Distributional and regularized radiation fields of non-uniformly moving straight dislocations, and elastodynamic Tamm problem

# Distributional and regularized radiation fields of non-uniformly moving straight dislocations, and elastodynamic Tamm problem

Heisenberg Research Group,
Department of Physics,
Hochschulstr. 6,

CEA, DAM, DIF,
F-91297 Arpajon, France
###### Abstract

This work introduces original explicit solutions for the elastic fields radiated by non-uniformly moving, straight, screw or edge dislocations in an isotropic medium, in the form of time-integral representations in which acceleration-dependent contributions are explicitly separated out. These solutions are obtained by applying an isotropic regularization procedure to distributional expressions of the elastodynamic fields built on the Green tensor of the Navier equation. The obtained regularized field expressions are singularity-free, and depend on the dislocation density rather than on the plastic eigenstrain. They cover non-uniform motion at arbitrary speeds, including faster-than-wave ones. A numerical method of computation is discussed, that rests on discretizing motion along an arbitrary path in the plane transverse to the dislocation, into a succession of time intervals of constant velocity vector over which time-integrated contributions can be obtained in closed form. As a simple illustration, it is applied to the elastodynamic equivalent of the Tamm problem, where fields induced by a dislocation accelerated from rest beyond the longitudinal wave speed, and thereafter put to rest again, are computed. As expected, the proposed expressions produce Mach cones, the dynamic build-up and decay of which is illustrated by means of full-field calculations.

Keywords: dislocation dynamics; non-uniform motion; generalized functions; elastodynamics; radiation; regularization.

## 1 Introduction

Dislocations are linear defects whose motion is responsible for plastic deformation in crystalline materials (Hirth and Lothe, 1982). To improve the current understanding of the plastic and elastic fronts (Clifton and Markenscoff, 1981) that go along with extreme shock loadings in metals (Meyers et al., 2009), Gurrutxaga-Lerma et al. (2013) recently proposed to make dynamic simulations of large sets of dislocations mutually coupled by their retarded elastodynamic field. Gurrutxaga-Lerma et al. (2014) review the matter and its technical aspects in some detail. This new approach is hoped to provide complementary insights over multi-physics large-scale atomistic simulations of shocks in matter (Zhakhovsky et al., 2011). If we leave aside the subsidiary (but physically important) issue of dislocation nucleation, dislocation-dynamics simulations involve two separate but interrelated tasks. First, one needs to compute the field radiated by a dislocation that moves arbitrarily. Second, given the past history of each dislocation, the current dynamic stress field incident on it due to the other ones, and the externally applied stress field (e.g., a shock-induced wavefront), the further motion of the dislocation must be determined by a dynamic mobility law. While some progress has recently been achieved in the latter subproblem —which involves scarcely explored radiation-reaction effects and dynamic core-width variations (Pellegrini, 2014)— the focus of the present paper is on the former —a very classical one.

Indeed, substantial effort has been devoted over decades to obtaining analytical expressions of elastodynamic fields produced by non-uniformly moving singularities such as point loads (Stronge, 1970; Freund, 1972, 1973), cracks, and dislocations. Results ranged, e.g., from straightforward applications to linear-elastic and isotropic unbounded media, to systems with interfaces such as half-spaces (Lamb’s problem) or layered media (Eatwell et al., 1982); coupled phenomena such as thermoelasticity (Brock et al., 1997) or anisotropic elastic media (Markenscoff and Ni, 1987; Wu, 2000), to mention but a few popular themes. Elastodynamic fields of dislocations have been investigated in a large number of works, among which (Eshelby, 1951; Kiusalaas and Mura, 1964, 1965; Mura, 1987; Nabarro, 1967; Brock, 1979, 1982, 1983; Markenscoff, 1980; Markenscoff and Ni, 2001a, b; Pellegrini, 2010; Lazar, 2011b, 2012, 2013a, 2013b). Early numerical implementations of time-dependent fields radiated by moving sources (Niazy, 1975; Madariaga, 1978) were limited to material displacements or velocities. As to stresses, Gurrutxaga-Lerma et al. (2014) based their simulations on the fields of Markenscoff and Clifton (1981) relative to a subsonic edge dislocation. Nowadays dynamic fields of individual dislocations or cracks are also investigated by atomistic simulations (Li and Shi, 2002; Tsuzuki et al., 2009; Spielmannová et al., 2009), or numerical solutions of the wave equation by means of finite-element (Zhang et al., 2015), finite-difference, or boundary-integral schemes (Day et al., 2005). Hereafter, the analytical approach is privileged so as to produce reference solutions.

