6524
Langmuir 2001, 17, 6524-6529
Adhesion of a Rigid Punch to an Incompressible Elastic Film Fuqian Yang and J. C. M. Li* Department of Mechanical Engineering, University of Rochester, Rochester, New York 14627 Received March 19, 2001. In Final Form: June 25, 2001 The adhesion between a rigid flat-end cylindrical punch and an incompressible elastic film deposited on a rigid substrate is studied. The contact interfaces between the punch and the elastic film and between the elastic film and the rigid substrate are either free to slide over one another (frictionless or slip boundary condition) or locked together (perfectly bonded or stick boundary condition). The solutions for these cases are the upper and lower bounds for the real situation in which the contact interfaces are neither frictionless nor perfectly bonded. By use of a thermodynamic method with the assumption that the detachment starts at the periphery so that the contact area remains circular all the time, the pull-off force to separate the rigid cylindrical punch from the elastic layer is obtained. The results show that the pull-off force strongly depends on the contact conditions at the interfaces.
Introduction Advances in instrumentation in the past decade now permit indentation techniques to be used in the measurement of adhesion between solid surfaces based on contact mechanics. For example, Veeramasuneni et al.1 used the atomic force microscope to measure the interaction forces between silica and R-alumina. Toikka et al.2 measured the pull-off force between iron oxide spheres and silica and then calculated the adhesion energy. Bowen et al.3 also used the atomic force microscope and measured adhesive forces between PS spheres and polymeric ultrafiltration membranes. To calculate the adhesion energy from the pull-off force, the popular equation used is the famous JohnsonKendall-Roberts (JKR)4 equation between a spherical indenter of radius R and a flat surface of a half space:
3 Fc ) - πγR 2
3πR(k1 + k2) [F + 3γπR + x6γπRF + (3γπR)2] 4 (2)
where
ki ) (1 - νi2)/πEi
Fc )
(1)
where Fc is the pull-off force and γ is the energy per unit area required to separate the two surfaces in contact. The relation between the load F and the contact radius a is
a3 )
involved in this equation are that γ is a constant over the contact area and does not change with either the time of contact or the load applied between the contact surfaces. These assumptions are usually not true, and when γ changes with both the load and the loading time, the JKR curves between load and contact area do not coincide for loading and unloading, namely, a contact hysteresis loop appears,5-8 even without any inelastic deformation of materials. This has caused a lot of trouble of using the JKR equation. To search for a better indentation method, a cylindrical indenter with a flat end came to mind. This problem has been analyzed by Kendall9 who gave the following pull-off force for the indentation of a half space:
i ) 1, 2
(3)
with 1 for the indenter, 2 for the half space, ν for the Poisson ratio, and E for Young’s modulus. The assumptions (1) Veeramasuneni, S.; Yalamanchilli, M. R.; Miller, J. D. Measurement of Interaction Forces between Silica and Alpha-alumina by Atomic Force Microscopy. J. Colloid. Interface Sci. 1996, 184, 504-600. (2) Toikka, G.; Hayes, R. A.; Ralston, J. Adhesion of Iron Oxide to Silica Studied by Atomic Force Microscopy. J. Colloid Interface Sci. 1996, 180, 329-338. (3) Bowen, W. R.; Hilal, N.; Lovitt, W. R.; Wright, C. J. A New Technique for Membrane Characterization: Direct Measurement of the Force of Adhesion of a Single Particle using an Atomic Force Microscope. J. Membr. Sci. 1998, 139, 269-274. (4) Johnson, K. L.; Kendall, K.; Roberts, A. D. Surface Energy and the Contact of Elastic Solids. Proc. R. Soc. London 1971, A324, 301313.
