A gauge theoretic approach to elasticity with microrotations

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by  Christian G. Boehmer

A GAUGE THEORETIC APPROACH TO ELASTICITY WITH
MICROROTATIONS
2 C.G.BO¨HMERANDYU.N.OBUKHOV
1
0
2 Abstract. Weformulateelasticitytheorywithmicrorotationsusingtheframe-
work of gauge theories, which has been developed and successfully applied
n
in various areas of gravitation and cosmology. Following this approach, we
a
demonstrate the existence of particle-like solutions. Mathematically this is
J
duetothefactthatourequationsofmotionareofSine-Gordontypeandthus
2
have solitontype solutions. Similarto Skyrmions and Kinks inclassical field
1
theory, wecanshowexplicitlythatthesesolutionshaveatopologicalorigin.

]
h
p
- 1. Introduction
h
t
a 1.1. Elasticity and gauge theories. In classical elasticity one assumes that the
m elements of a material continuum are structureless points which are characterized
[ bytheirpositionsonly,whereastheirinternalproperties(suchasorientation,defor-

mation etc) are disregarded. This picture is generalizedto the concept of continua
3
with microstructure when a material body or fluid is formed of a large number of
v
1 deformable particles whose (micro)elastic properties contribute to the macroscopic
5 dynamics of such a material medium. The corresponding mechanical model was
9 developedvery early by the Cosseratbrothers [8] who replaceda structureless ma-
5 terialpointbyapointthatcarriesatriadofaxes(anorientedframe,inthemodern
.
2 geometricallanguage). Thisideahasledtoarichvarietyofmodelswhichareknown
1 by various different names like orientedor multipolar medium, asymmetric elastic-
0
ity,micropolarelasticity,seee.g.[14,15,16,17,21,47,55,56,57,61,62,19]. This
1
field is also related to other areas of continuum mechanics, like the theory of fer-
:
v romagnetic materials,crackedmedia, liquid crystals [9], superfluids [65, 46, 58, 66]
i
X and granular media (see, for example, application of the Cosserat model to the
description of the grain size effects in metallic polycrystal [68]).
r
a In general case, the 3-frame attached to material elements is deformable, thus
introducing nine new degrees of freedom in addition to the classical elasticity (3
rotations,1volumeexpansion,and5sheardeformations);thisisusuallycalledami-
cromorphiccontinuum[19]. Aspecialcaseisthe microstretchcontinuumforwhich
the 3-frame can rotate and change volume (4 degrees of freedom: 3 microrotations
and 1 microvolume expansion). Finally, in the micropolar continuum, the struc-
tured material point is rigid, thus possessing 3 microrotation degrees of freedom.
ThelastisthesubjectoftheCosseratelasticitythatconsidersmediawhichcanex-
perience displacements and microrotations. In turn, this theory has two important
limiting cases, one with no microrotations which is the classical elasticity, and the
second case where one assumes that the medium only experiences microrotations
and no displacements. Models of the latter type have in fact a long history which
2000 Mathematics Subject Classification. (primary),(secondary).
1
2 C.G.BO¨HMERANDYU.N.OBUKHOV
canbe tracedbacktoMacCullaghin1839,see[67,64]. Recently[3,4]suchmodels
have been investigatedin the fully nonlinear setting and plane wave type solutions
were explicitly constructed. The existence of such solutions is a highly non-trivial
fact. In a linearised setting, similar solutions were investigated in [48, 49, 50].
Recent years have seen a revival of elasticity, in particular in the theory of
dislocations which has been analysed from a gauge theoretic point of view. The
gauge approachwas developed and successfully applied in high energy physics and
in the theories of gravity [25, 27, 28]. The gauge approach in elasticity also has a
long and rich history, see for example [29, 12, 42, 43, 32, 33, 34, 38, 39, 40, 41]. Of
particularinterest is the fact that the dislocations behave very muchlike particles.
