Tearing a neo-Hookean sheet. Part II: asymptotic analysis for crack tip fields

In this paper, we derive asymptotic crack tip fields during damage and tearing of a neo-Hookean sheet. The material damage near the crack tip is characterized by an exponential damage mode inspired by the damage distribution observed in a phase field model in Part I. The asymptotic governing equations consist of a quasi-linear eigenproblem for the deformed coordinates which reduce to the eigenproblem for a Laplace equation in the case of a purely elastic neo-Hookean sheet. The asymptotic fields are complemented by finite element results for a long-notched strip under uniaxial stretch. The asymptotic analysis indicates some interesting scaling relations and distribution features of the crack tip fields that are consistent with those observed in the phase field modelling. In particular, the stress component perpendicular to crack faces, σ22\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sigma }_{22}$$\end{document}, vanishes at the crack tip instead of approaching infinity as observed in a purely elastic neo-Hookean sheet. Its magnitude is proportional to the critical energy release rate Gf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${G}_{\mathrm{f}}$$\end{document} and inversely proportional to the internal length scale in a scaled coordinate (scaled by the internal length scale ℓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell$$\end{document}), i.e., σ22∼Gfℓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sigma }_{22}\sim \frac{{G}_{\mathrm{f}}}\ell$$\end{document}. The near-tip crack faces become more blunted in comparison to a purely elastic neo-Hookean sheet.