A simplified formulation of adhesion problems with elastic plates

Carmel Majidi , George G. Adams

Abstract

The solution of adhesion problems with elastic plates generally involves solving a boundary-value problem with an assumed contact area. The contact region is then found by minimizing the total potential energy with respect to the contact area (i.e. the contact radius for the axisymmetric case). Such a procedure can be extremely long and tedious. Here, we show that the inclusion of adhesion is equivalent to specifying a discontinuous internal bending moment at the contact region boundary. The magnitude of this moment discontinuity is related to the work of adhesion and flexural rigidity of the plate. Such a formulation can greatly reduce the algebraic complexity of solving these problems. It is noted that the related plate contact problems without adhesion can also be solved by minimizing the total potential energy. However, it has long been recognized that it is mathematically more efficient to find the contact area by specifying a continuous internal bending moment at the boundary of the contact region. Thus, our moment discontinuity method can be considered to be a generalization of that procedure which is applicable for problems with adhesion.

1. Introduction

Problems involving adhesion of elastic plates are important in a variety of applications. An important process in the semiconductor industry is wafer bonding. Wafers are typically bonded together by applying a load at the centre which causes the bond front to spontaneously propagate radially outward to the wafer edge (Turner & Spearing 2002), i.e. to zip shut. This process is driven by the combination of applied pressure and the surface interactions due to weak interatomic forces, such as van der Waals forces. A permanent bond is created by the formation of covalent bonds which normally occurs at a high temperature. Turner & Spearing (2002) modelled this process using elastic plate theory and minimizing the total potential energy with respect to the bond radius. An experimental investigation (Turner & Spearing 2005) verified the results of their earlier paper.

Nanomanufacturing is a relatively new field which is of great current interest. One potential method of nanomanufacturing uses self-assembly of nanoelements on a template followed by the transfer of these elements from the template to the device wafer (Pamp & Adams 2007). However, wafers typically have waviness or bow that can lead to surface deviations of many micrometres. Thus, an outside force, or pressure, is required to bring these surfaces into intimate contact to accomplish the transfer of the nanoelements. This process is similar to that used in wafer bonding. However, unlike with semiconductors, it is very important to avoid bonding of the wafers. Thus, the elastic energy stored during the deformation must be sufficient to separate the surfaces after the pressure is removed. Both cylindrical bending and axisymmetric deformation were considered by Pamp & Adams (2007). The axisymmetric problem was solved by minimizing the total potential energy. That procedure was extremely lengthy and required the use of a symbolic interpreter language to perform the algebra. In the analysis of cylindrical bending, it was shown that for the particular cases studied, the effect of adhesion is equivalent to a discontinuity in the internal bending moment at the contact boundary.

There is also considerable interest in bio-inspired adhesives, in particular in the peeling or attachment of plate-like spatulae or lamellae of wall-climbing insects and lizards. These latter structures are treated as a thin elastic plate in contact with a rigid, non-flat substrate (Persson & Gorb 2003; Majidi & Fearing 2008). In biology, cell adhesion is also an active area of study and has been addressed by various plate theories and adhesion models (Seifert 1991; Rosso et al. 2000; Wan & Liu 2001).

In general, the method used to solve adhesion problems with elastic plates involves solving a boundary-value problem with an assumed contact area. The actual contact region is then found by minimizing the total potential energy (which includes the work of adhesion) with respect to the contact area (i.e. the contact radius for an axisymmetric problem). This procedure can be extremely cumbersome and tedious. In this paper, we show that the inclusion of adhesion is equivalent to specifying a discontinuous internal bending moment at the contact region boundary. The magnitude of this moment discontinuity is shown to be equal to the square root of twice the product of the work of adhesion and the plate flexural rigidity. Such a formulation can lead to an enormous reduction in the algebraic complexity of solving adhesion problems with elastic plates.

It is noted that related contact problems without adhesion can also be solved by minimizing the total potential energy (which in this case does not include the work of adhesion). However it has long been recognized that it is mathematically more efficient to find the contact area by specifying a continuous internal bending moment at the boundary of the contact region (Timoshenko & Woinowsky-Kreiger 1959, pp. 308–313; Benson 1991). Thus, our moment discontinuity method (MDM) constitutes a generalization of that procedure which is applicable for contact problems with adhesion. The use of the method is illustrated with some example problems.

