Optimal Scaling in Solids Undergoing Ductile Fracture by Crazing
We derive optimal scaling laws for the macroscopic fracture energy of polymers failing by crazing. We assume that the effective deformation-theoretical free-energy density is additive in the first and fractional deformation-gradients, with zero growth in the former and linear growth in the latter. The specific problem considered concerns a material sample in the form of an infinite slab of finite thickness subjected to prescribed opening displacements on its two surfaces. For this particular geometry, we derive optimal scaling laws for the dependence of the specific fracture energy on cross-sectional area, micromechanical parameters, opening displacement and intrinsic length of the material. In particular, the upper bound is obtained by means of a construction of the crazing type.
© 2015 Springer-Verlag Berlin Heidelberg. Received October 21, 2014; Accepted June 15, 2015; Published online July 1, 2015. The work of SC was partially supported by the Deutsche Forschungsgemeinschaft through the Forschergruppe 797 Analysis and computation of microstructure in finite plasticity, project CO 304/4-2. Michael Ortiz gratefully acknowledges support from the Office of Naval Research through Grant N00014-11-1-0547, from the U.S. National Science Foundation through the Partnership for International Research and Education (PIRE) on Science at the Triple Point Between Mathematics, Mechanics and Materials Science, Award Number 0967140 and from the Institute for Applied Mathematics (IAM), University of Bonn, Germany.