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Aeronautics and Aerospace Open Access Journal

Technical Paper Volume 2 Issue 2

Exact solutions of one problem of nonlinear theory of elasticity for two potentials of energy of incompressible material

Andreeva Julia Yuryevna, Zhukov Boris Alexandrovich

Volgograd State Technical University, Volgograd

Correspondence: Zhukov Boris Alexandrovich, Volgograd State Technical University, Volgograd State Social and Pedagogical University, Volgograd

Received: December 11, 2017 | Published: April 16, 2018

Citation: Yuryevna AJ, Alexandrovich ZB. Exact solutions of one problem of nonlinear theory of elasticity for two potentials of energy of incompressible material. Aeron Aero Open Access J. 2018;2(2):88-93. DOI: 10.15406/aaoaj.2018.02.00036

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Abstract

In the nonlinear theory of elasticity, which is an apparatus for the study of rubber-like materials, at present there is no single equation of state for such materials, similar to Hooke's law in linear theory. The development of new rubber-like materials, the study of various biological materials leads to the need to create new models of response to external influences. In the framework of hyper elasticity, this leads to the appearance of new mathematical expressions for the deformation energy potential. In commercial packages designed to calculate the stress-strain state of structures from elastomers, the expressions for the deformation energy potential are rather rigidly defined. To accommodate new models, you have to create new application packages. Their verification is carried out using known exact solutions. In this paper, two exact solutions of one model problem for deformation energy potentials proposed by Fung1 and Gent et al.2, for which we could not find solutions in the literature. The practical significance of exact solutions is not limited to their application for the verification of numerical methods; they allow one to investigate effects not described by linear theory. It is shown that the difference in potentials leads not only to quantitative, but also to qualitative difference of solutions.

Keywords: final cylindrical deformation, hyper elastic incompressible material, deformation energy potentials, exact solutions

Introduction

The notation and terminology of the article follow the adopted in.3 Rubber is a unique structural material, due to which it has a lot of engineering applications.4 It has a high elasticity in a wide range of temperatures, resistance to aggressive media and hard radiation, which can be used as seals in space vehicles, chemical production. It has excellent damping properties, which contributes to use as shock absorbers, in particular cylindrical shock absorbers. Rubbers are used in the manufacture of solid fuels and in biomechanics. Biological materials5 have similar properties (high elasticity), therefore, rubber is widely used in prosthetics. Precise prediction of the mechanical behavior of rubber is an important task that is still far from full understanding. At the moment, many models are known which are in good agreement with experiment in one type of loading and poorly with another for specific materials. In6, 32 known models of hyper elastic isotropic incompressible materials are presented and new ones are proposed. In7 one can find 12 more models not mentioned in.6 Therefore, it is important to find exact solutions within the framework of different potentials for a reliable study of the properties of materials modeled by these potentials. Since the problem of antiplane deformation is one of the simplest, a lot of work has been devoted to it both by domestic and foreign authors. The results are outlined in the chapters of the monographs.8-10 We note that for incompressible materials a finite antiplane deformation is possible only in bodies with deformation energy potentials satisfying the conditions formulated.

It is always possible in materials with a generalized Neo-Hukonian potential. For potentials of the type, antiplane deformation is possible, for example, in the material of Mooney-Rivlin and some others. Here are the principal invariants of the right Cauchy-Green deformation tensor. In the general case, a cylindrical deformation is realized,11 that is, a complex of flat and antiplane deformations. The planar deformation acts in the transverse plane for the antiplane deformation of the plane. If external factors for plane deformation are absent, and it appears as a reaction to an antiplane shift, then it is called latent.12 If flat deformation is forbidden kinematically (as in shock absorbers), then a flat stress field appears.13,14 A review of the state of studies of finite antiplane deformation up to 1995 is contained in.15 A systematic investigation of the finite antiplane deformation begins with the works of Knowles JK.16-23 More recent works are associated with the names Gao,24,25 Hayes et al.26, Polignone et al.27, Pucci et al.28 Cylindrical deformation was considered in.13,14 In the present paper, two solutions are given, one for a material with a generalized non-wick potential deformation energy,1 for which the latent stress field in the transverse plane does not occur, and for a material with potentials,2 in which such a field arises. The effect of the latent stress field on the concentration of the deformation energy potential is discussed.

Statement of static problems for finite cylindrical deformation

The general statement of the problems of nonlinear elasticity is reduced to a form convenient for further consideration.11 We orient the spatial Cartesian coordinate system so that the radius vectors of the points in the current and reference configurations, respectively, have the form:

x=Xi+Yj+[ Z+w( X,Y ) ]k,X=Xi+Yj+Zk MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaaMc8 UaaGPaVlaahIhacqGH9aqpcaWGybGaaGPaVlaahMgacqGHRaWkcaWG zbGaaGPaVlaahQgacqGHRaWklmaadmaakeaajugibiaadQfacqGHRa WkcaWG3bWcdaqadaGcbaqcLbsacaWGybGaaiilaiaadMfaaOGaayjk aiaawMcaaaGaay5waiaaw2faaKqzGeGaaC4AaiaacYcacaaMc8UaaG PaVlaaykW7caWHybGaaGPaVlabg2da9iaadIfacaWHPbGaey4kaSIa amywaiaahQgacqGHRaWkcaWGAbGaaC4AaiaaykW7caaMc8UaaGPaVl aaykW7aaa@65D9@ (1.1)

where ( X,Y,Z ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWcdaqadaGcba qcLbsacaWGybGaaiilaiaadMfacaGGSaGaamOwaaGccaGLOaGaayzk aaaaaa@3C46@ - coordinates of points in the reference configuration, selected as material. { i,j,k } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWcdaGadaGcba qcLbsacaWHPbGaaiilaiaaykW7caWHQbGaaiilaiaaykW7caWHRbaa kiaawUhacaGL9baaaaa@4042@ - basis of the spatial Cartesian coordinate system. Expressions (1.1) ensure that the deformation is anti-flat.

Following,8 the stress state of a homogeneous isotropic hyper elastic incompressible material with a deformation energy potential W( I C ,I I C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGxb WcdaqadaGcbaqcLbsacaWGjbWcdaWgaaqaaKqzadGaam4qaaWcbeaa jugibiaacYcacaWGjbGaamysaSWaaSbaaeaajugWaiaadoeaaSqaba aakiaawIcacaGLPaaaaaa@41BA@ can be described with the aid of the Cauchy tensor of the true stresses T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWHub aaaa@3781@ in the form:

T=2{ [ W I I C ww+p 1 0 ]+[ W I C + W I I C ]( wk+kw )+ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWHub Gaeyypa0JaaGOmaSWaaiqaaOqaaSWaamWaaOqaaKqzGeGaeyOeI0Yc daWcaaGcbaqcLbsacqGHciITcaWGxbaakeaajugibiabgkGi2kaadM eacaWGjbWcdaWgaaqaaKqzadGaam4qaaWcbeaaaaqcLbsacqGHhis0 caWG3bGaaGPaVlabgEGirlaadEhacqGHRaWkcaWGWbGaaCymaSWaaS baaeaajugWaiaaicdaaSqabaaakiaawUfacaGLDbaajugibiabgUca RSWaamWaaOqaaSWaaSaaaOqaaKqzGeGaeyOaIyRaam4vaaGcbaqcLb sacqGHciITcaWGjbWcdaWgaaqaaKqzadGaam4qaaWcbeaaaaqcLbsa cqGHRaWklmaalaaakeaajugibiabgkGi2kaadEfaaOqaaKqzGeGaey OaIyRaamysaiaadMealmaaBaaabaqcLbmacaWGdbaaleqaaaaaaOGa ay5waiaaw2faaKqzGeGaaGPaVVWaaeWaaOqaaKqzGeGaey4bIeTaam 4DaiaaykW7caWHRbGaey4kaSIaaC4AaiabgEGirlaadEhacaaMc8oa kiaawIcacaGLPaaajugibiabgUcaRaGccaGL7baaaaa@78DB@

s [ W I C ww+p ]kk } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWcdaGacaGcba WcdaWadaGcbaWcdaWcaaGcbaqcLbsacqGHciITcaWGxbaakeaajugi biabgkGi2kaaykW7caWGjbWcdaWgaaqaaKqzadGaam4qaaWcbeaaaa qcLbsacqGHhis0caWG3bGaeyyXICTaey4bIeTaam4DaiabgUcaRiaa dchaaOGaay5waiaaw2faaKqzGeGaaGPaVlaaykW7caWHRbGaaC4Aaa GccaGL9baaaaa@521D@      (1.2)

Across 1 0 =ii+jj,1= 1 0 +kk MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaaCymaSWaaSbaaeaajugWaiaaicdaaSqabaqcLbsacqGH 9aqpcaWHPbGaaCyAaiabgUcaRiaahQgacaWHQbGaaiilaiaaykW7ca aMc8UaaCymaiabg2da9iaahgdalmaaBaaabaqcLbmacaaIWaaaleqa aKqzGeGaey4kaSIaaC4AaiaahUgaaaa@4FA1@  denotes the unit tensor, p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaamiCaaaa@3BC4@ -the hydrostatic pressure function =i/X+j/Y+k/Z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqGHhi s0cqGH9aqpcaWHPbGaaGPaVlabgkGi2kaac+cacqGHciITcaWGybGa ey4kaSIaaCOAaiaaykW7cqGHciITcaGGVaGaeyOaIyRaamywaiabgU caRiaahUgacaaMc8UaaGPaVlabgkGi2kaac+cacqGHciITcaWGAbaa aa@5110@ is the Hamiltonian operator in the reference configuration.