x
8πEa3γ 1 - ν2
(4)
where a is the radius of the rigid cylinder and E is Young’s modulus of the half space. Kendall9 demonstrated experimentally the relation between the pull-off force and the 3/2 power of a. Young’s modulus of the half space could be obtained from the same experiment by using the forcedisplacement relation for frictionless contact:
δ)
F(1 - ν2) 2aE
(5)
where δ is the displacement of the rigid cylinder relative to the position at zero force. If the half space is not that thick, it may become an elastic layer sandwiched between the indenter and a rigid (5) Silberzan, P.; Perutz, S.; Kramer, E. J.; Chaudhury, M. K. Study of Self-Adhesion Hysteresis of a Siloxane Elastomer Using the JKR Method. Langmuir 1994, 10, 2466-2470. (6) Ahn, D.; Shull, K. JKR Studies of Acrylic Elastomer Adhesion to Glassy Polymer Substrates. Macromolecules 1996, 29, 4381-4390. (7) Pertuz, S.; Kramer, E. J.; Baney, J.; Hui, C. Y.; Cohen, C. Investigation of Adhesion Hysteresis in Poly(dimethylsiloxane) Networks using the JKR Technique. J. Polym. Sci., Part B: Polym. Phys. 1998, 36, 2129-2139. (8) Luengo, G.; Pan, J.; Heuberger, M.; Israelachvili, J. Temperature and Time effects on the Adhesion Dynamics of Poly(butyl methacrylate)(PBMA) Surfaces. Langmuir 1998, 14, 3873-3881. (9) Kendall, K. The Adhesion and Surface Energy of Elastic Solids. J. Phys. D: Appl. Phys. 1971, 4, 1186-1195.
10.1021/la010409h CCC: $20.00 © 2001 American Chemical Society Published on Web 09/20/2001
Adhesion of a Rigid Punch to an Elastic Film
Langmuir, Vol. 17, No. 21, 2001 6525
coordinates (r,θ,z) are used such that the z-axis coincides with the axis of the punch, r is perpendicular to z, and θ is the angle between r and a reference line which is also perpendicular to z. The nonzero displacements corresponding to these coordinates (r,θ,z) are ur and uz, which satisfy the following equations:
ur
∇2ur -
2
r
∇2uz ) Figure 1. Adhesion of a cylindrical indenter to a thin film deposited on a rigid substrate.
Fc ) πa
x
2Kγ h
x
2Eγ h(1 - ν2)
where E is Young’s modulus and ν is Poisson’s ratio of the thin film. Equation 7 is derived in the Appendix and has been checked by finite element analysis against eq 6 using different Poisson ratios. The equation presented by Shull and Crosby10 (their eq 6) is also wrong. By using frictionless conditions between a rigid flatended cylinder and an elastic layer, Shull and Crosby10 obtained a semiempirical relation for the crack driving force by fitting the finite element results. The same problem was reported more recently by Lin et al.11 who did finite element calculations for both stick and slip boundary conditions at the cylinder/film interface. But the interface between the film and the substrate was assumed to be bonded perfectly. No further work has been done, and no analytical results are available for the effects of all the contact conditions between the film and the substrate and between the punch and the film. So it is the purpose of this work to analyze in detail the adhesion between a thin film and a rigid cylindrical punch by using the integral transform method. The goal is to find a relation between the interfacial energy and the pull-off force under four limiting boundary conditions as well as different ratios of the punch radius to the film thickness. Adhesion of a Rigid Punch to an Incompressible Elastic Layer General Formulation. As shown in Figure 1, a rigid punch of radius a is contacting an incompressible elastic film of thickness h and shear modulus µ. The film is supported on a rigid substrate. Here, cylindrical polar (10) Shull, K. R.; Crosby, A. J. Axisymmetric Adhesion Tests of Pressure Sensitive Adhesives. J. Eng. Mater. Technol. 1997, 119, 211215. (11) Lin, Y. Y.; Hui, C. Y.; Conway, H. D. A Detailed Elastic Analysis of the Flat Punch (Tack) Test for Pressure-Sensitive Adhesives. J. Polym. Sci., Part B: Polym. Phys. 2000, 38, 2769-2784.
(8) (9)
∂ur ∂r
(10)
ur r
(11)
∂uz ∂z
(12)
)
(13)
σθθ ) -p + 2µ
(6)
(7)
1 ∂p µ ∂z
σrr ) -p + 2µ
σzz ) -p + 2µ
where K is the bulk modulus of the thin film. Kendall9 showed experimentally the relation between the pull-off force and a and h. However, eq 6 is incorrect; the correct version is
Fc ) πa2
1 ∂p µ ∂r
The stress components in terms of displacements are
substrate. For a very thin layer of thickness h and frictionless contact interfaces, Kendall9 gave the following pull-off force: 2
)
σrz ) µ
(
∂ur ∂uz + ∂z ∂r
and
1 p ) - (σrr + σθθ + σzz) 3
(14)
which is the mean pressure. When an axial force F is applied to the punch, a displacement δ results as shown in Figure 1. Hence, the boundary conditions underneath the punch inside and outside the contact area are
uz ) δ 2π
for r e a
∫0a σzz(r,0)r dr ) F
σzz(r,0) ) σrz(r,0) ) 0
for r > a
(15) (16) (17)
Four special cases are considered here for the other boundary conditions. Case I: Frictionless at Both Contact Interfaces.
