Elastic Li\'{e}nard-Wiechert potentials of dynamical dislocations from tensor gauge theory in 2+1 dimensions

TL;DR

This paper develops a dual gauge theory framework to analyze the dynamics and interactions of dislocations in elastic media, deriving explicit potentials and forces, including non-reciprocal effects for moving dislocations.

Contribution

It introduces a systematic dual gauge theory approach to dislocation dynamics, deriving explicit potentials and extending classical force results to moving dislocations.

Findings

Derived explicit dual gauge potentials for moving dislocations.

Extended Peach-Koehler force to include non-reciprocal effects.

Provided a framework for analyzing defect interactions in elastic media.

Abstract

The dynamics of defect excitations in crystalline solids is necessary to understand the macroscopic low-energy properties of elastic media. We use fracton-elasticity duality to systematically study the defect dynamics and interactions in the linear isotropic medium. We derive the explicit expressions for the dual gauge potentials for moving dislocations and the resulting Jefimenko equations. We also compute stresses and strains. The paper includes two physical situations: when the vacancy number is fixed and when the number is fluctuating. If defects are present we show a constraint that needs to be satisfied by them when they climb perpendicularly to their Burgers vector. Next, we extend the classic result of Peach and Koehler for the force between two dislocations and show that, similarly, to moving charges in electrodynamics, it is non-reciprocal, when one dislocation is moving. We argue that our formalism can be extended beyond Cauchy's elasticity by exploiting the simplifications provided by the dual gauge formulation of elastic stresses.

Equations

$$S=\frac{1}{2}\int \left[({\partial }_t{u}_i{)}^2-C_{ijkl}{u}_{ij}{u}_{kl}\right]d^{2}xdt,$$

$$Z=\int \mathcal{D}{u}_j{e}^{-S},$$

$$S=\int \left[\frac{1}{2}{C}_{ijkl}^{-1}{\sigma }^{ij}{\sigma }^{kl}-\frac{1}{2}{\pi }^i{\pi }_i-{\sigma }^{ij}{u}_{ij}+{\pi }^i{\partial }_t{u}_i\right]d^{2}xdt.$$

$${\partial }_t{\pi }^i-{\partial }_j{\sigma }^{ij}=0.$$

$$B^i={ϵ}^{ij}{\pi }_j{E}_{\sigma }^{ij}={ϵ}^{ik}{ϵ}^{jl}{\sigma }_{kl}.$$

$${\partial }_t{B}_i+{ϵ}_{jk}{\partial }^j{E}_{\sigma }^{ki}=0.$$

$$E^{ij}={\tilde{C}}_{ijkl}^{-1}{E}_{\sigma }^{kl},$$

$$B^i={ϵ}_{jk}{\partial }^j{A}^{ki},E_{\sigma }^{ij}=-{\partial }_t{A}^{ij}-{\partial }_i{\partial }_j\varphi ,$$

$$A_{ij}\to A_{ij}+{\partial }_i{\partial }_{j}f(x_k,t),\varphi \to \varphi +{\partial }_{t}f(x_k,t),$$

$$S=\int \left[\frac{1}{2}{\tilde{C}}_{ijkl}^{-1}{E}_{\sigma }^{ij}{E}_{\sigma }^{kl}-\frac{1}{2}{B}^i{B}_i+\rho \varphi -J^{ij}{A}_{ij}\right]d^{2}xdt.$$

$$S=\int \left[\frac{1}{2}{\tilde{C}}_{ijkl}^{-1}{E}_{\sigma }^{ij}{E}_{\sigma }^{kl}-\frac{1}{2}{B}^i{B}_i+{\rho }^i{\varphi }_i-J^{ij}{A}_{ij}\right]d^{2}xdt.$$

$${\partial }_t{\rho }_j+{\partial }_i{J}_{ij}=0.$$

$$\mathcal{L}=\frac{1}{2}({\tilde{C}}_{ijkl}^{-1}{E}_{\sigma }^{ij}{E}_{\sigma }^{kl}-B^i{B}_i)-{\rho }^i{\varphi }_i-J^{ij}{A}_{ij},$$

$${\tilde{C}}_{ijkl}^{-1}=\left(\frac{1}{4\kappa }-\frac{1}{4\mu }\right){\delta }_{ij}{\delta }_{kl}+\frac{1}{4\mu }({\delta }_{ik}{\delta }_{jl}+{\delta }_{il}{\delta }_{jk}).$$

