On Boussinesq's problem

Transcription

1 Internatinal Jurnal f Engineering Science 39 (2001) 317±322 On Bussinesq's prblem A.P.S. Selvadurai * Department f Civil Engineering and Applied Mechanics, McGill University, 817 Sherbrke Street West, Mntreal, Quebec, Canada H3A 2K6 eceived 14 May 1999; accepted 9 July 1999 Abstract This nte presents an elementary prcedure fr btaining the slutin t Bussinesq's prblem fr the lading f an istrpic elastic halfspace by a cncentrated nrmal lad. Ó 2001 Published by Elsevier Science Ltd. All rights reserved. 1. Intrductin The prblem f determining the state f stress in an istrpic elastic halfspace which is subjected t a cncentrated frce nrmal t a tractin free surface was rst cnsidered by Bussinesq [1]. The slutin t this prblem can be btained by several methds. The rst apprach cnsists f reducing the prblem t a bundary value prblem in ptential thery. When the surface f the halfspace is subjected t nrmal tractins nly, the elasticity prblem is reduced t that f nding a single harmnic functin with all the characteristic features f a single layer distributed ver the plane regin with an intensity prprtinal t the applied nrmal tractins. The slutin t the cncentrated frce prblem is recvered as a special case f the general nrmal lading. The secnd apprach t the slutin f Bussinesq's prblem cmmences with Kelvin's slutin fr the pint frce acting at the interir f an in nite space and utilizes a distributin f cmbinatins f diples, which are equivalent t a distributin f centers f cmpressin alng an axis, t eliminate the shear tractins ccurring n the plane nrmal t the line f actin f the Kelvin frce, thereby recvering Bussinesq's slutin. A third apprach invlves the applicatin f integral transfrm techniques t the slutin f a gverning partial di erential equatin (e.g., fr Lve's strain functin) which can then be used t explicitly satisfy the tractin bundary cn- * Tel.: ; fax: address: apss@civil.lan.mcgill.ca (A.P.S. Selvadurai) /01/$ - see frnt matter Ó 2001 Published by Elsevier Science Ltd. All rights reserved. PII: S (00)

2 318 A.P.S. Selvadurai / Internatinal Jurnal f Engineering Science 39 (2001) 317±322 ditin's applicable directly t Bussinesq's prblem. Details f this prcedure are given by Sneddn [6]. These prcedures are well dcumented in classical treaties and papers by Michell, Lve, Westergaard, Sklnik, Lure, and Timshenk and Gdier [2±4,7±9]. While these appraches represent remarkably insightful prcedures fr btaining a slutin t Bussinesq's prblem, there is the questin as t whether there is a mre direct apprach via which Bussinesq's slutin can be btained. The bjective f this nte is t utline such a prcedure which is relatively elementary, and as far as the authr is aware, has nt been presented in the literature (e.g. [4]). 2. Gverning equatins We cnsider prblems which are symmetric abut the axis H ˆ 0 f a system f spherical plar crdinates ;#; H with 0; 1, # 0; 2p and H 0; p. The slutin f such axisymmetric prblems can be apprached either via the Lame strain ptential u ; H which satis es r 2 u ; H ˆ0 1 r a Lve strain functin U ; H which satis es r 2 r 2 U ; H ˆ0; 2 where r 2 is Laplace's peratr in spherical plar crdinates; i.e., r 2 ˆ ct H 2 H H 2 : 3 The displacement and stress cmpnents derived frm u ; H take the frms 2Gu ˆ u ; 2Gu H ˆ 1 u H 4 and r ˆ 2 u ; r 2 HH ˆ 1 r ## ˆ 1 u ct H 2 u 1 2 u 2 H ; 2 u H ; r H ˆ 2 H u ; respectively. The displacement and stress cmpnents derived frm U ; H are given by 2Gu ˆ cs H 2 1 m r 2 2 U sin H 1 U; 2 H 2Gu H ˆ sin H 2 1 m r U cs H 2 H 2 H 1 U 5 6

