CC BY
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Popov V.L. / Физическая мезомеханика 20 5 (2017) 73-76
73
УДК 539.612, 544.722.54
Contact properties and adhesion of incompressible power-law gradient media with high gradients
V.L. Popov123
1 Technische Universität Berlin, Berlin, 10623, Germany 2 National Research Tomsk Polytechnic University, Tomsk, 634050, Russia 3 National Research Tomsk State University, Tomsk, 634050, Russia
We discuss contact stiffness and adhesion of flat-ended cylindrical indenters with a graded material the elastic coefficient of which is a power-function of the depth with an exponent 1 < k < 3. So far, only graded materials with k < 1 have been considered in the literature as the stiffness of the medium becomes zero when k is approaching 1. However, it is known that the case of incompressible media is an exception. We argue that in this case the final stiffness can be defined up to values of k < 3. The interval 1 < k < 3, which has not been considered earlier occurs to be of special interest, since for k > 1 the adhesive properties of contacts change qualitatively from "brittle" to very tough even in the case of a purely elastic material.
Keywords: functionally graded materials, normal contact, adhesion
Контактные свойства и адгезия несжимаемых градиентных сред с высоким показателем степени градиентных свойств
В.Л. Попов1-2-3
1 Берлинский технический университет, Берлин, 10623, Германия
2 Национальный исследовательский Томский политехнический университет, Томск, 634050, Россия
3 Национальный исследовательский Томский государственный университет, Томск, 634050, Россия
Обсуждается контактная жесткость и адгезия плоских цилиндрических штампов с градиентной средой, упругий модуль которой является степенной функцией глубины с показателем степени 1 < k < 3. До сих пор в литературе рассматривались только градиентные среды с k < 1, поскольку контактная жесткость обращается в нуль при стремлении k к 1. Известно, однако, что несжимаемые среды представляют собой исключение. Мы показываем, что в этом случае контактная жесткость может быть однозначно определена вплоть до k < 3. Интервал 1 < k < 3, который ранее не рассматривался, представляет особый интерес в связи с тем, что при k > 1 адгезионные свойства среды качественно изменяются от «хрупких» к очень пластичным, с высокой работой разрушения, даже в случае чисто упругих материалов.
Ключевые слова: функционально градиентные материалы, нормальный контакт, адгезия
1. Introduction
In the last few decades, various methods have been developed for improvement of thermomechanical properties of components, many of them based on using layered structures such as coatings, plating techniques or layer lamination processes [1-3]. A logical generalization of this approach is the use of functionally graded materials (FGM), which became increasingly popular since the 1990s. The gradually varying composition and structure of FGM result in continuous changes in properties of materials, thus solving some of the typical problems of layered materials, such as poor interface strength and residual stresses. Gra-
dient media can be found in many biological structures such as skin, bones or bamboo trees [4]. Lots of work has been carried out for developing of manufacturing techniques for this kind of materials and for studying their behavior [57].
Contact problems involving gradient materials were initially studied by researchers in soil mechanics and geome-chanics [8-10]. In most cases the spatial variation of elastic modulus with depth was assumed to follow either an exponential or a power law; a detailed review can be found in the paper [7]. Analytical solutions for the general power law dependence of elastic modulus of an elastic half space
© Popov V.L., 2017
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was provided for different types of surface loading by Booker et al. [11, 12], and solutions for both exponential and power-law dependencies were given later in [13, 14]. For axisymmetric contacts with a power law elastic modulus, all the existing results can be reproduced very easily using the method of dimensionality reduction (MDR) [15]. Further developments have been reported in the conference paper [16]. A very compact review of the most important results related to the contact mechanics of power-law graded media (including tangential contact) can be found in [17].
2. Contact stiffness of a cylindrical rigid indenter in contact with power-law medium
2.1. Media with k < 1
Let us consider a contact of a rigid cylindrical indenter having the radius a with an elastic half-space whose Young modulus depends on the depth (coordinate z) according to
E * -Eo -
E ( z ) - Eo
Kcoj
(1)
where E0 is the characteristic Young modulus and c0 is a characteristic length.
The contact stiffness of a cylindrical punch with radius a has been found earlier (see e.g. [15]):
, 2E>(k, V) ak+1
kN (a) =-;--
N ck k+1
with
h( k, v) =
2(1 + k )cos (kn/ 2)r ((1 + k)/2)
(2)
_ VnC(k, v)p(k, v)sin (p(k,v)V2) r((1 + k)/2) ' C ( k, v ) -
- 21+k r((3 + k + p )/2) r((3 + k - p)/2) = nr(2 + k ) '
P(k,v)-, (1 + k)|1
kv 1 -v
(3)
(4)
(5)
Fig. 1. Dependency of h(k, v) on the exponent k. For all Poisson numbers with exception of V = 0.5 (incompressible medium), h(k, v) (and thus the stiffness) vanishes for k = 1. In the limiting case V = 0.5 it shows regular behavior with finite stiffness even with k = 1
1 -v
,2 '
The function h(k, v) which mainly determines the stiffness of the contact is shown in Fig. 1. From this figure one can see why only media with k < 1 have been investigated so far. When the exponent k tends to 1, the stiffness tends to zero. Physically this means that for k = 1 (as well as for larger values of k) the indentation depth at the given force is not defined (similarly to the situation for line contacts with homogeneous media). The only exception is the case when the Poisson number is exactly V = 0.5 (incompressible medium). In this case the stiffness remains finite at k =1 and this indicates that this point does not represent any singularity and the dependence can be continued for larger values of k.
