Normal contact problem between a cylindrical indenter and a half-space with long-range adhesion: study with the method of dimensionality reduction Текст научной статьи по специальности «Медицинские технологии»

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Dimaki A.V. / Физическая мезомеханика 18 4 (2015) 42-45

УДК 539.37, 539.61

Нормальный контакт между цилиндрическим индентором и полупространством с дальнодействующими адгезионными силами: моделирование в рамках метода редукции размерности

А.В. Димаки

Институт физики прочности и материаловедения СО РАН, Томск, 634055, Россия Национальный исследовательский Томский государственный университет, Томск, 634050, Россия Национальный исследовательский Томский политехнический университет, Томск, 634050, Россия

В рамках метода редукции размерности рассмотрена нормальная контактная задача с учетом экспоненциально спадающего потенциала адгезионных сил. Возможности развитой модели проиллюстрированы на примере решения контактной задачи с адгезией между цилиндрическим индентором и упругим полупространством. В целом построенная модель является общей и может быть применена для адгезионных потенциалов произвольного вида и различной геометрии инденторов.

Ключевые слова: адгезионные силы, редукция размерности, осесимметричный контакт

Normal contact problem between a cylindrical indenter and a half-space with long-range adhesion: study with the method of dimensionality reduction

A.V. Dimaki

Institute of Strength Physics and Materials Science SB RAS, Tomsk, 634055, Russia National Research Tomsk State University, Tomsk, 634050, Russia National Research Tomsk Polytechnic University, Tomsk, 634050, Russia

Adhesive contact with exponential adhesive interaction is simulated with the use of the method of dimensionality reduction. The developed procedure is illustrated with an example of adhesion of a cylindrical punch and an elastic half space. However, it is general and can be used for any form of interaction potential and any form of indenter.

Keywords: adhesion forces, dimensionality reduction, axially symmetric contact

1. Introduction

Adhesion forces play an important role in contacts, especially at submicroscale and nanoscale [1,2]. At the mic-roscale, the characteristic range of action of adhesion forces becomes comparable with the size of the contact patch or with the indentation depth. Under these conditions, it becomes necessary to take into account the range of adhesive potential explicitly, in contrast to well-known and widely-used classic models of adhesion [3-5]. One of the most well-known and pioneering works on the adhesion with finite range of interaction was performed by Maugis [6]. He used the simplest possible form of the interaction potential linearly decreasing on a given characteristic length. This potential, initially introduced by Dugdale [7], allows ob-

taining analytical solutions for stress-strain distributions in axially symmetric contacts [6].

In the present paper we generalize the approach of Maugis to an interaction potential of arbitrary shape and illustrate this approach with an example of an exponentially decaying potential without any cut-off distance. Such interaction potentials are much more realistic than the linearly decreasing model potential of Dugdale. The proposed approach is based on the solutions for axially-symmetric contacts presented by Sneddon in [8] (which however stems from Galin [9]). We use this approach in the formulation presented in [10] and known as method of dimensionality reduction (MDR) (see also the paper by Hess [11]).

The key idea of the method of dimensionality reduction is an exact mapping of axially symmetric contacts of an

© Dimaki A.V., 2015

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arbitrary shape to an abstract contact of a one-dimensional series of independent springs [10]. Results obtained with the model can be used for simulation of friction [12], wear [13] and many other contact mechanical problems. We would like to stress that—contrary to a common misunderstanding—the method of dimensionality reduction provides for axis-symmetrical contacts with compact contact area exact solutions and not "one-dimensional approximations"!

In the following paragraphs we describe the implemented model and provide some results of numerical simulation of contacts of an elastic half-plane with cylindrical indenter.

2. The model of tangential adhesive contact

Let us consider a contact of a three-dimensional rota-tionally symmetric indenter, having a profile z = f(r) and an elastic half-space. In the framework of the method of dimensionality reduction, the 3D profile of an indenter is first transformed into a one-dimensional profile g(x) according to the method of dimensionality reduction rule [ 10]

■ Jx1 fV)

g (x) = x

VX2"

rdr

(i)

0 v X - r

as illustrated in Fig. 1. The reverse transformation is given by the integral

f (r ) = - J n 0

g ( X )

VT2"

rdx.

(2)

In particular case of a cylindrical indenter (Fig. 1, a) with radius a0, we get

f0, r < a0,

f (r) = ] 0' (3)

r > a0.

The corresponding one-dimensional profile is

f 0, x < a0,

0 (4)

I x > a0.