Fromamoreformalpointofview,oneoftheopenquestionsinthisfieldofresearch
iswhether ornotitis possibleto find solitontype solutionswhichcouldjustify the
particle interpretation from a mathematical point of view. The main result of this
paper is that we are able to find soliton type solutions in a particular model of
elasticity motivated by gauge theories of gravity.
1.2. Geometry and elasticity. Our notation and conventions follow the book
[26] where the modern differential geometry of manifolds is presented. Here we
briefly summarize the notions and define objects needed for our current study.
We consider an elastic medium which occupies the whole of R3, and we can
identifymaterialpointswithpointsinspace. WeusetheLatinalphabettolabelthe
coordinate indices, i,j,k, = 1,2,3, say, we write xi for the spatial (holonomic)
···
coordinates of , a 3-dimensional simply connected manifold which we identify
M
with the material points of the continuum. This manifold occupies the entire 3-
dimensional Euclidean space R3. Greek indices refer to the (co)frame indices (also
tangent space or anholonomic indices), we use α,β, = 1,2,3 to label the frame
···
covectors(1forms)ϑα =hαdxi whichwecanrefertoasthedistortioninelasticity.
i
Wefollowthe Einsteinsummationconventionwherebywesumovertwicerepeated
indices. We denote the frame by e = hi ∂/∂xi, such that e ϑβ = hihβ = δβ.
α α α⌋ α i α
The e form the basis vectors of the tangent manifold T , and denotes the
α
M ⌋
interior product.
TheEuclideanmetricg isdefinedon asascalarproductg :T T R.
M M× M→
Asusual,thevaluesofthisscalarproductonthebasisvectorsintroducethemetric
tensor g =g(∂ ,∂ ), the components of which for the Cartesian coordinates read
ij i j
g =δ =diag(+1,+1,+1). The components of the metric g =g(e ,e ) in an
ij ij αβ α β
arbitrary basis are obviously related to the frame components via g = hiδ hj.
αβ α ij β
One can think of ϑα as a set of three mutually orthogonal and normal covectors
∗
whichformabasisatanypointp T . Eachsuchbasiscovectorhascomponents
∈ M
(ϑα) and thus we can also view the object hα as an orthogonal matrix. We can
i i
thinkofthemetricofthedeformedmediumastheCauchy-Greentensorinelasticity.
Technically,themetricisusedtoraiseandlowertheindices,forexampleweobtain
a covector-valued 1-form from a vector-valued coframe as ϑ = g ϑβ, etc. The
α αβ
componentsofthe coframecomprisethe 3 3matrixhα whichis inversetomatrix
× i
composedofthe framecomponentshi. Fromthe definitions wedirectlyobtainthe
α
following identity hα =gαβδ hj.
i ij β
One can draw a close analogy between gravityand elasticity, see the instructive
discussion [25], for example. The force and the hyperforce (or the couple-force),
as well as the stress (force per area) and the hyperstress (couple per area), caused
by the force and hyperforce, respectively, are the basic concepts in the continuum
A GAUGE THEORETIC APPROACH TO ELASTICITY WITH MICROROTATIONS 3
mechanics of structured media. The direct gravitational counterparts of the stress
and hyperstress are the energy-momentum current and the hypermomentum (that
includes spin, dilation and shear currents). The spacetime itself is geometrically
modelled as a continuum deformed under the action of the sources with nontriv-
ial “stress” (energy-momentum) and “hyperstress” (spin current, e.g.). Using this
correspondence, it is possible to put both gravity theory and elasticity theory into
the same differential-geometric framework in which the metric, coframe, and con-
nectionarethefundamentalfieldvariables. Wereferthereaderformoreformalism
and technical detail to the comprehensive review [28].
