Fundamental solutions for isotropic size-dependent couple stress elasticity

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1 undamenta soutons fo sotopc sze-dependent coupe stess eastcty A R. Hadjesfanda, Gay. Dagush ah@buffao.edu, gdagush@buffao.edu Depatment of Mechanca and Aeospace Engneeng, Unvesty at Buffao, State Unvesty of New Yok, Buffao, NY 460 USA ABSTRAT undamenta soutons fo two- and thee-dmensona nea sotopc sze-dependent coupe stess eastcty ae deved, based upon the decomposton of dspacement feds nto datatona and soenoda components. Whe sevea fundamenta soutons have appeaed pevousy n the teatue, the pesent veson s fo the newy deveoped fuy detemnate coupe stess theoy. Wthn ths theoy, the coupe stess tenso s skewsymmetca and thus possesses vectoa chaacte. The pesent devaton povdes soutons fo nfnte domans of eastc mateas unde the nfuence of unt concentated foces and coupes. Unke a pevous wok, unue soutons fo dspacements, otatons, foce-stesses and coupe-stesses ae estabshed, aong wth the coespondng foce-tactons and coupe-tactons. These fundamenta soutons ae centa n anayss methods based on Geen s functons fo nfnte domans and ae eued as kenes n the coespondng bounday ntega fomuatons fo sze-dependent coupe stess eastc mateas.. Intoducton. Backgound It has ong been suggested that the stength of mateas has sze-dependency n smae scaes. onseuenty, a numbe of theoes that ncude ethe second gadents of defomaton o exta otatona degees of feedom, caed mcootaton, have been

2 poposed to captue ths sze-effect. These deveopments have mpacts at the fnest contnuum scaes, such as mco- and nano-scaes, whch n tun can affect moden mco- and nano-technoogy. The ncuson of the above new measues of defomaton eues the ntoducton of coupe-stesses n these mateas, aongsde the tadtona foce-stesses. The possbe exstence of coupe-stess n mateas was ognay postuated by Vogt [], whe osseat and osseat [] wee the fst to deveop a mathematca mode to anayze mateas wth coupe-stesses. The osseats deveoped a genea theoy by consdeng mcootaton, ndependent of the cassca macootaton. At fst, expets n contnuum mechancs dd not see any necessty fo the exta atfca degees of feedom and consdeed the macootaton vecto as the soe degees of feedom descbng otaton. Toupn [], Mndn and Testen [4], Kote [5] and othes, foowng ths ne, consdeed the gadent of the otaton vecto, as the cuvatue tenso, the effect of the second gadent of defomaton n coupe stess eastc mateas. Howeve, thee ae some dffcutes wth these fomuatons. The moe notabe ones eate to the ndetemnacy of the spheca pat of the coupe-stess tenso and the appeaance of the body coupe n consttutve eatons fo the foce-stess tenso [4]. o sotopc eastc mateas, thee ae two sze dependent eastc constants eued n ths theoy. Howeve, the nea eubum euatons n tems of dspacements nexpcaby nvove ony one of these constants. Ths nconsstent theoy usuay s caed the ndetemnate coupe stess theoy n the teatue (Engen [6]) and ceay cannot be used as a bass fo eabe pedcton of matea behavo. As a esut of the nconsstences noted above, a numbe of atenatve theoes have been deveoped. One banch evves the dea of mcootaton and s caed mcopoa theoes (e.g., Mndn [7]; Engen [6]; Nowack [8]). Howeve, these theoes exhbt nconsstences, because mcootaton s not a tue contnuum mechanca concept. The effect of the dscontnuous mcostuctue of matte cannot be epesented mathematcay by an atfca contnuous mcootaton. The othe man banch, abeed second gadent theoes, avods the dea of mcootaton by ntoducng gadents of stan o otaton

3 (e.g., Mndn and Eshe [9]; Laza et a. [0]). Athough these theoes use tue contnuum epesentatons of defomaton, the esutng fomuatons ae not consstent wth coect bounday condton specfcatons and enegy conjugacy euements. Recenty, the pesent authos [] have esoved a of the above dffcutes and have deveoped the consstent coupe stess theoy fo sods. It has been shown that a consstent sze-dependent contnuum mechancs shoud nvove ony tue contnuum knematca uanttes wthout ecouse to any addtona atfca degees of feedom. By usng the defnton of admssbe bounday condtons, the pncpe of vtua wok and some knematca consdeatons, we have shown that the coupe-stess tenso has a vectoa chaacte and that the body coupe s not dstngushabe fom the body foce. The wok aso demonstates that the stesses ae fuy detemnate and the measue of defomaton coespondng to coupe-stess s the skew-symmetca mean cuvatue tenso. Ths deveopment can be extended ute natuay nto many banches of contnuum mechancs, ncudng, fo exampe, eastopastcty and pezoeectcty. Howeve, the fst step s the deveopment of nfntesma nea sotopc eastcty, whch nvoves ony a snge sze-dependent constant. Snce ths theoy s much moe compcated than auchy eastcty, anaytca soutons ae ae and, conseuenty, numeca fomuatons ae needed to sove moe genea szedependent coupe stess eastc bounday vaue pobems. Inteestngy, t seems the bounday eement method s a sutabe numeca too to sove a wde ange of coupe stess eastc bounday vaue pobems. Howeve, ths eues the fee space Geen s functons o fundamenta soutons as the eued kenes to tansfom the govenng euatons to a set of bounday ntega euatons. These fundamenta soutons ae the eastc soutons fo an nfntey extended doman unde the nfuence of unt concentated foces and coupes. The mpotance of these fundamenta soutons s enhanced futhe, when we notce that the fee space Geen s functons pay a dect oe n the souton of many pactca pobems fo nfnte domans.

