6570
Langmuir 1998, 14, 6570-6578
Contact Mechanics and Adhesion of Viscoelastic Spheres C.-Y. Hui* and J. M. Baney Department of Theoretical and Applied Mechanics, Kimball Hall, Cornell University, Ithaca, New York 14853
E. J. Kramer Department of Materials, Engineering II-1361C, University of California at Santa Barbara, Santa Barbara, California 93106 Received March 6, 1998. In Final Form: August 3, 1998 The extension of the Johnson-Kendall-Roberts (JKR) theory of contact to viscoelastic materials is addressed. Because the energy flow to the material at the moving edge of contact is coupled to the local bonding or debonding process for viscoelastic materials, any such extension must account for the micromechanical process of bonding or failure if it is to avoid a paradoxical prediction. A crack bonding theory due to Schapery that uses a very simple model of the crack closing process is applied to the loading phase of a JKR type experiment. Numerical simulations show that this theory predicts results that are in reasonably good agreement with the experimental results of Falsafi et al. There are, however, difficulties in applying this theory to the determination of the surface energy of a material, and an alternative method of characterizing adhesive properties is suggested.
1. Introduction The mechanics of contact between solid elastic hemispheres has been used extensively in studying the surface energy and tack of materials (for examples, see refs 1-11). An excellent account of the history of this problem has been given by Maugis,12 but here we simply state that the three main theories describing it are those of Hertz,13 Johnson, Kendall, and Roberts (JKR),14 and Derjaguin, Muller, and Toporov (DMT).15 The Hertz theory assumes that adhesion between the contacting spheres cannot be sustained, so that the stresses in the area of contact are always compressive. The JKR theory does allow for adhesion by taking into account the surface energies of the bodies, but it neglects adhesive forces that act outside of the contact zone. This approximation results in infinite tensile stresses at the edge of the contact zone, where they have the inverse square root singularity that is characteristic of linear elastic fracture mechanics (LEFM) problems. Indeed, the JKR approximation is conceptually equivalent to the small scale yielding (SSY) problem in * To whom correspondence should be addressed. (1) Roberts, A. D.; Thomas, A. G. Wear 1975, 33, 45. (2) Vallat, M. F.; Ziegler, P.; Vondracek, P.; Schultz, J. J. Adhes. 1991, 35, 95. (3) Rimai, D. S.; DeMejo, L. P.; Vreeland, W.; Bowen, R.; Gaboury, S. R.; Urban, M. W. J. Appl. Phys. 1992, 71, 2253. (4) Chaudhury, M. K.; Whitesides, G. M. Langmuir 1991, 7, 1013. (5) Silberzan, P.; Perutz, S.; Kramer, E. J.; Chaudhury, M. K. Langmuir 1994, 10, 2466. (6) Creton C.; Brown, H. R.; Shull, K. R. Macromolecules 1994, 27, 3174. (7) Ahn, D.; Shull, K. R. Macromolecules 1996, 29, 4381. (8) Horn, R. G.; Israelachvili, J. N.; Pribac, F. J. Colloid Interface Sci. 1987, 115, 480. (9) Maugis, D.; Barquins, M. J. Phys. D: Appl. Phys. 1978, 11, 1989. (10) Brown, H. Macromolecules 1993, 26, 1666. (11) Creton, C.; Leibler, L. J. Polym. Sci., Part B.: Polym. Phys. 1996, 34, 545. (12) Maugis, D. J. Colloid Interface Sci. 1992, 150, 243. (13) Hertz, H. J. Reine Angew. Math. 1882, 92, 156. (14) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. London, Ser. A 1971, 324, 301. (15) Derjaguin, B. V.; Muller, V. M.; Toporov, Yu. P. J. Colloid Interface Sci. 1975, 53, 314.