Whenconsideringdeformationsofanelasticmaterial,oneplacescertaincompat-
ibilityconditionsonthe inducedstress. ThesesocalledSaint-Venantcompatibility
conditionsareaformofintegrabilityconditionssothat the stresscanbe expressed
andthe symmetricderivativeofthe displacement. Inamoregeometricallanguage,
by deforming the medium, we do not want to induce any curvature into the mate-
rial. This means that one assumes the Riemann curvature tensor of the deformed
medium to vanish identically. It is well known that these two views are equiva-
lent, see for instance [1, 60]. In the continuum theory of media with defects [31],
the elastic properties of a crystal are described by means of the non-Riemannian
geometries in which the notion of curvature is supplemented by a new geometrical
quantitycalledtorsionwhichwasfirstintroducedbyCartanin1922. Therefore,we
willanalyseelasticmaterialsusingthe languageoftheRiemann-Cartangeometries
withtorsionandwe will stick to the wellestablishedrequirementofhavingvanish-
ing total curvature. The latter condition is sometimes called a second basic law in
thedifferentialgeometryofmediawithdefects[31],andithasthephysicalmeaning
thatthecrystalstructureisuniquelydefinedeverywhere,seealso[2,35,36]. Inthe
following subsection we will collect the most important facts of such geometries.
1.3. Geometries with vanishing Riemann curvature tensor. Letusdescribe
eachmaterialpointofourelasticmaterialbyacoframeϑα (avector-valued1-form)
and a connection Γ β (a matrix-valued 1-form). Recall that the latter arises from
α
the computation of the covariant derivative along a vector field v of the frame
v
∇
vectors, e = Γ α(v)e , see Sec. C.1 of [26]. The connection is assumed to
v β β α
∇
be metric-compatible which means the vanishing nonmetricity Dg = 0, hence
αβ
the connection 1-form is skew-symmetric, Γ = Γ (where Γ = Γ γg ).
αβ βα αβ α βγ
−
Thecoframespecifiestheorientationoftheorthonormalbasisvectorsatthispoint,
while the connection determines how an arbitrary vector is parallelly transported
near this point. The vanishing of the Riemann-Cartan curvature tensor (a matrix
valued 2-form) takes the form
(1.1) R β :=dΓ β +Γ β Γ σ 0,
α α σ α
∧ ≡
whered denotes the exteriorderivativeand is the exterior(alternating)product.
∧
In geometries where the Riemann-Cartan curvature identically vanishes the con-
nection is often referred to as a Weitzenbo¨ck connection. The condition R β 0
α
≡
is known in the literature by various names like teleparallel, fernparallel of distant
parallel geometry. Geometrically this condition means that the notion of paral-
lelism is no longer a local statement, but can be defined globally, this means on
theentiremanifold. Astraightlineconnectingtwomaterialpointsdefinesanabso-
lute notion of parallelism. Physically the condition (1.1) has the meaning that the
structureofanelasticmediumisuniquelydefinedeverywherein ,[35,36,31,2].
M
4 C.G.BO¨HMERANDYU.N.OBUKHOV
The only nonvanishinggeometricalobjectin suchgeometriesis Cartan’storsion(a
vector valued 2-form) defined by
(1.2) Tα =dϑα+Γ α ϑβ.
β
∧
Let ε = ϑ1 ϑ2 ϑ3 denote the volume 3-form, then the dual forms of the
∧ ∧
products of the coframe are defined by
(1.3) ε := ϑ =e ε,
α α α
∗ ⌋
(1.4) ε := (ϑ ϑ )=e ε ,
αβ α β β α
∗ ∧ ⌋
(1.5) ε := (ϑ ϑ ϑ )=e ε .
αβγ α β γ γ αβ
∗ ∧ ∧ ⌋
Theoperator ,introducedbytheformulasabove,iscalledtheHodgeoperator.The
∗
dual forms satisfy the following useful identities
(1.6) ϑβ ε =δβε,
∧ α α
(1.7) ϑβ ε =δβε δβε ,
∧ µν ν µ− µ ν
(1.8) ϑβ ε =δβε +δβε +δβε .