Disregarding couplings with other fields such as temperature, one might be tempted to believe that the simplest two-dimensional problem of the non-uniform motion of rectilinear dislocation lines in an unbounded, linear elastic, isotropic medium, leaves very little room for improvements over past analytical works. This is not so, and our present concerns are as follows:

(i) Subsonic as well as supersonic velocities. In elastodynamics, from the 70’s onwards, the method of choice for analytical solutions has most often been the one of Cagniard improved by de Hoop (Aki and Richards, 2009), whereby Laplace transforms of the fields are inverted by inspection after a deformation of the integration path of the Laplace variable has been carried out by means of a suitable change of variable (see above-cited references). However, to the best of our knowledge, no such solutions can be employed indifferently for subsonic and supersonic motions, in the sense that the supersonic case need be considered separately in order to get explicit results as, e.g., in (Stronge, 1970; Freund, 1972; Callias and Markenscoff, 1980; Markenscoff and Ni, 2001b; Huang and Markenscoff, 2011). Indeed, carrying out the necessary integrals usually requires determining the wavefront position relatively to the point of observation. To date, the supersonic edge dislocation coupled to both shear and longitudinal waves has not been considered, and existing supersonic analytical solutions for the screw dislocation have not proved usable in full-field calculations, except for the rather different solution obtained within the so-called gauge-field theory of dislocations (Lazar, 2009), which appeals to gradient elasticity. Thus, one objective of the present work is to provide ‘automatic’ theoretical expressions that do not require wavefront tracking, for both screw and edge dislocations, and can be employed whatever the dislocation velocity. To this aim, we shall employ a method different from the Cagniard–de Hoop one. This is not to disregard the latter but following a different route was found more convenient in view of the remaining points listed here.

(ii) Distributions and smooth regularized fields. For a Volterra dislocation in supersonic steady motion, fields are typically concentrated on Dirac measures along infinitely thin lines, to form Mach cones (Stronge, 1970; Callias and Markenscoff, 1980). Thus, the solution is essentially of distributional nature, and its proper characterization involves, beside Dirac measures, the use of principal-value and finite-parts prescriptions (Pellegrini and Lazar, 2015). Of course, in-depth analytical characterizations of wavefronts singularities can still be extracted out of Laplace-transform integral representations (Freund, 1972, 1973; Callias and Markenscoff, 1980). However their distributional character implies that the solutions cannot deliver meaningful numbers unless they are regularized by convolution with some source shape function representing a dislocation of finite width. Only by this means can field values in Mach cones be computed. Consequently, another objective is to provide field expressions for an extended dislocation of finite core width (instead of a Volterra one), thus taming all the field singularities that would otherwise be present at wavefronts and at the dislocation location, where Volterra fields blow up. In the work by Gurrutxaga-Lerma et al. (2013), a simple cut-off procedure was employed to get rid of infinities. Evidently, a similar device cannot be used with Dirac measures, which calls for a smoother and more versatile regularization. Various dislocation-regularizing devices have been proposed in the past, some consisting in expanding the Volterra dislocation into a flat Somigliana dislocation (Eshelby, 1949, 1951; Markenscoff and Ni, 2001a, b; Pellegrini, 2011). Such regularizations remove infinities, but leave out field discontinuities on the slip path (Eshelby, 1949). A smoother approach consisting in introducing some non-locality in the field equations has so far only be applied to the time-dependent motion of a screw dislocation. The one to be employed hereafter, introduced in (Pellegrini and Lazar, 2015), achieves an isotropic expansion the Volterra dislocation and smoothly regularizes all field singularities for screw and edge dislocations. In this respect, it resembles that introduced in statics by Cai et al. (2006). However, we believe it better suited to dynamics.

(iii) Field-theoretic framework. The traditional method of solution (Markenscoff, 1980) rests on imposing suitable boundary conditions on the dislocation path. It makes little contact with field-theoretic notions of dislocation theory such as plastic strain, or dislocation density and current used in purely numerical methods of solution (Djaka et al., 2015). Instead, we wish our analytical results to be rooted on a field-theoretic background. One advantage is that the approach will provide a representation of radiation fields where velocity- and acceleration-dependent contributions are clearly separated out, which is most convenient for subsequent numerical implementation. Again to the best of our knowledge, no such representation of the elastodynamic fields has been given so far. However, previous work in that direction can be found in (Lazar, 2011b, 2012, 2013a).