LEFM, where the dimension of the fracture process and plastic zone is assumed to be vanishingly small compared with all relevant dimensions in the problem. In other words, the JKR theory of contact assumes that the region outside the contact zone where cohesive forces act (i.e., the cohesive zone) is vanishingly small compared with the contact radius. On the other hand, the DMT theory takes such cohesive forces into account but assumes that they do not change the shape of the material outside the contact zone from the Hertzian profile. This approximation leads to adhesive stresses that are finite in the cohesive zone but zero in the contact zone. A unifying model of contact was proposed by Maugis,12 who used a cohesive zone model to connect the JKR and DMT approximations and to determine the applicability of each theory. Maugis used a Dugdale model16 to represent the intermolecular surface forces acting in the cohesive zone and determined the size of this zone by requiring that the stresses be finite at the edge of contact. The cohesive zone model thus eliminates the stress singularity of the JKR theory, and it also removes the stress discontinuity of the DMT theory. In this way a realistic description of the boundary conditions at the edge of contact can be obtained. Similar cohesive zone models are well-known in the fracture mechanics literature and have been used to describe both rate-independent and rate-dependent crack growth.17-23 While the JKR theory is usually applied to the adhesion of elastomers, the requirement of the theory that the materials be linearly elastic severely limits its application. (16) Dugdale, D. S. J. Mech. Phys. Solids 1960, 8, 100. (17) Kramer, E. J.; Hart, E. W. Polymer 1984, 25, 1667. (18) Budiansky, B.; Amazigo, J. C.; Evans, A. G. J. Mech. Phys. Solids 1988, 36, 167. (19) Riedel, H. Mater. Sci. Eng. 1977, 30, 187. (20) Schapery, R. A. Int. J. Fracture 1975, 11, 141. (21) Knauss, W. G. In Deformation and Fracture of High Polymers; Kausch, H. H., Hassell, J. A., Jaffee, R. I., Eds.; Plenum Press: New York, 1974; p 501. (22) Hui, C.-Y.; Xu, D.-B.; Kramer, E. J. J. Appl. Phys. 1992, 72, 3294. (23) Knauss, W. G.; Losi, G. U. J. Appl. Mech. 1993, 60, 793.
10.1021/la980273w CCC: $15.00 © 1998 American Chemical Society Published on Web 10/02/1998
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Most elastomers are viscoelastic; only those whose glass transition temperature lies well below the ambient (e.g., poly(dimethylsiloxane)) even approximately satisfy this requirement. For a linear elastic material there is a unique relation between the contact radius and the applied load, whereas for a viscoelastic material the size of the contact zone depends on the loading history. There is substantial literature on the contact problem for viscoelastic bodies (refs 24-27) focusing on the Hertzian contact problem, which assumes that adhesion between the bodies cannot be sustained. The extension of these methods to the case of nonzero adhesion is demonstrated in this work. Essentially, the results of this extension indicate that the stress and deformation can be determined if the contact zone size is known. However, the determination of the contact zone size is difficult and cannot be obtained using the correspondence principle of viscoelasticity. The reason is that in the JKR theory the contact radius is determined by the crack growth condition G ) W, where G is the energy flow to the crack tip per unit area of crack extension and W is the work of adhesion. For linear elastic materials G is completely determined by the singular part of the contact pressure, i.e.
