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Effect of work of adhesion on contact of a pressurized blister with a flat surface
International Journal of Adhesion & Adhesives 23 (2003) 207–214
Effect of work of adhesion on contact of a pressurized blister with a flat surface Raymond H. Plauta,*, Sally A. Whitea, David A. Dillardb a
Charles E. Via, Jr., Department of Civil and Environmental Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0105, USA b Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0219, USA Accepted 19 December 2002
Abstract The axisymmetric deflection of a circular thin film (or blister) pressurized on one side and coming into contact with a flat surface is considered. The edge of the blister is immovable. Attractive forces between the blister and surface cause an increase in the radius of the contact region. This problem is different from the usual blister test in which the film debonds at its edge as the pressure is increased. The blister is modeled here as a thin plate with no rotation at its edge, either excluding or including stretching resistance, and as a membrane with no bending resistance. Adhesion is modeled using both a JKR energy approach and a DMT approach. The effect of the work of adhesion on the contact radius is determined. r 2003 Elsevier Science Ltd. All rights reserved. Keywords: B. Boundary layers; B. Interfaces; D. Adhesion/non-stick; Blister
1. Introduction The blister test has been used to measure the adhesion and fracture toughness of a thin film bonded to a substrate [1–6]. Pressure is applied to a small region of initial delamination. The film deflects axisymmetrically and the debond radius grows. The energy release rate associated with the circular interface crack is of interest. In the constrained blister test, the transverse displacement of the film is limited to a certain value by a flat surface [7–10]. The present study considers the effect of the work of adhesion on a blister in a contact problem. The geometry resembles that of the constrained blister test, but debonding does not occur at the edge of the blister. Rather, the circular blister is immovable at its edge, which has a given radius. As pressure is applied on one side, the blister deflects and comes into contact with a flat, rigid surface parallel to the initial midplane of the blister. The attractive forces between the blister and the surface affect the radius of the circular contact area. The *Corresponding author. Tel.: +1-540-231-6072; fax: +1-540-2317532. E-mail address: [email protected] (R.H. Plaut).
relationship between the pressure, the adhesion forces, and the contact radius is investigated. A similar type of problem was treated by Shanahan [11], in which a pressurized spherical membrane (a ‘‘balloon’’) came into contact with a flat surface. The blister is assumed to be homogeneous, isotropic, and linearly elastic. A few studies of the blister test have included an initial tension [10,12], but it is not considered here. The displacements are assumed to be small, and three behavior models are analyzed: a thin plate in which stretching resistance is neglected (to be called the ‘‘linear plate’’); a thin plate in which both bending and stretching resistance are included (to be called the ‘‘nonlinear plate’’); and a membrane in which bending resistance is neglected. For the membrane model, as for the nonlinear plate, the two equilibrium equations are nonlinear and involve transverse and radial displacements. Friction between the flat surface and the blister is neglected, as well as the weight of the blister. Both JKR and DMT types of adhesion models are considered. In the following section, the linear plate is analyzed. Then the nonlinear plate is treated in Section 3, followed by the membrane in Section 4. Concluding remarks are presented in Section 5.
0143-7496/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0143-7496(03)00013-7
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2. Linear plate In this section, the blister is modeled as a thin plate in which the resistance to stretching of the midplane is neglected. The problem is axisymmetric and the transverse displacement W ðRÞ is a function of the radius R; with R ¼ R0 at the outer edge of the plate. The plate has thickness t; Young’s modulus E; Poisson’s ratio n; and flexural rigidity D ¼ Et3 =½12ð1 n2 Þ: The flat surface is at a height H above the plate, so that contact occurs when W ¼ H: The applied pressure is P and the contact radius of the blister with the surface is B: Equilibrium equations will involve the distance Y ¼ H W of the blister from the surface. The bending moment per unit length in the circumferential direction is Mr ; and the radial displacement (which will be included in subsequent sections) is UðRÞ: This study is conducted in terms of nondimensional variables. For the plate, these are W ; R0 H h¼ ; R0
w¼
Y ; R0 B b¼ ; R0 y¼
U R ; r¼ ; R0 R0 PR30 Mr R 0 ; mr ¼ : p¼ D D u¼
ð1Þ
The geometry of the plate along a diameter is depicted in Fig. 1. 2.1. No adhesion The behavior of the plate in the absence of adhesion between the plate and the surface is considered first. The linearized equilibrium equation in terms of yðrÞ has the general solution [13] yðrÞ ¼ A1 þ A2 ln r þ A3 r2 þ A4 r2 ln r
pr4 64
ð2Þ
in the region boro1: The plate is assumed to have no deflection or slope at r=1 (i.e., the edge is clamped). If po64 h, the plate does not contact the surface. If p>64 h, the quantities y, y0 , and y00 are zero at r ¼ b [14]. Application of the five boundary conditions to the solution in Eq. (2) leads to a transcendental equation in the contact radius b, and it is solved numerically [15]. The shape of the blister along a diameter is plotted in Fig. 2 for several values of the pressure p: The transverse
displacement is normalized by the height h of the surface. The contact radius b is almost zero for p=0.65 and is 0.079, 0.193, and 0.365 for p=1.2, 2, and 5, respectively. In Fig. 3, the lowest curve corresponds to a height h=0.02 and no adhesion (i.e., no van der Waals or dispersion interactions between the two materials). It illustrates how the contact radius b increases as the pressure p increases above p=1.28 for this case. 2.2. JKR type of analysis The JKR model of adhesion [16–18] involves an energy approach in which the influence of adhesion occurs within the contact region. The dimensional work of adhesion (surface energy per unit area) is denoted ðDgÞd and the nondimensional work of adhesion Dg is defined as Dg ¼ ðDgÞd R20 =D:
ð3Þ
For example, if an aluminum film (E=69 GPa, n=0.3) has thickness 0.1 mm and radius 20 mm, Dg ¼ 0:06ðDgÞd where ðDgÞd is in J/m2. This analysis is based on minimizing the total energy of the system. To nondimensionalize the energy, the dimensional energy is divided by D. For the linear plate, the total (nondimensional) energy is comprised of the strain energy VB due to bending, the potential VP of the pressure (i.e., the negative of the work done), and the surface energy VA of adhesion (i.e., the negative of the work of adhesion times the contact area). These energies can be written as Z 1 ðy0 Þ2 00 2 VB ¼ p rðy Þ þ dr; r b Z 1 yr dr; VA ¼ pb2 Dg: ð4Þ VP ¼ php þ 2pp b
The boundary conditions cause some of the usual terms in VB to be zero, and the nondimensional results for the linear plate do not depend on Poisson’s ratio n (but the dimensional results depend on n through D). The effect of the work of adhesion is modeled here by a bending moment mb per unit length along the edge of the contact region (i.e., at r ¼ b), which tends to increase the size of that region [19]. At the left end of the contact region in Fig. 1, the moment acts in a clockwise
Fig. 1. Geometry of plate along a diameter in nondimensional terms.
