Strain gradient plasticity: energetic or dissipative?

For an infinite slab of strain gradient sensitive material subjected to plane-strain tensile loading, computation established and analysis confirmed that passivation of the lateral boundaries at some stage of loading inhibits plastic deformation upon further loading. This result is not surprising in itself except that, remarkably, if the gradient terms contribute to the dissipation, the plastic deformation is switched off completely and only resumes at a clearly defined higher load, corresponding to a total strain εc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _c$$\end{document}, say. The analysis presented in this paper confirms the delay of plastic deformation following passivation and determines the exact manner in which the plastic flow resumes. The plastic strain rate is continuous at the exact point εc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _c$$\end{document} of resumption of plastic flow and, for the first small increment Δε=ε-εc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \varepsilon =\varepsilon -\varepsilon _c$$\end{document} in the imposed total strain, the corresponding increment in plastic strain, Δεp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \varepsilon ^\mathrm{p}$$\end{document}, is proportional to (Δε)2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\Delta \varepsilon )^2$$\end{document}. The constant A in the relation Δεp(0)=A(Δε)2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \varepsilon ^\mathrm{p}(0) = A(\Delta \varepsilon )^2$$\end{document}, where Δεp(0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \varepsilon ^\mathrm{p}(0)$$\end{document} denotes the plastic strain increment at the centre of the slab, has been determined explicitly; it depends on the hardening modulus of the material. The presence of energetic gradient terms has no effect on the value of εc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _c$$\end{document} unless the dissipative terms are absent, in which case passivation reduces the rate of plastic deformation but introduces no delay. This qualitative effect of dissipative gradient terms opens the possibility of experimental discrimination of their presence or absence. The analysis employs an incremental variational formulation that is likely to find use in other problems.