Chapter 3 Surface and Bulk Properties in Adherence of Elastic-Viscoelastic Solids J. C. Charmet, D. Vallet, and M. Barquins Laboratoire de Physique et de Mécanique des Milieux Hétérogènes, (UMR CNRS 7636) ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 05, France
The surface forces, of van der Waals type for rubber-like materials, are able to grandly modify the stress tensor provided by the contact of a blunt asperity applied against the flat and smooth surface of a rubber sample. It will be shown how the coupling of surface adhesion properties and bulk viscoelastic behavior of rubber-like material allows us to solve adherence problems. This will be illustrated through three examples: the spontaneous peeling due to the intervention of internal stresses; the no-rebound of balls on the smooth surface of a soft elastomer and the adhesive contact and rolling of a rigid cylinder under a smooth-surfaced sheet of rubber.
When a driver jams on the brakes he thinks usually that the deceleration of the vehicle results from this action, but the true reason why the vehicle stops is in fact the great force induced by friction at the tire-road interface. It will be shown that this friction force is the direct result of the coupling of surface adhesion properties and bulk viscoelastic behavior of rubber-like materials. It is now well-known that surface properties of solids are able to activate their bulk properties, and, as a consequence, strongly influence the mechanical behavior of the interface area of solids in contact. Our aim is to illustrate this matter through three examples: the no-rebound of balls on the smooth surface of a soft elastomer, the spontaneous peeling due to the intervention of internal stresses and the adhesive contact and rolling of a rigid cylinder under a smooth-surfaced sheet of rubber.
Preliminaries Solving a problem of adherence requires knowledge in thefieldsof contact mechanics, rupture mechanics and physics of surfaces. Therefore, as a prerequisite, the main results needed for a better understanding are described below.
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© 2000 American Chemical Society
43 The friction law. Thefirstworks on friction are those of Leonardo da Vinci (/) at the beginning of the XVI century and Guillaume Amontons (2) in 1669, but the taking into account of the influence of the surface properties of solids in contact is due to Charles Augustin de Coulomb (3). In 1785 he writes the famous relationship, which is known today as Coulomb'sfrictionlaw, connecting thefrictionresistant force Tto the normal load applied N:
In this relation the first term proportional to the applied load N is representative of the solids deformation and the coefficient \x is known today as the friction coefficient. The second term A represents the effect of the surface properties and is of great importance when the applied load N is small. This last term was introduced by Coulomb in order to explain the surprising result of previous experiments conducted by John Theophilus Desaguliers (-0 in 1734. Pressing firmly together two spheres made of lead, he showed that they remained « stuck » and that a non negligible traction load was necessary to separate them. The elastic contact Contact mechanics was initiated by the pioneer work of Heinrich Hertz (5) in 1882. He probably realized that forces induced in the contact area of two solids pressed together are not uniformly distributed. The applied force is transmitted through a few points of contact which can be compared to spherical caps characterized by their radius of curvature. So the real size of the contact area is much lower than its apparent size. For spheres of radii R and R of purely elastic materials, he demonstrates that the radius of the contact area a and the penetration depth 8 are related to the applied load F by: 9
1
and Req=
RR R + R!
where E E* are elastic Young's modules and v, • Poisson's ratios of the constitutive material of each sphere. This well known result is of current use in mechanics, however, it didn't take into account the adhesive properties of the surfaces in contact. In 1885 Joseph Boussinesq (6), trying to extend the validity of these results to the case of axi-symetrical rigid convex punches indenting a flat semi-infinite elastic medium, demonstrates that, without an adequate boundary condition, the size of the contact area is generally unknown. In order to overcome this difficulty, he imposes that normal stresses vanish on the border of the contact area. In other words, the profile of the distorted medium must be tangent to the surface of the punch on the border of the contact area. Note that this condition is the same as the condition presupposed by the Hertz's theory. With this assumption, the size OH of the contact area and the penetration depth 8H are completely defined (Figure 1). An observed penetration depth 8*8H rewritten as 8=8H+(8-8H) corresponds to the addition of a vertical displacement 8-8H at constant radius of contact an, that is to say to the addition of the vertical movement of a flat rigid axi-symetrical punch inducing a strong divergence of the normal stresses on the border of the contact area. 9
44
H J
8
^
^
V
^
^
Figure 1. Comparison of distorted surface profiles of the pure elastic contact (Hertz) and of the adhesive one between a spherical punch and a flat plane and smooth surface of an highly elastic substrate for the same radius an of the contact area.