The principal invariants of the right Cauchy-Green deformation tensor are obtained in the form:

I C =I I C =3+ww,II I C =1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaamysaSWaaSbaaeaajugWaiaadoeaaSqabaqcLbsacqGH 9aqpcaWGjbGaamysaSWaaSbaaeaajugWaiaadoeaaSqabaqcLbsacq GH9aqpcaaIZaGaey4kaSIaey4bIeTaam4DaiabgwSixlabgEGirlaa dEhacaGGSaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGjbGaam ysaiaadMealmaaBaaabaqcLbmacaWGdbaaleqaaKqzGeGaeyypa0Ja aGymaaaa@5CF8@    (1.3)                        

Equilibrium equations in the absence of mass forces will be written in the form:

{ p=( W I I C ww ) [ ( W I C + W I I C )w ]=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWcdaGabaqcLb saeaqabOqaaKqzGeGaey4bIeTaamiCaiabg2da9iabgEGirlabgwSi xVWaaeWaaOqaaSWaaSaaaOqaaKqzGeGaeyOaIyRaam4vaaGcbaqcLb sacqGHciITcaWGjbGaamysaSWaaSbaaeaajugWaiaadoeaaSqabaaa aKqzGeGaey4bIeTaam4DaiabgEGirlaadEhaaOGaayjkaiaawMcaaa qaaKqzGeGaey4bIeTaeyyXIC9cdaWadaGcbaWcdaqadaGcbaWcdaWc aaGcbaqcLbsacqGHciITcaWGxbaakeaajugibiabgkGi2kaadMealm aaBaaabaqcLbmacaWGdbaaleqaaaaajugibiabgUcaRSWaaSaaaOqa aKqzGeGaeyOaIyRaam4vaaGcbaqcLbsacqGHciITcaWGjbGaamysaS WaaSbaaeaajugWaiaadoeaaSqabaaaaaGccaGLOaGaayzkaaqcLbsa cqGHhis0caWG3baakiaawUfacaGLDbaajugibiabg2da9iaaicdaaa GccaGL7baaaaa@6F53@    (1.4)

Eliminating the hydrostatic pressure function in the first equation from (1.4), we obtain

×( W I I C ww )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqGHhi s0cqGHxdaTcqGHhis0cqGHflY1lmaabmaakeaalmaalaaakeaajugi biabgkGi2kaadEfaaOqaaKqzGeGaeyOaIyRaamysaiaadMealmaaBa aabaqcLbmacaWGdbaaleqaaaaajugibiabgEGirlaadEhacqGHhis0 caWG3baakiaawIcacaGLPaaajugibiabg2da9iaaicdaaaa@5058@    (1.5)

                                                        

Thus, the strain energy potential must ensure the consistency of the second equation in (1.4) and equation (1.5). The conditions for choosing the potential are given in.16 It is obvious that these conditions are satisfied by the generalized Neo-Hooke potential, since for it W/I I C =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaeyOaIyRaam4vaiaac+cacqGHciITcaWGjbGaamysaSWa aSbaaeaajugWaiaadoeaaSqabaqcLbsacqGH9aqpcaaIWaaaaa@4542@ .

When cylindrical deformation (including anti-flat), the cylindrical lateral surface again passes into a cylindrical surface with a unit vector of the normal < n= n x i+ n y j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaaCOBaiabg2da9iaad6galmaaBaaabaqcLbmacaWG4baa leqaaKqzGeGaaCyAaiabgUcaRiaad6galmaaBaaabaqcLbmacaWG5b aaleqaaKqzGeGaaCOAaaaa@475C@ , so the force boundary condition on the lateral surface 2n[ p 1 0 W I I C ww ]= f pl MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaaIYa GaaCOBaiabgwSixVWaamWaaOqaaKqzGeGaamiCaiaahgdalmaaBaaa baqcLbmacaaIWaaaleqaaKqzGeGaeyOeI0YcdaWcaaGcbaqcLbsacq GHciITcaWGxbaakeaajugibiabgkGi2kaadMeacaWGjbWcdaWgaaqa aKqzadGaam4qaaWcbeaaaaqcLbsacqGHhis0caWG3bGaaGPaVlabgE GirlaadEhaaOGaay5waiaaw2faaKqzGeGaaGPaVlabg2da9iaahAga lmaaBaaabaqcLbmacaWGWbGaamiBaaWcbeaaaaa@59C0@ will be written in the form:

2n[ p 1 0 W I I C ww ]= f pl MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaaIYa GaaCOBaiabgwSixVWaamWaaOqaaKqzGeGaamiCaiaahgdalmaaBaaa baqcLbmacaaIWaaaleqaaKqzGeGaeyOeI0YcdaWcaaGcbaqcLbsacq GHciITcaWGxbaakeaajugibiabgkGi2kaadMeacaWGjbWcdaWgaaqa aKqzadGaam4qaaWcbeaaaaqcLbsacqGHhis0caWG3bGaaGPaVlabgE GirlaadEhaaOGaay5waiaaw2faaKqzGeGaaGPaVlabg2da9iaahAga lmaaBaaabaqcLbmacaWGWbGaamiBaaWcbeaaaaa@59C0@            (1.6)

2n[ W I 1 + W I 2 ]w= f z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaaIYa GaaCOBaiabgwSixVWaamWaaOqaaSWaaSaaaOqaaKqzGeGaeyOaIyRa am4vaaGcbaqcLbsacqGHciITcaWGjbWcdaWgaaqaaKqzadGaaGymaa WcbeaaaaqcLbsacqGHRaWklmaalaaakeaajugibiabgkGi2kaadEfa aOqaaKqzGeGaeyOaIyRaamysaSWaaSbaaeaajugWaiaaikdaaSqaba aaaaGccaGLBbGaayzxaaqcLbsacaaMc8Uaey4bIeTaam4DaiaaykW7 cqGH9aqpcaWGMbWcdaWgaaqaaKqzadGaamOEaaWcbeaaaaa@5865@     (1.7)

Here, the density of external forces f= f pl + f z k, f pl k=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaaCOzaiabg2da9iaahAgalmaaBaaabaqcLbmacaWGWbGa amiBaaWcbeaajugibiabgUcaRiaadAgalmaaBaaabaqcLbmacaWG6b aaleqaaKqzGeGaaC4AaiaacYcacaaMc8UaaGPaVlaaykW7caWHMbWc daWgaaqaaKqzadGaamiCaiaadYgaaSqabaqcLbsacqGHflY1caWHRb Gaeyypa0JaaGimaaaa@5659@ />is calculated per unit area of ​​the lateral surface in the deformed state (for an antiplane deformation, the area of ​​the lateral surface does not change). In expression (1.6), the quantity f pl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaaCOzaSWaaSbaaeaajugWaiaadchacaWGSbaaleqaaaaa @3F09@  gives the density of external forces in the transverse plane necessary to maintain the regime of antiplane deformation. If this regime is maintained kinematically, then the force density f pl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaaCOzaSWaaSbaaeaajugWaiaadchacaWGSbaaleqaaaaa @3F09@ manifests itself as a reaction from the rigid clips, between which a deformable cylindrical body is enclosed. In the case of a generalized Neo-Hooke potential, the function is assumed to be zero and there are no stresses in the transverse plane. The kinematic boundary conditions reduce to setting the longitudinal shear on the outer lateral surface.

Model problem of the finite longitudinal displacement of a circular cylindrical sleeve between rigid concentric rings

As a problem for which exact solutions are sought, the problem of the finite longitudinal shear of a circular cylindrical sleeve between rigid concentric rings is considered. On the one hand, this is one of the simplest tasks, on the other hand it has an applied value, since it is one of the designs of a shock absorber. The inner cage is stationary, and the outer casing is shifted along the symmetry axis by Δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaeuiLdqeaaa@3C35@ . The counting and deformed sleeve configurations are shown in Figure 1.

As material we will use the coordinates of the cylindrical coordinate system ( r,θ,z ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWcdaqadaGcba qcLbsacaWGYbGaaiilaiabeI7aXjaacYcacaWG6baakiaawIcacaGL Paaaaaa@3D57@ with a unit basis:

e 1 =cosθi+sinθj MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWHLb WcdaWgaaqaaKqzadGaaGymaaWcbeaajugibiabg2da9iGacogacaGG VbGaai4CaiabeI7aXjaahMgacqGHRaWkciGGZbGaaiyAaiaac6gacq aH4oqCcaWHQbaaaa@4725@ , e 2 =sinθi+cosθj MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWHLb WcdaWgaaqaaKqzadGaaGOmaaWcbeaajugibiabg2da9iabgkHiTiGa cohacaGGPbGaaiOBaiabeI7aXjaahMgacqGHRaWkciGGJbGaai4Bai aacohacqaH4oqCcaWHQbaaaa@4813@ , e 3 =k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaaMc8 UaaCyzaSWaaSbaaeaajugWaiaaiodaaSqabaqcLbsacqGH9aqpcaWH Rbaaaa@3DC7@ ;and an axis directed along the axis of symmetry of the sleeve. The Hamilton operator in this basis is obtained in the form: = e 1 /r+ r 1 e 2 /θ+k/z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqGHhi s0cqGH9aqpcaWHLbWcdaWgaaqaaKqzadGaaGymaaWcbeaajugibiaa ykW7cqGHciITcaGGVaGaeyOaIyRaamOCaiabgUcaRiaadkhalmaaCa aabeqaaKqzadGaeyOeI0IaaGymaaaajugibiaahwgalmaaBaaabaqc LbmacaaIYaaaleqaaKqzGeGaaGPaVlabgkGi2kaac+cacqGHciITcq aH4oqCcqGHRaWkcaWHRbGaaGPaVlabgkGi2kaac+cacqGHciITcaWG 6baaaa@5A76@ . In this case, the antiplane deformation is also axisymmetric, that is w=w( r ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWG3b Gaeyypa0Jaam4DaSWaaeWaaOqaaKqzGeGaamOCaaGccaGLOaGaayzk aaaaaa@3CD0@ and p=p( r ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaamiCaiabg2da9iaadchalmaabmaakeaajugibiaadkha aOGaayjkaiaawMcaaaaa@40ED@ ,