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Fig. 2. Blister shapes for linear plate with no adhesion, h ¼ 0:01:
Fig. 3. Effect of pressure on contact length for linear plate, JKR analysis, h=0.02.
direction, and at the right end it acts in a counterclockwise direction. If one considers the force distribution of the Lennard–Jones potential [18] and allows it to act over a small radial distance, one can imagine that it is replaced here by a moment (and a force which is absorbed into the shear force at r ¼ b). In accordance with the JKR approach, there is no effect of adhesion outside the contact region. The boundary condition y00 =0 at r ¼ b; used in Section 2.1, is replaced by y00 ¼ mb : The four boundary conditions on deflection and slope are applied to yðrÞ in Eq. (2) and the energies in Eq. (4) are then obtained in terms of b: For specified values of the height h; pressure p; and work of adhesion Dg; the value of b which minimizes the total energy is determined numerically using Mathematica [15,20]. The corresponding value of y00 ðbÞ is then mb : Results for several values of the work of adhesion are presented in Fig. 3 when h=0.02. As expected, at a constant pressure, the contact radius is greater if the work of adhesion is greater. 2.3. DMT type of analysis In the DMT approach [18,21–23], the influence of adhesion occurs in a small annular cohesive zone just outside the contact region. A Dugdale model is adopted, in which (in nondimensional terms) the attractive pressure is assumed to have the constant value f0 if the
Fig. 4. Geometry of half of plate cross section for DMT analysis.
gap yðrÞ is less than the value a: This is depicted in Fig. 4, where c denotes the outer radius of the cohesive zone. The general solution for the deflection in the region boroc has the same form as Eq. (2) except that p is replaced by p þ f0 : At r ¼ c; the five transition conditions are y ¼ a and continuity of deflection, slope, bending moment, and shear force. With the use of these conditions and the five boundary conditions described in Section 2.1, the eight unknown coefficients in the formulas for yðrÞ are eliminated and the resulting two coupled transcendental equations in b and c are solved numerically [15]. Fig. 5 shows curves of contact radius b versus pressure p for several values of f0 when a=106 and the height is h=0.01. The lowest curve corresponds to the case of no adhesion. Dashed portions of the two leftmost curves denote unstable equilibrium states, which exist when the curve has a negative slope. On those two curves, as the pressure is increased from zero, the plate reaches the surface and the contact radius increases smoothly from zero until a certain pressure is attained. Then the contact radius jumps to a larger value, given by the point on the curve above the lower vertical tangent. Upon further increase of pressure, the contact radius increases smoothly. If the pressure is then reduced, the path is followed downward until the upper vertical tangent is reached, and the contact radius then jumps to a smaller value (on the lower portion of the
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nonlinear equilibrium equations involving the transverse and radial displacements. 3.1. No adhesion The von Ka! rma! n assumptions are used, in which the radial displacements are smaller than the transverse displacements, and u0 ðrÞ and ½w0 ðrÞ2 are neglected in comparison with unity [13,26]. A thinness parameter x is defined by 2 R0 Et 3 2 x ¼ 12ð1 n Þ ¼ p: ð5Þ PR0 t Fig. 5. Effect of pressure on contact length for linear plate, DMT analysis, h=0.01, a=106.
For example, with n=0.3, the values x=20, 25, and 30 correspond to the ratios R0/t=27, 38, and 50, respectively. The equilibrium equations in uðrÞ and yðrÞ for the case of no adhesion are u u0 ð1 nÞ 0 2 ðy Þ y0 y00 u00 ¼ 2 ð6Þ r 2r r and 2 y00 y0 x3 y0000 ¼ y000 þ 2 3 p þ r r r ð1 n2 Þ nu0 y0 nuy00 uy0 nðy0 Þ3 ðy0 Þ2 y00 u0 y00 þ þ þ 2 þ þ ð7Þ r r r 2r 2
Fig. 6. Comparison of results for linear plate, h ¼ 0:03:
curve). In some cases, this vertical tangent might be at a value of p lower than the bifurcation value at b=0, and then the blister would jump off the surface. A comparison of some results for the linear plate is presented in Fig. 6. In this case the height of the surface is h=0.03. The lower curve shows the relationship between contact radius and pressure when adhesion is neglected. The other two curves show the results using the JKR approach with Dg=0.01 and using the DMT approach with a=105 and f0=1000. The area f0 a in the plot of the adhesive force versus the gap y represents the work of adhesion in the DMT model [24,25] and has the same value as Dg in Fig. 6, so that the two curves can be correlated. They are very close to each other except at small values of b:
3. Nonlinear plate The blister is again modeled as a thin plate in this section, but the analysis includes resistance to stretching as well as bending. Instead of a single linear equilibrium equation involving the transverse displacement and having an analytical solution, there are two coupled,
in the region boro1: Eq. (6) and an integrated version of Eq. (7) with no pressure are given in terms of U and W by Eq. (231) in Ref. [27]. In the contact region 0orob; with friction neglected, y=0 and the solution to Eq. (6) is u ¼ br where b is a positive constant. The edge r=1 is clamped. Therefore the boundary conditions can be written as follows: y ¼ 0; y0 ¼ 0; y00 ¼ 0; u ¼ bu0 at r ¼ b; y ¼ h; y0 ¼ 0; u ¼ 0 at r ¼ 1:
ð8Þ
Eqs. (6) and (7) are put in first-order form and solved numerically as an initial-value problem, using a shooting method and Mathematica. The values of n; h; b; and x are chosen, and the unknown values of y000 ðbÞ; u0 ðbÞ; and p are varied until the boundary conditions at r=1 are satisfied with sufficient accuracy. The results for the nonlinear plate and (in Section 4) for the membrane depend on Poisson’s ratio, and the value n=0.3 will be used in all the following numerical examples. For the nonlinear plate with no adhesion, the lowest curve in Fig. 7 shows a plot of the contact radius b versus the pressure p when h=0.05 and x=30. 3.2. JKR type of analysis In addition to the energies in Eq. (4), the total energy for the nonlinear plate includes the strain energy VS due
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almost bilinear, and naturally b increases as Dg increases. 3.3. DMT type of analysis
Fig. 7. Effect of pressure on contact length for nonlinear plate, JKR analysis, h ¼ 0:05; x=30.
