Normal Contact with Adhesion
3.2 Solution of the Adhesive Normal Contact Problem
The two most well-known theories of adhesion by JKR and DMT both lead to an adhesive force which is independent of the elastic properties of the contacting media. To avoid any misunderstanding, it should be expressly noted that this simple property only applies to parabolic contacts. In no way can this lack of dependency on the elastic moduli be generalized to arbitrary adhesive contacts.
The consideration of the tangential stresses in the adhesive contact requires an-other general remark. Both JKR and DMT theory are based on the Hertzian theory offrictionless contact. It is surely a valid and self-consistent model assumption.
From the physical point of view, on the contrary, this assumption is rather ques-tionable. Physically, the JKR limiting case implies an infinitely strong yet infinitely short-ranged interaction. This means that the surfaces of the adhesive JKR contact are pressed together under infinitely low ranged but infinitely strong stress, which undermines the notion of a “frictionless” contact. However, for practical applica-tions, the difference between frictional or frictionless adhesive contact is relatively limited and can be considered negligible in most cases.
The following two sections will present two alternative approaches to the solu-tion of the adhesive normal contact problem. The first approach is the reducsolu-tion of the adhesive normal contact problem to the non-adhesive one, and the second approach is the direct solution utilizing the MDR.
Furthermore, we will assume the solution of the non-adhesive contact problem, particularly the relationships between the indentation depth, the contact radius, and the normal force. Any quantity of the triplefFN; a; dguniquely defines the others.
It will prove advantageous to describe the normal force and the indentation depth as functions of the contact radius:
FN DFN;n.a..a/; d Ddn.a..a/: (3.5)
The indices “n.a.” indicate that these are the solutions of the non-adhesive con-tact problem. These equations also imply that the dependency of the force on the indentation depth is known. We obtain the incremental contact stiffnesskn.a. by differentiating the force with respect todand the elastic energyUn.a.by integrating with respect tod. These quantities can also be rewritten as functions of the contact radius:
kn.a.Dkn.a..a/; Un.a.DUn.a..a/: (3.6) In the following all functions for (3.5) and (3.6) are considered to be known.
Let us now indent the profile to a contact radiusa. The elastic energy of this state isUn.a..a/, the indentation depthdn.a..a/, and the forceFN;n.a..a/. In the second step, we lift the profile byl without changing the contact radius. The stiffness (only dependent on the radius) remains constant for this process and equalskn.a..a/. The force is then given by:
FN.a/DFN;n.a..a/kn.a..a/l; (3.7)
and the potential energy is:
U.a/DUn.a..a/FN;n.a..a/lCkn.a..a/l2
2 : (3.8)
The new indentation depth at the end of the process equals:
d Ddn.a..a/l: (3.9)
Obtainingl from (3.9) and by substituting it into (3.8), we obtain the potential energy:
U.a/DUn.a..a/FN;n.a..a/.dn.a..a/d /Ckn.a..a/.dn.a..a/d /2
2 : (3.10)
The total energy (under consideration of the surface energy) now equals:
Utot.a/DUn.a..a/FN;n.a..a/.dn.a..a/d / Ckn.a..a/.dn.a..a/d /2
2 a2: (3.11)
The equilibrium value of the contact radius follows from the total energy minimum condition (for constant indentation depthd):
@Utot.a/
@a D@Un.a..a/
@a @FN;n.a..a/
@a lFN;n.a..a/@dn.a..a/
@a C@kn.a..a/
@a l2
2 Ckn.a..a/l@dn.a..a/
@a 2a
D
@Un.a..a/
@a FN;n.a..a/@dn.a..a/
@a
@FN;n.a..a/
@a lkn.a..a/l@dn.a..a/
@a C@kn.a..a/
@a l2
2 2a
D0: (3.12)
The terms in parentheses disappear and the equation takes the form:
@kn.a..a/
@a l2
2 D2a: (3.13)
It follows that:
l D
s4a
@kn.a..a/
@a
.for the general axially symmetric case/: (3.14) Substituting this quantity into (3.9) and (3.7) yields an equation for determining the relationship between the indentation depth, the contact radius, and the normal force:
d Ddn.a..a/
s4a
@kn.a..a/
@a
.for an arbitrary medium/; (3.15) FN.a/DFN;n.a..a/kn.a..a/
s4a
@kn.a..a/
@a
.for an arbitrary medium/: (3.16) It becomes apparent that the three functions which directly (and in its entirety) de-termine the solution of the adhesive contact problem are the three dependencies of the non-adhesive contact: indentation depth as a function of the contact radius, nor-mal force as a function of the contact radius, and therefore the incremental stiffness as a function of the contact radius. The latter quantity equals the stiffness for the indentation by a circular cylindrical indenter of radiusa. It should be noted that these are general equations and not subject to the homogeneity of the medium (nei-ther in-depth nor in the radial direction). As such, they also apply to layered or functionally graded media.The sole condition for the validity of (3.15) and (3.16) is the conservation of rotational symmetry during indentation.
For a homogeneous medium, the equations can be simplified even further. Here, the stiffness is given bykn.a.D2Ea, and we obtain:
l.a/D
r2a
E .for a homogeneous medium/: (3.17) The determining equations (3.15) and (3.16) take on the form:
d Ddn.a..a/
r2a
E .for a homogeneous medium/;
FN.a/DFN;n.a..a/p
8Ea3 .for a homogeneous medium/: (3.18)
Naturally, the pressure distribution in an adhesive contact and the displacement field outside the contact area are also composed of the two solutions of the non-adhesive contact problem: the solutions for the non-adhesive indentation bydn.a..a/and the subsequent rigid retraction byl. Let us denote the stress distribution and the dis-placement field for the non-adhesive contact problem by .rIa/n.a.andw.rIa/n.a., respectively. With the stress and the displacement field for a rigid translation given by (2.22), the stress distribution and the displacement for the adhesive contact prob-lem are then represented by the following equations:
.rIa/D .rIa/n.a.C El p
a2r2; r < a;
w.rIa/Dw.rIa/n.a.2l
arcsina r
; r > a; (3.19) or after inserting (3.17):
.rIa/D .rIa/n.a.C
r2Ea
p 1
a2r2; r < a .for a homogeneous medium/;
w.rIa/Dw.rIa/n.a.
r8a
E arcsina r
;
r > a .for a homogeneous medium/: (3.20)
Equations (3.18) and (3.20) completely solve the adhesive normal contact problem.
The magnitude of theforce of adhesion is of particular interest. We will de-fine it as the maximum pull-off force required to separate bodies. In mathematical
terms, this means the maximum pull-off force for which there still exists a stable equilibrium solution of the normal contact problem.
Another important quantity to consider is the force in the last possible state of stable equilibrium, after which the contact dissolves entirely. This force depends on the character of the loading conditions. We commonly distinguish between the limiting cases of force-controlled and displacement-controlled trials. The stability condition for force-control is given by:
dFN
da ˇˇˇˇaDa
c
D0; (3.21)
and for displacement-control case by:
dd da
ˇˇˇˇaDa
c
D0: (3.22)
The conditions (3.21) and (3.22) can be consolidated into the condition
ddn.a..a/
da ˇˇˇˇaDa
c
D s
2Eac; D
(3; force-control;
1; displacement-control; (3.23)
from which we can determine the critical contact radius, where the contact detaches (see Sect.3.3for a full derivation). The respective critical values of the indentation depth and normal force are obtained by substituting this critical radius into (3.18).
3.3 Direct Solution of the Adhesive Normal Contact Problem