$$2C_2{\partial }_t{\partial }_i{A}_{ij}+C_2{\partial }_i^2{\varphi }_j+C_1{\partial }_t{\partial }_j{A}_{kk}+C_3{\partial }_j{\partial }_k{\varphi }_k-{\rho }_j=0,$$

$$2C_{2}\partial_{t}^{2}A_{ij}+2C_{2}\partial_{t}\partial_{i}\phi_{j}+C_{1}\big{[}\partial_{t}^{2}A_{kk}+\partial_{t}\partial_{k}\phi_{k}\big{]}\delta_{ij}-\partial_{k}^{2}A_{ij}+$$

$$+{\partial }_i{\partial }_k{A}_{kj}+J_{ij}=0,$$

$$C_1=\frac{(\mu -\kappa )}{4\mu \kappa },C_2=\frac{1}{4\mu },C_3=\frac{1}{4\kappa }.$$

$${\partial }_i{A}_{ij}+\frac{1}{2\mu }{\partial }_t{\varphi }_j=0.$$

$$\square {\varphi }_j+\left(\frac{\mu -\kappa }{\mu +\kappa }\right){\partial }_t{\partial }_j{A}_{kk}-\left(\frac{4\mu \kappa }{\mu +\kappa }\right){\rho }_j=0,$$

$$\square A_{ij}-\left(\frac{\mu -\kappa }{4\kappa \mu }\right)\left[{\partial }_t^2{A}_{kk}-2\mu {\partial }_m{\partial }_n{A}_{mn}\right]{\delta }_{ij}-J_{ij}=0,$$

$$s={ϵ}_{ij}{\partial }_i{\partial }_j\theta ={ϵ}_{ij}{ϵ}_{kl}{\partial }_i{\partial }_j{\partial }_k{u}_l=\rho ,$$

$$b_j={\oint }_{\gamma }{\partial }_i{u}_{j}d{r}_i.$$

$$\rho =q\delta (\mathbf{r}-{\mathbf{r}}^{\prime })-\mathbf{p}\cdot \nabla \delta (\mathbf{r}-{\mathbf{r}}^{\prime })+...$$

$${\rho }_i({\mathbf{r}}^{\prime },t)={ϵ}_{ij}{b}_j\delta ({\mathbf{r}}^{\prime }-{\mathbf{r}}_s(t^{\prime })),$$

$$J_{ij}({\mathbf{r}}^{\prime },t)={ϵ}_{(ik}{\upsilon }_{j)}{b}_k\delta ({\mathbf{r}}^{\prime }-{\mathbf{r}}_s(t^{\prime })),$$

$$\square {\varphi }_j-2\mu {\rho }_j=0,{\upsilon }_{\varphi }=\sqrt{2\mu },$$

$$\square A_{ij}-J_{ij}=0,{\upsilon }_A=\sqrt{2\mu }.$$

$$D_{ii}=-{\int }_V{d}^{2}x({\rho }_i{x}_i+2E_{ii})=\text{const.}\text{vspace}2mm$$

$${\varphi }_j$$

Full text

I Introduction

Stresses induced by dislocations play an important role in many fundamental physical phenomena in crystals and thin crystalline films. This role of topological defects is manifested for example in plastic, electronic or optical characteristics. In order to have a good understanding of phenomena involving dislocations one needs to shed light on their dynamics. This is a classic problem in the theory of elasticity that was initiated at its early days and subsequently explored by many works (see e.g. eshelby_uniformly_1949 ; kiusalaas_elastic_1964 ; weertman_stress_1967 ; brock_dynamic_1982 ; markenscoff_nonuniformly_1981 ; markenscoff_accelerating_1982 ; markenscoff_dislocation_1983 ; brock_non-uniform_1983 ; pellegrini_dynamic_2010 ; lazar_elastic_2011 ; lazar_elastodynamic_2012 ; lazar_retardation_2013 ; lazar_distributional_2016 ; hirth_theory_1992 ; kazinski_self-interaction_2022 ). In order to fully solve this problem one must account for both external stresses, and self-stresses due to the movement of a dislocation and a direct progress using equations of elasticity has been obstructed by many difficulties in handling the Navier’s equations for displacements.

