Cavity Growth from Crack-like Defects in Soft Materials

摘要

Failure of pressure sensitive adhesives are often preceded by the growth of interfacial or internal voids. The growth of these voids create a highly fibrillated micro-structure which is desirable since it enhances the toughness of the adhesive. There is a large literature on how cavities deform in a rubbery material, for example, see and the references within. The majority of these works focus on the growth of spherical or cylindrical cavities subjected to internal or external pressure since the problems can be reduced to a one dimensional one and therefore amendable to exact analysis. A well known result is that a spherical cavity in an infinite media consisting of an ideal rubber (i.e., a Neohookean material) will grow without bound when the applied hydrostatic tension P reaches a critical value of 2.5 μ, where μ is the infinitesimal shear modulus. However, this result seems to be at odd with experiments which showed that the critical hydrostatic tension depends on the cavity size. Specifically, smaller cavities require higher tension for rapid growth, whereas large cavities fail at a stress of approximately 2.5 μ. Since the Neohookean model typically underestimates the amount of strain hardening, one may argue that the higher critical stress required for rapid growth is due to hardening. This can not be the case for the following reason: since the cavity radius is the only length scale in the problem, the critical hydrostatic tension for cavity stability is independent of cavity size. In other words, strain hardening will increase the critical tension irrespective of cavity size. Therefore, it alone cannot explain why rapid growth of smaller cavities requires higher negative pressure. Gent suggested that the rapid growth of small cavities is not a deformation controlled process but a fracture process. Specifically, surface energy provides an additional restraint upon expansion, which becomes significant for small cavities with relative high surface areas. He started with the hypothesis that cavities grow from initial crack like defects. Gent's insight can be quantified by dimensional analysis: the energy release rate of a circular crack in an infinite medium must necessarily scales with the initial crack radius. Therefore, smaller defects have a smaller energy release rate for the same applied load. However, Griffith's fracture criterion dictates that crack growth occurs when the energy release rate reaches the fracture toughness G_c, which is a material constant. This explains the size dependence. In Gent's analysis, various approximations are made in order to obtain closed form expression for the energy release rate. In this work, we evaluate the energy release rate free of these approximations.