for constructing efficient analytic solutions. Its implementation is demonstrated by an example.
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Received: 31 May 2019
DOI: 10.1002/mma.6107
Contact problems for bodies with complex coatings
Kirill Kazakov1Svetlana Kurdina2
1Laboratory of Modeling in Solids
Mechanics, Ishlinsky Institute for
Problems in Mechanics of the Russian
Academy of Sciences, Moscow, Russia
2Department of Applied Mathematics,
Bauman Moscow State Technical
University, Moscow, Russia
Correspondence
Kirill E. Kazakov, Ishlinsky Institute for
Problems in Mechanics of the Russian
Academy of Sciences, prosp. Vernadskogo
101-1, Moscow, 119526 Russia.
Email: kazakov-ke@yandex.ru
Communicated by: D. Zeidan
The paper deals with various statements and mathematical models of contact
and contact-wear problems for bodies with coatings. It is shown that the
mathematical models for a number of such problems can be represented
as a mixed integral equation or a system of mixed integral equations with
additional conditions. It is also shown that these equations contain rapidly
varying or even discontinuous functions in the case of interaction between
bodies of complex shape and with some surface properties. Therefore, it is
necessary to use a special approach for constructing efficient analytic solutions.
Its implementation is demonstrated by an example.
KEYWORDS
contact, linear integral equations, nonuniform coatings, projection method
MSC CLASSIFICATION
45A05 (Linear integral equations); 74M15 (Contact)
1INTRODUCTION AND HISTORICAL REVIEW
Apparently, the studies in contact interaction mechanics began in 1882 with the work of H. Hertz,1where he described
the solution of a contact problem for two elastic bodies with curved surfaces. The results of these studies, as well as the
results of the later work in2are considered to be classical and have not still lost their value despite several serious restrictive
assumptions in the original formulation. The contact mechanics is also related to fundamental problems such as the
problem of action of a concentrated force on a half-space, which was solved by J. Boussinesque in3and the equilibrium
problem for an elastic half-space under the action of a distributed load, solved by A. Flaman in 1892.4It should be noted
that, apparently, the first flat contact problem was posed and solved by S.A. Chaplygin in 1900. He considered the general
problem of pressure of a cylinder on an elastic soil. However, the work was not published and was found in archival
documents5; therefore, the contact problem for an elastic half-space and a rigid punch with flat base is usually called the
M.A. Sadowsky problem.6Until the 1930s, the comprehensive experimental theoretical studies confirmed Hertz's theory
and made contributions to the development of its applications in engineering. No fundamental research in the field of
contact mechanics was practically carried out since necessary mathematical tools were absent.
In the 1930s, Academician N.I. Muskhelishvili and his followers began to develop efficient methods in the theory of
function of a complex variable.7,8 They obtained fundamental results in the field of integral equations, methods of integral
transforms, and potential theory. The results of these investigations allowed one to solve new problems in the elasticity
theory and, in particular, to obtain solutions of contact problems. The solution of the basic mixed problem for a half-plane
was obtained in the most general form. A. Liapounoff9also made a significant contribution to the development of the
theory of contact interactions. These results were used, for example, by A.Ia. Shtaerman.10 Using the new mathematical
apparatus, V.A. Florin proposed an approximate solution of the contact problem for a punch rigidly connected with the
base.11 The results in the field of integral transforms obtained by I.N. Sneddon12 gave another important mathematical
tool. They allowed one to solve complex boundary value problems of elasticity.
New mathematical methods allowed one to carry out investigations in continuum mechanics, elasticity theory, and, in
particular, contact interaction mechanics. Therefore, in the middle of the 20th century, there appeared a large number
Math Meth Appl Sci. 2020;43:7692–7705.wileyonlinelibrary.com/journal/mma© 2019 John Wiley & Sons, Ltd.