Based on Fig. 4, Eqs. (6) and (7) are valid for coro1; whereas the pressure p in Eq. (7) is replaced by p þ f0 for boroc: The quantities y; y0 ; y00 ; u; and u0 are continuous at r ¼ c; with y ¼ a: In the shooting procedure, new coordinate systems are chosen with their origin at r ¼ c: One of the horizontal coordinates goes to r ¼ b and the other to r ¼ 1: Their lengths are scaled to be unity so that a single coordinate T with range 0oTo1 can be used. Values of n; h; x; f0 ; a; and p are chosen. For the 12 first-order equations, unknown initial conditions on y0 ; y00 ; y000 ; u; and u0 at r ¼ c are varied, along with b and c; until the boundary conditions in Eq. (8) are satisfied [15]. The relationship between the contact radius and the pressure is plotted in Fig. 9 for h ¼ 0:05; x=30, a=105, and several values of f0 ; including the case of no adhesion. With adhesion, the curves are almost linear and parallel. Fig. 10 presents a comparison of JKR and DMT results for the nonlinear plate, with the same values of
Fig. 8. Effect of work of adhesion on contact length for nonlinear plate, JKR analysis, h ¼ 0:05; x=30.
to stretching, which is given by [28] Z 1 2 px3 u ðy0 Þ2 0 þ 2nu u þ VS ¼ ð1 n2 Þ b r 2 ) 2 ðy0 Þ2 px3 b2 b2 : þr u0 þ dr þ 2 ð1 nÞ
ð9Þ
As in Section 2.2, no condition is placed on y00 ðbÞ: In the shooting method, the values of n; h; b; x; Dg; and p are specified, and y00 ðbÞ; y000 ðbÞ; and u0 ðbÞ are varied until the boundary conditions at r=1 in Eq. (8) are satisfied. Then the total energy is computed, with b ¼ u0 ðbÞ in VS ; and the procedure is repeated for different values of b until a local minimum of the total energy is obtained [15]. For the case h=0.05 and x=30, results are presented in Figs. 7 and 8. For the curves of b versus p shown in Fig. 7, the blister jumps onto or off the surface, respectively, as the pressure is increased or decreased past threshold values. Similar behavior occurs for the linear plate model if h=0.01 and Dg is sufficiently large [15]. Such jumps due to adhesion have been observed in other problems, e.g., between contacting spheres [29]. In Fig. 8, the contact radius b is plotted versus the work of adhesion Dg for fixed values of pressure. The curves are
Fig. 9. Effect of pressure on contact length for nonlinear plate, DMT analysis, h=0.05, x=30, a=105.
Fig. 10. Comparison of results for nonlinear plate, h ¼ 0:03; x ¼ 30:
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Dg; a; and f0 as in Fig. 6. Here h=0.03 and x=30 (whereas Fig. 6 corresponds to the case x=0). The correlation between the curves for the JKR and DMT types of analysis in Fig. 10 is not as good as in Fig. 6. For the JKR curve, the blister jumps onto the surface as the pressure is increased, whereas the behavior is smooth for the DMT model. Fig. 11. Blister shapes for membrane with no adhesion.
4. Membrane Finally, the bending resistance is neglected and the blister is modeled as a membrane. At the edge r ¼1, the blister is immovable, but rotation can occur. The same smallness assumptions on u0 and w0 are made as for the nonlinear plate, and there are two coupled, nonlinear equilibrium equations in the transverse and radial displacements. For the membrane, it is convenient to use a different nondimensionalization for the displacements, surface height, and work of adhesion: W Et 1=3 Y Et 1=3 wm ¼ ; ym ¼ ; R0 PR0 R0 PR0 U Et 2=3 H Et 1=3 ; hm ¼ ; um ¼ R0 PR0 R0 PR0 ðDgÞm ¼ ðDgÞd
ðEtÞ1=3 ðPR0 Þ4=3
:
ð10Þ
The use of the quantities in Eq. (10) enables the pressure to be eliminated from the governing equations, which are then given in terms of um ðrÞ and ym ðrÞ as
and ry0m u00m
um u0m ð1 nÞ 0 2 ðym Þ 2r r2 r
þ num þ
ru0m
ð11Þ
3 0 2 00 þ rðym Þ ym 2
¼ ð1 þ nÞu0m y0m
ð14Þ
which follows from Eqs. (1), (5), and (10). Membrane shapes for four values of b are shown in Fig. 11. As b approaches zero (i.e., the case of the pressurized membrane with no contact), hm approaches 0.65344, which is consistent with the value 0.6534 listed in Ref. [31]. The relationship between the contact radius and the pressure for h=0.01 and x=30 is plotted as the lowest curve in Fig. 12. If x or h increases, this curve shifts rightward. 4.2. JKR type of analysis
ðy0m Þ3
þ ð1 n2 Þr ð12Þ 2 for boro1 [14,30]. One can ‘‘solve’’ Eqs. (11) and (12) for u00m and y00m in terms of lower-order terms, and then put the resulting equations in first-order form. In the contact region (0orob), ym ¼ 0 and um ¼ br: The boundary conditions are ym ¼ 0; y0m ¼ 0; um ¼ bu0m at r ¼ b; ym ¼ h; um ¼ 0 at r ¼ 1:
ym ð1Þ; and the ratio of p to ðxhÞ3 is obtained from p ¼ ðxh=hm Þ3
4.1. No adhesion
u00m þ y0m y00m ¼
Fig. 12. Effect of pressure on contact length for membrane, JKR analysis, h ¼ 0:01; x ¼ 30:
ð13Þ
As for the nonlinear plate, a shooting method is used to obtain numerical solutions. Here the values of n and b are chosen, and the unknown value of u0m ðbÞ is varied until the solution yields um ð1Þ ¼ 0: Then hm is given by
In this case, the dimensional energy is multiplied by 10=3 ðEtÞ1=3 P4=3 R0 ; and the resulting total nondimensional energy is comprised of VS from Eq. (9) and VA and VP from Eq. (4) if x and p are replaced by unity and if the subscript m is added to y; u; h; and Dg: Since the membrane has no bending resistance, the effect of adhesion in this JKR model will not be represented by a moment, but by an upward line force along the edge of the contact region ðr ¼ bÞ: This allows a discontinuity of slope, so that y0 ðbÞ will be positive. To obtain numerical solutions, the values of n; h; b; x; p; and Dg are chosen, hm and ðDgÞm are computed from Eq. (10), and y0m ðbÞ and u0m ðbÞ are varied until the conditions at r=1 in Eq. (13) are satisfied. Then b is