In this work the aim is to address this problem using the so-called fracton-elasticity dualities, which depart from displacement formulation of elasticity (for recent reviews see grosvenor_space-dependent_2022 ; Gromov:2022cxa ). The dualities originate from the works on two-dimensional crystals for which the low-energy elastic degrees of freedom can be written down as gauge variables with two-indices kleinert_duality_1982 ; kleinert_double_1983 ; beekman_dual_2017-1 ; beekman_dual_2017 . More recently this original framework of elastic dualities was refined, and topological defects, in the absence of crystal vacancies, interpreted as fractons, i.e. excitations with mobility constraints pretko_fracton-elasticity_2018 ; pretko_crystal–fracton_2019 ; radzihovsky_fractons_2020 ; radzihovsky_quantum_2020 ; gromov_duality_2020 ; surowka_dual_2021 ; hirono_effective_2022 ; caddeo_emergent_2022 ; zaanen_crystal_2022 . Such excitations act as sources for the tensor gauge fields. The most important part of this construction are convenient low-energy degrees of freedom together with symmetry principles that precisely capture the dynamics of defects and, allow one to get insight based purely on symmetry considerations pretko_the_2018 ; gromov_towards_2019 . A successful application of this line of reasoning has been in the quest of constructing hydrodynamic theories of kinematically constrained constituents gromov_fracton_2020 ; grosvenor_hydrodynamics_2021 ; Glorioso_breakdown_22 ; burchards_coupled_2022 ; Glodkowski:2022xje ; Glorioso:2023chm .

The dualities reformulate elasticity in the language of gauge theories. In the simplest incarnation a moving topological charge is therefore a generalization of the moving electromagnetic charge in U(1) gauge theories. This problem was solved by Liénard and Wiechert who computed electromagnetic potentials of an arbitrarily moving relativistic electric charge (see e.g. zangwill_modern_2013 ). Later Jefimenko generalized these results to give electric and magnetic fields due to a distribution of electric charges and electric currents in space jefimenko_electricity_1989 . Therefore, a dual gauge theory formulation of elasticity offers a unique opportunity to understand the dynamics and stresses generated by a distribution of elastic defects. As we will see separate solutions can be obtained in materials with vacancies and without vacancies. In the former case the dislocation is not constrained, however, their movement perpendicular to the Burgers vector is tightly connected to the distribution of vacancies in the material. In the later case the dislocations are kinematically restricted to move along their Burgers vector.

Modern studies of defect dynamics go much beyond original crystalline systems. Examples include topological defects in rheology furst_microrheology_2017 or active matter bowick_symmetry_2022 ; shankar_topological_2022 . In rheology viscoelastic properties for incompressible systems can be derived by the correspondence principle upon changing the real elastic parameters to complex transport coefficients pipkin_lectures_1986 . Therefore, a good understanding of elasticity is a convenient starting point to understand more complex viscoelastic materials. Finally defects in active matter naturally emerge in fluids that form from collections of bacterial suspensions sokolov_emergence_2019 or other macroscopics ensembles of living organisms duclos_topological_2017 ; tan_odd_2022 . More recently topological defects appeared in the context of active solids and metamaterials braverman_topological_2021 .

In this work we show that the dynamics of topological defects in isotropic elasticity on a plane can be simplified considerably if elasticity is formulated as a gauge theory. Exploiting this simplification, we derive the stresses that follow from the dynamics of arbitrarily moving dislocation. We also include time dependence into the classic result of Peach-Koehler force that captures the interactions and external stresses acting on a dislocation peach_forces_1950 . The strength of our results lies in the fact that we do not attempt an analytic or numerical tour-de-force but use the simplification provided by the fraction-elasticity duality. In consequence our formalism can be generalized to study more complex theories of elasticity or viscoelasticity.

Our manuscript is organized as follows. In Sec. II we review the fracton-elasticty duality. In Sec. III we derive equations of motion in for the dual gauge potentials. Sec. IV introduces the notion of defects. In Sec. V we present the solutions and discuss physical implications of the defect motion. Sec. VI is devoted to interactions between dislocations. We derive the expression for the dynamical Peach-Koehler force between two dislocations. Finally in Sec. VII we close with conclusions and discussion.