(iv) Integrals in closed form. In (Gurrutxaga-Lerma et al., 2013) the numerical implementation of the results by Markenscoff and Clifton (1981), where the retarded fields are expressed in terms of an integral over the path abscissa, is not fully explicit. Indeed, this integral is split over path segments, and each segment is integrated over numerically — a tricky matter, as pointed out by the former authors. By contrast, and dealing with time intervals instead of path segments, the sub-integrals will be expressed hereafter in closed form by means of the key indefinite integrals obtained in (Pellegrini and Lazar, 2015).

(v) Arbitrary paths. Results will be given in tensor form, with the dislocation velocity as a vector. Thus, they can be applied immediately to arbitrary dislocation paths parametrized by time. Using the time variable as the main parameter is a natural choice, and does not require computing so-called ‘retarded times’. Although we must leave such applications to further work, this makes it straightforward to investigate radiative losses in various oscillatory motions, e.g., (lattice-induced) periodic oscillations in the direction transverse to the main glide plane during forward motion, which space-based parametrizations such as in the procedure outlined by Brock (1983) make harder to achieve.

Accordingly, our work is organized as follows. First, we begin by computing in Section 2 general forms for the elastic fields of non-uniformly moving screw and edge dislocations using the theory of distributions, starting from the most general field equations in terms of dislocation densities and currents. Our approach relies on Green’s functions [e.g., Barton (1989); Mura (1987)]. In Section 2.1, inhomogeneous Navier equations for the elastic fields are derived as equations of motion, with source terms expressed in terms of the fields that characterize the dislocation (dislocation density tensor and dislocation current tensor). The Cauchy problem of the Navier equations is then addressed in Section 2.2, where the solutions for the elastic fields are written as the convolution of the retarded elastodynamic Green-function tensor —interpreted as a distribution— with the dislocation fields. As a result, the mathematical structure of the latter is partly inherited from the former. Some connections with past works are made in Section 2.3. Second, we specialize the obtained field expressions to Volterra dislocations: the fields themselves become distributions. In Section 3, the structure of the Green tensor and of the elastodynamic radiation fields is revealed and analyzed in terms of locally-integrable functions and pseudofunctions (namely, singular distributions that require a ‘finite part’ prescription). The Volterra screw (Section 3.1) and edge (Section 3.2) dislocations are addressed separately for definiteness. The expressions reported are mathematically well-defined, and cover arbitrary speeds including faster-then-wave ones, which is the main difference with classical approaches. Third, since distributional fields, although mathematically correct, cannot in general produce meaningful numbers unless being applied to test functions, we turn the formalism into one suitable to numerical calculations by means of the isotropic-regularization procedure alluded to above, where the relevant smooth test function represents the dislocation density. The procedure is introduced in Section 4.1, and regularized expressions for the elastic fields are obtained in integral form in Section 4.2, after the regularized Green tensor has been defined. Next, a numerical implementation scheme that involves only closed-form results is proposed in Section 5, based on the key integrals of Pellegrini and Lazar (2015). As a first illustration, the particular case of steady motion for the edge dislocation is discussed in detail, with emphasis on faster-than-wave motion. Finally, the procedure is applied in Section 6 to the numerical investigation of the elastodynamic equivalent of the Tamm problem, where fields induced by a dislocation accelerated from rest beyond the longitudinal wave speed, and thereafter put to rest again, are computed and analyzed. Section 7 provides a concluding discussion, which summarizes our approach and results, and points out some limitations. The most technical elements are collected in the Appendix.

## 2 Basic geometric equations and field equations of motion

### 2.1 Field identities and equations of motion

In this Section, the equations of motion of the elastic fields produced by moving dislocations are derived in the framework of incompatible elastodynamics (see, e.g., Mura (1963, 1987); Kosevich (1979); Lazar (2011b, 2013b)). An unbounded, isotropic, homogeneous, linearly elastic solid is considered. In the theory of elastodynamics of self-stresses, the equilibrium condition is333We use the usual notation and .

 ˙pi−σij,j=0, (1)

where and are the linear momentum vector and the stress tensor, respectively. For incompatible linear elastodynamics, the momentum vector and the stress tensor can be expressed in terms of the elastic velocity (particle velocity) vector and the incompatible elastic distortion tensor by means of the two constitutive relations

 pi =ρvi, (2a) σij =Cijklβkl, (2b)

where denotes the mass density, and the tensor of elastic moduli or elastic tensor. It enjoys the symmetry properties . For isotropic materials, the elastic tensor reduces to

 Cijkl=λδijδkl+μ(δikδjl+δilδjk), (3)

where and are the Lamé constants. If the constitutive relations (2a) and (2b) are substituted into Eq. (1), the equilibrium condition expressed in terms of the elastic fields and may be written as

 ρ˙vi−Cijklβkl,j=0. (4)

The presence of dislocations makes the elastic fields incompatible, which means that they are not anymore simple gradients of the material displacement vector . In the eigenstrain theory of dislocations (e.g., Mura (1987)) the total distortion tensor consists of elastic and plastic parts444Note, however, that the tensors and defined by Mura are the transposed of the ones used in the present work. The same goes for .