The one-dimensional profile (1) is pressed to a given indentation depth d into an elastic foundation consisting of independent springs with spacing Ax (Fig. 1, b) which normal and tangential stiffness are given by

g ( x) =

kz = E Ax, where E is the effective elastic modulus:

_1_ E*

1 -v2 1 -v2

Ei

(5)

(6)

The vertical displacement of an individual spring in contact area is given by

uz (x) = d - g(x), x < a (7)

and the resulting contribution of elastic forces into the normal force is equal to

Fz (x) = E\z (x) = E*Ax(d - g(x)). (8)

The linear density of elastic forces is therefore

qz (x) = ^ = E(d - g(x)). Ax

(9)

We introduce adhesive forces by means of considering an "adhesion pressure" between interacting bodies in the following manner:

Padh(r) = Padh(Az(r)) = ac exp(-Az(r)/P), (10) Az(r) > 0,

where Az(r) = f (r) - d + uz (r) is a distance between surfaces of indenter and a half-space, b denotes a characteristic length of the potential.

With three-dimensional "adhesion pressure" distribution (12) we can determine the corresponding linear density of adhesive forces by means of the following transformation:

( ) 2 a0 rPadh(r) d

qadh,z(x) = 2 J 12 2 dr.

x Sir

(11)

¡r - X

This one-dimensional force density will lead to displacements uz (x ) of the springs of the elastic foundation. The displacements in the original three-dimensional system then are calculated according to [10] by means of the following procedure:

dx

^ 1 a° duz ( x )_

uz( r ) = --J —d--n-2

n r dx Vx2 - r2

uz (r) = 2 J Uz.(x). dx, d < 0.

d >0,

(12)

n oVT

Fig. 1. The 3-dimensional body of revolution (a) and the corresponding one-dimensional MDR-transformed profile in a contact with the elastic foundation (b)

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Dimaki A.V. / &u3unecKan Me30MexaHum 18 4 (2015) 42-45

V^no

2

Fig. 2. The dependencies of normal reaction force (a) and contact radius (b) on vertical displacement of cylindrical indenter

The first transformation in (12) represents the "internal" Abel's transformation while the second transformation is the "external" one.

Equilibrium between adhesive forces and elastic forces determines vertical positions of one-dimensional springs outside of contact area as follows:

uz (x) = -qadh (x)/E*, x > a. (13)

The force with which the points of the contacting bodies are pressed to each other inside the contact region with the contact radius a is given by the equation

^conu (x) = A x ( qadh,z (x) + qz (x)) =

= Ax (qadh,z (x) + E* (d - g(x))). (14)

The contact radius a is given by the equation AFcont,z (a) = = 0:

malized to the value of "characteristic adhesive force"

21- rPadh(r) dr + e- g (a )) = o.

a 2 2

a Vr - a

(15)

According to the method of dimensionality reduction rules, the distribution of normal pressure p in the three-dimensional problem can be calculated using the following integral transformation [14]:

p ( r )=-1 j^üxL dx=-L J X)

2 rcAx Jr/r2 --2

dx. (16)

-r r yx - r

The normal reaction force Fn for a cylindrical indenter

can be estimated as follows:

Fn = 2E*J°(d - g(x))dx. (17)

o

The procedure described above allows calculation of normal components of stresses and deformations for a normal contact of an axially symmetric indenter with adhesion for a given indentation depth. Note that all above results obtained by the method of dimensionality reduction represent exact solution of the corresponding three-dimensional problem.

The developed numerical procedure has been applied for the simulation of indentation of a cylindrical indenter into an elastic half-space. The results of simulation are shown in Fig. 2. Here the resulting reaction force is nor-

Fn0

^ na Gc, radius of contact is normalized to the value

of radius of cylinder a0 and indentation depth is normalized to the value of characteristic length of the potential p. Note that here we suggest the reaction force is "purely elastic", so that at zero indentation depth the reaction force is zero.

In contrast to the classic JKR model [3], the developed procedure allows explicitly take into account adhesive forces occurring after complete loss of "mechanical" contact between an indenter and foundation. We have to note that for more complicated shapes of indenters the transformation (12) must take into account convex and depressed parts of a profile of an elastic half-space and matching conditions of these parts.

3. Conclusions

We have studied the normal contact problem of a cylindrical indenter in the presence of exponential adhesion forces. A dependence for normal reaction force versus vertical displacement in adhesive contact of cylindrical indenter with a half-space has been provided. Since the method of dimensionality reduction allows obtaining exact solutions for arbitrary axially symmetric profiles of indenter, the presented model is applicable for a wide range of adhesive normal contact problems with an arbitrary long-range potential of interaction between contacting bodies.

Acknowledgements

Author acknowledges fruitful discussions with V.L. Popov and financial support from the Deutscher Akademischer Austauschdienst and the Program of Basic Scientific Research of the State Academies of Sciences for 2013-2020 (Russia).

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Поступила в редакцию 14.05.2015 г.

Сведения об авторе

Dimaki Andrey Victorovich, Cand. Sci. (Engng.), Researcher of ISPMS SB RAS, Senior Researcher of Tomsk State University, [email protected]