Computational assessment of cracks under strain-gradient plasticity

With introducing the strain gradient into plasticity theory, the crack field singularity at the crack tip changes. Characterization of the crack in strain-gradient dependent materials has not found a common consensus. In the present work the crack tip field is systematically investigated within the frame of the strain-gradient plasticity using the element-free Galerkin method. Under both small scale yielding and finite yielding conditions the crack field in strain-gradient plasticity consists of three zones: The elastic K-field, the known plastic HRR-field dominated by the known energy release rate, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal G}$$\end{document}, and the hyper-singular stress field which cannot be characterized by the known fracture mechanics parameters. Effects of the finite strains have been considered for both small scale yielding and finite cracked specimens. The computational results show that the finite strains do not change the characterization of the crack tip fields under the strain-gradient plasticity. In all investigated cases the hyper-singular zone is small (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${r < \mathcal G/\sigma_0}$$\end{document}) and well contained by the HRR zone under the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal G}$$\end{document} dominance condition. The hyper-singular zone is in the size of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${r/(\mathcal G/\sigma_0)}$$\end{document} and may grow with the material length, but decrease with applied load. The known elastic-plastic fracture mechanics parameter, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal G}$$\end{document}, can be directly applied to characterize the crack under strain-gradient plasticity, within the intrinsic material length much smaller than the overall component size, l < 0.1%L0. In this case the strain-gradient term in the gradient plasticity will not affect description of cracks.