∧ αµν α µν µ να ν αµ
The object ε is the totally antisymmetric Levi-Civita symbol with ε =1.
αβγ 123
1.4. Elastic invariants. Since Tα is the only nontrivial geometric object, it is
thereforetheonlynon-trivialobjectcharacterisingthedeformationsofthemedium.
It is natural to decompose it into its irreducible pieces which will serve as our
buildingblocksofelasticinvariants. ThetorsionTα = 1Tα ϑµ ϑν isrepresented
2 µν ∧
asarank3tensorwhichisskew-symmetricinonepairofindices,Tα = Tα . It
µν νµ
has 9 independent components. Since we work in R3, it is also natural t−o consider
the Hodge dual of torsion which can be regarded as a 3 3 matrix with no a
×
priori symmetries. At any point we can view a 3 3 matrix as an element of a 9
×
dimensionalvector space V. Performingrotations which leave the metric invariant
will change the components of this matrix, thus the group SO(3) acts on V and
we are interested in identifying invariant subspaces. Those give rise to the so-
calledirreducible pieces ofeither this matrix,orits Hodgedual, the torsiontensor.
Therefore, let us define α := Tα.
T ∗
The three irreducible pieces of torsion are defined by
1
(1.9) (2)Tα = ϑα (e Tβ),
β
2 ∧ ⌋
1
(1.10) (3)Tα = (ϑ Tβ)εα,
β
3 ∗ ∧
(1.11) (1)Tα =Tα (2)Tα (3)Tα.
− −
As we already explained, the metric is used for the raising and lowering of the
indices, i.e., εα =gαβε and ϑ =g ϑγ.
β β βγ
The irreduciblepieces(1.9)-(1.11)canbe convenientlyunderstoodintensorlan-
guage as the skew-symmetric part, the trace part, and the traceless symmetric
part of the matrix, see the discussion below and in particular the explicit formulas
(2.8)-(2.10).
It is important to note that there is a gauge freedom in choosing the variables
ϑα,Γ β . All coframes are related to each other by means of local rotations.
α
{ }
A GAUGE THEORETIC APPROACH TO ELASTICITY WITH MICROROTATIONS 5
Therefore, the same elastic medium can be described equivalently by another pair
ϑ′α,Γ′ β which is related to the former by means of the gauge transformation
{ α }
(1.12) ϑ′α =Lα ϑσ,
σ
(1.13) Γ′ α =Lα Γ σ(L−1)ρ +Lα d(L−1)σ ,
β σ ρ β σ β
where Lα is an arbitrary 3 3 rotation matrix, this means L SO(3). The
β
× ∈
covariant condition R β 0 is preserved by this gauge transformations, whereas
α
torsion Tα transforms cov≡ariantly, T′α =Lα Tσ.
σ
Making use of the local transformations (1.12)-(1.13), we can eliminate either
of the variables in the pair ϑα,Γ β . The ability to do this is closely to our
α
{ }
assumption of a globally flat manifold (1.1). In particular, in [3], the teleparallel
connectionisgaugedawayΓ β =0,andthe coframeϑα isleftasthe onlyvariable.
α
Alternatively in [4] the coframe is gaugedto the standardconstant values hα =δα
i i
and the only variable is the connection, in which case once can write
(1.14) Γ α =Λα d(Λ−1)σ .
β σ β
Here Λ SO(3) parametrises a teleparallelconnection. We will refer to this choice
∈
of variables as the connection gauge since the only dynamical variable is the con-
nection.
2. Basic equations of the model
2.1. The potential energy. The natural potential energy is based on the sum of
the three quadratic microrotational elastic invariants constructed from the nonva-
nishing torsion
3
(2.1) V = c (i)Tα T ,
i α
∧∗
Xi=1
where c , i=1,2,3,are non-negative constants which are physically interpreted as
i
elastic moduli. The non-negativity of the material constants c is a consequence of
i
the conditionofthe positivesemi-definiteness ofthe microrotationalelasticenergy,
V 0, in complete agreement with [39].
