Improved Muller approximate solution of the pull-off of a sphere from a viscoelastic substrate
aa r X i v : . [ c ond - m a t . m t r l - s c i ] D ec Improved Muller approximate solution of the pull-off ofa sphere from a viscoelastic substrate
M. Ciavarella
Politecnico di BARI. DMMM department. Viale Gentile 182, 70126 [email protected]
Abstract
The detachment of a sphere from a viscoelastic substrate is clearly afundamental problem. In the case viscoelastic dissipation is concentrated atthe contact edge, and the work of adhesion follows a quite popular simplifiedmodel, Muller has suggested an approximate solution, which however is basedon an empirical observation. We revisit Muller’s solution and show it leadsto very poor fitting of the actual full numerical results, particularly for theradius of contact at pull-off, and we suggest an improved fitting of the pull-offwhich works extremely well over a very wide range of withdrawing speeds,and correctly converges to the JKR value at very low speeds.
Keywords:
Viscoelasticity, Adhesion, JKR theory, soft matter
1. Introduction
The problem of viscoelastic dissipation during crack growth or contactpeeling has attracted much interest due to its fundamental importance inmany areas of science and technology. Many authors have applied fracturemechanics concepts and made extensive measurements (Gent and Schultz,1972, Barquins and Maugis 1981, Gent, 1996, Gent & Petrich 1969, Andrews& Kinloch, 1974, Barber et al , 1989, Greenwood & Johnson, 1981, Maugis& Barquins, 1980, Persson & Brener, 2005) postulating peeling involves aneffective work of adhesion w as the product of adiabatic value w and afunction of velocity of peeling of the contact/crack line and temperature,namely w = w [1 + k ( a T v p ) n ] (1) Preprint submitted to journal December 14, 2020 here k, n are constants of the material, with n in the range 0 . − . a T is the WLF factor (Williams, Landel & Ferry, 1955) which permits totranslate results at various temperatures T from measurement at a certainstandard temperature. The details of the derivation from crack models in-volving cohesive Barenblatt zones or models ”truncating” or ”blunting” cracktip dissipation (Barber Donley and Langer 1989, Greenwood and Johnson,1981, Persson & Brener, 2005) vary, but the form (1) remains the most pop-ular simple choice, and therefore a baseline for comprehension of possiblemechanics of contact and crack problems.In the case of adhesive contact of the fundamental spherical geometry,various authors (Barquins & Maugis, 1981, Greenwood & Johnson, 1981,Muller, 1999) have attempted to apply the fracture mechanics formulationwith the model (1), and some approximate results have been given in terms ofexplicit dependences of the pull off force or work, contact radius and approachat pull-off see ref. (Muller, 1999), which we shall revisit here in comparisonwith full numerical simulation, finding very significant discrepancies, andsuggesting some improved fitting of the numerical results, at least for thepull-off force which is the quantity of greater interest.
2. Spherical contact mechanics theory
The fracture mechanics formulation for the adhesive contact problem for asphere is classic, and we shall revisit here only the essentials. We consider thestress intensity factor at the contact edge is due to the difference between P ,the load required to maintain a contact radius a in the absence of adhesion P ( a ) = 43 E ∗ R a (2)where E ∗ = E/ (1 − ν ) is the plane strain elastic modulus ( E being Young’smodulus and ν Poisson’s ratio) and P , the smaller load to maintain thesame contact radius in the presence of adhesion. So we find the strain energyrelease rate as G ( a, P ) = K ( a, P ) E ∗ = ( P ( a ) − P ) πE ∗ a (3) The factor 2 which is missing in Muller (1999) comes from the fact that strain energyexists only in one material, assuming the other is rigid. For two identical materials, E ∗ = E ∗ and we return to the standard LEFM case with G ( a ) = K ( a ) E ∗ .
