Locating the Focus of a Starting Earthquake

Transcription

1 Intenational Jounal of Geosciences, 01, 5, Published Online Septembe 01 in SciRes. Locating the Focus of a Stating Eathquake Alexande Ivanchin Institute of Monitoing of Climatic and Ecological Systems (IMCES SB RAS), Tomsk, Russia alex_ivanchin@mail.u Received 5 July 01; evised August 01; accepted 15 Septembe 01 Copyight 01 by autho and Scientific Reseach Publishing Inc. This wok is licensed unde the Ceative Commons Attibution Intenational License (CC BY). Abstact This aticle descibes a method of locating the focus of a stating eathquake based on the use of the elastic inteaction enegy. The method allows detemining the focus location and its enegy class as well as evaluating the stesses caused by it and obseving its evolution. Keywods Eathquake Foecasting, Locating the Focus of Eathquake, Stesses of Eathquake 1. Intoduction An eathquake is always an unexpected phenomenon. Moden science is not able to pedict the time o the place o the eathquake stength. The poblem of locating the focus of a stating eathquake has not even been set due to the poo level of undestanding the pocesses peceding its stat. At pesent the main eathquake hypothesis is the explosive elaxation of the high elastic stesses accumulated in the lithosphee. A seious objection to the above hypothesis is the stess elaxation caused by the plastic flow whose ate gows exponentially with tempeatue and stess [1]. In the lithosphee the tempeatue and the pessue incease with the depth and, theefoe, models of the eathquake focus with no elastic stess concentations, such as the Inetial Eathquake Focus (IEF) model, ae moe pobable []-[]. Nowadays, geophysical measuements can be pefomed only on the suface o in the thin subsuface laye of the Eath, so all the eathquake hypotheses ae puely theoetical. Although some data can be obtained fom seismic waves, thei amount is fa fom sufficient, since the waves ae long and povide little infomation. Theefoe, thee emains only one way of locating the focus of a stating eathquake to develop a method using the elastic stess field of a stating eathquake focus. This method is suggested in the pesent wok.. Inetial Eathquake Focus In the lithosphee tectonic plates move and tun elative to each othe. Due to the Eath s otation each plate possesses the moment of momentum M paallel to the axis of otation of the Eath s body [5] How to cite this pape: Ivanchin, A. (01) Locating the Focus of a Stating Eathquake. Intenational Jounal of Geosciences, 5,

2 M = IΩ. Hee I is the moment of inetia, Ω is the vecto of the angula velocity of the Eath s otation. Thee can be two types of change of the angula velocity of otation illustated in Figue 1 by the example of a otating cylinde. One is pecession, when by the action of the extenal moment, that is pependicula to the axis of otation zz, the oientation of the axis of otation in space changes, wheeas the angula velocity value does not change in magnitude. The diection of the axis of otation changes fom the vetical oientation in Figue 1(а) into the hoizontal one, as is shown in Figue 1(b), its position elative to the mateial points of the cylinde does not change, and the extenal moment does not pefom wok. The othe type of change of the angula velocity of otation is shown in Figue 1(c), whee the position of the axis of otation elative to the mateial points of the cylinde changes. Fo a cylinde to change the oientation of the otation axis elative to the points of the body, it is necessay, fist, in the hoizontal otation position of a cylinde (b) to apply the extenal moment of foce diected opposite to the angula velocity vecto Ω 1 and stop the otation. Then, it is necessay to apply the extenal moment of foce diected along the axis zz and make the cylinde otate aound and along the axis (Figue 1(c)). Fo simplicity, let us conside the IEF in the fom of a sphee. As a esult of tectonic movement, the diection of axis zz will deviate by an angle fom the Eath s axis of otation. If the tun occued in a vacuum, the diection of the angula velocity would deviate by the angle and its value would be Ω 1. At the same time, the equality Ω 1 =Ω would be satisfied. Howeve, the IEF is located in the solid lithosphee and otates togethe with the Eath, so the angula velocity cannot diffe in value o diection fom the Eath s angula velocity Ω. Hee thee occus a change in the position of the axis of otation elative to the IEF points. Theefoe, fist we stop the IEF otation applying elastic stesses with the moment of foce K inclined at the angle of π + to the Eath s otation axis, with the IEF kinetic enegy wholly tuning into the potential enegy of the elastic stesses occuing aound it in the lithosphee. Then we make the IEF otate eaching the angula velocity Ω applying to it the elastic stess with the moment of foce K diected along the Eath s otation axis. The kinetic enegy of otation completely tuns into the potential enegy of the elastic stesses and vice vesa, which means that the following equality will be satisfied K = K (.1) The esulting vecto is K = K K (.) hence K = Ksin (.) Thus, the IEF gets its own moment of foce K. As is known, the condition of equilibium in the elasticity theoy equies that all the concentated moments be zeo [6]. Howeve, the above condition applies only to in- Ω Ω z Ω 1 z z (a) (b) (c) Figue 1. The oientation change of the otating axis is displayed. 118

3 etial systems, wheeas the otating Eath is not such a system. If the Eath did not otate, thee would be no moment of foce. The stess field of the moment of foce K cannot be educed by plastic elaxation without tuning the IEF as a whole etuning to the initial position. As the angle inceases, the stesses gow and can achieve a citical value, the IEF tuns abuptly deceasing the angle and the stess field, and an eathquake takes place.. The Elastic Field of the Moment of Foce The equation of elastic equilibium fo the displacement vecto U is [6] The defomation tenso is and the stess tenso is ν ( U) ( U ) = 0 (.1) 1 ( ν ) 1 U U i j εij = + xj x i σ = µ ν ε + ( ε + ε + ε ) δ ν i j ij xx yy zz i j Hee µ is the shea modulus, ν is the Poisson coefficient, δ ij is the Konekke symbol. Fo the sphee S of the adius R with the cente in the oigin of coodinates the foce is and the moment of foce is (.) F = σ ds (.) S K = ( σ n)ds (.) S Hee is the adius-vecto, n is the unitay nomal to S. If the moment (.) is not zeo, the egion will be efeed to as the Toque Region (TR). Substituting (.) into (.) we deive ν K = µ ( ε n) ds + µ ( )( ) ds = µ ( ε ) ds ν U n n (.5) S S S since fo a sphee with the cente in the oigin of coodinates the vecto and n ae paallel. Suppose that U is a solenoidal vecto, that is, thee exists the vecto P such that Then (.1) is educed to the equation U = P (.6) P =0 (.7) since the fist tem in (.1) becomes zeo. Fo the moment of foce K in (.5) to be independent of R, it is necessay that U ~1. Theefoe, let us take the vecto P in the fom Hee A is the abitay constant. The vecto (.8) is the solution (.7), then 1 P = A 0,0, (.8) A A U = { yx,,0} = { 0,0,sinϑ} (.9) Hee the displacement vecto is witten fist in the Catesian system of coodinates and then in the spheical one. The sequence of witing the spheical vecto components is as follows: adial U zenith U ϑ and asymuthal U ϕ. The Catesian coodinates ae elated to the spheical ones in the following way x = sinϑcos ϕ, y = sinϑsin ϕ, z = cosϑ 119

4 Diffeentiating (.9) we obtain that U =0 (.10) The potential vecto U poduces a defomation tenso having two components not equal to zeo in the spheical coodinates A ε ϕ = εϕ = sin ϑ (.11) The stess tenso has the same µ A σ ϕ = σ ϕ = µε ϕ = sinϑ (.1) The values (.9), (.11), (.1) ae not zeo only outside the IEF. Let us choose a system of coodinates so that the moment of foce had only one component along the axis z not equal to zeo, then hence π π K = d ϕ σ sin ϑ d ϑ = π µ A 0 0 ( ϕ ) (.1) K A = (.1) π µ If a souce is poduced by inetial effects, that is the IEF, then multiplying (.1) by sin ( ) we deive Similaly, The elastic enegy density will be witten as K = µ (.15) 6π Asin A A U = sin { yx,,0} = sin { 0,0,sinϑ} (.16) A ε ϕ = εϕ = sin ϑ sin (.17) µ A σ ϕ = σ ϕ = µε ϕ = 6 sinϑsin (.18) sin ϑ w= µε ϕ = 18µ A sin 6 The elastic enegy of the stesses of the moment of foce (.) is A W = wdv = d d w sin d = 16π sin V R π π ϕ ϑ ϑ µ 0 0 R (.19) The unknown constant A can be detemined fom two conditions. The fist condition is when the moment of foce is specified, and then the value of A is found accoding to (.1). The second condition is when A is detemined accoding to a specified kinetic enegy of otation. The enegy density of the elastic field (.9) is as follows 9 sin ϑ w= µ A 6 and the enegy will be A wdv V π R (.0) W = = The constant A is detemined fom the condition of the complete change of the kinetic enegy 110