σrz(r,0) ) 0
for r e a
σrz(r,h) ) 0 and uz(r,h) ) 0
(18) (19)
Case II: Frictionless between the Punch and the Elastic Film and Perfect Bonding between the Elastic Film and the Substrate.
σrz(r,0) ) 0
for r e a
uz(r,h) ) ur(r,h) ) 0
(20) (21)
Case III: Perfect Bonding between the Punch and the Elastic Film and Frictionless between the Elastic Film and the Substrate.
ur(r,0) ) 0
for r e a
σrz(r,h) ) 0 and uz(r,h) ) 0
(22) (23)
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Yang and Li
Case IV: Perfect Bonding at Both the Contact Interfaces.
ur(r,0) ) 0 for r e a
(24)
uz(r,h) ) ur(r,h) ) 0
(25)
In all of the four cases, no separation between the elastic film and the rigid substrate is assumed, that is, uz(r,h) ) 0. Before these equations are solved, the following dimensionless parameters are introduced:
(u/r ,
(r*, z*) ) (r, z)/a
u/z )
) (ur, uz)/a
{
f(ξh*) ) ξh* + exp(-ξh*) sinh(ξh*) for case I ξh* + cosh(ξh*) sinh(ξh*) ξh* + (ξh*)2 + exp(-ξh*) cosh(ξh*) for case II (ξh*)2 + cosh2(ξh*) and B2 satisfies the following equations:
B2 ) g(τ) +
∇2p ) 0
(27)
Due to the axisymmetry of the indentation problem, p can be expressed as
p)
∫0
∞
(40)
h* ) h/a (26)
It is known that12
P(ξ,z*)ξJ0(ξr*) dξ
2δ ∫01 [K(t + τ) + K(t - τ)]g(t) dt ) - πa
K(x) )
∫0∞ f(ξh*) cos(ξx) dξ
(43)
Perfect Bonding between the Punch and the Elastic Film (Stick Condition). Using the boundary conditions for cases III and IV, we have
(A1, A2) )
[
ψ(ξ,z*) )
∫0∞ u/z(r*,z*)r*J0(ξr*) dr*
(30)
for case III and
(A1, A2) )
u/r (r*,z*) )
∫0∞ φ(ξ,z*)ξJ1(ξr*) dξ
(31)
u/z (r*,z*) )
∫0∞ ψ(ξ,z*)ξJ0(ξr*) dξ
(32)
By substitution of eqs 28, 31, and 32 into eqs 8 and 9 and use of the incompressibility condition,
1 (B1, B2) (ξh*)2 -ξh* - sinh(ξh*)/2 cosh2(ξh*) sinh(ξh*)/2 - ξh* -sinh2(ξh*)
[
P(ξ,z*) ) 2ξµ[A1 cosh(ξz*) + A2 sinh(ξz*)] (34) φ(ξ,z*) ) -(A2ξz* + A1 + B2) cosh(ξz*) (A1ξz* + A2 + B1) sinh(ξz*) (35) ψ(ξ,z*) ) (A1ξz* + B1) cosh(ξz*) + (A2ξz* + B2) sinh(ξz*) (36) where A1, A2, B1, and B2 are functions of ξ to be determined by different interfacial boundary conditions. Frictionless between the Punch and the Elastic Film (Slip Condition). Using the boundary conditions for cases I and II, we have
B 1 + A2 ) -
( )
g(τ) + $(τ)
∫
1
0
∫01$(t) sin(ξt) dt
1 ξ
]( )
[
(45)
(46)
()
K11(τ,t) K12(τ,t) g(τ) 2δ 1 dt ) K21(τ,t) K22(τ,t) $(τ) πa 0
(47)
which are the Fredholm integral equations of the second kind. The kernels are defined as follows, for case III,
