On the characterization of geometrically necessary dislocations in finite plasticity


PAOLO CERMELLI

Dipartimento di Matematics
Universit`a di Torno
I-10123 Torino, Italy
cermelli@dm.unito.it

and

MORTON E. GURTIN
Department of Mathematical Sciences
Carneige Mellon University
Pittsburgh, PA 15213
USA

Abstract. We develop a general theory of geometrically necessary dislocations based on the decomposition $\mbox{\boldmath$F$ } = \mbox{\boldmath$F$ }^e \mbox{\boldmath $F$ }^p$. The incompability of $\mbox{\boldmath$F$ }^e$ and that of $\mbox{\boldmath$F$ }^p$ are characterized by a single tensor $\mbox{\boldmath$G$ }$ giving the Burgers vector, measured and reckoned per unit area in the microstructural (intermediate) configuration. We show that G may be expressed in terms of Fpand the referential curl of Fp, or equivalently in terms of Fe-1 and the spatial curl of Fe-1. We derive explicit relations for $\mbox{\boldmath$G$ }$ in terms of Euler angles for a rigid-plastic material and -- without neglecting elastic strains -- for strict plane strain and strict antiplane shear. We discuss the relationship between G and the distortion of microstructural planes. We show that kinematics alone yields a balance law for the transport of geometrically necessary dislocations.

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