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y^K 539.62
On the reduction method of dimensionality: The exact mapping of axisymmetric contact problems with and without adhesion
M. Hefi
Berlin University of Technology, Berlin, 10623, Germany
Starting from the classical theory of contact mechanics it is shown that the relationship between load, penetration and contact radius of any axisymmetric contact can be mapped exactly on a one-dimensional system, thus the reduction method of dimensionality is valid for conforming and non-conforming contacts. Furthermore the reduction method has been successfully extended to adhesive contact problems. The mapping of the classical theory of Johnson, Kendall and Roberts derived for spherical contacts as well as its application to axisymmetric contacts of arbitrary shape is possible; all results are reproduced precisely.
Keywords: method of reduction of dimensionality, axisymmetric Boussinesq problem, adhesive contact, JKR-theory
1. Introduction
The classical theory of contact mechanics started with the work of Hertz [1], who solved the frictionless, nonadhesive contact problem between two elastic spheres. Nearly hundred years later Johnson, Kendall, and Roberts [2] extended the Hertz theory to adhesive contact using a balance of elastic and surface energy (JKR-theory), which is, strictly speaking, only valid to large and soft spheres. Nevertheless, up to date those classical theories are probably the most widely used since they provide the basis for many modern contact problems. Especially the contact between rough surfaces is generally represented by multi-asperity contact models, which are mostly based on Hertz’s theory.
Many technical and biological contacts are not spherical, but display a variety of shapes, in particular rotational symmetry. The latter (non-adhesive) case called axisym-metric Boussinesq problem. The methods of potential theory was used by Boussinesq [3] to derive a solution of the indentation problem of an elastic half-space by a rigid axisymmetric frictionless punch. The solution in terms of Hankel transforms has been provided by Sneddon [4]. His theory plays a central role in the measurement of hardness and elastic modulus by instrumented indentation [5]. By using Sneddon’s equations Maugis and Barquins [6-8] showed that the JKR-theory is consistent with the linear elastic fracture mechanics and they generalized its application to any axisymmetric contact. Only in 2006, Yao and Gao [9] have used this theory to analyze the effects of
different shapes of contacting surfaces on the pull-off force and the adhesion strength. Their results explain the shapes commonly found in biological systems and provide basic informations on the development of novel adhesive materials for engineering sciences.
In the present letter it is demonstrated that all the above mentioned theories can be mapped exactly by a one-dimensional system. This is one of the main goals of the reduction method of dimensionality proposed by Geike and Popov [10]. They succeeded in exactly mapping the three-dimensional Hertzian contact on a one-dimensional model. Based on that, they developed a one-dimensional system to the contact problem between two three-dimensional bodies with randomly rough surfaces without loss of essential contact properties. This meant a huge reduction of computational time. A detailed description can be found in [11, 12], further applications in [13].
The following considerations are restricted to the exact mapping of the relationship between load, penetration and contact radius of arbitrary shaped axisymmetric contacts taking into account the adhesion. A variety of other possible mapping theorems including the pressure distribution inside the contact area as well as the stresses and displacements inside the half-space were proved in [14].
2. Theoretical background: Sneddon’s equations
The indentation problem of the linear elastic half-space by rigid axisymmetric punches of arbitrary profile has been
© Hefi M., 2012
solved by Sneddon, who used the theory of Hankel transforms and dual integral equations.
The so-called axisymmetric Boussinesq problem have the mixed boundary conditions uz (r, 0) = 8- f (r), 0 < r < a,
azz (r, 0) = 0, r > a,
(1)
Trz (r, 0) = 0,
where 8 is the penetration depth of the punch, a is the radius of contact and z = f (r) is the shape function; note that /(0)= 0 and the shape function has to conform with the surface of the elastic half-space inside the contact area. Moreover, we distinguish between the penetration depth 8 and the contact depth 8c = f (a), see Fig. 1. Sneddon derived the following simple expressions, which only depend on the shape of the punch:
) 4T-
P = nE* a Jx(t )dt,
a zz (r,0) = E
2a
% V t2 - r2 Vi - r
r < 1,
uz (r,0) A-^dt, r > 1,
0 V r -1
(2)
(3)
(4)
(5)
where r = r/a, E = e/(1 -v2). The function x(t) can be deduced from
,5 , 6
(6)
x(t)= -
n
^ 0 V t2 - x'
by knowing the shape function.
3. Axisymmetric contacts without adhesion
Let us apply these formulas to some special cases without adhesion.