2. Model

Suppose that an elastic, thin-walled, axisymmetric body (e.g. hollow sphere, a bowl or a circular disk) makes point contact with a rigid, axisymmetric surface and that the axis of symmetry for the two bodies coincide. As illustrated in figure 1a, point contact is only possible if the local curvature of the plate is more positive than that of the surface.

Figure 1

An initially curved plate adhering to a rigid surface under the action of line forces and moments. (a) Natural configuration; and (b) equilibrium configuration.

Interfacial adhesion and externally applied loads cause the contact to grow from a point to a circle of radius a, as shown in figure 1b. External loads can be applied as surface tractions (not shown in the figure) or line forces and moments at the edge of the plate. Moreover, the edge of the plate may be subject to kinematic constraints that prevent displacement and/or rotation. As with the geometry, the external loads and constraints are axisymmetric.

(a) Kinematics

Let the coordinate s∈[0,L] denote the radial arclength of a point on the midplane, where L is the radial arclength of the plate. Next, define the rotation angle θ=θ(s) of the undeformed midplane with respect to horizontal, as illustrated in figure 1a. At equilibrium, the rotation angle becomes θ+ϕ, where ϕ=ϕ(s) is the change in angle induced by adhesion and external loads. At equilibrium, points may have also displaced tangentially by an amount us=us(s). These deformations lead to elastic strains that also depend on the change in curvature ϕ′=dϕ/ds and tangential stretch Embedded Image.

Let x=x(s) denote the radius of the plate at s. As shown in figure 1b, the radius becomes x+ux at equilibrium. Both x(s) and ux(s) are evaluated by integrating over the interval [0,s]:Embedded Image(2.1)Similarly, y=y(s) and uy(s) denote the initial height and vertical displacement of a point at s,Embedded Image(2.2)

Because the undeformed plate is axisymmetric, it will not only have a curvature θ′+ϕ′ in the radial direction, but also a curvature κ in the hoop direction. At equilibrium, the curvature becomes κ+ζ. From geometry it follows thatEmbedded Image(2.3)

At equilibrium, the plate contacts the substrate over the interval [0,a], where the radial arclength a of the contact zone is less than or equal to L. Although ϕ and u are continuous over the entire domain [0,L], their derivatives ϕ′ and Embedded Image will generally have a jump at s=a. Hence, it is convenient to defineEmbedded Image(2.4)

(b) Boundary conditions

Both θ and the surface profile of the substrate are prescribed and so it is straightforward to determine the angle change ϕα required for contact. If the interface is non-slip, then uα may also be prescribed (e.g. uα=0). However, if the contact is frictionless then uα is unknown and must be solved for. Regardless, the plate is subject to the following boundary conditions:Embedded Image(2.5)There may also be a kinematic constraint at the edge s=L that leads to the boundary conditionEmbedded Image(2.6)Lastly, there could be end constraints of the formEmbedded Image(2.7)andEmbedded Image(2.8)Equations (2.7) and (2.8) constrain the displacement of the edge of the plate in the vertical and radial directions respectively.

(c) Energy functional

The total potential energy of the system Π is composed of the elastic strain energy, the work of the applied forces, the virtual work of the reaction forces associated with the end contraints (2.7) and (2.8), and the energy of adhesion. For a von Kármán plate, the elastic energy density per unit area isEmbedded Image(2.9)where D=EH3/12(1−ν2) is the flexural rigidity; E is the elastic modulus; H is the plate thickness; and ν is Poisson's ratio. The work of the line forces and moment at s=L areEmbedded Image(2.10)The virtual work associated with the end constraints areEmbedded Image(2.11)where the Lagrangian multipliers λx and λy represent the reaction forces necessary to maintain (2.7) and (2.8) (Lanczos 1970). The work of the surface tractions are obtained by integratingEmbedded Image(2.12)andEmbedded Image(2.13)over the interval [0,L]. Lastly, γ denotes the energy of adhesion per unit area of contact.

