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.
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Notes
Originally proposed by Bilby et al. [5].
Note that \(\varvec{\mathcal {R}}\) is different from the rotation tensor \(\mathbf {R}\) obtained from polar decomposition of \(\mathbf {F}\).
This measure for GND density is same as the one derived in Cermelli and Gurtin [9].
The Gram-Schmidt procedure does not specify the 1 coordinate direction nor the 12 coordinate plane, based upon which the other coordinate directions are determined. Therefore, one can potentially have a non-unique set of bases for an experimenter’s frame of reference. This issue is resolved by re-indexing the deformation gradient using a strategy whereby the selected 1 direction undergoes the least amount of transverse shear, and the selected 12 coordinate plane experiences least amount of in-plane shear. This re-indexed deformation gradient is denoted by \(\varvec{\mathcal {F}}\), whose construction is developed in a paper soon to be published. In fact, the re-indexed deformation gradient \(\varvec{\mathcal {F}}\) is decomposed into \(\varvec{\mathcal {R}}\) and \(\varvec{\mathcal {U}}\). This is merely a problem of selecting the ‘correct’ set of bases. Deformation gradients \(\mathbf {F}\) and \(\varvec{\mathcal {F}}\) essentially contain the same information.
Although it is indeed possible for bodies made up of a rigid plastic material to undergo a plastic only deformation, such a constitutive relation is too restrictive.
Such cases appear whenever the deformation gradient is upper-triangular, e.g., a uniaxial or a biaxial tension.
Note that in this case, elongation along the 1 direction is \(a=1\).
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Paul, S., Freed, A.D. Characterizing geometrically necessary dislocations using an elastic–plastic decomposition of Laplace stretch. Z. Angew. Math. Phys. 71, 196 (2020). https://doi.org/10.1007/s00033-020-01420-7
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DOI: https://doi.org/10.1007/s00033-020-01420-7