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Adhesion of an axisymmetric elastic body: Ranges of validity of monomial approximations and a transition model
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Adhesion of an axisymmetric elastic body: Ranges of validity of monomial approximations and a transition model Fouad Oweiss, George G. Adams n Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA 02115, USA
art ic l e i nf o
a b s t r a c t
Article history: Received 25 August 2015 Received in revised form 3 November 2015 Accepted 10 February 2016
The Johnson–Kendall–Roberts and the Maugis models have been used to model the adhesion of spherical elastic bodies (represented by paraboloids). Recent work has investigated adhesion of higher-order monomial shapes. These results differ significantly from those of paraboloids. Given that any practical shape will not be “exactly” either a paraboloid or a higher-order monomial, the question arises as to the ranges of validity of these models and what is the requirement, if any, for a transition model. In this investigation the ranges of validity of these models are established by considering a two-term transition model. Furthermore the need for using this transition model is shown to depend not only on the geometry of the bodies but also on material properties. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Adhesion: Contact, Forces Contact: Elastic Model: Analytical
1. Introduction The elastic contact with adhesion of spherical elastic bodies (whose surface profiles are approximated by a single second-order term) was investigated by Johnson et al. (JKR model,) [1] and Derjaguin et al. (DMT model) [2]. Both models add to the Hertz contact model the effect of the Dupré energy of adhesion w, which is defined as the work per unit area needed for the reversible, isothermal separation of two solids. This quantity, also known as the work of adhesion, is given by w ¼ γ 1 þ γ 2 γ 12 , where γ 1 and γ 2 are the surface energies of the two bodies in contact and γ 12 is the interfacial energy. If the bodies are identical then, w ¼ 2γ . The JKR model adds the adhesion effect by minimizing the total potential energy which includes the Dupré energy of adhesion, giving a pull-off force of 32π wR. The DMT model adds the adhesive stresses outside the contact region while maintaining the Hertz stress distribution inside the contact area, thereby obtaining a pull-off force of 2π wR. Using the Dugdale model from fracture mechanics, Maugis introduced a model (also known as the M-D model) [3] which demonstrates a continuous transition between the JKR and DMT theories based on a parameter λ that is closely related to the Tabor parameter m [4]. The Tabor parameter is a measure of the ratio of the elastic deformation to the range of surface forces. The Maugis parameter can be expressed in terms of Tabor parameter as λ ¼ 1:16μ. Maugis showed that as λ-0 the DMT model is n
Corresponding author. E-mail address: [email protected] (G.G. Adams).
applicable whereas when λ-1 the JKR model is called for. For practical purposes λ less than about 0.1 is DMT and λ greater than about 3 is JKR. Although the three models mentioned above assume that the bodies in contact are spherical, their surface geometries are in fact approximated by a single second-order term. In their research on the friction force between an atomic force microscope (AFM) tip and a nominally flat surface, Carpick et al. introduced an extended JKR model applicable to axisymmetric elastic bodies in contact with a surface profile described by a single n-th order term, i.e. Cr n [5,6]. Later, Zheng et al. developed an analytical model to extend the M-D theory to asperities with such power-law geometries, called the M-D-n model [7]. Grierson et al. then performed a finite element analysis and experimental measurements which agreed well with the analytical model [8]. Some questions naturally arise in deciding whether to use a second- or higher-order approximation of a surface profile [10]. For example, if the real surface profile is “exactly” either a 2ndorder or 4th-order shape then it is clear which shape to use in calculating the pull-off force, the force vs. the contact area, and the force vs. the penetration. However, a realistic shape could no doubt be approximated to different degrees of accuracy by either a 2nd-order or a 4th-order shape. The choice of which approximation to use is expected to depend upon how close the actual profile is to each of these shapes, but does it also depend upon a parameter involving the material properties? Furthermore, under what conditions does the shape need to be described by more than one term, i.e. a combination of a 2nd-order term and a 4th-order order term? Another issue is the choice of using the JKR and its extended
http://dx.doi.org/10.1016/j.triboint.2016.02.012 0301-679X/& 2016 Elsevier Ltd. All rights reserved.
Please cite this article as: Oweiss F, Adams GG. Adhesion of an axisymmetric elastic body: Ranges of validity of monomial approximations and a transition model. Tribology International (2016), http://dx.doi.org/10.1016/j.triboint.2016.02.012i
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2
model as opposed to the more complicated Maugis and its extended model. For a pure 2nd-order surface profile these regimes are reasonably clear. However do these regimes differ for a two-term approximation? By extending JKR and M-D mo`dels to a surface profile with two terms, we hope to answer these questions.
The value of the exponent n in the above pressure distribution depends on the surface profile of the elastic bodies in contact. For example n ¼ 12 represents the pressure distribution between two spherical elastic solids in frictionless contact without adhesion as obtained by Hertz (1882), with a maximum contact stress p1 and can be written in terms of the resultant force P 1 as pðr Þ ¼
2. JKR model extension for two terms 2.1. Problem formulation If a pressure distribution is applied to a circular region of an elastic half space of radius a, a closed-form solution for the normal component of surface displacement can be found. Consider a cylindrical coordinate system (r; θ; z) and apply on z ¼ 0 the pressure distribution given by n r2 roa ð1Þ pðr Þ ¼ pm 1 2 a where the maximum contact pressure is pm , which occurs at r ¼ 0. Following the procedure of Johnson [9] and referring to Fig. 1, the displacement of the surface at point B can be found using a local polar coordinate system s; ϕ with origin at point B. At a distance s from B, the pressure p s; ϕ that acts on a small element of area corresponds to a force with magnitude p s; ϕ sdsdϕ. The displacement at B resulting from the pressure distributed on the whole area is then uz ¼
1 ν2 ∬ p s; ϕ dsdϕ πE A
ð2Þ
where ν is Poisson's ratio and E is Young's modulus. The resultant force P is obtained by integration of the pressure over the circular region Za P¼
2π rpðr Þdr
ð3Þ
0
1 3P 1 r2 2 1 2π a2 a2
roa
ð4Þ
The displacement expressed in terms of the resultant force P 1 is 3P 1 r2 0or oa ð5Þ uz ðr; 0Þ ¼ 2 2 8E a a uz ðr; 0Þ ¼
3P 1 4π aE
" 2
rffiffiffiffiffiffiffiffiffiffiffiffiffi!# r2 r2 1a þ 1 sin r a2 a2
r 4a
ð6Þ
where E is the composite Young's modulus defined by 1 1 ν21 1 ν22 ¼ þ E E1 E2
ð7Þ
where E1 and E2 are the elastic Young's moduli of elastic bodies 1 and 2 respectively, and ν1 and ν2 are Poisson's ratios of bodies 1 and 2 respectively. A pressure distribution in which n ¼ 12 results in a uniform normal displacement of the circular region ðr o aÞ and corresponds to the pressure, p2 , exerted by a flat-ended, frictionless, rigid punch pressed against an elastic half space with contact radius a [9]. In this case, the displacement at the surface can be expressed in terms of the resultant force P 2 as uz ðr; 0Þ ¼ uz ðr; 0Þ ¼
P2 2E a P2
π aE
0 or oa sin 1
a r
ð8Þ
r 4a
ð9Þ
By using the fracture mechanics concept of the stress intensity factor K I at the edge of the circular region, the force needed to separate the punch from the half-space may be obtained as a function of the Dupré energy of adhesion w pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð10Þ K I ¼ limpðrÞ 2π ða rÞ r-a
By setting K I equal to its critical value pffiffiffiffiffiffiffiffiffiffiffiffi P2 pffiffiffiffiffiffiffiffi ¼ 2wE 3 2 πa
ð11Þ
is obtained. The displacement inside the circular region is therefore: rffiffiffiffiffiffiffiffiffiffiffiffiffi 2π wa 0oroa ð12Þ uz ðr; 0Þ ¼ E If the exponent valuen of the pressure distribution is equal to 32, with a maximum contact pressure p3 and a resultant force P 3 , the corresponding displacements are 15P 3 r2 3 r4 0or oa ð13Þ uz ðr; 0Þ ¼ 1 2 þ 8 a4 16aE a and uz ðr; 0Þ ¼
15P 3 8π aE
" 1
rffiffiffiffiffiffiffiffiffiffiffiffiffi! # r2 3 r4 r2 3 3 r2 1a ; þ þ 1 sin r 4 8 a2 a2 8 a4 a2
r 4a
ð14Þ A superposition of these three pressure distributions results in the following surface displacements Fig. 1. Schematic of axisymmetric elastic body with center O showing point B, at distance r from the center, used to calculate vertical displacement uz .