II Fracton-Elasticity Duality

We start the theoretical study with the two-dimensional low-energy elasticity theory action:

[TABLE]

where $u_{ij}=\frac{1}{2}(\partial_{i}u_{j}+\partial_{j}u_{i})$ is the linear part of the symmetric strain tensor, with $u_{j}$ the displacement field and $C_{ijkl}$ is the 1st-gradient elasticity matrix, with its components determined by the type of medium at hand. Since we will only be working with the defects of the theory, we are interested in the singular part $u_{ij}^{s}$ of the strain tensor field $u_{ij}=u_{ij}^{s}+\frac{1}{2}(\partial_{i}\overline{u}_{j}+\partial_{i}\overline{u}_{j})$, with the smooth part later integrated out to provide a constrain of the theory. In order now to introduce the dual fields of the theory, we perform a Hubbard-Stratonovich transformation, by completing the square in the functional:

[TABLE]

to receive:

[TABLE]

where $\sigma_{ij}$ is elasticity stress-tensor with the symmetry $\sigma_{ij}=\sigma_{ji}$, and $\pi_{i}$, the elastic momentum vector. We employ Einstein’s summation convention. The details of inversion of the elastic tensor are presented in App. (A). If the smooth and singular part of the symmetric strain tensor are now plugged in the equation above, the smooth part is then integrated out, giving the well known Hooke’s law:

[TABLE]

We now introduce the two rotated fields:

[TABLE]

which along with the two-dimensional Levi-Civita identity $\epsilon^{ij}\epsilon_{jk}=\delta^{i}_{k}$ transform the above equation into Faraday’s Law:

[TABLE]

It must be stated that the lower case index $\sigma$ on the rank-2 tensor electric field, signifies the relation with the rotated stress field. This field is actually the elastic counterpart of the usual electric field used in EM theory and works the same way as the electric displacement field. The relation connecting one another is given through the formula:

[TABLE]

where $\tilde{C}_{ijkl}^{-1}=\epsilon_{im}\epsilon_{jn}\epsilon_{ko}\epsilon_{lp}C_{ijkl}^{-1}$ is the rotated inverted elasticity tensor.

Faraday’s equation can be solved by introducing the generalization of the EM theory into a rank-2 gauge theory through:

[TABLE]

where $\phi$ is the usual scalar fractonic field, and $A_{ij}$ a symmetric tensor gauge field. Of course Faraday’s equation stays invariant under the gauge transformation:

[TABLE]

where $f(x_{k},t)$ is an arbitrary space-time function. Using the gauge fields now, we can rewrite the action of our theory, to also include currents and sources

[TABLE]

The above equation describes isolated charges of the theory in the form disclinations. Since our main concern are dislocation defects, we introduce the vector fractonic potential $\phi_{i}=\partial_{i}\phi$ and the dislocation charge density $\rho=-\partial_{i}\rho^{i}$. By performing integration by parts on (9) we get the required action describing gauge fields sourced by dislocations

[TABLE]

We have successfully managed to map the low energy action of the elasticity theory containing two gapless phonon modes (longitudinal and transverse motion) into another U(1) vector charge gauge theory containing two gapless gauge modes. With the usage of the 1st (Gauss) and the 4th (Ampère) Maxwell law for the duality, we can extract the continuity equation for the vector charge theory

[TABLE]

III Equations of motions for isotropic materials in two dimensions

We start by writing the Lagrangian for elasticity in the language of gauge fields:

[TABLE]

where for the case of an isotropic material we have the formula for the inverse elasticity tensor as a function of the elastic bulk modulus $\kappa$ and the shear modulus $\mu$,

[TABLE]

The above tensor obeys the Maxwell-Betti reciprocity relations $\tilde{C}_{ijkl}^{-1}=\tilde{C}_{jikl}^{-1}=\tilde{C}_{ijlk}^{-1}=\tilde{C}_{jilk}^{-1}$. By plugging Eq. (7) in Eq. (12) and using the Euler-Lagrange equations, we extract the two equations of motion

[TABLE]

[TABLE]

[TABLE]

where the coefficients depend on the elastic moduli in the following way

[TABLE]

Both Eq.(14) and (15) are coupled, in analogy to classical electrodynamics. In addition the trace of the tensor gauge field $A_{ij}$ is present, signifying its relation to the motion of dislocations as we show later on.

In order to simplify the equations we need to fix a gauge. We note that in the analogous problem in electrodynamics it is convenient to choose the Lorenz gauge in order to decouple the equations. Therefore we intend to generalize the usual Lorenz gauge to the case of tensor gauge theories. This can be done as follows

[TABLE]

With this choice the second term in (15) cancels with the contribution of the fifth term. Notice that in all the equations, the density $\rho_{d}=1$. Generically in the gauge condition the second term in (16) shall be multiplied by $\rho_{d}$.

