Weitzenböck多様体によるらせん転位のモデル化と数値解析

抄録

This study conducts the modeling and numerical analysis of a screw dislocation in a three-dimensional continuous medium. Our modeling is based on the differential geometry of Weitzenböck manifold, i.e., a Riemann-Cartan manifold which equips itself with non-zero torsion in the affine connection. Following to the standard framework of geometrical elasto-plasticity, we introduce the three smooth manifolds representing the reference R, intermediate B and current S configurations and express the kinematics using the diffeomorphisms between them. Our primary concern is the geometrical construction of the intermediate configuration B. For a given dislocation density τ, we calculated the plastic distorsion F_p through the integration of τ by homotopy operator. This analysis yields the dual frame ϑ of the Cartan moving frame, which satisfies the first structure equation and Bianchi identity, simultaneously. The current configuration S is obtained by embedding of B to the conventional Euclidean space R³ so as to minimize the strain energy functional. The variational problem is solved numerically using the isogeometric analysis; Galerkin method with non-uniform rational B-spline basis functions. Present analysis revealed that far-field stresses around a screw dislocation agree quantitatively well with those of the Volterra dislocation. A notable difference is non-singularity at the dislocation core. Another remarkable feature is the emergence of hydrostatic pressure due to the geometrical nonlinearity. We also found that surface displacements include a vortex centered at the dislocation line. This is a realization of Eshelby twist.