Thus, the expressions (1.1) - (1.4) go over into the following (the prime denotes the derivative with respect to r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaamOCaaaa@3BC6@ : x=X+w(r)k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWH4b Gaeyypa0JaaCiwaiabgUcaRiaadEhacaGGOaGaamOCaiaacMcacaWH Rbaaaa@3EAD@ ,

T=2{ [ p( r )( e 1 e 1 + e 2 e 2 ) W 2 ( r ) w 2 ( r ) e 1 e 1 ] +( W 1 ( r )+ W 2 ( r ) ) w ( r )( e 1 k+k e 1 )+ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaaCivaiabg2da9iaaikdalmaaceaakeaalmaadmaakeaa jugibiaadchalmaabmaakeaajugibiaadkhaaOGaayjkaiaawMcaaS WaaeWaaOqaaKqzGeGaaCyzaSWaaSbaaeaajugWaiaaigdaaSqabaqc LbsacaWHLbWcdaWgaaqaaKqzadGaaGymaaWcbeaajugibiabgUcaRi aahwgalmaaBaaabaqcLbmacaaIYaaaleqaaKqzGeGaaCyzaSWaaSba aeaajugWaiaaikdaaSqabaaakiaawIcacaGLPaaajugibiabgkHiTi aadEfalmaaBaaabaqcLbmacaaIYaaaleqaamaabmaakeaajugibiaa dkhaaOGaayjkaiaawMcaaKqzGeGabm4DayaafaWcdaahaaqabeaaju gWaiaaikdaaaWcdaqadaGcbaqcLbsacaWGYbaakiaawIcacaGLPaaa jugibiaahwgalmaaBaaabaqcLbmacaaIXaaaleqaaKqzGeGaaCyzaS WaaSbaaeaajugWaiaaigdaaSqabaaakiaawUfacaGLDbaaaiaawUha aKqzGeGaey4kaSYcdaqadaGcbaqcLbsacaWGxbWcdaWgaaqaaKqzad GaaGymaaWcbeaadaqadaGcbaqcLbsacaWGYbaakiaawIcacaGLPaaa jugibiabgUcaRiaadEfalmaaBaaabaqcLbmacaaIYaaaleqaamaabm aakeaajugibiaadkhaaOGaayjkaiaawMcaaaGaayjkaiaawMcaaKqz GeGabm4DayaafaWcdaqadaGcbaqcLbsacaWGYbaakiaawIcacaGLPa aalmaabmaakeaajugibiaahwgalmaaBaaabaqcLbmacaaIXaaaleqa aKqzGeGaaC4AaiabgUcaRiaahUgacaWHLbWcdaWgaaqaaKqzadGaaG ymaaWcbeaaaOGaayjkaiaawMcaaKqzGeGaey4kaScaaa@8F42@    

[ W 1 ( r ) w 2 ( r )+p( r ) ]kk } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaSWaaiGaaOqaaSWaamWaaOqaaKqzGeGaam4vaSWaaSbaaeaajugW aiaaigdaaSqabaWaaeWaaOqaaKqzGeGaamOCaaGccaGLOaGaayzkaa qcLbsaceWG3bGbauaalmaaCaaabeqaaKqzadGaaGOmaaaalmaabmaa keaajugibiaadkhaaOGaayjkaiaawMcaaKqzGeGaey4kaSIaamiCaS WaaeWaaOqaaKqzGeGaamOCaaGccaGLOaGaayzkaaaacaGLBbGaayzx aaqcLbsacaWHRbGaaC4AaaGccaGL9baaaaa@5318@ , W 1 ( r )= W I C , W 2 ( r )= W I I C MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaam4vaSWaaSbaaeaajugWaiaaigdaaSqabaWaaeWaaOqa aKqzGeGaamOCaaGccaGLOaGaayzkaaqcLbsacqGH9aqplmaalaaake aajugibiabgkGi2kaadEfaaOqaaKqzGeGaeyOaIyRaamysaSWaaSba aeaajugWaiaadoeaaSqabaaaaKqzGeGaaiilaiaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaadEfalmaaBaaabaqcLbmacaaIYaaa leqaamaabmaakeaajugibiaadkhaaOGaayjkaiaawMcaaKqzGeGaey ypa0ZcdaWcaaGcbaqcLbsacqGHciITcaWGxbaakeaajugibiabgkGi 2kaadMeacaWGjbWcdaWgaaqaaKqzadGaam4qaaWcbeaaaaaaaa@6567@      (2.1)

I 1 = I 2 =3+ w 2 ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaamysaSWaaSbaaeaajugWaiaaigdaaSqabaqcLbsacqGH 9aqpcaWGjbWcdaWgaaqaaKqzadGaaGOmaaWcbeaajugibiabg2da9i aaiodacqGHRaWkceWG3bGbauaalmaaCaaabeqaaKqzadGaaGOmaaaa lmaabmaakeaajugibiaadkhaaOGaayjkaiaawMcaaaaa@4BC2@  

{ p ( r )= 1 r [ r W 2 ( r ) w 2 ( r ) ] [ r( W 1 ( r )+ W 2 ( r ) ) w ] =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWcdaGabaqcLb saeaqabOqaaKqzGeGabmiCayaafaWcdaqadaGcbaqcLbsacaWGYbaa kiaawIcacaGLPaaajugibiabg2da9SWaaSaaaOqaaKqzGeGaaGymaa GcbaqcLbsacaWGYbaaaSWaamWaaOqaaKqzGeGaamOCaiaadEfalmaa BaaabaqcLbmacaaIYaaaleqaamaabmaakeaajugibiaadkhaaOGaay jkaiaawMcaaKqzGeGabm4DayaafaWcdaahaaqabeaajugWaiaaikda aaWcdaqadaGcbaqcLbsacaWGYbaakiaawIcacaGLPaaaaiaawUfaca GLDbaalmaaCaaabeqaaKqzGeGamai8gkdiIcaaaOqaaSWaamWaaOqa aKqzGeGaamOCaSWaaeWaaOqaaKqzGeGaam4vaSWaaSbaaeaajugWai aaigdaaSqabaWaaeWaaOqaaKqzGeGaamOCaaGccaGLOaGaayzkaaqc LbsacqGHRaWkcaWGxbWcdaWgaaqaaKqzadGaaGOmaaWcbeaadaqada GcbaqcLbsacaWGYbaakiaawIcacaGLPaaaaiaawIcacaGLPaaajugi biqadEhagaqbaaGccaGLBbGaayzxaaWcdaahaaqabeaajugibiadac VHYaIOaaGaeyypa0JaaGimaaaakiaawUhaaaaa@6FFB@      (2.2)

< p>Equation (1.5) is a consequence of the second equation in (2.2), that is, an axisymmetric version of the antiplane deformation exists for any deformation energy potential.

The kinematic boundary conditions are written in the form:

w( R 1 )=0,w( R 2 )=Δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWG3b WcdaqadaGcbaqcLbsacaWGsbWcdaWgaaqaaKqzadGaaGymaaWcbeaa aOGaayjkaiaawMcaaKqzGeGaeyypa0JaaGimaiaacYcacaaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaadEhalmaabmaakeaajugibiaadkfa lmaaBaaabaqcLbmacaaIYaaaleqaaaGccaGLOaGaayzkaaqcLbsacq GH9aqpcqqHuoaraaa@50AA@      (2.3)

We find the first integral of the second equation in (2.2):

[ W 1 ( r )+ W 2 ( r ) ] w ( r )= c r ,cconst MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWcdaWadaGcba qcLbsacaWGxbWcdaWgaaqaaKqzadGaaGymaaWcbeaadaqadaGcbaqc LbsacaWGYbaakiaawIcacaGLPaaajugibiabgUcaRiaadEfalmaaBa aabaqcLbmacaaIYaaaleqaamaabmaakeaajugibiaadkhaaOGaayjk aiaawMcaaaGaay5waiaaw2faaKqzGeGabm4DayaafaWcdaqadaGcba qcLbsacaWGYbaakiaawIcacaGLPaaajugibiabg2da9SWaaSaaaOqa aKqzGeGaam4yaaGcbaqcLbsacaWGYbaaaiaacYcacaaMc8UaaGPaVl aaykW7caWGJbGaeyOeI0Iaam4yaiaad+gacaWGUbGaam4Caiaadsha aaa@5BC6@    (2.4)

The unit vector of the outer normal to the outer lateral surface of the sleeve with equation r= R 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaamOCaiabg2da9iaadkfalmaaBaaabaqcLbmacaaIYaaa leqaaaaa@3FC4@ equals , therefore, the shearing force calculated per unit length of the sleeve according to (1.7), (2.1) and (2.4) is obtained as:

Q/h= 0 2π e 1 T( R 2 ) R 2 dθ= 0 2π [ T 11 ( R 2 ) e 1 + T 13 ( R 2 )k ] R 2 dθ= 2π T 13 ( R 2 ) R 2 k=4π R 2 [ W 1 ( R 2 )+ W 2 ( R 2 ) ] w ( R 2 )k=4πc MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGceaqabeaajugibi aahgfacaGGVaGaamiAaiabg2da9SWaa8qCaOqaaKqzGeGaaCyzaSWa aSbaaeaajugWaiaaigdaaSqabaqcLbsacqGHflY1caWHubWcdaqada GcbaqcLbsacaWGsbWcdaWgaaqaaKqzadGaaGOmaaWcbeaaaOGaayjk aiaawMcaaKqzGeGaamOuaSWaaSbaaeaajugWaiaaikdaaSqabaaaba qcLbmacaaIWaaaleaajugWaiaaikdacqaHapaCaKqzGeGaey4kIipa caWGKbGaeqiUdeNaeyypa0ZcdaWdXbGcbaWcdaWadaGcbaqcLbsaca WGubWcdaWgaaqaaKqzadGaaGymaiaaigdaaSqabaWaaeWaaOqaaKqz GeGaamOuaSWaaSbaaeaajugWaiaaikdaaSqabaaakiaawIcacaGLPa aajugibiaahwgalmaaBaaabaqcLbmacaaIXaaaleqaaKqzGeGaey4k aSIaamivaSWaaSbaaeaajugWaiaaigdacaaIZaaaleqaamaabmaake aajugibiaadkfalmaaBaaabaqcLbmacaaIYaaaleqaaaGccaGLOaGa ayzkaaqcLbsacaWHRbaakiaawUfacaGLDbaaaSqaaKqzadGaaGimaa WcbaqcLbmacaaIYaGaeqiWdahajugibiabgUIiYdGaamOuaSWaaSba aeaajugWaiaaikdaaSqabaqcLbsacaWGKbGaeqiUdeNaeyypa0dake aajugibiaaikdacqaHapaCcaWGubWcdaWgaaqaaKqzadGaaGymaiaa iodaaSqabaWaaeWaaOqaaKqzGeGaamOuaSWaaSbaaeaajugWaiaaik daaSqabaaakiaawIcacaGLPaaajugibiaadkfalmaaBaaabaqcLbma caaIYaaaleqaaKqzGeGaaC4Aaiabg2da9iaaisdacqaHapaCcaWGsb WcdaWgaaqaaKqzadGaaGOmaaWcbeaadaWadaGcbaqcLbsacaWGxbWc daWgaaqaaKqzadGaaGymaaWcbeaadaqadaGcbaqcLbsacaWGsbWcda WgaaqaaKqzadGaaGOmaaWcbeaaaOGaayjkaiaawMcaaKqzGeGaey4k aSIaam4vaSWaaSbaaeaajugWaiaaikdaaSqabaWaaeWaaOqaaKqzGe GaamOuaSWaaSbaaeaajugWaiaaikdaaSqabaaakiaawIcacaGLPaaa aiaawUfacaGLDbaajugibiqadEhagaqbaSWaaeWaaOqaaKqzGeGaam OuaSWaaSbaaeaajugWaiaaikdaaSqabaaakiaawIcacaGLPaaajugi biaahUgacqGH9aqpcaaI0aGaeqiWdaNaam4yaaaaaa@B96E@    

Hence c=Q ( 4πh ) 1 ,Q=| Q | MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGJb Gaeyypa0JaamyuaSWaaeWaaOqaaKqzGeGaaGinaiabec8aWjaadIga aOGaayjkaiaawMcaaSWaaWbaaeqabaqcLbmacqGHsislcaaIXaaaaK qzGeGaaiilaiaaykW7caaMc8UaaGPaVlaadgfacqGH9aqplmaaemaa keaajugibiaahgfaaOGaay5bSlaawIa7aaaa@4E70@ . Thus, we obtain:

σ rz = T R13 = 2c r = Q 2πhr MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaeq4Wdm3cdaWgaaqaaKqzadGaamOCaiaadQhaaSqabaqc LbsacqGH9aqpcaWGubWcdaWgaaqaaKqzadGaey4bIeTaamOuaiaaig dacaaIZaaaleqaaKqzGeGaeyypa0ZcdaWcaaGcbaqcLbsacaaIYaGa am4yaaGcbaqcLbsacaWGYbaaaiabg2da9SWaaSaaaOqaaKqzGeGaam yuaaGcbaqcLbsacaaIYaGaeqiWdaNaamiAaiaaykW7caWGYbaaaaaa @5623@  

In what follows we turn to dimensionless quantities:

δ= Δ R 1 ,κ= R 2 R 1 ,q= Q 2πh R 1 μ ,v( ρ )= w( R 1 ρ ) R 1 ,ρ= r R 1 ,1ρκ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaeqiTdqMaeyypa0ZcdaWcaaGcbaqcLbsacqqHuoaraOqa aKqzGeGaamOuaSWaaSbaaeaajugWaiaaigdaaSqabaaaaKqzGeGaai ilaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaeqOUdSMaeyypa0Zc daWcaaGcbaqcLbsacaWGsbWcdaWgaaqaaKqzadGaaGOmaaWcbeaaaO qaaKqzGeGaamOuaSWaaSbaaeaajugWaiaaigdaaSqabaaaaKqzGeGa aiilaiaaykW7caaMc8UaamyCaiabg2da9SWaaSaaaOqaaKqzGeGaam yuaaGcbaqcLbsacaaIYaGaeqiWdaNaamiAaiaadkfalmaaBaaabaqc LbmacaaIXaaaleqaaKqzGeGaeqiVd0gaaiaaykW7caaMc8Uaaiilai aaykW7caaMc8UaamODaSWaaeWaaOqaaKqzGeGaeqyWdihakiaawIca caGLPaaajugibiabg2da9SWaaSaaaOqaaKqzGeGaam4DaSWaaeWaaO qaaKqzGeGaamOuaSWaaSbaaeaajugWaiaaigdaaSqabaqcLbsacqaH bpGCaOGaayjkaiaawMcaaaqaaKqzGeGaamOuaSWaaSbaaeaajugWai aaigdaaSqabaaaaKqzGeGaaiilaiaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaeqyWdiNaeyypa0ZcdaWcaaGcbaqcLbsacaWGYbaakeaaju gibiaadkfalmaaBaaabaqcLbmacaaIXaaaleqaaaaajugibiaacYca caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGymai abgsMiJkabeg8aYjabgsMiJkabeQ7aRbaa@A5D4@

The correspondence of the notation for the derivatives is obtained in the form:

dw( r ) dr = dv( ρ ) dρ w ( r )= v ˙ ( ρ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtg9frVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaSWaaSaaaOqaaKqzGeGaamizaiaadEhalmaabmaakeaajugibiaa dkhaaOGaayjkaiaawMcaaaqaaKqzGeGaamizaiaadkhaaaGaeyypa0 ZcdaWcaaGcbaqcLbsacaWGKbGaamODaSWaaeWaaOqaaKqzGeGaeqyW dihakiaawIcacaGLPaaaaeaajugibiaadsgacqaHbpGCaaGaeyi1HS Tabm4DayaafaWcdaqadaGcbaqcLbsacaWGYbaakiaawIcacaGLPaaa jugibiabg2da9iqadAhagaGaaSWaaeWaaOqaaKqzGeGaeqyWdihaki aawIcacaGLPaaaaaa@5A35@

With these relations in mind, the expression for the shear stress can be rewritten

σ rz ( ρ )= μq ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaeq4Wdm3cdaWgaaqaaKqzadGaamOCaiaadQhaaSqabaWa aeWaaOqaaKqzGeGaeqyWdihakiaawIcacaGLPaaajugibiabg2da9S WaaSaaaOqaaKqzGeGaeqiVd0MaamyCaaGcbaqcLbsacqaHbpGCaaaa aa@4B27@     (2.5)

Expression (2.5) does not depend on the deformation energy potential and coincides with the solution of the linear theory.

If it is necessary to find the stress state in the transverse plane, the hydrostatic pressure function is found from the first equation in (2.2):

p= R 1 r 1 t d dt [ t W 2 ( t ) w 2 ( t ) ] dt+μ p 0 = 1 ρ 1 t d dt [ t W 2 ( t ) v ˙ 2 ( t ) ] dt+μ p 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGWb Gaeyypa0ZcdaWdXbGcbaWcdaWcaaGcbaqcLbsacaaIXaaakeaajugi biaadshaaaWcdaWcaaGcbaqcLbsacaWGKbaakeaajugibiaadsgaca WG0baaaSWaamWaaOqaaKqzGeGaamiDaiaadEfalmaaBaaabaqcLbma caaIYaaaleqaamaabmaakeaajugibiaadshaaOGaayjkaiaawMcaaK qzGeGabm4DayaafaWcdaahaaqabeaajugWaiaaikdaaaWcdaqadaGc baqcLbsacaWG0baakiaawIcacaGLPaaaaiaawUfacaGLDbaaaSqaaK qzadGaamOuaWWaaSbaaeaajugWaiaaigdaaWqabaaaleaajugWaiaa dkhaaKqzGeGaey4kIipacaaMc8UaamizaiaadshacqGHRaWkcqaH8o qBcaWGWbWcdaWgaaqaaKqzadGaaGimaaWcbeaajugibiabg2da9SWa a8qCaOqaaSWaaSaaaOqaaKqzGeGaaGymaaGcbaqcLbsacaWG0baaaS WaaSaaaOqaaKqzGeGaamizaaGcbaqcLbsacaWGKbGaamiDaaaalmaa dmaakeaajugibiaadshacaWGxbWcdaWgaaqaaKqzadGaaGOmaaWcbe aadaqadaGcbaqcLbsacaWG0baakiaawIcacaGLPaaajugibiqadAha gaGaaSWaaWbaaeqabaqcLbmacaaIYaaaaSWaaeWaaOqaaKqzGeGaam iDaaGccaGLOaGaayzkaaaacaGLBbGaayzxaaaaleaajugWaiaaigda aSqaamaaCaaameqabaqcLbmacqaHbpGCaaaajugibiabgUIiYdGaaG PaVlaadsgacaWG0bGaey4kaSIaeqiVd0MaamiCaSWaaSbaaeaajugW aiaaicdaaSqabaaaaa@8D18@