for 0 o r o a
Please cite this article as: Oweiss F, Adams GG. Adhesion of an axisymmetric elastic body: Ranges of validity of monomial approximations and a transition model. Tribology International (2016), http://dx.doi.org/10.1016/j.triboint.2016.02.012i
F. Oweiss, G.G. Adams / Tribology International ∎ (∎∎∎∎) ∎∎∎–∎∎∎
# rffiffiffiffiffiffiffiffiffiffiffiffiffi 2π wa 3P 1 15P 3 3P 1 15P 3 45P 3 2 4 þ þ 5 r 3þ 3 r þ E 4E a 16E a 128E a 8E a 16E a
ð15Þ and for r 4a 2 h uz ðr; 0Þ ¼
1 6 4 π aE
sin
i 3 2 r2 3 r4 P 2 þ 34P 1 2 ar 2 þ 15 8 P 3 1 a2 þ 8 a4 7 h qffiffiffiffiffiffiffiffiffiffiffi
i 5 r2 3 15 3 3 r2 1 P þ P þ 4 1 8 3 4 8 a2 a2
1a r
ð16Þ Now if the surface profile of the elastic body in contact is described by a combination of a 2nd- and a 4th-order terms, then the surface displacement is given by r2 r2 1þϵ 2 0or oa ð17Þ uz ðr; 0Þ ¼ δ 2R R where δ is the interference and R is the composite radius of curvature expressed as 1 1 1 ¼ þ R R1 R2
P
π wR
ð20Þ
ð22Þ
ε0 ¼ ε
2 2 6π w 3 5 E R
ð23Þ
δ¼
!13 16E2 δ 9π 2 w 2 R
ð24Þ
The non-dimensional load and non-dimensional approach may now be written as pffiffiffiffiffiffiffiffi
ð25Þ P ¼ 6a 3 þ a 3 1 þ ε0 a 2
δ¼
rffiffiffiffiffiffi 8 5 a þ a 2 1 þ ε0 a 2 3 6
2.0
ε′ =0.1 ε′ =0.5
1.5
1.0
ð26Þ
In summary Eqs. (25) and (26) can be used to relate the dimensionless force to contact radius and approach for various values of ε0 . 2.2. Results and discussion In Fig. 2, the dimensionless contact radius vs. the dimensionless load is plotted for different values of ε0 . Note that ε0 as defined by Eq. (23) depends not only on the relative magnitude of the fourthorder term compared to the second-order term, but it also
ε′ =0
ε′ =0.25
ε′ =0
0.5
ε′ =10
-3
-2
-1
0
ε′ =3
1
ε′ =1
2
3
4
Total dimensionless load Fig. 2. The dimensionless contact radius vs. dimensionless load for different values of ε0 . The solid lines represent a surface profile described by a combination of a 2ndand 4th-order terms, whereas the dashed lines represent a surface profile described by a single 4th-order term.
1.0 ε′ =3
0.5
Both of the above equations can be simplified by introducing non-dimensional parameters given by 13 4E a¼ a ð21Þ 3π wR2 P¼
ε′ =0.1
0.0 -4
ð18Þ
where R1 and R2 are the radii of curvatures of bodies 1 and 2 respectively. By matching the terms in Eqs. (15)–(17), the total resultant force P is obtained by summing forces P 1 ; P 2 ; P 3 and is given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4E a3 8ε 1 þ 2 a2 ð19Þ P ¼ 8π wE a3 þ 3R 5R whereas the total interference is " rffiffiffiffiffiffiffiffiffiffiffiffiffi # 2π wa a2 4εa4 1þ δ¼ þ E R 3R3
Dimensionless contact radius
2.5
Total dimensionless load
" uz ðr; 0Þ ¼
3
ε′ =10
ε′ =1
0.0
ε′ =0 ε′ =0.1
-0.5 -1.0
ε′ =0.25
ε′ =0
ε′ =0.5
-1.5 -2.0 -2.5 -1.5
ε′ =0.5 ε′ =0.1
ε′ =0.25
-1.0
-0.5
0.0
0.5
1.0
1.5
Total dimensionless approach Fig. 3. The dimensionless load vs. dimensionless approach for different values of ε0 . The solid lines represent a surface profile described by a combination of a 2nd- and 4th-order terms, whereas the dashed lines represent a surface profile described by a single 4th-order term.
depends on w/E*R. The solid lines in Fig. 2 represent the results for a surface profile described by a combination of 2nd- and 4th-order terms, whereas the dashed lines represent a pure 4th-order surface profile. It is noted that for a given positive (compressive) load, the resulting contact radius decreases as the weight of the 4thorder term increases compared to the 2nd-order term (i.e. an increase of ε0 ). For ε0 as low as about 0.05 (not shown), the effect of the 4th-order term is important. Also it is not until ε0 is as large as about 10 that the pure 4th-order description is accurate. Thus the range 0:05 o ε0 o10 describes the transition in which a two-term approximation is needed. In Fig. 3, the dimensionless force vs. the dimensionless approach (interference) is plotted for different values of ε0 . Again, the solid lines represent a surface profile described by a combination of 2nd- and 4th-order terms, whereas the dashed lines represent a pure 4th-order surface profile. The range 0:05 o ε0 o 10 still describes the transition region in which a two-term approximation is needed. For a geometry described by two terms, separation of the bodies in contact (which occurs at the maximum tensile load) is accompanied by a dimensionless negative interference (stretching) ranging from approximately δ ¼ 0:9 at ε0 ¼ 0 to around δ ¼ 0:75 at ε0 ¼ 10. In Fig. 4, the dimensionless pull-off force P off (i.e. the maximum tensile force) is plotted as a function of ε0 . Also shown is the curve-
Please cite this article as: Oweiss F, Adams GG. Adhesion of an axisymmetric elastic body: Ranges of validity of monomial approximations and a transition model. Tribology International (2016), http://dx.doi.org/10.1016/j.triboint.2016.02.012i
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4
Dimesnionless pull-off force
2.0
δε ¼
1.5
Data scatter 1.0
0.0
ð33Þ
3R3
for this pure 4th-order shape without adhesion. The gap resulting from the pressure distribution due to a 4th-order shape becomes 4 4εa4 2 3 r 1a ½uzε ¼ cos 1 r 8 a π 3R3 " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! #
4 2 8εa r 1 3 r 2 þ þ 1 ð34Þ a 4 8 a 3π R3
Single 4th-order term
0.5
4εa4
Curve-fit
0
2
4
6
8
10
ε′ Fig. 4. Data scatter and curve-fit for the variation in dimensionless pull-off force with ε0 for a surface profile described by a combination of 2nd- and 4th-order terms. The dashed curve represents the pull-off force for a surface profile described by a single 4th-order term.
fit equation which is expressed in the form P off ¼ 0:7905e 0:05618ε þ 0:7094e 1:39ε 0
0
ð27Þ
The root mean squared error between the curve-fit and the data is about 1.7%, with a maximum error of about 4.8%. It is noted that the results reduce to the original JKR model when ε0 ¼ 0.