 βTij :=ui,j=βij+βPij, (5)

but . Here is the plastic distortion tensor or eigendistortion tensor. The plastic distortion is a well-known quantity in dislocation theory and in Mura’s theory of eigenstrain. Nowadays, this field can be understood as a tensorial gauge field in the framework of dislocation gauge theory (Lazar and Anastassiadis, 2008; Lazar, 2010).

For dislocations, the incompatibility tensors are the dislocation density and dislocation current tensors (e.g., Holländer (1962); Kosevich (1979); Lazar (2011a)). The dislocation density tensor and the dislocation current tensor are classically defined by (e.g., Kosevich (1979); Landau and Lifschitz (1986))

 αij =−ϵjklβPil,k, (6a) Iij =−˙βPij, (6b)

or they read in terms of the elastic fields

 αij =ϵjklβil,k, (7a) Iij =˙βij−vi,j. (7b)

Eqs. (6a) and (6b) are the fundamental definitions of the dislocation density tensor and of the dislocation current tensor, respectively, whereas Eqs. (7a) and (7b) are geometric field identities. Originally, Nye (1953) introduced the concept of a dislocation density tensor, and the definition (6a) of goes back to Kröner (1955, 1958) and Bilby (1955) (see also Kröner (1981)). The tensor was introduced by Kosevich (1962) under the name ‘dislocation flux density tensor’ —a denomination used by Kosevich (1979); Teodosiu (1970), and Lardner (1974)— and by Holländer (1962) as the ‘dislocation current’ (see also Kosevich (1979); Landau and Lifschitz (1986); Teodosiu (1970)). We adopt hereafter the latter denomination. Both and have nine independent components. Moreover, they fulfill the two dislocation Bianchi identities (see also Landau and Lifschitz (1986); Lazar (2011a))

 αij,j =0, (8a) ˙αij+ϵjklIik,l =0, (8b)

which are geometrical consequences due to the definitions (6a)–(7b). Thus,if the dislocation density tensor and dislocation current tensor are given in terms of the elastic fields and plastic fields according to Eqs. (6a)–(7b), then the two dislocation Bianchi identities (8a) and (8b) are satisfied automatically. Conversely, if the two dislocation Bianchi identities (8a) and (8b) are fulfilled, then the dislocation density tensor and the dislocation current tensor can be expressed in terms of elastic and plastic fields according to Eqs. (6a)–(7b) using the additive decomposition (5). Therefore, the dislocation Bianchi identities (8a) and (8b) are a kind of compatibility conditions for the dislocation density tensor and dislocation flux tensor or ‘dislocation conservation laws’ (see also Kosevich (1979)).

From the physical point of view, Eq. (8a) states that dislocations do not end inside the body and Eq. (8b) shows that whenever a dislocation moves or the dislocation core changes its structure and shape, the dislocation current is nonzero. Thus, the dislocation density can only change via the dislocation current, which means that the evolution of the dislocation density tensor is determined by the curl of the dislocation flux tensor .

From the equilibrium condition (4), uncoupled field equations for the elastic fields and produced by dislocations may be derived as equations of motion (see, e.g., Lazar (2011b, 2013b)). They read

 Likβkm =ϵnmlCijklαkn,j+ρ˙Iim, (9a) Likvk =CijklIkl,j, (9b)

where stands for the elastodynamic Navier differential operator

 Lik=ρδik∂tt−Cijkl∂j∂l. (10)

Substituting Eq. (3) into Eq. (10), its isotropic form reads

 Lik=ρδik∂tt−μδikΔ−(λ+μ)∂i∂k (11)

where denotes the Laplacian. Eq. (9a) is a tensorial Navier equation for and Eq. (9b) is a vectorial Navier equation for , where the dislocation density and current tensors act as source terms.