The next step into deriving the wave equations uses the fact that the vector potential $\phi_{i}$ is given by the gradient of the scalar field $\phi$, meaning that permutation of the indices in the derivatives is allowed $\partial_{j}(\partial_{i}\phi_{i})=\partial_{i}^{2}\phi_{j}.$ The final form of the wave equations for the vector field $\phi_{i}$ and the tensor gauge field $A_{ij}$ that contain the trace reads

[TABLE]

[TABLE]

where $\Box=\nabla^{2}-(1/\upsilon^{2})\partial_{t}^{2}$, with the velocity for the $\phi_{j}$ field being equal to $\upsilon_{\phi}=\sqrt{\frac{\mu(\kappa+\mu)}{\kappa}}$ and for the $A_{ij}$ field equal to $\upsilon_{A}=\sqrt{2\mu}$. The result is not surprising but a few clarifications need to be made. These velocities are the dual of the ones that appear in the regular elasticity theory of a two-dimensional isotropic solid. The phonon fields there propagate with velocities equal to $\upsilon_{L}=\sqrt{\frac{\kappa+\mu}{\rho}}$ and $\upsilon_{T}=\sqrt{\frac{\mu}{\rho}}$ for a longitudinal and transverse wave propagation respectively. The fracton vector charge density is now multiplied with a factor containing the elastic constants although that is not true for the currents of our theory. In the next section we briefly introduce how defects can be included in the theory.

IV Defects

Topological defects such as disclinations or dislocations, act as sources for the singular part of strain tensor $u_{ij}$. The disclination density is given by

[TABLE]

where $\theta=\epsilon_{kl}\partial_{k}u_{l}$ is the bond angle. Disclinations represent the isolated fracton charges of our theory that cannot move. However, two opposite disclinations of the same magnitude would formulate the dipole that can move albeit with mobility restrictions dubbed dislocations. Their density is represented as a function of a particular lattice vector, called the Burgers vector

[TABLE]

This lattice vector is always perpendicular to the vector joining the two disclination defects. The line element $dr_{i}$ encircles a specific area on the solid that encloses the defect, forming the Burgers circuit $\gamma$.

In general the system containing monopoles, dipoles and even higher combinations of particles can be represented by a charge density of the form:

[TABLE]

Since we are interested in the motion of a defect in the medium, we focus on a single dislocation, dropping both higher order terms and the immobile monopole. By comparing the above density with the vector density in our Lagrangian we can write

[TABLE]

where $\epsilon_{ij}b_{j}=p_{i}$ is the dipole moment. It is set to be equal to the Burgers vector but pointing perpendicular to it. This gives a current of the form

[TABLE]

where $\upsilon_{i}$ is the dislocation velocity and $\mathbf{r}_{s}(t^{\prime})$ is the dislocations trajectory as seen by the observer at retarded time $t^{\prime}$. The trace of the current $J_{ii}$ is related to dislocation climb, i.e. dislocation moving perpendicular to its Burgers vector and not parallel, and is equal to 0 in the case where the are no vacancies in the crystal. We discuss the static solutions of the field equations corresponding to dislocations and disclinations in App. (B). The dynamical solutions will be studied in the next section.

V Liénard-Wiechert potentials and Jefimienko’s equations

V.1 Traceless theory

We start with the simplest exampled dubbed traceless gauge theory pretko_generalized_2017 . We note that imposing the condition can be done following different conventions beekman_dual_2017 . In this work we follow the convention used by Pretko that automatically forces the theory to obey the Ehrenfest constraint. We demand that the trace of the electric tensor field $E_{ii}$ is equal to zero and the intermediate terms of (17) and (18) containing contributions arising from it, drop out. The equations we have then turn out to be the regular two-dimensional wave equations for the two potentials

[TABLE]

[TABLE]

It follows from the equations that the two wave velocities turn out to be the same. Before we move on to discuss the solutions, we give a physical meaning to setting the trace equal to zero. According to Pretko’s theory, the vector charge case is governed by the two conservation laws of charge and dipole moment, easily seen through the first Gauss law. Dislocations of this theory are proven to be able to move only along their Burgers vector (longitudinal motion) and not transverse. The explanation given has to do with the vacancies/interstitials present on the solid. Going to the analysis of the quadrupole moment, the trace is given by:

[TABLE]

where from the duality formulas we have: $E^{ii}=(1/2\kappa)E_{\sigma}^{ii}=\partial_{i}u_{i}=n_{d}=0$, with the smooth part taken to be very small. Here $n_{d}$ represents the difference between vacancies and interstitial defects. The above requirement with the physical meaning of having a component of quadrupole moment related to the trace preserved, implies that the defects of the theory only move parallel to their Burgers vector (fracton dipoles moving only transverse to the dipole moment).