3 and A.P.S. Selvadurai / Internatinal Jurnal f Engineering Science 39 (2001) 317± r ˆ cs H 2 r HH ˆ cs H mr2 1 sin H H r ## ˆ cs H sin H r H ˆ cs H H 1 sin H m r 2 2 U sin H H 2 2 m r 2 3 H m r m r 2 1 U 2 mr 2 2 H H 2 1 m r U H 2 ; U; 2 U; U; 7 2 H 2 respectively. In (4)±(7), G and m are, respectively, the shear mdulus and Pissn's rati. We further nte that 2 u ; H is biharmnic. 3. Bussinesq's prblem We start with Kelvin's prblem fr the pint frce f magnitude P K acting at the interir f an istrpic elastic in nite space. A basic bservatin is that since P K is a pint frce and since the medium is f in nite extent, there is n natural length scale assciated with Kelvin's prblem. Yet the use f either a Lame ptential functin r a Lve strain functin shuld yield, thrugh apprpriate di erentiatins with respect t, expressins fr stresses which are f rder 1= 2 t generate the crrect dimensins fr stress (i.e., di erentiatin f u ; 0 twice with respect t and the di erentiatin f U ; H thrice with respect t ). Als the chice f an `exterir' slutin shuld be such that the stresses are nite within the regin (excluding the rigin) and shuld reduce t zer as!1. The axial cmpnent f tractins acting n any clsed surface which enclses the pint f applicatin f the Kelvin frce (r includes it n the bundary f the surface) shuld be identically equal t P K. This invariance requirement als pint t the fact that the dimensins f the stresses shuld be f rder 1= 2 (Michell [4] crrectly makes this bservatin; see als [5]). The exterir slutin fr u ; H is C=, where C is a cnstant. Frm (5) it is clear that the `exterir' Lame slutin will nt yield the required rder 1= 2 fr the stress distributin. Lve's strain functin derived frm this exterir slutin U ; H ˆC 8 will prvide the crrect rder in fr the stress cmpnents. Aviding details, the displacements and stresses applicable t Kelvin's slutin take the frms

4 320 A.P.S. Selvadurai / Internatinal Jurnal f Engineering Science 39 (2001) 317±322 2Gu ˆ 4C 1 m cs H C 3 4m sin H ; 2Gu H ˆ 9 and 2C 2 m cs H r ˆ ; 2 r HH ˆ r ## ˆ C 1 2m cs H 2 ; r H ˆ C 1 2m 2 sin H; 10 where C ˆ PK 1 m : 8p 11 If we cnsider the halfspace regin z P 0 assciated with the slutin t Kelvin's prblem, it is clear that the plane z ˆ 0 is subjected t the stresses r HH ; p 2 ˆ 0; r H ; p ˆ 2 C 1 2m sin H 2 : 12 Cnsider Bussinesq's prblem where the surface f the halfspace is subjected t a cncentrated nrmal frce P B at ˆ 0. Here again, there is n natural length parameter assciated with the prblem and the slutins derived frm either the Lame ptential u ; H r Lve's strain functin U ; H shuld yield the crrect frm f the rder 1= 2 in the apprpriate derivatives t prvide a dimensinally cnsistent measure f the stresses. We have already emplyed the exterir slutin fr u ; H t generate the Lve strain functin fr Kelvin's prblem. Cnsequently a biharmnic slutin cannt be expected t prvide a slutin with the crrect rder 1= 2 fr the variatin in stress. We therefre seek a slutin f the Lame strain ptential which shuld be f a frm such that when di erentiated twice with respect t, the resulting expressin shuld be f rder 1= 2. The required slutin shuld thus be f the frm u ; H ˆA ln f H Š; where A is a cnstant and f H is an arbitrary functin. Substituting (13) in (1) we btain d dh sin H f df dh sin H ˆ 0: The slutin f (14), btained via successive integratins has the fllwing frm: f H ˆexp Z H 0 cs / 1 sin / d/ ˆ 1 cs H : The Lame strain ptential