4 Mndn and Testen [4] have gven the necessay potenta functons fo obtanng the thee-dmensona dspacement fundamenta soutons n the ndetemnate sotopc eastcty. howdhuy and Gockne [] povde anaogous functons by a matx nveson technue fo steady state vbaton. Inteestngy, we fnd that ou nea eubum euatons n tems of dspacements n the detemnate theoy ae dentca to those of Mndn and Testen [4]. Ths means the above dspacement soutons ae vad n ou detemnate theoy. Howeve, due to the ndetemnacy of stesses, appeaance of body coupes ndependent of body foce and exstence of two sze-dependent eastc constants, the coespondng stesses wee not obtanabe wthn pevous coupe stess theoy. In the pesent wok, we obtan a two- and thee-dmensona dspacement and stess fundamenta soutons fo ou consstent sotopc coupe stess eastcty. Wthn ths theoy, eveythng s fuy detemnate and depends on ony a snge sze-dependent matea constant. o the thee-dmensona case, we deve the dspacement kenes decty by a decomposton method and then detemne the coespondng stesses. Two-dmensona stess fundamenta soutons have been pesented n Hugo [] ony fo concentated foce, based on the ndetemnate coupe-stess theoy deveoped by Mndn [4,5]. Refeence [] shows that some of these deveopments fo pane pobems eman usefu n ou detemnate coupe stess eastcty. Howeve, we shoud emembe n Mndn s theoy, thee s an exta matea constant, aong wth ndetemnacy n the spheca pat of the coupe-stess tenso. Hee we deve the compete two-dmensona fundamenta soutons fo pont foce and pont coupe, ncudng dspacements, foce- and coupe-stesses wth a method sma to that used n the thee-dmensona case. Befoe contnung wth the deveopment fo coupe stess eastcty, we shoud menton the wok by Hashma and Tomsawa [6], Sandu [7] and Khan et a. [8] to deveop fundamenta soutons wthn the famewok of mcopoa eastcty. Athough the esuts have some smates to ou fundamenta soutons, we shoud emphasze once 4

5 agan that mcootatons ae not a fuy consstent contnuum mechancs concept. onseuenty, these soutons ae of mted pactca utty.. Basc Euatons Let us assume the thee-dmensona coodnate system x xx as the efeence fame wth unt vectos e, e and e. onsde an abtay pat of the matea contnuum occupyng a voume V encosed by bounday suface S. In a contnuum mechanca theoy fo sze-dependent coupe stess mateas, the euatons of eubum become whee σ j and σ j, j + = 0 () μ j, j + ε jkσ jk = 0 () μ j ae foce- and coupe-stess tensos, and s the body foce pe unt voume of the body. Wthn ths theoy, the foce-stess tenso s geneay nonsymmetc and can be decomposed as σ = σ + σ () j ( j ) [ j] whee σ ( j) and σ [ j] ae the symmetc and skew-symmetc pats, espectvey. In [], we have shown that n a contnuum mechanca theoy, the coupe-stess tenso s skewsymmetca. Thus, μ = (4) j μ j Theefoe, the coupe-stess vecto μ dua to the tenso μ j can be defned by whee we aso have μ = ε jkμkj (5) ε μ = μ (6) jk k j Then, the angua eubum euaton gves the skew-symmetc pat of the foce-stess tenso as σ = μ (7) [ j ] [, j] 5

6 We can consde the axa vecto s dua to σ [ j], whee whch aso satsfes = [ j k ] (8) s ε jkσ, ε = σ (9) s jk k [ j] It s seen that by usng Es. (7) and (8), we obtan whch can be wtten n the vectoa fom = (0) s ε jkμk, j s = μ () Inteestngy, t s seen that s = 0 () The foce-tacton vecto at a pont on a suface wth unt noma vecto n can be expessed as t ( n) = σ n () j j Smay, the moment-tacton vecto can be wtten ( n) m = μ n = ε n μ (4) j j jk j k whch can aso be wtten n the vectoa fom ( n) m = n μ (5) Refeence [] shows that n coupe stess mateas, the body coupe s not dstngushabe fom the body foce. The body coupe tansfoms nto an euvaent body foce ε jk k, j n the voume and a foce-tacton vecto ε jk jnk boundng suface. In vectoa fom, ths means on the 6

