Andrzej Trzesowski

Global invariance and Lie-algebraic description in the theory of dislocations

The relationships between the Lie-algebraic description of continuously distributed dislocations and the global affine invariance of their tensorial density are studied. The affinely-invariant Lagrange description of static self-equilibrium distributions of dislocations is proposed and field equations describing distributions of internal stresses and couple stresses are formulated. The analogy between the proposed theory of dislocations and “3-fold electrodynamics” is formulated.

Gauge theory of dislocations

In this continuation of work by the author the notion of the distortion of an ideal crystal structure is generalized and the gauge field is defined, fundamental states (“vacuum configurations”) of which are the crystal structure elementary distortions due to dislocations. The form of the structure equations of the connection form defined by this gauge field is discussed.

Geometry of crystal structure with defects. II. Non-Euclidean picture

Continuously distributed defects of crystal structure are considered. The starting point is the Euclidean geometry of the ideal crystal lattice and the topological description of the distortion of the crystal structure. It is shown how the non-Euclidean geometry of distorted crystal structure, as well as the basic assumptions of the phenomenological plasticity theory concerning the deformation of a continuum, are related to those theories. A form for an affine connection describing continuously distributed dislocations is proposed.

Dislocations and internal length measurement in continuized crystals. I. Riemannian material space

Distributions of dislocations creating point defects are considered. These point defects are described by a metric tensor, which supplements a Burgers field responsible for dislocations. The metric tensor depends on the distribution of dislocations and defines a Riemannian geometry of the material space of a continuized crystal and thus an internal length measurement in this crystal. The dependence of the distribution of dislocations on the existence of point defects created by these dislocations is modeled by treating the Burgers field as a field defined on the Riemannian material space. Field equations, following from geometric identities, are formulated as balance equations on this Riemannian space and their source terms, responsible for interactions of dislocations and point defects, are identified.

Congruences of Dislocations in Continuously Dislocated Crystals

The time-dependent congruences of Volterra-type dislocations are investigated based on the generalized formulas of Frenet in a Riemannian space. The analysis is applied to the description of congruences of edge and mixed dislocations consistent with a continuous distribution of dislocations for which its material space is an equidistant Riemannian space. In particular, the principal congruences of dislocations are considered. The kinematics of congruences of mixed dislocations endowed with univocally defined local slip planes is discussed. It is shown that the geometry of such congruences of dislocations admits a class of nonlinear evolution equations describing the curvature and torsion of a congruence of curves in a Riemannian space. Additional conditions imposed on the derived system of equations in order to describe the evolution of curvature and torsion of congruences of edge dislocations are proposed. In the static case, an expression is given for shear stresses required to bend prismatic edge dislocations of torsion zero located on the totally geodesic crystal surfaces. It follows that the congruence of these dislocations is endowed with a finite self-energy function.

Locally equilibrium diffusion processes. I. Geometry of local thermodynamic equilibrium states

The geometric description of local thermodynamic equilibrium states in a locally homogeneous body is formulated. The considered geometry is defined on the basis of the condition that the gauge procedure of the partial derivative operations should preserve the form of the equations describing the local thermodynamic equilibrium state of the homogeneous body with diffusion, and be consistent with the description of diffusion as a Markovian diffusion process with different mean arriving and starting velocities. The statistical entropy of the diffusion process is examined.

Nonlinear diffusion and Nelson-Brown movement

The nonlinear diffusion process, which can be described as the Nelson-Brown motion, is considered. The obtained equation becomes the classical linear diffusion equation for small relaxation times, and for long relaxation times it is transferred into the Schrödinger-like equation. The possible nonequilibrium stationary states are discussed.

On the Isothermal Geometry of Corrugated Graphene Sheets

Variational geometries describing corrugated graphene sheets are proposed. The isothermal thermomechanical properties of these sheets are described by a 2-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 3-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.

Self-Balance Equations and Bianchi-Type Distortions in the Theory of Dislocations

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.

Kinematics of edge dislocations. I. Involutive distributions of local slip planes

The geometry of continuous distributions of dislocations and secondary point defects created by these distributions is considered. Particularly, the dependence of a distribution of dislocations on the existence of secondary point defects is modeled by treating dislocations as those located in a time-dependent Riemannian material space describing, in a continuous limit, the influence of these point defects on metric properties of a crystal structure. The notions of local glide systems and involutive distributions of local slip planes are introduced in order to describe, in terms of differential geometry, some aspects of the kinematics of the motion of edge dislocations. The analysis leads, among others, to the definition of a class of distributions of dislocations with a distinguished involutive distribution of local slip planes and such that a formula of mesoscale character describing the influence of edge dislocations on the mean curvature of glide surfaces is valid.