General Solution of the Stress Potential Function in Lekhnitskii s Elastic Theory for Anisotropic and Piezoelectric Materials

Transcription

1 dv. Studies Teor. Pys. Vol. 007 no General Solution o te Stress Potential Function in Lenitsii s Elastic Teory or nisotropic and Pieoelectric Materials Zuo-en Ou StateKey Laboratory o Explosion Science and Tecnoloy eiin Institute o Teconloy P. R. ina cou@bit.edu.cn Yi-Hen en Scool o erospace te Laboratory o MSSV Xian Jiaoton niversity P. R. ina ycen@mail.xtu.edu.cn bstract It is ound tat te oriinal Lenitsii eneral solution Lenitsii 9 is ar rom bein completeness because some important terms were lost. New types o complete eneral solution or anisotropic as well as pieoelectric materials are ten developed analytically. Te new eneral solutions include te oriin one completely. nlie te oriinal one te new eneral solution can be proved to deenerate normally to te Muselisvili teory or isotropic materials. Hence te Lenitsii and te Muselisvili teory are uniied and te well-nown imperect in te oriinal Lenitsii teory wic perplexed researcers or a lon time is avoided. Keywords: Lenitsii s teory; General solution; Pieoelectric materials; nisotropic materials; ompleteness

2 8 Zuo-en Ou and Yi-Hen en. Introduction It is well nown tat bot te elastic mecanics and te racture mecanics o anisotropic as well as pieoelectric materials are very important in teory analyses and practical enineerin applications in devisin some structures and components. s a oundation stone o te elastic mecanics and te racture mecanics Lenitsii s teory Lenitsii 9 as been widely used in dealin wit various inds o problems in te anisotropic and pieoelectric elastic mecanics over te past several decades. ased on te eneral solution o te stress potential unction in te teory a lot o researc wor as been done. However as pointed out by some previous researcers Tin 99; Yin 000 Lenitsii s teory as a serious questionable point i.e. it can not deenerate bac to te Muselisvili teory Muselisvili 9 or isotropic elastic mecanics. Let us see it in more details and see wat will appen. In Muselisvili s teory te iry stress unction M x y or a plane problem is expressed in te orm well-nown Goursat ormulism M x y Re[ χ ] were Re represents te real part o te related complex unction and te overbar means te transpose; and χ are arbitrary analytical unctions bot o wic are deined on te wole -plane and x iy is a complex variable. In Lenitsii s teory or a plane strain problem in anisotropic materials te iry stress unction L x y is expressed as x y Re[ ] L were and are arbitrary analytical unctions deined on te - and -plane respectively. x μ y and x μ y are two complex variables and μ μ wit teir conuates are te our caracteristic roots o te related eienvalue problem. omparin Eqs. and sows clearly tat or isotropic materials and tus μ μ Eq. can not be induced directly into Eq. and wic or a lon time as been seein as a imperect in Lenitsii s teory. Tin 99 and Yin 000 discussed tis problem in scarcely way and presented some solutions or so-called deenerate and extra-deenerate materials includin isotropic materials. However up to now altou muc o suc wor is done te correctness or in oter words te completeness o te Lenitsii teory is not veriied and some questions are still existed: wy te Lenitsii teory can not be deenerated into te

3 General solution o stress potential unction 9 Muselisvili teory wic is an obvious act in view o te pysical point? Wy te expression or te iry stress unction in anisotropic materials wic is relatively more complicated is simpler tan tat in isotropic materials and is it really available? Eq. sows tat te iry stress unction in isotropic materials is not an analytical unction. It is so strane tat te iry stress unction or an anisotropic material wic is muc more complicate tan te isotropic one can o so ar as to be an analytical unction wy? One o reasonable replies to above questions sould be clearly tat some mistaes must be existed in te oriinal Lenitsii teory wic is ust an important problem solved in tis paper. Moreover recently more and more contradictions between experimental data and teoretical predication in te pieoelectric racture mecanics are observed wic as perplexed researcers or a lon time and it also be a promotion to re-examine te teoretical tools adopted in tose analyses in more details altou some oter researcers maybe interestin in contributin tese contradictions to some microscope actors suc as electric domain switc or te nonlinearity o crac-tip ield. In te ollowin sections a detailed teoretical manipulation is developed and a new type o te eneral solution o te stress potential unction in Lenitsii s teory or anisotropic materials is obtained. Some sort discussions are ten ollowed. dditionally te eneral potential unction solution under eneralie plane strain in a pieoelectric material is also presented.. asic ormulism Te Manipulation about te new eneral potential unction solution is so tedious tat or te brieness only te simplest case as an example is presented ere. onsiderin an x-y plane elastic problem under plane strain state or an anisotropic material rom Lenitsii s ormula Lenitsii 9 te eneral solution o te stress potential unction will be derived by solvin directly te dierential equation as ollow: D D D D 0 were te dierential operator D / y μ / x and μ are te caracteristic roots in Lenitsii s teory; x y is iry s stress unction wic is deined by

4 0 Zuo-en Ou and Yi-Hen en σ xx σ yy σ xy y x x y and σ xx σ yy σ xy are correspondin components o te stress tensor. Eq. is equivalent to te ollowin roup o dierential equations D D D D 0. In order to develop te solution o Eq. enerally by usin complex variables x μ y one can transorm dierential operators D into te ollowin orm: D μ μ D μ μ and it can be easily sown tat d d d ; 7a were μ μ μ μ. 7b μ μ μ μ Equation sows tat any complex variable can be represented as a unction o oter two independent variables and wic implies tat an arbitrary complex unction o variable and can be transerred directly into te unction o te variables and and tis will be very important in te ollowin manipulations. Lenitsii 9 sowed tat te caracteristic roots μ must be complex numbers so tat tey can be expressed as μ μ μ μ and μ μ. Tus Eqs. and can be rewritten respectively in te orm and D D D D 0 8 D D D D 0. 9