K11(τ,t) ) -
K12(τ,t) )
K21(τ,t) )
2 π
∫0∞
ξh* + exp(-ξh*) sinh(ξh*) ξh* + sinh(2ξh*)/2 cos(ξτ) cos(ξt) dξ (48)
ξh* cos(ξτ) cos(ξt) dξ ∫0∞ ξh* + sinh(2ξh*)/2
2 π
(49)
ξh* sin(ξτ) sin(ξt) dξ ∫0∞ ξh* + sinh(2ξh*)/2
2 π
(50)
(37)
ctanh(ξh*) for case I [tanh(ξh*) - ξh* sech2(ξh*)]-1 for case II (38) -1 (39) B2 ) -B1(1 + f(ξh*))
]
for case IV. Here,
(33)
the formal solutions of eqs 28-30 are
]
-cosh2(ξh*) sinh(ξh*)/2 1 (B1, B2) ξh* -sinh(ξh*)/2 sinh2(ξh*) (44)
(29)
{
(42)
and
∫0∞ u/r(r*,z*)r*J1(ξr*) dr*
A2 ) -B1
(41)
1 π
φ(ξ,z*) )
/ / 1 ∂(rur ) ∂uz + )0 r* ∂r* ∂z*
∫01 g(t) cos(ξt) dt
(28)
By application of the Hankel transforms of the first order and the zeroth order on u/r (r*,z*) and u/z (r*,z*), respectively,
A 1 ) B1
in which
K22(τ,t) ) -
2 π
∫0∞
and for case IV,
ξh* - exp(-ξh*) cosh(ξh*) ξh* + sinh(2ξh*)/2 sin(ξτ) sin(ξt) dξ (51)
Adhesion of a Rigid Punch to an Elastic Film
2 π
K11(τ,t) ) -
∫0∞
Langmuir, Vol. 17, No. 21, 2001 6527
ξh* + (ξh*)2 + exp(-ξh*) sinh(ξh*)
g(t) ∝ δ
(ξh*)2 + cosh2(ξh*)
By substitution of eq 61 into eqs 59 and 60, the total energy of the system is
cos(ξτ) cos(ξt) dξ (52)
∫0
K21(τ,t) )
(ξh*)2
∞
2 K12(τ,t) ) π
cos(ξτ) cos(ξt) dξ
(ξh*)2 + cosh2(ξh*) (ξh)2
∫0
∞
2 K22(τ,t) ) π
(54)
(ξh*)2 + cosh2(ξh*)
The same problem has been solved numerically by Conway et al.13 for cases I and III except for a general elastic layer with Poisson ratios of 0 and 0.3. The stress distribution at the contact interface between the indenter and the film was calculated for three a/h ratios, 1/4, 1, and 4, and compared with the limiting case of a half space or a/h ) 0 given by Boussinesq. In the limiting case, the boundary condition at the interface is immaterial. Detachment of the Rigid Punch from the Elastic Film. On the basis of eqs 12, 28, and 32, the normal stress underneath the punch is
(∫ x r*
g′(t)
dt -
t2 - r*2
g(1)
)
x1 - r*2
for r* < 1 (56)
By use of the force balance and integration of eq 56 over the area of contact, the magnitude of the applied load F is
∫01 σzz(r*,0)r* dr* ) 4πµa2∫01 g(t) dt
F ) -2πa2
(57)
(58)
UE )
∫0δ F dδ ) 4πµa2∫0δ ∫01 g(t) dt dδ
(59)
∫01 g(t) dt
4πγ ∂a | a ∂δ F
(60)
From eqs 42, 47, and 57, we have (12) Fung, Y. C. Foundations of Solid Mechanics; Prentice-Hall: Englewood Cliffs, NJ, 1977; p 157. (13) Conway, H. D.; Vogel, S. M.; Farnham, K. A.; So, S. Normal and Shearing Contact Stesses in Indented Strips Slabs. Int. J. Eng. Sci. 1966, 4, 343-359.