Of course, it would be interesting to investigate in detail how the transition to the limiting case of incompressible media does occur. Using analogy with line contacts we can suggest that the singular transition at exactly V = 0.5 will be regularized by the finite size of the considered system. At this point we do not carry out a formal mathematical analysis but just assume that there are some physical reasons for the nonsingularity of the solution given by Eq. (2) for larger values of k. We investigate only the question what consequences this assumption has. A rigorous mathematical proof still has to be done in a later publication.
2.2. Media with k > 1
Evaluation of (3) shows that the function h (k, V = 0.5 ) (and the contact stiffness) have no peculiarities at the value k = 1. It becomes equal to zero only at k = 3. While the complete analytical expression for h (k,v = 0.5 ) is very cumbersome, it can be approximated by a simple two-term expression (Fig. 2):
h(k, V = 0.5) = 1.3666sin(0.9010k + 0.4997) +
+ 0.4727sin (1.8884k + 0.7664 ). (7)
Fig. 2. Dependence of h(k, v) on the exponent k for v = 0.5 (according to Eq. (3)). For incompressible media, stiffness vanishes at k = 3. The exact analytical dependence is well approximated by Eq. (7)
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3. Adhesion of flat-ended indenters with power-law graded media
Let us consider the adhesive contact between a rigid body with plane surface and an elastic half-space in a state with a contact radius a and the indentation depth d. The differential stiffness of a contact with the radius a is given by Eq. (2). The elastic energy stored in the medium is equalto Uel = 1/2 kN (a)d2 and the adhesion energy Uad = = -y12na2, where y12 is the work of adhesion per unit area. Thus, the total energy of the system is equal to
Utot = Uel + Uad = 2kN (a) d2 - Yl2n«2 =
= E**h(k, V) O^ 2 2
= ck k +1d № •
(8)
One can easily see that the total energy has qualitatively different structure for k smaller and larger than 1.
3.1. Media with k < 1
For k < 1, the total energy has only one maximum at the critical radius ac is determined by the condition
dUto,
da
whence
ac =
Eoh(k,V) k,2 ~ n n
-Ocd - 2Yi2nOc = 0,
( 2Yi2ncok
\1/(k-1)
(9)
(10)
E0h(k, v)d
Thus, there exists only one equilibrium state, and this state is unstable. If the initial value of a exceeds ac (10), the contact spreads to infinity (or, if the plane is finite, to the maximum possible radius). For any initial a smaller ac it shrinks to zero, and the bodies lose the contact. For a cylindrical indenter with finite radius a, Eq. (10) gives the relation between the radius and the critical value of d in the moment of detachment, whence
(
dc =-
2Yi2nck
Y/2
k-1
(11)
E0h(k, v)aK"
where we take the negative solution, as only in this case detachment is (geometrically) possible. The normal force in this critical state is
Fn = kN(a) dc =
1 |8nE0y12h(k, v)a'
k+3
k +1
k c0
The corresponding "force of adhesion" Fa lute value of this force:
F, =■
1
8nE0 Y12h(k, v ) a
k+3
k +11
(12)
is just the abso-
(13)
3.2. Media with 1 < k < 3
Qualitatively different behavior have media with high exponents 1 < k < 3. In this case, potential energy has only one minimum. Thus, again only one equilibrium state
exists, but this state is stable. The equilibrium value of the radius of the remaining contact is still given by the Eq. (10). This equation now gives the radius of the adhesive contact as a function of the distance d between planes. When the distance increases, the equilibrium contact radius is decreasing. In particular, the detachment of flat-ended cylindrical punches will occur in a different way:
- for k < 1, the detachment occurs at once after reaching the critical distance (11);
- for 1 < k < 3, the detachment starts at the distance (11). With increasing distance, the force decreases but the contact remains stable at any distance.
The described difference in behavior is most obvious if the experiment is carried out under conditions of "controlled distance" (with a very stiff testing machine). In the opposite case of controlled force (very soft testing machine), the formally calculated force of adhesion is still given in both cases by the same Eq. (13). However, the work of adhesion up to the complete breakdown of the contact is zero in the case k < 1 and continuously increases with increasing k. Thus, adhesive contacts with media having over-critical k have much higher "fracture toughness".
4. Conclusion
We considered an extension of the existing theory of power-law graded media for the case of large exponents 1< k < 3 which has not been considered previously. This extension is only possible for incompressible media (Poisson number v = 0.5). Here we just used the existing solutions and have shown that they are formally defined up to values of k = 3. We did not carry out a more detailed analysis of the structure of stresses and displacements in the medium. Neither did we present any analysis of the detailed mechanisms of the limiting transition to the case v = 0.5. Thus, a formal mathematical proof of the correctness of the results remains to be done.
Proceeding from the existing solution, which we extended to the case of higher powers k, we have shown that exactly this new interval of powers is of very high physical interest as in this interval a qualitative change of adhesive behavior occurs: while adhesive contacts (without consideration of viscoelasticity) are brittle for k < 1, they become increasingly tough when k is increasing within the interval 1 < k < 3. This shows a possible way of increasing the fracture toughness of adhesive joints.
Note that for more complicated shapes of the cross-section of a flat-ended indenter, the stabilizing effect of increasing k can be observed even for k < 1. This interval of powers was investigated in detail in [18].
Finally, let us note that it would be interesting to generalize the recently developed formulation of the boundary element method in application to adhesive contacts of arbitrary shape [19], which at the present is only valid for k <1, for the extended interval 1 < k < 3.
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Acknowledgments
The author is thankful for useful discussions of highexponent gradient media with M. Hess and Q. Li. This paper has been partially supported by the program of competitiveness of the Tomsk State University.
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Поступила в редакцию 06.04.2017 г.
Сведения об авторе
Valentin L. Popov, Prof., Technische Universität Berlin, Germany; Prof., TPU; Prof., TSU, v.popov@tu-berlin.de
