The T(3)-Gauge Model, the Einstein-Like Gauge Equation, and Volterra Dislocations with Modified Asymptotics

Annals of Physics 286, 249277 (2000) doi:10.1006aphy.2000.6088, available online at http:www.idealibrary.com on

The T(3)-Gauge Model, the Einstein-Like Gauge Equation, and Volterra Dislocations with Modified Asymptotics C. Malyshev V. A. Steklov Institute of Mathematics at St. Petersburg, Fontanka 27, St. Petersburg 191011, Russia E-mail: malyshevpdmi.ras.ru Received November 16, 1999; revised June 18, 2000

Three-dimensional Lagrangian field theory is investigated as the T(3)-gauge model of static defects in continuous solids. The gauge Lagrangian is proposed in the HilbertEinstein form, and the non-conventional incompatibility law is given by an Einstein-like gauge equation with the elastic stress tensor as ``matter'' source. The stress function method is used, and two solutions to the linearized gauge equation (i.e., two modified stress functions) are obtained which are short-ranged in comparison with the Airy and Prandtl stress functions of classical Volterra dislocations. The modified stress functions obtained enable two ways of influencing the asymptotical behaviour of the edge and screw dislocations. Namely, the classically known rule 1\ is changed either at spatial extension (in favour of more fast decay of the modified stress components), or within the defect's cores (linear stresses of background Volterra defects become smoothed out due to formation of a core region).  2000 Academic Press

1. INTRODUCTION 1.1. Reviewing Remarks Analogies between defects in solids and certain gravitational (especially, threedimensional) models [14], as well as lower-dimensional gravity [5, 6] itself, have been very popular in the last years. For instance, dimension-independent formalism of exterior calculus is used in [1] to propose a geometrized Lagrangian gauge approach for continuously distributed defects. However, the elastic Lagrangian is omitted in [1], and the solution found for continuum line defects is a purely differential-geometric one for a space of constant curvature. The ``elastic'' Lagrangian is used in [2] to study a special structure due to the Poincare translation gauge field on the d=3+1 space-time manifold. The Euclidean version of gravitational solutions for point particles [5] is considered in [3] for static line defects in three dimensions. Specifically, the geometry of defects is governed in [3] by the Euclidean Einstein equation with the singular source given by the 249 0003-491600 35.00 Copyright  2000 by Academic Press All rights of reproduction in any form reserved.

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energymomentum tensor of point particles. The elastic Lagrangian is also, practically, left aside in [3]. Another gauge-invariant approach to continual dislocations is formulated in [7]. Statistical mechanics of lattice models and defects has been extensively studied in [8] (which continues [9]). That is, crystalline solids and associated phase transitions, in which line defects condense and destroy the spatially ordered states, have been investigated in [8]. In particular, considerable attention has been paid in [8] to discrete arrays of line defects. In this respect, distributions supported on lines and surfaces have appeared in [8] as mathematical tools to describe defect densities and their gauge-transformation properties. Correspondence between differential geometry of continuized defects and that of gravity with torsion is also discussed in [8]. Path integration formalism has been developed in [10] to describe propagation of non-relativistic point particles in geometrically nontrivial spaces with curvature and torsion (i.e., in the presence of singular defect densities). Apparent attention has been also paid in the last years to various applications of the ISO(3) gauge model of defects in solids proposed in [11] and, as a more _ elaborated version, [12]. Here ISO(3) denotes the semi-direct product T(3) # SO(3) of translation and rotation groups of d=3 Euclidean space; i.e., ISO(3) is the group of symmetry of a continuous rigid body. The gauge theory of continuum damage [13], the gauge theory of materials exhibiting a relaxation phenomenon [14], and the gauge theory of a ``plastically incompressible'' non-dissipative medium [15] have been considered, as well as the problem of electronic states in a solid with defects [16, 17]. Other developments of the gauge approach [11, 12] can be found in [18, 22]. References [1315, 17] have been mainly concerned with the translational sector of [11, 12] governed by the translational Lagrangian L, . The Lagrangian L, has been chosen in [11, 12] in the special form quadratic in T(3)-gauge field strength. Notice also the SO(3)-gauge approach for the dislocation and disclination continuum elaborated in [19] with the internal energy chosen also quadratic in the defect densities and strains. The applications found look promising for the translational models with quadratic Lagrangians like L, . In its turn, the conventional approach [20] to defects in solids is widely acknowledged. The approach [20] deals with Volterra dislocations and disclinations, as well as with their arrays, and it admits a number of reliable solutions for mechanical problems of solid state physics (one should consult, say, [21] for numerous references). Besides, some of the solutions [20] have also been adapted in [22] (sound propagation and vibrational spectra) and [23] (particle diffusion). Discrete arrays of Volterra defects and discontinuous plastic fields [20] have been intensively used in [8]. For more recent developments in the realm of phase transitions in NambuGoldstone systems with emphasis on the role of singular gauge fields one should refer to [24]. To sum up, rather conventional usage [8, 20, 21] of the Volterra-type singular solutions is actively developed and is progressing together with applications of the less conventional non-linear gauge approach [11, 12].

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1.2. The Subject of the Paper Although a continuous description for defects only approximates the discrete nature of ordering in real solids, it certainly provides us with numerous technical advantages: for instance, variational methods (the first example for gauge dislocations is given in [25]; see also [13, 7, 11, 12, 19]) and exterior calculus combined with the differential geometry tools (after the pioneer papers [26, 27]), etc., can be successfully involved. Translational and rotational gauge fields in defected solids can be governed, formally, by various gauge-invariant Lagrangians. However, a continuous Lagrangian description can be considered as reliable one, provided core energy terms [24] quadratic in the defect densities (i.e., in curvature and torsion) are included into the action functional apart from elastic energies [8]. In this respect, the following problem appears apparently to call attention: whether the continuous gauge Lagrangian formalism admits, say, as a special limit, conventional Volterra defects, like dislocations and disclinations, or their modifications. For instance, single defects and their arrays are considered in [20] at the same footing since it is allowed to replace continuous defect densities by discontinuous (i.e., $-like) ones. It has been claimed [11, 12] (see also [28]) that the use of the quadratic L, allows one to reproduce (``to replicate,'' the terminology of [28]) both the edge and screw dislocations as special limits of certain linear gauge solutions. Moreover, the last statement stimulated [17] to follow [11] to calculate second order corrections to the stress field of the screw dislocation. However, unsatisfactory treatment [11] of the static edge dislocation has been pointed out in [29]. With the aim of avoiding the unpleasant implication concerning the edge defect, a different version of the gauge-translational model is proposed in the present paper. Let us briefly note the ``translational'' situation in [11, 12]. The equilibrium equation of a medium is fulfilled by the stress tensor in the form of the Kroner ansatz [30]. In its turn, the translational gauge equation (the gauge equation, for brevity) specifies stress functions (stress potentials) which parametrize this ansatz. As to the screw defect, the gauge equation results in the Helmholtz equation instead of the standard Poisson equation. The Helmholtz equation provides the modified Bessel function K 0( \) as that solution which reproduces the Prandtl stress function [30]. Thus, the screw dislocation appears in [11, 12] as a special limit. The situation is less satisfactory for the edge dislocation: the gauge equation constraints appropriate stress functions so that the stress _ 33 obtained is equal, in the replication limit, to & &1(_ 11 +_ 22 ) instead of &(_ 11 +_ 22 ) as it must be for the defect along the third axis [20]. Such incorrectness is rather due to the inappropriate gauge equation caused by the specific L, than due to the choice of the Kroner ansatz. It turns out that the required replacement & W & &1 can be achieved by means of another gauge equation with the double curl differential operator [29]. Therefore, improvement of the translational sector of the model can be given. The present paper is to complete [29] and to propose a 3-dimensional Lagrangian gauge-translational model which describes, though modified, static dislocations.