The problem at hand parallels the one of finding Liénard-Wiechert potentials in three space dimensions, however, since we are on the plane the exact form of the solution differs (see boito_maxwells_2020 ; watanabe_integral_2015 for an analogous problem in electrodynamics). The Green’s function related to the above differential equations is given as the Heaviside step-function with a relativistic factor included, giving rise to a phenomenon called ”afterglow” lazar_elastic_2011 ; lazar_elastodynamic_2012 ; dai_origin_2013 .

The afterglow phenomenon is a direct sequence of Huygens principle not being valid in 2+1 dimensions and generically in space-times in odd number of dimensions. This can be seen from the Green’s function describing the equations of motion. In 3+1 dimensions the function is proportional to a Dirac’s delta function, instantly having a pulsating effect and vanishing. Here the Heaviside $\Theta$ function is present, meaning that even though at times $t<t^{\prime}+|\mathbf{r}-\mathbf{r_{s}^{\prime}}|/\upsilon_{s}$ the contribution is zero, the pulse emitted exactly at $t=t^{\prime}$ has an everlasting effect, with the denominator playing the role of an attenuation factor, eventually disappearing at large values of $t$. The full wave describing this effect can be constructed as a superposition of wave modes with velocity values ranging from 0 to $\upsilon_{s}$ (velocity of the outermost wave - the so-called Huygens surface). This is known as ”tail” of the Green’s function leading to a non-sharp wave propagation, in contrast to the (3+1) dimensions case dai_origin_2013 .

Using two-dimensional Green’s function we get

[TABLE]

where $\upsilon_{s}$ is the speed of sound, and $t^{\prime}$, $\mathbf{r^{\prime}}$ the retarded coordinates. By using the definitions of the density and current from the previous section and performing the integration in space, we arrive at:

[TABLE]

[TABLE]

The above integrals have considerable complexity which only in some simple cases yield results of elementary functions in a closed form. We will thus be using the formalism of near-field approximation adopted by Lazar lazar_elastic_2011 ; lazar_elastodynamic_2012 ; lazar_retardation_2013 , where the lower time limit solution is that of regular elastostatics with the defect immobile until $t=0$. The lower limit of the above integral is switched from $-\infty$ to [math] and the upper one has the relativistic boundary value of $t-|\mathbf{r}-\mathbf{r^{\prime}}|/\upsilon_{s}$. By assuming that the defect does not move too far from the observational point, we can take $|\mathbf{r}-\mathbf{r^{\prime}}|\approx d$. The final result is

[TABLE]

[TABLE]

As it can be seen, just like the regular charged particle case a connection between the fields potentials is given in the form of $A\propto\upsilon^{-1}\cdot\phi$. This time the usual relation is inverted because we worked with $\tilde{C}^{-1}_{ijkl}$ in our equations of motion.

By making use of Eq. (7), we find the elastic analogue of Jefimenko’s equations for a dislocation

[TABLE]

[TABLE]

where in the above $n_{i}$ is the unitary vector component of $\mathbf{r}-\mathbf{r_{s}}$ and we have also introduced the relativistic factors $\beta_{i}=\upsilon_{i}/\upsilon_{s}$, $A(t,d)=1+d^{2}/[B(t,d)(B(t,d)-\upsilon_{s}t)]$ and $B(t,d)=\upsilon_{s}t+\sqrt{\upsilon_{s}^{2}t^{2}-d^{2}}$.

We can now invert the duality in order to find the dynamical time dependent stress-tensors components to be

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

From the static limit of the theory and by requiring a symmetric stress tensor, another condition extracted is $b_{x}n_{x}=-b_{y}n_{y}$. Both of these are a result of our initial symmetry of $\partial_{i}\phi_{j}=\partial_{j}\phi_{i}$, since the the field $\phi_{j}$ is the gradient of the scalar fracton field $\phi$.

The strains now can be easily given by making use of the $u_{ij}=C_{ijkl}^{-1}\sigma_{kl}$. We present the full expression in App. C. One can see that the components containing velocities of the dislocation contribute the less to stresses and strains, the higher the numerical values of the elastic moduli are.