G ) KI2/2E*
(1)
where KI is the mode I stress intensity factor that results from considering the contacting bodies as forming an external crack and E* is an effective modulus defined by eq 6 below. Equation 1 is a well-known result in fracture mechanics and is valid when the size of the contact zone is small compared with the contact radius (i.e., SSY). The problem is that the amount of energy flow to the crack tip depends on the viscoelastic energy dissipation, so that eq 1 is meaningless (in the sense that a unique modulus E* cannot be defined) when one or both of the materials in contact are viscoelastic, even if the SSY condition is satisfied. A simple way to see that eq 1 is inapplicable for contact between viscoelastic materials is to consider a paradox that arises in viscoelastic fracture mechanics. An excellent summary of this paradox and its solution is given by Rice.28 The problem is how to express the energy flow to the crack tip G for a viscoelastic material in terms of the applied loading (represented by KI) in a method analogous to (1). The answer (given by ref 28) if the crack tip is assumed to be perfectly sharp (or the size of the cohesive zone is identically zero) is
G ) KI2/2E0*
(2)
where E0* is an effective instantaneous modulus of the material defined by (8) below and analogous to E* above. Equation 2 can be understood via the following argument: For a continuously growing crack, material elements near the crack tip experience an infinite rate of loading due to the approach of the singular stress field. Because of this infinite loading rate, the material must respond in a manner that depends only on its instantaneous elastic properties, and hence E0* is appropriate in (2). The crack growth criterion G ) W would therefore imply a complete lack of crack speed dependence in the propagation of the crack (assuming that W is rate independent). (24) Ting, T. C. J. Appl. Mech. 1968, 35, 248. (25) Lee, E. H.; Radok, J. R. M. J. Appl. Mech. 1960, 27, 438. (26) Graham, G. A. C. Int. J. Eng. Sci. 1967, 5, 495. (27) Yang, W. H. J. Appl. Mech. 1968, 35, 379. (28) Rice, J. R. J. Appl. Mech. 1968, 35, 379.
This paradoxical prediction of a complete lack of crack speed dependence of the growth criterion is physically unrealistic. Imagine the loading history is such that the crack is moving so slowly that the material is completely relaxed, and hence the system responds with the long time modulus E∞*. In this limit one anticipates that the energy release rate G should be given by
G ) KI2/2E∞*
(3)
but the previous argument would imply that (2) is correct as long as the crack is moving. This paradox is wellknown in viscoelastic fracture and is resolved by the work of Schapery20 and Knauss.21 The resolution comes from the fact that the cohesive zone region where the intermolecular forces act cannot be exactly zero. This means that the stress must be finite everywhere, so that the argument that the material elements near the crack tip must respond in a manner depending only on their instantaneous elastic properties is not always true. Because of the finite stresses, the rate of energy flow to the crack tip must be dependent on crack speed so that the growth criterion is rate dependent. When this reasoning is applied to the contact of viscoelastic materials, it shows that a cohesive zone theory analogous to that of Maugis12 must be used. The above discussion indicates that the rate of energy flow to the crack tip G is coupled to the details of the intermolecular forces acting in the cohesive zone. In this paper we discuss the problems associated with erroneously attempting to model viscoelastic contact without taking these forces into account and then apply a cohesive zone model due to Schapery29 to the loading phase of a displacement controlled viscoelastic contact test. Numerical simulation results from this model are then presented and compared with experimental results from Falsafi et al.30 2. Notation In the following discussion of contact mechanics, we shall assume that both bodies are spherical so that the contact area is a circle of radius a. Materials 1 and 2 have radii R1 and R2, respectively, and the effective radius R is defined by
1 1 1 ≡ + R R1 R2
(4)
As in the JKR theory, R is assumed to be much greater than a, so that the deformation of each sphere is well approximated by the deformation of a half-space. Elastic materials can be described by bulk moduli (k1, k2) and shear moduli (G1, G2), respectively. A material property that appears often in elastic contact mechanics is
κ≡
1 - ν2 3κ + 4G 1 ) 6κ + 2G 2G E
(5)
where E and ν are Young’s modulus and Poisson’s ratio, respectively. E* in (1) is defined by 2
2
1 - ν1 1 - ν2 1 ≡ + E* E1 E2
(6)
(29) Schapery, R. A. Int. J. Fracture 1989, 39, 163. (30) Falsafi, A.; Deprez, P.; Bates, F. S.; Tirrell, M J. Rheol. 1997, 41, 1349.
6572 Langmuir, Vol. 14, No. 22, 1998
Hui et al.
where F is given by
F ) x(x - x′)2 + (y - y′)2,
(11)
and the operator * is defined by
C1*p ≡
Figure 1. The geometry of contacting spheres before and after load is applied. The dimensions are such that a , R1, R2. δ is the load point displacement.