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The geometric background of [11, 12] is revised here, and an equivalent description in terms of the metric tensor and affine connection is proposed. Thus, the underlying geometry is, basically, that of a RiemannCartan manifold with Euclidean signature. The gauge Lagrangian is chosen in the HilbertEinstein form LHE which is equivalent to a specific combination of terms quadratic in torsion (teleparallel geometry). The ``matter'' Lagrangian L! implies potential energy of the elastic body. A new gauge equation is obtained by the Lagrangian method as a 3-dimensional Einstein-like non-linear equation with the elastic stress tensor as ``matter'' source. The stress function approach [30] is accepted below, and the Einstein gauge equation is used instead of the conventional incompatibility law to specify unknown stress functions. Replication of both (i.e., edge and screw) dislocations is investigated below in linear approximation. The main point consists in that the approach presented leads to the ``modified stress functions'' which decay in comparison with the classical ones which unboundly increase at spatial infinity. Therefore, new defects, though keeping a connection with Volterra solutions, demonstrate a modified asymptotical behaviour. A new gauge equation should be appropriate to find out second order corrections to the classical linear solutions. The present paper is concerned only with T(3)-gauging which is treated here as specialization of the framework of ISO(3)-gauging. It was argued in [31] that the gauge approach can be developed for some basic models of classical mechanics so that a distinctive role is played by translational gauging. Various gauge transformations inherent to discrete line defects are discussed also in [8]. Below it is understood that dislocations are rather naturally related to inhomogeneity of the translation group since they are traditionally concerned with non-integrable displacements; i.e., gauging of T(3) is considered here just closely to the transparent ``compensating'' ideology of [11, 12]. We are restricted with statics because even statics of conventional defects demonstrates a group of gauge transformations [32]; before switching on time one should be convinced that the statics itself of the model proposed is self-contained. The paper is written in seven sections. Section 1 is introductory, Section 2 is concerned with difficulties of the replication of the static edge dislocation in [11], and a modified linear model (i.e., another translational gauge equation, in fact) is proposed in Section 3. Section 4 is devoted to the Lagrangian derivation of the corresponding non-linear model, and it suggests the gauge Lagrangian in the HilbertEinstein form. Section 5 is devoted to solutions found for the (modified) screw and edge dislocations, and Section 6 provides another interpretation for the solutions found in Section 5. Discussion in Section 7 concludes the paper. The consideration is time independent; Greek indices run from 0 to 3, Latin ones from 1 to 3, and repeated indices mean summation. In addition, Latin letters a, b, c, ... denote curvilinear indices of a deformed configuration, and i, j, k, ... denote the Cartesian frame of the initial one. Bold-face letters denote second rank tensors (matrices). Certain details and comments on the motives of the ISO(3)-gauging for defects in solids can be found in [11, 12].

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2. THE EDGE DISLOCATION FROM THE QUADRATIC LAGRANGIAN Let us briefly note the basic elements of the theory of the translational gauge field , a : assuming that the gauge-rotational degrees of freedom of the whole ISO(3)-model [11] are ``frozen.'' The gauge field , a : plays a compensating role as the Abelian non-compact group T(3)rR 3 acts by inhomogeneous shifts on the current configuration variable ! a, and the corresponding compensated derivative has the form B a : # : ! a +, a : , where the  : are partial derivatives x : , and x 0 implies time. The gauge transformation rules for ! a and , a : have the form ! a Ä ! a +' a, , a : Ä , a : & : ' a, while B a : is thus translationally gauge invariant. The current configuration ! a (x , x 0 ) can be written in the form ! a =$ ai x i +u a (x , x 0 ) with respect to a Cartesian coordinate system so that x implies an initial undeformed configuration, and u is displacement field. Appearance of B a : instead of the pure gradient  : ! a implies that ! a itself ceases to be an everywhere adequate variable to label points. As far as replication of static dislocations is the focus of the present paper (see [20] for the standard results), it will be assumed that  0 #0, , a 0 #0, so that the model is governed by the equilibrium equation,  i _ a i =0,

(1.1)

and by the translational gauge equation,  j ( j, a i & i, a j )= j  j, a i & i ( j , a j )=(2s) &1 _ a i ,

(1.2)

where _ a i =B aj _ ij is the stress field, _ ij is given by _ ij =*$ ijE k k +2+E ij,

(2)

* and + are the Lame constants, and E ij is the strain field 2E ij =B k i B kj &$ ij (see Subsection 4.1 for the usage of up and down indices). It is seen that (1.1) ensures integrability of (1.2). The gauge equation (1.2) is due to the following choice of the spatial part of the quadratic gauge-translational Lagrangian, L, =(&2s)  [i , c j]  [i, c j],

(3)

where square brackets imply antisymmetrization, and the parameter s is the ``coupling'' constant. It has been claimed in [28] that the gauge equations (1.2) ``are what replace the compatibility conditions of linear elasticity theory.'' In other words, (1.2) is proposed to play in the gauge approach the role equivalent to that of the traditional incompatibility law.

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Let us recall the approach [11, 12] to the linear dislocation problem. A purely integrable contribution to B bi is assumed zero, and since B bi is thus $ bi +, bi , (2) can be linearized in , bi as _ ij =2+ , (ij ) +*$ ij , kk ,

(4)

where , (ij ) =(12)(, ij +, ji ) (in the linear case we do not distinguish up and down indices). Further, conditions , 12 =, 21 , , 13 =, 31 =0, , 23 =, 32 =0 are imposed, and  3 #0 is supposed to adjust the axial orientation. It can be realized that the choice of , ij in the symmetric form , ij =, (ij ) should provide us with the appropriate solution of (1.1) and (1.2) as a linear elastic strain tensor, while (4) gets the status of the constitutive relation of an isotropic body (the Hooke law). The way to handle (1.1) and (1.2) is as follows: the ansatz postulated for the stress field of the edge dislocation along the third axis Ox 3 ,

\

 222 f

+ &1 _= & 212 f 0

& 212 f 0  211 f 0

+

0 , p

(5)

fulfills (1.1), while (1.2), expressed by means of (4) and (5), specifies the parametrizing functions f = f (x 1 , x 2 ) and p= p(x 1 , x 2 ). Specifically, (1.2) results in (1&a) 2f &ap=} 2f, 2

(1&a) 2p&a 22f=} p,

(6) (7)

where a=*(3*+2+) &1, } 2 #+s (s is positive [11]), and 2 denotes the 2-dimensional Laplacian  211 + 222 . Let us exclude p from (7) with the help of (6). We assume here the limit } Ä 0 proposed in [11] to obtain conventional solutions. In this case, 22f =0 arises to define the limiting form of f, while (6) provides p=

1 1&a 2f# 2f a &

(8)

as the limiting form of p. The fourth order differential equation, which arises to govern f at arbitrary }, looks troublesome for analytical solution. Thus, with regard to 22f =0, it has been guessed in [11] that f should exist at arbitrary } in such form that the Airy function asymptotics [30] are valid for it at } Ä 0. But since 22f =0 is to indicate at the correct limit for f, the same reasons (e.g., linearity of the substitution of p) imply that (8) defines the limiting form of p. However, (8) contradicts the plane problem condition as follows. Indeed, the strain (33)-component , 33 should be expected to become zero, at least as a limit of an ``extended'' solution. Such vanishing of , 33 would just express the famous plane problem requirement which is necessary to discuss the classical

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THE T(3)-GAUGE MODEL

edge dislocation. Therefore, the constraint & 2f =p has to be expected at } Ä 0, since _ 33 =&(_ 11 +_ 22 ), where &=*2(*++) is the Poisson ratio, implies vanishing of the strain (33)-component. The relation (5) with p=& 2f but, formally, with opposite sign at f, is nothing but the standard ansatz of the theory of dislocations [30]. Airy's stress function f has been found in [30] as the biharmonic potential  2( \ 2 log \) (or  1( \ 2 log \)), \ 2 =x 21 +x 22 , and it enables us to obtain from (5) all the stress tensor components of the edge defect. Thus, in spite of the satisfactory equation for f, Eqs. (6) and (8) are telling us that the correct _ 33 of the edge dislocation would not appear at }{0 nor at } Ä 0. Notice, it has been recognized in [11] that only for }{0 does the new _ 33 differ from the standard one. To summarize, the joint use of (1.2) and (5) implies that it is impossible to get f as the biharmonic potential and to fulfill, at the same time, p=& 2f. One gets either only _ 33 is incorrect due to (8), or f is not the Airy function at all [29]. Let us calculate , using (4) inverted and _ (5) with f =& f A and p (8), where fA implies the Airy function fA =

b 1  2(\ 2 log \) 4? 1&&

(9)

corresponding to the edge dislocation along Ox 3 with the Burgers vector in the x 1 -direction [20]. In this case it can be argued [29] that the two-dimensional displacement field obtained from , is just missing the famous logarithmic contribution responsible for the ``closure failure'' of the edge defect. Hence, in spite of the statement in [11, 12], it is plausible to conclude that the edge dislocation is missing when (1.2) and (5) are used together. Another gauge equation will be proposed in the next section which enables us to exchange a and 1&a in (8).