≥
In general, V = V(Tα) can depend arbitrarily on the torsion, but the choice of
the quadratic function (2.1) yields the linear constitutive law, [39, 40, 41]. When
our medium occupies the whole of R3, the three quadratic terms in (2.1) are not
independent because of the teleparallel condition (1.1). Recall that there is an
identity (see Eq. (5.9.18) of [28]) that is valid in n-dimensional space:
(2.2)
1
R˜αβ ε =Rαβ ε [(1)Tα (n 2)(2)Tα (3)Tα] T d(2ϑα T ).
αβ αβ α α
∧ ∧ − − − − 2 ∧∗ − ∧∗
The tilde denotes the Riemannian geometric objects, i.e. those constructed from
the Christoffel symbols of the corresponding Riemannian metric of the manifold.
Taking into account that the Riemann-Cartan curvature vanishes due to the
teleparallelconstraint(1.1)andthatbyassumptiontheelasticmediumisembedded
in a flat Euclidean space with R˜αβ = 0, for n = 3 the identity (2.2) allows us to
express the square of the first (tensor) irreducible part in terms of the trace and
axial trace squares:
1
(2.3) (1)Tα T =(2)Tα T + (3)Tα T d(2ϑα T ).
α α α α
∧∗ ∧∗ 2 ∧∗ − ∧∗
6 C.G.BO¨HMERANDYU.N.OBUKHOV
In tensor language, using (1.9), (1.10), and (1.6), (1.7), we have
1
(2.4) (2)Tα T = Tν T αµε,
α αν µ
∧∗ 2
1
(2.5) (3)Tα T = (T εµρσ)2ε.
α µρσ
∧∗ 12
It is straightforward to establish relation between the irreducible parts of the
torsion 2-form Tα and its dual 1-form α. We have α = αβϑ , denote the
β
T T T
matrix αβ by T, and we find explicitly the decomposition of the second rank
T
tensor
(2.6) αβ =(1) αβ +(2) αβ +(3) αβ,
T T T T
(2.7) T=(1)T+(2)T+(3)T,
into the trace, antisymmetric and traceless symmetric parts:
1 1
(2.8) (3) αβ = T εµρσgαβ, (3)T= (trT)I,
µρσ
T 6 3
1
(2.9) (2) αβ = εαβµTν , (2)TT = (2)T
µν
T 2 −
1 1
(2.10) (1) αβ = T(α εβ)µν T εµρσgαβ, (1)TT =(1)T, tr(1)T=0.
µν µρσ
T 2 − 6
2.2. The choice of the gauge. In the recent papers [3, 4], the propagation of
waves was studied in the structured medium under consideration. The results
obtained describe the elastic waves in the two complementary pictures which arise
duetothetwodifferentchoicesofthevariablesallowedbythegaugefreedom(1.12)-
(1.13). In [3], the coframe gauge is used when the connection is zero (completely
gauged out). The components of the coframe are the only dynamical variables.
Geometrically, they represent an arbitrary orthogonal 3 3 matrix. Since the
×
connection is trivial in this gauge, the torsion 2-form reduces to Tα = dϑα. Using
the components of the coframe and frame, ϑα =hαdxi, e =hi∂ , we have for the
i α α i
holonomic torsion
(2.11) Ti =hiTα =hid(hα) dxj =hi∂ hαdxk dxj.
α α j ∧ α k j ∧
The dual 1-form then reads
(2.12) i = i dxj =hi∂ hαεkl dxj.
T T j α k l j
Here εkl = hkhlhγǫαβ is the Levi-Civita tensor with respect to the holonomic
j α β j γ
coordinate basis.