The dimensionless integration constant p 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaamiCaSWaaSbaaeaajugWaiaaicdaaSqabaaaaa@3DE3@ is found from the condition that the principal vector on the end surface vanishes. Since the transverse plane is deplanated in the case of an antiplane deformation, it is easiest to take into account the vanishing of the principal vector in this plane by means of the first Piola-Kirchhoff T R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtg9frVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaaCivaSWaaSbaaeaajugWaiaahkfaaSqabaaaaa@3DBD@ , stress tensor obtained from the Cauchy tensor by using the relation T R =T X T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtg9frVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaaCivaSWaaSbaaeaajugWaiaahkfaaSqabaqcLbsacqGH 9aqpcaWHubGaeyyXICTaey4bIeTaaCiwaSWaaWbaaeqabaqcLbmaca WGubaaaaaa@4714@ , written for an incompressible material. The vector of the unit normal to the cross section in the reference configuration is the vector e 3 =k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtg9frVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaaCyzaSWaaSbaaeaajugWaiaaiodaaSqabaqcLbsacqGH 9aqpcaWHRbaaaa@4039@ , therefore we have:

R 1 R 2 0 2π T R krdrdθ = R 1 R 2 0 2π [ e 1 T R13 +k T R 33 ]rdrdθ = MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtg9frVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaSWaa8qCaOqaaSWaa8qCaOqaaKqzGeGaaCivaSWaaSbaaeaajugi biaahkfaaSqabaqcLbsacaaMc8UaeyyXICTaaGPaVlaahUgacaaMc8 UaaGPaVlaadkhacaWGKbGaamOCaiaadsgacqaH4oqCaSqaaKqzadGa aGimaaWcbaqcLbmacaaIYaGaeqiWdahajugibiabgUIiYdaaleaaju gWaiaadkfammaaBaaabaqcLbmacaaIXaaameqaaaWcbaqcLbmacaWG sbaddaWgaaqaaKqzadGaaGOmaaadbeaaaKqzGeGaey4kIipacqGH9a qplmaapehakeaalmaapehakeaalmaadmaakeaajugibiaahwgalmaa BaaabaqcLbmacaaIXaaaleqaaKqzGeGaamivaSWaaSbaaeaajugWai aahkfacaaIXaGaaG4maaWcbeaajugibiabgUcaRiaahUgacaWGubWc daWgaaqaaKqzadGaaCOuaaWcbeaadaWgaaqaaKqzadGaaG4maiaaio daaSqabaqcLbsacaaMc8oakiaawUfacaGLDbaajugibiaaykW7caWG YbGaamizaiaadkhacaWGKbGaeqiUdehaleaajugWaiaaicdaaSqaaK qzadGaaGOmaiabec8aWbqcLbsacqGHRiI8aaWcbaqcLbmacaWGsbad daWgaaqaaKqzadGaaGymaaadbeaaaSqaaKqzadGaamOuaWWaaSbaae aajugWaiaaikdaaWqabaaajugibiabgUIiYdGaeyypa0daaa@934A@    

2π R 1 R 2 T R 33 rdr k=4π R 1 2 1 κ [ p W 2 ( ρ ) v ˙ ( ρ ) ]ρdρk= MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtg9frVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaaGOmaiabec8aWTWaa8qCaOqaaKqzGeGaamivaSWaaSba aeaajugWaiaahkfaaSqabaWaaSbaaeaajugWaiaaiodacaaIZaaale qaaKqzGeGaaGPaVlaadkhacaWGKbGaamOCaaWcbaqcLbmacaWGsbad daWgaaqaaKqzadGaaGymaaadbeaaaSqaaKqzadGaamOuaWWaaSbaae aajugWaiaaikdaaWqabaaajugibiabgUIiYdGaaC4Aaiabg2da9iaa isdacqaHapaCcaWGsbWcdaqhaaqaaKqzadGaaGymaaWcbaqcLbmaca aIYaaaaSWaa8qCaOqaaSWaamWaaOqaaKqzGeGaamiCaiabgkHiTiaa dEfalmaaBaaabaqcLbmacaaIYaaaleqaamaabmaakeaajugibiabeg 8aYbGccaGLOaGaayzkaaqcLbsaceWG2bGbaiaalmaabmaakeaajugi biabeg8aYbGccaGLOaGaayzkaaaacaGLBbGaayzxaaqcLbsacqaHbp GCcaWGKbGaeqyWdiNaaC4Aaiabg2da9aWcbaqcLbmacaaIXaaaleaa jugWaiabeQ7aRbqcLbsacqGHRiI8aaaa@7CE7@   

4πk R 1 2 1 κ [ 1 ρ 1 t d dt [ t W 2 ( t ) v ˙ 2 ( t ) ] dt+μ p 0 W 2 ( ρ ) v ˙ ( ρ ) ] ρdρ=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtg9frVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaaGinaiabec8aWjaahUgacaWGsbWcdaqhaaqaaKqzadGa aGymaaWcbaqcLbmacaaIYaaaaSWaa8qCaOqaaSWaamWaaOqaaSWaa8 qCaOqaaSWaaSaaaOqaaKqzGeGaaGymaaGcbaqcLbsacaWG0baaaSWa aSaaaOqaaKqzGeGaamizaaGcbaqcLbsacaWGKbGaamiDaaaalmaadm aakeaajugibiaadshacaWGxbWcdaWgaaqaaKqzadGaaGOmaaWcbeaa daqadaGcbaqcLbsacaWG0baakiaawIcacaGLPaaajugibiqadAhaga GaaSWaaWbaaeqabaqcLbmacaaIYaaaaSWaaeWaaOqaaKqzGeGaamiD aaGccaGLOaGaayzkaaaacaGLBbGaayzxaaaaleaajugWaiaaigdaaS qaamaaCaaameqabaqcLbmacqaHbpGCaaaajugibiabgUIiYdGaaGPa VlaadsgacaWG0bGaey4kaSIaeqiVd0MaamiCaSWaaSbaaeaajugWai aaicdaaSqabaqcLbsacqGHsislcaWGxbWcdaWgaaqaaKqzadGaaGOm aaWcbeaadaqadaGcbaqcLbsacqaHbpGCaOGaayjkaiaawMcaaKqzGe GabmODayaacaWcdaqadaGcbaqcLbsacqaHbpGCaOGaayjkaiaawMca aaGaay5waiaaw2faaaWcbaqcLbmacaaIXaaaleaajugWaiabeQ7aRb qcLbsacqGHRiI8aiaaykW7caaMc8UaeqyWdiNaamizaiabeg8aYjab g2da9iaaicdaaaa@8CC6@    (2.6)

We obtained an equation for finding.

Equation (2.4) can be rewritten in the form

[ W 1 ( ρ )+ W 2 ( ρ ) ] v ˙ ( ρ )= μq 2ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWcdaWadaGcba qcLbsacaWGxbWcdaWgaaqaaKqzadGaaGymaaWcbeaadaqadaGcbaqc LbsacqaHbpGCaOGaayjkaiaawMcaaKqzGeGaey4kaSIaam4vaSWaaS baaeaajugWaiaaikdaaSqabaWaaeWaaOqaaKqzGeGaeqyWdihakiaa wIcacaGLPaaaaiaawUfacaGLDbaajugibiqadAhagaGaaSWaaeWaaO qaaKqzGeGaeqyWdihakiaawIcacaGLPaaajugibiabg2da9SWaaSaa aOqaaKqzGeGaeqiVd0MaamyCaaGcbaqcLbsacaaIYaGaeqyWdihaaa aa@5581@    (2.7)

This non-linear differential equation is used to find the cross-sectional de-planing function w( r )= R 1 v( r/ R 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaam4DaSWaaeWaaOqaaKqzGeGaamOCaaGccaGLOaGaayzk aaqcLbsacqGH9aqpcaWGsbWcdaWgaaqaaKqzadGaaGymaaWcbeaaju gibiaadAhalmaabmaakeaajugibiaadkhacaGGVaGaamOuaSWaaSba aeaajugWaiaaigdaaSqabaaakiaawIcacaGLPaaaaaa@4BE7@ .

Figure 1 The counting and deformed sleeve configurations.

Choice of models of isotropic incompressible hyper elastic material

This problem was solved by many authors for various potentials.29 Let us dwell on some potentials from,1 for which we could not find a solution in the literature. Since for homogeneous first-order materials W 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYlg9frVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaam4vaKqbaoaaBaaaleaajugWaiaaigdaaSqabaaaaa@3E0A@ and W 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYlg9frVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaam4vaKqbaoaaBaaaleaajugWaiaaikdaaSqabaaaaa@3E0B@ are functions of only v ˙ ( ρ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceWG2b Gbaiaalmaabmaakeaajugibiabeg8aYbGccaGLOaGaayzkaaaaaa@3B9F@ some parameters, when substituting a specific expression for the deformation energy potential into equation (2.7), it becomes an algebraic or transcendental equation with respect to v ˙ ( ρ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceWG2b Gbaiaalmaabmaakeaajugibiabeg8aYbGccaGLOaGaayzkaaaaaa@3B9F@ . The success in obtaining an explicit expression for an exact solution v ˙ ( ρ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceWG2b Gbaiaalmaabmaakeaajugibiabeg8aYbGccaGLOaGaayzkaaaaaa@3B9F@ depends on the possibility of obtaining an explicit expression for solving this equation with respect to v ˙ ( ρ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsaceWG2b Gbaiaalmaabmaakeaajugibiabeg8aYbGccaGLOaGaayzkaaaaaa@3B9F@ . Obtaining such a solution is always possible for algebraic equations of degree not higher than the fourth. For transcendental equations, obtaining explicit expressions for solving is problematic.