### 2.2 Green tensor and integral solutions

We now turn to the solution of the retarded field problem of Eqs. (9a) and (9b). For this purpose we use Green functions (e.g., Barton (1989)). Let denote the Dirac delta function and denote the Kronecker symbol. The elastodynamic Green tensor is the solution, in the sense of distributions,555The ‘plus’ superscript serves to distinguish this distribution from the associated function to be introduced in Sec. 5. of the (anisotropic) inhomogeneous Navier equation with unit source

 LikG+km(r−r′,t−t′)=δimδ(t−t′)δ(r−r′), (12)

subjected to the causality constraint

 G+ij(r−r′,t−t′)=0fort

The following properties hold in the equal-time limit (Appendix A):

 limτ→0+G+ij(r,τ)=0,limτ→0+∂tG+ij(r,τ)=ρ−1δijδ(r). (14)

Now we consider the Cauchy problem of the inhomogeneous Navier equation, expressed by Eqs. (9a) and (9b). For an unbounded medium, its solutions are (see also Eringen and Suhubi (1975); Barton (1989); Vladimirow (1971))

 βim(r,t) =ϵnml∫tt0dt′∫CjkplG+ij(r−r′,t−t′)αpn,k(r′,t′)dr′ +∫tt0dt′∫ρG+ij(r−r′,t−t′)˙Ijm(r′,t′)dr′ +∫G+ij(r−r′,t−t0)˙βjm(r′,t0)dr′ +∫˙G+ij(r−r′,t−t0)βjm(r′,t0)dr′ (15a) and vi(r,t) =∫tt0dt′∫CjklmG+ij(r−r′,t−t′)Ilm,k(r′,t′)dr′ +∫G+ij(r−r′,t−t0)˙vj(r′,t0)dr′ +∫˙G+ij(r−r′,t−t0)vj(r′,t0)dr′, (15b)

where integrals over are over the whole medium, and where the following functions have been prescribed as initial conditions at throughout the medium:

 β(r,t0),˙β(r,t0),v(r,t0),˙v(r,t0). (16)

Because the elastodynamic Navier equation is a generalization of the wave equation, Eqs. (15) and (15) are similar to the Poisson formula for the latter (Vladimirow, 1971).

Since and vanish as , Eqs. (15) and (15) can be represented as convolutions of the Green tensor with the sources of the inhomogeneous Navier equations, only (Mura, 1963; Lazar, 2011b). Letting thus the solutions for and reduce to

 βim(r,t) =ϵnml∫t−∞dt′∫CjkplG+ij(r−r′,t−t′)αpn,k(r′,t′)dr′ +∫t−∞dt′∫ρG+ij(r−r′,t−t′)˙Ijm(r′,t′)dr′, (17a) vi(r,t) =∫t−∞dt′∫CjklmG+ij(r−r′,t−t′)Ilm,k(r′,t′)dr′, (17b)

or equivalently

 βim(r,t) =ϵnml∫t−∞dt′∫CjkplG+ij,k(r−r′,t−t′)αpn(r′,t′)dr′ +∫t−∞dt′∫ρ˙G+ij(r−r′,t−t′)Ijm(r′,t′)dr′, (18a) vi(r,t) =∫t−∞dt′∫CjklmG+ij,k(r−r′,t−t′)Ilm(r′,t′)dr′. (18b)

Eqs. (17)–(18b) are valid for general dislocation distributions (continuous distribution of dislocations, dislocation loops, straight dislocations). Later on, we shall specialize to straight dislocations.

Let the velocity of a moving dislocation be some given function of time. Then, the following relation holds between its associated dislocation density and current tensors:

 Iij=ϵjknVkαin. (19)

This relation means that the current is caused by the moving dislocation density . Thus, is a convection dislocation current (Günther, 1973; Lazar, 2013b). Substituting Eq. (19) into relation (8b), the Bianchi identity (8b) reduces to the following form in terms of the dislocation density tensor and the dislocation velocity vector

 ˙αij=−ϵjkl(ϵkmnVmαin),l=(Vjαil),l−(Vlαij),l. (20)

Sometimes the Bianchi identity (20) is called dislocation density transport equation (see, e.g., Djaka et al. (2015)).