Acomplementaryapproachisdevelopedin[4],wheretheonlydynamicalvariable
is the connection, whereas the coframe is fixed to its trivial constant value ϑα =
δαdxi. The torsion (1.2) reduces in this gauge to Tα =Γ α ϑβ, and since in the
i β ∧
Weitzenbo¨ck space the connection is given by (1.14), we have
(2.13) Ti =δi Tα =δiΛα d(Λ−1)σ δβdxj.
α α σ β j
Denoting ui :=δiΛα , we thus have
σ α σ
(2.14) Ti =uid(uα) dxj =ui∂ uαdxk dxj.
α j ∧ α k j ∧
This is completely equivalent to (2.11), with the difference that the orthogonal
matrix hi, representing the coframe, is replaced with the orthogonal matrix u :=
α
ui, representing the connection.
α
A GAUGE THEORETIC APPROACH TO ELASTICITY WITH MICROROTATIONS 7
In [4], there are two technical deviations as compared to the model developed
in [3]. The first deviation is the different choice of the variables mentioned above.
Noticing that (uT∂u)i = ui∂ uα is skew-symmetric in i,j (being an element of
kj α k j
the Lie algebra of the orthogonal group), in [4] one chooses as a basic variable a
new 3 3 matrix
×
(2.15) A=⋆(uT∂u),
whichhasnoapriorisymmetries. This⋆denotesthedualizationoperatorwhichre-
latesskewsymmetricmatricesto vectors,similartotheHodge operator. Writing
∗
out the indices explicitly, this matrix can be written in the following way
1
(2.16) A = ε jui∂ uα.
lk 2 li α k j
Thisquantityisknowninthepolarelasticitytheoryundervariousdifferentnames:
theNyetensor[30],thewrynesstensor[13],thesecondCosseratdeformationtensor
[19], the third order right micropolar curvature tensor [51], or torsion-curvature
tensor (note that this is somewhat a misnomer since it neither directly relates to
torsion nor curvature in the sense of differential geometry). The inverse of (2.16)
yields the components of the torsion tensor (2.14)
(2.17) ui∂ uα =A εli .
α k j lk j
Substituting this into (2.14), we can find a relation between the the second rank
tensor T (which we can view as a matrix in R3) to the matrix A, so that
(2.18) T=A tr(A)I.
−
The second deviation of [4] from [3] is a different structure of the Lagrangian.
Namely, the Lagrangian of a rotationally elastic medium is assumed in [4] to be a
function of orthogonal matrix only.
2.3. KineticenergyandtotalLagrangian. TheLagrangian4-formV =Ldt ε
∧
isconstructedbytakingL=L(u,∂ u,∂ u)asaquadraticfunctionoftheirreducible
0 i
parts of A . In addition, the kinetic term is chosen to be
lk
1 1
(2.19) Lkin = tr(u˙Tu˙)= (∂ ui)(∂ uα).
2 2 t α t i
This form of the kinetic energy is well motivated, since when linearised it yields
angular kinetic energy or rotation energy. Strictly speaking, in general the kinetic
energy should be constructed by taking the microinertia into account [19, 44].
Introducing,analogouslyto (2.16), the velocityofthe deformationsofthe mate-
rial continuum
1
(2.20) A = ε jui∂ uα,
lt 2 li α t j
and denoting Alt =glkA , we can recast (2.19) as
kt
1
(2.21) Lkin = (∂ ui)(∂ uα)=A Alt.
2 t α t i lt
Taking into account the identity (2.3), the potential energy contains just two
quadratic invariants. As a result, the general Lagrangianreads
(2.22) L=Lkin(A ) Lpot(A ), Lpot(A )=λ (Ak )2+λ A A[lk].
lt lk lk 1 k 2 [lk]
−
Thecouplingconstantsλ = 4(c +1c )andλ =c +c areconstructedfromthe
1 3 3 2 1 2 1 2
elastic moduli (2.1).