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varied until the total energy has a local minimum [15]. Curves of b versus p are presented in Fig. 12 for h=0.01 and x=30. Except for very small values of Dg; the curves have unstable portions and the jump behavior occurs as the pressure is increased or decreased past threshold values. 4.3. DMT type of analysis The DMT analysis for the membrane is similar to that in Section 3.3. For boroc; the last term in Eq. (12) is multiplied by ð1 þ fm Þ and c is the value of r when ym ¼ am ; where f0 Et 1=3 ax fm ¼ ; am ¼ a ¼ 1=3 : ð15Þ PR0 p p Values of n; h; x; f0 ; a; and p are chosen, and fm and am are computed from Eq. (15). For the eight first-order equations, unknown initial conditions on y0m ; um ; and u0m at r ¼ c are varied, along with b and c; until the boundary conditions in Eq. (13) are satisfied [15]. Fig. 13 depicts curves of contact radius versus pressure for h=0.05, x=30, a=105, and several values of f0 ; including the case of no adhesion. Unlike the curves for the similar adhesion cases in Fig. 9, the ones here start with negative slopes and unstable equilibrium states, and jump phenomena occur. A comparison between three membrane curves is shown in Fig. 14, with h ¼ 0:03 and x ¼ 30: The correlation of the JKR and DMT curves, with f0 a ¼ Dg; is very good.
Fig. 13. Effect of pressure on contact length for membrane, DMT analysis, h ¼ 0:05; x ¼ 30; a=105.
Fig. 14. Comparison of results for membrane, h ¼ 0:03; x ¼ 30:
5. Concluding remarks The axisymmetric problem of a pressurized, linearly elastic, circular blister making contact with a flat surface has been analyzed. The thin film was modeled as a linear plate (with stretching resistance neglected), as a nonlinear plate (including both bending and stretching resistance), and as a membrane (with bending resistance neglected). The first of these models has analytical solutions for the transverse displacement and the total energy, while the other two involve coupled nonlinear differential equations (in the transverse and radial displacements) which were solved numerically using a shooting method. The edge of the blister was assumed to be immovable, and also to have no rotation in the plate analyses. Comparisons between results for the three blister models are shown in Figs. 15–17 when there is no adhesion. Fig. 15 depicts the shapes along a diameter for a fixed value of contact radius (b ¼ 0:1) when the surface height h is 0.03 and the thinness parameter x defined in Eq. (5) is 10. The required pressures are different for each model. The nonlinear plate, with both bending and
Fig. 15. Blister shapes with no adhesion when b ¼ 0:1; h ¼ 0:03; x ¼ 10:
Fig. 16. Blister shapes with no adhesion when p ¼ 3; h ¼ 0:03; x ¼ 10:
stretching resistance, exhibits the smallest transverse displacement outside the contact region. The membrane rotates at its edge and has the largest transverse displacement. For the same values of h and x; shapes are compared in Fig. 16 for a given applied pressure (p ¼ 3). Here the two plate models exhibit almost the same transverse
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very good, whereas the curves in Fig. 10 for the nonlinear plate are not too close in the range shown. The determination of surface and interfacial energies is challenging, especially when the materials involved are not soft. The problem analyzed in this paper represents a new type of blister test that might prove useful in determining the work of adhesion between two materials, even when neither is an elastomer or other soft material.
Fig. 17. Effect of pressure on contact length, no adhesion, h ¼ 0:05; L: linear plate; NP: nonlinear plate; M: membrane; 1: x ¼ 20; 2: x ¼ 25; 3: x ¼ 30:
displacement, while the membrane has much larger displacement and contact radius. For h ¼ 0:05; the effect of x is illustrated in Fig. 17. The curve denoted L corresponds to the linear plate model, for which x ¼ 0: The symbols M and NP denote membrane and nonlinear plate, respectively, and the numbers 1, 2, and 3 are associated with the values x=20, 25, and 30, respectively. As x increases, the curves shift to the right, i.e., the contact radius decreases. However, the nondimensionalization of the pressure (as well as the definition of x) involves the blister thickness. In dimensional terms, for a fixed pressure, as the blister thickness gets smaller, the contact radius tends to increase. For example, if x increases from 20 to 25 (i.e., from curve M1 to M2, or NP1 to NP2, in Fig. 17), the thickness decreases by a factor of 0.72 and the value of p increases by a factor of 2.73. Two models of the adhesion between the flat surface and the membrane were considered, based on the JKR and the DMT concepts. The JKR approach involves the total energy, including the surface energy of adhesion within the contact region and at its edge. The DMT approach involves forces acting in a small annular region just outside the contact area, and the forces were assumed to be uniformly distributed in that cohesive zone. It was shown that in some cases the blister ‘‘jumps’’ onto the surface when the pressure reaches a threshold value, and jumps off when the pressure is then decreased. In some other cases, the blister may initiate contact smoothly with an increase of pressure, but then jump to a larger contact radius at a certain value of the pressure. A similar jump to a smaller contact radius may occur as the pressure is decreased. One application of this work would be to determine the work of adhesion based on measurement of the contact radius. Attention was focused on the relationship of the contact radius and the applied pressure. The correlations between the JKR and DMT results in Fig. 6 for the linear plate and in Fig. 14 for the membrane are
Acknowledgements This research was supported by the National Science Foundation under Grant No. CMS-9713949.