The total potential energy is obtained by subtracting the work and adhesion energy from the elastic energy (Kendall 1971). This yields a functional of the formEmbedded Image(2.14)where the Lagrangian densities areEmbedded Image(2.15)Embedded Image(2.16)Here, hα and hβ are defined as Embedded Image and Embedded Image, respectively, where h is the integrand that corresponds to the sum of work of the applied line forces, reaction forces and surface tractions,Embedded Image(2.17)For convenience, the constant terms that do not contain ϕ, ϕ′, u or u′ are omitted from the energy functional.

3. Analysis and results

At equilibrium, the energy functional Π must be stationary with respect to kinematically admissible variations of the formEmbedded Image(3.1)These conditions lead to the differential and boundary forms of the balance laws for both linear and angular momentum as well as a jump condition at the interface s=a. The main results are presented in §3ac.

(a) Moment balance

Infinitesimal variations in ϕα and ϕβ lead to a variation δΠϕ in the total potential energy of the system. At equilibrium, δΠ must vanish for any arbitrary functions δϕα and δϕβ that satisfy the kinematic boundary, i.e. the variations must be kinematically admissible. As shown in appendix A, δΠ=0 if and only if the following conditions are satisfied:Embedded Image(3.2)Embedded Image(3.3)Equations (3.2) and (3.3) represent the differential and boundary form of the moment balance, respectively.

(b) Force balance

Similarly, infinitesimal variations in uα and uβ result in a variation δΠu that vanishes at equilibrium for arbitrary but kinematically admissible functions δuα and δuβ. This implies (see appendix A)Embedded Image(3.4)Embedded Image(3.5)Embedded Image(3.6)Embedded Image(3.7)Equations (3.4), (3.5) and (3.6), (3.7) are equivalent to the differential and boundary forms of the linear force balance, respectively.

(c) Adhesive boundary condition

Lastly, varying the arclength a of the adhesive contact zone by an infinitesimal amount δa results in a variation of Π that has the form δΠa=(dΠ/da)δa. At equilibrium, dΠ/da must equal 0, which results in the following jump condition at s=a:Embedded Image(3.8)where the last term is dropped if uα is not prescribed. Details of the derivation for (3.8) are presented in appendix B. In brief, the derivation involves Leibniz's integration rule and makes use of the boundary conditions and balance laws.

The adhesive boundary condition (3.8), which is derived using the calculus of variations, represents a special case of the second Weierstrass–Erdmann Corner Condition or the variable endpoint (free horizon) problem (Seifert 1991; Troutman 1996). In general, it can also regarded as an Eshelbian energy momentum balance or material (configurational) force balance at the edge (s=a) of the adhesive zone. In this respect, it may be possible to derive the adhesive boundary condition (3.8) using the J-integral method or configurational mechanics (Majidi 2007). In §4, we show that for a frictionless contact (such that uα is not prescribed), the adhesive boundary condition can also be regarded as a discontinuity in the internal bending moment.

4. Moment discontinuity method

Suppose that the interface between the plate and substrate is frictionless. Since uα is not prescribed, the last term in (3.8) must be dropped. Substituting the expressions for Embedded Image and Embedded Image into (3.8) and noting that x, u, u′ and ϕ are continuous through s=a, the adhesive boundary condition becomesEmbedded Image(4.1)Using the expression of ψ in (2.9), it follows that Embedded Image. The curvature may be represented in terms of the internal momentEmbedded Image(4.2)which implies Embedded Image. According to the MDM, the right-hand side may be treated as an adhesion-induced singular moment Embedded Image that is applied at s=a. The negative sign is chosen because the effect of adhesion is to produce an applied moment in the negative θ direction (figure 1a). If the rigid surface were on the top of the plate then the sign of this adhesion-induced moment would be positive. The jump condition can be expressed asEmbedded Image(4.3)where Mo is the discontinuity in the internal bending moment. Again if the rigid surface was above the plate then the sign of Mo would be reversed. An extension of this MDM into two plates can be found in Majidi & Wan (2009) in which the square root of the sum of the squares of the discontinuities in bending moments is equal to Mo.

5. Examples

By replacing adhesion with a line moment Mo at the contact boundary, the MDM allows plate adhesion problems to be studied entirely within the framework of classical plate theories. This is in contrast to the energy approach used in §3, the J-integral method, or the material (configurational) force balance, which all require insights and analyses that are outside the scope of Newtonian balance laws or moment-curvature constitutive models. For many one-dimensional and axisymmetric adhesion problems, MDM greatly simplifies the analysis and reduces derivation of the contact length a to only a few lines of algebra. This is demonstrated in the following examples.

(a) Adhesion to a cylinder

As illustrated in figure 2, a rectangular plate of length 2L is pressed onto a cylinder of radius R by a vertical line force Fy=−P applied at its two ends. Both the adhesion and applied loads lead to a contact zone of length 2a≤2L between the plate and cylinder. By symmetry only half of this configuration is analysed. At the edge of the contact zone, the plate has internal moments Mα=−D/R and Mβ=−P(La) just inside and outside of the zone, respectively. These internal moments must balance the singular moment Embedded Image induced at the edge:Embedded Image(5.1)It is noted that MDM was developed for an axisymmetric geometry. However, it is straightforward to show that this method also works for cylindrical bending. In the absence of a load P, the plate will either adhere completely (a=L) or not at all (a=0) depending on whether the magnitude of Mα is less than or greater than Mo. It is straightforward, though significantly more tedious, to obtain a using an energy minimization argument.

Figure 2

(a) Adhesion of a rectangular plate of length 2L to a rigid cylinder of radius R; (b) adhesion of a circular plate of radius L to a rigid sphere of radius R. (c) Both plates are subject to a line force Fy=−P along their edge.

(b) Adhesion to a sphere

Now suppose that a circular plate of radius L is pressed into a sphere of radius R. Referring to figure 2, the edge of the plate is subject to a vertical line force Fy=−P. According to MDM, the radius aL of the contact zone must satisfy the moment balance Mβ=Mα+Mo. The internal moment inside but near the edge of the contact zone is Mα=−(1+ν)D/R while the internal moment just outside of the contact zone is Embedded Image.

The end force P results in a line shear load Q(s)=PL/s. Therefore, from eqn (54) in Timoshenko & Woinowsky-Kreiger (1959, p. 53) and the boundary conditions (2.5)1 and (3.3), it follows that ϕβ=ϕβ(s) is the solution toEmbedded Image(5.2)

Alternatively, (5.2) may be derived by substituting the Lagrangian density obtained from §2c into the moment balance (3.2). Noting that the plate is initially flat (θ=0), it follows from (2.1) and (2.3) that x=s, ux≈0 and ζϕβ/s. Ignoring elastic stretching (u and u′)Embedded Image(5.3)An algebraic solution is obtained in Maple 12 and is used to evaluate Embedded Image. Substituting this into the moment balance Mβ=Mα+Mo:Embedded Image(5.4)The contact radius at equilibrium is determined by solving the above equation for a.

The adhesion of a circular plate to a sphere was previously studied by Majidi & Fearing (2008). In their analysis, a bound on a was obtained by ignoring strain energy in the non-contacting portion (i.e. ψβ=0) and performing an energy minimization. Their equilibrium condition (2.8) is equivalent to the condition γ=ψα(a) that results from (4.1) for ψβ=0. Equation (5.4) represents a significant improvement on this result, since it allows for an applied force P and furnishes an explicit value for a at equilibrium rather than a bound.

(c) Adhesion under a uniform surface traction

Finally, consider the adhesion of a circular plate of radius L to a rigid flat surface under a uniform surface traction ty=−t. As illustrated in figure 3, the plate is initially concave up with a slight spherical bow. The initial angle of rotation along the arclength is thus θ=s/R, where R is the natural radius of curvature. For RL and assuming that the deflections are small compared to the thickness, the small-angle theorem may be employed and effects of the induced u′ can be neglected. Here, the line shear load is Q(s)=L2/2ss/2 and so from eqn (54) in Timoshenko & Woinowsky-Kreiger (1959, p. 53), it follows that:1Embedded Image(5.5)which is subject to the boundary conditions ϕβ(a)=−a/R, Mβ(a)=−D(1+ν)/R+Mo and Mβ(L)=0. The solution of (5.5) subject to these boundary conditions givesEmbedded Image(5.6)which relates the contact radius (a); the applied pressure (t); and work of adhesion (γ). This result is far simpler than that obtained by Pamp & Adams (2007), but gives the same numerical results.