8 C.G.BO¨HMERANDYU.N.OBUKHOV
Onecanusedifferentparametrizationsoftheorthogonalmatrices. Forexample,
in [3], the spinor parametrization was used. Here we find it more convenient to
describe an arbitrary orthogonal matrix with the help of the three real functions
βi which is based on the identification of SO(3) with the three-sphere. Fromthese
independent constituents, an orthogonalmatrix is constructed as follows
(2.23) ui = δi +2β βi 2δiβ2+2αβkεi δj.
γ j j − j jk γ
(cid:0) (cid:1)
Here βi =δijβ , α2+β2 =1 (with β2 =δ βiβj). The inverse matrix reads
j ij
(2.24) uγ = δj +2βjβ 2δjβ2 2αβkεj δγ.
i (cid:16) i i− i − ik(cid:17) j
Substituting (2.23) and (2.24) into (2.16) and (2.20), we find
(2.25) A = 2 ε βi∂ βj +β ∂ α α∂ β ,
lk lij k l k k l
−
(2.26) A = 2(cid:0)ε βi∂ βj +β ∂ α α∂ β (cid:1).
lt lij t l t t l
−
(cid:0) (cid:1)
For small β, the above formulas can be linearised, so that A 2∂ β , A
lk k l lt
≈ − ≈
2∂ β , and the linearised model is described by the Lagrangian
t l
−
(2.27) L 4β˙2 4λ (divβ)2 2λ (curlβ)2.
1 2
≈ − −
Here as usual divβ =∂ βi, and (curlβ) =∂ β ∂ β , etc.
i 1 2 3 3 2
−
2.4. The nonlinear equations of motion. The complete nonlinear equations
are more nontrivial. Denote the derivatives
∂Lpot ∂Lkin
(2.28) Hlk = , Hlt = .
∂A ∂A
lk lt
Then the field equations read
(2.29) (∂ Hit ∂ Hik)P +2(HitQ HikQ )=0,
t k ij tij kij
− −
where we introduced
1
(2.30) P =ε βl+ (δ δ β2+β β ),
ij ijl ij ij i j
α −
1
(2.31) Q = ε (δ β δ β ) ∂ βl,
tij ijl ij l il j t
(cid:20) − α − (cid:21)
1
(2.32) Q = ε (δ β δ β ) ∂ βl.
kij ijl ij l il j k
(cid:20) − α − (cid:21)
The equations (2.29) are valid for any Lagrangian L = Lkin(A ) Lpot(A )
lt lk
−
with an arbitrary dependence on the variables A ,A . However, for the specific
lt lk
choice of the quadratic Lagrangian(2.21) and (2.22), we have explicitly
(2.33) Hit =2Ait, Hik =2λ Al δik+2λ A[ik].
1 l 2
The tensor (2.30) is invertible and the inverse reads
(2.34) (P−1)jk =αδjk εjknβ .
n
−
One can easily check that P (P−1)jk = δk. Multiplying (2.29) with this inverse,
ij i
we obtain another convenient form of the fields equations:
(2.35) ∂ Hit ∂ Hik+2(HjtG i HjkG i)=0,
t k tj kj
− −
A GAUGE THEORETIC APPROACH TO ELASTICITY WITH MICROROTATIONS 9
where we have introduced
(2.36) G i = Q (P−1)li =ε il∂ (αβ )+βi∂ β β ∂ βi,
tj tjl j t l t j j t
−
(2.37) G i = Q (P−1)li =ε il∂ (αβ )+βi∂ β β ∂ βi.
kj kjl j k l k j j k
−
Note that both objects are antisymmetric in i,j.
The resulting system of nonlinear differential equations (obtained after substi-
tuting (2.33), (2.30)-(2.32) and (2.25), (2.26) into (2.29)) is quite nontrivial. It is
possible, however,to find simple solutions under the additional assumptions.
3. Spherically-symmetric solutions – the soliton
Letus nowlookforthe configurationswith a3-dimensionalsphericalsymmetry.