3. Maugis model extension to two terms The model introduced by Maugis [3], which was developed for spherical bodies in contact, will now be extended to a combined 2nd- and 4th-order shape. The first step is to determine the solution due to a single term 4th-order shape without adhesion; it corresponds to a summation of pressure distributions with n ¼ 12 and n ¼ 32. This solution will then be superposed onto the Maugis solution in such a manner as to account for the fact that the added pressure distribution changes the displacement in the separation region and hence affects the adhesion condition. For a pressure distribution given by 1 3 r2 2 r2 2 pðr Þ ¼ p1 1 2 þ p3 1 2 a a
roa
ð28Þ
by using Eq. (2), the resulting displacement is for 0 or oa uz ðr; 0Þ ¼
1 r2 3 r4 ð 12P þ 15P Þ ð 6P þ15P Þ þ ð 15P Þ 1 3 1 3 3 16aE 8 a4 a2
ð29Þ
for r 4 a 2 h 1 6 6 uz ðr; 0Þ ¼ 8π aE 4
sin
2
2 4
i 3 6P 1 2 ar þ 15P 3 1 ar þ 38 ar 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 r 2
5 3 3 r 2 1 6P 1 þ 15P 3 4 8 a þ a 1a r
ð30Þ
Now considering the results of Maugis, the relative normal displacement in the separation region for a parabolic shaped profile with a constant adhesive stress (σ o ) is at r ¼ c pffiffiffiffiffiffiffiffiffiffiffiffiffiffii a2 hpffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 1 þ m2 2 tan 1 m2 1 ho ¼ πR i pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4σ o ah pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð35Þ m2 1 tan 1 m2 1 þ 1 m þ πE where m ¼ ac : Note that in the Maugis model and its extension a constant adhesive stress is assumed at points in the separation region for which the local separation is less than ho , where ho σ o ¼ w defines the value of ho . Maugis shows that ho is approximately equal to the atomic equilibrium separation ðZ o Þ of two half-spaces. The stress due to adhesion is assumed to be a constant σ o in this region and zero elsewhere. Thus for the combined 2nd-and 4th-order model, the total air gap ½uz at r ¼ c can be expressed as: "" # # pffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 m2 2 8εa4 38m4 1 1 2 1 ho ¼ m þ tan πR 3π R3 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 4 a 2εa εa þ þ 3 m2 m2 1 þ 3 π R 3π R π R i pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4σ o ah pffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 1 tan 1 m2 1 þ 1 m þ ð36Þ πE pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ¼ tan 1 m2 1 has been used,. in which the identity cos 1 m Now the Maugis dimensionless parameter is introduced
λ¼
2σ o
1 2 3
ð37Þ
π wK R
and Eq. (36) becomes " !pffiffiffiffiffiffiffiffiffiffiffiffiffiffi λa 2 5ε0 a 2 2 2 1þ m þ m2 1 3 2 8 # ! pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 5ε0 a 2 3 4 1 2 m 1 þ m 2 þ m 1 tan 8 3 i pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4 2 h pffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 1 tan 1 m2 1 þ 1 m ¼ 1 þ λ a 3
ð38Þ
which reduces to equation (6.17) in [3] if ε0 ¼ 0. Eqs. (32) and (33) can be rewritten using dimensionless parameters as P ε ¼ ε0 a 5
ð39Þ
5 6
δ ε ¼ ε0 a 4
ð40Þ
ð31Þ
The total dimensionless load can be obtained by adding Eq. (39) to the dimensionless load obtained by Maugis [3] (equation 6.18) and is hpffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffii ð41Þ P ¼ ε0 a 5 þ a 3 λ a 2 m2 1 þ m2 tan 1 m2 1
then, by matching Eqs. (29) and (31), the corresponding total load (P ε ) and interference (δε ) are given by
The dimensionless approach given by Eq. (40) can be added to Eq. (6.19) in [3] to give the total dimensionless approach:
If the surface profile of the elastic body in contact is described by a single 4th-order term, i.e. r4 uz ðr; 0Þ ¼ δ ϵ 3 2R
Pε ¼ P1 þ P3 ¼
0 or oa
32εE a5 15R
3
ð32Þ
5 6
4 3
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
δ ¼ ε0 a 4 þ a 2 a λ m2 1
ð42Þ
Please cite this article as: Oweiss F, Adams GG. Adhesion of an axisymmetric elastic body: Ranges of validity of monomial approximations and a transition model. Tribology International (2016), http://dx.doi.org/10.1016/j.triboint.2016.02.012i
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2.5
ε′ =0.1
ε′ =0
ε′ =0
Dimensionless contact radius
Dimensionless contact radius
2.5
ε′ =0.25
2.0
1.5
1.0
0.5
0.0 -4
ε′ =3
ε′ =1
ε′ =0.5
ε′ =10
-3
-2
-1
0
1
2
3
2.0
1.5 ε′ =0
1.0
0.5 ε′ =3
0.0 -4
4
ε′ =0.1
-3
-2
Dimensionless load
3
4
ð43Þ
0.5 ε′ =3
0.0
ε′ =1
ε′ =0 ε′ =0.1
-1.0 -1.5
-2.5 -1.5
ε′ =0.25
ε′ =0.5
ε′ =0.5
ε′ =0.25
ε′ =0.1
-1.0
-0.5
ð44Þ ð45Þ
The total air gap at r ¼ c can be expressed as 4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi εa4 εa 8 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 1 2 2 1 þ 2 1 tan m m ho ¼ m m þ 3 3 3 π R3 π R h pffiffiffiffiffiffiffiffiffiffiffiffiffiffi i pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4σ o a 1 m2 1 tan m2 1 þ 1 m þ ð46Þ π E Using Maugis dimensionless parameter from Eqs. (37) and (46) becomes pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5ε0 λ a 4 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 m2 1 þ m4 tan 1 m2 1 m2 þ 3 3 16 i pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4 2 h pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 ð47Þ m 1 tan þ λ a m 1 þ 1 m ¼ 1 3 By adding the dimensionless load and dimensionless approach obtained by Maugis, the total dimensionless load for a pure 4thorder shape with adhesion becomes hpffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffii m2 1 þ m2 tan 1 m2 1 P ¼ ε0 a 5 λ a 2 ð48Þ and the total dimensionless approach for a pure 4th-order shape with adhesion is given by 5 4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi δ ¼ ε0 a 4 a λ m2 1 ð49Þ 6 3 In Figs. 5 and 6, the dimensionless contact radius is plotted vs. the dimensionless load for λ ¼ 0:5 and λ ¼ 1 respectively. Also, the dimensionless load is plotted vs. the dimensionless approach in Figs. 7 and 8 at λ ¼ 0:5 and λ ¼ 1 respectively. Based on these results, a single 4th-order term can be used to describe the surface profile of an elastic body when ε0 4 10, while a single 2nd-order
ε′ =10
-0.5
-2.0
0.0
0.5
1.0
1.5
Dimensionless approach Fig. 7. Dimensionless load vs. dimensionless approach at λ ¼ 0:5 at different values of ε0 . The solid lines represent a surface profile described by a combination of a 2ndand 4th-order terms, whereas the dashed lines represent a surface profile described by a single 4th-order term.