Figure 3

Adhesion of a spherically bowed circular plate of radius L to a flat rigid substrate and subject to a uniform surface traction t.

6. Conclusion

In this paper, we have shown that the inclusion of adhesion, in a contact problem with an elastic plate, is equivalent to specifying a discontinuous internal bending moment at the contact region boundary. The magnitude of this moment discontinuity is equal to the square root of twice the product of the work of adhesion and the flexural rigidity of the plate. This formulation greatly reduces the algebraic complexity of solving adhesion problems with plates. Our moment discontinuity method can be considered a generalization of the well-known use of a continuous bending moment to solve plate contact problems without adhesion. The method has been implemented on some sample problems.

Appendix A. Derivation of moment and force balance

The first two variations in (3.1) lead to the following variation in the total potential energy of the system:Embedded Image(A1)By the chain rule (Lanczos 1970),Embedded Image(A2)Following the same argument for the other integral in (A 1), leads to the result:Embedded Image(A3)According to the boundary condition that ϕα is prescribed, δϕα must equal 0 for all s∈[0,a]. Moreover, δϕβ(a)=0 since ϕβ(a) is constrained to be equal to the fixed value ϕα(a). Also, if ϕβ is prescribed at s=L then δϕβ(L)=0 as well. For s between a and L, δϕβ is arbitrary and so δΠ vanishes only ifEmbedded Image(A4)If ϕβ(L) is not prescribed then δϕβ(L) is also arbitrary and so the condition δΠ=0 also requires Embedded Image.

Using a similar argument, it can be shown that variations of the third and fourth terms in (3.1) yieldEmbedded Image(A5)The condition δΠu=0 requires thatEmbedded Image(A6)Also, if uα is not prescribed then δΠu=0 only holds ifEmbedded Image(A7)and Embedded Image. Also, if uβ(L) is not prescribed, then equilibrium requires Embedded Image.

Appendix B. Derivation of adhesive boundary condition

The variation of the fifth term in (3.1) corresponds toEmbedded Image(B1)By the balance law (3.2) and the chain rule,Embedded Image(B2)

The variation δΠa is simplified by following the same argument for the other integrals and noting that dϕα/da=0 and (duα/da)s=0=0. Also, if uβ(L) and ϕβ(L) are prescribed then duβ/da=dϕβ/da=0 at s=L. Otherwise, it follows from (3.3) and (3.7) that Embedded Image. In either case,Embedded Image(B3)Hence,Embedded Image(B4)If uα is not prescribed, then u=u(s) is smooth at s=a since both uα and uβ are solutions to the same balance equations and subject to the boundary conditions uα(a)=uβ(a) and (3.6). Hence Embedded Image and so in light of (3.6) this implies that the last two terms of (B 4) cancel each other out. However, if uα is prescribed then Embedded Image and the second to last term will remain.

In order to determine dϕβ/da and duβ/da at s=a, consider a variation of the form a=a*+δa, where a* is the arclength of the contact zone at equilibrium and δa represents an infinitesimal variation from a* (Seifert 1991). This results in variations of the form Embedded Image and Embedded Image. According to the boundary conditions (and noting that the variations must be kinematically admissible), Embedded Image, ϕβ(a)=ϕα(a), Embedded Image and uβ(a)=uα(a), where both ϕα and uα are prescribed. Moreover, according to the fundamental theorem of calculus, for any function Χ=Χ(s), Embedded Image. Therefore,Embedded Image(B5)which, according to the boundary condition ϕβ(a)=ϕα(a), is equivalent toEmbedded Image(B6)Noting that Embedded Image and solving for Embedded Image implies that Embedded Image. Similarly, Embedded Image. Hence, the condition δΠa=0 impliesEmbedded Image(B7)If uα is not prescribed then the last term must be dropped.

Footnotes

  • Equation (5.5) can also be obtained from the arguments in §2c. As in the previous example, we may approximate x=s, ux=0 and ζ=ϕβ/s.

    • Received February 4, 2009.
    • Accepted March 31, 2009.

References

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