5 General solution o stress potential unction Let us now solve Eq.9 step by step. Firstly in virtue o Eq. te last equation in Eq.9 ives out 0 0 were μ μ are complex constants wic are determined completely by material constants. One ence reaces tat remember tat as described above can be seen as a unction o te variables and now were is an arbitrary unction. Secondly by substitutin Eq. into te tird equation in Eq.9 one as. Tus wit te second equation in Eq.7a te unction can be interated as: d d d were anoter arbitrary unction is introduced and we denote tat d. Tirdly rom te second equation in Eq.9 and te expression o te unction as described in Eq. one arrives to. Similarly tis ives te unction as ollowin:

6 Zuo-en Ou and Yi-Hen en d d d d and some oter arbitrary unctions and are introduced and denoted as d d. 7 t last by substitutin Eq.7 into te irst equation in Eq.9 one as 8 and ten te eneral potential unction solution can be obtained by solvin above equation as 9 were te newly introduced arbitrary unction are expressed as: d d d 0 Moreover rom te symmetrical nature o te caracteristic roots Eq.9 can be written or example as 0 D D D D a and ten te correspondin equation roup is 0 D D D D. b Tus or tis equation roup one can et its solution in te same manner as described above as:

7 General solution o stress potential unction. Here and ollowin all appeared new unctions are arbitrary. Similarly oter symmetrical solution o te potential unction can be derived in te ollowin orms: ; a ; b y tain te conuation o above our solutions 9 and and summin all o tem up one can derive te needed eneral potential unction solution as { } Re ψ ψ ψ χ χ χ were χ a 9 χ b 8 9 χ c

8 Zuo-en Ou and Yi-Hen en ψ d ψ 7 e ψ Discussion new type o te eneral solution o te stress potential unction in Lenitsii s elastic teory o a plane strain problem in anisotropic materials is obtained analytically as sown in Eq.. It is seen tat te new eneral solution not only includes completely te oriinal one wen te terms concern wit unctions χ χ ψ and ψ are nelected derived by Lenitsii 9 imsel but also be muc dierent rom te oriinal one. Tis means tat te oriinal eneral solution is ar rom bein completeness wit some important terms nelected. Moreover it sould be noticed tat or isotropic materials μ μ and and ten 0 rom Eq.7b and ten χ ψ 0 rom Eqs. a and d wic implies tat Eq. induces directly into Eq. i.e. te new complete Lenitsii solution can be deenerated to Muselisvili solution. Te Muselisvili teory is tus included in te new complete Lenitsii teory and te two teories are uniied now. Tis also provides an indirect veriication o te validity o te present solution. It can also be seen clearly rom te manipulatin process o te new solution described above tat te orm o te eneral stress potential unction solution is dependin on te practical problem considered. For instance in treatin a eneralied plane problem in an anisotropic material in wic te control dierential equation is in six order and its eneral potential unction solution is o te orm

9 General solution o stress potential unction Re { } ψ ψ ψ 0 ψ. Namely in a eneral potential unction solution te iest order o te variable is equal to al te order o te control dierential equation. Hence or a eneralied plane problem in a pieoelectric material te eneralied solution will tae te orm o Re { } ψ ψ ψ ψ 0 ψ. 7 From te new orm o a eneral potential unction solution expressed by Eqs. and 7 or correspondin materials under certain conditions it can be observed tat te newly derived solutions are muc more eneral as well as complicate comparin wit te oriinal one. nder te new solution accordinly te expressions o some related pysical quantities suc as stress strain and displacement as well as electrical quantities suc as electric ield and electric displacement etc must cane in comparison wit te existed one. Hence it can be expected tat under te new eneral potential unction solution some new teoretical results in anisotropic and pieoelectric racture mecanics must be arrived and wic may rewritte te existed teoretical conclusions reatly. On te oter and te well nown Stro teory Stro 98 9 wic as also been used widely in anisotropic and pieoelectric elastic and racture mecanics sould be reconsidered too in details since its eneral solution was reaced by adoptin a inverse metod eiter in wic some terms in te solutions maybe inored too. Finally it sould be pointed out tat weter te new solution is a complete one in matematics is not sure certainly. Only can be sown is tat te more completed eneral potential unction solution comparin wit te existed one is reaced in tis paper. cnowledements Tis wor was supported by te inese National Science Foundation Grant No. 070 and te Proram or anian Scolars and Innovative Researc Team in university. Te autors also wis to tan te National asic Researc Proram o ina trou Grant No or inancial support o tis wor.

10 Zuo-en Ou and Yi-Hen en Reerences [] S.G. Lenitsii Teory o elasticity o an anisotropic elastic body Holden-Day New Yor 9. [] N.I. Muselisvili Some basic problems o te matematical teory o elasticity Noordo Leyden 9. [].N. Stro Dislocations and cracs in anisotropic elasticity Pil. Ma [].N. Stro Steady state problems in anisotropic elasticity J. Mat. Pys [] T..T. Tin nisotropic elasticity teory and applications Oxord niversity Press New Yor 99. [] W.L. Yin Deconstructin plane anisotropic elasticity Part I: Te latent structure o Lenitsii's ormalism Int. J. Solids Struct Received: pril 007