Solution for potential [1]

Here and below is the shear modulus of the linear theory of elasticity. Although the equation (2.7) for the potential [1] W=( μ/( 2β ) )[ exp( β( I C 3 ) )1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGxb Gaeyypa0ZcdaqadaGcbaqcLbsacqaH8oqBcaGGVaWcdaqadaGcbaqc LbsacaaIYaGaeqOSdigakiaawIcacaGLPaaaaiaawIcacaGLPaaalm aadmaakeaajugibiGacwgacaGG4bGaaiiCaSWaaeWaaOqaaKqzGeGa eqOSdi2cdaqadaGcbaqcLbsacaWGjbWcdaWgaaqaaKqzadGaam4qaa WcbeaajugibiabgkHiTiaaiodaaOGaayjkaiaawMcaaaGaayjkaiaa wMcaaKqzGeGaeyOeI0IaaGymaaGccaGLBbGaayzxaaaaaa@549A@ , v ˙ ( ρ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGabmODayaacaWcdaqadaGcbaqcLbsacqaHbpGCaOGaayjk aiaawMcaaaaa@3FCA@ :

e β ( v ˙ ( ρ ) ) 2 v ˙ ( ρ )= q ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaamyzaSWaaWbaaeqabaqcLbmacqaHYoGymmaabmaaleaa jugWaiqadAhagaGaaWWaaeWaaSqaaKqzadGaeqyWdihaliaawIcaca GLPaaaaiaawIcacaGLPaaammaaCaaabeqaaKqzadGaaGOmaaaaaaqc LbsaceWG2bGbaiaalmaabmaakeaajugibiabeg8aYbGccaGLOaGaay zkaaqcLbsacqGH9aqplmaalaaakeaajugibiaadghaaOqaaKqzGeGa eqyWdihaaaaa@545A@    (3.1);

it is not less than that, the solution of the differential equation is found and expressed through the function of Lambert30 and the exponential integral

Ei( z,a )= 1 e tz t a dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaqGfb GaaeyAaSWaaeWaaOqaaKqzGeGaamOEaiaacYcacaWGHbaakiaawIca caGLPaaajugibiabg2da9SWaa8qCaOqaaKqzGeGaamyzaSWaaWbaae qabaqcLbmacqGHsislcaWG0bGaamOEaaaajugibiaadshalmaaCaaa beqaaKqzadGaeyOeI0IaamyyaaaajugibiaadsgacaWG0baaleaaju gWaiaaigdaaSqaaKqzadGaeyOhIukajugibiabgUIiYdaaaa@532B@   

v( ρ )=q{ e A( ρ ) e A( 1 ) + 1 2 [ Ei( 1,A( ρ ) )Ei( 1,A( 1 ) ) ] } A( ρ )= 1 2 LambertW( 2β q 2 ρ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO abaeqabaqcLbsacaWG2bWcdaqadaGcbaqcLbsacqaHbpGCaOGaayjk aiaawMcaaKqzGeGaeyypa0JaamyCaSWaaiWaaOqaaKqzGeGaamyzaS WaaWbaaeqabaqcLbmacqGHsislcaWGbbaddaqadaWcbaqcLbmacqaH bpGCaSGaayjkaiaawMcaaaaajugibiabgkHiTiaadwgalmaaCaaabe qaaKqzadGaeyOeI0IaamyqaWWaaeWaaSqaaKqzadGaaGymaaWccaGL OaGaayzkaaaaaKqzGeGaey4kaSYcdaWcaaGcbaqcLbsacaaIXaaake aajugibiaaikdaaaWcdaWadaGcbaqcLbsaciGGfbGaaiyAaSWaaeWa aOqaaKqzGeGaaGymaiaacYcacaWGbbWcdaqadaGcbaqcLbsacqaHbp GCaOGaayjkaiaawMcaaaGaayjkaiaawMcaaKqzGeGaeyOeI0Iaciyr aiaacMgalmaabmaakeaajugibiaaigdacaGGSaGaamyqaSWaaeWaaO qaaKqzGeGaaGymaaGccaGLOaGaayzkaaaacaGLOaGaayzkaaaacaGL BbGaayzxaaaacaGL7bGaayzFaaaabaqcLbsacaWGbbWcdaqadaGcba qcLbsacqaHbpGCaOGaayjkaiaawMcaaKqzGeGaeyypa0ZcdaWcaaGc baqcLbsacaaIXaaakeaajugibiaaikdaaaGaaeitaiaabggacaqGTb GaaeOyaiaabwgacaqGYbGaaeiDaiaabEfalmaabmaakeaalmaalaaa keaajugibiaaikdacqaHYoGycaWGXbWcdaahaaqabeaajugWaiaaik daaaaakeaajugibiabeg8aYTWaaWbaaeqabaqcLbmacaaIYaaaaaaa aOGaayjkaiaawMcaaaaaaa@8F2E@     (3.2)

This verification is verified by the direct substance (3.2) in (3.1). For the connection I take a relationship30

dL dz = e L [ 1+L ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipD0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWcdaWccaGcba qcLbsacaWGKbGaamitaaGcbaqcLbsacaWGKbGaamOEaaaacqGH9aqp lmaaliaakeaajugibiaadwgalmaaCaaabeqaaKqzadGaeyOeI0Iaae itaaaaaOqaaSWaamWaaOqaaKqzGeGaaGymaiabgUcaRiaadYeaaOGa ay5waiaaw2faaaaaaaa@455C@ , L=LambertW( z ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipD0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacaWGmb Gaeyypa0JaciitaiaacggacaGGTbGaaiOyaiaacwgacaGGYbGaaiiD aiaacEfalmaabmaakeaajugibiaadQhaaOGaayjkaiaawMcaaaaa@4272@ .

From (3.2) we obtain the hardness curve of the resinometallic amortizator

δ=q{ e A( κ ) e A( 1 ) + 1 2 [ E i ( 1,A( κ ) ) E i ( 1,A( 1 ) ) ] } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaeqiTdqMaeyypa0JaamyCaSWaaiWaaOqaaKqzGeGaamyz aSWaaWbaaeqabaqcLbmacqGHsislcaWGbbaddaqadaWcbaqcLbmacq aH6oWAaSGaayjkaiaawMcaaaaajugibiabgkHiTiaadwgalmaaCaaa beqaaKqzadGaeyOeI0IaamyqaWWaaeWaaSqaaKqzadGaaGymaaWcca GLOaGaayzkaaaaaKqzGeGaey4kaSYcdaWcaaGcbaqcLbsacaaIXaaa keaajugibiaaikdaaaWcdaWadaGcbaqcLbsacaWGfbWcdaWgaaqaaK qzadGaamyAaaWcbeaadaqadaGcbaqcLbsacaaIXaGaaiilaiaadgea lmaabmaakeaajugibiabeQ7aRbGccaGLOaGaayzkaaaacaGLOaGaay zkaaqcLbsacqGHsislcaWGfbWcdaWgaaqaaKqzadGaamyAaaWcbeaa daqadaGcbaqcLbsacaaIXaGaaiilaiaadgealmaabmaakeaajugibi aaigdaaOGaayjkaiaawMcaaaGaayjkaiaawMcaaaGaay5waiaaw2fa aaGaay5Eaiaaw2haaaaa@70D2@    (3.3)

For β0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaeqOSdiMaeyOKH4QaaGimaaaa@3F17@ potential [1] it is aimed at the non-zero potential β0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaeqOSdiMaeyOKH4QaaGimaaaa@3F17@ , and the expression (3.3) is about to be known δ=ln(κ)q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azcqGH9aqpciGGSbGaaiOBaiaacIcacqaH6oWAcaGGPaGaamyCaiaa ykW7aaa@40BF@ . Figures 2 are represented by curves (3.3) for W=μ( I C 3 )/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaam4vaiabg2da9iabeY7aTTWaaeWaaOqaaKqzGeGaamys aSWaaSbaaeaajugWaiaadoeaaSqabaqcLbsacqGHsislcaaIZaaaki aawIcacaGLPaaajugibiaac+cacaaIYaaaaa@47D0@ (curve 1) and β=0.9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipD0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqaHYo GycqGH9aqpcaaIWaGaaiOlaiaaiMdaaaa@3AFC@ (curve 2). The linear direct corresponds to the non-zero potential in all cases β=0.9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipD0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqaHYo GycqGH9aqpcaaIWaGaaiOlaiaaiMdaaaa@3AFC@ . It is clear that the dependencies are non-linear and approach with decrease β=0.9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipD0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqaHYo GycqGH9aqpcaaIWaGaaiOlaiaaiMdaaaa@3AFC@ to the non-dependent dependence of the lower; they are described by more severe materials.

Poskol'ku potential [1] relates to the class of generalized non-magnetic potentials, this voltage state does not drive in the transverse plane. Introducing interest is a consideration of the potential, which depends on both main algebraic invariants β=0.9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipD0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqaHYo GycqGH9aqpcaaIWaGaaiOlaiaaiMdaaaa@3AFC@ .In this case, a field of voltage is applied in the lateral plane.

