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.

在本文中，使用拉普拉斯拉伸\（\ varvec {\ mathcal {U}} \）的弹塑性分解研究了几何位错密度张量和Burgers向量。拉普拉斯拉伸是由变形梯度的\（\ mathbf {QR} \）分解产生的，并且非常有用，因为可以通过执行实验来直接，明确地测量其分量。几何位错密度张量\（{\ tilde {\ mathbf {G}}} \）是在合适的配置\（\ tilde {\ kappa _p_）中使用Burgers电路故障的经典论点获得的，其中变形身体的完全归因于位错的运动。空间\（\ tilde {\ kappa} _p \）的几何特征被探索。结果表明，导出的几何位错张量与\（\ tilde {\ kappa} _p \）的扭转有关，这是该空间中不相容性的度量。此外，\（{\ tilde {\ mathbf {G}}} \）仅在空格\（\ tilde {\ kappa} _p \）时消失。兼容。考虑到位错通量和源位错的影响，得出了几何位错的平衡定律。塑性拉普拉斯拉伸的物理含义以及由此得出的几何位错张量的物理意义被证明在位错的分类中特别有用。最后，讨论了位错密度张量的意义。导出的几何位错密度张量在开发应变梯度和尺寸相关的可塑性理论中可能特别有用。