(64)
Critical Detachment Force for Thick and Thin Films Detachment of the Punch from an Elastic Half Space (a/h f 0). The normal stress underneath the punch is
σzz(r*,0) ) -
4µδ 1 πa x 1 - r*2
for r* < 1
(65)
which relates the applied load to the punch displacement as
∫01 σzz(r*,0)r* dr* ) 8aµδ
F ) -2πa2
for a , h (66)
By use of eq 63, the pull-off force is
for a , h
(67)
which agrees with eq 4 for 0.5 Poisson ratio, and so E ) 3µ. Detachment of the Punch from an Elastic Thin Film (h/a f 0). For h/a f 0, we need to consider four different boundary conditions. Case I: Frictionless at Both Contact Interfaces. The normal stress underneath the punch is
4µδ h
for r* < 1
(68)
which gives the following applied load:
F)
The potential energy due to the external load is
UP ) -Fδ ) -4πµa2δ
σc ) -
σzz(r*,0) ) -
The strain energy stored in the thin film is
(63)
where Fc is the tensile force to initiate detachment of the punch from the elastic film. The mean punch stress, σc, applied onto the punch to separate the punch from the elastic film is
Fc ) 4ax2πaµγ
The total energy, UT, of the system consists of the interfacial energy, US, between the punch and the thin film, the stored strain energy, UE, in the deformed thin film, and the potential energy, UP, of the punch in the impression/tension process. They are as follows: The interfacial energy is
US ) -πa2γ
)
g(t) dt - γ (62) δ
Fc∂δ | ) -4πaγ ∂a F
ξh* - (ξh*)2 - exp(-ξh*) cosh(ξh*)
1
∫01
The equilibrium contact between the punch and the film requires
sin(ξτ) sin(ξt) dξ (55)
σzz(r*,0) ) 2µ
(
Fδ ) 2
πa2 -2µδ2
(53)
∫0∞ (ξh)2 + cosh2(ξh)sin(ξτ) sin(ξt) dξ
2 π
UT ) US + UE + UP ) -πa2γ -
(61)
4πµa2δ h
for a . h
(69)
By use of eqs 63 and 69, the critical load to detach the punch from the thin film is
x8µγh
Fc ) πa2
for a . h
(70)
which is proportional to a2 and inversely proportional to the square root of the film thickness. Equation 70 agrees with eq 7 for ν ) 0.5.
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Yang and Li
Cases II and III: Frictionless between the Punch and the Elastic Film and Perfect Bonding between the Elastic Film and the Substrate or Perfect Bonding between the Punch and the Elastic Film and Frictionless between the Elastic Film and the Substrate. The normal stress underneath the punch (r* < 1) is
σzz(r*,0) ) -
3µδ (1 - r*2) 4h*3a
(71)
which gives the critical load to separate the punch from the thin film as
Fc )
x3µγ 2h
πa3 2h
for a . h
(72)
which is proportional to a3 and inversely proportional to h3/2. Case IV: Perfect Bonding at Both the Contact Interfaces. The normal load applied to the punch is
F)
4
3πµa δ 2h3
for a . h
Figure 2. Dependence of the stored strain energy on the film thickness.
(73)
By use of eqs 63 and 73, the critical load to detach the punch from the thin film is
Fc )
x3µγ 2h
πa3 h
for a . h
(74)
which is twice that required for Cases II and III. Numerical Calculations. The adhesion between the punch and the elastic film is represented by the Fredholm integral equations of the second kind as shown in eqs 42 and 47. The function K(x), eq 43, the Fourier integral, is first calculated numerically. For a given function K(x), the functions (g(τ) and $(τ)) are solved numerically in the form of the Chebyshev series of N terms (N g 5).15 Then, the load applied onto the punch is obtained by calculating the integral in eq 57, which is used to calculate the stored strain energy in eq 59. In the calculations, the number of terms N in the Chebyshev series that approximate the solutions of g(τ) and $(τ) is taken to be 10 when h/a > 0.5. For values of h/a < 0.5, the number of terms is increased to obtain accurate values for g(τ) and $(τ). Figure 2 shows the dependence of the stored strain energy (UE ) 0 for the stress-free state) on the film thickness. The ratio of UE/4aµδ2 starts at 1 for a/h ) 0 which corresponds to the punching of a half space. Then, it starts to increase with a/h and approaches the results of the four limiting cases for the thin film. For the same punch displacement, the largest stored strain energy is for the case of the perfect bond or stick condition on both the contact interfaces (case IV) and the smallest is for the case of the frictionless or slip condition on both the contact interfaces (case I). Note that for the half space, the slip or stick condition at the punch/film interface is immaterial. The dependence of the critical detachment force on a/h is shown in Figure 3. The critical load to separate the punch from the film increases with a/h. A higher load is required to detach the punch from the thin film compared to the case of a half space. For films of the same thickness, the highest load required to separate the punch from the film is for the perfect bond or stick condition on both the contact interfaces (case IV) and the lowest is for the
Figure 3. Effect of the film thickness on the critical detachment force.
frictionless or slip condition on the two contact interfaces (case I). Note that for the half space, the slip or stick condition at the punch/film interface is immaterial. As shown in Figure 3, to use the equation for the half space, eq 67, the a/h should be less than only about 0.5. For a/h greater than about 10, the equations for thin films can be used, namely, eq 70 for the slip condition on both contact interfaces, eq 74 for the stick condition on both the contact interfaces, and eq 72 for one stick and one slip condition on the two contact interfaces. Because of the large difference between eqs 70 and 72 for thin films, it is possible to use the pull-off force to detect debonding at the film/substrate interface, provided that the punch/film contact interface is frictionless. If the punch/film interface is perfect bonding, the difference between the stick and slip boundary conditions at the film/substrate interface is small. Still, it is possible to detect debonding at the film/substrate interface by precise pull-off force measurements. Experiments
(14) Timoshenko, S. P.; Goodier, J. N. Theory of Elasticity; McGrawHill: New York, 1970; p 70. (15) Erdogan, F. SIAM J. Appl. Math. 1969, 17, 1041.
