Structural Geology (The Journal of the Tectonic Research Group of Japan), No. .0, ,3ῌ-,, ,**,

Orowan’s Formula, Di#erential Geometry
and Four-dimensional Material Space

C along the curve : A῔B῔C as follows
A῏dA῏d῍A῏dA῎.
A῔B

Kazuhito Yamasaki
Received ,2 February ,**,.
Department of Earth and Planetary Sciences, Faculty ofNext, we map point A to the Euclid tangent
plane in point C along the curve : A῔D῔C asfollows
A῏dA῏d῍A῏dA῎.

Science, Kobe University, Kobe, 0/1ῐ2/*+, JapanA῔DKey words : Owowan’s formula, Di#erential geometry, dislocation, disclination, Four-dimen-sionals spaceB῔CD῔CTherefore, when we map point A to the Euclid
tangent plane in point A again along the closed
curve : A῔B῔C῔D῔A, we obtain (Fig. + (b))
A῏dA῏d῍A῏dA῎ῐ῍A῏dA῏d῍A῏dA῎῎A῔B῔CIntroduction
In this paper, we explain the di#erential geometry description of continuum mechanics
paid attention by theoretical geosciences inrecent years (e.g., Yamashita and Teisseyre, +33.
; Teisseyre, +33/ ; Takeo and Ito, +331 ; Yamasaki and Nagahama, +333 a ; Nagahama and Teisseyre, ,**+). To explain the theory concisely,
we do not expand the general frame of thetheory. Instead, we take up the concrete phenomenon described by the well-known equationsuch as the rheology of rocks described by
Orowan’s formula :.
ῌeῒrbv
where e is strain, r is scalar density of dislocations, b is Burgers vector and v is velocity ofdislocations. The purpose of this paper is toderive Orowan’s formula from a viewpoint of
the di#erential geometry.
Di#erential geometry descriptionof defect ﬁeld
One of the most famous correspondences between continuum mechanics and di#erential geometry is the deﬁnition of strain that is given by
the di#erence of metric tensors : eijῒ(dijῐgij)/,.
It has been a center problem in di#erential geometry analysis of continuum mechanics toextend this correspondence even to the defectﬁeld. In this section, we explain concisely thisresult by considering the virtual circuit in crystal lattice.
Consider the closed curve : A῔B῔C῔D῔A,
where AῒA(u), BῒA(u῏du), CῒA(u῏du῏du῏ddu), DῒA(u῏du) and dduῒddu (Fig. + (a)). We
map point A to the Euclid tangent plane in pointC῔D῔AῒddAῐddAῒDA.
῍Now, when the base vector is set to be Ai, weobtain dAῒAi dui. Because dAi is also vector anddAi῔* for dui῔*, it can be expand by using basevectors as follows : dAjῒG ijk duk Ai, where G ijk is
a$ne connection. In this case, ῍ becomesDAῒddAῐddAῒd῍Ai dui῎ῐd῍Ai dui῎ῒ῍ddui῏G ijk duj duk῎Aiῐ῍ddui῏G ijk duj duk῎Ai
ῒ῍G ijk-G ikj῎duj duk Ai.
Because G ijkῒG ikj in Euclid space (or Riemannspace), the discrepancy, DA, is zero. On the other
hand, in the space with torsion, we have G ijkῐ
G ikjῒ,G i[jk]ῒ,Sijkΐ*, where Sijk is called torsion
tensor. Therefore, the discrepancy is expressedas῎DAῒSijk emjk dSm Aijk
where d῕m (ῒemjk du du /,) is inﬁnitesimal area
enclosed with closed curve. ekmn is Levi-Civitatensor. We have ekmnῒ* whenever two of theindicies coincide, and otherwise is given by ekmn
ῒῑ+ if sgn (k, m, n)ῒῑ+. For instance, e++,ῒ*, e+,ῒ+ and e+-,ῒῐ+. The discrepancy vector in thedirection of l is given by῏DAῒSljk emjk dSm.
In crystal lattice, we can obtain similar discrepancy called Burgers vector. The curve thatcloses in a perfect lattice (Fig. + (c)) does not
close in the lattice where the dislocation exists(Fig. + (d)). (When strictly saying, the discrepancy is deﬁned by the naturalization in which
non-Euclid material space with dislocations ismapped to Euclid material space without dislocations (Fig. ,).) This discrepancy increases inproportion to the dislocation density and thearea enclosed with the curve. Therefore, Burgers vector, bl, is given by the product of dislocation density tensor, aml, and inﬁnitesimal areaῌ 29 ῌ

Kazuhito Yamasaki

Fig. ,
Consider the material space with a dislocation.
First, take out the element that is small enough to be freefrom the dislocation. Second, the element is transformedelasticity to the eigenstrain-free state by cutting o# fromthe surrounding and releasing it from the constraints ofthe surrounding. Finally, we reconstruct the material
space by using these eigenstrain-free elements.
This
virtual process is called naturalization. Geometrically, thisprocess is equivalent to the mapping of the non-Euclid
space with torsion into the Euclid space. Burgers vector isdeﬁned as the discrepancy due to the naturalization.