45 With a great insight, Boussinesq noticed that this singularity exists only if the solids in contact exhibit adhesion. This proposal is now interpreted in the opposite way: adhesion of solids create a singularity in the stress distribution near the border of the contact area. Griffith's theory of brittle fracture. The connection between bulk and surface properties of solids appears for the first time in the work of Griffith (7). In 1920 he writes his famous condition of crack propagation: the work 2ydA consumed by the creation of a new surface of crack dA is deducted from the elastic energy stored in the solid by the stresses or strains applied to it. So, in addition to the Young's modulus E and the Poisson ratio v characterizing the behavior of intermolecular forces when the atoms are slightly displaced from their equilibrium position, we must introduce the surface energy y which gives the work necessary to break bonds per unit area along an imaginary plan and to reversibly separate the two parts of the solid of isothermal manner. This surface energy y is not different from the bond energy insuring the cohesion of solids. Dimensionally y is the product of the Young's elastic modulus E by an interatomic spacing. In order to establish the equilibrium condition of a crack, we consider a thermodynamically isolated system of constant total energy U composed by a solid containing a crack of surfaced and its loading apparatus. Its energy is the sum of three terms: U^UP+UE+US where Up is the potential of the applied forces, UE the elastic energy stored and U the surface energy associated to the crack. With W? =-C/p, work of the applied forces, and UUF-WV+UE the mechanical energy of the system, the equilibrium condition dt/=0 of the crack is given by application of the principle of virtual work: S
dA ~ dA
dA
7
G,firstletter of Griffith name, is the « strain energy release rate ». G is only a function of the solid and crack geometry and of the loading conditions. It can be calculated by solving the elastic linear equations of the problem. At the oppositely is a material property of the solid. As usual, the equilibrium stability is given by the sign of the second derivative of the total energy: G2y crack closing open crack equilibrium crack propagation at velocity V ~ ~ 0 spontaneous G -2y is the motive energy of crack extension. In the Griffith model (Figure 2), the surface energy vanishes on the crack and rises steeply to the value 2y at the crack tip, inducing a singularity of the stress t
46
a
interaction energy
p
b
S
Figure 2. Interaction energy, opening displacement 5 and stress distribution o between the two sides of a crack as a function of the distance p from the crack tip. In Griffith's model, there is no interaction between the two sides of the crack and the surface energy w=2y appears as a step function (Figure a). The shape of the crack tip is parabolic (Figure b, dotted line) and the stress distribution, null inward the crack, is singular outward (Figure c, dotted line) varying as Kl^lxp. In Barenblatt's model, the cohesive forces act on a finite but short distance inward the crack tip and the surface energy, null on the most part of the crack, increases progressively over this distance, reaching the value w at crack tip (Figure a). The sides of the crack are elastically distorted, giving the shape of the crack tip (Figure b, heavy line). There is no singularity of stress at the crack tip (Figure c, heavy line) and the maximum stress is equal to the theoretical stress Oth deducedfromthe strength of the interatomic bond.
47 distribution varying as
,
where r is the distance to the crack tip. This non-
physical singularity is characterized by the factor K named « stress intensity factor ». As Irwin (8) showed, in plane strain opening mode of the crack K and G are linked by E the relation ^ G=K. This relation emphasizes the equivalence of the global energetic approach (rate G) and the local analysis of stress distribution (factor K). In the sixties Barenblatt (9) and Dugdale (10) were interested in the problem of the distribution of cohesive forces in the crack tip. The Griffith equation of crack equilibrium G=2y don't allow us to understand the role played by the attractive molecular forces. In 1962 Barenblatt demonstrated the existence of cohesive forces acting on a short length of the crack end. In brittle solids all the resistant energy of the crack is concentrated in this small zone. Analyzed in term of stresses, this energy distribution is associated to a singular stress distribution of stress intensity factor K characterizing the material. Without external loading, these cohesive forces tend to close the crack. At equilibrium, this factor exactly compensates the factor K produced by the external loading, the stresses takefinitevalues and the two surfaces of the crack join tangentially at the crack tip as physically expected (Figure 2). The stresses are recognized elastic outwardfromthe crack tip and cohesive inward. 7
m
Adhesion and rupture. We now interpret adhesion phenomena from the point of view of the rupture mechanics. When dissimilar bodies are in intimate contact, attractive forces exist also and, trying to separate them, an energy equal to wdA must be supplied to achieve their separation on an area of size dA. This energy is of the same nature as the surface energy 2ydA, however, we must take into account the elastic energy stored in the interface during the initial stage when the two solids was sticked together. So M/=yi+y2-Yi2, called the Dupr6 adhesion energy, is the sum of the surface energies yi and of each solids reduced by the interfacial energy recovered Y12.