Self-adhesion of PDMS [poly(dimethylsiloxane)] was studied using stainless steel cylindrical indenters of 0.79,
Adhesion of a Rigid Punch to an Elastic Film
1.19, 1.59, 2.38, and 3.18 mm radii coated at the ends with a thin film of PDMS of about 10 µm thick. Each cylinder was pushed into a PDMS layer of about 5 mm thick at several different temperatures. Since a/h was between 0.16 and 0.84, an inspection of Figure 3 shows that eq 4 or eq 67 can be used. First, we checked the linear relation between the load and the displacement and the reversibility between loading and unloading. From the slope, we calculated the Young modulus of PDMS. Then, we checked the 3/2 power dependence of the radius of the cylinder on the pull-off force. From the pull-off force, we calculated the interface energy at the instant of pull-off. This technique gave us an opportunity to study the change of interface energy with contact time. The interface energy increases with the square root of contact time indicating a diffusional process involved in self-healing or bond formation. However, we must realize that the interface energy we measured is really the fracture toughness. For a brief description of experiments, please read ref 16, and for a complete report, please read ref 17. Conclusion The adhesion problem between a rigid flat-ended punch and an incompressible elastic film deposited on a rigid substrate under four limiting boundary conditions has been studied by using an integral transform method. The four boundary conditions are either stick or slip at the punch/film interface and the film/substrate interface. It is found that the critical detachment load depends on the square root of the shear modulus and that of interfacial energy. This dependence is unaffected by the film thickness. The film can be considered as half space if the ratio of punch radius to film thickness a/h is less than about 0.5, and the pull-off force is given by eq 4 for any film and eq 67 for an incompressible film. For a/h larger than about 10, the film can be regarded as very thin and the pull-off force is given by eqs 70, 72, and 74 depending on the boundary conditions at the contact interfaces. For a/h between 0.5 and 10, Figure 3 must be used. Some experiments are described elsewhere. Because of the large difference between the slip and stick boundary conditions at the film/substrate interface, it is possible to detect debonding at such an interface by simply measuring the local pull-off force. Appendix Adhesion of a Rigid Punch to an Elastic Thin Film over a Rigid Substrate. Derivation of Equation 7. First, (16) Yang, F.; Zhang, X.; Li, J. C. M. Adhesion of a Rigid Cylinder to an Incompressible Film. MRS Symposium, Fall, 2000, to appear. (17) Zhang, X.; Li, J. C. M. Self-Adhesion of PDMS by Using a Cylindrical Indenter. To be published.
Langmuir, Vol. 17, No. 21, 2001 6529
let us cut out a thin disk of radius a from the thin film under the punch. By application of a compressive force F or a stress
σ ) F/πa2
(A1)
axially over the surface of the disk by the punch, the disk will expand in the radial direction and increase its radius by νaσ/E. Because of the slip condition on both the contact interfaces, the strain and stress in the disk are uniform. Now put the enlarged disk back to the hole in the film. A pressure p will develop around the periphery of the disk. This pressure p will shrink the disk in the radial direction by14
ud ) -ap(1 - ν)/E
(A2)
and enlarge the hole by14
uh ) ap(1 + ν)/E
(A3)
To fit the disk back into the hole, we must have
uh - ud ) νaσ/E
(A4)
p ) νσ/2
(A5)
which gives
This pressure will make the disk a little thicker so the depth of penetration or the displacement of the punch is
δ)
(
)
ν2 ν2 hσ hσ 1- ) (1 - ν2) E 2 2 E
(A6)
from which we can calculate the stored energy in the system:
UE )
h(1 - ν2)F2
(A7)
2πa2E
and the potential energy due to the applied force
UP ) -Fδ ) -
h(1 - ν2)F2
(A8)
πa2E
Maximization of the total (free) energy
UT ) -πa2γ + UE + UP ) -πa2γ at constant F with respect to a gives eq 7. LA010409H
Fδ 2
(A9)