1.0 0.5
ε′ =1 ε′ =3 ε′ =10
Dimensionless load
5 6
δ 0 ¼ ε0 a 4 a 2
2
1.0
Using the same previous procedures, the corresponding 0 dimensionless total load (P 0 ) and dimensionless interference (δ ) without adhesion, for the shape in Eq. (43) are given by P 0 ¼ ε0 a 5 a 3
ε′ =10 0 1
Fig. 6. Dimensionless contact radius vs. dimensionless load at λ ¼ 1 at different values of ε0 . The solid lines represent a surface profile described by a combination of a 2nd- and 4th-order terms, whereas the dashed lines represent a surface profile described by a single 4th-order term.
Dimensionless load
It is noted that when ε0 ¼ 0 Eqs. (41) and (42) reduce to the original equations obtained by Maugis. To obtain the values of dimensionless load and the dimensionless approach relevant to a pure 4th-order shape, the surface profile of the elastic body in contact is assumed so as to suppress the second order term when superposed to the original M-D model, i.e. 0or oa
-1
ε′ =1 ε′ =0.5 ε′ =0.25
Dimensionless load
Fig. 5. Dimensionless contact radius vs. dimensionless load at λ ¼ 0:5 at different values of ε0 . The solid lines represent a surface profile described by a combination of a 2nd- and 4th-order terms, whereas the dashed lines represent a surface profile described by a single 4th-order term.
r2 r4 uz ðr; 0Þ ¼ δ þ ϵ 3 2R 2R
5
0.0 ε′ =0
-0.5
ε′ =0.1 ε′ =0.25
-1.0
ε′ =0.5
-1.5 -2.0 -2.5 -1.5
ε′ =0.1
ε′ =1
ε′ =0.25
ε′ =0.5
-1.0
-0.5
0.0
0.5
1.0
1.5
Dimensionless approach Fig. 8. Dimensionless load vs. dimensionless approach at λ ¼ 1 at different values of ε0 . The solid lines represent a surface profile described by a combination of a 2ndand 4th-order terms, whereas the dashed lines represent a surface profile described by a single 4th-order term.
term may be used if ε0 o 0:05. In the transition range where 0:05 o ε0 o 10, the surface profile should be described by two terms. Fig. 9 highlights the decrease in the pull-off force as ε0 increases for a variety of λ: At large values of ε0 (Fig. 9a) the trend is clear that greater values of λ lead to greater values of the pull-off force. However for smaller values of ε0 (Fig. 9b) the trend is much more
Please cite this article as: Oweiss F, Adams GG. Adhesion of an axisymmetric elastic body: Ranges of validity of monomial approximations and a transition model. Tribology International (2016), http://dx.doi.org/10.1016/j.triboint.2016.02.012i
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6
Dimensionless approach at pull-off
0.0
Dimensionless pull-off force
2.0
1.5
λ =1 and JKR
1.0
0.5 λ =0.05 λ =0.1
0.0
0
λ =0.2
2
4
6
8
-0.2
λ =0.1
-0.3
λ =0.2
-0.4 JKR λ =3
-0.5
λ =0.5
-0.6 λ =1
0
2
4
6
8
10
ε′
10
Fig. 11. Dimensionless approach at pull-off vs. ε0 for a surface profile described by a combination of second- and fourth-order terms at different values of λ. The dashed curve represents the dimensionless approach at pull-off force vs. ε0 obtained by the extended JKR model for a surface profile described by a combination of second- and fourth-order terms.
2.0
Dimensionless pull-off force
λ =0.05
-0.7
λ =0.5
ε′
1.8
4. Conclusions λ =0.5
1.6
λ =1 λ =3
1.4
1.2 λ =0.05
1.0
0
λ =0.1
0.1
λ =0.2
JKR 0.3
0.2
0.4
0.5
0.6
ε′ Fig. 9. a and b Dimensionless pull-off force vs. ε0 for a surface profile described by a combination of second- and fourth-order terms at different values of λ. The dashed curve represents the dimensionless pull-off force vs. ε0 obtained by the extended JKR model for a surface profile described by a combination of second- and fourthorder terms.
Dimensionless contact radius at pull-off
-0.1
The extension of the JKR and Maugis models to two terms has been accomplished. The results demonstrate the conditions under which using a single 2nd-order term, a single 4th-order term, or the two-term approximation is needed. This determination is based upon a parameter ε0 which represents a combination of the relative weight of the 4th-order term compared to the 2nd-order term and also on the ratio of the work of adhesion to the product of the composite Young’s modulus and the radius of curvature of the body. From these results, the surface profile of an elastic body can be approximated to a single 2nd-order form if ε0 is less than about 0.05, whereas a single 4th-order term approximation is applicable for ε0 greater than about 10. Therefore the range 0:05 o ε0 o 10 is the transition regime in which a two-term approximation is called for.
1.4
References
1.2 1.0 0.8
JKR λ= 3
0.6 0.4
λ =1 λ =0.05, 0.1
0.2 0.0
0
2
λ =0.5
λ = 0.2
4
6
8
10
ε′ Fig. 10. Dimensionless contact radius at pull-off vs. ε0 for a surface profile described by a combination of second- and fourth-order terms at different values of λ. The dashed curve represents the dimensionless contact radius at pull-off vs. ε0 obtained by the extended JKR model for a surface profile described by a combination of second- and fourth-order terms.
[1] Johnson KL, Kendall K, Roberts AD. Surface energy and the contact of elastic solids. Proc R Soc Lond 1971;A324:301–13. [2] Derjaguin BV, Muller VM, Toporov YP. Effect of contact deformations on the adhesion of particles. J Colloid Interface Sci 1975;53:314–26. [3] Maugis D. Adhesion of spheres: the JKR-DMT transition using a Dugdale model. J Colloid Interface Sci 1992;150:243–69. [4] Tabor D. Surface forces and surface interactions. J Colloid Interface Sci 1976;58:2–13. [5] Carpick RW, Agraït N, Ogletree DF, Salmeron M. Measurement of interfacial shear (friction) with an ultrahigh vacuum atomic force microscope. J Vac Sci Technol B 1996;14:1289–95. [6] Carpick RW, Agraït N, Ogletree DF, Salmeron M. Erratum: measurement of interfacial shear (friction) with an ultrahigh vacuum atomic force microscope. J Vac Sci Technol B 1996;14:1289–95. [7] Zheng Z, Yu J. Using the Dugdale approximation to match a specific interaction in the adhesive contact of elastic objects. J Colloid Interface Sci 2007;310:27–34. [8] Grierson DS, Liu J, Carpick RW, Turner KT. Adhesion of nanoscale asperities with power-law profiles. J Mech Phys Solids 2013;61:597–610. [9] Johnson KL. Contact mechanics; 1985. p.56–62. [10] Oweiss F, Adams G. Adhesion of an axisymmetric elastic body with a surface profile described by combined second- and fourth-order Terms. Boston, MA, USA: Northeastern University Library; 2015.