3. THE DOUBLE CURL GAUGE EQUATION AND ITS SOLUTIONS The equilibrium equation (1.1) can be satisfied identically provided the stress field _ is chosen as a double curl of a twice differentiable symmetric tensor potential / which is called [30] the tensor field of second order stress functions: _ ij =(inc /) ij #&= ikl= jmn 2km / ln .

(10)

Both particular ansatz proposed in [11, 12] to obtain the edge and screw dislocations appear as specializations of the general representation (10) (though (10) itself is not emphasized in [11, 12]). The stress function method based on (10) has been developed in [33, 34] to approach non-linear dislocation problems. Equation (10) has been also discussed in [8, 30, 35] as an implication of a torsion-free stress space. Therefore (5), as the specification of (10) to the plane problem, looks more fundamental than (1.2) with regard to the unsatisfactory p (8). If so, another

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candidate should be attempted instead of the master Eq. (1.2) for the translational field. It turns out that the most direct way to get p consistent with _ 33 =&(_ 11 +_ 22 ) at } Ä 0 is to choose the L.H.S. of (1.2) in another form, namely, (inc , Sym ) ij #&= ikl= jmn 2km , (ln) =(2s) &1 _ ij ,

(11)

where superscript Sym implies the tensor symmetrized, = ikl is the permutation symbol, and _ ij is given by (4). When the R.H.S. of (11) is zero, this looks like the compatibility law of linearized elasticity provided , ij is viewed as distortion. Equation (11) replaces traditional incompatibility law in the sense that its R.H.S. is non-trivial while the L.H.S. looks habitual. In the next section Eq. (11) will be obtained as an Einstein-type equation. Let us use (11) instead of (1.2). However, it is more appropriate to rewrite it through unknown second order stress functions. To this end, one should express , Sym in the L.H.S. of (11) using both the Hooke law (4) inverted and (10). The stress _ (10) itself takes its place in the R.H.S. of (11). Eventally, the fourth order differential equation appears, 2 (3) 2 (3)/ ij +a D ij 2 (3)/+((1&a)  2ij +a$ ij 2 (3) )  2kl / kl &2 (3)( 2ik / jk + 2jk / ik ) =} 2(2 (3)/ ij +D ij /+$ ij  2kl / kl & 2ik / jk & 2jk / ik ),

(12)

where / implies the trace of /, a and } are defined in (6), (7), and we have denoted the differential operators 2 (3) =$ ij  2ij and D ij = 2ij &$ ij 2 (3). In its turn, (12) can be considerably simplified if we replace / by another symmetric potential /$ as

\

/ ij =2+ /$ij +

& $ /$ , 1&& ij

+

where /$ fulfills  i /$ij =0. Thus we get

\

2 (3) 2 (3)/$ij =} 2 2 (3)/$ij +

1 D ij /$ . 1&&

+

To clarify the suggestive character of (11), let us assume that (1.2) is also written in terms of , =, Sym. It can be further re-expressed through /, and the equation thus resulting appears to be completely similiar to (12) except for only one thing: a and 1&a become exchanged. In other words, the Poisson ratio &=a(1&a) and its inverse & &1 become exchanged. Therefore it can be guessed that the problem related to p (8) should disappear when (11) is used instead of (1.2). In order to specialize (12) to the edge dislocation, we introduce two independent functions p and f as f#/ 33 ,

p#& 222 / 11 & 211 / 22 +2 212 / 12 ,

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THE T(3)-GAUGE MODEL

while the other / ij are zero. The plane problem is adjusted by  3 #0, and thus 2 (3) becomes 2 (see Section 2 for 2). It can be verified that now (10) generates (5), and six equations (12) are reduced [29] to a 2f+(1&a) p=} 2f,

(13)

2

(1&a) 22f+a 2p=} p.

(14)

It is seen that changing f to &f and a W 1&a one obtains (6), (7) from (13), (14), accordingly. Therefore Eq. (11) provides the remarkable opportunity to convert the embarrassing ratio (1&a)a in (8) into a(1&a), and p (13) gets around the obstacle discussed in the previous section. However, Eq. (13) indicates that deviation is expected at }{0 from the condition specifying the plane problem because , 33 acquires the value (} 22) f by means of (4) and (5). This is close to the statement of [11, 12] that the limit } 2f Ä 0 at } Ä 0 is necessary to reproduce the standard solutions. We shall turn to this again in Section 5. Notice that choosing f #& 1 / 23 + 2 / 31 , one can deduce from (12) a single equation for the screw dislocation, 2f=} 2f,

(15)

which is the same as in [11, 12]. It is clear that this coincidence is because (15) does not contain the Poisson ratio. Solutions to (13)(15) (and, generally, to (12)) should be called ``modified stress functions'' to distinguish them from the classical (harmonic, biharmonic) potentials. In what follows, we shall be concerned with the solution to (15) considered in [11, 12]: f s( \)=(b2?) K 0(}\). As (13) defines p, the second equation governing f is in the form (2&M 2 )(2+N 2 ) f=0,

N 2 #M 2

1 , 1&2a

(16)

(1&2a is positive at positive * and +). For a correspondence with [11], we use M 2 instead of } 2 when (16) and its solution are considered. Equation (16) simply differs from the corresponding Eq. (4.6.27) in [11]: only 2a&1 and 1&2a are interchanged under a W 1&a. Equation (16) has the Bessel and Neumann functions, J 0(N\) and Y 0(N\), and the modified Bessel functions, I 0(M\) and K 0(M\), as four angle-independent (\# |x | ) basic solutions [36]. As  2 commutes with 2, the modified stress function for the edge dislocation, which solves (16), can be written as [29] f e(x 1 , x 2 )=G  2 F( \) , N ? F( \)=log J 0(N\)& Y 0(N\)&K 0(M\), M 2

b 1 G# . 2? M 2

(17)

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C. MALYSHEV

Since the plane problem requires M Ä 0, it is appropriate to estimate F (17) at M\< <1, F(\)&

(M\) 2 # M\ &1 , log 2(1&&) 2

\ \

+ +

(18)

where # is the Euler constant [36]. It is seen from (18) that f e (17) results in f A (9) at M\< <1. Eventually, we are expecting all components of _ to be those of the edge dislocation because p=&2(& f ) now. In the opposite case, M\> >1, the radial part of f e (17) is expressed as a linear combination of (N\) &12 sin(N\) and (N\) &12 cos(N\). Recall that the solution f s is fast and monotonic decreasing as (}\) &12 e &}\. Asymptotically decreased potentials f s and f e should be referred to as ``shortranged'' ( f s is truly short-ranged) since they replace the Prandtl (tlog \) and Airy (t\ log \) stress functions unlimited at infinite \. Let us also recall [37], where a ``mass'' of defects is discussed as an implication of the translational model [11] (specifically, as implication of the corresponding time-dependent gauge equation). This is because the KleinGordon equation with the mass parameter } 2 is possible for the translational gauge field in the Lorentzlike gauge. It is seen from above that the structure of the gauge equation (12) leads to the analogous } 2-terms in (15), (16) also. In turn, it is why the solutions f s and <1 and the decreased fe are possible with the prescribed asymptotics at }\< character at }\> >1. Thus it is suggestive (and preferable) to relate the ``mass'' effect, i.e., the existence of such scale } &1 that the modified stress potentials decay at \> >} &1, to the gauge equation (12) also.