5 W k IΩ = = π ρr Ω 15 into the potential enegy of the elastic stesses (.1). The inetial moment of a sphee is As a esult, we obtain A 5 5 8πρR I = (.1) 15 ρ 15µ Substituting (.) into (.15) we deive the value of the moment of foce The stess field accoding to (.18) looks like K = Ω R (.) ρµ 15 = 6π Ω R sin (.) ΩR 1ρµ σ ϕ = σ ϕ = µε ϕ = sinϑsin. 5 Substituting (.)into (.19) we obtain an expession fo the IEF On the IEF suface the stesses ae σ ϕ W 16 π R ρ sin 15 5 = Ω (.) 1ρµ = Ω R sinϑsin (.5) 5 As is shown in [7], in ode to deive a geneal solution fo the elasticity poblem, a nonpotential solution should be added the potential solution (.16). Howeve, in the Appendix it is shown that the inteaction enegy of the potential and nonpotential solutions is zeo. Addition of the nonpotential solution to the potential one (.16) will incease the system enegy, and fo this eason it is excluded fom consideation.. Inteaction of Focuses. Locating an Eathquake Focus If thee exist two toque egions, thee is enegy of elastic inteaction between them, that can be used to locate the focus of a stating eathquake. Fo this pupose, it is necessay to ceate an atificial TR in the lithosphee applying shea stesses to the suface of the atificial cavity. Let us designate the values elating to the atificial TR by one point at the top (futhe efeed to as the fist TR), and the IEF by two points (futhe efeed to as the second TR). The centes of these focuses ae located on the abscissa at point a fo the fist TR and + a fo the second TR. The plane xxxx is vetical and passes though the Eath s cente, y -coodinate is nomal to it and foms the angle π ψ with the Eath s axis. The diection of y -coodinate is chosen so that the system of coodinates was ight. The vecto potentials of the moment of foce will be P A A = { sin ϑcos ϕ,sin ϑsin ϕ,cos ϑ}, = { sin ϑcos ϕ,sin ϑsin ϕ,cos ϑ} P (.1) Hee ( ), ( ) = x a + x + y = x+ a + x + y. The zenith angles ϑ and ϑ ae measued fom z axis, and the angles ϕ and ϕ fom that of abscissa. The moments K and K coincide in diection with P and P. The laboatoy Catesian system with the oigin in the cente of the fist focus is oiented in the following way (the vaiables elated to it ae designated by the sign ^): the axis ẑ coincides with the vetical at the given point and is diected upwad, the axis ˆx is diected along the meidian to the noth and the axis ŷ is diected along the latitude to the west. The plane xz ˆˆ 111

6 is meidianal, the angle between the nomals of the planes xz ˆˆ and xz, that is between the axes y and ŷ, is ψ. The change to the laboatoy Catesian system of coodinates is implemented by tuning aound axis ẑ by the angle ψ and is descibed by the matix cosψ sinψ 0 Λ = sinψ cosψ The displacement vectos in the laboatoy system of coodinates can be epesented like this ycos ϑ+ zsin ϑsin ( ϕ+ ( x a) ϑ z ϑ ( ϕ ϑ y ( ϕ+ x ( ϕ+ A U = ( ΛP ) = + cos sin cos +, sin cos sin ycos ϑ+ zsin ϑsin ( ϕ+ ( x a) ϑ z ϑ ( ϕ ϑ y ( ϕ x ( ϕ A = ( U ΛP ) = sin cos sin cos +. sin cos + sin + The displacements (.) detemine the defomation tensos A εxx = 5 ( x + a ) y cos ϑ z sin sin ( ), ϑ ϕ+ ψ A εyy = 5 y ( x + a ) cos ϑ+ z sin cos ( ), ϑ ϕ+ ψ A εzz = sin cos 5 z ϑ y ( ) ( x a ) sin ( ), ϕ+ ψ + + ϕ+ ψ A εxy = 5 { ( x+ a) + y cos ϑ+ zsin ϑ ( x+ a) cos( ϕ+ ysin ( ϕ+ }, A εxz = cos 5 yz ϑ ( ) ( ) (( ) ) ( ) { + sin ϑ y x+ a cos ϕ+ ψ + x+ a z sin ϕ ψ + }, A εyz = 5 { ( x+ a) zcos ϑ+ sin ϑ ( z y ) cos( ϕ+ + y( x+ a) sin ( ϕ+ }. A εxx = sin ( x a) ycos ϑ zsin sin 5 ( ), ϑ ϕ+ ψ A εyy = sin y 5 ( x a) cos ϑ+ zsin cos ( ), ϑ ϕ+ ψ A εzz = sin z ycos 5 ( ϕ ( x a) sin ( ϕ, A εxy = 5 sin ( x a) + y cos ϑ+ z sin { ϑ ( x a) cos ( ϕ+ ysin ( ϕ+ }, A εxz = 5 sin { yz cos ϑ+ sin ϑ ( x a) y cos ( ϕ+ ( ( x a) z ) sin ( ϕ+ }, A εyz = 5 sin { ( x a) z cos ϑ+ sin ϑ ( z y ) cos ( ϕ+ + ( x a) y sin ( ϕ+ }. The total defomation tenso is The total elastic enegy density is witten as (.) (.) (.) ε = ε + ε (.5) 11