7 + n V (6) and ( n) ( n) t + n t on S (7) Ths s the fst mpotant esut eued fo the deveopment of the fuy detemnate coupe stess theoy. In addton, thee s a need to ntoduce the appopate knematca and consttutve eatons. As an nta step, the dspacement gadents ae decomposed nto symmetca and skew-symmetca components, such that whee u = e + ω (8), j j j (, j ) = ( u, j + u j ) e j u, = (9) [, j] = ( u, j u j ) ω j = u, (0) Snce the otaton tenso ω j s skew-symmetca, one can ntoduce a dua otaton vecto, such that ω = εjkωkj () In the usua nfntesma auchy eastcty, ony the symmetc stan tenso e j contbutes to the stoed enegy. Howeve, n the sze-dependent coupe stess eastc theoy, mean cuvatues κ j aso pay a oe, whee [, j] = ( ω, j ω j ) κ j = ω, () om E. (7), one can ecognze that the mean cuvatue tenso s skew-symmetca and thus can be ewtten n tems of a dua mean cuvatue vecto [], whee 7

8 κ = εjkκkj () Refeence [] deves the genea consttutve eatons fo an eastc matea, when stoed enegy s expessed n tems of the symmetca stan tenso e j and the mean cuvatue vecto κ. Inteestngy, fo nea sotopc eastc meda, the foowng consttutve eatons can be wtten fo the foce-stess and coupe-stess, espectvey σ = λe δ + μe (4) ( j) kk j j μ = 8 (5) ηκ and theefoe σ = μ = 8ηκ = ηε ω (6) [ j] [ j] [ j, ] jk k, Hee λ and μ ae the usua Lamé eastc modu, whe η s the soe addtona paamete that accounts fo coupe stess effects n an sotopc matea. It s seen that these eatons ae sma to those n the ndetemnate coupe stess theoy (Mndn, Testen [4]; Kote [5]), when the two sze-dependent popetes have the eaton η = η. Thus, n [], we have deved coupe stess theoy wth ony one snge szedependent constant n whch a fome toubes wth ndetemnacy dsappea. Thee s no spheca ndetemnacy and the second coupe stess coeffcent η depends on η, such that the coupe-stess tenso becomes skew-symmetc. Inteestngy, the ato η μ = (7) specfes a chaactestc matea ength, whch s absent n auchy eastcty, but s fundamenta to the sma defomaton sze-dependent eastcty theoy unde consdeaton hee. It shoud be aso notced that ν λ = μ (8) ν whee ν epesents the usua Posson ato. Theefoe, E. (4) can be wtten as 8

9 ν σ ( ) = j μ ekkδ j + ej ν (9) and the tota stess tenso becomes ν σ j = μ ekkδ j + ej + μ ε jk ωk ν (0) By usng the eatons n Es. (9) and (), ths tenso can be wtten n tems of dspacements as ( u u ) ν σ j = μ u k, k δ j + u, j + u j, η, j j, () ν Afte substtutng E. () nto E. (), we can ewte the govenng dffeenta euatons n tems of the dspacement and body foce densty feds as o n vectoa fom as ( ) ukk u λ+ μ+ η, + ( μ η ) + = 0 () ( + μ + η ) ( u) + ( μ η ) u + = 0 λ () Note that Mndn and Testen [4] pevousy deved an dentca fom wthn the context of the ndetemnate coupe stess theoy. Howeve, eca that the stesses n the Mndn-Testen fomuaton not ony nvove two paametes η and η, but aso ae ndetemnate. The genea souton fo the dspacement n the ndetemnate theoy has been deved by Mndn and Testen [4]. Ths can be wtten ( ) ( ν ) u = B B + B 0 4 B (4) whee the vecto functon B and scaa functon B 0 satsfy the eatons ( ) μ B= (5) μ B 0 = (6) 9

10 Athough, emakaby, these eman vad fo the detemnate coupe stess theoy, we nstead use the dect decomposton method n the pesent wok to deve the dspacement fundamenta soutons. Aftewads, we contnue by deveopng the coespondng otatons, cuvatues and fuy-detemnate foce- and coupe-stesses and tactons, whch has pevousy not been possbe. Ths, n tun, enabes the fomuaton of new bounday ntega epesentatons and the deveopment of bounday eement methods to sove a boad ange of coupe stess eastostatc bounday vaue pobems n both two- and thee-dmensons.. undamenta Soutons fo Thee-Dmensona ase In ths secton, we deve the fundamenta soutons fo the thee-dmensona case. As was mentoned, these ae the eastc soutons of an nfntey extended doman unde the nfuence of unt concentated foces and coupes.. oncentated oce Assume that n the nfntey extended matea, thee s a unt concentated foce at the ogn n an abtay decton specfed by the unt vecto a. Ths concentated foce can be epesented as a body foce ( ) ( ) ( ) ( ) = δ x a (7) whee δ x s the Dac deta functon n thee-dmensona space. It s known that By appyng the vectoa eaton fo the vecto 4π δ ( ) ( ) x = (8) 4π ( G) = ( G) G (9) a, we decompose the body foce as a a = a = (40) 4π 4π 4π 0