We moreover obtain from Eqs. (9a) and (9b) the field equations of motion in the form

 Likβkm =ϵnml[Cijklαkn,j+ρ∂t(Vlαin)], (21a) Likvk =ϵnmlCijkm(Vlαkn),j, (21b)

where the sources are given in terms of the dislocation density tensor and the dislocation velocity vector. Obviously, the validity of Eq. (20) is conditioned by the assumptions that underlie Eq. (19). Thus, Eq. (19) makes sense only for a discrete dislocation line with rigid core, since it neglects changes with time of its core shape. However, by imposing a suitable parameterization of the dislocation density or of the plastic eigenstrain (e.g., Pellegrini (2014)), an additional term in the current tensor related to core-width variations could easily be derived from Eq. (6b). Such effects are neglected in the present study. Accordingly, from Eq. (20) and using for the problem considered, we deduce with the help of the Bianchi identity (8a) that for one single rigid dislocation

 ˙αij =−Vkαij,k. (22)

### 2.3 Remarks

It is worthwhile pointing out that although we insisted, for better physical insight, on deriving Eq. (18) from field equations with sources expressed in terms of dislocation density and current, the latter equation is fully consistent with the perhaps more familiar writing of the elastic distortion in terms of the plastic distortion and the second derivatives of the Green tensor as (e.g., Mura (1987))666 Indeed, ignoring our present emphasis on the distributional character of the Green tensor, Eq. (18) is nothing but Eq. (38.36) on p. 351 of Mura’s treatise. This is realized upon comparing Mura’s Eq. (38.19) with the above definition (6b) of , bearing in mind the transposed character of our dislocation tensors with respect to Mura’s (see note 3).

 βij(r,t) =−∫t−∞dt′∫dr′G+ik,jl(r−r′,t−t′)CklmnβPmn(r′,t′)−βPij(r,t). (23)

Also, the issue of the upper boundary of the time-integration in the integral solutions deserves some comments. It is sometimes read in treatises on Green functions, e.g., (Barton, 1989), that the upper boundary should lie slightly above , which is usually denoted by . Such a device helps one to easily check that the integral formulas are indeed solutions of the equation of motion they derive from. Because of the causality constraint, the upper time-integration boundary can as well be taken as . However, it is less recognized that the boundary can as well be chosen slightly below , which we denote as . This is possible because of the two limiting properties (14), the first of which ensuring that removing the interval from the integration interval makes no difference on the final result. The second property in (14) allows us to show —in Appendix B— that solutions written with integrals over satisfy the equation of motion as well. Since the solution is unique, all these formulations give identical results. However, the use of , which amounts to eliminating the immediate vicinity of the point from time integrals, is much more convenient for numerical and analytical purposes, as will be shown in Section 5.2. This device has already been employed in (Pellegrini, 2011, 2012, 2014; Pellegrini and Lazar, 2015), but was introduced there without any detailed justification. Until Section 4, we continue denoting the upper boundary as in general formulas, for simplicity.

## 3 Straight Volterra dislocations in the framework of distributions

In this Section, the elastodynamic fields produced by the non-uniform motion of straight screw and edge Volterra dislocations are studied using the theory of distributions or generalized functions (Schwartz, 1950/51; Gel’fand and Shilov, 1964; Kanwal, 2004). The field equations of motion are solved by means of Green functions. The problem is two-dimensional, of anti-plane strain or plane strain character.

### 3.1 Screw dislocation

We address first the anti-plane strain problem of a Volterra screw dislocation in non-uniform motion at time along some arbitrary path prescribed in advance in the time range . The dislocation line and the Burgers vector are parallel to the -axis. The dislocation velocity has two non-vanishing components: , . The dislocation density and dislocation current tensors are

 αzz=bzℓzδ(R(t)),Izj=bzϵjkzVk(t)ℓzδ(R(t)), (24)

where , is a unit vector in -direction and . The index is a fixed index (no summation).

Eqs. (21a) and (21b) simplify enormously for the nonvanishing components , , and . Using Eq. (24) and , we obtain from Eqs. (21a) and (21b) the following equations of motion of a screw dislocation:

 Lzzβzm =ϵzml[Czjzlαzz,j+ρ(˙Vlαzz−VlVkαzz,k)], (25a) Lzzvz =ϵzmlCzjzmVlαzz,j, (25b)

where is the dislocation acceleration. With the dynamic elastic tensor for non-uniform motion, namely,

 ˜Cijkl(V) =Cijkl−ρVjVlδik (26)

and using the property of the differentiation of a convolution, the appropriate solution may be written as the convolution integrals

 βzm(r,t) +ρG+zz(r−r′,t−t′)˙Vl(t′)}αzz(r′,t′)dr′, (27a) vz(r,t) (27b)

The dynamic elastic tensor (26) was originally introduced by Sáenz (1953) for uniformly moving dislocations (see also Bacon et al. (1979), who use a different index ordering), and employed in elastodynamics by Wu (2000) with the same index ordering as in Eq. (26). It possesses only the major symmetry .