7 µν w = ε + ε + ε + µ ε + ε + ε + ε + ε + ε ν ( xx yy zz ) ( xx yy zz xy xz yz ) Substituting hee (.5) we find that the total density enegy is the sum of the following components: the density of the enegy of the fist focus the enegy density of the second focus µν w = ε + ε + ε + µ ε + ε + ε + ε + ε + ε ν ( ) ( xx yy zz xx yy zz xy xz yz ) µν w = ε + ε + ε + µ ε + ε + ε + ε + ε + ε ν and the density of the inteaction enegy of the focuses Integating w and is ( ) ( xx yy zz xx yy zz xy xz yz ) µν wi = ε xx ε xx + ε yy ε yy + ε zz ε zz + ε xx ε yy + ε xx ε zz + ε yy ε zz ν ( ) ( xx xx yy yy zz zz xy xy xz xz yz yz ) + µ ε ε + ε ε + ε ε + ε ε + ε ε + ε ε. (.6) w we obtain the focus enegies in the fom (.19). The inteaction enegy of the focuses W = w dv (.7) I V I Fo integating tun to the bipola coodinates in (.7). The elastic enegy density is a scala, and the change to the bipola coodinates is pefomed by eplacing the Catesian coodinates by the bipola ones (.7) using the fomulas [8]: sinh sin cos sin sin sin x = a τ, y = a σ ϕ, z = a σ ϕ, Y = a σ coshτ cosσ coshτ cosσ coshτ cosσ coshτ cosσ The coodinate suface fo τ = const is the adius sphee ( ) (.8) a R = (.9) sinhτ with the cente on the abscissa at the point acoshτ. At τ > 1 the value of coshτ is close to unity and the cente of the coodinate sphee, in fact, coincides with that of the IEF. Fo instance, aleady fo τ = 1.5 the value of coshτ = Theefoe, one can expect with sufficiently geat accuacy that the centes of the focuses and the centes of the coodinate sphees coincide. Fo this eason, the limits of integation ove τ, in accodance with (.9), ae as follows a a τ1 = acsinh, τ = acsinh R R The elastic inteaction enegy (.7) in the bipola coodinates is witten as I τ τ1 π π W = µ dτ dσ wydϕ Hee it is designated 0 0 I. ( ) ( ) πµ AA 1 1 = ηsin 1+ ζ ln ζ + 1+ ζ 1+ ζ + ln ζ ζ. 8a ζ ζ (.10) R R η = cosϑcosϑ cos ( ϕ ϕ) + cos ( ϕ+ ϕ+ sinϑsin ϑ, ζ =, ζ = (.11) a a Since the fist TR is an atificial souce of the moment of foce, the distance to the focusawill always be much lage than its size R, theefoe, 11