11 If we consde the decomposton of the esutng dspacement whee () u and It s seen that If we ntoduce two vectos ( ) ( ) u = u + u u as ( ) u ae the datatona and soenoda pat of dspacement () ( λ + μ) ( ) (4) u satsfyng u = 0 (4) ( ) u = 0 (4) u = (44) 4π a ( ) ( ) η u μ u = (45) 4π a () A and u ( ) A, such that ( ) ( ) u ( A ) = ( A ) ( ) ( ) ( A ) A = (46) ( ) ( ) = (47) It s seen that they satsfy the condtons n Es. (4) and (4), espectvey, and theefoe () + A = a (48) 4π ( λ μ) ( ) ( ) η A μ A = (49) 4π a It s obvous that the soutons shoud be n the fom ( ) A = ϕa ( ) A = ψa (50) (5) whee ϕ and ψ ae scaa functons of havng ada symmety. Theefoe, ϕ = (5) 4π ( λ + μ) η ψ μ ψ = (5) 4π

12 whee educes to s the apacan opeato n thee dmensons. Because of ada symmety, t d d d + = d d d d d (54) The egua soutons to the Es. (5) and (5) ae ϕ = (55) 8π ( λ + μ) Theefoe, we have η ψ = 4πμ A () e / ( λ + μ) a 8πμ (56) = (57) 8π A ( ) / e = a a 4πμ 8πμ (58) It s seen that fom E. (46) () xx u = δ a 8π( λ+ μ) By usng the eaton n E. (8), we have ν λ + μ = μ (60) ν and E. (59) can be wtten as () ν xx u = δ a 6πμ ( ν ) (59) (6) It s aso seen that fom E. (47) Theefoe u e ( A ) / + a ( ) ( ) = 4πμ (6)

13 ( ) / xx / u = e e δ a πμ xx + + δ a 8πμ (6) and fo the tota dspacement, we have () ( ) xx ( 4 u = u + u = ν) δ + a 6πμ ( ν ) / xx / + e e δ a 4πμ It shoud be notced that the fst pat of ths eaton s the dspacement fom auchy eastcty. Howeve, n ou detemnate theoy, we can deve detemnate stesses. These foce- and coupe-stesses ae deveoped n the foowng. (64) As a fst step n that decton, we fnd that the gadent of dspacement fom E. (64) s u x δ x + δ x x x x ( 4ν) 6 πμ( ν ) j j j j, j= δ+ a / xx jx e + a 4 4πμ / δjx + δ jx + δxj e + + e 4πμ / j x a δ (65) Theefoe, the stan tenso becomes

14 e j = u(, j) xjδ + xδ j δjx + δ jx + xjδ xxjx = ( 4ν ) + a 6 πμ( ν ) xx x / j e 4 / jx + jx + xj e + 4πμ δ δ δ / x j x e δ + jδ + + a 8πμ a (66) Now, t s easy seen that ν x = a (67) 8πμ( ν ) ekk uk, k = Theefoe, the symmetc pat of foce-stess tenso becomes ν σ μ δ ( j) = ekk j + ej = ν xjδ + xδ j xδj xxjx ( ν ) + a 8 π( ν) xx x / j e + 4 π / δjx + δ jx + δxj e + + e 4π / xδ + x δ j j a a (68) o the otaton vecto, we have ω = u = u ( ) e = 4πμ / a (69) whch gves 4

15 ω / px e ε = 8πμ + p a (70) It s seen that the mean cuvatue vecto s κ ε ω = jk k, j xx = πμ 6πμ / / e a e δ a and theefoe, fo the coupe-stess vecto, we have μ = 8μ κ xx = π π / / + + e a + + e δ a (7) (7) It s seen that the skew-symmetc pat of the foce-stess tenso becomes / xδ σ[ j] μ[, j] e = = 4π + x δ j j a (7) Theefoe, the tota foce-stess tenso s xjδ + xδ j xδj xxjx σ j = σ( j) + σ[ j] = ( ν) + a 8 π( ν) / xx jx e + a 4 π / δjx + δ jx + δxj e + π + e / xδ j a (74) Inteestngy, the foce-tacton vecto becomes 5

16 ( ) t n σ jnj = = xn j jδ + xn xn xxx j ( ν ) + n j a 8 π( ν) / xx jx e n j + a 4 π / nx + nx + δxn j j e / + xn e + a π and the moment-tacton vecto s ( n) / xn j j xn j xp m = ε n μ = e ε ε a + + π / e εj n j a jk j k p jp + π + (75) (76) nay, we can consde the foowng eatons whee U, Ω, Σ j, Μ, T and u Ua = (77) Ωa ω = (78) j ja σ = Σ (79) μ = Μ a (80) ( n) t = T a (8) ( n) m = M a (8) M epesent the coespondng dspacement, otaton, foce-stess, coupe-stess, foce-tacton and moment-tacton, espectvey, at x due to a unt concentated foce n the -decton at the ogn. It s seen that, these Geen s functons ae 6

17 U xx ( 4 = ν) δ + 6πμ ( ν ) / xx / + e e δ πμ Ω / εpx = e 8πμ + p (8) (84) Σ j xjδ + xδ j xδj xxjx = ( ν ) + a 8 π( ν) Μ / xx jx e + a 4 π / δjx + δ jx + δxj e / x jδ + + e a π xx = δ π π / / e e (85) (86) ( ) T xnδ + xn xn xxx ( ν ) nj 8 π( ν) n j j j = + / xx xn j j e + 4 π / nx + nx + δxn j j e / xn + + e π M x n xn x π / e + εj n j π + / j j j p = + + e ε p ε jp (87) (88) 7