V.2 Vacancies present in the crystal

The next physical system we are interested in studying is a solid with vacancies present. In order to get a better intuition about the solution we note that elastodynamics is described by a system of field equations representing the transmission of elastic waves. This can be easily appreciated in the language of displacements since taking the curl of Navier’s equations one obtains two wave equations for dilatational and rotational disturbances eringen_elastodynamics_1975 . This is not so easy to implement in the stress formulation of elastodynamics, that parallels our gauge theory formulation, and in consequence the stress approach is less developed (see e.g. ostoja-starzewski_ignaczak_2019 ). Our strategy to obtain wave solutions from (17) and (18) requires that, on top of fixing a gauge, we need to impose an additional constraint

[TABLE]

The physical interpretation of this constraint is that the climb motion of a topological defect requires the addition or removal of a particle from the regular lattice. In other words the only way to move a dislocation is to excite an interstitial. Eq. 32 constitutes a general condition how this process happens since the trace, related to non-topological defects is now a part of the solution. In order to have a better intuition we can rewrite (32) using the definition of electric field (7)

[TABLE]

In consequence $C$ parameterizes the configuration of non-topological defects in the theory.

In the next step we calculate the distribution of fields. The equations of motion read

[TABLE]

[TABLE]

We identify two wave equations characterized by different velocities. One important consequence of the above equations is that the vector charge density is coupled to the bulk modulus as opposed to the case without vacancies or interstitials. The dipoles in the theory are now allowed to move parallel and perpendicular to the Burgers vector.

Solutions for the $\phi_{i}$ and the off-diagonal component of $A_{ij}$ of course have the same form as before. The differential equation that is left to compute is taken by tracing Eq.(35) to get

[TABLE]

The final solution for this case will be mixing contributions for two waves propagating with two different velocities. We have:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

VI Dynamical Peach-Koehler force

In this section we focus on the interaction between dislocations (for a recent review of the developments in this subject in the conventional formulation of elasticty see lubarda_dislocation_2019 ). We derive the formula describing the force between two dislocations for the traceless theory. An analogous computation for vacancies present is straightforward. The Lorentz force exerted by one dipole on another reads:

[TABLE]

The first term represents the usual Peach-Koehler force, which will have contributions from both gauge fields, which account for relativistic corrections depending on time and the defect’s velocity, along with the usual term that encapsulates the material’s elastic properties. The second term is the Biot-Savart analogue for the elastic force-interaction and it purely depends on the velocity. The full force written as a function of the Burgers vectors and the velocity reads:

[TABLE]

The force contains also a contribution from the acceleration (if present) of the defect just like in electrodynamics. The first term on the right hand side of Eq. (VI) is the regular ”electrostatic” component in the low velocity limit. This can be seen if the relativistic correction is equal to unity. It happens when one interchanges $b_{i}\longleftrightarrow d_{i}$ and then fixes $n_{i}\rightarrow-n_{i}$. The result will be the force on the first dislocation produced by the second one, $F^{PK}_{2\rightarrow 1}$. The contributions from the first terms will cancel each other out, but this is not the case for the rest of terms containing corrections from velocities (i.e. magnetic part). This parallels results in electrodynamics. We know that two moving charged particles exert forces that are not reciprocal. The explanation being that the difference in rate of change of the system’s momentum is carried out as electromagnetic radiation. In our case the stress field carries the disturbance and as a consequence of the elastic radiation from a moving dislocation.

VII Discussion

In this work we show that a dual gauge theory formulation of elasticity simplifies the analysis of defect dynamics. Using this simplification, we extend existent results focused on edge dislocations. We construct solutions to the equations of tensor gauge theory, dual to Cauchy’s isotropic elasticity in a gauge, that is analogous to the Lorenz gauge in electrodynamics, for an arbitrarily moving dislocation. We use this to determine elastic stresses and strains due to the moving defect. We discus separately two physical scenarios, when non-topological defects are absent and when they are present. The former case corresponds to dislocations obeying the glide constraint, which in the static limit has been studied. We generalize this result to account for vacancies and interstitials. This leads to a new constraint that both topological and non-topological defects must satisfy, that we formulate as a relationship between gauge potentials. We interpret this result as a physical mechanism that forces that dislocations to excite non-topological defects if they move perpendicularly to their Burgers vector. Finally, we use the duality to extend the formula of Peach and Koehler for the interactions between dislocations. We give an analytic formula that describes the dynamical Peach-Koehler force for two dislocations with different Burgers vectors or charges.