4. THE EINSTEIN-LIKE GAUGE EQUATION AND ITS DERIVATION 4.1. Geometric Preliminaries Before proceeding with the Lagrangian derivation of (11), let us briefly present the geometrical apparatus which underlies our consideration. The idea of using non-Euclidean geometric tools to discuss crystallographic defects takes its origin in the pioneer works by Kondo [26] and Bilby et al. [27] (see [30, 38] for historical remarks). Especially, the basic identification of the dislocation density as the Cartan torsion has to be mentioned. For compactness, it will be assumed that [8] should be inspected for basics of geometry of the RiemannCartan spaces in the form accommodated to describe defects. Other useful references can also be found in [39]. Motives of the gauge approach to crystallographic defects, the definition of a continuized Bravais crystal, and speculations on the origin of affine connection can be obtained up from [38, 40]. As to gauging of the group ISO(3) which is important for our work, a formally close gauging of the Poincare group (which also is a semi-direct product of translation and pseudo-orthogonal rotation

THE T(3)-GAUGE MODEL

259

groups though of 4-dimensional Minkowskian space-time) has already been extensively developed in the realm of gravitational physics [41, 42]. Although we are restricted to T(3)-gauging, it is more appropriate to admit temporarily a more general framework of the ISO(3)-gauging. This means that the principal bundle of affine frames [42] is assumed as a geometrical background of the model in question. Here the couple of the Cartan structure equations, R a b, ij = i | a b, j & j | a b, i +| a c, i | c b, j &| a c, j | c b, i , T a , ij = i B a j & j B a i +| a c, i B c j &| a c, j B c i ,

(19)

appears as a basic differentialgeometric relation. These equations relate coefficients of differential forms which define the geometry of so-called RiemannCartan spaces: antisymmetric in i, j coefficients R a b, ij and T a , ij determine curvature and torsion 2-forms, accordingly R a b and T a. The coefficients | a b, i and B a i define connection and co-frame 1-forms, | a b and B a, respectively. Indices a, b demonstrate R a b , | a b as elements of the Lie algebra of the group SO(3), while T a, B a belong to the Lie algebra of the group T(3). All fields are considered locally in open domains of a 3-dimensional manifold with Euclidean signature. It is known [42, 43] that when gauging ISO(3), one immediately is concerned with iso(3)-valued connection and curvature differential forms which are split into ``linear'' and ``translational'' parts owing to the semi-direct sum structure of iso(3) (iso(3) is the Lie algebra of ISO(3)). However, the translational curvature is transformed non-covariantly under the ISO(3) gauge transformation. In order to extract an appropriate quantity which is transformed gauge-covariantly, it is proposed to use the auxiliary field ! a (the section of an associated vector bundle) [4245]. As a result, the Cartan structure equations in the ISO(3) gauging are still given by (19) though the B a i have the form B a i =, a i + i ! a +| a b, i ! b.

(20)

The coefficients | a b, i and , a i in (20) are just the linear and translational parts of an iso(3)-valued connection 1-form. Apart from B a i , it is also appropriate to define their reciprocals (i.e., triads or frames) B b j as follows: B a i B b i =$ ab , B a i B a j =$ ij . It is crucial that the auxiliary field ! a (20) has been identified in [11, 12] as a deformed configuration variable (see [43] for other interpretations of ! a ) as !: x i Ä ! a (x i ),

! &1: ! a Ä x i (! a ).

This means that the group indices a, b, c get ``material'' meaning since the triples [! a ] label points in deformed configuration. The coordinate indices i, j, k get the status of Cartesian ones in the initial (undeformed) configuration. In particular, B a i and B a i are useful to pass from one set of indices to another. The conventional theory of defects [20] can be naturally considered [32] as the Abelian gauge model with the additive gauge group iso(3)rR 6 [46]. In this case the motor [47] of disclination and dislocation loop densities plays the role of

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C. MALYSHEV

a 6-component Abelian gauge potential, and the motor of the defect densities is that of the Abelian gauge field strength [48]. The Cartan equations (19) have been reduced [48] to the conventional relationship between the motor of the defect densities and the motor of the defect loop densities. The field ! a enabled this truncation so that Schaefer's exterior differentiations inherent to [20] naturally appeared. Regarding the material interpretation of triples [x i ] and [! a ], let us define now the Green deformation tensor g ij , g ij =' ab B a i B b j ,

(21)

and the corresponding Lagrangian strain tensor E ij , 2E ij = g ij &' ab $ ai $ bj ,

(22)

where B a i is given by (20), and ' ab is flat metric since the length element in a deformed configuration is ds 2 =' ab d! a d! b [4951]. The metric g ij and its inverse can be used for raising and lowering the indices i, j, while ' ab is for a, b, c. Now we are in a position to discuss about covariant differentiation which enables us to compare arbitrary, say, vector fields in infinitesimally neighbouring points. Covariant differentiation requires the corresponding affine connection coefficients, and we shall denote them either as 1 kij (differentiation with respect to local coordinate basis), or as | a b, i . Both representations must be adjusted to each other in the sense that the covariant derivative of the field B a k (indices of both types) vanishes,  i B a j &1 kij B a k +| a b, i B b j =0.

(23)

The metric g ij (21) is covariantly constant due to (23), i.e.,  i g jk &1 lij g lk &1 lik g jl =0.

(24)

In its turn, (24) can be solved for 1 kij as 1 lik g lj =1 ik, j =[ik, j]+K ikj , where the Christoffel symbol of first kind and contortion are expressed, respectively, as: [ik, j]= 12 ( i g kj + k g ij & j g ik ),

K ikj = 12 (T ik, j +T ji, k &T kj, i ) ,

with T ij, k #1 ij, k &1 ji, k . Using (23) to express | a b, i , | a b, i =Bb l(1 kil B a k & i B a l ),

261

THE T(3)-GAUGE MODEL

we obtain for the RiemannCartan curvature R a b, ij (19): [ ]

[ ]

R a b, ij =Bb m B a k R kmij +Bb m B a k ( {i K jm k & {j K im k +K in kK jm n &K jn kK im n ) , (25) where R k mij = i

k k k & j + jm im in

n k & jm jn

n im

{ = { = { ={ = { ={ =

k is the Riemann(-Christoffel) part of the curvature, [ jm ] = g kl [ jm, l ] is the [ ]

Christoffel symbol of second kind, and {i means covariant differentiation with respect to the torsion-less part of 1 kjm . Since we are interested only in the gauge-translational defects, the situation of primary interest for us is given by | a b, i #0, i.e., by R a b, ij #0, while the torsion T a , ij = i B a j & j B a i remains nontrivial. Although R a b, ij vanishes, the Riemann Christoffel and contortion parts of (25) can differ from zero cancelling each other. This is just the framework of teleparallel geometry [43] which is the most suitable for dislocations [30]. As the last step, we define the scalar curvature R which is an important geometric invariant as follows: R=R i i , where R ij =R k ijk is the Ricci tensor. More generally, we can obtain from (25) the following remarkable identity [3], 1 2 1 R+ T ijk T ijk & T ijk T kij &T i T i &  m(BT m )=R, 4 2 B

(26)

where R is also the scalar curvature but obtained by contraction of the Cartan curvature tensor (more about (26) and numerous related references can be found in [43]). Hence, when | a b, i is zero, the scalar R is completely equivalent (up to the surface term) to the specific combination of squared torsion components. The scalar R and Eq. (26) are just what we need to discuss derivation of (11) in the Lagrangian approach. The variable B a i (20) is translationally gauge invariant [11, 12], and the same is true for the model build up with its various combinations, say, like (21), (22). 4.2. Lagrangian Derivation This section is devoted to Lagrangian derivation of the gauge equation which admits (11) as a weak field limit. The present paper is concerned with gauging for static arrays of defects since the dynamical problem for them is rather complicated. It is doubtful that it can be solved in the way [1] close to general relativity where space and time constitute a (3+1)-dimensional continuum. However, the problematic character of the (3+1)-metric is recognized in [1] for as far as dynamics of

262

C. MALYSHEV

defects. Thus, we consider the 3-dimensional field theory governed by the action principle with the Euclidean action S= d 3xL. Six compatibility equations of defectless elasticity are the vanishing conditions of six independent components of the RiemannChristoffel tensor R i jkl . Instead, the conventional approach to dislocations implies vanishing of R a b, ij , while the torsion T a , ij (local dislocation density [26, 27, 30]) is postulated. In the presence of dislocations the RiemannChristoffel curvature becomes nontrivial, expressing that the compatibility law is broken by the corresponding incompatibility caused by T a , ij . However, another opportunity will be exploited here. Namely, in our approach, which is to describe (modified) translational defects, the RiemannChristoffel curvature will be given by the corresponding Einstein-like gauge equation with the elastic stress tensor in its R.H.S. Indeed, due to 3-dimensionality, the second rank Einstein tensor G ij can be used instead of R i jkl as follows: G ij =(14) e ikle jmn R klmn , where e ijk =- g = ijk, and g is det( g ij ). Linearizing G ij in deviations of g ij from a flat metric one gets a double curl expression like the L.H.S. of (11). That is, (11) can be viewed as a weak field approximation of an Einstein-type non-linear equation since its L.H.S. looks like linearization of G ij, while the stress _ ij in its R.H.S. can acquire higher powers of E ij as a non-linear constitutive relation ``stressstrain.'' More generally, (11) is reminiscent of a gravitational equation which relates the Einstein tensor to a matter energymomentum tensor. Thus, both sides of (11) can be extended, and the resulting equation admits a standard Lagrangian derivation. In three dimensions the most general geometric Lagrangian, which provides gauge equations no more than of second order, must contain six invariants quadratic in components of curvature and torsion, the scalar curvature term, and ``cosmological'' term [3]. In the teleparallel situation this general Lagrangian yields the HilbertEinstein one, 1 LHE =s R, B

(27)

where B#det(B a i ), and R is the RiemannChristoffel scalar curvature which depends only on B a i (the ``cosmological'' term is set equal to zero). According to (26), LHE (27) is equivalent to the Lagrangian, say L& , which is purely quadratic in torsion. Notice that only a part of such a quadratic L& was used in [11] as L, . Gauge theories of lattice stresses and defects have been considered in [5254] as a base to describe statistical mechanics of defect fluctuations and melting transitions in solids. It has been suggested in [53, 54] that continuous approximation for the lattice models is reliable provided extra core energies of defects are included into the total free energy. In other words, which are quadratic in the defect densities should be present to generate a complete non-linear gauge description for continuized defects. Due to its obviously special status in gauge-geometric models, LHE (27) itself fairly deserves consideration as the gauge-translational Lagrangian. Fortunately, with regard to (26), usage of LHE correlates seemingly with the

THE T(3)-GAUGE MODEL

263

proposal by [53, 54] (see also [8]) concerning quadratic terms. Variation of LHE reads (up to terms irrelevant by the Stokes theorem) R 1 $LHE i B i #2sG a i , a =2s R a & B $B i 2 a

\

+

(28)

where G a i is the Einstein tensor. Since the field ! a is an important constituent of more general ISO(3)-formalism, it also should be governed by an appropriate Lagrangian L! . If so, L! becomes responsible for the source in the gauge equation. Our consideration is static, and therefore we choose (&1B) L! in the form of potential energy W of the isotropic non-linear elastic continuum. Variation of L! acquires the form 1 $L! =&7 a i #&_ a i &B a iW. B $B a i

(29)

By definition, the stress field _ ij =_ a iB aj is given as $W$E ij =_ ij, and the second term in 7 a i is due to variation of B. Putting together (28) and (29), we obtain the Einstein-like gauge equation G a i =(2s) &1 7 a i ,

(30)

which generalizes (11). Rejecting in (30) higher terms we obtain (11); s tending to infinity we just recover the general compatibility law, G ij =0. Another equation, to govern ! a, can be readily obtained because of ``covariant conservation'' of the Einstein tensor by the Bianchi identity [8]. For the first time in the context of gauge dislocations, an equation similiar to (30) has been obtained by the variational approach in [25], but without any further elaboration. Let us specialize the potential energy W as [55] &1 4 * W= I 21(E)++I 1(E 2 )+ I 31(E)+& 2 I 1(E) I 1(E 2 )+ & 3 I 1(E 3 ), 2 6 3

(31)

where E implies the gauge invariant strain (22), and the invariant function I 1( . . . ) implies the trace of tensor argument, while * and + are the Lame constants of second order, and & 1, 2, 3 of third orders. The stress tensor acquires the form

_

_ ij =$ ij *I 1(E)+

&1 2 I (E)+& 2 I 1(E 2 ) +2E ij [ ++& 2 I 1(E)]+4& 3 E ikE kj , 2 1

&

where I 1(E)=E k k and I 1(E 2 )=E kl E kl. When & 1, 2, 3 are zero, we obtain _ (2) which coincides with _ (4) in the weak field limit. The problem of second order corrections has been actively investigated for dislocations by the stress function method [33, 34] and by the displacement function method [56, 57]. As to other ways to handle non-linear dislocations, one should

264

C. MALYSHEV

refer to [50, 51, 58]. The paper [59] devoted to a wedge disclination provides an example of further development in the field of second order corrections to defect solutions. The common practice is to take W in the form (31) when contributions beyond linear elasticity are required.

5. (MODIFIED) SCREW AND EDGE DISLOCATIONS The gauge equation (30) is assumed to play an important role in the translational gauge model of defects proposed in the present paper. However, specific solutions to its linearization (11) should be investigated first in order to understand the most essential implications of the approach developed. Therefore, the present section is concerned with the solutions to (13), (14) (edge dislocation), and (15) (screw dislocation). 5.1. Screw Dislocation First, it is appropriate to consider (15) and its solution f s #(b2?) K 0(}\), which describe the modified screw dislocation. Here the co-frame matrix arises as B=$+,, where

,=

1 2

\

0 0  2 f &2| 1 0 0 & 1 f +2| 2 0  2 f +2| 1 & 1 f &2| 2

+

(32)

(the Kroner ansatz and the Hooke law are used). In (32), f denotes f s , and two auxiliary functions | 1 and | 2 are used to express the antisymmetric part of ,. It should be stressed that f s and f e are, in fact, the fundamental solutions to the corresponding gauge equations; i.e., (15) and (16) acquire $-functions in their R.H.S. provided f s and f e are differentiated as distributions [60]. Notice that only the symmetric part of ,, i.e., , Sym can be found by means of (32) provided only f s is known. However, the special character of f s , as well as the choice of orientation of the defect, allows us to restore B=$+, completely so that the corresponding torsion and Riemann curvature parts are mutually cancelled accordingly to (25) (teleparallelism). For the screw dislocation along Ox 3 , we choose | 1 , | 2 so that , 1 3 and , 2 3 (32) are zero. Thus, , 3 1 =(&b2?) } sin ,K 1(}\),

, 3 2 =(b2?) } cos ,K 1(}\),

where K 1 is the modified Bessel function [36], and the corresponding torsion component appears, as distribution, in the form T 3 , 12 =&2f =b $(x 1 ) $(x 2 )&

b 2 } K 0(}\). 2?

(33)

265

THE T(3)-GAUGE MODEL

Solutions to (13)(15), as well as other relations that result, depend on the radial variable through the product }\, and thus it is suggestive to introduce the characteristic length 1}. Appearance of the (dislocation) length scale 1} is also emphasized in [12]: the parameter } 2 =+s appears in the R.H.S. of the gauge equation, and it plays the role of relative measure of the contributions from LHE (or L, ) and L! into the total potential energy. Here the Lame constant + characterizes the order of magnitude of the elastic potential energy, while s characterizes the static gauge Lagrangian. It is appropriate to point out two important limits: }\< <1 and }\> >1. In this respect, let us consider two concentric cylindric domains, (I) and (II). One of them, (I), is a hollow cylinder restricted by two surfaces: inner, at some <1 for any \ between \ C and \ I . \=\ C >0, and outer, at \=\ I >\ C , so that }\< When } is much smaller than unity, i.e., }< <1, and can approach zero as } Ä 0, the bound radius \ I is allowed to increase, roughly speaking, so that the transverse cross-section of (I) coincides asymptotically with the infinite plane containing a hole. It will be seen that within (I) the gauge solutions replicate the conventional ones. Another domain, (II), constitutes exteriority of the cylinder bounded at \=\ E > >\ I , so that }\> >1 for any \\ E . When } varies from zero to a finite value, asymptotical behaviour of the gauge solutions varies also because }\> >1 comes to play for sufficiently large \ and finite }{0. Thus, the gauge solutions based on the Lagrangian including both elastic and gauge parts imply confinement of conventional defects inside a cylindric body of finite transverse extension. The stress function f s can be expressed in the series form (see the Appendix). Within (I), it approximately becomes the Prandtl stress function [30], f s & f P # &(b2?) log \+const, and , 3 1 and , 3 2 acquire their conventional form [20] due to K 1(}\) & 1}\. In lowest order we estimate the defect density, T 3 , 12 &