3.1. The flat-ended cylindrical punch
Let us consider a flat-ended rigid cylindrical punch of radius a, which penetrates the elastic half-space. In this special case, the penetration depth 8 doesn’t depend on a
and the contact depth 8c vanishes. The shape function being given by f (r) = 0, so we get
X(t) = —8 = X(1), n
P = 2 E*a8>, a zz (r, 0) = -- P
(7)
(8)
2 , 1 . (9)
2na ^1 - (r/a)2 There is a square-root stress singularity and a discontinuity of displacement at the edge of the contact (Fig. 2). Differentiating the load P from Eq. (8) with respect to 8 leads to the contact stiffness
k = — = 2 E * a, d8
(10)
which is proportional to the radius (and not the area) of contact.
3.2. Punch with a power-law shape profile
Except for the flat-ended punch, the radius of contact is generally unknown beforehand. Boussinesq [3] pointed out, that if the punch has a smooth profile, the normal stress vanishes at the edge of the contact. Sneddon [4] used this additional condition which made it possible to determine the area of contact and showed that
X(1) = 0. (11)
In the following, we consider an axisymmetric punch with a power-law shape profile
f (r) = cnrn, n e R a n > 0, (12)
where cn denotes a shape coefficient. In 1939, Steuermann [15] used a similar monomial approach but restricted to positive even degrees. A Taylor series expansion for a spherical punch has been investigated by Segedin [16]. Substitution of Eqs. (11) and (12) into Eq. (2) leads to the penetration depth
8 nr(n/2) n /i
8 =------------—cna , (13)
2 r((n +1)/2) n ’ v '
which can be expressed by the contact depth
8c = f (a) = cnan (14)
in the way
Fig. 1. Indentation of an elastic half-space by a rigid axisymmetric punch
r/a
Fig. 2. Indentation by a flat-ended circular punch
8 = K(n)8c with K(n) =
•x/rc n r(n/ 2)
(15)
2 r((n +1)/2)'
Equation (15) will play an important role in the subsequent modeling ofthe system of reduced dimensionality. The same applies to the following equation. Using (11) and (12) equation (3) yields after some simple manipulations
P = 2 E>(n) [f (a) - f (r)]dr. (16)
0
The integrand contains only the relative displacements measured from the edge of the contact (not from the undeformed surface). The one-dimensional model presented below, shows the same characteristic.
Replacing f (r) by its expression given in (12), we obtain
P = E*Jk
iT(n/2 +1) c „+1 =
= 2 E
r((n + 3)/2) n K(n)cnan+1.
n +1
(17)
By differentiating Eq. (13) and (17) with respect to the contact radius a and comparing the results it can be easily shown, that the contact stiffness given by Eq. (10) derived for the contact between a flat-ended cylindrical punch and the elastic half-space still holds in the case of a punch with a power-law shape profile:
7 dP »
k = — = 2 E a. d8
Pharr et al. [17] proved that this relationship is perfectly valid for any indenter whose geometry can be described as a solid of revolution of smooth function. The generality of contact stiffness meant a great advance in the measurement of hardness and elastic modulus by instrumented indentation (see [5]).
4. The reduction method of dimensionality: Mapping theorems
In the previously discussed three-dimensional contact problems the displacement at any point of the surface depends on the entire pressure distribution inside the contact area. This difficulty is avoided in a so-called Winkler foundation. Therefore, many researchers used similar models to provide approximate solutions of half-space theory; only in a few problems of soil mechanics considering inhomoge-neous media exact solutions have been recovered [18].
Now, let us consider a one-dimensional Winkler foundation of depth l0 and width Ay for which the normal (surface) displacement uz (x) of any point is directly proportional to the contact pressure p(x) at that point, thus
p(x) = EwUz (xX l0
(18)
where EW denotes the elastic modulus of the foundation. The corresponding mechanical model is a one-dimensional layer of closely spaced, independent, linear springs of stiffness AkN. The elastic parameters are connected by
Fig. 3. Contact between a rigid flat-ended punch and a one-dimensional Winkler foundation
A xAy
AkN= , EW
l
- Ax,
(19)
where Ay is the constant width of the layer perpendicular to the viewing plane of Fig. 3, i.e. the normal contact stiffness increases linearly with the contact length. Remember, that any axisymmetric contact problem shows the same characteristic, see Eq. (10).
In the case in which the layer is indented by a flat-ended rigid punch as shown in Fig. 3, all springs are deflected by the penetration depth 8, so we easily obtain the normal load by
j Ak
P = 2 £ AkN 8 = 2 —Na8, (20)
i=1 A x
wherein a is the semi-contact width and j = a/Ax. By comparing Eq. (20) with Eq. (8), we obtain
AkN = E Ax. (21)
Notice that if we choose the stiffness of springs according to Eq. (21) the model exactly predicts the macroscopic behavior of contact between a rigid flat-ended cylindrical punch and the elastic half-space.