For contact between two isotropic viscoelastic materials, the elastic compliance in (5) should be replaced by the creep compliance functions C1(t) and C2(t) for materials 1 and 2. Here we assume that the behavior of each material can be well described by linear viscoelasticity under the loads that are encountered in an experiment. The loads are also assumed to cause small strains and displacements, and in particular excessive bulk flow is excluded so that R remains constant. We define
C(t) ≡ C1(t) + C2(t)
(7)
w(r, t) )
E10
+
1 - (ν20)2 0
E2
≡
1 (8) E0*
1C1*p dA Amax π F
∫∫
(13)
where Amax is the maximum area of contact that has been reached. For monotonically nondecreasing contact the maximum area is always equal to the current area, so
Amax ) A(t)
(14)
The surface displacement of material 2 is obtained by replacing C1 in (10) by C2. The kinematic condition for contact is 2
r dA ) δ(t) ∫ ∫A(t) π1 C*p F 2R
0 2
1 - (ν1 )
(12)
We have assumed that the bodies are undisturbed when t < 0, and the 0- in (12) takes the possibility of sudden step loading into account. The total surface deflection w at a radial distance r from the center of contact due to the contact pressure p(r,t) is, by superposition
The short- and long-term limits correspond to the instantaneous and long time elastic responses so that
C0 ≡ C(0) ) C1(0) + C2(0) )
dτ ∫0-t C1(t - τ) ∂p ∂τ
(15)
where δ is the displacement at the remote point where the load is applied and with C given by (7). It can be shown that the solution of (15) must be of the form
C*p(r, t) ) [b0(t)(a2(t) - r2)1/2 - b1(t)(a2(t) - r2)-1/2] × H(a(t) - r) (16)
and
C∞ ≡ C(∞) ) C1(∞) + C2(∞) )
1 - (ν1∞)2 E1∞
+
1 - (ν2∞)2 ∞
E2 ∞,
where a is the contact radius, which has been assumed to be much less than R, and H(t) is the Heaviside step function defined by
≡
1 (9) E∞*
∞
The constants ν1 E1 ν1 and E1 denote the short- and long-time Poisson’s ratios and Young’s moduli of material 1. Since most elastomers are nearly incompressible, C1(t) and C2(t) are proportional to the extensional creep compliance, with the constant of proportionality being 1 - ν2 ) 3/4. In general, C1(t) and C2(t) are the creep compliances in a plane strain tension test. 0,
0,
3. Mechanics of Contact of Viscoelastic Spheres In this section we formulate the problem of viscoelastic contact without considering the cohesive zone, and we discuss the implications of the results. The derivation below extends the argument of Yang27 in his discussion of Hertzian contact of viscoelastic bodies. The geometry of the contacting bodies being considered is shown in Figure 1. The surface deflection dw due to a force p dA applied at a point (x,y) on the surface z ) 0 of a viscoelastic half space (material 1) is
1 C1*p dA dw ) π F
(10)
H(t) )
{
0, t < 0 1, t > 0
(17)
b0 and b1 are determined by substituting (16) into the left-hand side of (17) and equating the resulting integral to the right-hand side. This procedure results in
b0(t) ) b0 )
2 πR
(18)
and
δ(t) )
a2(t) π b0[a(t)]2 - πb1(t) ) - πb1(t) 2 R
(19)
Note that b0 is independent of time. Equation (16) implies that
p(r, t) ) C-1 * {[b0(a2(t) - r2)1/2 - b1(t)(a2(t) r2)-1/2] H(a(t) - r)} (20) where C-1 is the inverse operator to C and is defined by
C*φ ) ψ S φ ) C-1*ψ
(21)
The determination of C-1 in terms of C can be obtained using the Laplace transform. Indeed, it is well-known
Contact to Viscoelastic Spheres
Langmuir, Vol. 14, No. 22, 1998 6573
that
s2L(C-1)L(C) ) 1
(22)
where L is the Laplace transform of a function and s is the transform variable. For the special case where the two bodies are the same material, C-1 is equal to two-thirds of the relaxation modulus of the material under uniaxial extension. The total force F(t) is found by integrating (16) over the contact region, resulting in
