Energy budget during the propagation of a dynamic crack in elastomers

Abstract

The theory of Linear Elastic Fracture Mechanics (LEFM) limits the speed of a dynamic crack in a material to the Rayleigh wave speed (and hence to be smaller than the shear wave speed) in mode-I. Experiments performed on elastomers, for instance on Polyurethane sheets, reveal that the cracks in these materials can travel faster than the shear wave speed (and hence, the Rayleigh wave speed). Two main hypotheses can be found in the literature to explain this observation. One hypothesis relies on the hyperelastic stiffening of elastomers - the upturn in the stress-strain curve at higher stretch values resulting in higher stiffness and higher wave speeds at those stretches. The strain state of the material at the vicinity of the tip is higher when compared to the material that is far as a consequence of strain concentration, and hence, is stiffer than the material that is far. This was thought to result in crack speeds that exceed the shear wave speed corresponding to the bulk material. The second hypothesis relies on the viscoelastic stiffening of the material. Since the strain rates in the material at the vicinity of the tip are higher, the viscoelastic stiffening of the material was thought to result in higher crack speeds. The earlier study by the authors examines these two hypotheses and concludes the viscoelastic stiffening to be the reason for Transonic cracks in elastomers. That analysis has been performed using the Finite Linear Viscoelastic (FLV) model to describe the viscoelastic behavior of the bulk material. The FLV model uses viscous overstress as an internal variable and does not have an expression for the strain energy density or dissipation. Whether the FLV model is thermodynamically consistent is also not known. Hence, the Finite viscoelastic (FV) model has been used in this study to estimate the dissipation in the material as the crack passed through. It shall be noted that other viscoelastic models that are thermodynamically consistent are available. The FV model has been chosen since it can model the possible shift in the relaxation times of the material with the applied strain. Using this, it has been concluded that a significant portion of the energy stored in the body is lost through viscoelastic dissipation in the bulk material.