Optimal scaling in solids undergoing ductile fracture by crazing (Q5962867)

The authors of this interesting paper consider the engineering model of crazing proposed by \textit{S. Heyden} et al. [``A micromechanical damage and fracture model for polymers based on fractional strain-gradient elasticity'', J. Mech. Phys. Solids 74, 175--195 (2015; \url{doi:10.1016/j.jmps.2014.08.005})]. Crazes can be described as thin layers of highly localized tensile deformation. The craze surfaces are bridged by numerous fine fibrils. In the above mentioned model, the fracture of polymers results from ``a competition between distributed damage, due to progressive chain failure, and fractional strain-gradient elasticity''. Note that their model generalizes the conventional theories of strain-gradient elasticity. It is assumed in accordance with the theory 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''. In the case of conventional strain-gradient elasticity the second derivative of the deformation field (denoted by \(y\)) is assumed to be integrable, i.e. \(\int\limits_{\Omega }|D^2y|dx< \infty \). Then, it follows that only deformation mappings in the Sobolev space \(W^{ 2,1}\) are admissible. It turns out that this energy is rigid in order to allow for crazing and, hence, is incompatible with experimental observations of fracture in polymers. Here, the authors give an experimental picture of scale fibrils which are topologically like cylinders connecting the two crack flanks. By continuous embedding of \(W^{ 2,1}(\mathbb{R}^2)\) in \(L^{\infty }\) it is shown that the restriction of any deformation mapping \(y\in W_{\mathrm{loc}}^{ 2,1}(\mathbb{R}^3;\mathbb{R}^3)\) to planes orthogonal to the crack plane is continuous. Then, it is concluded that ``the model with deformations in the Sobolev space \(W^{ 2,1}\), as in conventional strain-gradient elasticity, would negate the possibility of crazing through the formation of fibrils''. It is noteworthy that the fractional strain-gradient model does not suffer the drawback and it is compatible with the formation of fibrils. The main accent here is on the approach to derive rigorous optimal scaling laws for the macroscopic fracture energy of polymers from the above stated micromechanical model. The considered specific problem concerns the model of an infinite slab of finite thickness subjected to opening displacements on its two surfaces. After analyzing these special structures the authors establish 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.