8 hence ζ 1 ( ) ζ ln ζ + 1+ ζ ζ ζ Fo the second TR the following elation can be tue ζ ~1, howeve, in both cases, we deal with the equality As a esult, we deive fom (.10) The total enegy will be witten as ( ) ζ ln ζ + 1+ ζ ζ ζ KΩR η sin ρ WI = 8πaR 15µ K ΩR η sin ρ W = W + WI + W = W + + W 8πaR 15µ (.1) Hee, instead of a, the value of s = a is used, which is the distance between the centes of the focuses. Diffeentiating the enegy W with espect to s we obtain a genealized foce acting between the focuses in the fom dw KΩR η ρ f = = sin ds πrs 15µ (.1) If f < 0, then the focuses ae attacted to each othe, and if f > 0, then they epel. The sign depends on the coefficient η fom (.11) The following unknowns ae included hee: R is the IEF size, sin, the distance between the focuses s, and η 5.The values R, ρ, Ω and µ ae known, they ae specified duing an expeiment. Afte measuing the value f fo vaious oientations of the vecto K, we detemine the oientation facto η, that is the IEF oientation. The above method does not allow defining the value R sepaately fom sin ( ). Howeve, we can suppose that the angle will not be vey small, othewise the enegy of the focus of a stating eathquake will not be vey small too, so it may be that sin ( ) will be of the ode of ~1, then the size of the focus of a stating eathquake R and its enegy ae evaluated accoding to (.) and (.1). Making measuements fo vaious oientations one can define the unknown paametes ϑ, ϕ and ψ and hence the IEF oientation. The foce f acting on the fist focus can be consideed as the concentated foce. The displacement field of the concentated foce can be found fom the following consideations. As the value of the concentated foce does not depend on the adius of the integation sphee in (.), so the stess field must decease with the distance fom the point of its application as 1,and then the displacement field will decease as 1. Conside that the concentated foce is diected along the axis z, theefoe, let us take the displacement vecto such as B u = 0,0, x + y + z Hee B is the abitay constant. The above vecto does not satisfy the equation of elastic equilibium (.1), wheeas a check poves that the following equality is satisfied We have ( u) u = = u Bz ( x + y + z ) 11

9 is The potential Ψ satisfying the equation The vecto Ψ = Ψ= ( ν) ( ν) Bz ( x + y + z ) Bz x + y + z 1 B u = u 1 Ψ = 81 + { cosϕsin ϑ,sinϕsin ϑ,7 8ν cos ϑ} is the solution of the Equation (.1). Hee the vecto u is witten in the Catesian system of coodinates. The foce (.) caused by the field u does not depend on the size of the integation sphee S and is f = πµ B (.1) This foce is diected along the line connecting the IEF centes. Equating the foces (.1) and (.1) we obtain the value of the abitay constant B The displacement vecto component witten as KΩR η ρ sin B =. 16π Rs 15µ u z is diected along the line connecting the IEF centes and can be 7 8ν cosϑ u z = B+ B ( ν) ( ν) It consists of two tems. The fist tem is the displacement of the adius sphee as a whole without defomation. The second tem is the defomation component. Substituting the coefficient B with the change of = R into the fist tem, we get the displacement of the fist TR in the fom 7 8ν K ΩR η ρ γ = sin. 81 ν 16π Rs 15µ ( ) Measuing γ fo vaious oientations and at seveal points we obtain a set of equations to detemine the angles ϑ, ϕ, ψ the distance s and the size of the focus R. In this way we detemine the focus location, its oientation and enegy. Fo instance, fo the following paametes: Ω= ad/s, R = 0.1m, a = m, η = 1, ρ = 000 kg/m, µ = 10 N/m, ν = 0., R = 0000 m (such a value of R coesponds to the focus enegy of the ode of J ), we obtain γ ~ 17 mm. The diection of the displacement vecto γ is always oiented to the focus of a stating eathquake. Depending on the sign of the oientation facto η the displacement will be eithe towad IEF o fom it. Measuing γ at seveal points one can define the IEF coodinates and evaluate the enegy level by the value γ. Accoding to (.) and (.5), even fo the most intensive eathquakes of an enegy of the ode of 10 J, when R = m, the maximum stess will be 7 ~ 10 N/m. In the IEF thee is no stess concentation. To locate an IEF, it is necessay to detemine the displacement of an atificial TR with an accuacy up to a millimete and bette. In ode to ceate the moment of foce, one can use pecession. Between the moment of foce K nomal to the moment of momentum M and the angula velocity of pecession ω thee exists the elation [5] K = [ ω M ] The above method can be the simplest one. It allows achieving an abitay oientation of the moment of foce, which is necessay to pefom measuements. 5. Conclusions In the pesent wok a method of locating the focus of a stating eathquake is suggested. The existence of the 115

