Short hstorcal notes Dslocatons: V. Volterra (905) K. Kondo (95) J. F. Nye (953) B. Blby,. Bullough, E. Smth (955) E. Kröner, I. Dzyaloshnsk, G. Volovk, M. Kléman, I. Kunn, J. Madore, A. Kadć, D. Edelen, H. Klenert, E. Bezerra de Mello, F. Moraes,. Malyshev, M. Lazar,... Dsclnatons: F. Frank (958) I. Dzyaloshnsk, G. Volovk (978) J. Hertz (978) Torson: E. artan (9) Torson n gravty: A. Ensten, E. Schrödnger, H. Weyl, T. Kbble, D. Scama,. Fnkelsten, F.Hehl, P. von der Heyde, G. Kerlch, J. Nester, Y. Ne eman, J. Ntsch, J. Mcrea, J. Melke, Yu. Obukhov M. Blagojecć, I. Nkolć, M. Vaslć, K. Hayash, T. Shrafuj, E. Sezgn, P. van Neuwenhuzen, I. Shapro,...
3, y =,,3 δ u ε σ j ( ) = ( u + u ) j j j j Notatons Geometrc Theory of Defects M. O. Katanaev Steklov Mathematcal Insttute, Moscow Katanaev, Volovch Ann. Phys. 6(99); bd. 7(999)03 Katanaev Theor.Math.Phys.35(003)733; bd. 38(004)63 Phscs Uspekh 48(005)675. - contnuous elastc meda = Eucldean three-dmensonal space - Eucldean metrc - artesan coordnates - dsplacement vector feld - stress tensor - stran tensor Elastcty theory of small deformatons j σ + f = j 0 j j k j k σ = λδ ε + ε - Newton s law - Hooke s law f ( ) λ, - densty of nonelastc forces - Lame coeffcents ( f = 0)
Dfferental geometry of elastc deformatons y = y + u ( ) y ( y) - dffeomorphsm: y δ 3 3 j g j k l y y gj ( ) = kl j uj ju j j j δ δ = δ ε Γ jk = ( ) 0 g jk + jgk k gj - hrstoffel s symbols - nduced metrc (*) l l m l = Γ Γ Γ ( j) = 0 jk jk k jm - curvature tensor j jk = Γ k - etremals (geodescs) l jk = 0 - Sant-Venant ntegrablty condtons of (*) T = Γ Γ = k k k j j j 0 - torson tensor 3
Dslocatons Lnear defects: 3 Edge dslocaton b 3 b Screw dslocaton b - Burgers vector Pont defects: Vacancy u ( ) s contnuous = elastc deformatons s not contnuous = dslocatons 4
(*) b Edge dslocaton d u = d y = b, =,, 3 y ( ) b = d e = d d ν ( e e ) ν ν S j ν = ν ω ν j ( ν) - torson T e e j j k j ν ν ν k e ( ) y = lm = ω ω ω ( ν) y - arbtrary curvlnear coordnates - s not contnuous! - outsde the cut - on the cut -curvature - Burgers vector n elastcty ω j = ω j (*) - trad feld (contnuous on the cut) SO(3)-connecton ν ν b = d d T - defnton of the Burgers vector n the geometrc theory Back to elastcty: f j j ν = 0 then ω 0 5
Ferromagnets Dsclnatons n ( ) n0 j - unt vector feld - fed unt vector n = n0 Sj ( ω) j S SO(3) j j ω = ω so(3) jk ω = εjkω - orthogonal matr - Le algebra element (spn structure) - rotatonal angle ( ε = ) εjk - totally antsymmetrc tensor 3 Eamples j j Ω = Θ = ε jk d ω Ω jk - Frank vector (total angle of rotaton) Θ = π Θ = 4π Θ = ΘΘ 6
More eamples Nematc lqud crystals n n Θ = π Θ = 3π Model for a spn structure: k j j j j k S = cos + ωε sn + ωω ( cos ) SO(3), = l ω ( ) so(3) - basc varable δ ω ω ω ω ω ω ω ω j k j = ( S ) Sk - trval SO(3)-connecton (pure gauge) j l = 0 - prncpal chral SO(3)-model 7
Frank vector j ω ( ) - s not contnuous! ω j ( ) j ω = j lm ω - outsde the cut - on the cut - SO(3)-connecton (contnuous on the cut) j j ν j j ν ν Ω = d ω = d d ( ω ω ) j j k j ν ν ν k = ω ω ω ( ν) - the Frank vector -curvature j Ω = d ν d j ν - defnton of the Frank vector n the geometrc theory Back to the spn structure: f n then SO(3) SO() 8