Fig. +(a) Consider the closed curve ABCDA in the spacewith torsion. The area enclosed with this closed curve isinﬁnitesimal enough, and we set AB῏CD῏d and BC῏DA
῏d. (b) When this closed curve is mapped to the Euclid
space, it does not close as shown in this ﬁgure. This is a
result of the path dependency in the space with torsion,that is, the mapping result is di#erent according towhether the path ABC passes or the path ADC passes even
if the destination is the same point C. (c) In the perfect
crystal where the dislocation does not exist, the closed
curve can be drawn in an arbitrary region. (d) When thedislocation exists, there is a region where the closed curve
cannot be drawn. This quantity of the discrepancy iscalled a Burgers vector.

enclosed with the curve, dsm :῏bl῏aml dsm.
mlFirst and the second index of a is the directionof dislocation line and Burgers vector, respectively. Therefore, if l῏m, aml is a screw dislocation density, and if lῐm, aml, is an edge dislocation density. From the comparison between ῎and ῏, we can obtain the relations such as DAlῌbl, Sljk emjkῌaml and dSmῌdsm. In a word, thedislocation density is expressed geometrical by
the torsion of the material space.
Geometrical description of dislocation density
was derived by the mapping of the point. Next,
we consider the mapping of the vector (Fig. - (a))
in order to derive the geometrical description ofdisclination density. Here, a disclination is ageneric name of rotating defects (a dislocations
is a generic name of translational defects.). By
the similar procedure to derive ῍, we haveῐDAj῏ddAj῎ddAj῏Rijkh duk duh Ai
where Ri jkh (῏(h G i jk ῎(k G i jh ῍G l jk G i lh ῎G l jh G i lk)
is curvature tensor (Fig. - (b)). In crystal lattice,
we can obtain similar discrepancy of vector
called Frank vector. In the parallel shift in theperfect crystal, the direction of the vector turns
to the same direction in the starting point andthe terminal (Fig. - (c)). On the other hand, it is
not for the same direction when the disclination
exists (Fig. - (d)). This discrepancy of the vector
direction is called Frank vector. This discrepancy is proportional to the disclination density andthe area of the closed surface as well as theBurger vector. In addition, it is also proportional to the size of the vector. Then, we have
ῑfj῏qjim Ai dSm
imwhere fj is Frank vector, qj is disclination density. From the comparison between ῐ and ῑ,
we have emkh Rijkhῌqjim, that is, the disclination
density is expressed geometrical by the curvature of the material space. In the ordinal analysis, the degenerated disclination density, eiml qjim῏qjl, is used.
Neither the dislocation density nor the disclination density are independent as known in
the analysis of the liquid crystal. Next, weconsider what kind of form the relation shouldbe. The point of the above-mentioned analysis
was to have expressed the physical quantities
such as dislocations and disclinations by thegeometrical quantity such as torsion and curvature of the material space. The curvature andthe gradient of the torsion are in not independence but the following relations :ῒRa[bcd]῏([b Sacd]
where we ignore the higher order terms of thea$ne connection. ῒ is called the ﬁrst Bianchiidentity that is one of the basic relation in thedi#erential geometry. The substitution of Sacd
ebcd῏aba and ebcd Rabcd῏qbab into ῒ leads toΐ
qbab῏(b aba.
This means that the divergence of dislocations

ῌ 30 ῌ

Structural Geology No. .0

Orowan’s Formula, Di#erential Geometry and Four-dimensional Material SpaceFig. (a) Consider the parallel shift of the vector along
the closed curve in the space with curvature. (b) When
this closed curve is mapped to the Euclid space, theparallel shift of the vector is accompanied by thediscrepancy in the terminal point. (c) In the parallel shiftof the perfect crystal where the disclination does not exist,
the direction of vector turns to the same. (d) When thedisclination exists, the direction is not the same. This
discrepancy of the vector direction is called a Frankvector.