The general case is slightly more complicated since the deflections of springs depend on the distance x from the centre of contact. Notice that a three-dimensional axisym-metric contact means the contact between solids of revolution. In our one-dimensional model the shape of the punch is symmetric about the z-axis and shall be defined by f (x) = cnxn, n e R a n > 0. (22)
As can be seen from Fig. 4, in the one-dimensional model we don’t need to distinguish between contact depth and penetration depth, they are identical. The force in a spring at distance xi is
AP( x) = AkN [f (a) - f (| x;-1)], (23)
which may be written by using Eqs. (21) and (22) as
AP(x) = E*(cnan - cn \xt\")Ax. (24)
By adding up all forces of springs and then considering the limit Ax ^ 0, we obtain
P = E* J [f (a) - f (|x|)]dx =
-a
= 2 E* J [f (a) - f (x)]dx,
0
Fig. 4. Contact between an axisymmetric, rigid punch and an elastic half-space (a); indentation of a one-dimensional Winkler foundation by a rigid punch whose shape is symmetric about the z-axis (b)
where cn in Eqs. (24) and (25) denotes the shape coefficient of the (analogous) punch in the one-dimensional system.
Now, by comparing Eq. (25) with Eq. (16) derived for the real axisymmetric contact problem, we get
Cn = K(n) Cn (26)
. vn n r(n/2)
with K(n) = ---------v ' ’ .
2 r((n + 1V2)
If we choose the shape coefficient of the (analogous) indenter according to Eq. (26) and take into account the dependency of elastic parameters given by Eq. (21), then the relationship between load P, penetration depth 8 and contact radius a of the real contact problem will be exactly reproduced by the one-dimensional system.
Equation (26) provides the necessary geometric transformation between the two shape functions f% and f. It implies a simple vertically stretching of the real indenter shape by the factor K(n), which physically represents the ratio of penetration depth and contact depth. This process can be easily understood by means of Fig. 4. The shape factor k as a function of the exponent n has been displayed in Fig. 5. The well-known cases of the conical and the parabolic indentation are marked separately. In the latter, for example, we have
C2 =K(2)c2 = 2c2, (27)
which may be written as
C2 = 2— = -, (28)
2 2 R R
wherein R denotes the radius of curvature of the (real) sphere. If we introduce a radius of curvature R1D of the
corresponding one-dimensional model and replacing C2 by 1/ (2 R1D), equation (28) leads to
Rm = 2 R. (29)
Equation (29) implies that if we bring a rigid cylinder of radius R1D in contact with our one-dimensional chain of springs, then it provides exactly the same behavior as the contact between a rigid sphere of a two times larger radius R and the elastic half-space. These results fully agree with those from Popov and Geike [10].
The previously presented theory generalizes the reduction method of dimensionality. Since the principle of superposition holds, its applicability to shapes represented by power series is obvious. Hence, it can be applied to any non-adhesive, axisymmetric contact problem. The invariance of contact properties is restricted to the relationship between load, penetration and radius of contact.
5. The implementation of adhesion
In the following section let us discuss the exact mapping of axisymmetric contact problems with adhesion in the context of the JKR-theory. Johnson, Kendall and Roberts [2] modified the elastic contact theory of Hertz by including the surface energies of the solids. Their theory is based on an energy balance and leads to the macroscopic relations
. a 2naY
5(o) = R1—■
P(a) =
4 E_a_
3 R
--y/snE
a3 Y,
(30)
(31)
Fig. 5. Dependence of shape factor k on the exponent n of shape function
where Y denotes the relative surface energy of the interface. The first terms on the right-hand sides of Eqs. (30) and (31) indicate the solutions of the non-adhesive contact problem. We call them 8na and Pj.a. For example, Pna stands for the apparent load in the non-adhesive case which would lead to the same contact radius a as that observed under the load P including the adhesion, see Fig. 6. From Eqs. (30) and (31) we get
Pi.a - P = vBrcEVY = 2E*a(8n.a -8), (32)
respectively
P = Pi.a - 2E*a(8n.a -8). (33)
Equation (33) indicates the main idea of the JKR-theory.
r/ac
Fig. 6. Interpretation of the first part of Eq. (31): non-adhesive indentation due to apparent load Pn a, which leads to the same radius a as that under the load P in the adhesive case
The adhesive contact arises from the contact without adhesion plus a rigid body translation.