18 . oncentated oupe Assume that n an nfntey extended coupe stess eastc matea thee s a unt concentated coupe at the ogn n the abtay decton specfed by the unt vecto a. Theefoe, ths concentated coupe can be epesented as a body coupe ( ) ( ) = δ x a (89) As was mentoned pevousy, the effect of a body coupe s euvaent to the effect of a body foce epesented by o = = δ ε jkδ, j ( ) ( x) a (90) = ( x) ak (9) wth a vanshng suface effect at nfnty. Ths shows that the dspacement fed of the concentated coupe = δ ( ) ( ) x a s euvaent to the otaton fed of the concentated foce = δ ( ) ( ) x a. Theefoe, the soutons of the two pobems ae eated, such that whch gves u ε u = ω = jk k, j / εpxp u = e a 8πμ + (9) (9) om ths t s seen that the gadent of the dspacement s xx u e a e a 8πμ 8πμ / p j /, j = + + ε p + + ε j (94) Theefoe, the stan tenso becomes / (, ) xj x x p e = u j = + + e ε ε a + 6πμ j p jp (95) 8

19 It s nteestng to note that e = kk uk, k = 0 (96) whch means the defomaton fed of a concentated coupe = δ ( ) euvoumna. Theefoe, the symmetc pat of foce-stess tenso s x a s / xx j p xx p ( j) = e = e a π σ μ ε ε j p jp (97) o the otaton vecto, we have ω = εjkuk, j xx = 6πμ 8πμ / / e δ + + a + e δ a (98) uthemoe, the mean cuvatue vecto s κ = ε ω, = e πμ + ε x / p p jk k j a and theefoe the coupe-stess vecto becomes μ 4π ε x / p p = 8μ κ = e a + (99) (00) The skew-symmetc pat of μ s, j / xx j p xx p μ[, j] = e a + + ε ε 8π + e 4π + / ε j a p jp (0) Theefoe, the skew-symmetc pat of foce-stess tenso becomes 9

20 / xx j p xx p σ[ j] = μ[, j] = e ε ε a + + 8π e 4π + / ε j a p jp (0) and the tota foce-stess tenso becomes xx xx σ ε ε 8π 8π j p / p j = pa e jpa e 4π + / ε j a (0) Then, the foce-tacton vecto becomes ( ) t = σ jnj xx xx = 8π 8π j p / p ε pna j e ε jpna j e 4π + / ε n a j j and the coupe-tacton vecto s gven by n ( ) ( ) / xδ j xjδ m = ε n μ = e n a 4π + jk j k j (04) (05) Theefoe, we can consde u Ua = (06) Ωa ω = (07) j ja σ = Σ (08) μ = Μ a (09) ( n) t = T a (0) ( n) m = M a () 0

21 whee U, Ω, Σ j, Μ, T and M epesent the coespondng dspacement, otaton, foce-stess, coupe-stess, foce-tacton and moment-tacton, espectvey, at x due to a unt concentated coupe n the decton at the ogn. Theefoe, t s easy seen that these nfnte space Geen s functons ae U Ω e 8πμ / = = ε + p x p () xx Ω e e 6πμ 8πμ / / = δ δ T Σ j xx = 8π j p ε p xx π 4π + xx 8π j p = ε pnj / p / e ε jp e ε j () (4) / pxp e ε Μ = 4π + (5) xx π 4π + / p / e ε jpnj e ε jnj M / xn = e 4π + x nδ j j (6) (7). undamenta Soutons fo Two-Dmensona ase In ths secton, we fst pesent the govenng euatons of sze-dependent coupe stess eastcty n two-dmensons unde pane stan condtons. Then, we deve the compete two-dmensona fundamenta soutons n a sma method used n the thee-dmensona case. Refeence [9] uses these fundamenta soutons wthn a two-dmensona bounday eement method.

22 . Govenng Euatons fo Two Dmensons We suppose that the meda occupes a cyndca egon, such that the axs of the cynde s paae to the x -axs. uthemoe, we assume the body s n a state of pana defomaton paae to ths pane, such that u, = 0, u = 0 n V (8a,b) whee a Geek ndces, hee and thoughout the emande of the pape, w vay ony ove (,). Aso, et ( ) V and the x x -pane and ts boundng edge n that pane. ( ) S epesent, espectvey, the coss secton of the body n As a esut of these assumptons, a uanttes ae ndependent of x. Then, thoughout the doman and ω = 0, e = e = 0, κ = 0 (9a-c) σ σ 0, μ = μ 0 (0a,b) = = = Intoducng the abdged notaton = = () ω ω ε β uβ, whee ε β s the two-dmensona atenatng symbo wth ε = ε =, ε = ε 0 () = Now, t s seen that the non-zeo components of the cuvatue vecto ae = () κ εβω, β Theefoe, the non-zeo components of stesses ae wtten and σ σ μ 4ηεβω, β = (4) = λe δ + μe (5) ( β ) γγ β β [ βε ] μ[, β ] = ηε β ω = (6)