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Effect of work of adhesion on contact of a pressurized blister with a flat surface Raymond H. Plauta,*, Sally A. Whitea, David A. Dillardb a
Charles E. Via, Jr., Department of Civil and Environmental Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0105, USA b Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0219, USA Accepted 19 December 2002
Abstract The axisymmetric deflection of a circular thin film (or blister) pressurized on one side and coming into contact with a flat surface is considered. The edge of the blister is immovable. Attractive forces between the blister and surface cause an increase in the radius of the contact region. This problem is different from the usual blister test in which the film debonds at its edge as the pressure is increased. The blister is modeled here as a thin plate with no rotation at its edge, either excluding or including stretching resistance, and as a membrane with no bending resistance. Adhesion is modeled using both a JKR energy approach and a DMT approach. The effect of the work of adhesion on the contact radius is determined. r 2003 Elsevier Science Ltd. All rights reserved. Keywords: B. Boundary layers; B. Interfaces; D. Adhesion/non-stick; Blister
1. Introduction The blister test has been used to measure the adhesion and fracture toughness of a thin film bonded to a substrate [1–6]. Pressure is applied to a small region of initial delamination. The film deflects axisymmetrically and the debond radius grows. The energy release rate associated with the circular interface crack is of interest. In the constrained blister test, the transverse displacement of the film is limited to a certain value by a flat surface [7–10]. The present study considers the effect of the work of adhesion on a blister in a contact problem. The geometry resembles that of the constrained blister test, but debonding does not occur at the edge of the blister. Rather, the circular blister is immovable at its edge, which has a given radius. As pressure is applied on one side, the blister deflects and comes into contact with a flat, rigid surface parallel to the initial midplane of the blister. The attractive forces between the blister and the surface affect the radius of the circular contact area. The *Corresponding author. Tel.: +1-540-231-6072; fax: +1-540-2317532. E-mail address: [email protected] (R.H. Plaut).
relationship between the pressure, the adhesion forces, and the contact radius is investigated. A similar type of problem was treated by Shanahan [11], in which a pressurized spherical membrane (a ‘‘balloon’’) came into contact with a flat surface. The blister is assumed to be homogeneous, isotropic, and linearly elastic. A few studies of the blister test have included an initial tension [10,12], but it is not considered here. The displacements are assumed to be small, and three behavior models are analyzed: a thin plate in which stretching resistance is neglected (to be called the ‘‘linear plate’’); a thin plate in which both bending and stretching resistance are included (to be called the ‘‘nonlinear plate’’); and a membrane in which bending resistance is neglected. For the membrane model, as for the nonlinear plate, the two equilibrium equations are nonlinear and involve transverse and radial displacements. Friction between the flat surface and the blister is neglected, as well as the weight of the blister. Both JKR and DMT types of adhesion models are considered. In the following section, the linear plate is analyzed. Then the nonlinear plate is treated in Section 3, followed by the membrane in Section 4. Concluding remarks are presented in Section 5.
0143-7496/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0143-7496(03)00013-7
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2. Linear plate In this section, the blister is modeled as a thin plate in which the resistance to stretching of the midplane is neglected. The problem is axisymmetric and the transverse displacement W ðRÞ is a function of the radius R; with R ¼ R0 at the outer edge of the plate. The plate has thickness t; Young’s modulus E; Poisson’s ratio n; and flexural rigidity D ¼ Et3 =½12ð1 n2 Þ: The flat surface is at a height H above the plate, so that contact occurs when W ¼ H: The applied pressure is P and the contact radius of the blister with the surface is B: Equilibrium equations will involve the distance Y ¼ H W of the blister from the surface. The bending moment per unit length in the circumferential direction is Mr ; and the radial displacement (which will be included in subsequent sections) is UðRÞ: This study is conducted in terms of nondimensional variables. For the plate, these are W ; R0 H h¼ ; R0
w¼
Y ; R0 B b¼ ; R0 y¼
U R ; r¼ ; R0 R0 PR30 Mr R 0 ; mr ¼ : p¼ D D u¼
ð1Þ
The geometry of the plate along a diameter is depicted in Fig. 1. 2.1. No adhesion The behavior of the plate in the absence of adhesion between the plate and the surface is considered first. The linearized equilibrium equation in terms of yðrÞ has the general solution [13] yðrÞ ¼ A1 þ A2 ln r þ A3 r2 þ A4 r2 ln r
pr4 64
ð2Þ
in the region boro1: The plate is assumed to have no deflection or slope at r=1 (i.e., the edge is clamped). If po64 h, the plate does not contact the surface. If p>64 h, the quantities y, y0 , and y00 are zero at r ¼ b [14]. Application of the five boundary conditions to the solution in Eq. (2) leads to a transcendental equation in the contact radius b, and it is solved numerically [15]. The shape of the blister along a diameter is plotted in Fig. 2 for several values of the pressure p: The transverse
displacement is normalized by the height h of the surface. The contact radius b is almost zero for p=0.65 and is 0.079, 0.193, and 0.365 for p=1.2, 2, and 5, respectively. In Fig. 3, the lowest curve corresponds to a height h=0.02 and no adhesion (i.e., no van der Waals or dispersion interactions between the two materials). It illustrates how the contact radius b increases as the pressure p increases above p=1.28 for this case. 2.2. JKR type of analysis The JKR model of adhesion [16–18] involves an energy approach in which the influence of adhesion occurs within the contact region. The dimensional work of adhesion (surface energy per unit area) is denoted ðDgÞd and the nondimensional work of adhesion Dg is defined as Dg ¼ ðDgÞd R20 =D:
ð3Þ
For example, if an aluminum film (E=69 GPa, n=0.3) has thickness 0.1 mm and radius 20 mm, Dg ¼ 0:06ðDgÞd where ðDgÞd is in J/m2. This analysis is based on minimizing the total energy of the system. To nondimensionalize the energy, the dimensional energy is divided by D. For the linear plate, the total (nondimensional) energy is comprised of the strain energy VB due to bending, the potential VP of the pressure (i.e., the negative of the work done), and the surface energy VA of adhesion (i.e., the negative of the work of adhesion times the contact area). These energies can be written as Z 1 ðy0 Þ2 00 2 VB ¼ p rðy Þ þ dr; r b Z 1 yr dr; VA ¼ pb2 Dg: ð4Þ VP ¼ php þ 2pp b
The boundary conditions cause some of the usual terms in VB to be zero, and the nondimensional results for the linear plate do not depend on Poisson’s ratio n (but the dimensional results depend on n through D). The effect of the work of adhesion is modeled here by a bending moment mb per unit length along the edge of the contact region (i.e., at r ¼ b), which tends to increase the size of that region [19]. At the left end of the contact region in Fig. 1, the moment acts in a clockwise
Fig. 1. Geometry of plate along a diameter in nondimensional terms.
