Geometrical frustration in nonlinear mechanics of screw dislocation

Abstract: The existence of stress singularities and reliance on linear approximations pose significant challenges in comprehending the stress field generation mechanism around dislocations. This study employs differential geometry and calculus of variations to mathematically model and numerically analyse screw dislocations. The kinematics of the dislocation are expressed by the diffeomorphism of the Riemann--Cartan manifold, which includes both the Riemannian metric and affine connection. The modelling begins with a continuous distribution of dislocation density, which is transformed into torsion $\tau$ through the Hodge duality. The plasticity functional is constructed by applying the Helmholtz decomposition to bundle isomorphism, which is equivalent to the Cartan first structure equation for the intermediate configuration $\mathcal{B}$. The current configuration is derived by the elastic embedding of $\mathcal{B}$ into the standard Euclidean space $\mathbb{R}^3$. The numerical analysis reveals the elastic stress fields effectively eliminate the singularity along the dislocation line and exhibit excellent conformity with Volterra's theory beyond the dislocation core. Geometrical frustration is the direct source of dislocation stress fields, as demonstrated through the multiplicative decomposition of deformation gradients. By leveraging the mathematical properties of the Riemann--Cartan manifold, we demonstrate that the Ricci curvature determines the symmetry of stress fields. These results substantiate a long-standing mathematical hypothesis: the duality between stress and curvature.