Substituting the dislocation density (24) into Eqs. (27) and (27b), and performing the integration over , we obtain

 βzm(r,t) =bzℓzϵzml∫t−∞{G+zz,k(r−s(t′),t−t′)˜Czkzl(V(t′))+ρG+zz(r−s(t′),t−t′)˙Vl(t′)}dt′ (28a) vz(r,t) =bzℓzϵzml∫t−∞CzkzmG+zz,k(r−s(t′),t−t′)Vl(t′)dt′, (28b)

where is the retarded Green function (distribution) of the anti-plane problem defined by

 LzzG+zz=(ρ∂tt−μΔ)G+zz=δ(t)δ(r). (29)

If the material is infinitely extended, the two-dimensional elastodynamic Green-function distribution of the anti-plane problem, which is nothing but the usual Green function of the two-dimensional scalar wave equation (e.g. Morse and Feshbach (1953); Barton (1989)), interpreted as a distribution, reads (see, e.g., Eringen and Suhubi (1975); Kausel (2006))

 (30)

with the velocity of transverse elastic waves (shear waves, also called S-waves)

 cT=√μ/ρ, (31)

and where is the Heaviside unit-step function that restricts this causal solution to positive times. In this writing, the generalized function , defined as (see, e.g., Schwartz (1950/51); Gel’fand and Shilov (1964); Kanwal (2004); de Jager (1969))

 xλ+={0for x<0xλfor x>0 , (32)

has been used. The derivative of is given by

 (x1/2+)′=12x−1/2+. (33)

The derivative of gives a pseudofunction (see Schwartz (1950/51); Gel’fand and Shilov (1964); Zemanian (1965)):

 (x−1/2+)′=−12Pf x−3/2+. (34)

The symbol Pf in Eq. (34) stands for pseudofunction. In general, pseudofunctions are distributions generated by Hadamard’s finite part of a divergent integral. They arise naturally when certain distributions are differentiated. In Eq. (34) the regular distribution was differentiated. Using Eqs. (33) and (34), the derivative of the Green function (30) is expressed as the pseudofunction

 (35)

Finally, the elastic fields of a non-uniformly moving screw Volterra dislocation read, in distributional form

 βzx(r,t) =bzℓz2πc2T∫t−∞(˙Vy(t′)[¯t2−R2(t′)/c2T]−1/2+ +((1−V2y(t′)c2T)Ry(t′)−Vx(t′)Vy(t′)c2TRx(t′))Pf[¯t2−R2(t′)/c2T]−3/2+)dt′, (36a) βzy(r,t) =−bzℓz2πc2T∫t−∞(˙Vx(t′)[¯t2−R2(t′)/c2T]−1/2+ +((1−V2x(t′)c2T)Rx(t′)−Vx(t′)Vy(t′)c2TRy(t′))Pf[¯t2−R2(t′)/c2T]−3/2+)dt′, (36b) vz(r,t) =bzℓz2πc2T∫t−∞(Vy(t′)Rx(t′)−Vx(t′)Ry(t′))Pf[¯t2−R2(t′)/c2T]−3/2+dt′, (36c)

where .

The fields given by Eqs. (36)–(36c) clearly consist of two parts: (i) Fields depending on the dislocation velocities and alone and proportional to the pseudofunction distribution of power —dislocation velocity-dependent fields or near fields, built from the gradient of the Green tensor; (ii) Fields depending on the dislocation accelerations and and proportional to the regular distribution of power —dislocation acceleration-dependent fields or far fields, built on the Green tensor itself. The velocity field (36c) possesses no acceleration part.

It should be mentioned that it seems to be hard to find a measurement which can distinguish between the acceleration- and velocity-depending fields. Such a decomposition is basically conceptual. In a natural way, we may separate into two parts, one which involves the dislocation acceleration and goes to zero for , and one which involves only the dislocation velocity and yields the static field for a dislocation with . Dislocations at rest or in steady motion do not generate elastodynamic waves. Only non-uniformly moving dislocations emit elastodynamic radiation.

Some historical remarks are in order. In the 1950s already, Sauer (1954, 1958) emphasized the interest of introducing the theory of distributions in supersonic aerodynamics. In particular, in gas dynamics and wing theory, pseudofunctions of power have been used in the framework of distribution theory, e.g., Sauer (1954, 1958); Dorfner (1957) (see also de Jager (1969)).