23 σ β = λe δ + μe ηε ω (7) γγ A the othe stesses ae zeo, apat fom σ and μ, whch ae gven as β β β σ =νσ γγ (8) μ 4ηω, = (9) It shoud be notced that these stesses n Es. (8) and (9), actng on panes paae to the x x -pane, do not ente decty nto the souton of the bounday vaue pobem. o the pana pobem, the stesses must satsfy the thee eubum euatons σ, = 0 (0) β β + μ ε σ 0 () β, β + β β = wth the obvous euement = 0. The moment euaton can be wtten as σ, = () [ β ] μ[ β ] whch actuay gves the non-zeo components σ σ = μ = () [ ] [ ] [,] We aso notce that the foce-tacton educes to ( ) σ β nβ t n = (4) and the moment-tacton has ony one component m. Ths can be convenenty denoted by the abdged symbo m, whee ( n) ( n) ω m = m = ε β μ nβ = 4η (5) n. oncentated oce Assume that thee s a ne oad on the x axs wth an ntensty of unty pe unt ength n the abtay decton specfed by the unt vecto a. Ths dstbuted oad can be epesented by the body foce ( ) ( ) ( ) ( ) = δ x a (6) whee δ x s the Dac deta functon n two-dmensona space. It s known that

24 n π = δ ( ) ( x) (7) By appyng the vectoa eaton n E. (9) fo the vecto body foce as n a, we decompose the π n n n a a = a = (8) π π π If we consde the decomposton of dspacement whee () u and ( ) ( ) u = u + u u as (9) ( ) u ae the datatona and soenoda pat of the dspacement vecto u satsfyng then t s seen that η () ( λ + μ) u u ( ) ( ) μ u ( ) u = 0 (40) ( ) u = 0 (4) n = π a n = π a (4) (4) If we ntoduce two vectos () A and u ( ) A such that ( ) = ( A ) ( ) ( ) ( A ) A ( ) ( ) u = A (44) ( ) ( ) = (45) whch satsfy the condtons of Es. (40) and (4), we obtan n + = a (46) π () ( λ μ) A ( ) ( ) n η A μ A = a (47) π 4

25 Then, the soutons ae n the fom ( ) A = ϕa ( ) A = ψa (48) (49) whee ϕ and ψ ae scaa functons of havng two-dmensona ada symmety. Theefoe ϕ = n (50) π ( λ + μ) n η ψ μ ψ = (5) π In two-dmensons, wth ada symmety, the apacan educes to d d + (5) d The egua soutons to Es. (50) and (5) ae ( ) ( n ) λ + μ ϕ = 8π (5) ( n ) ψ = K0 n + πμ + 8πμ (54) whee K 0 s the modfed Besse functon of zeoth ode. Theefoe, () A = n 8 + a (55) ( ) ( ) μ π λ ( ) = K0 + n + ( n ) A πμ a a (56) 8πμ Then, fom E. (44) and fom E. (45) () ν xxρ u = + ( n ) δ a 6πμ ( ν ) ρ ρ ( ) (57) ( ) ( ) u = A K0 + n πμ a (58) 5

26 By usng the eatons K K 0 = K = K 0 K whee K s the modfed Besse functon of fst ode, we obtan ( ) xxρ u =+ ( n ) δρ aρ 8πμ + xxρ + K0 + K a K 0 + K δ a πμ πμ ρ ρ ρ (59) (60) (6) It shoud be notced that thee ae gd body tansaton tems n u () and u whch cannot affect stess dstbutons. These tems can be negected n ths Geen s functon fo stess anayss. Theefoe, by gnong these gd body tems and usng we obtan ( ) ( 4 ) xxρ u = ν nδ a 8πμ ν ( ) ( ) u ρ ρ ( ) u = u + (6) x x + K + K a K + K a πμ πμ ρ 0 ρ 0 δ ρ ρ (6) o the gadent of dspacements, we have xβδρ xδβρ + xρδβ xxβxρ uβ, = ( 4ν ) + a 8πμ ( ν ) δjx + δ jx + xjδ xxβxρ + K0 + K 4 a πμ xβδρ xxβxρ + K a πμ (64) Theefoe, the stan tenso becomes 6

27 xβδρ + xδβρ δβxρ xxβxρ u( β, ) = eβ = ( ν ) + a 8πμ ( ν ) ρ δβ xρ + δβρ x + δρ xβ x xβ xρ + K0 + K 4 a πμ δβρx + δρxβ x xβ xρ + K a 4πμ ρ ρ (65) Ths eaton shows that ν x = a (66) 4πμ ( ρ e ) () γγ eγγ = ρ ( ν ) Theefoe, the symmetc pat of the foce-stess tenso s ν σ μ δ ( β ) = eγγ β + eβ = ν δβρx + δρxβ δβ xρ x xβ xρ ( ν ) + a 4π( ν) δβ xρ + δβρ x + δρ xβ x xβ xρ + K0 + K 4 a π δβρx + δρxβ x xβ xρ + K a π ρ (67) Next, we consde otatons and note that the ony non-zeo n-pane component s ω 4πμ = εβu β, = K ερ x a ρ (68) Then, the mean cuvatue components ae xxρ = β, β = K 0 + K a ρ κ ε ω 8πμ + K 0 + K δ a ρ 8πμ ρ (69) 7

28 Theefoe, the coupe-stess vecto becomes μ = 8μ κ x x = K + K a K + K a π π ρ 0 ρ 0 δ ρ ρ (70) and the skew-symmetc pat of the foce-stess tenso becomes σ xδ = μ = K π [ β ] [, β ] ρ βρ x δ β ρ a (7) wth non-zeo components xa xa σ [ ] = σ [ ] = K (7) π Theefoe, the tota foce-stess tenso s σβ = σ( β ) + σ[ β] = δβρx + δρxβ δβ xρ x xβ xρ ( ν ) + a 4π( ν) δβ xρ + δβρ x + δρ xβ 4x xβ xρ + K0 + K a π δβρx x xβ xρ + K a π ρ (7) and fo the foce-tacton vecto, we have ( ) t n σ βnβ = = nρx + δρxβnβ nxρ xxρxβnβ ( ν ) + a 4π( ν) nxρ + nρx + δρxβnβ 4xxρxβnβ + K0 + K a π nρx xxρxβnβ + K a π ρ ρ (74) 8

29 whe fo moment-tacton, we obtan ( n) ε β x xρnβ m = ε μ n = K0 + K a π β β ρ K0 + K ε n a π βρ β ρ (75) nay we can consde the foowng eatons u = U a (76) ρ Ω ρ ρ ω = a (77) ρ β βρaρ σ = Σ (78) μ = Μ a (79) ρ ρ ( n) t m ( n) = T a (80) ρ ρ M = a ρ ρ (8) whee the fundamenta soutons U ρ, Σ βρ, Μ ρ and T ρ epesent the coespondng dspacement, foce-stess, coupe-stess and foce-tacton, espectvey, at x due to a unt concentated foce n the ρ -decton at the ogn. uthemoe, the Geen s functons Ω ρ and M ρ epesent the coespondng otaton and moment-tacton, espectvey, at x. om the above eatons, one can estabsh ( ) ( ) xxρ Uρ = 4ν nδρ 8πμ ν xxρ + K0 + K K 0 + K δ πμ πμ K πμ 4 Ω ρ = ε x ρ ρ (8) (8) 9

30 δβρx + δρxβ δβ xρ x xβ xρ Σβρ = ( ν ) + 4π( ν) δβxρ + δβρx + δρxβ 4xxβxρ + K0 + K π δβρx x xβ xρ + K π (84) x x ρ Μ ρ = K 0 + K K 0 + K δ (85) ρ π π nρx + δρxβnβ nxρ xxρxβnβ Tρ = ( ν ) + 4π( ν) n xρ + nρx + δρxβnβ 4xxρxβnβ + K0 + K π nρx xxρxβnβ + K π ε x xn Mρ = K + K π β ρ β 0 K0 + K ε n βρ π β (86) (87). oncentated oupe Now, assume that thee s a ne dstbuton of coupe oad aong the x axs wth unt ntensty pe unt ength. Ths dstbuted oad can be epesented by a body coupe ( ) ( ) = δ x e (88) As we mentoned eae, sma to thee-dmensona case, the effect of a body coupe n an nfnte doman s euvaent to the esut of a body foce epesented by = = δ = ε β δ ( ) ( ) ( ) x e, β e (89) Theefoe, t s seen that the soutons of the two pobems of concentated foce = δ ( ) ( ) x a and concentated coupe = δ ( ) ( ) x e ae eated, such that 0

31 o u u = εβu γ, βeγ (90) = U (9) ε γβ γ, β Thus, we have u = K πμ 4 and we can wte the gadent of dspacement as 4πμ ε x x ε x γ u γ γ β, β = K + K + K 0 γ 4πμ ε β (9) (9) Theefoe, the stan tenso s e β ε x x + ε x x = K 0 + K γ γ β βγ γ 8πμ (94) It s nteestng to note that e = γγ uγ, γ = 0 (95) whch means the defomaton fed of a concentated coupe s euvoumna, a popety shaed by the thee-dmensona case. Then, the symmetc pat of the foce-stess tenso s σ 4π ( ) e K0 K γ γ β β = μ β = + ε x x + ε βγ x γ x (96) We aso have fo n-pane otaton, 0 ω = ε β u β = K 8πμ (97) and the mean cuvatue vecto s

32 κ ε ω = 6πμ = β, β εβ x K β (98) Theefoe, the coupe-stess vecto becomes ε x β β μ = 8 μ κ = K (99) π o the skew-symmetc pat of foce-stess tenso, we have σ = μ = 4π K ε [ β ] [, β ] 0 β (00) wth the non-zeo components σ [ ] = σ [ ] = K 0 (0) 4π Theefoe, fo the tota foce-stess tenso, we have σβ = σ( β ) + σ[ β] εγ xx γ β + εβγ xx γ = K 0 + K K + 0 ε 4π 4π β (0) The foce-tacton vecto then becomes ( n ) t = T = σ n β β εγ xxn γ β β + εβγ xxn γ β = K 0 + K K + 0 εβn 4π 4π and the moment-tacton s gven by m ( n) ε μ n xβ n = K π = β β β β (0) (04) nay, we have a of the necessay Geen s functons o nfuence functons fo the twodmensona case, whch can be wtten