The subject of adhesive contact mechanics may be said to have started when Kendall (11), solving the problem of the adhesive contact of arigidflatcylinder punch indenting the smooth plane surface of an elastic medium, demonstrated that the border of the contact area can be considered as a crack tip. The more complex problem of a spherical punch was solved in 1971 by Johnson, Kendall and Roberts (12). The JKR theory predicts the existence of contact area greater then that given by the elastic contact Hertz's theory. The molecular attractive forces are responsible for this increase and, even in the absence of external compressive loading, the contact area has afinitesize. Separating the two solids requires the application of an adherence force despite the existence of infinite normal stresses in the border of the contact area. The energy balance of Griffith's theory cannot give information about the stability of the system that depends upon the second derivative of the total energy. That is the reason why Maugis and Barquins (13) were led to reintroduce the concepts of fracture mechanics such as the strain energy release rate G in order to study the stability according to the sign of the derivative of G. This approach has the advantage of enabling one to study the kinetics of crack propagation and to predict the evolution of the system whatever the geometry of contact and loading conditions. Therefore,
48 adding knowledge of contact mechanics, rupture mechanics and physics of surface is a necessity to obtain pertinent answers to solid adherence problems. As adhesive forces are generally weak, the strains induced remain very small for most of the currently used materials. For this reason most of the experiments have been conducted on elastomers because the surface forces, of van der Waals type for these rubber-like materials, are able to greatly increase the stress tensor provided by the contact of a blunt asperity applied against the flat and smooth surface of rubber samples. As a consequence they have the great advantage of giving large visible deformations. The viscoelastic losses. The equilibrium G=w may be disrupted by a change the applied load or displacement. As soon as G>w the crack supports a force G-w per unit length. If the solid is purely elastic, the crack accelerates at fixed G-w until its propagation speed reaches the Rayleigh velocity of the surface waves. For real bodies with internal losses the crack takes instantaneously a velocity such that the losses balance exactly the motive energy G-w, further evolution of the crack speed V depends upon the variation of G-w with the length of the crack. In adhesive contacts, when G>w, the two bodies begin to separate and their separation can be seen as the propagation of a crack in the opening mode, the contact area decreasing as the crack grows. The modified energy balance can be written as G-w=W(V) where *F is the dissipation function associated to the internal losses (14). For rubber-like materials, the viscoelastic losses vary with the strain rate and the temperature. The principle of «time-temperature » equivalence proposed in 1955 by Williams et al. (75) allows us to superimpose the experimental curves obtained at different temperatures through the known translation factor a of the WLF transformation. As a consequence, at fixed geometry, the adherence forces provoking crack extensions at different speeds V can be studied as a function of the reduced parameter ajV. As suggested by peeling experiments (16), its is usually assumed that viscoelastic losses are proportional to w and are only localized at the crack tip. As a consequence, Maugis and Barquins (73) have proposed in 1978 to rewrite the dissipation function *¥(V) as: T
y
¥(Vy=w^(a V) 1
where the second term is the viscous drag resultingfromthe losses limited to the crack tip. \) With these conditions, this relation which can be represented by a master curve q^^KQ), always provides the critical point, under static and in rolling conditions, with the help of the double equality q^Q^l. The corresponding critical values b and P being these characteristic of the static case multiplied by a whole power of the rolling parameter u (with u£l): AR=6 W and PK=P U , this last relation clearly showing that the rolling of a cylinder, whose weight per unit axial length is greater than the ultimate static value, can be observed. This result can seem obvious a posteriori because the contact area broken at the trailing edge during the rolling motion is rebuilt at the leading edge. Nevertheless, we think that it was useful to demonstrate this effect. 2
C
4
C