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Fig. 2. Blister shapes for linear plate with no adhesion, h ¼ 0:01:
Fig. 3. Effect of pressure on contact length for linear plate, JKR analysis, h=0.02.
direction, and at the right end it acts in a counterclockwise direction. If one considers the force distribution of the Lennard–Jones potential [18] and allows it to act over a small radial distance, one can imagine that it is replaced here by a moment (and a force which is absorbed into the shear force at r ¼ b). In accordance with the JKR approach, there is no effect of adhesion outside the contact region. The boundary condition y00 =0 at r ¼ b; used in Section 2.1, is replaced by y00 ¼ mb : The four boundary conditions on deflection and slope are applied to yðrÞ in Eq. (2) and the energies in Eq. (4) are then obtained in terms of b: For specified values of the height h; pressure p; and work of adhesion Dg; the value of b which minimizes the total energy is determined numerically using Mathematica [15,20]. The corresponding value of y00 ðbÞ is then mb : Results for several values of the work of adhesion are presented in Fig. 3 when h=0.02. As expected, at a constant pressure, the contact radius is greater if the work of adhesion is greater. 2.3. DMT type of analysis In the DMT approach [18,21–23], the influence of adhesion occurs in a small annular cohesive zone just outside the contact region. A Dugdale model is adopted, in which (in nondimensional terms) the attractive pressure is assumed to have the constant value f0 if the
Fig. 4. Geometry of half of plate cross section for DMT analysis.
gap yðrÞ is less than the value a: This is depicted in Fig. 4, where c denotes the outer radius of the cohesive zone. The general solution for the deflection in the region boroc has the same form as Eq. (2) except that p is replaced by p þ f0 : At r ¼ c; the five transition conditions are y ¼ a and continuity of deflection, slope, bending moment, and shear force. With the use of these conditions and the five boundary conditions described in Section 2.1, the eight unknown coefficients in the formulas for yðrÞ are eliminated and the resulting two coupled transcendental equations in b and c are solved numerically [15]. Fig. 5 shows curves of contact radius b versus pressure p for several values of f0 when a=106 and the height is h=0.01. The lowest curve corresponds to the case of no adhesion. Dashed portions of the two leftmost curves denote unstable equilibrium states, which exist when the curve has a negative slope. On those two curves, as the pressure is increased from zero, the plate reaches the surface and the contact radius increases smoothly from zero until a certain pressure is attained. Then the contact radius jumps to a larger value, given by the point on the curve above the lower vertical tangent. Upon further increase of pressure, the contact radius increases smoothly. If the pressure is then reduced, the path is followed downward until the upper vertical tangent is reached, and the contact radius then jumps to a smaller value (on the lower portion of the
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nonlinear equilibrium equations involving the transverse and radial displacements. 3.1. No adhesion The von Ka! rma! n assumptions are used, in which the radial displacements are smaller than the transverse displacements, and u0 ðrÞ and ½w0 ðrÞ2 are neglected in comparison with unity [13,26]. A thinness parameter x is defined by 2 R0 Et 3 2 x ¼ 12ð1 n Þ ¼ p: ð5Þ PR0 t Fig. 5. Effect of pressure on contact length for linear plate, DMT analysis, h=0.01, a=106.
For example, with n=0.3, the values x=20, 25, and 30 correspond to the ratios R0/t=27, 38, and 50, respectively. The equilibrium equations in uðrÞ and yðrÞ for the case of no adhesion are u u0 ð1 nÞ 0 2 ðy Þ y0 y00 u00 ¼ 2 ð6Þ r 2r r and 2 y00 y0 x3 y0000 ¼ y000 þ 2 3 p þ r r r ð1 n2 Þ nu0 y0 nuy00 uy0 nðy0 Þ3 ðy0 Þ2 y00 u0 y00 þ þ þ 2 þ þ ð7Þ r r r 2r 2
Fig. 6. Comparison of results for linear plate, h ¼ 0:03:
curve). In some cases, this vertical tangent might be at a value of p lower than the bifurcation value at b=0, and then the blister would jump off the surface. A comparison of some results for the linear plate is presented in Fig. 6. In this case the height of the surface is h=0.03. The lower curve shows the relationship between contact radius and pressure when adhesion is neglected. The other two curves show the results using the JKR approach with Dg=0.01 and using the DMT approach with a=105 and f0=1000. The area f0 a in the plot of the adhesive force versus the gap y represents the work of adhesion in the DMT model [24,25] and has the same value as Dg in Fig. 6, so that the two curves can be correlated. They are very close to each other except at small values of b:
3. Nonlinear plate The blister is again modeled as a thin plate in this section, but the analysis includes resistance to stretching as well as bending. Instead of a single linear equilibrium equation involving the transverse displacement and having an analytical solution, there are two coupled,
in the region boro1: Eq. (6) and an integrated version of Eq. (7) with no pressure are given in terms of U and W by Eq. (231) in Ref. [27]. In the contact region 0orob; with friction neglected, y=0 and the solution to Eq. (6) is u ¼ br where b is a positive constant. The edge r=1 is clamped. Therefore the boundary conditions can be written as follows: y ¼ 0; y0 ¼ 0; y00 ¼ 0; u ¼ bu0 at r ¼ b; y ¼ h; y0 ¼ 0; u ¼ 0 at r ¼ 1:
ð8Þ
Eqs. (6) and (7) are put in first-order form and solved numerically as an initial-value problem, using a shooting method and Mathematica. The values of n; h; b; and x are chosen, and the unknown values of y000 ðbÞ; u0 ðbÞ; and p are varied until the boundary conditions at r=1 are satisfied with sufficient accuracy. The results for the nonlinear plate and (in Section 4) for the membrane depend on Poisson’s ratio, and the value n=0.3 will be used in all the following numerical examples. For the nonlinear plate with no adhesion, the lowest curve in Fig. 7 shows a plot of the contact radius b versus the pressure p when h=0.05 and x=30. 3.2. JKR type of analysis In addition to the energies in Eq. (4), the total energy for the nonlinear plate includes the strain energy VS due
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almost bilinear, and naturally b increases as Dg increases. 3.3. DMT type of analysis
Fig. 7. Effect of pressure on contact length for nonlinear plate, JKR analysis, h ¼ 0:05; x=30.