Summary of the geometrc approach (physcal nterpretaton) Meda wth dslocatons and dsclnatons = 3 wth a gven emann-artan geometry Independent varables = e j ω - trad feld - SO(3)-connecton j ν ν ν j T = e ω e ( ν) j j k j ν ν ν k = ω ω ω ( ν) - torson (surface densty of the Burgers vector) - curvature (surface densty of the Frank vector) Elastc deformatons: j ν ν = 0, T = 0 Dslocatons: j ν ν = 0, T 0 Dsclnatons: j ν ν 0, T = 0 Dslocatons and dsclnatons: j ν ν 0, T 0 9
The free energy 3 S = d el, e= dete 4 4 jk kj j k 3 L= κ T ( βt + β T + β T δ ) jk jkl klj k jl 3 + ( γ + γ + γ δ ) +Λ jkl k ν j = j j = Tj T e e T T k = = jk Postulate: j k ν The result:,... - trace of torson - cc tensor - scalar curvature equatons of equlbrum admt solutons - transformaton of ndces L= κ γ e () [ j] (, e ω) [ j] = 0, T 0 0, T = 0 = 0, T = 0 [ j] κ, β, β, β 3 γ, γ, γ, Λ 3 - the Hlbert-Ensten acton - couplng constants - only dslocatons - only dsclnatons - elastc deformatons - antsymmetrc part of the cc tensor 0
Elastc gauge ( σ ) Δ u + u = 0 e j σ = δ j λ ( λ + ) u - the elastcty equaton - Posson rato - the lnear appromaton ( σ ) e + e = 0 - the elastc gauge (fes dffeomorphsms) Lorentz gauge j ω = 0 - the Lorenz gauge (fes SO(3)-nvarance) If there are no dsclnatons j ν = 0, then j j ω ( ) k j = l = S Sk j l = 0 - prncpal chral SO(3)-model pure gauge
y Wedge dslocaton n elastcty theory z πθ r, ϕ, z u - cylndrcal coordnates { u(),() r v r ϕ,0} = - dsplacement vector Boundary condtons: u = 0, u = 0, u = r, u = 0 r r= 0 ϕ ϕ= 0 ϕ ϕ= π πθ r r r= vr () u r( r ru) = D - elastcty equatons r σ D = θ, σ - Posson rato σ D c u = rln r+ cr +, c, = const - a general soluton r σ r σ r dl = + θ ln dr + r + θ ln + θ dϕ σ σ σ nduced metrc θ, r = θ r θ - defct angle
Wedge dslocaton n the geometrc theory y πθ θ - defct angle α = + θ dl = df + f dϕ - metrc for a concal sngularty α (eact soluton of 3D Ensten eqs.) Where s the Posson rato σ??? The elastc gauge: ( σ ) e + e = 0 e = u For t reduces to elastcty equatons: ( σ ) Δ u + u = 0 j j ( n ) r α r dl = dr + dϕ n - eact soluton of the Ensten equatons n the elastc gauge n = θσ + θ σ + 4( + θ)( σ) ( σ ) 3
omparson of the elastcty theory wth the geometrc model ( n ) r α r dl = dr dϕ + n - the geometrc model σ θ, n + θ ( σ ) σ r σ r dl = + θ ln dr + r + θ ln + θ dϕ σ σ σ - the elastcty theory The result of the elastcty theory s vald only for small defct angles and near the boundary θ r The result of the geometrc model s vald for allθ and everywhere lnduced metrc components defne the deformaton tensor and can be measured epermentally 4
oncluson Geometrc theory of defects????? Elastc gauge Lorenz gauge Elastcty theory Prncpal chral SO(3)-model ) The geometrc theory of defects n solds appears to be a fundamental theory of defects. ) It descrbes sngle defects as well as contnuous dstrbuton of defects. 3) It provdes a unfed treatment of defects n meda (dslocatons) and n spn structures (dsclnatons). 4) In the absence of defects t reduces to the elastcty theory for the dsplacement vector feld and to the prncpal chral SO(3)-model for spn structures. 5