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Chapter summaries for "Computational Contact Mechanics" by P. Wriggers 2nd edition, Springer Verlag, 2006 Chapter 2: Introduction to Contact Mechanics The basic methodology and di�culties related to contact mechanics are discussed in this introductory chapter by means of simple contact problems. This includes besides the general formulation also algorithmic implications. All examples are one-dimensional and stem from di�erent areas like static, thermal or dynamic processes. Key words: introduction, one-dimensional problems, contact, impact. Chapter 3: Continuum Solid Mechanics and Weak Forms This chapter summarizes the main equations which govern the nonlinear deformation of solids. This includes the description of basic kinematical relations for large deformations, the statement of equations of balance and several constitutive equations. Furthermore weak forms and linearizations are presented which are needed within the algorithmic treatment of the nonlinear continuum formulation. 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Chapter 5: Constitutive Equations for Contact Interfaces Two main lines can be followed to impose contact conditions in normal direction: (1) the formulation of the non-penetration condition as a purely geometrical constraint (the normal contact stresses follow then from the constraint equation) and (2) the development of constitutive laws for the micromechanical approach within the contact area which yields a response function for the normal stresses. For the tangential direction one has the same situation for stick in the contact interface where either (1) a geometrical constraint equation can be formulated or a (2) constitutive law for the tangential relative micro displacements between the contacting bodies can be used. For tangential sliding between the bodies constitutive equations for friction are derived based on an evolution equation. All di�fferent approaches and associated relations are discussed in this chapter. Key words: non-penetration condition, micromechanical approach, constitutive contact laws, stick, sliding, evolution equations. Chapter 6: Contact Boundary Value Problem and Weak Form The additional terms due to contact when formulating the general contact boundary value problem are discussed in detail in this chapter. This results in non-linear boundary value problems for which { in view of the �nite element method { also weak formulations have to be developed. for contact problems. The main concern of this chapter is the incorporation of the constraint equations formulated for frictionless contact and of interface laws related to stick and sliding in the contact interface. However one of the major problems in contact mechanics is the non-di�erentiability of normal contact and friction terms. To overcome these di�fficulties di�erent formulations are developed. Key words: weak form, frictionless contact, stick, sliding, non-di�fferentiability, contact formulations. Chapter 7: Discretization of the Continuum The discretization of the domain contributions of the bodies being in contact is treated in this chapter for problems undergoing large strains. The use of isoparametric elements is discussed for formulations with respect to the initial and current con�guration of the bodies. For these all matrices and vectors are defi�ned, needed to compute the residuals and the tangent sti�ness matrices of the discretized continua. Key words: initial con�guration, current con�guration, Chapter 8: Discretization, Small Deformation Contact For the case of two deformable bodies being in contact the contact discretization is derived in this chapter. For this purpose di�erent possible formulations are discussed including node-to-node contact, isoparametric discretizations with surface-to-surface contact and segment-to-segment contact. For the latter mortar and the Nietsche method are introduced. Key words: contact discretization, node-to-node contact, surface-to-surface contact, segment-to-segment contact, mortar method, Nietsche method. Chapter 9: Discretization, Large Deformation Contact In this section the discretization techniques for large deformation contact are discussed. Here sliding of a contacting node or element over several elements of the other body is allowed. To describe such process properly a master-slave concept in the current con�guration is introduced. The master segment will be a line in two-dimensional situations and a surface in three-dimensional contact problems. Discretizations are based on the isoparametric concept and on descriptions of the contact surfaces by C1-continuous approximations. Furthermore, mortar discretization schemes are developed. Key words: isoparametric formulation, large deformation, master-slave concept, C1-continuous discretization, moratr method. Chapter 10: Solution Algorithms In this chapter the algorithms which are essential for the treatment of contact problems are considered. These are applied to the discretized problem which are derived using the formulations in the previous chapters. The algorithms include general contact detection, local gap computations, general solution procedures and local integration of interface laws. For these procedures di�erent possibilities are discussed in the light of robustness and e�ciency. Key words: contact detection, integration of interface laws, general solution procedures, robustness, e�ciency. Chapter 11: Thermo-Mechanical Contact In case of thermo-mechanical contact problems two �elds { deformation and temperature { interact and thus have to be considered within the formulation. In the general setting these �elds are coupled since the constitutive parameters depend on the temperature. Furthermore the evolution of the thermal �field is related to the deformation and heat can be generated by dissipative mechanisms like plastic deformations or frictional forces. For simulations in this area the �finite element discretizations is developed as well as the associated algorithms based on staggered schemes. Key words: thermo-mechanical contact, coupled �elds, staggered schemes, temperature, frictional dissipation. Chapter 12: Beam Contact In this chapter engineering problems which involve contact of beams undergoing large displacement are discussed. The beams can be either in contact in the initial con�guration or get into contact during the motion. Hence special relations for kinematics and constitutive behavior at the contact interface have to be formulated for beam contact. These are derived and also presented in matrix form for the discretization process. Special search procedures are developed. Key words: beam contact, large displacements, search procedures. Chapter 13: Adaptive Finite Element Methods for Contact Problems This chapter is related to a method in the area of �nite element techniques which ensures a successive improvement of the numerical solution via an adaptive mesh re�nement. Here di�erent error estimation techniques are discussed for contact problems which include residual based and dual error estimation as well as Zienkiewicz, Zhu error indication procedures. Adaptive mesh re�nement strategies are discussed for frictionless and frictional contact problems. The latter include also history data transfer techniques. Key words: transfer of history data, adaptive �nite element methods, error estimators, error indicators, frictionless contact, frictional contact. Chapter 14: Computation of Critical Points with Contact Constraints Post-critical behavior of contact problems is studied in this chapter. In this case classical path-following algorithms with branch-switching and contact formulations have to be modi�ed and combined since constraints associated with contact are inequalities. Here �rst a general formulation of the problem is given. After that special algorithms based on the direct computation of critical point are applied. Key words: post-critical behavior, critical points, branch-switching, path- following methods, direct computation of critical points.
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On the indentation of a system of punches into a layered foundation
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Contact problem for a system of rigid rough punches and layered foundation with nonuniform upper layer is considered. Mathematical model of the problem is compiled. It’s effective solution is constructed by using Manzhirov projection method. Qualitative conclusions are presented. © 2019, International Association of Engineers. All rights reserved.
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Modeling the Contact Interaction between a Nonuniform Foundation and a Rough Punch
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A contact problem for a double foundation and a rigid punch is considered. It is assumed that the surface nonuniformity of the thin upper layer and the shape of the punch base can be described by complex rapidly changing functions. A projection method is developed that allows us to accurately solve the equation, which is not possible by the known methods. An algorithm of the numerical-analytical calculation is described. A model example is presented.
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Axisymmetric problem of fretting wear for a foundation with a nonuniform coating and rough punch
Conference Paper
  • May 2018
The axisymmetric contact problem with fretting wear for an elastic foundation with a longitudinally nonuniform (surface nonuniform) coating and a rigid punch with a rough foundation has been solved for the first time. The case of linear wear is considered. The nonuniformity of the coating and punch roughness are described by a different rapidly changing functions. This strong nonuniformity arises when coatings are deposited using modern additive manufacturing technologies. The problem is reduced the solution of an integral equation with two different integral operators: a compact self-adjoint positively defined operator with respect to the coordinate and the non-self-adjoint integral Volterra operator with respect to time. The solution is obtained in series using the projection method of the authors. The efficiency of the proposed approach for constructing a high-accuracy approximate solution to the problem (with only a few expansion terms retained) is demonstrated.
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On Indentation of a System of Irregularly Shaped Rigid Punches into a Coated Foundation
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  • Sep 2017
We consider the contact problem of interaction between a coated viscoelastic foundation and a system of rigid punches in the case where the punch shape is described by rapidly varying functions. A system of integral equations is derived, and possible versions of the statement of the problem are given. The analytic solution of the problem is constructed for one of the versions.
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Contact problem with wear for a foundation with a surface nonuniform coating
Article
  • Jul 2017
The plane contact problem with wear for an elastic foundation with a longitudinally nonuniform (surface nonuniform) coating and a rigid punch with a flat foundation has been solved for the first time. The case of linear wear is considered. The nonuniformity of the coating is described by a rapidly changing function. This strong nonuniformity arises when coatings are deposited using modern additive manufacturing technologies. The problem is reduced to the solution of an integral equation with two different integral operators: a compact self-adjoint positively defined operator with respect to the coordinate and the non-selfadjoint integral Volterra operator with respect to time. The solution is obtained in series using author’s projection method. The efficiency of the proposed approach for constructing a high-accuracy approximate solution to the problem (with only a few expansion terms retained) is demonstrated. A simple engineering formula for estimating the contact stresses under a punch for large values of times is proposed.
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