Note that the second part resembles the indentation by a flat-ended cylindrical punch; the contact area remains constant. It is illustrated by Fig. 7. Since it was already shown that both parts can be exactly mapped by the one-dimensional model, the same holds for the superposition and therefore the classical JKR-theory.
The validity of Eq. (33) for any axisymmetric contact problem can easily be proved by using the analogy between JKR-theory and the Griffith theory of linear elastic fracture mechanics. Based on Sneddon’s Eqs. (2)-(6), Maugis and Barquins [6-8] analyzed the normal stresses and displacement in the neighborhood of the edge of the contact. They showed that the appearing stress singularities and the discontinuity of displacement are those of fracture mechanics:
E* X(1)
a zz (r = a(1 -e), 0) =--
(34)
2a V2e ’
Auz(r = a(1 + e), 0) = -x(1)V2e, (35)
where e << 1 and Auz represents the gap outside of the contact. Therefore, it was possible to regard the edge of the contact area as a mode I crack and the rigid body translation x(1) may be expressed by the stress intensity factor K j established by Irwin [19]:
r/ac
Fig. 7. The second part of Eq. (31): unloading resembles the indentation by a flat-ended punch; the contact radius is maintained constant at a; the special case of maximum pull-off force (force of adhesion) is shown, where ac denotes the critical radius of contact
K j(a) = -^ X(1). 2V a
(36)
By introducing the elastic energy release rate G, equation (36) becomes
G = nL X(1)2.
8a
(37)
By considering the energetic fracture criterion of Griffith [20]
G = Y (38)
Eq. (37), after some further manipulations, yields
X(1) = -
8aY
tcE*
(39)
Substituting the rigid body translation according to Eq. (39) in Eqs. (2)-(4) gives
8=J fxL dx - i2aKY
(40)
=8
P = 2 E*aJ
8n.a - t J
f X x) JF-x1
rdx
dt-yj8nEc*, (41)
=P
a (r-0) = El 1 X(t) dt + Pn.a - P_L
°zz(r,0) 2a4422444^ + 2na2 ^1-
(42)
= an
Equations (40) and (41) generalize the JKR-theory to the adhesive contact between a rigid, axisymmetric punch of arbitrary shape and the elastic half-space. Its application to contacts between two elastic solids with axisymmetric surface profiles is possible, if we define the effective elastic modulus by
1
E*
1 -v1 1 -v,
E1 E2
(43)
By comparing (40) and (41) the validity of Eq. (33) has been proved. These results date back to Maugis and Bar-quins (for more details see also [21]) and has been applied to power law profiles by Yao and Gao [9].
The distribution of normal stresses in the area of contact is given by Eq. (42). It consists of two parts: a (non-adhesive) compression part due to the apparent load Pna and an attractive part, whose distribution is equal to that observed under a flat-ended cylindrical punch with a stress singularity at the edge of contact. The unloading from Pn a to P proceeds, while maintaining the radius of contact constant at a and increasing the energy of adhesion from zero up to its value Y-
The realization of those two parts in the context of the reduction method can be easily understood. In the first step we modify the profile of punch according to Eq. (26) and press it into the one-dimensional Winkler-foundation, whose stiffness has to be chosen as given by Eq. (21). Then we
Pn.a
Fig. 8. Exact mapping of generalized JKR-theory: loading and pull-off in the one-dimensional model exemplary shown for a conical indentation problem
reduce the load while all contacting springs adhere to the punch, the contact width remains constant. If the deflection of outer springs reaches a critical value Almax, the equilibrium state of adhesive contact has been found, see Fig. 8.
The critical value between stick and separation — the pull-off condition—may be calculated from Eq. (40), which gives
Almax (a) =^pa. (44)
The critical deflection of springs is proportional to the square-root of semi-contact width a, which is necessary to note in the context of numerical implementation.
6. Conclusions
In the present paper it were mathematically proved some mapping theorems within the framework of the reduction method of dimensionality. With this fundamental analysis, it is possible to map any frictionless normal contact between two elastic solids with arbitrary-shaped, axisymmetric surface profiles on a one-dimensional system. It was considered non-adhesive and adhesive contact problems; in both cases all results of the three-dimensional contact theory have been exactly reproduced by the reduction method. The attention is focused on the exact mapping of the relationship between load, penetration and radius of contact. However, some possibilities to map the pressure distribution as well as the stresses and displacements inside the half-space can be found in [10, 14].
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HeB Markus, Dr., Berlin University of Technology, Germany, markus.hess@tu-berlin.de