complicated. The variation of the dimensionless contact radius at pull-off vs. ε0 is shown in Fig. 10 for various values of λ. Finally, Fig. 11 shows the dimensionless approach at pull-off (stretching, since it is negative) vs. as ε0 for different values of λ:
Please cite this article as: Oweiss F, Adams GG. Adhesion of an axisymmetric elastic body: Ranges of validity of monomial approximations and a transition model. Tribology International (2016), http://dx.doi.org/10.1016/j.triboint.2016.02.012i
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Adhesion of an axisymmetric elastic body: Ranges of validity of monomial approximations and a transition model Fouad Oweiss, George G. Adams n Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA 02115, USA
art ic l e i nf o
a b s t r a c t
Article history: Received 25 August 2015 Received in revised form 3 November 2015 Accepted 10 February 2016
The Johnson–Kendall–Roberts and the Maugis models have been used to model the adhesion of spherical elastic bodies (represented by paraboloids). Recent work has investigated adhesion of higher-order monomial shapes. These results differ significantly from those of paraboloids. Given that any practical shape will not be “exactly” either a paraboloid or a higher-order monomial, the question arises as to the ranges of validity of these models and what is the requirement, if any, for a transition model. In this investigation the ranges of validity of these models are established by considering a two-term transition model. Furthermore the need for using this transition model is shown to depend not only on the geometry of the bodies but also on material properties. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Adhesion: Contact, Forces Contact: Elastic Model: Analytical
1. Introduction The elastic contact with adhesion of spherical elastic bodies (whose surface profiles are approximated by a single second-order term) was investigated by Johnson et al. (JKR model,) [1] and Derjaguin et al. (DMT model) [2]. Both models add to the Hertz contact model the effect of the Dupré energy of adhesion w, which is defined as the work per unit area needed for the reversible, isothermal separation of two solids. This quantity, also known as the work of adhesion, is given by w ¼ γ 1 þ γ 2 γ 12 , where γ 1 and γ 2 are the surface energies of the two bodies in contact and γ 12 is the interfacial energy. If the bodies are identical then, w ¼ 2γ . The JKR model adds the adhesion effect by minimizing the total potential energy which includes the Dupré energy of adhesion, giving a pull-off force of 32π wR. The DMT model adds the adhesive stresses outside the contact region while maintaining the Hertz stress distribution inside the contact area, thereby obtaining a pull-off force of 2π wR. Using the Dugdale model from fracture mechanics, Maugis introduced a model (also known as the M-D model) [3] which demonstrates a continuous transition between the JKR and DMT theories based on a parameter λ that is closely related to the Tabor parameter m [4]. The Tabor parameter is a measure of the ratio of the elastic deformation to the range of surface forces. The Maugis parameter can be expressed in terms of Tabor parameter as λ ¼ 1:16μ. Maugis showed that as λ-0 the DMT model is n
Corresponding author. E-mail address: [email protected] (G.G. Adams).
applicable whereas when λ-1 the JKR model is called for. For practical purposes λ less than about 0.1 is DMT and λ greater than about 3 is JKR. Although the three models mentioned above assume that the bodies in contact are spherical, their surface geometries are in fact approximated by a single second-order term. In their research on the friction force between an atomic force microscope (AFM) tip and a nominally flat surface, Carpick et al. introduced an extended JKR model applicable to axisymmetric elastic bodies in contact with a surface profile described by a single n-th order term, i.e. Cr n [5,6]. Later, Zheng et al. developed an analytical model to extend the M-D theory to asperities with such power-law geometries, called the M-D-n model [7]. Grierson et al. then performed a finite element analysis and experimental measurements which agreed well with the analytical model [8]. Some questions naturally arise in deciding whether to use a second- or higher-order approximation of a surface profile [10]. For example, if the real surface profile is “exactly” either a 2ndorder or 4th-order shape then it is clear which shape to use in calculating the pull-off force, the force vs. the contact area, and the force vs. the penetration. However, a realistic shape could no doubt be approximated to different degrees of accuracy by either a 2nd-order or a 4th-order shape. The choice of which approximation to use is expected to depend upon how close the actual profile is to each of these shapes, but does it also depend upon a parameter involving the material properties? Furthermore, under what conditions does the shape need to be described by more than one term, i.e. a combination of a 2nd-order term and a 4th-order order term? Another issue is the choice of using the JKR and its extended
http://dx.doi.org/10.1016/j.triboint.2016.02.012 0301-679X/& 2016 Elsevier Ltd. All rights reserved.
Please cite this article as: Oweiss F, Adams GG. Adhesion of an axisymmetric elastic body: Ranges of validity of monomial approximations and a transition model. Tribology International (2016), http://dx.doi.org/10.1016/j.triboint.2016.02.012i
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2
model as opposed to the more complicated Maugis and its extended model. For a pure 2nd-order surface profile these regimes are reasonably clear. However do these regimes differ for a two-term approximation? By extending JKR and M-D mo`dels to a surface profile with two terms, we hope to answer these questions.
The value of the exponent n in the above pressure distribution depends on the surface profile of the elastic bodies in contact. For example n ¼ 12 represents the pressure distribution between two spherical elastic solids in frictionless contact without adhesion as obtained by Hertz (1882), with a maximum contact stress p1 and can be written in terms of the resultant force P 1 as pðr Þ ¼
2. JKR model extension for two terms 2.1. Problem formulation If a pressure distribution is applied to a circular region of an elastic half space of radius a, a closed-form solution for the normal component of surface displacement can be found. Consider a cylindrical coordinate system (r; θ; z) and apply on z ¼ 0 the pressure distribution given by n r2 roa ð1Þ pðr Þ ¼ pm 1 2 a where the maximum contact pressure is pm , which occurs at r ¼ 0. Following the procedure of Johnson [9] and referring to Fig. 1, the displacement of the surface at point B can be found using a local polar coordinate system s; ϕ with origin at point B. At a distance s from B, the pressure p s; ϕ that acts on a small element of area corresponds to a force with magnitude p s; ϕ sdsdϕ. The displacement at B resulting from the pressure distributed on the whole area is then uz ¼
1 ν2 ∬ p s; ϕ dsdϕ πE A
ð2Þ
where ν is Poisson's ratio and E is Young's modulus. The resultant force P is obtained by integration of the pressure over the circular region Za P¼
2π rpðr Þdr
ð3Þ
0
1 3P 1 r2 2 1 2π a2 a2
roa
ð4Þ
The displacement expressed in terms of the resultant force P 1 is 3P 1 r2 0or oa ð5Þ uz ðr; 0Þ ¼ 2 2 8E a a uz ðr; 0Þ ¼
3P 1 4π aE
" 2
rffiffiffiffiffiffiffiffiffiffiffiffiffi!# r2 r2 1a þ 1 sin r a2 a2
r 4a
ð6Þ
where E is the composite Young's modulus defined by 1 1 ν21 1 ν22 ¼ þ E E1 E2
ð7Þ
where E1 and E2 are the elastic Young's moduli of elastic bodies 1 and 2 respectively, and ν1 and ν2 are Poisson's ratios of bodies 1 and 2 respectively. A pressure distribution in which n ¼ 12 results in a uniform normal displacement of the circular region ðr o aÞ and corresponds to the pressure, p2 , exerted by a flat-ended, frictionless, rigid punch pressed against an elastic half space with contact radius a [9]. In this case, the displacement at the surface can be expressed in terms of the resultant force P 2 as uz ðr; 0Þ ¼ uz ðr; 0Þ ¼
P2 2E a P2
π aE
0 or oa sin 1
a r
ð8Þ
r 4a
ð9Þ
By using the fracture mechanics concept of the stress intensity factor K I at the edge of the circular region, the force needed to separate the punch from the half-space may be obtained as a function of the Dupré energy of adhesion w pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð10Þ K I ¼ limpðrÞ 2π ða rÞ r-a
By setting K I equal to its critical value pffiffiffiffiffiffiffiffiffiffiffiffi P2 pffiffiffiffiffiffiffiffi ¼ 2wE 3 2 πa
ð11Þ
is obtained. The displacement inside the circular region is therefore: rffiffiffiffiffiffiffiffiffiffiffiffiffi 2π wa 0oroa ð12Þ uz ðr; 0Þ ¼ E If the exponent valuen of the pressure distribution is equal to 32, with a maximum contact pressure p3 and a resultant force P 3 , the corresponding displacements are 15P 3 r2 3 r4 0or oa ð13Þ uz ðr; 0Þ ¼ 1 2 þ 8 a4 16aE a and uz ðr; 0Þ ¼
15P 3 8π aE
" 1
rffiffiffiffiffiffiffiffiffiffiffiffiffi! # r2 3 r4 r2 3 3 r2 1a ; þ þ 1 sin r 4 8 a2 a2 8 a4 a2
r 4a
ð14Þ A superposition of these three pressure distributions results in the following surface displacements Fig. 1. Schematic of axisymmetric elastic body with center O showing point B, at distance r from the center, used to calculate vertical displacement uz .