Given the loading history F(t), the unknowns are δ(t), a(t), and b1(t). However, there are only two equations, (19), and (23), so that one more equation is needed. In other words, if we assume KI(t) is known, then (19), (23), and (28) imply that
a3(t) )
[
2 3R 3Rπ(C0KI) + (C*F) + 4 2
C0KI
4a3(t) - 2πb1(t)a(t) C*F ) 3R
(23)
∫∫A(t) C
-1
KI )
∫∫A(t)ψ dA
*ψ dA ) C *
It is worthwhile to compare (23) with its counterpart in the JKR theory of elastic contact, which is
4a3 F ) - 2πd1a E* 3R
(24)
where E* is given by (6) and d1 is an unknown constant. In the elastic JKR theory d1 is independent of the loading history (whereas b1 in (23) is history dependent) and is determined by the equilibrium crack propagation criterion
G)W
(25)
where W is the work of adhesion between the two surfaces. This condition results in
d1 )
KIxa
(26)
xπE*
The problem with extending (26) to viscoelasticity is that G cannot be simply expressed in terms of the local stress field alone as in (1). The micromechanics of the local bonding or failure process must be taken into account, as discussed in the Introduction. Note that (23) is the equation that couples the global response of the viscoelastic material (through the term C*F) to the local micromechanics, since the force history is related to b1, and b1 will provide the boundary conditions to the local micromechanics problem. From (20) we can determine the behavior of the pressure at points just inside the contact zone. We find that as r f a-
p f C-1*
[
-b1(t)
x2a(t) x(a(t) - r)
]
-C0-1b1(t)
x2a(t) x(a(t) - r)
≡
-KI
x2π(a(t) - r)
(27)
where KI is the mode I stress intensity factor. Equation (27) implies that the pressure distribution has an inverse square root singularity as the edge of the contact zone is approached. The minus sign indicates that the stress is tensile. Furthermore, KI(t) is related to b1(t) by
KI(t) )
C0-1b1(t)
xa(t)/π
(28)
]
+ 3πR(C*F) (29)
4a3 - 3R(C*F)
(30)
6xπRC0a3/2
In the linear elastic case, the extra equation comes from (1). The familiar JKR equation can then be obtained by setting C(t) ) 1/E* in (29), yielding
E*a3 )
3R [F + 3πRW + x9π2R2W2 + 6πRWF] (31) 4
The question is, what is the corresponding additional equation in the viscoelastic case? Consider the crack propagation criterion KI(t) ) Kc, where Kc is assumed to be a material constant. Note that if the material is linear elastic so that E* is independent of time, this criterion is equivalent to (1) since Kc ) (2E*W)1/2. However, for a viscoelastic material the rate of energy flow to the crack tip is not given by (1) as discussed above. Although the rate-independent, stressbased condition KI(t) ) Kc may not be at all realistic, it is worthwhile to see how this crack propagation criterion affects the analysis. Equation (29), together with KI(t )) Kc, implies that
a3(t) )
[
2 3R 3Rπ(KcC0) + (C*F) + 4 2
KcC0
x[
]
3πR KC 2 c 0
2
]
+ 3πR(C*F) (32)
and
δ(t) )
a2(t) - KcC0 xπa(t) R
(33)
Recently, Falsafi et al.30 proposed the following viscoelastic extension of the JKR equation:
C-1*a3 )
)
]
2
or equivalently
where we have use of the fact that -1
x[
3πR C0KI 2
3R [F + 3πRW + x9π2R2W2 + 6πRWF] 4
(34)
This equation, presumably obtained from the linear elastic case (31) by replacing E* by the operator C-1, is incorrect as the “correspondence principle” has been incorrectly applied. From our analysis it is clear that the operator C cannot be factored from the right-hand side of (32). Note that both (32) and (34) imply that the energy release rate G is completely determined by the loading and geometry parameters and the work of adhesion, without consideration of the details of the adhesive process. For example, the condition KI(t) ) Kc implies a lack of crack speed dependence in the propagation criterion and leads to the paradox that was discussed in the Introduction.
