- The paper develops a translational gauge theory framework to derive dynamic field equations for screw dislocations, regularizing classical core singularities.
- It employs symmetry arguments to decouple the field equations into Klein–Gordon-type equations and constructs solutions using retarded Green functions.
- The study distinguishes between near-field velocity effects and far-field radiation, linking core width with inertial response and energy dissipation.
Introduction and Theoretical Framework
This work systematically develops the translational gauge theory of dislocations and applies it to the dynamic problem of a nonuniformly moving screw dislocation in infinite media. Traditional treatments of dislocation dynamics—whether in elastodynamics, nonlocal elasticity, or gradient elasticity—either introduce ad hoc assumptions regarding scale and dynamical response or fail to address core singularity structure rigorously. By contrast, translational gauge theory constructs dislocation fields as canonical gauge fields resulting from spontaneously broken translation symmetry. The field equations and constitutive laws are derived from a Lagrangian, yielding dynamic equations of motion based on momentum and force stress balances, with the physical field sources naturally emerging from the canonical framework. This approach inherently regularizes core singularities for both subsonic and supersonic dislocation motion and introduces characteristic length (core-width) and time scales into the theory, something absent from classical elasticity.
The state variables for gauge-theoretical dislocation dynamics are the incompatible elastic distortion tensor and the physical velocity, from which the dislocation density and current tensors are derived. The latter fulfill translational Bianchi identities. Constitutive laws are formulated using isotropic and linear assumptions for momentum, stress, dislocation momentum flux, and pseudomoment stress tensors, with stability enforced by positivity conditions on the material moduli.
Dynamical Field Equations and Analytical Reduction
Reduction to the specific case of a moving screw dislocation is achieved through symmetry arguments. The equations of motion are derived for an infinitely long screw dislocation parallel to the z-axis, moving in the xy-plane. This results in a system of coupled partial differential equations for the non-zero components of the elastic velocity and distortion tensors and the dislocation density and current tensors. Strategic assumptions, specifically the identification of static and dynamic characteristic length scales, allow for the decoupling of the field equations into Klein–Gordon-type equations for the involved fields. The analysis reveals that, within the gauge-theoretical description, the dynamic dislocation fields satisfy inhomogeneous (1+2)-dimensional Klein–Gordon equations—a result that explicitly encapsulates the 'massiveness' of the gauge field associated with the dislocation. This automatically regulates the classical core singularity.
It is shown that, for a screw dislocation, the core-regularizing length scale directly determines the inertial response, establishing a connection between gauge-theoretical core width and kinetic terms in the field equations.
Explicit integral solutions for all non-vanishing field quantities are constructed using retarded Green functions of the corresponding Klein–Gordon (and, in the case of the elastic fields, Bopp–Podolsky-type) equations. The dislocation density and current tensors are expressed as convolutions of the source term (corresponding to the moving dislocation line) with these Green functions. Retardation reflects the 'memory' of the system: the dislocation at a point (x,y,t) generates fields that depend on its entire past trajectory, up to the retarded time determined by the field propagation speed cT and the characteristic gauge-theory core length. This precisely encodes the non-Huygens character of 2D wave propagation and the history-dependent response first emphasized by Eshelby.
The elastic fields themselves (velocity and distortion components) are obtained in terms of 'constrained Liénard–Wiechert potentials' analogous to potentials for massive vector fields in electrodynamics. These incorporate the entire time-dependent motion of the dislocation through retarded trajectory contributions, and their structure is determined by the convolution with the Bopp–Podolsky Green function, which regularizes the core singularity via the characteristic length l1 of the gauge theory.
Radiation Fields and Core Structure Evolution
A major analytical result is the separation of the elastic field solutions into two distinct contributions: (a) fields depending exclusively on dislocation velocity (the near field/velocity field), and (b) fields involving acceleration (the far field/radiation field), which are singular in the classical theory but remain finite here due to regularization by the gauge length scale. The elastic velocity is history-dependent but contains no explicit acceleration contributions; instead, all acceleration effects manifest in the elastic distortion (strain/radiation) field components.
Explicit expressions demonstrate that the core field structure changes dynamically with dislocation motion, particularly when the dislocation accelerates or approaches transonic speeds. Radiation is manifested in the field components proportional to the inverse of the retarded distance, structurally analogous to electromagnetic radiation but fundamentally regulated by the effective mass parameter and core width of the gauge field.
Theoretical and Practical Implications
This paper establishes that screw dislocations, dynamically described using gauge theory, are massive fields. This regularizes their core structure and their radiation, making the theory more physically realistic than classical elasticity, which otherwise predicts pathologies in the core region and in the emission of elastic waves upon acceleration. The results have immediate implications for mesoscale modeling of crystal plasticity, where accurate treatment of dislocation self-energy, core interactions, and radiation damping is crucial. By framing dislocation motion within the field-theoretic language, connections to other gauge field theories become manifest, providing natural extensions toward relativistic regimes and the incorporation of anisotropy, finite temperature, and coupling to other field defects (e.g., disclinations).
Furthermore, the explicit classification of the field contributions provides a route to compute energy loss and effective equations of motion for accelerating dislocations, relevant for high-speed plastic deformations, seismic rupture dynamics, and the interpretation of dynamic transmission electron microscopy.
Conclusion
The translational gauge theory of dislocations enables a rigorous, non-singular description of the dynamics of nonuniformly moving screw dislocations. All fundamental field quantities—elastic fields, dislocation density, and current, and radiation fields—are constructed explicitly, exhibiting dependence on retarded times and regularized by characteristic gauge-theory length and time scales. The theoretical structure resolves classical pathologies and allows for a consistent treatment of dislocation radiation and inertia. This approach provides a robust basis for the further development of generalized continuum theories of defect dynamics and for computational applications in materials modeling at the mesoscale.
