Institute of Fundamental Technological Research

Variational geometries describing corrugated graphene sheets are proposed. The isothermal thermomechanical properties of these sheets are described by a two-dimensional Weyl space. The equation that couples the Weyl geometry with isothermal distributions of the temperature of graphene sheets, is formulated. This material space is observed in a three-dimensional orthogonal configurational point space as regular surfaces which are endowed with a thermal state vector field fulfilling the isothermal thermal state equation. It enables to introduce a non-topological dimensionless thermal shape parameter of non-developable graphene sheets. The properties of the congruence of lines generated by the thermal state vector field are discussed.

We deals with two aspects of the geometric description of continuized Bravais monocrystals with many dislocations. First, a triad of vector fields is distinguished constituting a basis for the C∞-module of smooth vector fields tangent to the body. This moving frame defines its object of anholonomity as well as an intrinsic (“material”) Riemannian metric of the body. Second, these geometric objects are used to define both the notions of principal congruence of Volterra-type effective dislocations and principal local Burgers vector associated with these congruences. The main topics discussed are (i) self-balance equations of dislocations and secondary point defects generated by distributions of these dislocations; (ii) a linkage of the Bianchi classification of three-dimensional real Lie algebras with the physical classification of principal local Burgers vectors.

self-balance equations, Bianchi-type distortions, dislocations