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Publication 04-CNA-09

A gradient theory of small-deformation isotropic plasticity that accounts for the Burgers vector

Morton E. Gurtin
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213, USA
mg0c@andrew.cmu.edu

Abstract: This study develops a gradient theory of small-deformation viscoplasticity based on: a system of microforces consistent with its peculiar balance; a mechanical version of the second law that includes, via the microforces, work performed during viscoplastic flow; a constitutive theory that accounts for the Burgers vector through a free energy dependent on curl H$p$, with H$p$ the plastic part of the elastic-plastic decomposition of the displacement gradient. The microforce balance and the constitutive equations, restricted by the second law, are shown to be together equivalent to a nonlocal flow rule in the form of a coupled pair of second-order partial differential equations. The first of these is an equation for the plastic strain-rate $\dot{\mbox{\bf E }}^p$ in which the stress T plays a basic role; the second, which is independent of T, is an equation for the plastic spin $\dot{\mbox{\bf$W$}p}$. A consequence of this second equation is that the plastic spin vanishes identically when the free energy is independent of curlH$^p$, but not generally otherwise. A formal discussion based on experience with other gradient theories suggests that suffciently far from boundaries solutions should not differ appreciably from classical solutions, but close to microscopically hard boundaries, boundary layers characterized by a large Burgers vector and large plastic spin should form.

Because of the nonlocal nature of the flow rule, the classical macroscopic boundary conditions need be supplemented by nonstandard boundary conditions associated with viscoplastic .ow. As an aid to solution, a variational formulation of the flow rule is derived.

Finally, we sketch a generalization of the theory that allows for isotropic hardening resulting from dissipative constitutive dependences on $\Delta \dot{\mbox{\bf E}}^p$.

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