Characterizing geometrically necessary dislocations using an elastic–plastic decomposition of Laplace stretch

Abstract

In this paper, the geometric dislocation density tensor and Burgers vector are studied using an elastic–plastic decomposition of Laplace stretch $\varvec{\mathcal {U}}$. The Laplace stretch arises from a $\mathbf {QR}$ decomposition of the deformation gradient and is very useful, as one can directly and unambiguously measure its components by performing experiments. The geometric dislocation density tensor ${\tilde{\mathbf {G}}}$ is obtained using the classical argument of failure of a Burgers circuit in a suitable configuration $\tilde{\kappa }_p$ where the deformation of a body is solely due to the movement of dislocations. The geometric features of space $\tilde{\kappa }_p$ are explored. It is shown that the derived geometric dislocation tensor is related to the torsion of $\tilde{\kappa }_p$, which serves as a measure of incompatibility in this space. Additionally, ${\tilde{\mathbf {G}}}$ vanishes only when the space $\tilde{\kappa }_p$ is compatible. A balance law for geometric dislocations is derived taking into account the effect of the dislocation flux and source dislocations. The physical meaning of the plastic Laplace stretch, and consequently, of the derived geometric dislocation tensor proves to be particularly useful in the classification of dislocations. Finally, the significance of the dislocation density tensor is discussed. The derived geometric dislocation density tensor could be specifically useful in developing a strain-gradient and size-dependent theory of plasticity.