The corresponding ansatz reads
xi
(3.1) βi = cosw, α=sinw,
r
where the scalar function w = w(t,r) depends on time and on the radial variable
r=√x xi.
i
Then (2.25) and (2.26) yield
xi x x δ x x
(3.2) A = 2 ε cos2w+ l k w′ sinwcosw lk l k ,
lk (cid:20) lik r2 r2 − (cid:18) r − r3 (cid:19)(cid:21)
x
l
(3.3) A = 2 w˙.
lt
r
Hereafter the dot and the prime denote the derivatives with respect to time and
radius, respectively.
Accordingly, the trace and the skew-symmetric parts of (3.2) are
sinwcosw
(3.4) Ak = 2w′ 4 ,
k
− r
xn
(3.5) A = 2ε cos2w.
[lk] − lkn r2
As a result, we find
4xi
(3.6) Hit = w˙,
r
sinwcosw x
(3.7) Hik = 2λ δik 2w′ 4 4λ εikn n cos2w.
1 (cid:18) − r (cid:19)− 2 r2
It is straightforwardto see that
4xi
(3.8) ∂ Hit = w¨, HjtG i =0,
t tj
r
′
sinwcosw x x sinwcosw
G i = ε i +ε il l k
kj kj r j r2 (cid:18) r (cid:19)
cos2w
(3.9) +(xiδ x δi) .
kj − j k r2
Using these results, we find that the field equations (2.35) reduce to
4xi 2 1
′′ ′
(3.10) w¨ λ w + w U(w) =0,
r (cid:20) − 1(cid:18) r (cid:19)− r2 (cid:21)
10 C.G.BO¨HMERANDYU.N.OBUKHOV
where
(3.11) U(w)=sin(2w)[(λ λ )+(λ 2λ )cos(2w)].
2 1 2 1
− −
SincethetwotermsintheroundbracketsaretheLaplacianinsphericalcoordinates,
thisequationofmotioncanalsobe writtenintheneatformw¨ λ ∆w =U(w)/r2.
1
−
Let us introduce a new function ϕ := wr, and next let us rescale r √λ t,
1
7→
ϕ √λ ϕ, then (3.10) becomes
1
7→
U(ϕ/r)
′′
(3.12) ϕ¨ ϕ + =0.
− λ3/2r
1
The resulting equation is closely related to the spherical ‘sine-Gordon’ equation,
seeforinstance[10,53]andreferencestherein. Itisworthwhiletomentionthatthe
model of Kontorova-Frenkel [37, 20, 5] was the first theory discussing media with
dislocations where the ‘sine-Gordon’ equation and its soliton solutions naturally
appeared. The main difference between previously studied spherical sine-Gordon
equationsandourequation(3.12)is thatthe nonlinearitycarriesanextrafactorof
1/r. ThisissimilartoangularmomentumwhenstudyingthesphericalSchr¨odinger
equation or Newton’s equation. This additional factor has some interesting impli-
cations. Atlargedistancesfromthecentrethistermbecomesnegligibleandasymp-
totically we recoverthe wave equations. This analysisrenders a strong support for
the existence of soliton solutions in our model with a localised configuration near
the centre. Below we confirm this by numerical integration.
wHrL
1.0
0.8
0.6
0.4
0.2
r
0 10 20 30 40 50
Figure 1. Soliton solution of the rotational elasticity.
In the static case, the equations of motion reduce to the single ODE
(3.13) λ (r2w′)′+U(w)=0.
1
For λ =2λ this further reduces to (r2w′)′+sin(2w)=0, whereas when λ =λ
2 1 2 1
one is left with (r2w′)′ 1sin(4w) = 0. The qualitative analysis of the equation
− 2
(3.13) reveals the existence of static solutions that vanish at r = 0 and approach
asymptotically π/4 at infinity for r .
→∞