6574 Langmuir, Vol. 14, No. 22, 1998
Hui et al.
Figure 3. A continuum description of the cohesive surface forces that act near the edge of contact. σ represents the normal stress acting on the material inside the cohesive zone, and ∆ is the air gap outside of the contact region. The Dugdale model is shown in (a) while (b) represents a physically realistic model.
Figure 2. A schematic diagram of the JKR experiment showing (a) the outer problem of contact mechanics and (b) the inner problem of the local adhesive bonding. L is the size of the cohesive zone and is assumed to be much less than a. In (a) the contact region grows under an applied force F, whereas in (b) bonding occurs at the edge of contact in response to the boundary condition of the local stress field. In the case of the Dugdale model the cohesive stress is constant and is denoted by σ0. For viscoelastic materials the two problems are intimately coupled, so that knowledge of the bonding process in (b) is necessary to calculate the contact area in (a).
4. Connection between Contact Mechanics and Adhesion The example at the end of the previous section illustrates the error of decoupling the local micromechanics of adhesion from the mechanics of contact. As long as the contact radius is nondecreasing, the contact mechanics of the viscoelastic bodies are completely described by (29). What we have shown is that the additional equation containing the criterion for crack propagation must be based on the local adhesive mechanism. This can be done using the SSY assumption, as illustrated in Figure 2. Essentially, the problem is broken into an outer problem of viscoelastic contact mechanics (Figure 2a) and an inner problem of adhesion mechanics (Figure 2b). The outer problem is governed by (19), (23), and (28), which couples the outer problem to the inner one. In the inner problem the crack is assumed to be infinite compared to the size of the cohesive zone, so that the condition for crack propagation will come from the solution to the problem of the healing of a semi-infinite cohesive zone crack in a viscoelastic material. Equations (34) and (32) can both be thought of as being incorrect because of an incorrect solution to this inner problem. During the loading phase of a JKR experiment the contact area is nondecreasing, which corresponds to the healing of the crack faces outside the contact zone. This problem differs from the case of unloading, which corresponds to crack growth as pointed out by Greenwood and Johnson.31 The problem of crack healing in viscoelastic fracture is a difficult one and has been considered by Schapery,29 who like Maugis used the Dugdale model16 to represent the cohesive forces that act across the crack (31) Greenwood, J. A.; Johnson, K. L. Philos. Mag., A 1981, 43, 697.
faces. In this model, which is illustrated in Figure 3a, the adhesive stress σ0 is independent of the crack opening (or air gap) ∆ until a critical gap ∆c is reached. When ∆ ) ∆c the interface fails, as it can no longer support tensile load. The Dugdale model is a mathematically simple approximation to a realistic force-displacement law such as that shown in Figure 3b. Studies of cohesive zone laws (e.g., ref 32) indicate that the important parameters are the total work of adhesion and the maximum allowable stress, and it is expected that with the correct parameters the Dugdale model can approximate a realistic force law reasonably well. Additionally Rice28 noted that the work of adhesion for any rate-independent cohesive zone model is the area under the σ-∆ curve of Figure 3, so that for the Dugdale model we have
W ) σ0∆c
(35)
In this section we use Schapery’s solution to the aforementioned inner problem to study the loading phase of the JKR test for viscoelastic materials. For a crack receding (contact area growing) at a rate a˘ which is slow relative to the sound velocity of the materials, the relationship between the crack tip stress field and a˘ is given by29
W ) σ0∆c )
4 2 σ L [C0 + C1cmγm (L/a˘ )m] π 0
(36)
where L is the length of the cohesive zone and the compliance function is assumed to be of the form
C(t) ) C0 + C1tm,
0 < m
