Adhesion Mechanics of A Spherical Shell Jiayi Shi , Sinan Müftü and Kai-tak Wan Mechanical Eng, Northeastern Univ, Boston, MA 02115, USA (
[email protected]) Introduction
dβ / d φ 0 R
sin φ0 − sin φ r0
(2)
Nφ = K (εφ + νεθ )
Nθ = K (εθ + νεφ )
(3)
M φ = D (κφ + νκθ )
M θ = D (κθ + νκφ )
(4)
κφ =
Adhesion between two solid spheres is extensively investigated in many branches of science and technology, especially in terms of the celebrated models by Johnson-Kendall-Roberts (JKR) and Derjaguin-Muller-Toporov (DMT). These models are, however, invalid in membranous spherical capsules as in biological cells and drug delivery microcapsules. In this poster, we consider adhesive contact of a spherical shell compressed by two parallel plates. We introduce adhesion into nonlinear shell model by balancing the thermodynamic energy of elastic energy, potential energy, and surface energy. This model couples bending and stretching components and allows both large deflection and edge rotation. Interrelationship between applied load, approach distance, contact radius, and deformed profile, as well as the “pull-off” phenomenon, are derived.
d (rV ) / dφ0 + r α pV = 0
(5a)
d (rH ) / dφ0 − α N θ + r α pH = 0
(5b)
d (rM φ ) / dφ0 + dr / dφ0 (M φ − M θ ) − r α Q = 0
(5c)
Numerical solution is obtained by employing finite difference method to solve the two-point boundary-value problem. The vertical displacement and applied force can be determined accordingly as
P
h
δ = w + R (1 − cos φ*0 )
(6)
P = − 2 πRV sin φ *0
(7)
2
2δ
2a
u r0
δ0
P0 = 0
-2
-4
δ2 P2 20
40
60
80
100
2
Contact area, πa (mm )
A linear elastic and isotropic spherical shell with elastic modulus, E, Poisson’s ratio, v, thickness, h, radius of curvature, R, compressed by two parallel rigid plates is shown in figure 1. Mechanical models employ Reissner’s theory for arbitrarily large deflection. Though the shell experiences large deformation, the linear elastic strain remains small. The governing equations of classical theory are given by the followings, in which the shell meridian angle, φ0, is employed as the only independent variable. εθ =
δ1
2
Theory
⎞ ⎟⎟ − 1 ⎠
P
+
0
⎛ du / dφ 0 ⎜⎜1 + dr0 / dφ 0 ⎝
*
P
-6
Figure 1. Schematic of a deformed spherical shell with thickness of h and an original radius R. Surface forces and external force P compels the shell in to an adhesive contact with radius a and approach distance 2δ.
∗
δ
Pull off at fixed P
Energy, UT (μJ)
R
cos φ 0 cos φ
δ (mm) P (mN)
+
δ 0
εφ =
κθ =
(1)
Figure 2. Variation of the total energy UT as a function of contact area. By decreasing the contact area, equilibrium is stable for both fix load and fix displacement up to P*(δ*), between δ* and δ+, the equilibrium is unstable at fix load but stable at fix displacement; beyond δ+, equilibrium is always unstable. To solve the adhesive contact problem, we adopt the classical JKR approach of thermodynamic energy balance of the stored elastic energy in the elastic shell, UE, potential energy of applied load, UP, and surface energy, US. For the total energy of the system, UT = US + UE + UP, equilibrium is obtained by setting dUT / da = 0. Elastic energy stored is the area under the force-displacement curve F(δ). The potential energy of external load UP is evaluated at fixed load and fixed grips conditions. The energy needed to create new surface is given by US = − π a2 γ . Figure 2 shows UT as a
The model can be generalized by normalization of P, δ, and a, by putting P = P(Eh2/R4γ4)1/3, δ = δ(E2h4/R5γ2)1/3, and a = a(Eh2/R4γ)1/3, respectively. Figure 4 shows a(P) with the error bars denoting the numerical errors in the computational model. The other relations a(δ) and P(δ) can be similarly deduced. The normalization constants contrast that of the classical JKR spheres derived by Maugis. Here, the extra parameter of shell thickness, h, is present. The critical “pull-off” force is found to be P* = –12 ± 0.5, thus 1/ 3
⎛ γ 4 R4 ⎞ ⎟ P = P ⎜⎜ 2 ⎟ ⎝ Eh ⎠
G E
(8)
γ (mJ/m ) γ = 200 γ = 100 γ = 70
4
γ=0
D
Pull off
2
-10
-5
0
5
Applied load, P (mN)
10
15
20
Figure 3. A typical case (E = 2.0MPa, h = 200 μm, R = 15 mm) showing contact radius as a function of external load with effect of surface energy. Grey line shows the critical pull off trajectory.
1.5
Contact radius a = a [Eh2/ γR4)]1/3
The adhesion-delamination mechanical response is obtained by measuring the experimentally measurable quantities, namely, P, δ, and a. Figure 3 shows a(P) for a range of adhesion energy. In the presence of adhesion energy, non-zero contact radius is predicted even when there is no external load, and stronger adhesion leads to larger contact. A tensile force is needed to shrink the adhesion contact area. At a critical load, P *, the contact reduces to a * and the shell snaps from the substrate or “pull-off” (shown as grey circles in Fig. 3).
*
2
K
0 -15
Results
*
(a) 6
Contact radius, a (mm)
function of the contact area πa2. Equilibrium is achieved at ∂UT / ∂(πa2) = 0. For fixed load (dotted curves), equilibrium is stable at large contact radius, until critical load P* is reached, where [ ∂2UT / ∂(πa2) ]P = 0. Spontaneous pull-off occurs at P*. Beyond P*, equilibrium is always unstable for fixed load. For fixed grips (dash curves), equilibrium is stable until δ+ is reached, where [ ∂2UT / ∂(πa2) ]δ = 0. It is interesting to note that for δ+ < δ < δ*, equilibrium is stable in fixed grips but unstable in fixed load. A fluctuation from an unstable equilibrium can lead to spontaneous pull-off.
(a)
Tension Compression
1.0
0.5
Pull off
0.0 -20
-10
0
10
20
30
Applied load, P = P [ Eh2 / γ4R4)]1/3
Figure 4. Contact radius as a function of external load in terms of normalized parameter. (With standard deviation)
Within numerical errors, the contact radius reduces to zero at pull-off under fixed grips.
Acknowledgements
Conclusions
This work is supported by the National Science Foundation CMMI # 0757140.
The pseudo JKR adhesion of a spherical shell compressed between rigid plates is obtained by linear elasticity and thermodynamic energy balance. “Pull-off” phenomenon occurs at finite contact radius, and the interrelationship between applied load P, approach distance δ and contact radius a, are derived. Behavior of a shell is distinctly different from the conventional solid spheres, and finds many applications such as contact lens, microcapsules of drug delivery, tissue formation from cells, biofilm formation, and flexible electronic devices etc.
References 1. K.L. Johnson, K. Kendall, and A.D. Roberts, "Surface energy and the contact of elastic solids," Proceedings of the Royal Society of London, 301-313, 1971. 2. D. P. Updike and A. Kalnins, "Axisymmetric behavior of an elastic spherical shell compressed between rigid plates," Journal of Applied Mechanics, 37, 635-640 (1970).