Based on Fig. 4, Eqs. (6) and (7) are valid for coro1; whereas the pressure p in Eq. (7) is replaced by p þ f0 for boroc: The quantities y; y0 ; y00 ; u; and u0 are continuous at r ¼ c; with y ¼ a: In the shooting procedure, new coordinate systems are chosen with their origin at r ¼ c: One of the horizontal coordinates goes to r ¼ b and the other to r ¼ 1: Their lengths are scaled to be unity so that a single coordinate T with range 0oTo1 can be used. Values of n; h; x; f0 ; a; and p are chosen. For the 12 first-order equations, unknown initial conditions on y0 ; y00 ; y000 ; u; and u0 at r ¼ c are varied, along with b and c; until the boundary conditions in Eq. (8) are satisfied [15]. The relationship between the contact radius and the pressure is plotted in Fig. 9 for h ¼ 0:05; x=30, a=105, and several values of f0 ; including the case of no adhesion. With adhesion, the curves are almost linear and parallel. Fig. 10 presents a comparison of JKR and DMT results for the nonlinear plate, with the same values of
Fig. 8. Effect of work of adhesion on contact length for nonlinear plate, JKR analysis, h ¼ 0:05; x=30.
to stretching, which is given by [28] Z 1 2 px3 u ðy0 Þ2 0 þ 2nu u þ VS ¼ ð1 n2 Þ b r 2 ) 2 ðy0 Þ2 px3 b2 b2 : þr u0 þ dr þ 2 ð1 nÞ
ð9Þ
As in Section 2.2, no condition is placed on y00 ðbÞ: In the shooting method, the values of n; h; b; x; Dg; and p are specified, and y00 ðbÞ; y000 ðbÞ; and u0 ðbÞ are varied until the boundary conditions at r=1 in Eq. (8) are satisfied. Then the total energy is computed, with b ¼ u0 ðbÞ in VS ; and the procedure is repeated for different values of b until a local minimum of the total energy is obtained [15]. For the case h=0.05 and x=30, results are presented in Figs. 7 and 8. For the curves of b versus p shown in Fig. 7, the blister jumps onto or off the surface, respectively, as the pressure is increased or decreased past threshold values. Similar behavior occurs for the linear plate model if h=0.01 and Dg is sufficiently large [15]. Such jumps due to adhesion have been observed in other problems, e.g., between contacting spheres [29]. In Fig. 8, the contact radius b is plotted versus the work of adhesion Dg for fixed values of pressure. The curves are
Fig. 9. Effect of pressure on contact length for nonlinear plate, DMT analysis, h=0.05, x=30, a=105.
Fig. 10. Comparison of results for nonlinear plate, h ¼ 0:03; x ¼ 30:
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Dg; a; and f0 as in Fig. 6. Here h=0.03 and x=30 (whereas Fig. 6 corresponds to the case x=0). The correlation between the curves for the JKR and DMT types of analysis in Fig. 10 is not as good as in Fig. 6. For the JKR curve, the blister jumps onto the surface as the pressure is increased, whereas the behavior is smooth for the DMT model. Fig. 11. Blister shapes for membrane with no adhesion.
4. Membrane Finally, the bending resistance is neglected and the blister is modeled as a membrane. At the edge r ¼1, the blister is immovable, but rotation can occur. The same smallness assumptions on u0 and w0 are made as for the nonlinear plate, and there are two coupled, nonlinear equilibrium equations in the transverse and radial displacements. For the membrane, it is convenient to use a different nondimensionalization for the displacements, surface height, and work of adhesion: W Et 1=3 Y Et 1=3 wm ¼ ; ym ¼ ; R0 PR0 R0 PR0 U Et 2=3 H Et 1=3 ; hm ¼ ; um ¼ R0 PR0 R0 PR0 ðDgÞm ¼ ðDgÞd
ðEtÞ1=3 ðPR0 Þ4=3
:
ð10Þ
The use of the quantities in Eq. (10) enables the pressure to be eliminated from the governing equations, which are then given in terms of um ðrÞ and ym ðrÞ as
and ry0m u00m
um u0m ð1 nÞ 0 2 ðym Þ 2r r2 r
þ num þ
ru0m
ð11Þ
3 0 2 00 þ rðym Þ ym 2
¼ ð1 þ nÞu0m y0m
ð14Þ
which follows from Eqs. (1), (5), and (10). Membrane shapes for four values of b are shown in Fig. 11. As b approaches zero (i.e., the case of the pressurized membrane with no contact), hm approaches 0.65344, which is consistent with the value 0.6534 listed in Ref. [31]. The relationship between the contact radius and the pressure for h=0.01 and x=30 is plotted as the lowest curve in Fig. 12. If x or h increases, this curve shifts rightward. 4.2. JKR type of analysis
ðy0m Þ3
þ ð1 n2 Þr ð12Þ 2 for boro1 [14,30]. One can ‘‘solve’’ Eqs. (11) and (12) for u00m and y00m in terms of lower-order terms, and then put the resulting equations in first-order form. In the contact region (0orob), ym ¼ 0 and um ¼ br: The boundary conditions are ym ¼ 0; y0m ¼ 0; um ¼ bu0m at r ¼ b; ym ¼ h; um ¼ 0 at r ¼ 1:
ym ð1Þ; and the ratio of p to ðxhÞ3 is obtained from p ¼ ðxh=hm Þ3
4.1. No adhesion
u00m þ y0m y00m ¼
Fig. 12. Effect of pressure on contact length for membrane, JKR analysis, h ¼ 0:01; x ¼ 30:
ð13Þ
As for the nonlinear plate, a shooting method is used to obtain numerical solutions. Here the values of n and b are chosen, and the unknown value of u0m ðbÞ is varied until the solution yields um ð1Þ ¼ 0: Then hm is given by
In this case, the dimensional energy is multiplied by 10=3 ðEtÞ1=3 P4=3 R0 ; and the resulting total nondimensional energy is comprised of VS from Eq. (9) and VA and VP from Eq. (4) if x and p are replaced by unity and if the subscript m is added to y; u; h; and Dg: Since the membrane has no bending resistance, the effect of adhesion in this JKR model will not be represented by a moment, but by an upward line force along the edge of the contact region ðr ¼ bÞ: This allows a discontinuity of slope, so that y0 ðbÞ will be positive. To obtain numerical solutions, the values of n; h; b; x; p; and Dg are chosen, hm and ðDgÞm are computed from Eq. (10), and y0m ðbÞ and u0m ðbÞ are varied until the conditions at r=1 in Eq. (13) are satisfied. Then b is