Solution for potential [2]

The expression for the potential has the form:

β=0.9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipD0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqaHYo GycqGH9aqpcaaIWaGaaiOlaiaaiMdaaaa@3AFC@  

brought into forms, for which the equation of state with this potential in small deformations is transferred to the Guka law.

Correlation (2.7) for the potential [2] is equivalent to the third-order algebraic equation relative to v ˙ ( ρ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGabmODayaacaWcdaqadaGcbaqcLbsacqaHbpGCaOGaayjk aiaawMcaaaaa@3FCA@ :

( 2β3 )ρ [ v ˙ ( ρ ) ] 3 +3q [ v ˙ ( ρ ) ] 2 9ρ v ˙ ( ρ )+9q=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaSWaaeWaaOqaaKqzGeGaaGOmaiabek7aIjabgkHiTiaaiodaaOGa ayjkaiaawMcaaKqzGeGaeqyWdi3cdaWadaGcbaqcLbsaceWG2bGbai aalmaabmaakeaajugibiabeg8aYbGccaGLOaGaayzkaaaacaGLBbGa ayzxaaWcdaahaaqabeaajugWaiaaiodaaaqcLbsacqGHRaWkcaaIZa GaamyCaSWaamWaaOqaaKqzGeGabmODayaacaWcdaqadaGcbaqcLbsa cqaHbpGCaOGaayjkaiaawMcaaaGaay5waiaaw2faaSWaaWbaaeqaba qcLbmacaaIYaaaaKqzGeGaeyOeI0IaaGyoaiabeg8aYjqadAhagaGa aSWaaeWaaOqaaKqzGeGaeqyWdihakiaawIcacaGLPaaajugibiabgU caRiaaiMdacaWGXbGaeyypa0JaaGimaaaa@6726@

The following equation has significant coefficients, that is to say, whether it is three important corners, whether two complexally connected and one of the pure root. In our case for values 0q<0.8,0β<3/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYlg9frVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaaGimaiabgsMiJkaadghacqGH8aapcaaIWaGaaiOlaiaa iIdacaGGSaGaaGPaVlaaykW7caaMc8UaaGPaVlaaicdacqGHKjYOcq aHYoGycqGH8aapcaaIZaGaai4laiaaikdaaaa@4F33@ the second option is implemented. The only real root is:

v ˙ ( ρ )= i[ ( i 3 ) q 2 +2iq H 1/3 +( i+ 3 ) H 2/3 +3( i 3 ) ρ 2 ( 2β3 ) ] 2 H 1/3 ( 2β3 )ρ , i 2 =1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYlg9frVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGabmODayaacaqcfa4aaeWaaOqaaKqzGeGaeqyWdihakiaa wIcacaGLPaaajugibiabg2da9KqbaoaalaaakeaajugibiaadMgaju aGdaWadaGcbaqcfa4aaeWaaOqaaKqzGeGaamyAaiabgkHiTKqbaoaa kaaakeaajugibiaaiodaaSqabaaakiaawIcacaGLPaaajugibiaadg hajuaGdaahaaWcbeqaaKqzadGaaGOmaaaajugibiabgUcaRiaaikda caWGPbGaamyCaiaadIeajuaGdaahaaWcbeqaaKqzadGaaGymaiaac+ cacaaIZaaaaKqzGeGaey4kaSscfa4aaeWaaOqaaKqzGeGaamyAaiab gUcaRKqbaoaakaaakeaajugibiaaiodaaSqabaaakiaawIcacaGLPa aajugibiaadIeajuaGdaahaaWcbeqaaKqzadGaaGOmaiaac+cacaaI ZaaaaKqzGeGaey4kaSIaaG4maKqbaoaabmaakeaajugibiaadMgacq GHsisljuaGdaGcaaGcbaqcLbsacaaIZaaaleqaaaGccaGLOaGaayzk aaqcLbsacqaHbpGCjuaGdaahaaWcbeqaaKqzadGaaGOmaaaajuaGda qadaGcbaqcLbsacaaIYaGaeqOSdiMaeyOeI0IaaG4maaGccaGLOaGa ayzkaaaacaGLBbGaayzxaaaabaqcLbsacaaIYaGaamisaKqbaoaaCa aaleqabaqcLbmacaaIXaGaai4laiaaiodaaaqcfa4aaeWaaOqaaKqz GeGaaGOmaiabek7aIjabgkHiTiaaiodaaOGaayjkaiaawMcaaKqzGe GaeqyWdihaaiaacYcacaaMc8UaaGPaVlaadMgajuaGdaahaaWcbeqa aKqzadGaaGOmaaaajugibiabg2da9iabgkHiTiaaigdaaaa@95A6@    (3.4)

H= q 3 9q( 2 β 2 5β+3 ) ρ 2 +3( 2β3 )ρ q 4 +3( 3 β 2 6β+2 ) q 2 ρ 2 +3( 32β ) ρ 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYlg9frVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaamisaiabg2da9iabgkHiTiaadghajuaGdaahaaWcbeqa aKqzadGaaG4maaaajugibiabgkHiTiaaiMdacaWGXbqcfa4aaeWaaO qaaKqzGeGaaGOmaiabek7aILqbaoaaCaaaleqabaqcLbmacaaIYaaa aKqzGeGaeyOeI0IaaGynaiabek7aIjabgUcaRiaaiodaaOGaayjkai aawMcaaKqzGeGaeqyWdixcfa4aaWbaaSqabeaajugWaiaaikdaaaqc LbsacqGHRaWkcaaIZaqcfa4aaeWaaOqaaKqzGeGaaGOmaiabek7aIj abgkHiTiaaiodaaOGaayjkaiaawMcaaKqzGeGaeqyWdixcfa4aaOaa aOqaaKqzGeGaamyCaKqbaoaaCaaaleqabaqcLbmacaaI0aaaaKqzGe Gaey4kaSIaaG4maKqbaoaabmaakeaajugibiaaiodacqaHYoGyjuaG daahaaWcbeqaaKqzadGaaGOmaaaajugibiabgkHiTiaaiAdacqaHYo GycqGHRaWkcaaIYaaakiaawIcacaGLPaaajugibiaadghajuaGdaah aaWcbeqaaKqzadGaaGOmaaaajugibiabeg8aYLqbaoaaCaaaleqaba qcLbmacaaIYaaaaKqzGeGaey4kaSIaaG4maKqbaoaabmaakeaajugi biaaiodacqGHsislcaaIYaGaeqOSdigakiaawIcacaGLPaaajugibi abeg8aYLqbaoaaCaaaleqabaqcLbmacaaI0aaaaaWcbeaaaaa@8CC2@

By (3.4) there is an expression for v( ρ )= 1 ρ v ˙ ( t )dt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYlg9frVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaamODaKqbaoaabmaakeaajugibiabeg8aYbGccaGLOaGa ayzkaaqcLbsacqGH9aqpjuaGdaWdXbGcbaqcLbsaceWG2bGbaiaaaS qaaKqzadGaaGymaaWcbaqcLbmacqaHbpGCaKqzGeGaey4kIipajuaG daqadaGcbaqcLbsacaWG0baakiaawIcacaGLPaaajugibiaadsgaca WG0baaaa@51A6@ and a rigid curve amortizator (dependence from) 

δ= 1 κ v ˙ ( ρ )dρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYlg9frVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaeqiTdqMaeyypa0tcfa4aa8qCaOqaaKqzGeGabmODayaa caaaleaajugWaiaaigdaaSqaaKqzadGaeqOUdSgajugibiabgUIiYd qcfa4aaeWaaOqaaKqzGeGaeqyWdihakiaawIcacaGLPaaajugibiaa dsgacqaHbpGCaaa@4EC7@     (3.5)

When β0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaeqOSdiMaeyOKH4QaaGimaaaa@3F17@ the potential [2] tends to the Neo-Hooke potential. Figure 3 shows the curves (3.5) for β=1.3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipD0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqaHYo GycqGH9aqpcaaIXaGaaiOlaiaaiodaaaa@3AF7@ curve 1) and β=1.0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipD0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqaHYo GycqGH9aqpcaaIXaGaaiOlaiaaicdaaaa@3AF4@ (curve 2). The shaded line corresponds to the Neo-Hooke potential. In all cases κ=2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipD0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqaH6o WAcqGH9aqpcaaIYaaaaa@399A@ . It can be seen that the non-linear dependences approach the decrease β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVC0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqaHYo Gyaaa@37A7@ to the Neo-Hukonian dependence from above, that is, they describe the softer materials.

It is of interest to compare the distribution of tangential stresses σ rz MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtg9frVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaeq4Wdm3cdaWgaaqaaKqzadGaamOCaiaadQhaaSqabaaa aa@3FBE@ that are not taken into account by the linear theory, with shear stresses from longitudinal shear .