5 A.P.S. Selvadurai / Internatinal Jurnal f Engineering Science 39 (2001) 317± u ; H ˆA ln cs H 16 gives the stress cmpnents r HH ; H ˆ A cs H 2 1 cs H ; r H ; H ˆ A sin H 2 1 cs H : 17 The result (17) can nw be cmbined with the stresses derived fr the Kelvin prblem, (10), t satisfy the zer shear tractin bundary cnditin required fr Bussinesq's slutin. This gives A ˆ C 1 2m 18 and C can be determined by evaluating the resultant f axial tractins acting n a hemispherical surface, f arbitrary radius a, centered abut the rigin, i.e., P B 2p Z p=2 0 r cs H r H sin HŠa2 sin H dh ˆ 0; 19 where r and r H refer t the stress state btained by cmbining (10) and (17). This gives C ˆ PB 2p : 20 The displacement and stress cmpnents take frms 2Gu ˆ PB 4 1 m cs H 1 2m Š 2p 2Gu H ˆ PB sin H 1 2m 3 4m 2p 1 cs H 21 and r ˆ PB 1 2m 2 2 m cs HŠ; 2p2 r HH ˆ PB 1 2m cs 2 H 2p 2 1 cs H ; r ## ˆ PB 1 2m 2p 2 r H ˆ PB 1 2m 2p 2 sin H cs H 1 cs H : cs H sin 2 H 1 cs H ; 22 As is evident, when m ˆ 1, bth Bussinesq's slutin fr the nrmal lading f the surface f a 2 halfspace and Kelvin's slutin fr the interir lading f an in nite space by a cncentrated frce reduce t the same result, where the state f stress is purely radial.

6 322 A.P.S. Selvadurai / Internatinal Jurnal f Engineering Science 39 (2001) 317± Cncluding remarks The expsitins f the derivatin f the slutin t Bussinesq's classical prblem cncerning the surface lading f a halfspace by a cncentrated nrmal frce given in the literature range frm the use f results f ptential thery, superpsitin schemes invlving Kelvin's slutin and the applicatin f integral transfrm techniques. The frmer tw prcedures are largely based n familiarity with the apprpriate mathematical analgy and the ingenius chice f superpsitin f cncentrated frce slutins assciated with the in nite space. The Hankel integral transfrm prcedure is mre frmal and readily yields the slutin t Bussinesq's prblem. It is shwn that the slutin t Bussinesq's prblem can als be btained thrugh the use f a Lame ptential, the frm f the slutin f which is guided by dimensinal cnsideratins. Acknwledgements This wrk was cmpleted during the tenure f an Erskine Fellwship at the University f Canterbury, Christchurch, New Zealand. The authr is grateful t Prfessr.O. Davis, Department f Civil Engineering, University f Canterbury fr helpful cmments and fr the kind hspitality during the visit. eferences [1] J. Bussinesq, Applicatin des Ptentiels a L'etude de l'equilibre et due Muvement des Slides Elastique, Gauthier Villars, Paris, [2] A.E.H. Lve, A Treatise n the Mathematical Thery f Elasticity, Cambridge University Press, Cambridge, [3] A.I. Lure, Three-dimensinal Prblems in the Thery f Elasticity, Wiley, New Yrk, [4] J.H. Michell, Sme elementary distributins f stress in three-dimensins, Prc. Lnd. Math. Sc. 32 (1900) 23±35. [5] L.I. Sedv, Mechanics f Cntinuus Media, vl. 1, Wrld Scienti c, New Jersey, 1997, pp. 532±534. [6] I.N. Sneddn, Furier Transfrms, McGraw-Hill, New Yrk, 1951, pp. 450±486. [7] I.S. Sklnik, Mathematical Thery f Elasticity, McGraw-Hill, New Yrk, [8] S.P. Timshenk, J.N. Gdier, Thery f Elasticity, McGraw-Hill, New Yrk, [9] H.M. Westergaard, Thery f Elasticity and Plasticity, Wiley, New Yrk, 1952.