33 T whee Σ β β = σ = U u = K πμ 4 = Ω εγ x γ (05) = ω = K0 (06) 8πμ εγ xx γ β + εβγ xx γ K 0 + K K + 0 ε 4π 4π ( n ) = t = β (07) x εβ β Μ = μ = K (08) π εγ xxn γ β β + εβγ xxn γ β K 0 + K K + 0 εβn 4π 4π U, Σ β, Μ and M m ( n) = K π = x β n β β (09) (0) T epesent, espectvey, the coespondng dspacement, foce-stess, coupe-stess and foce-tacton at x caused by a unt n-pane concentated coupe at the ogn. Meanwhe, Ω and M epesent the espectve coespondng otaton and moment-tacton at x due to ths unt n-pane concentated coupe at the ogn. 4. oncusons We have deved the thee- and two-dmensona fundamenta soutons fo sotopc szedependent coupe stess eastcty, based upon the fuy detemnate coupe-stess theoy. Reca that ths new theoy esoves a of the dffcutes pesent n pevous attempts to constuct a vabe sze-dependent eastcty. uthemoe, snce n ths theoy, body coupes do not appea n the consttutve eatons and eveythng depends on ony a snge sze-dependent matea constant, a expessons fo the fundamenta soutons ae eeganty consstent and ute usefu n pactce. In patcua, these soutons can be used decty as nfuence functons to anayze nfnte doman pobems o as kenes n ntega euatons fo numeca anayss. o exampe, a bounday eement method fo

34 pane pobems of coupe stess eastcty s deveoped n [9], based upon these fundamenta soutons. utue wok w ncude the fomuaton of bounday eement methods fo nea eastc factue mechancs and fo thee-dmensona coupe stess pobems. Refeences [] W. Vogt, Theoetsche Studen fbe de Eastztatsvehtnsse de Kstae (Theoetca Studes on the Eastcty Reatonshps of ystas), Abh. Gesch. Wssenschaften 4, 887. [] E. osseat,. osseat, Théoe des ops Défomabes (Theoy of Defomabe Bodes), A. Hemann et s, Pas, R, 909. [] R.A. Toupn, Eastc Mateas wth oupe-stesses, Achve fo Ratona Mechancs and Anayss, (96) [4] R.D. Mndn, H.. Testen, Effects of oupe-stesses n Lnea Eastcty, Achve fo Ratona Mechancs and Anayss, (96) [5] W.T. Kote, oupe Stesses n the Theoy of Eastcty, I and II, Poceedngs of the Konnkjke Nedeandse Akademe van Wetenschappen. Sees B. Physca Scences, 67 (964) [6] A.. Engen, Theoy of mcopoa eastcty, actue, vo, ed. H. Lebowtz, Academc Pess, New Yok, (968 ) [7] R.D. Mndn, Mco-Stuctue n Lnea Eastcty, Achve fo Ratona Mechancs and Anayss, 6 (964) [8] W. Nowack, Theoy of Asymmetc Eastcty, Pegamon Pess, Oxfod, UK, 986. [9] R.D. Mndn, N.N. Eshe, On st Stan-Gadent Theoes n Lnea Eastcty, Intenatona Jouna of Sods and Stuctues. 4 (968) [0] M. Laza, G.A. Maugn, E.. Afants, On Dsocatons n a Speca ass of Geneazed Eastcty, Phys. Status Sod (b), 4 (005) [] A.R. Hadjesfanda, G.. Dagush, oupe Stess Theoy fo Sods, Intenatona Jouna of Sods and Stuctues, 48 (0) [] K.L. howdhuy, P.G. Gockne, Matx Invesons and Sngua Soutons n the Lnea Theoes of Eastcty, Jouna of Eastcty, 4 (974) 5-4. [] R.R. Hugo, On the oncentated oce Pobem fo Two-dmensona Eastcty wth oupe Stesses, Intenatona Jouna of Engneng Scence, 5 (967) 8-9. [4] R.D. Mndn, Infuence of oupe-stesses on Stess-oncentatons, Expementa Mechancs, (96) -7. 4

35 [5] R.D. Mndn, ompex Repesentaton of Dspacements and Stesses n Pane Stan wth oupe-stesses, Poceedngs of the Intenatona Symposum on Appcaton of the Theoy of unctons n ontnuum Mechancs, Tbs, USSR (96) [6] K-I. Hashma, S. Tomsawa, undamenta Soutons fo Stess oncentaton Pobems of Two- Dmensona osseat Eastc Body, Theoetca and Apped Mechancs, 6 (977) [7] N. Sandu, On Some Pobems of the Lnea Theoy of the Asymmetc Eastcty, Intenatona Jouna of Engneeng Scence, 4 (966) [8] S.M. Khan, R.S. Dhawa, K.L. howdhuy, Sngua Soutons and Geen's Method n Mcopoa Theoy of Eastcty, Apped Scentfc Reseach, 5 (97) [9] A.R. Hadjesfanda, G.. Dagush, Bounday Eement omuaton fo Pane Pobems n oupe Stess Eastcty, Intenatona Jouna fo Numeca Methods n Engneeng, (0) DOI: 0.00/nme. 5