for 0 o r o a
Please cite this article as: Oweiss F, Adams GG. Adhesion of an axisymmetric elastic body: Ranges of validity of monomial approximations and a transition model. Tribology International (2016), http://dx.doi.org/10.1016/j.triboint.2016.02.012i
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# rffiffiffiffiffiffiffiffiffiffiffiffiffi 2π wa 3P 1 15P 3 3P 1 15P 3 45P 3 2 4 þ þ 5 r 3þ 3 r þ E 4E a 16E a 128E a 8E a 16E a
ð15Þ and for r 4a 2 h uz ðr; 0Þ ¼
1 6 4 π aE
sin
i 3 2 r2 3 r4 P 2 þ 34P 1 2 ar 2 þ 15 8 P 3 1 a2 þ 8 a4 7 h qffiffiffiffiffiffiffiffiffiffiffi
i 5 r2 3 15 3 3 r2 1 P þ P þ 4 1 8 3 4 8 a2 a2
1a r
ð16Þ Now if the surface profile of the elastic body in contact is described by a combination of a 2nd- and a 4th-order terms, then the surface displacement is given by r2 r2 1þϵ 2 0or oa ð17Þ uz ðr; 0Þ ¼ δ 2R R where δ is the interference and R is the composite radius of curvature expressed as 1 1 1 ¼ þ R R1 R2
P
π wR
ð20Þ
ð22Þ
ε0 ¼ ε
2 2 6π w 3 5 E R
ð23Þ
δ¼
!13 16E2 δ 9π 2 w 2 R
ð24Þ
The non-dimensional load and non-dimensional approach may now be written as pffiffiffiffiffiffiffiffi
ð25Þ P ¼ 6a 3 þ a 3 1 þ ε0 a 2
δ¼
rffiffiffiffiffiffi 8 5 a þ a 2 1 þ ε0 a 2 3 6
2.0
ε′ =0.1 ε′ =0.5
1.5
1.0
ð26Þ
In summary Eqs. (25) and (26) can be used to relate the dimensionless force to contact radius and approach for various values of ε0 . 2.2. Results and discussion In Fig. 2, the dimensionless contact radius vs. the dimensionless load is plotted for different values of ε0 . Note that ε0 as defined by Eq. (23) depends not only on the relative magnitude of the fourthorder term compared to the second-order term, but it also
ε′ =0
ε′ =0.25
ε′ =0
0.5
ε′ =10
-3
-2
-1
0
ε′ =3
1
ε′ =1
2
3
4
Total dimensionless load Fig. 2. The dimensionless contact radius vs. dimensionless load for different values of ε0 . The solid lines represent a surface profile described by a combination of a 2ndand 4th-order terms, whereas the dashed lines represent a surface profile described by a single 4th-order term.
1.0 ε′ =3
0.5
Both of the above equations can be simplified by introducing non-dimensional parameters given by 13 4E a¼ a ð21Þ 3π wR2 P¼
ε′ =0.1
0.0 -4
ð18Þ
where R1 and R2 are the radii of curvatures of bodies 1 and 2 respectively. By matching the terms in Eqs. (15)–(17), the total resultant force P is obtained by summing forces P 1 ; P 2 ; P 3 and is given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4E a3 8ε 1 þ 2 a2 ð19Þ P ¼ 8π wE a3 þ 3R 5R whereas the total interference is " rffiffiffiffiffiffiffiffiffiffiffiffiffi # 2π wa a2 4εa4 1þ δ¼ þ E R 3R3
Dimensionless contact radius
2.5
Total dimensionless load
" uz ðr; 0Þ ¼
3
ε′ =10
ε′ =1
0.0
ε′ =0 ε′ =0.1
-0.5 -1.0
ε′ =0.25
ε′ =0
ε′ =0.5
-1.5 -2.0 -2.5 -1.5
ε′ =0.5 ε′ =0.1
ε′ =0.25
-1.0
-0.5
0.0
0.5
1.0
1.5
Total dimensionless approach Fig. 3. The dimensionless load vs. dimensionless approach for different values of ε0 . The solid lines represent a surface profile described by a combination of a 2nd- and 4th-order terms, whereas the dashed lines represent a surface profile described by a single 4th-order term.
depends on w/E*R. The solid lines in Fig. 2 represent the results for a surface profile described by a combination of 2nd- and 4th-order terms, whereas the dashed lines represent a pure 4th-order surface profile. It is noted that for a given positive (compressive) load, the resulting contact radius decreases as the weight of the 4thorder term increases compared to the 2nd-order term (i.e. an increase of ε0 ). For ε0 as low as about 0.05 (not shown), the effect of the 4th-order term is important. Also it is not until ε0 is as large as about 10 that the pure 4th-order description is accurate. Thus the range 0:05 o ε0 o10 describes the transition in which a two-term approximation is needed. In Fig. 3, the dimensionless force vs. the dimensionless approach (interference) is plotted for different values of ε0 . Again, the solid lines represent a surface profile described by a combination of 2nd- and 4th-order terms, whereas the dashed lines represent a pure 4th-order surface profile. The range 0:05 o ε0 o 10 still describes the transition region in which a two-term approximation is needed. For a geometry described by two terms, separation of the bodies in contact (which occurs at the maximum tensile load) is accompanied by a dimensionless negative interference (stretching) ranging from approximately δ ¼ 0:9 at ε0 ¼ 0 to around δ ¼ 0:75 at ε0 ¼ 10. In Fig. 4, the dimensionless pull-off force P off (i.e. the maximum tensile force) is plotted as a function of ε0 . Also shown is the curve-
Please cite this article as: Oweiss F, Adams GG. Adhesion of an axisymmetric elastic body: Ranges of validity of monomial approximations and a transition model. Tribology International (2016), http://dx.doi.org/10.1016/j.triboint.2016.02.012i
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4
Dimesnionless pull-off force
2.0
δε ¼
1.5
Data scatter 1.0
0.0
ð33Þ
3R3
for this pure 4th-order shape without adhesion. The gap resulting from the pressure distribution due to a 4th-order shape becomes 4 4εa4 2 3 r 1a ½uzε ¼ cos 1 r 8 a π 3R3 " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! #
4 2 8εa r 1 3 r 2 þ þ 1 ð34Þ a 4 8 a 3π R3
Single 4th-order term
0.5
4εa4
Curve-fit
0
2
4
6
8
10
ε′ Fig. 4. Data scatter and curve-fit for the variation in dimensionless pull-off force with ε0 for a surface profile described by a combination of 2nd- and 4th-order terms. The dashed curve represents the pull-off force for a surface profile described by a single 4th-order term.
fit equation which is expressed in the form P off ¼ 0:7905e 0:05618ε þ 0:7094e 1:39ε 0
0
ð27Þ
The root mean squared error between the curve-fit and the data is about 1.7%, with a maximum error of about 4.8%. It is noted that the results reduce to the original JKR model when ε0 ¼ 0.