is accompanied by disclinations. Disclinations
does not exist in the ordinal material, so we havethe well-known preservation law of the dislocation : (b abaῑ*. Note that ῎ is only the physicalexpression of the basic relation in di#erential
geometry ῍.
Orowan’s formula and Cartan’s structure equation in Four-dimensional material space
The ﬁrst Bianchi identity ῍ can be derived by
the Cartan’s structure equation. Abstract expression of this equation is῍torsion tensor῎ῑ῍di#erential operator῎῍metric῎.
Because the torsion and the metric corresponds
to dislocation density and strain, this becomes῏aijῑeikl (k bli.
For detailed proof, refer to Yamasaki and Nagahama (+333 b) etc. ῏ mean that the dislocation
density is given by the rotation of the strain. If
the distortion is integrable, it is given by thegradient of the displacement, then we have aijῑeikl (k (luiῑ*, that is, the dislocation density vanishes. In other words, the dislocation density isgiven by the non-integral term (non-holonomyterm) of the strain. In the same way, the disclination density is given by the non-holonomy

term of the bend-twist that is the gradient ofrotational displacement. (In Taylor-Bishop-Hillmodel, all the non-holonomic terms vanish.)Because ῏ is the basic equation in describing
dislocation ﬁeld, it plays an important role incondensed matter physics (e.g., Edelen and Lagoudas, +322 ; Kleinert, +323), seismology (e.g.,Yamashita and Teisseyre, +33. ; Takeo and Ito,+331) and geodesy (e.g., Yamasaki and Nagahama, +333 a). However, ῏ does not include thee#ect of time, so we cannot derive the evolutional equation such as Orowan’s formula ῌ.
Then, in this section, we try to add time base to
the coordinate system where the dislocationﬁeld is described. As known by special theory ofrelativity, if the velocity of light is multiplied bytime base, it becomes the dimension of the space,so the spacetime can be described as the fourmanifold. Similarly, by multiplying the velocityof elastic wave by time base in the material
space, the four-dimensional description of deformation ﬁeld becomes possible. In the dislocationﬁeld, the dislocation current corresponds to thevelocity, so we haveῑῐDiῑIiῒdT῏ai
where Di is the four-dimensional dislocation ,-form, Ii(ῑIiA dxA) is the dislocation current +-form, ai(ῑaiA dSA) is the dislocation density ,-form and ῒ is the wedge product. In this paper,
we do not go deep into the explanation of thedi#erential formalism. For detailed proof, refer
to Edelen and Lagoudas (+322). In the same way,
we can deﬁne the four-dimensional distortion(strain):ῒῐBiῑniῒdT῏biwhere Bi is four-dimensional distortion +-form, ni
is the velocity *-form and bi(ῑbiA dxA) is thedistortion +-form. It is followed from ῏ that thedi#erentiation of distortion is accompanied by
the dislocation density. Then, we derive fourdimensional dislocation density as the di#erentiation of four-dimensional distortion :ΐῐDiῑdBi.
ΐ is done by comparing(The strict deriving of ῐthe form numbers of the distortion and the dislocation. See Yamasaki and Nagahama (+333 b).)ῑ and ῐῒ, the concrete form of ῐΐ is given
From ῐby῔ῐ
IiῒdT῏aiῑ῍dS niῐ(t bi῎ῒdT῏dS bi
where dῑdtῒ(t῏dS and dS is the di#erentiation

ῌ 31 ῌ

Kazuhito Yamasaki

῏ is dividof space. By comparing coe$cients, ῎
ed into the spatial component and the time component :ai῍dS biῐ῎ῑ῎Ii῍dS ni-(t bi.
Because the di#erentiation of +-form correῐ issponds to the rotation in vector analysis, ῎another expression of ῍. Dislocation current isgiven by the product among scalar density, dislocation velocity and Burgers vector. Therefore,
it is understood that the special one that theῑgradient of the velocity is disregarded in ῎
corresponds to Owowan’s formula ῌ. That is,
Orowan’s formula is nothing but the equationthat looks at the physical expression of Cartan’s
structure equation from time base.

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Acknowledgements

theory of defects and gravity anomaly. Acta Geophys. Pol., .1, ,-3ῒ,/1.

The motive for writing this manuscript is recommended of K. Michibayashi. I wish to express
my sincere thanks to him.

Yamasaki, K. and Nagahama, H., +333 b, Hodge dualityReferrences

Yamashita, K. and Teisseyre, R., +33., Continuum

and continuum theory of defects. J. Phys. A :Math. Gen., -,, L.1/ῒL.2+.
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