For the potential [2] W 2 =μβ [ v ˙ 2 ( ρ )+3 ] 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtg9frVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaam4vaSWaaSbaaeaajugWaiaaikdaaSqabaqcLbsacqGH 9aqpcqaH8oqBcqaHYoGylmaadmaakeaajugibiqadAhagaGaaSWaaW baaeqabaqcLbmacaaIYaaaaSWaaeWaaOqaaKqzGeGaeqyWdihakiaa wIcacaGLPaaajugibiabgUcaRiaaiodaaOGaay5waiaaw2faaSWaaW baaeqabaqcLbmacqGHsislcaaIXaaaaaaa@516B@ and equation (2.6) can be rewritten in the form

p 0 = 2 1 κ 2 1 κ [ 1 ρ 1 t d dt [ βρ v ˙ 2 ( t ) [ v ˙ 2 ( t )+3 ] ] dt β [ v ˙ 2 ( ρ )+3 ] v ˙ ( ρ ) ] ρdρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtg9frVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaamiCaSWaaSbaaeaajugWaiaaicdaaSqabaqcLbsacqGH 9aqplmaalaaakeaajugibiaaikdaaOqaaKqzGeGaaGymaiabgkHiTi abeQ7aRTWaaWbaaeqabaqcLbmacaaIYaaaaaaalmaapehakeaalmaa dmaakeaalmaapehakeaalmaalaaakeaajugibiaaigdaaOqaaKqzGe GaamiDaaaalmaalaaakeaajugibiaadsgaaOqaaKqzGeGaamizaiaa dshaaaWcdaWadaGcbaWcdaWcaaGcbaqcLbsacqaHYoGycaaMc8Uaeq yWdiNaaGPaVlqadAhagaGaaSWaaWbaaeqabaqcLbmacaaIYaaaaSWa aeWaaOqaaKqzGeGaamiDaaGccaGLOaGaayzkaaaabaWcdaWadaGcba qcLbsaceWG2bGbaiaalmaaCaaabeqaaKqzadGaaGOmaaaalmaabmaa keaajugibiaadshaaOGaayjkaiaawMcaaKqzGeGaey4kaSIaaG4maa GccaGLBbGaayzxaaaaaaGaay5waiaaw2faaaWcbaqcLbmacaaIXaaa leaajugWaiabeg8aYbqcLbsacqGHRiI8aiaaykW7caWGKbGaamiDai abgkHiTSWaaSaaaOqaaKqzGeGaeqOSdigakeaalmaadmaakeaajugi biqadAhagaGaaSWaaWbaaeqabaqcLbmacaaIYaaaaSWaaeWaaOqaaK qzGeGaeqyWdihakiaawIcacaGLPaaajugibiabgUcaRiaaiodaaOGa ay5waiaaw2faaaaajugibiqadAhagaGaaSWaaeWaaOqaaKqzGeGaeq yWdihakiaawIcacaGLPaaaaiaawUfacaGLDbaaaSqaaKqzadGaaGym aaWcbaqcLbmacqaH6oWAaKqzGeGaey4kIipacaaMc8UaaGPaVlabeg 8aYjaadsgacqaHbpGCaaa@99D0@    (3.6)

Substituting (3.4) into (3.6) we obtain an expression for finding,that is not given because of the unwieldiness.

For example, Figure 4 shows the dependences p 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtg9frVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaamiCaSWaaSbaaeaajugWaiaaicdaaSqabaaaaa@3DB4@ of q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtg9frVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaamyCaaaa@3B96@ for β=1.3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtg9frVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaeqOSdiMaeyypa0JaaGymaiaac6cacaaIZaaaaa@3F71@ (curve 1) and β=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtg9frVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaeqOSdiMaeyypa0JaaGymaaaa@3E01@ (curve 2). In both cases κ=2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtg9frVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaeqOUdSMaeyypa0JaaGOmaaaa@3E14@ .

For the tangential stress,

σ φφ =2p=2μ[ 1 ρ 1 t d dt [ βρ v ˙ 2 ( t ) [ v ˙ 2 ( t )+3 ] ] dt+ p 0 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtg9frVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaeq4Wdm3cdaWgaaqaaKqzadGaeqOXdOMaeqOXdOgaleqa aKqzGeGaeyypa0JaaGOmaiaadchacqGH9aqpcaaIYaGaeqiVd02cda WadaGcbaWcdaWdXbGcbaWcdaWcaaGcbaqcLbsacaaIXaaakeaajugi biaadshaaaWcdaWcaaGcbaqcLbsacaWGKbaakeaajugibiaadsgaca WG0baaaSWaamWaaOqaaSWaaSaaaOqaaKqzGeGaeqOSdiMaaGPaVlab eg8aYjaaykW7ceWG2bGbaiaalmaaCaaabeqaaKqzadGaaGOmaaaalm aabmaakeaajugibiaadshaaOGaayjkaiaawMcaaaqaaSWaamWaaOqa aKqzGeGabmODayaacaWcdaahaaqabeaajugWaiaaikdaaaWcdaqada GcbaqcLbsacaWG0baakiaawIcacaGLPaaajugibiabgUcaRiaaioda aOGaay5waiaaw2faaaaaaiaawUfacaGLDbaaaSqaaKqzadGaaGymaa WcbaqcLbmacqaHbpGCaKqzGeGaey4kIipacaaMc8Uaamizaiaadsha cqGHRaWkcaWGWbWcdaWgaaqaaKqzadGaaGimaaWcbeaaaOGaay5wai aaw2faaaaa@7B05@   (3.7)

Substituting (3.4) into (3.7) we obtain the final expression. Figure 5 shows the distributions of the relative stresses σ rz /μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtg9frVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaeq4Wdm3cdaWgaaqaaKqzadGaamOCaiaadQhaaSqabaqc LbsacaGGVaGaeqiVd0gaaa@42B6@ c(curve 1) and σ φφ /μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtg9frVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaeq4Wdm3cdaWgaaqaaKqzadGaeqOXdOMaeqOXdOgaleqa aKqzGeGaai4laiabeY7aTbaa@443A@ (curve 2) for β=1.3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtg9frVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaeqOSdiMaeyypa0JaaGymaiaac6cacaaIZaaaaa@3F71@ and κ=2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtg9frVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaeqOUdSMaeyypa0JaaGOmaaaa@3E14@ . It is seen that the concentration of tangential stresses is quite comparable to the concentration of tangential stresses, although lower.

Figure 6 shows the distribution of the specific energy of deformation energy W/μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8urps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaajugibi aadEfacaGGVaGaeqiVd0gaaa@3B9E@ across the cross section of the sleeve, for W/μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8urps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaajugibi aadEfacaGGVaGaeqiVd0gaaa@3B9E@ and different β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8urps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaajugibi abek7aIbaa@39FA@ . At β=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8urps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaajugibi abek7aIjabg2da9iaaicdaaaa@3BBA@ (Neo-Hooke potential) we have curve 1. In this case, there is no latent stress field in the transverse plane. The values ​​of β=1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8urps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaajugibi abek7aIjabg2da9iaaigdaaaa@3BBB@ and β=1.2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8urps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaajugibi abek7aIjabg2da9iaaigdacaGGUaGaaGOmaaaa@3D29@ correspond to curves 2 and 3. It is seen that with increasing transverse stress field not taken into account by the linear theory of elasticity, the concentration of the deformation energy potential increases and exceeds the corresponding values ​​for the Neo-Hooke potential. This may be important, since there are experimental data31 that the destruction of rubber products does not begin in the zones of maximum stresses, but in the zones of maximum values ​​of the deformation energy potential.

Figure 2 Representation of curves (3.3) for β=2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipD0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqaHYo GycqGH9aqpcaaIYaaaaa@3989@  (curve 1) and β=0.9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipD0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqaHYo GycqGH9aqpcaaIWaGaaiOlaiaaiMdaaaa@3AFC@ (curve 2).

Figure 3 Representation of curves (3.5) for β=1.3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipD0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqaHYo GycqGH9aqpcaaIXaGaaiOlaiaaiodaaaa@3AF7@  (curve 1) and β=1.0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipD0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqcLbsacqaHYo GycqGH9aqpcaaIXaGaaiOlaiaaicdaaaa@3AF4@ (curve 2).

Figure 4 The dependences p 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtg9frVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaamiCaSWaaSbaaeaajugWaiaaicdaaSqabaaaaa@3DB4@ of q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtg9frVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaamyCaaaa@3B96@ for β=1.3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtg9frVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaeqOSdiMaeyypa0JaaGymaiaac6cacaaIZaaaaa@3F71@  (curve 1) and β=1.3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtg9frVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaeqOSdiMaeyypa0JaaGymaiaac6cacaaIZaaaaa@3F71@  (curve 2).

Figure 5 The distributions of the relative stresses σ rz /μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtg9frVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaeq4Wdm3cdaWgaaqaaKqzadGaamOCaiaadQhaaSqabaqc LbsacaGGVaGaeqiVd0gaaa@42B6@  (curve 1) and σ φφ /μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtg9frVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaeq4Wdm3cdaWgaaqaaKqzadGaeqOXdOMaeqOXdOgaleqa aKqzGeGaai4laiabeY7aTbaa@443A@ (curve 2) for β=1.3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtg9frVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaeqOSdiMaeyypa0JaaGymaiaac6cacaaIZaaaaa@3F71@ and κ=2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYtg9frVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaO qaaKqzGeGaeqOUdSMaeyypa0JaaGOmaaaa@3E14@ .

Figure 6 The distribution of the specific energy of deformation energy W/μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8urps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaajugibi aadEfacaGGVaGaeqiVd0gaaa@3B9E@ across the cross section of the sleeve, for q=1,κ=2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8urps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaajugibi aadghacqGH9aqpcaaIXaGaaiilaiaaykW7caaMc8UaeqOUdSMaeyyp a0JaaGOmaaaa@424A@ and different β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8urps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciaacaGaaeqabaWaaeaaeaaakeaajugibi abek7aIbaa@39FA@ .

Conclusion

New exact analytical solutions of one problem of nonlinear elasticity theory for two strain energy potentials of incompressible material are obtained. As a model, the well-known problem of the antiplane axisymmetric deformation of a cylindrical sleeve between rigid concentric rings was considered. The nonlinear effect of the stress state in the transverse plane under the action of a finite longitudinal shear is investigated; its effect on the concentration of the deformation energy potential is elucidated.

Acknowledgement

None.

Conflicts of Interest

None.

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