3. Maugis model extension to two terms The model introduced by Maugis [3], which was developed for spherical bodies in contact, will now be extended to a combined 2nd- and 4th-order shape. The first step is to determine the solution due to a single term 4th-order shape without adhesion; it corresponds to a summation of pressure distributions with n ¼ 12 and n ¼ 32. This solution will then be superposed onto the Maugis solution in such a manner as to account for the fact that the added pressure distribution changes the displacement in the separation region and hence affects the adhesion condition. For a pressure distribution given by 1 3 r2 2 r2 2 pðr Þ ¼ p1 1 2 þ p3 1 2 a a
roa
ð28Þ
by using Eq. (2), the resulting displacement is for 0 or oa uz ðr; 0Þ ¼
1 r2 3 r4 ð 12P þ 15P Þ ð 6P þ15P Þ þ ð 15P Þ 1 3 1 3 3 16aE 8 a4 a2
ð29Þ
for r 4 a 2 h 1 6 6 uz ðr; 0Þ ¼ 8π aE 4
sin
2
2 4
i 3 6P 1 2 ar þ 15P 3 1 ar þ 38 ar 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 r 2
5 3 3 r 2 1 6P 1 þ 15P 3 4 8 a þ a 1a r
ð30Þ
Now considering the results of Maugis, the relative normal displacement in the separation region for a parabolic shaped profile with a constant adhesive stress (σ o ) is at r ¼ c pffiffiffiffiffiffiffiffiffiffiffiffiffiffii a2 hpffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 1 þ m2 2 tan 1 m2 1 ho ¼ πR i pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4σ o ah pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð35Þ m2 1 tan 1 m2 1 þ 1 m þ πE where m ¼ ac : Note that in the Maugis model and its extension a constant adhesive stress is assumed at points in the separation region for which the local separation is less than ho , where ho σ o ¼ w defines the value of ho . Maugis shows that ho is approximately equal to the atomic equilibrium separation ðZ o Þ of two half-spaces. The stress due to adhesion is assumed to be a constant σ o in this region and zero elsewhere. Thus for the combined 2nd-and 4th-order model, the total air gap ½uz at r ¼ c can be expressed as: "" # # pffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 m2 2 8εa4 38m4 1 1 2 1 ho ¼ m þ tan πR 3π R3 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 4 a 2εa εa þ þ 3 m2 m2 1 þ 3 π R 3π R π R i pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4σ o ah pffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 1 tan 1 m2 1 þ 1 m þ ð36Þ πE pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ¼ tan 1 m2 1 has been used,. in which the identity cos 1 m Now the Maugis dimensionless parameter is introduced
λ¼
2σ o
1 2 3
ð37Þ
π wK R
and Eq. (36) becomes " !pffiffiffiffiffiffiffiffiffiffiffiffiffiffi λa 2 5ε0 a 2 2 2 1þ m þ m2 1 3 2 8 # ! pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 5ε0 a 2 3 4 1 2 m 1 þ m 2 þ m 1 tan 8 3 i pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4 2 h pffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 1 tan 1 m2 1 þ 1 m ¼ 1 þ λ a 3
ð38Þ
which reduces to equation (6.17) in [3] if ε0 ¼ 0. Eqs. (32) and (33) can be rewritten using dimensionless parameters as P ε ¼ ε0 a 5
ð39Þ
5 6
δ ε ¼ ε0 a 4
ð40Þ
ð31Þ
The total dimensionless load can be obtained by adding Eq. (39) to the dimensionless load obtained by Maugis [3] (equation 6.18) and is hpffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffii ð41Þ P ¼ ε0 a 5 þ a 3 λ a 2 m2 1 þ m2 tan 1 m2 1
then, by matching Eqs. (29) and (31), the corresponding total load (P ε ) and interference (δε ) are given by
The dimensionless approach given by Eq. (40) can be added to Eq. (6.19) in [3] to give the total dimensionless approach:
If the surface profile of the elastic body in contact is described by a single 4th-order term, i.e. r4 uz ðr; 0Þ ¼ δ ϵ 3 2R
Pε ¼ P1 þ P3 ¼
0 or oa
32εE a5 15R
3
ð32Þ
5 6
4 3
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
δ ¼ ε0 a 4 þ a 2 a λ m2 1
ð42Þ
Please cite this article as: Oweiss F, Adams GG. Adhesion of an axisymmetric elastic body: Ranges of validity of monomial approximations and a transition model. Tribology International (2016), http://dx.doi.org/10.1016/j.triboint.2016.02.012i
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2.5
ε′ =0.1
ε′ =0
ε′ =0
Dimensionless contact radius
Dimensionless contact radius
2.5
ε′ =0.25
2.0
1.5
1.0
0.5
0.0 -4
ε′ =3
ε′ =1
ε′ =0.5
ε′ =10
-3
-2
-1
0
1
2
3
2.0
1.5 ε′ =0
1.0
0.5 ε′ =3
0.0 -4
4
ε′ =0.1
-3
-2
Dimensionless load
3
4
ð43Þ
0.5 ε′ =3
0.0
ε′ =1
ε′ =0 ε′ =0.1
-1.0 -1.5
-2.5 -1.5
ε′ =0.25
ε′ =0.5
ε′ =0.5
ε′ =0.25
ε′ =0.1
-1.0
-0.5
ð44Þ ð45Þ
The total air gap at r ¼ c can be expressed as 4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi εa4 εa 8 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 1 2 2 1 þ 2 1 tan m m ho ¼ m m þ 3 3 3 π R3 π R h pffiffiffiffiffiffiffiffiffiffiffiffiffiffi i pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4σ o a 1 m2 1 tan m2 1 þ 1 m þ ð46Þ π E Using Maugis dimensionless parameter from Eqs. (37) and (46) becomes pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5ε0 λ a 4 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 m2 1 þ m4 tan 1 m2 1 m2 þ 3 3 16 i pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4 2 h pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 ð47Þ m 1 tan þ λ a m 1 þ 1 m ¼ 1 3 By adding the dimensionless load and dimensionless approach obtained by Maugis, the total dimensionless load for a pure 4thorder shape with adhesion becomes hpffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffii m2 1 þ m2 tan 1 m2 1 P ¼ ε0 a 5 λ a 2 ð48Þ and the total dimensionless approach for a pure 4th-order shape with adhesion is given by 5 4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi δ ¼ ε0 a 4 a λ m2 1 ð49Þ 6 3 In Figs. 5 and 6, the dimensionless contact radius is plotted vs. the dimensionless load for λ ¼ 0:5 and λ ¼ 1 respectively. Also, the dimensionless load is plotted vs. the dimensionless approach in Figs. 7 and 8 at λ ¼ 0:5 and λ ¼ 1 respectively. Based on these results, a single 4th-order term can be used to describe the surface profile of an elastic body when ε0 4 10, while a single 2nd-order
ε′ =10
-0.5
-2.0
0.0
0.5
1.0
1.5
Dimensionless approach Fig. 7. Dimensionless load vs. dimensionless approach at λ ¼ 0:5 at different values of ε0 . The solid lines represent a surface profile described by a combination of a 2ndand 4th-order terms, whereas the dashed lines represent a surface profile described by a single 4th-order term.