### 3.2 Edge dislocation

We next turn to the straight edge Volterra dislocation in the plane-strain framework. Its associated dislocation density and current tensors read, respectively,

 αij=biℓjδ(R(t)),Iij=biϵjklVk(t)ℓlδ(R(t)), (37)

where and . Using , we obtain from Eqs. (21a) and (21b) the following equations of motion:

 Likβkm =ϵnml[Cijklαkn,j+ρ(˙Vlαin−VlVkαin,k)], (38a) Likvk =ϵnmlCijklVlαkn,j, (38b)

where . Using the property of the differentiation of a convolution, the corresponding solutions of Eqs. (38a) and (38b) are given in convolution form

 βim(r,t) +ρG+ij(r−r′,t−t′)˙Vl(t′)αjn(r′,t′))dr′, (39a) vi(r,t) =ϵnml∫t−∞dt′∫CjkpmG+ij,k(r−r′,t−t′)Vl(t′)αpn(r′,t′)dr′, (39b)

where the two-dimensional (distributional) Green tensor is defined as the retarded solution of Eq. (12), with (11).

Substituting Eq. (37) into (39) and (39b) and performing the -integration, we find

 βim(r,t) =ϵnml∫t−∞(G+ij,k(r−s(t′),t−t′)˜Cjkpl(V(t′))bpℓn +ρG+ij(r−s(t′),t−t′)˙Vl(t′)bjℓn)dt′ (40a) vi(r,t) =ϵnml∫t−∞CjkpmG+ij,k(r−s(t′),t−t′)Vl(t′)bpℓndt′. (40b)

Using the distributional approach, the two-dimensional retarded Green tensor is given by (see Eason et al. (1956); Eringen and Suhubi (1975); Kausel (2006) for the Green tensor in the classical approach)

 G+ij(r,t) =θ(t)2πρ{xixjr4[t2(t2−r2/c2L)−1/2++(t2−r2/c2L)1/2+−t2(t2−r2/c2T)−1/2+−(t2−r2/c2T)1/2+] −δijr2[(t2−r2/c2L)1/2+−t2(t2−r2/c2T)−1/2+]}. (41)

It consists of regular distributions of power and . The shear velocity is defined in (31), and is the velocity of the longitudinal elastic waves (P-wave) expressed in terms of the Lamé constants as

 cL=√(2μ+λ)/ρ. (42)

It is noted that , Eq. (30), is twice the spherical part of in (41). Using Eqs. (33) and (34), the derivative of the Green tensor (41) is obtained as

 G+ij,k(r,t) =θ(t)2πρ{(δikxj+δjkxir4−4xixjxkr6)[t2(t2−r2/c2L)−1/2++(t2−r2/c2L)1/2+ −t2(t2−r2/c2T)−1/2+−(t2−r2/c2T)1/2+] +2δijxkr4[(t2−r2/c2L)1/2+−t2(t2−r2/c2T)−1/2+] +xixjxkr4[t2c2LPf(t2−r2/c2L)−3/2+−1c2L(t2−r2/c2L)−1/2+ −t2c2T% Pf(t2−r2/c2T)−3/2++1c2T(t2−r2/c2T)−1/2+] +δijxkr2[1c2L(t2−r2/c2L)−1/2++t2c2TPf(t2−r2/c2T)−3/2+]}, (43)

which involves pseudofunctions of power in addition to distributions of power and . Eqs. (41) and (3.2) display and in expanded form for clarity. However, more compact expressions for these distributions that emphasize the occurrence of solely via well-defined groups containing either or can be found in (Pellegrini and Lazar, 2015) [see also Eq. (A.3) below].

Substituting Eqs. (41) and (3.2) into Eqs. (40) and (40b), we obtain the elastic distortion tensor of the non-uniformly moving Volterra edge dislocation as

 βim(r,t) =12πρϵnml∫t−∞dt′(˜Cjkpl(V(t′))bpℓn{(δikRj(t′)+δjkRi(t′)R4(t′)−4Ri(t′)Rj(t′)Rk(t′)R6(t′)) ×[¯t2(¯t2−R2(t′)/c2L)−1/2++(¯t2−R2(t′)/c2L)1/2+ −¯t2(¯t2−R2(t′)/c2T)−1/2+−(¯t2−R2(t′)/c2T)1/2+] +2δijRk(t′)R4(t′)[(¯t2−R2(t′)/c2L)1/2+−¯t2(¯t2−R2(t′)/c2T)−1/2+] +Ri(t′)Rj(t′)Rk(t′)R4(t′)[¯t2c2LPf(¯t2−R2(t′)/c2L)−3/2+−1c2L(¯t2−R2(t′)/c