# b 2 } log }\ , 2? 2

\ +

\ # (I ),

(34)

where # is Euler's constant [36], and it tends to zero at } Ä 0. Let us consider a circle C of radius R (or, generally, any homotopically equivalent closed loop) which encloses the region \\ C and belongs entirely either to (I) or (II). Integration of the 1-form , 3 #, 3 1 dx 1 +, 3 2 dx 2 along C yields



C

B 3=



, 3 =b}RK 1(}R) C

(}R) 2 1 # &log }R 2 2 2

_ \ ? &b \2 }R+ e

& b 1&

\ ++& & b,

R # (I ),

12

&}R

< <1,

R # (II ).

(35)

266

C. MALYSHEV

For any circuit C # (I), the integral (35) behaves typically for conventional screw dislocation since it acquires, in leading approximation, a constant value. The equivalent circuit situated inside (II) does not allow us to indicate the defect inside the loop since the integral vanishes rapidly. Eventually, let us turn to the stress field _. All the stress components are zero now except _ ,z =(b+2?) }K 1(}\). It is known that the circular boundary of the cylindric body (containing defect) is stress-free provided the surface traction is zero, i.e., T k =_ kl n l =0 on the boundary, where n \ =1, n , =n z =0 [51]. In other words, the stresses _ \\ , _ \, , _ \z , are zero on the bound surface. Clearly, for the modified screw dislocation any co-axial circular boundary is stress-free. The standard result of the screw dislocation is valid for _ ,z inside (I) (see the Appendix), whereas inside (II) we estimate [36] _ ,z &

b+

}

2 - 2? \ \ +

12

1 e &}\ < < . \

(36)

Stress components of dislocations in infinite bulk behave as 1\ at infinity. Estimation (36) demonstrates that the modified solution is short-ranged in comparison with the conventional one in the sense that _ ,z penetrates weakly outside \ & 1}. Formally, certain points of this section resemble the result of [11, 12]. First, however, this is only for the linear case. As can be seen comparing (1.2) and (30), the whole situation is different: e.g., second order effects from (30) could be different. Besides, according to [11], the integral  C B 3 is less twice than the result (35) (e.g., it is b2 instead of b at R # (I)) owing to neglection of the antisymmetric part of ,. Such a loss of correct normalization for  C B 3 witnesses that the classical limit [11] does not look perfect even for the screw dislocation. 5.2. Edge Dislocation Let us now turn to the edge dislocation described by _ (5) and f e (17). Fewer explanations will be given below since the gauge solution will be investigated along the lines of the previous section. Now the co-frame matrix B includes

, = 12

\

& 222 f +a(2f &p)  212 f +2| 0

 212 f &2| 0 2 & 11 f +a(2f &p) 0 , 0 a 2f +(1&a) p

+

(37)

while the torsion components are T 1 , 12 = 1 |+

1&& &M 2  2 2f+  2 f, 2 2

1&& &M 2 T 2 , 12 = 2 |&  1 2f&  1 f, 2 2

(38)

267

THE T(3)-GAUGE MODEL

where | accounts for the antisymmetric part of , (37), f implies f e , and p is given by (13). Let us calculate, say T 1 , 12 at arbitrary M assuming, to determine |, that T 2 , 12 =0. Thus we obtain: T 1 , 12 =b $(x 1 ) $(x 2 )+

b (N 2 F( \)+(N 2 &M 2 ) K 0(M\)). 4?

The function F is adjusted to reproduce within (I) the Airy function (9), and so it is straightforward to obtain the estimate: T 1 , 12 & &

# b & M 2 log M\ , 2? 1&& 2

\

+

\ # (I ),

which is similiar to T 3 , 12 (34). Just as in Subsection 5.1, let us integrate 1-forms , 1 and , 2 along the circuit C of radius R. We obtain



B 1=

C



,1=

C

_

& b 1&

&

b N ? J 1(NR)& Y 1(NR) +MRK 1(MR) NR log 2 M 2

_ \

+

& 1 # (MR) 2 log MR & 2(1&&) 2 2

\ \

b (NR) 12 8 +(R), (2?) 12

+ +& & b,

&

R # (I ),

R # (II ),

(39)

while  C B 2 =0. Here and below the following two linear combinations of trigonometric functions are defined for convenience:

8+

N

?

?

?

\M + cos \N\& 4+ & 2 sin \N\& 4+ , N ? ? ? (\)=log \M + sin \N\& 4+ + 2 cos \N\& 4+ .

8 &(\)=log

These combinations arise when F( \) and F$(\) are considered for large arguments [36]. Again, we approximately obtain from (39) the known result,  C B 1 & b, for any circuit C # (I). Circuit integrals (35) and (39) estimated at long distances look similar to each other except for the oscillating factor 8 + for (39) instead of the decreased exponent for (35). This difference reflects the different asymptotical behaviour of the potentials f s and f e . Notice that &N 2 cannot be obtained instead of +N 2 in (16) because 1&2a<0 requires *, + to be of opposite signs. This is unacceptable from an elastic-theoretical point of view. Thus, it is just the structure of (16) which obviously forbids to combine the stress function we are interested in

268

C. MALYSHEV

using, say K 0(M\) and K 0(N\) (both exponentially decreased), since K 0(N\) is not the solution. Instead, J 0(N\) and Y 0(N\) appear as particular solutions to (16), and their long-distance behaviour dominates so that f e decreases less fast and oscillates, while K 0(M\) is negligible within (II). The integral (39) indicates that we are still concerned, even asymptotically, with a (complicated) defect which distorts the (x 1 , x 2 )-plane since the closure failure remains nonzero and behaves complicatedly in (II). Let us obtain the stress field corresponding to f e (17). It is convenient to do this in the cylindrical coordinates (Oz#Ox 3 ) as _ \\ =

b+ sin ,P( \), 2?

_ \, =&

b+ cos ,P( \), 2?

_ ,, +_ \\ =

b+ sin ,Q$( \), 2?

(40)

_ zz =&(_ \\ +_ ,, )++(1+&) M 2f e , while _ \z =_ ,z =0. Here we have denoted two auxiliary functions, P( \)#

Q( \) 2 F $(\)+ , (M\) 2 \

Q(\)#

N2 (F( \)+K 0(M\))+K 0(M\). M2

Equations (1.1) are fulfilled by (40). Up to a factor, Eqs. (40) also represent the appropriate components of the Einstein tensor. Within (I) we estimate Q$ and P, # & 1 1& (M\) 2 log M\ &1 (1&&) \ 1&& 2

_ \ \ + +& , # & 1 2 (M\) log M\ & 1& . Q$ & & _ \ \ + (1&&) \ 1&& 2 2 +& P& &

(41)

2

It is clear that neglecting terms in (41) which are small to unity we obtain from (40) the stress components of the edge dislocation along Ox 3 with the Burgers vector in the x 1 -direction: _ \\ , _ ,, & &

b + sin , , 2? 1&& \

_ \, &

b + cos , . 2? 1&& \

(42)

Using (18) and (41) we estimate _ 33 &&(_ \\ +_ ,, ). Thus, the conventional edge dislocation appears within (I) as a leading approximation. Let us consider the domain (II). Neglecting small exponents, we obtain _ \\ &qN

sin ,8 &( \) , (N\) 32

sin ,8 +(\) _ zz , _ \\ +_ ,, & &qN , (N\) 12

_ \, & &qN

cos ,8 &( \) , (N\) 32

+(1+&) b q# . ?(2?) 12 1&&

(43)

THE T(3)-GAUGE MODEL

269

Recall that for the screw dislocation any co-axial cylindric surface is stress-free, while the modified defect itself is ``localized'' since everything decays exponentially for it. Now the situation is different: the arbitrary cylindric surface belonging to (II) is not stress-free, though the surface traction T k =_ kl n l on it calculated with _ \\ , _ \, (43) turns out to be weaker than for the classical solution (42) because N(N\) 32 < <1\. Although f e decays in (II), and the surface traction is rather weak, the modified edge dislocation cannot be viewed as localized so perfectly like the screw one, because the closure failure does not vanish in (II). Thus, a self-consistent defect solution is obtained in the framework of the field theory in question. As to its connection with plane elasticity, it is seen that the corresponding requirement is, formally, violated by the second term in _ zz (40), and the triad component , zz gets the form , zz =(M 22) f e =(b4?) sin , F $( \). The function F $ can be estimated inside (I) by (18), whereas inside (II) F$& &

2 N8 +( \)

? (N\)

12

.