1.0 0.5
ε′ =1 ε′ =3 ε′ =10
Dimensionless load
5 6
δ 0 ¼ ε0 a 4 a 2
2
1.0
Using the same previous procedures, the corresponding 0 dimensionless total load (P 0 ) and dimensionless interference (δ ) without adhesion, for the shape in Eq. (43) are given by P 0 ¼ ε0 a 5 a 3
ε′ =10 0 1
Fig. 6. Dimensionless contact radius vs. dimensionless load at λ ¼ 1 at different values of ε0 . The solid lines represent a surface profile described by a combination of a 2nd- and 4th-order terms, whereas the dashed lines represent a surface profile described by a single 4th-order term.
Dimensionless load
It is noted that when ε0 ¼ 0 Eqs. (41) and (42) reduce to the original equations obtained by Maugis. To obtain the values of dimensionless load and the dimensionless approach relevant to a pure 4th-order shape, the surface profile of the elastic body in contact is assumed so as to suppress the second order term when superposed to the original M-D model, i.e. 0or oa
-1
ε′ =1 ε′ =0.5 ε′ =0.25
Dimensionless load
Fig. 5. Dimensionless contact radius vs. dimensionless load at λ ¼ 0:5 at different values of ε0 . The solid lines represent a surface profile described by a combination of a 2nd- and 4th-order terms, whereas the dashed lines represent a surface profile described by a single 4th-order term.
r2 r4 uz ðr; 0Þ ¼ δ þ ϵ 3 2R 2R
5
0.0 ε′ =0
-0.5
ε′ =0.1 ε′ =0.25
-1.0
ε′ =0.5
-1.5 -2.0 -2.5 -1.5
ε′ =0.1
ε′ =1
ε′ =0.25
ε′ =0.5
-1.0
-0.5
0.0
0.5
1.0
1.5
Dimensionless approach Fig. 8. Dimensionless load vs. dimensionless approach at λ ¼ 1 at different values of ε0 . The solid lines represent a surface profile described by a combination of a 2ndand 4th-order terms, whereas the dashed lines represent a surface profile described by a single 4th-order term.
term may be used if ε0 o 0:05. In the transition range where 0:05 o ε0 o 10, the surface profile should be described by two terms. Fig. 9 highlights the decrease in the pull-off force as ε0 increases for a variety of λ: At large values of ε0 (Fig. 9a) the trend is clear that greater values of λ lead to greater values of the pull-off force. However for smaller values of ε0 (Fig. 9b) the trend is much more
Please cite this article as: Oweiss F, Adams GG. Adhesion of an axisymmetric elastic body: Ranges of validity of monomial approximations and a transition model. Tribology International (2016), http://dx.doi.org/10.1016/j.triboint.2016.02.012i
F. Oweiss, G.G. Adams / Tribology International ∎ (∎∎∎∎) ∎∎∎–∎∎∎
6
Dimensionless approach at pull-off
0.0
Dimensionless pull-off force
2.0
1.5
λ =1 and JKR
1.0
0.5 λ =0.05 λ =0.1
0.0
0
λ =0.2
2
4
6
8
-0.2
λ =0.1
-0.3
λ =0.2
-0.4 JKR λ =3
-0.5
λ =0.5
-0.6 λ =1
0
2
4
6
8
10
ε′
10
Fig. 11. Dimensionless approach at pull-off vs. ε0 for a surface profile described by a combination of second- and fourth-order terms at different values of λ. The dashed curve represents the dimensionless approach at pull-off force vs. ε0 obtained by the extended JKR model for a surface profile described by a combination of second- and fourth-order terms.
2.0
Dimensionless pull-off force
λ =0.05
-0.7
λ =0.5
ε′
1.8
4. Conclusions λ =0.5
1.6
λ =1 λ =3
1.4
1.2 λ =0.05
1.0
0
λ =0.1
0.1
λ =0.2
JKR 0.3
0.2
0.4
0.5
0.6
ε′ Fig. 9. a and b Dimensionless pull-off force vs. ε0 for a surface profile described by a combination of second- and fourth-order terms at different values of λ. The dashed curve represents the dimensionless pull-off force vs. ε0 obtained by the extended JKR model for a surface profile described by a combination of second- and fourthorder terms.
Dimensionless contact radius at pull-off
-0.1
The extension of the JKR and Maugis models to two terms has been accomplished. The results demonstrate the conditions under which using a single 2nd-order term, a single 4th-order term, or the two-term approximation is needed. This determination is based upon a parameter ε0 which represents a combination of the relative weight of the 4th-order term compared to the 2nd-order term and also on the ratio of the work of adhesion to the product of the composite Young’s modulus and the radius of curvature of the body. From these results, the surface profile of an elastic body can be approximated to a single 2nd-order form if ε0 is less than about 0.05, whereas a single 4th-order term approximation is applicable for ε0 greater than about 10. Therefore the range 0:05 o ε0 o 10 is the transition regime in which a two-term approximation is called for.
1.4
References
1.2 1.0 0.8
JKR λ= 3
0.6 0.4
λ =1 λ =0.05, 0.1
0.2 0.0
0
2
λ =0.5
λ = 0.2
4
6
8
10
ε′ Fig. 10. Dimensionless contact radius at pull-off vs. ε0 for a surface profile described by a combination of second- and fourth-order terms at different values of λ. The dashed curve represents the dimensionless contact radius at pull-off vs. ε0 obtained by the extended JKR model for a surface profile described by a combination of second- and fourth-order terms.
[1] Johnson KL, Kendall K, Roberts AD. Surface energy and the contact of elastic solids. Proc R Soc Lond 1971;A324:301–13. [2] Derjaguin BV, Muller VM, Toporov YP. Effect of contact deformations on the adhesion of particles. J Colloid Interface Sci 1975;53:314–26. [3] Maugis D. Adhesion of spheres: the JKR-DMT transition using a Dugdale model. J Colloid Interface Sci 1992;150:243–69. [4] Tabor D. Surface forces and surface interactions. J Colloid Interface Sci 1976;58:2–13. [5] Carpick RW, Agraït N, Ogletree DF, Salmeron M. Measurement of interfacial shear (friction) with an ultrahigh vacuum atomic force microscope. J Vac Sci Technol B 1996;14:1289–95. [6] Carpick RW, Agraït N, Ogletree DF, Salmeron M. Erratum: measurement of interfacial shear (friction) with an ultrahigh vacuum atomic force microscope. J Vac Sci Technol B 1996;14:1289–95. [7] Zheng Z, Yu J. Using the Dugdale approximation to match a specific interaction in the adhesive contact of elastic objects. J Colloid Interface Sci 2007;310:27–34. [8] Grierson DS, Liu J, Carpick RW, Turner KT. Adhesion of nanoscale asperities with power-law profiles. J Mech Phys Solids 2013;61:597–610. [9] Johnson KL. Contact mechanics; 1985. p.56–62. [10] Oweiss F, Adams G. Adhesion of an axisymmetric elastic body with a surface profile described by combined second- and fourth-order Terms. Boston, MA, USA: Northeastern University Library; 2015.
complicated. The variation of the dimensionless contact radius at pull-off vs. ε0 is shown in Fig. 10 for various values of λ. Finally, Fig. 11 shows the dimensionless approach at pull-off (stretching, since it is negative) vs. as ε0 for different values of λ:
Please cite this article as: Oweiss F, Adams GG. Adhesion of an axisymmetric elastic body: Ranges of validity of monomial approximations and a transition model. Tribology International (2016), http://dx.doi.org/10.1016/j.triboint.2016.02.012i