Rolling on and under an inclined surface. Experiments were effected with the help of simple apparatus (37). A thick (16mm) block of a soft and transparent natural rubber (Young's modulus £=0.89 MPa and Poisson's ratio v=0.5) with a flat, smooth and clean surface, is glued under an inclined rigid plate made of PMMA, with a slope angle 0 variablefrom0 to 90° with regard to the horizontal. A cylinder (radius R = 30mm and length L = 40mm), also made of PMMA, presenting a lateral perfectly smooth surface, is equipped with a ball-bearing system withoutfrictionwhere different loads can befixedin order to change the load per unit axial length P of the cylinder for the same contact geometry. All the experiments, in which dand P could be changed as required, were recorded with the help of a video system in order to measure more accurately, the rolling speed V and the associated width 2b of the contact area. Taking into account that the rolling of a rigid cylinder on an elastic half-space can be seen, with regard to the dissipated energy, as a n/2 peel experiment at the trailing edge of the contact area (38) the strain energy release rate G is equal to the tangential rolling force F per unit axial length, i.e. the linear weight P applied to the cylinder with a correction due to the inclination Oof the surface: G^Psinft As expected, G varies as a power function of Fin a large range of velocities. G=kV" with /?=0.55, which is in perfect agreement with previous kinetics of adherence of punches, rolling and rebound results obtained with the same rubber-like material (22,24,32,39,40). It should be pointed out that, as already observed, rolling upon and under the rubber surface occurs with the same speed for the same inclination, as in the case where the load per unit axial length of the cylinder is smaller than the absolute value of the elastic adherence force Pc. The most important result that should be emphasized is the magnitude of observed rolling parameters. For instance, the cylinder is able to roll under the inclined rubber surface without falling off, even if the tensile force is 50 times greater than the critical load in static conditions (elastic adherence force P This situation corresponds to a very large interfacial rupture energy of Gnu«=130 J.m* with an associated rolling speed ^ =33cm" , which is the maximum value that can be recorded with our c
2
1
nax
63 experimental system. For comparison, the equilibrium contact upon and under the horizontal rubber surface (F=0). corresponds to G=w » 50 mJ.m" for common environmental conditions in laboratory rooms (ambient temperature T=22 °C and relative humidity RH = 60%). 2
Width of the rolling contact area. The rolling speed depends on the mass per unit axial length M and or the inclination 0 of the rubber surface with regard to the horizontal, this is the same for the contact width 2b. From the following relations: P=Mgcosd and F==Mgsin#, g being the intensity of the gravity, the study of variations of the parameter Q, as a function of M and V allows us to predict the theoretical contact width of the rolling contact and to compare it with experimental data deduced from video recordings. In this case, Q=-P/(3Psu ) can be written: 9
A
3
QHYcos #{4-
4
x
3
sin & +1 } with XrMglw and £ =256W/(27TI£*#)
Two main particular cases exist: in the first, 0=0, i.e. %cos0 =0, that corresponds to Mg=0 or (Hi/2, when rolling occurs vertically (P=0); in the second, Q*Q % (necessarily greater than one) remains proportional to Q sin 6y(4 cos 0. In this case, Q can be written in the simple form: Q=^^sin^/tan^ so that the contact half-width * is equal to: b=b }[X&DWF4 &(Q). 3
2
3
3
9
s
3
If 0=7i/2. b is always proportional to F" , so that it is expected that the contact area width will vary with the w/3=0.55/3=0.183 power function of the rolling speed V. The value of &(Q) being limited by ^ ( O ) ^ and ^(1)=1, the widths of contact area for inclinations different from O = must verify the proportionality to as soon as the rolling speed V is high enough in order that variations of 3?(Q) may be neglected before those of V**. Figure 10 clearly shows that experimental data (symbols) are in very good agreement with computed curves drawn for the three inclinations tested: #=TC/2, £=22.75° and £=50.5°. The particular case of rolling down a vertical rubber surface ((hn/2 and P=0), as the rolling speed is necessarily high, hence F / w » l , leads to the very simple result dna=b/2. Taking into account that the width of the contact area is divided into two unequal parts a=(6-d)/2 and $=(b+d)/2 (Figure 9), the two particular values can be deduced: a^rblA and P*/2=36/4, which correspond exactly to the observations and 273
2
allow us to write bnj2-\l%R F /frE *, so that the rolling force H/2
#=7t/2, rigorously varies as P , and the variation of bna with
for the inclination is found.
Conclusion Due to the intervention of molecular attraction forces, the interface between a rigid solid and a rubber-like material can sustain stresses, so that bulk properties are involved. For this reason, shear delaminations are not instantaneous and kinetics are measured, also phenomena as no-rebound of balls on a rubber surface and rolling of a cylinder under a rubber sheet are observed.
64
10
2b (mm)
V (mm/s)
M 0.01
0.1
10
• I ......I 100
Figure 10. Width 2b of the rolling contact area versus the rolling speed V. The width 2b varies as F ° in a large range of rolling velocities. Experimental data (symbols: x (9=90°); o (0=50.5°); A (0=22.75°) and theoretical curves (heavy line ($=90°). 183
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