Therefore, , zz is small at M< <1 inside both (I) and (II), and , zz Ä 0 at M Ä 0. For the intermediate region, F $ can also be considered as small because of the differentiation rule, dd\=Ndd(N\). Thus, geometrically, the picture on the entire (x 1 , x 2 )-plane (except for \<\ C ) still looks realistic at M< <1 in spite of the slight deviation of , zz from zero. However, when M increases, the direct connection of the modified defect with plane elasticity is kept only at those \ which respect M\< <1. The solution obtained in this section is valid for the classically represented _ (5). Although for the screw dislocation the classical ansatz suits the modified gauge equation (11), the ansatz (5) could require, with regard to (11), a certain generalization, say, due to inclusion of 3-dimensionality outside (I). Such a possibility should be investigated separately. The central task of our investigationto demonstrate that the use of the gauge equation (30) allows, in the linear approximation, the edge dislocationis achieved. More generally, we have succeeded in finding the RiemannEinstein type Lagrangian gauge model which enables both the classical defect solutions as limits of the modified ones.

6. MICROSCOPICALLY CONSISTENT INTERPRETATION In the previous sections the self-consistent gauge-translational model given by L=LHE +L! was discussed as a direct combination of d=3 Euclidean Einstein gravity with isotropic elasticity. Essentially LHE and L! both are constructed from the same translationally invariant variables (21), (22), and therefore the stress 7 (30) (polynomial in strainmetric) turns out to be a perturbation of geometric

270

C. MALYSHEV

configurations governed by G (metric differentiated). Indeed, when }< <1 (i.e., +< >1) spatial contraction. The latter is especially transparent for the modified screw dislocation. Owing to equivalence between LHE and the purely quadratic Lagrangian L& , after the microscopic studies [8, 24, 53, 54] it is suggestive to consider s (27) as a scale of dislocation core energy, = core . If so, the contraction of classical solutions demonstrated above looks unnatural at +> >= core . Then, a straightforward modification of the model can immediately be proposed as follows. Indeed, defects are considered above as created due to deviation of geometry from an initially perfect (i.e., defectless) configuration. In order to account for equivalence st= core , the model should be treated as describing final configurations as influenced by an effect of, say, the material ``response,'' which consists in formation of the core region; i.e., final configurations should be considered as deviations, due to core formation, from background ones given, for instance, by classical Volterra defects. Let us express that the initial state is not perfect, by adding the external source (1B) Lbg =E kl _ kl V to the total Lagrangian, where _ V is the a priori known stress of the classical Volterra defect. This term is gauge invariant due to invariance of E ij , and it looks like the work of an external traction. Variation over B a i does not affect _ V , and therefore (30) acquires now the difference 7&7 V instead of 7 in its R.H.S., while the equilibrium equation is not changed. In other words, not the total stress but only its deviation from the background one is assumed to influence the final geometric configurations. In linear approximation, 7 V is reduced to (4), and for the screw dislocation the latter can be expressed by means of (32), where f is f P =(&b2?) log \. In this case the gauge equation gets the form (2&} 2 )( f& f P )=&2f P =b $ (2)(x),

(44)

and it is solved by f &f P =&f s , i.e., f =f P & f s . The minus at f s is dictated by plus at b$ (2)(x), and vice versa, i.e., the distributional aspect of the solution becomes essential (in Section 5, the choice of the sign for f s , f e is a matter of tacit consent). Further, the total stress reads _ ,z =

1 b+ } &K 1(}\) . 2? }\

\

+

(45)

Within (I), _ ,z (45) tends to zero at }\ Ä 0 (see the Appendix) because the logarithms are mutually cancelled in f P & f s . The singularity 1\ of the classical stress thus disappears. Within (II), _ ,z is that of the screw dislocation due to exponential decay of K 1 . Besides, T 3 , 12 =&2( f P & f s )=

b 2 } K 0(}\). 2?

(46)

271

THE T(3)-GAUGE MODEL

The defect density is less singular at the origin since the $-function does not appear. The circuit integral  C B 3 is b for C # (II), while it goes to zero in (I), accordingly to (35). Hence, the domain (I) ( \ C can be set equal to zero) acquires the sense of a core region where the celebrated singularity of the Volterra solution is smoothed out due <+) implies the weak core's to core formation. Now the limit }> >1 (i.e., = core < influence, and the domain restricted by \ & 1} is strongly reduced. The opposite limit, }< <1, implies the strong core's influence, and the domain restricted by \& 1} is extended. Let us turn to the edge dislocation. Here the couple of the gauge equations is

(2&M 2 )(2+N 2 )( f& f A )=&22f A =&

2b  2 $ (2)(x). 1&&

(47)

p=&& 2f+(1+&) M 2( f& f A ) Equations (47) are solved by f &f A =&f e , i.e., f =f A & f e , and the stress components are now given by _ \\ =&

1 b+ sin , +P( \) , 2? (1&&) \

\

+

_ \, =

1 b+ cos , +P( \) , 2? (1&&) \

\

+

(48) _ ,, +_ \\ =&

2 b+ sin , +Q$( \) , 2? (1&&) \

\

+

_ zz =&(_ \\ +_ ,, )&+(1+&) M 2f e .

Let us discuss the total configuration (48). The surface traction calculated on an arbitrary cylindric surface belonging to (II) behaves, in leading approximation, as that of the edge dislocation, since the classical part of _ \\ , _ \, (48) dominates over P. However, the non-conventional part of _ ,, (48) exceeds that from the background since 1\< >1. Thus, the influence of the core is not completely suppressed, and the modified defect asymptotically differs from the edge dislocation. Within (I), however, the classical singularity of _ V is perfectly smoothed out. In its turn,  C B 1 tends to zero within (I), while it oscillates about b with slowly increasing amplitude in (II). Hence, (48) describes a complicated gauge defect which respects the equilibrium equation (1.1), and which is characterized geometrically by the closure failure (39). As to its connection with plane elasticity, the deviation of (48) from the plane problem is small at M< <1 (the strong core's influence). In this case the core region is extended, and the external asymptotics of the modified solution clearly describe the closest exterior vicinity of the core region. Our choice of the Kroner ansatz for the modified edge dislocation simply implies that the core's influence still penetrates outside \ &1M#1}, and the defect obtained looks complicated near its core. It

