MAC-solutions of the nonexistent solutions of mathematical physics

Transcription

1 Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences - Finite Elements - Finite Volumes - Boundry Elements MAC-solutions of the nonexistent solutions of mthemticl physics IGO NEYGEBAUE University of Dodom Deprtment of Mthemtics nd Sttistics P.O.Box 259, Dodom TANZANIA newigor52r@yhoo.com Abstrct: The method of dditionl conditions or MAC is pplied to the boundry vlue problems of mthemticl physics, where the clssicl solution does not exist or nonphysicl generlized solution is obtined. The Dirichlet problem for circulr elstic membrne is considered s the simplest exmple of the problem with nonexistent solution. The constnt boundry conditions re given in the center of the membrne nd on the finite rdius. The scheme of Dugdle-Brrenblt in Griffith s liner elstic crck problem is used to obtin the MAC-solution. The dditionl pressure on membrne is pplied ner the center, the conditions of smoothness re used, the line-integrl, which follows from the second Green s identity is considered nd it is tken s mesure of the introduced correction of the initil Dirichlet problem for plce eqution. The pplied conditions crete n unique solution, which t lest corresponds to the physicl sitution nd esily cn be proofed experimentlly. The similr situtions re briefly considered, where MAC pproch seems to be useful: circulr elstic plte, symmetric vibrtions of membrne, Kirsch problem, me problem. Key Words: Singulrity, Mthemticl Physics Equtions, Method of Additionl Conditions 8 Mrch 20 Introduction Some clssicl boundry vlue problems from elsticity will be considered. It is esy to solve these problems nd obtin the generl solution of the differentil equtions. The problem is to stisfy the prescribed boundry conditions. We will see tht the solutions of some problems does not exist. In this cse it is possible to crete generlized solution, using limit of existing solutions. These solutions cn be esily verified in experiments. Experiment show, tht the physicl solution exist nd differs from the obtined generlized solution. Then the MAC or the method of dditionl conditions cn be pplied. This method llows to trnsform the obtined generlized solution to the physiclly cceptble form. The MAC ws introduced bout yers go in scheme of Dugdle-Brrenblt [, 3] in frcture mechnics. This scheme ws pplied to the liner elstic crck problem. The liner elstic solution hs singulrity ner the tip of crck. To void this singulrity Dugdle nd Brrenblt introduced dditionl yield stresses ner the tip. The pplied nonsingulr condition gve the size of the zone, there the stresses re pplied. This scheme ws developed in [5], where the second dditionl condition of zero J-integrl ws introduced. The MAC solution for the plce eqution in n ngle ws considered. The principle of superposition ws used to crete the MAC model for the membrne in [6]. This new condition gve the vlue of the pplied dditionl stresses, which re 6 times more then the given stresses t infinity. This stress concentrtion fctor corresponds to the experiments of Griffith s nd Inglis [3]. The obtined the liner elstic field ner the tip of crck mkes it possible to use lot of nlyticl, numericl methods to nlyze the stress field in different situtions. The usul strength criteri, which re used in the elstic stress fields without singulrities, cn be used here. et us pply the method of dditionl conditions MAC to consider the displcements of n elstic membrne. ISBN:

2 Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences - Finite Elements - Finite Volumes - Boundry Elements 2 Circulr elstic membrne 2. Nonexistent solution et us consider circulr elstic membrne with symmetric Dirichlet boundry conditions. Then it is resonble to consider the symmetric solution. The plce s eqution in this 2D cse is: r We consider domin Ω: r u The boundry conditions re: = 0; 0 r ; 2 u0 = u 0 0; 3 u = 0; 4 The generl solution of the eqution is u = A + B ln r ; 5 where A, B - re rbitrry constnts. It follows from the first boundry condition t r = 0, tht u0 = A = u 0 ; B = 0 ; 6 Then the solution cn not stisfy the second boundry condition, becuse we hve u = A = u 0 0; 7 It is esy to see in the physicl model of membrne, tht the solution must exist. 2.2 Generlized solution We cn obtin the generlized solution in the following wy. et us consider the membrne s ring: 0 < r ; 8 The plce eqution in symmetric 2D cse will be r r u = 0 9 The boundry conditions re u = u 0 0; 0 u = 0 The solution of the boundry vlue problem 9, 0, is u = u 0 ln ln + u 0 ln lnr 2 If 0, then the generlized solution of the initilly given problem, 2, 3, 4 is u = u 0 for r = 0, u = 0 for 0 < r. This solution is not physicl, becuse s we cn esy see in experiment, tht the solution hs finite derivtive t r = 0 nd is not zero in the domin. 2.3 MAC-solution 2.3. The chnged boundry conditions Now we will consider the MAC solution of our problem with nonexistent solution. It is not importnt, but convenient, to chnge slightly the boundry conditions 3, 4 to the following ones u0 = 0; 3 u = u 0; Generlized solution The the generlized solution in this cse will tke the form u0 = 0, 5 ur = u, 0 < r Algorithm to obtin the MACsolution To crete the MAC-solution we use the following lgorithm: The second Green s identity gives the invrint integrl like J-integrl in frcture mechnics. The generlized solution is used. The pplied externl pressure ner the center of the membrne in the domin Ω : 0 r is constnt. Solution hs not singulrity in grdient t r = 0. The domin Ω is divided in two prts: Domin Ω : 0 r ; Domin Ω 2 : < r. Solution u is from the clss C. The mximum of the J- integrl in the first domin must be equl to tht constnt vlue, which is obtined on the clssicl generlized solution. ISBN:

3 Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences - Finite Elements - Finite Volumes - Boundry Elements Additionl condition in the form of the line integrl To obtin the dditionl condition in the form of line integrl let us consider the clssicl second Green s identity [4]: Ω v u u v dω = Ω v u u v ds, 7 where the functions u nd v ɛ C 2 Ω. Consider ny two solutions of plce eqution u, v then we hve: u = 0; 8 v = 0. 9 Then we obtin for ny closed contour which is piecewise smooth nd belongs to Ω v u u v ds = Consider = + 2, where the contours, 2, re : r = ; 2 where 2 : r = 2 ; 22 0 < 2 ; 23 nd the contour follows in the opposite clockwise direction. We cn split our integrl 20 in two prts f ds = f ds + f ds = 0, 24 2 where f is the following function From 24 we obtin f ds = f = v u u v f ds. 26 If we chnge the direction of integrtion in the integrl long, then we obtin n invrint integrl long ny curve : r =, where the rdius is rbitrry nd stisfies the condition 0 <. We hve J = f ds = v u u v ds = C, 27 where C is some constnt. The formul 27 is right for ny closed piecewise smooth curve round the origin O. It tkes the following form in cse of circumference : r = : J = v u u v r dϕ = :r= 2 π r v u u v = C The vlue of the J-integrl et us clculte this constnt, tking two solutions of the plce eqution. The first solution u = u is prt of the generlized solution 5, 6. The second solution is the generl solution of the plce eqution : v = A + B lnr, 29 where A, B re rbitrry constnts nd they will be determined below. Then we get from 28: J = 2 π u B = C. 30 And invrint integrl J is equl in this cse J = 2 π u B = C. 3 The constnt B will be determined lter The vlue J0 for physicl solution The experiments with membrne show tht it is resonble to suggest tht the rel physicl solution hs the following properties: the deflection is smooth nd its derivtive is bounded t the origin nd in its vicinity. If we denote the MAC-solution s ũ, then we obtin in the smll enough vicinity of the origin r = 0: ũ C, 32 ũ C r, 33 where C is some positive constnt. It is esy now to estimte the vlue of function Jr ccording to 27 or 28 t r 0, where the generlized solution u is replced by the physicl MAC-solution ũ. We obtin in the vicinity of the origin Jr = 2 π r v ũ ũ v 2 π r v ũ + ũ v. 34 ISBN:

4 Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences - Finite Elements - Finite Volumes - Boundry Elements Using 32, 33 the inequlity 34 tkes the form Jr 2 π r A + B ln r C + C r B r, or we cn write 35 Jr 2 π r C A + B ln r + B. 36 It follows from the inequlity 36 tht we cn define the vlue of J0 s J0 = lim r 0 Jr = If we tke function v ccording to 29 such tht B 0. Then we obtin from 3 tht the constnt C 0. It mens tht the condition 37 is not fulfilled in cse of generlized solution 5, 6. emrk on the J integrl The vlue of the J integrl obtined from the second Green s identity depends on the solution being considered: the generlized solution or the physicl MACsolution. It will be shown below. The first Green s identity does not possess this property. et us consider this identity [4]: u v u dω + Ω x v x + u y v dω = y Ω = Ω v u ds. 38 It is esy to see tht if the generlized solution u 7 is tken then ll integrls in 39 re zero becuse the function u = const for ny smooth enough function v. As we hve seen the second Green s identity hs creted the nonzero line integrl MAC-solution in the domin Ω Consider the domin Ω : 0 r nd the Poisson eqution: r r u = p, 39 where the constnt p is unknown pressure, tht must help to trnsform the nonphysicl generlized solution to the physicl MAC-solution. The prmeter p will be determined some lter. The generl solution of the eqution 39 is u = C 2 + C 3 ln r + p r2 4, 40 where C 2 nd C 3 re rbitrry constnts. We hve to stisfy the boundry condition 3, therefore we obtin u0 = C 2 = 0, 4 C 3 = 0, 42 becuse we hve to void singulrity t r = 0. Then the MAC-solution in the domin Ω is u = p r The prmeter p will be determined lter MAC-solution in the domin Ω 2 Consider the domin Ω 2 : r nd plce eqution: r r u = The generl solution of the eqution 44 is u = A + B lnr, 45 where A, B re rbitrry constnts. The boundry condition 3 gives the reltion between the constnts A nd B : u = A + B ln. 46 Then the MAC-solution 45 cn be written in the form: r u = u + B ln. 47 There re three prmeters to determine: p, B,. We will use two smoothness conditions nd the properties of the function Jr 28 to obtin the unique vlues of these prmeters Smoothness conditions Consider the smoothness conditions t r = on the boundry between domins Ω nd Ω 2. We require tht the MAC-solution is smooth, therefore u 0 = u + 0, 48 u u 0 = If we use the MAC-solutions 43 nd 47, then the smoothness conditions 48 nd 49 give the reltions: p 2 4 = u + B ln, 50 p 2 = B. 5 ISBN:

5 Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences - Finite Elements - Finite Volumes - Boundry Elements The equtions 50 nd 5 llow to get the prmeters p, B s the functions of prmeter. We obtin 2 u B = + 2 ln 52, 4 u p = ln 53. The remining prmeter will be found using the function Jr Extremum of Jr Consider the function Jr ccording to 28, where the function u is the MAC-solution: 43 in Ω nd 47 in Ω 2 nd the function v in the whole domin Ω equls to the function 47. Then it is esy to see tht J0 = 0 nd the property 37 is stisfied. Moreover the function Jr = 0 in the domin Ω 2 becuse the functions u nd v coincide in this domin ccording to the choice of these functions. This function Jr is not zero in the domin Ω nd there we cn consider its extremum. If the function u is the generlized solution of the stted membrne problem 6 for 0 < r then the function Jr is constnt ccording to 3 nd equls Jr = 2 π u B, 54 Introducing the MAC-solution we men to connect the physicl zero vlue J0 with the vlue 54. Then we continue the dditionl pressure to the position where it will be possible smoothly to connect the MAC-solutions in two domins Ω nd Ω 2. Therefore it is resonble to require the following condition: for 0 < r < extrjr = 2 π u B, 55 The function Jr from 28 hs the expression r Jr = 2πr u + B ln p r 2 B pr2. 4 r 56 Or we hve Jr = p π r 2 r u + B ln B We re looking for the points of extremum. Therefore we need the derivtive of the function Jr. The derivtive of the function Jr?? is J r r = 2 p π r u + B ln. 58 The necessry condition for n extremum is J r = 0 59 nd cretes two points. The first one is r = 0 nd Jr = 0. And the second point of extremum stisfies the eqution: r2 u + B ln = 0, 60 where the constnt B is tken from 52. Then we obtin from 52 nd 60 the eqution 2 u r2 u + ln = ln It follows form 6 tht r2 + 2 ln = 0, 62 nd this eqution 62 hs solution for the second extremum r 2 = e. 63 Then the vlue of Jr 2 ccording to 57 nd 63 will be Jr 2 = p π 2 e u + B ln e The expression 64 cn be simplified using 52 s follows nd finlly we obtin the extremum of the function Jr inside the domin Ω : Jr 2 = 2 p u π 2 e + 2 ln dius Consider the condition 55. It mens the equlity of extremum of the function Jr inside the domin Ω 65to the vlue of the J-integrl for the generlized solution 54. This condition is very importnt if we consider the vibrtion of the membrne. The condition 55 is 2 p u π 2 e + 2 ln = 2 π u p 2 2, 66 where the expression for B from 5 is pplied. The nonzero solution of the eqution 66 is = e 2 e. 67 The vlue 67 of is outside the intervl 0 < nd the mximum of Jr 2 65 will be t = nd we hve to ccept this vlue of. It mens tht we hve only domin Ω nd the domin Ω 2 does not exist in considered problem. ISBN:

6 Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences - Finite Elements - Finite Volumes - Boundry Elements Expression for MAC-solution The MAC-solution of the problem, 2, 3 nd 4 is given in the form 43 nd 53 t = s follows r 2 u = u 68 in the domin Ω emrks to the MAC-solution The obtined MAC-solution 68 shows tht if we mesure the function nd its derivtive on the boundry then we cn determine the inclusion with given potentil of ny size. It is evident if we consider n experiment with membrne: ny nonzero displcement in the center of membrne cn be determined through the mesurements on the boundry. If the generlized solution 5, 6 is used then we hve to conclude tht it is impossible to determine the point inclusion in the middle of membrne. It could be interesting to consider this effect of two theories in experiment with n electrosttic field. The clssicl pproch nd the theory is presented in []. 3 Eigenfrequencies of circulr membrne Consider the symmetric vibrtions of circulr membrne in the domin 0 r. The differentil eqution of membrne is r r u = c 2 2 u t The Dirichlet boundry conditions re u0 = 0, u = It is esy to see tht the eigenfrequencies in this problem does not exist. It is clssicl solution of the problem. If we consider the generlized solution tking the domin r nd the first boundry condition in 70 s u = 0, 7 then the eigenfrequencies exist. If 0 then tht eigenfrequencies re the sme s in membrne which does not hve ny hole nd the first boundry condition in 70. Similr result is obtined in more generl problem bout eigenvlues of the plce opertor [2], where the miniml eigenvlue of bll does not depend on very smll internl hole in form of bll. The clssicl nd generlized solutions does not correspond to relity. It cn be esily shown experimentlly. The MAC-solution cn be obtined using similr dditionl condition in form of line integrl 27 nd the sme lgorithm s before. It is evident tht we hve possibility to stisfy the experimentl dt. 4 Conclusion Some problems of mthemticl physics with nonexistent solutions cn hve the MACsolutions. These MAC-solutions re corresponding nd explining the rel physicl situtions. Therefore it is importnt to consider not only clssicl solutions, which cn hve nonphysicl singulrities or even sometimes do not exist, but lso the MAC-solutions of tht problems. The problems in liner elsticity bout stress concentrtion fctors, contct problems, crck problems cn crete very interesting MAC-solutions. The similr problems cn be esily found in other boundry vlue problems for the different prtil differentil equtions. eferences: [] H. Ammri, H. Kng, Polriztion nd Moment Tensores With Applictions to Inverse Problems nd Effective Medium Theory, Springer, 2007, pp. 32. [2] M. H. C. Anis, A.. Aithl, On two functionls connected to the plcin in clss of doubly connected domins in spce-forms, Proc. Indin Acd. Sci. Mth. Sci., Vol. 5, No., 2005, pp [3] V. I. Astfjev, J. N. dev,. V. Stepnov, Nonliner frcture mechnics, Smr, Publisher Smr University, [4] G. Gilbrg, N. S. Trudinger, Elliptic prtil differentil equtions of second order, Springer-Verlg, Berlin, Heidelberg, New York, Tokyo, 983. [5] I. Neygebuer, The method to obtin the finite solutions in the continuum mechnics, Fundmentl reserch t the Sint Petersburg Stte Polytechnic University, [6] I. Neygebuer, MAC solution for rectngulr membrne, Journl of Concrete nd Applicble Mthemtics, Vol.8,No.2, 200, pp ISBN: