Elliptical adhesive contact under biaxial stretching

Abstract

Adhesive contact of the Hertzian indenter with an incompressible elastic substrate bi-directionally stretched along the indenter principal planes of curvature is considered in the Johnson-Kendall-Roberts theoretical framework. An approximate model is constructed by examining energy release rate conditions only on the edges of the minor and major axes of the contact ellipse. The effect of weak coupling between fracture modes I and II is introduced using a phenomenological mode-mixity function. This study was motivated by the need to model a passive-adhesive mechanism in cell mechanics on stretchable substrates.

Keywords: adhesion; elliptical contact; fracture mode mixity; incompressible substrate.

Figure 1.

Schematic of the adhesive contact between a Hertzian probe and an elastic half-space under bi-directional stretch. (Online version in colour.)

Figure 2.

Schematic of the adhesive contact between a spherical probe and an elastic half-space under unidirectional stretch.

Figure 3.

Variation of the contact area eccentricity (a) and the contact area major axis (b) as functions of the substrate uniaxial stretch in the case of zero normal loading. (Online version in colour.)

Figure 4.

Variation of the critical contact area eccentricity (a) and the critical contact area major axis (b) versus the compressive normal load. (Online version in colour.)

Figure 5.

Variation of the critical substrate stretch versus the compressive normal load. (Online version in colour.)

Figure 6.

Variation of the pull-off force versus the substrate stretch. (Online version in colour.)

Figure 7.

Variation of the critical contact area eccentricity (a) and the critical contact area major axis (b) at the moment of detachment versus the substrate stretch. (Online version in colour.)

Figure 8.

(a) Variation of the dimensionless major semi-axis of the contact area $\tilde{a}$ as a function of the dimensionless contact force $\tilde{P}$ for different levels of the substrate uniaxial stretch (the red dashed line corresponds to the critical state). (b) Variation of $\tilde{a}$ as a function of the reduced substrate uniaxial stretch ξε_y (the dot-dashed line corresponds to the critical state, while the double-dot-dashed line corresponds to the unstable critical state for the pull-off force). (Online version in colour.)

Figure 9.

Variation of the contact area eccentricity (a) and the contact area major axis (b) as functions of the substrate uniaxial stress in the case of zero normal loading. (Online version in colour.)

Figure 10.

Variation of the contact area in the strain-controlled (a) and stress-controlled (b) cases for zero normal loading. (Online version in colour.)

Figure 11.

Variation in the contact area in the strain-controlled (red solid lines), stress-controlled (blue solid lines) and axisymmetric (dashed lines) cases for zero normal loading. (Online version in colour.)

Figure 12.

Schematic of the bonded contact between an elliptical inextensible patch and an elastic half-space under unidirectional tensile stress. (Online version in colour.)