272

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is hopeful that a form more complex, than (5) of the Kroner ansatz could improve the situation with localization of the non-conventional mode in _ ,, (48). Thus, the initial configuration contains now a classical defect, and the following variable can be defined, B ij =$ ij +, ij +, ij , where it is understood that the gauge potential is divided into two parts so that , ij implies background, say, , bg, while , means fluctuation around it. Let us define 2E ij =B ai B a j &$ ij as a strain of background configuration, and 2E ij =B ai B a j &B ai B a j as a measure of deviation of final configuration from initial one. The total strain is E tij =E ij +E ij . In lowest approximation we get E ij =, (ij) +, k(i , k j) + 12 , ki , k j r , (ij) , and so E tij r, (ij) +, (ij) . In leading approximation the strain E tij is given by superposition of , (ij) and , (ij ) , but coupling between , and , bg is expected to become important in the next order. Therefore, the present section demonstrates the reasonable first-order situation, and predicts interesting couplings in next order. Strictly speaking, the translational gauge equations found in [12] are not (13)(15), but rather (44), (47). More accurately, the distinction between our (44), (47) and their equivalent forms [12] is simply notational and since a and 1&a are exchanged. A concept of, so-called, nucleation fields which are to generate background configurations modifying the sources in the gauge equations is advanced in [12]. This is just the case why equivalents of (44), (47) appear in [12]. However, (44) is still reduced in [12] to the previously known form (15), while a solution of (47) is left as a home exercise. Only recently was the approach [12] properly used in [61] to obtain the stress potential of the modified screw dislocation, i.e., b [K 0(}\)+log \+const], 2? instead of f s =(b2?) K 0(}\). Reference [61] provides formulas coinciding with (44)(46). Unfortunately, (gauge) edge dislocation is again missed in [61]. Modification of certain gravitational effects due to translational gauge fields governed by the quadratic in the field strength gauge-translational Lagrangian is discussed in [2]. The translational Lagrangian [2] corresponds to the quadratic form in (26) rather than to (3). The isotropic quasi-elastic Lagrangian also appears in [2], and it enables a short-ranged character of the correction to the background Newton's law; i.e., the situation [2] looks like that of the modified screw dislocation in the present section and in [61].

7. DISCUSSION Three-dimensional Lagrangian field theory is proposed above as a gauge model of translational defects in solids. An attempt [11, 12] has already been made to

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incorporate Abelian gauge features of [20] into a more general non-Abelian framework. Traditional Volterra solutions for dislocated continua look, in many respects, satisfactory, and therefore [11, 12] should be tested for correspondence with them. Unfortunately [29], conventional edge dislocation does not fit in [11, 12]. Nevertheless, a self-contained gauge-translational description for dislocations (and their ensembles) still merits attention [14, 7, 8, 18, 19, 2225]. For instance, the broadly elaborated realm of low-dimensional gravity [5, 6] seems to be of much help in formulating a geometrized model(s) of defects. The present paper completes [29] and demonstrates that edge and screw Volterra dislocations turn out to be associated with the gauge model proposed, which directly combines d=3 Euclidean Einstein gravity with a model of elastic continuum. In other words, both classical solutions are shown to appear provided the traditional incompatibility law is replaced by the Einstein-like gauge equation with the elastic stress tensor in its R.H.S. Actually, linearization of the Einstein-type gauge equation is considered, and axial bulk geometry is used to capture infinitely long line defects. Specifically, two limits are studied: }\< <1 (with possible \ Ä  at } Ä 0) and }\> >1, where } 2 =+s measures a relative contribution from the gauge and elastic Lagrangians, and \ is the cylindrical radial coordinate. The model admits both conventional defects in infinite bulk at } Ä 0. When }{0 and \ is sufficiently large, }\> >1 becomes valuable, and the asymptotics of the solutions obtained loose its classical feature. Namely, for the screw defect all the relevant quantities decay exponentially thus demonstrating truly short-ranged character. The modified stress potential of the edge dislocation is decaying weaker, and thus we are still concerned with the incompatible strain (i.e., with a defect) even outside \ & 1}. However, geometrically, this defect does not look ``elementary.'' Notice that namely due to } 2-terms in the gauge equations unboundly increasing classical stress functions are replaced by those decreased at spatial infinity. Our consideration shows that the HilbertEinstein Lagrangian LHE is suitable for conventional dislocations, since it is responsible for the classical limit of the solutions obtained, while L! influences the asymptotical suppression of them. But further studies are needed to go beyond the simple picture presented, say, to make the spatial contraction within the cylinder of radius \ &1} more realistic in the sense of the free boundary condition. Although LHE is more adequate than L, (3) for Volterra defects, the latter can be more appropriate in such problems as [1316, 18]. Further, it is seen that the stress function method can be successfully used in our Einstein-type gauge approach. Although, for the screw dislocation, the traditional choice of the Kroner ansatz looks appropriate, a certain elaboration is seemingly still required for the edge dislocation. Provided teleparallel geometry is chosen for dislocated continuum, the most general gauge Lagrangian, LHE (the ``cosmological'' term is absent), turns out to be equivalent to a Lagrangian quadratic in torsion. Hence, the usage of LHE seems to implement the proposal by [8, 53, 54] concerning the importance for the continual approach of the contributions quadratic in the defect densities. To account for

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equivalence st= core , the model proposed should be re-considered in a slightly different way, namely, as a model which describes a material ``response'' (i.e., core formation after inclusion of = core ) to a background configuration given by the classical Volterra defect. The re-considered version results in the configurations which agree with the classical Volterra defects at sufficiently far distances, while the celebrated axial discontinuities of the linear stresses are smoothed out within the core regions. Thus, the important suggestion by [8] gets confirmation since the length scale 1}, which specifies the core region, arises in the continual Lagrangian model. For the screw dislocation, the core region is perfectly localized, since outside \ & 1} the stress field looks classical. For the edge dislocation, the closure failure does not look ``elementary'' outside \ & 1}, since the core's influence penetrates outside the core region. However, surface traction on an arbitrary cylindric surface of sufficiently large radius is, mainly, that of the edge defect. Existence of two possibilities to influence the behaviour of classical Volterra dislocations by means of practically the same Lagrangian model is not surprizing. Technically, both of them occur similarly because the modified stress functions are governed by Helmholtz-like operators 2\} 2 instead of the two-dimensional Laplacian 2. However, it is our formal freedom to consider the gauge Lagrangian either on the rather formal ground or under the microscopically motivated assumption, which enables us to treat the solutions obtained either on their own rights or as deviations from background configurations. In their turn, the theoretical possibilities of modification of gravitational Newton's law either at large or small distances by means of the inhomogeneous Helmholtz equation have their own history [62], and still continue to attract attention (see [2, 63] for references). In this respect, the modifications of the classical stress potentials considered here and in [61] resemble (just in the formal sense of the origin and implication of } 2-terms in the modified gauge equations) certain issues in general relativity, though for lower spatial dimensionality. Thus we are faced with another aspect of ``parallelism'' between crystallographic defects and gravity. Gauge gravity with torsion has still continued to attract considerable attention during the last decade. For instance, propagation of a quantum particle in geometrically non-trivial spaces induced by non-holonomic mappings has been considered in [6469]. Appropriate novel gauge invariance found in [6469] can become useful for further studies of gauge geometry of defects, since, according to [69], it gives hints on how to construct the proper field equations in gravity with torsion. The geometric approach [70] to scattering of elastic waves on parallel dislocations should also be mentioned. To summarize, the Lagrangian T(3)-gauge approach is proposed which enables the gauge solutions which look like Volterra dislocations modified at distances either far (towards finitness of the bulk's cross-section) or short (linear stresses are smoothed out within the core). The calculation presented can be considered as a first order approximation. Incorporation of second order contributions to the gauge

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equation andor extension to the ISO(3)-case is needed to decide about further perspectives of T(3)-ISO(3)-gauging for defects in solids. With regard to the great attention to translational gauge models with quadratic gauge Lagrangians [2, 8, 1119, 24, 53, 54] (partially, to interpret gauge-translational degrees of freedom in gravity in the spirit of continuized crystallographic translational defects), it is hoped that the work presented will stimulate a more adequate understanding of the problem discussed.

APPENDIX The following series are known for K 0 and K 1 [36], # (}\) 2 (}\) 2 # K 0(}\)=&log &log(}\) 1+ 1&log + } } } , + }}} + 2 4 4 2

\

K 1(}\)=

+

\

+

1 }\ (}\) 2 }\ # 1&2 log + } } } . + log(}\) 1+ + }}} & }\ 2 8 4 2

\

+

\

+

ACKNOWLEDGMENTS The author is grateful to A. E. Romanov and L. A. Turski for numerous illuminating discussions on defects. The author thanks V. M. Babich for discussing Section 3, and F. W. Hehl for reading a draft of the manuscript. Part of the paper was written at the Center for Theoretical Physics in Warsaw. The research described was partially supported by the Russian Foundation for Fundamental Research, Projects 960100807, 980100313, and by Grant from the J. Mianowski Foundation for Science Promotion (Poland).

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