2n the adhesionless conditions, the remote approach is α ( a ) = a R , so inthe adhesive condition we have to decrease this by an amount given by a flatpunch displacement ∆ α = P − P E ∗ a (since in moving from the adhesionless tothe adhesive solution we keep the contact area constant) giving the generalresult for approach α ( a, P ) = a R − P ( a ) − P E ∗ a (4)from which we can obtain P ( a, α ) using (2) P ( a, α ) = P ( a ) + 2 E ∗ aα ( a, P ) − E ∗ a R = 2 E ∗ aR (cid:18) Rα ( a, P ) − a (cid:19) (5)which corresponds to Muller (1999) equation 10, whereas using (3) G ( a ) = ( P ( a ) − P ) πE ∗ a = E ∗ πaR (cid:0) Rα ( a ) − a (cid:1) (6)which corresponds to Muller (1999) equation 15 except for a factor 2 misprint.For the elastic case, JKR (Johnson, Kendall & Roberts, 1971) theory isobtained by using (6) and (4) P = 43 E ∗ R a − p πw E ∗ a (7)Putting ζ = (cid:16) πw RE ∗ (cid:17) / (8)we have at P = 0 from (7) and (5) a = (cid:18) πR w E ∗ (cid:19) / = 3 Rζ (9) α = a R = 3 Rζ (10)where there is a factor 3 misprint in Muller (1999) equation 19.3 . Viscoelasticity Now, for a viscoelastic material, the material dissipation at the cracktip/contact edge requires that energy balance imposes the velocity of crackaccording to (1). Further, we can write the velocity of the contact edge as v p = − dadt = v dadα (11)where v is the remote pull-off rate imposed by the loading equipment. Thecondition G ( a ) = w therefore defines a differential equation for a = a ( α )obtained using (6, 11)1 k /n a T v (cid:20) E ∗ πaR w (cid:0) Rα ( a ) − a (cid:1) − (cid:21) /n = dadα (12)By using we the JKR values at zero load (9,10) and the JKR values forpull-off for P = πRw , and finally the adiabatic work of adhesion for G ,we obtain the dimensionless variables G ′ = Gw ; P ′ = PP ; a ′ = aa ; α ′ = αα (13)If we now remove the (’) for simplicity in the following equation s, werewrite (12) as dadα = β − (cid:20) a (cid:16) α a − (cid:17) − (cid:21) /n (14)where we have introduced the only dimensionless factor in the problem, apartfrom n , namely β = (cid:18) RE ∗ πw (cid:19) / (cid:18) k (cid:19) /n a T v (15)The latter two equation s correspond to Muller (1999) equation 24,23.The differential equation (14) can be solved for initial conditions startingfrom a point on the loading curve , which is the JKR curve which in thisdimensionless notation and in parametric form is P ( a ) = 4 (cid:0) a − a / (cid:1) (16) Strictly speaking, during loading adhesion is reduced with respect to the adiabaticvalue at zero speed, but we neglect this effect, or else we consider that loading occurs nearthermodynamic equilibrium. α ( a ) = 3 a − a / (17)After a ( α ) is obtained, we can compute the load which in dimensionlessform is obtained from P ( a, α ) = 2 a (cid:0) α − a (cid:1) (18)Notice that the strain energy release rate in dimensionless form is G = 94 a (cid:16) α a − (cid:17) (19) Muller (1999) in searching for the pull-off as the minimum of the P ( α )curve, postulates that this is close to the minimum of P ( α ) + G ( α ) which isalso 0 in the minimum. There is no fundamental reason for this mix of thedimensionless load with the dimensionless strain energy release rate to haveany special property, and indeed we found the two minima are not necessarilyvery close. Muller’s postulate anyway leads to radius of contact, approachand load at pull-off, a m = κβ q (20) α m = − κ β q (21) P m = | P min | = 4 κ β q (22)where q = n/ ( n + 3) and κ = (cid:16) / n (cid:17) / ( n +3) . Notice obviously that this resultat zero velocity would give incorrect results as all values go to zero, ratherthan the asymptotic values of JKR theory for thermodynamic equilibrium.Remark that the actual velocity of the crack line (recall a and α aredimensionless here, not to be confused with equation 11) v p v = 1 ζ (cid:18) dadα (cid:19) m = 1 ζ a m (23)and given a m ∼ ζ <<
1, it is clear that v p v >>
4. Numerical results and fittings
Here we report some results of the numerical solution of the differentialequation , comparison with Muller’s approximate solution, and some im-proved fitting results for the pull-off, which is (perhaps) the most importantquantity.From Fig.1 we see the withdrawing curves for an example case of low n = 0 .
