Geometrical frustration in nonlinear mechanics of screw dislocation

The stress singularities and reliance on linear elasticity pose significant challenges in comprehending the mechanism of the formation of stress fields around dislocations. In this study, we use differential geometry and calculus of variations for the mathematical modelling and numerical analyses for the mechanics of screw dislocations. We express the kinematics of dislocations by using a diffeomorphism of the Riemann-Cartan manifold, which equips the Riemannian metric and affine connection. We formulate the function of the Cartan first structure equation for solving plastic deformation of dislocations, while we employ the stress equilibrium equations for nonlinear elasticity. To solve this nonlinear problem, we conduct isogeometric analysis with NURBS basis functions. We implement this using C++. The analysis results obtained by this implementation show that the stress fields effectively eliminate the singularity along the dislocation line and exhibit excellent agreement with Volterra's theory outside the dislocation core. Furthermore, by utilising the mathematical properties of the Riemann-Cartan manifold and smoothness of the NURBS functions used in the isogeometric analysis, we show that geometrical frustration is the direct source of dislocation stress fields, and the Ricci curvature determines the symmetry of stress fields. These results demonstrate the duality between stress and curvature, a mathematical hypothesis posed in previous studies.