  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.