25, and (b) an example showing that initial conditions seem to veryweakly affect the actual pull-off, as Muller had remarked. From Fig.2 wesee that the contact radius at pull-off is very poorly predicted by Muller’sapproximate solution (20), and it is much more weakly dependent on β . Inparticular, at high β , Muller’s solution predicts very large a m which do notmake much sense. Indeed, as we have seen there is not much dependenceon the initial condition, we expect a m < a i = 1. An exception, where we see a m > P i = 5 of fig.2a,c). At low β ,Muller’s prediction underestimates the radius at pull-off, particularly at high β . Also not very good predictions, but perhaps better than for contact ra-dius, are those for the approach at pull-off (fig.3). Here, the actual resultstend to be higher than Muller’s prediction (21), at all speeds, and start offwith a value near α m = − . - - - - - - - Α P n = Α i = (a) - - - - - Α P n = (b)Fig.1 - Dimensionless load P dimensionless approach α (a) for various β = 2 × − ∗ i , ( i = 1 ,
10) and for n = 0 . . The inner black curve is theadiabatic JKR curve. (b) very weak dependence of pull-off on initialconditions (initial load P = 0 ,
5) for an example case β = 0 . * * * * * * * * Β a m n = * P i = (a) * * * * * * * * * Β a m n = (b) * * * * * * * * * Β a m n = * P i = (c)Fig.2 - Dimensionless contact radius at pull-off a m for n = 0 .
25 (a) n = 0 . n = 0 .
75 (c) as a function of the dimensionless speed factor β. (initialload in the figure P = 0 or 5) * * * * * * * * * Β Α m n = * P i = (a) ** * * * * * * * * Β Α m n = * P i = (b)8 * ** * * * * * * Β Α m n = * P i = (c)Fig.3 - Dimensionless absolute value of approach at pull-off | α m | for n = 0 .
25 (a) n = 0 . n = 0 .
75 (c) as a function of the dimensionlessspeed factor β. (initial load in the figure P = 0 or 5)Considering these poor performances on a m and α m , the results for thepull-off load vs Muller’s prediction (see Fig.4) are relatively good (blue linevs the markers of the numerical simulations), which is probably why he wassatisfied in his paragraph ”comparison with exact calculation” where he hasonly comparison with pull-off load or work for pull-off, but still we find themonly rough ”estimates”. It is easy to obtain much better fit of the results,considering we have only two independent dimensionless parameters, n and β of course, so we improve Muller’s prediction in two respects:1) we add a crossover towards the JKR value P = 1, by adding ”1” toMuller’s equation (22) the JKR load;2) we improve the power law exponent at large β with a corrective factorto Muller’s equation (22) in the form P m = | P min | = 1 + 4 κ β q/c ( n ) (24)where c ( n ) = 1 . n/ .
65 (25)This improvement shows clearly a much better fit with respect to detailednumerical calculations in the entire range of realistic values for n and of β covering 10 orders of magnitude in β which is probably more than enoughconsidering the other approximations made in the model, namely the form ofthe work of adhesion, that there is no viscoelasticity in the bulk, no thermaleffects, and so on.Notice that Violano and Afferrante (2019) have numerically solved theMuller equation s, and found good correlation with experimental results.9his suggests that our solution would be very valuable for an analyticalfitting of experiments such as those of Violano & Afferrante (2019). ** * * * * * * * ** * * * * * * * * ** * * * * * * * * * Β P m n = * P i = * P i = * P i = ___ proposal ___ Muller (a) ** * * * * * * * ** * * * * * * * * ** * * * * * * * * Β P m n = * P i = * P i = * P i = ___ proposal ___ Muller (b) * * * * * * * * * ** * * * * * * * ** * * * * * * * * * Β P m n = * P i = * P i = * P i = ___ proposal ___ Muller (c)Fig.4 - Absolute value of the dimensionless load at pull off P m for n = 0 . n = 0 . n = 0 .
75 (c) as a function of the dimensionless speed factor β. (initial load as indicated by different colors in the markers in the figure P i = 0 ,
5. Conclusions
We have revisited the Muller approximate solution for the pull-off ofsphere from a flat viscoelastic material, finding significant errors in the ap-proximate solution, which stem from the rather arbitrary assumption thatthe pull-off condition occurs when the sum of a dimensionless load and adimensionless strain energy release rate has a minimum. We have added a”cross-over” towards the JKR solution for very low velocities, and corrected10he power law enhancement of pull-off with velocity of withdrawal. The so-lution can be useful for quick estimates of the effect of viscoelasticity on theincrease of adhesion in spherical geometries.
6. Acknowledgements
MC acknowledges support from the Italian Ministry of Education, Univer-sity and Research (MIUR) under the program ”Departments of Excellence”(L.232/2016).
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