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varied until the total energy has a local minimum [15]. Curves of b versus p are presented in Fig. 12 for h=0.01 and x=30. Except for very small values of Dg; the curves have unstable portions and the jump behavior occurs as the pressure is increased or decreased past threshold values. 4.3. DMT type of analysis The DMT analysis for the membrane is similar to that in Section 3.3. For boroc; the last term in Eq. (12) is multiplied by ð1 þ fm Þ and c is the value of r when ym ¼ am ; where f0 Et 1=3 ax fm ¼ ; am ¼ a ¼ 1=3 : ð15Þ PR0 p p Values of n; h; x; f0 ; a; and p are chosen, and fm and am are computed from Eq. (15). For the eight first-order equations, unknown initial conditions on y0m ; um ; and u0m at r ¼ c are varied, along with b and c; until the boundary conditions in Eq. (13) are satisfied [15]. Fig. 13 depicts curves of contact radius versus pressure for h=0.05, x=30, a=105, and several values of f0 ; including the case of no adhesion. Unlike the curves for the similar adhesion cases in Fig. 9, the ones here start with negative slopes and unstable equilibrium states, and jump phenomena occur. A comparison between three membrane curves is shown in Fig. 14, with h ¼ 0:03 and x ¼ 30: The correlation of the JKR and DMT curves, with f0 a ¼ Dg; is very good.
Fig. 13. Effect of pressure on contact length for membrane, DMT analysis, h ¼ 0:05; x ¼ 30; a=105.
Fig. 14. Comparison of results for membrane, h ¼ 0:03; x ¼ 30:
5. Concluding remarks The axisymmetric problem of a pressurized, linearly elastic, circular blister making contact with a flat surface has been analyzed. The thin film was modeled as a linear plate (with stretching resistance neglected), as a nonlinear plate (including both bending and stretching resistance), and as a membrane (with bending resistance neglected). The first of these models has analytical solutions for the transverse displacement and the total energy, while the other two involve coupled nonlinear differential equations (in the transverse and radial displacements) which were solved numerically using a shooting method. The edge of the blister was assumed to be immovable, and also to have no rotation in the plate analyses. Comparisons between results for the three blister models are shown in Figs. 15–17 when there is no adhesion. Fig. 15 depicts the shapes along a diameter for a fixed value of contact radius (b ¼ 0:1) when the surface height h is 0.03 and the thinness parameter x defined in Eq. (5) is 10. The required pressures are different for each model. The nonlinear plate, with both bending and
Fig. 15. Blister shapes with no adhesion when b ¼ 0:1; h ¼ 0:03; x ¼ 10:
Fig. 16. Blister shapes with no adhesion when p ¼ 3; h ¼ 0:03; x ¼ 10:
stretching resistance, exhibits the smallest transverse displacement outside the contact region. The membrane rotates at its edge and has the largest transverse displacement. For the same values of h and x; shapes are compared in Fig. 16 for a given applied pressure (p ¼ 3). Here the two plate models exhibit almost the same transverse
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very good, whereas the curves in Fig. 10 for the nonlinear plate are not too close in the range shown. The determination of surface and interfacial energies is challenging, especially when the materials involved are not soft. The problem analyzed in this paper represents a new type of blister test that might prove useful in determining the work of adhesion between two materials, even when neither is an elastomer or other soft material.
Fig. 17. Effect of pressure on contact length, no adhesion, h ¼ 0:05; L: linear plate; NP: nonlinear plate; M: membrane; 1: x ¼ 20; 2: x ¼ 25; 3: x ¼ 30:
displacement, while the membrane has much larger displacement and contact radius. For h ¼ 0:05; the effect of x is illustrated in Fig. 17. The curve denoted L corresponds to the linear plate model, for which x ¼ 0: The symbols M and NP denote membrane and nonlinear plate, respectively, and the numbers 1, 2, and 3 are associated with the values x=20, 25, and 30, respectively. As x increases, the curves shift to the right, i.e., the contact radius decreases. However, the nondimensionalization of the pressure (as well as the definition of x) involves the blister thickness. In dimensional terms, for a fixed pressure, as the blister thickness gets smaller, the contact radius tends to increase. For example, if x increases from 20 to 25 (i.e., from curve M1 to M2, or NP1 to NP2, in Fig. 17), the thickness decreases by a factor of 0.72 and the value of p increases by a factor of 2.73. Two models of the adhesion between the flat surface and the membrane were considered, based on the JKR and the DMT concepts. The JKR approach involves the total energy, including the surface energy of adhesion within the contact region and at its edge. The DMT approach involves forces acting in a small annular region just outside the contact area, and the forces were assumed to be uniformly distributed in that cohesive zone. It was shown that in some cases the blister ‘‘jumps’’ onto the surface when the pressure reaches a threshold value, and jumps off when the pressure is then decreased. In some other cases, the blister may initiate contact smoothly with an increase of pressure, but then jump to a larger contact radius at a certain value of the pressure. A similar jump to a smaller contact radius may occur as the pressure is decreased. One application of this work would be to determine the work of adhesion based on measurement of the contact radius. Attention was focused on the relationship of the contact radius and the applied pressure. The correlations between the JKR and DMT results in Fig. 6 for the linear plate and in Fig. 14 for the membrane are
Acknowledgements This research was supported by the National Science Foundation under Grant No. CMS-9713949.
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