10 IEF is the effect of the Eath s otation. Its analogue is the Coiolis foce. Cyclones and anticyclones ae caused by the Eath s otation. In the lithosphee they ae analogous to the IEF. Undoubtedly, the IEF evolution takes much longe than a cyclone, and the ates of gas flows in the atmosphee ae by odes of magnitude highe than tectonic ates. Howeve, it should not lead to denying o ignoing inetial effects in the lithosphee. The existence of tectonic plates with localization of elastic stesses is the main hypothesis of eathquake focuses. The easons fo stess localization can be diffeent as well as thei space dependence. If thee exist two egions with a simila space distibution of stesses, then between them thee is the enegy of elastic inteaction. Using the above enegy it is possible to locate the focus of a stating eathquake ceating atificially a egion with a specified stess pofile. Mathematical calculation of othe models of eathquake focuses will be somewhat diffeent fom the one discussed hee, which is not a poblem. Refeences [1] Lothe, J. and Hith, J.P. (1967) Theoy of Dislocations. McGaw-Hill Book Company, New Yok. [] Vikulin, A.V. and Ivanchin, A.G. (1997) Seismic Pocess Model. Computing Technologies,, 5-5. [] Vikulin, A.V. and Ivanchin, A.G. (1998) Rotational model of Seismic Pocess. Pacific Geology, 17, [] Vikulin, A.V., Ivanchin, A.G., et al. (01) Data-Pocessing and a Computing System of Simulation of Seismic and Volcanic Pocesses as a Basis fo Studying Wave Geodynamic Phenomena. Compute Technologies,, -5. [5] Landau, L.D. and Lifshitz, E.M. (1960) Mechanics. Pegamon Pess Ltd., Oxfod/London [6] Landau, L.D. and Lifshits, E.M. (1986) Theoy of Elasticity. d Edition, Elsevie Buttewoth-Heimenann, Oxfod. [7] Ivanchin, A. (010) Potential. Solution of Poisson s Equation, Equation of Continuity and Elasticity. [8] Kon, T.M. and Kon, G.M. (1968) Mathematical Handbook fo Scientists and Enginees. McGaw-Hill Book Company, New Yok. [9] Ivanchin, A. (008) Nonpotential Solution of the Electon Poblem

11 Appendix. Nonpotential Solution As is known, along with the potential solution thee can also be a nonpotential one [7] [9]. A geneal solution will be thei linea combination, whose coefficients ae detemined fom the condition of the total enegy minimum. To deive the nonpotential solution take the vecto The oto is One must find the solenoidal vecto G, such that then the vecto U + = 0,0, z + z U = 5 { yx,,0} (6.1) + z G = U = 5 { y, x,0}, is the solution (.1). Fo simplification, let us integate (6.1) ove z : Hee it is designated U + 1 = U ν G (6.) + 1 Ψ = U d z = { y, x,0} (6.) { } Ψ = Ψx, Ψ y,0 = G dz. Since G is the solenoidal vecto, that is G 0, the following equality takes place G = G. Poceeding fom (6.) it is necessay to solve two equations Integate (6.) ove y and (6.5) ove Hee it is designated In the spheical coodinates we have Its solution is y Ψ x = (6.) x Ψ y = (6.5) x. As a esult, we will deive, pactically, the same equation 1 1 Φ 1 =, Φ = Φ = Ψ d y, Φ = Ψ dx 1 x 1 Φ1 1 1 Φ 1 =, = Φ =, Φ = Diffeentiating we obtain the components of the vecto Ψ in the fom Φ1 y Φ x Ψ x = =, Ψ y = =, Ψ z = 0 y x y (6.6) 117

12 The vecto divegence is Ψ 0. which means that the vecto Ψ is solenoidal. Diffeentiating (6.6) ove z we find hence z G = { y, x,0}, z G = 5 { xyz,, }. The nonpotential solution (6.) is witten in Catesian and spheical coodinates as ( ν ) ( ) ( ν ) { x( z ), y( z ), z ( ν 1) z } + 1 z U = U G = 5 ν 1 1 ν + ν 5 cosϑ =,sin ϑ,0. 1 In the defomation tenso fo the displacement U the component ε ϕ = 0 and in the defomation tenso (.17) only this component is othe than zeo. It means that the inteaction enegy of the potential and nonpotential solutions is zeo. Hence, the elastic enegy minimum is achieved using the potential solution (.9). 118

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