The motion of defects in crystal lattices is essential to our understanding of material strength and plasticity. Nevertheless, despite much effort dedicated to the subject, solving the defect dynamics in conventional formulation of elasticity has led only to partial results. On top of that isotropic Cauchy’s elasticity describes only the most basic crystalline solid. This renders those results unfeasible to be generalized to more complicated systems involving anisotropy, rotational degrees of freedom or incommensurate lattices. On the other hand, the simplicity of our approach allows one to extend results presented here to ask more detailed questions in crystals, such as radiation or problems related to fast-moving dislocations as well as understand defect dynamics in more general theories of elasticity that can be formulated in the language of gauge theories.

Finally we note that the problem of defects has recently reappeared in the context of active solids. Generalizing our results to the case of non-Hermitian elasticity could shed light on the dynamics of defects in active media. This will be useful in understanding recent experiments on active solids with odd elasticty.

Acknowledgements.

PS acknowledges support form the Polish National Science Centre (NCN) Sonata Bis grant 2019/34/E/ST3/00405. LT was supported in part by the Deutsche Forschungsgemeinschaft through the cluster of excellence ct.qmat (Exzellenzcluster 2147, Project 390858490).

Appendix A Inversion of the elasticity tensor

We present here the supplementary theory regarding the inversion of the elasticity tensor. The most general case represented in the form of the three projector operators, is given by

[TABLE]

We define

[TABLE]

[TABLE]

[TABLE]

where $p^{1}$, $p^{2}$, $p^{3}$ are functions of the elastic parameters $\kappa$ and $\mu$. The projectors satisfy the closure identity $P_{ijkl}^{0}+P_{ijkl}^{1}+P_{ijkl}^{2}=\delta_{ij}\delta_{kl}$ and $P_{ijkl}^{a}P_{klmn}^{b}=P_{ijmn}^{a}$ if $a=b$ and otherwise. The inversion then is easily performed giving

[TABLE]

For the case of an isotropic solid, we require the system to be invariant under the improper rotations group O(2) satisfying the relation $P_{ijkl}^{1}\sigma_{ij}=2\omega_{kl}=0$, where $\omega_{ij}$ is the angular velocity. Using the above formulas the elasticity tensor and its inverted counterpart for this case are given by

[TABLE]

[TABLE]

For the rotated inverted tensor now, we have

[TABLE]

proving Eq.(13) used in deriving the equations of motion.

Appendix B Fracton Elastostatics

B.1 Disclination

In the case of no time dependence in the equations of motion, the differential equation for the fractonic vector field is

[TABLE]

The scalar field $\phi$ acts as an Airy’s stress function and the differential equation at hand is a non-homogeneous biharmonic equation in two dimensions, appearing often in linear elasticity and linearized fluid mechanics. The solution is given as special case of the Mitchell solution with no radial dependence, for $\rho=q\delta^{2}(r-r_{s})$. $q$ represents the disclination charge. A simple and easy way of solving the above is by noticing that $\nabla^{2}\ln(r-r_{s})=2\pi\delta^{2}(\mathbb{r}-\mathbb{r}_{s}^{\prime})$. This gives the Poisson differential equation:

[TABLE]

The solution is given by having the potential drop to zero at the boundary $R$ of the material

[TABLE]

where again the logarithmic behaviour is expected. The vector field $\phi_{i}$ has components

[TABLE]

Upon inversion of Eq.(4) one can extract the stress and electric tensors:

[TABLE]

where, $\sigma_{xx}=E_{yy}$, $\sigma_{yy}=E_{xx}$. The above equations are in full agreement with those in braverman_topological_2021 . The strain tensor is given by making use of $u_{ij}=C_{ijkl}^{-1}\sigma_{kl}$, to get

[TABLE]

B.2 Dislocation

For the case of a dislocation now, we assume a density of the form $\rho_{j}=b_{j}\delta^{2}(\mathbb{r}-\mathbb{r_{s}})$ and get a solution

[TABLE]

For the electric field and the stress tensor we calculate

[TABLE]

where $\sigma_{xx}=E_{yy}$, $\sigma_{yy}=E_{xx}$, $\sigma_{yx}=\sigma_{xy}=-E_{xy}=-E_{yx}$. Finally we obtain the expressions for the strains

[TABLE]

Appendix C Strain

The strain field extracted by using the relation $u_{ij}=C_{ijkl}^{-1}\sigma_{kl}$ reads

[TABLE]

[TABLE]

[TABLE]