Markus Lazar

On dislocations in a special class of generalized elasticity

In this paper we consider and compare special classes of static theories of gradient elasticity, nonlocal elasticity, gradient micropolar elasticity and nonlocal micropolar elasticity with only one gradient coefficient. Equilibrium equations are discussed. The relationship between the gradient theory and the nonlocal theory is discussed for elasticity as well as for micropolar elasticity. Nonsingular solutions for the elastic fields of screw and edge dislocations are given. Both the elastic deformation (distortion, strain, bend-twist) and the force and couple stress tensors do not possess any singularity unlike ‘classical’ theories.

Dislocations in gradient elasticity revisited

In this paper, we consider dislocations in the framework of first as well as second gradient theory of elasticity. Using the Fourier transform, rigorous analytical solutions of the two-dimensional bi-Helmholtz and Helmholtz equations are derived in closed form for the displacement, elastic distortion, plastic distortion and dislocation density of screw and edge dislocations. In our framework, it was not necessary to use boundary conditions to fix constants of the solutions. The discontinuous parts of the displacement and plastic distortion are expressed in terms of two-dimensional as well as one-dimensional Fourier-type integrals. All other fields can be written in terms of modified Bessel functions.

The fundamentals of non-singular dislocations in the theory of gradient elasticity: Dislocation loops and straight dislocations

Abstract The fundamental problem of non-singular dislocations in the framework of the theory of gradient elasticity is presented in this work. Gradient elasticity of Helmholtz type and bi-Helmholtz type are used. A general theory of non-singular dislocations is developed for linearly elastic, infinitely extended, homogeneous, and isotropic media. Dislocation loops and straight dislocations are investigated. Using the theory of gradient elasticity, the non-singular fields which are produced by arbitrary dislocation loops are given. ‘Modified’ Mura, Peach–Koehler, and Burgers formulae are presented in the framework of gradient elasticity theory. These formulae are given in terms of an elementary function, which regularizes the classical expressions, obtained from the Green tensor of the Helmholtz–Navier equation and bi-Helmholtz–Navier equation. Using the mathematical method of Green’s functions and the Fourier transform, exact, analytical, and non-singular solutions were found. The obtained dislocation fields are non-singular due to the regularization of the classical singular fields.

Non-singular dislocation loops in gradient elasticity

Abstract Using gradient elasticity, we give in this Letter the non-singular fields produced by arbitrary dislocation loops in isotropic media. We present the ‘modified’ Mura, Peach–Koehler and Burgers formulae in the framework of gradient elasticity theory.

Peach-Koehler Forces within the Theory of Nonlocal Elasticity

We consider dislocations in the framework of Eringen’s nonlocal elasticity. The fundamental field equations of nonlocal elasticity are presented. Using these equations, the nonlocal force stresses of a straight screw and a straight edge dislocation are given. By the help of these nonlocal stresses, we are able to calculate the interaction forces between dislocations (Peach-Koehler forces). All classical singularities of the Peach-Koehler forces are eliminated. The extremum values of the forces are found near the dislocation line.

On the correspondence between a screw dislocation in gradient elasticity and a regularized vortex

We show the correspondence between a screw dislocation in gradient elasticity and a regularized vortex. The effective Burgers vector, nonsingular distortion and stress fields of a screw dislocation and the effective circulation, smoothed velocity and momentum of a vortex are given and discussed.