On Nye’s lattice curvature tensor

•The lattice curvature tensor is represented by a triplet of vectors.•A relation between material lattice curvature and contorsion tensor is derived.•A relation between material lattice curvature tensor and Nye curvature is derived.•Linearized Einstein tensor is expressed using linearized lattice curvature tensor. We revisit Nye’s lattice curvature tensor in the light of Cartan’s moving frames. Nye’s definition of lattice curvature is based on the assumption that the dislocated body is stress-free, and therefore, it makes sense only for zero-stress (impotent) dislocation distributions. Motivated by the works of Bilby and others, Nye’s construction is extended to arbitrary dislocation distributions. We provide a material definition of the lattice curvature in the form of a triplet of vectors, that are obtained from the material covariant derivative of the lattice frame along its integral curves. While the dislocation density tensor is related to the torsion tensor associated with the Weitzenböck connection, the lattice curvature is related to the contorsion tensor. We also show that under Nye’s assumption, the material lattice curvature is the pullback of Nye’s curvature tensor via the relaxation map. Moreover, the lattice curvature tensor can be used to express the Riemann curvature of the material manifold in the linearized approximation.