Couple stress theories: Theoretical underpinnings and practical aspects from a new energy perspective

Transcription

1 Couple stress theores: Theoretcal underpnnngs and practcal aspects fro a new energy perspectve Abstract Al R. Hadesfandar Gary F. Dargush Departent of Mechancal and Aerospace Engneerng Unversty at Buffalo The State Unversty of New York Buffalo NY 460 USA ah@buffalo.edu gdargush@buffalo.edu Noveber 5 06 In ths paper we exane theoretcal and practcal aspects of several versons of couple stress theory. Ths ncludes ndeternate Mndln-Tersten-Koter couple stress theory (MTK-CST) ndeternate syetrc odfed couple stress theory (M-CST) and deternate skew-syetrc consstent couple stress theory (C-CST). We observe that MTK-CST and M-CST not only suffer fro nconsstences these theores also cannot descrbe properly several eleentary deforatons such as pure torson of a crcular bar and pure bendng of a plate. By usng an energy ethod we also deonstrate another aspect of the nconsstency of the ndeternate MTK-CST and M-CST for elastc solds. Ths s acheved by dervng the governng equlbru equatons for elastc bodes n MTK-CST M-CST and C-CST. Ths developent shows that the drect nzaton of the total potental energy for MTK-CST and M-CST volates the dvergence free copatblty condton of the rotaton vector feld that s 0. On the other hand the drect nzaton of the total potental energy for C-CST satsfes ths copatblty condton autoatcally. Ths result deonstrates another aspect of the nner consstency of C-CST. Keywords: Couple stress theores; Sze-dependent echancs; Indeternaces; Curvature tensors; Varatonal ethods; Lagrange ultplers

2 . Introducton Mndln and Tersten (96) and Koter (964) developed the ntal verson of the couple stress theory based on the Cosserat contnuu theory (Cosserat and Cosserat 909) n whch the deforaton s copletely specfed by the contnuous dsplaceent feld u. Therefore the kneatcal quanttes such as the rotaton vector and easures of deforaton such as the stran tensor e and rotaton gradent tensor are derved fro ths dsplaceent feld. As a result Mndln-Tersten-Koter couple stress theory (MTK-CST) s based on the rgd body porton of oton of nfntesal eleents of atter at each pont of the contnuu (Hadesfandar and Dargush 05a). In these portant developents Mndln Tersten and Koter correctly establshed that fve geoetrcal and fve echancal boundary condtons can be specfed on a sooth surface. In MTK-CST the couple-stress tensor s energetcally conugate to the gradent of rotaton vector whch s taken as the curvature tensor. Although these researchers ade a sgnfcant step forward for contnuu echancs the fnal MTK-CST suffers fro soe serous nconsstences and dffcultes wth the underlyng forulatons (Hadesfandar and Dargush 05b). The three an nconsstences of MTK-CST for sotropc lnear elastc aterals are:. The ndeternacy n the sphercal part of the couple-stress tensor and as a result n the skew-syetrc part of the force-stress tensor;. The nconsstency n boundary condtons snce the noral coponent of the coupletracton vector appears n the forulaton; 3. The appearance of two couple-stress elastc coeffcents and for lnear elastc sotropc ateral although only one of these elastc coeffcents appears n the fnal governng equatons when wrtten n ters of dsplaceents. The appearance of an arbtrary sphercal couple stress coponent n the couple-stress tensor s the result of the devatorc or trace free 0 character of the bend-twst tensor n ths theory. The dsturbng character of ths sphercal coponent s that t does not create any deforaton n the body whch eans ts effect s equvalent to a zero loadng condton. Ths sphercal part of

3 the couple-stress tensor reans ndeternate when the rotaton feld s prescrbed on the whole boundary and cannot be sply gnored. It cannot also be deterned n a consstent systeatc way n ost cases when the noral couple-tracton vector s specfed on the whole or soe part of the boundary surface. Erngen (968) realzed ths nconsstency as a aor atheatcal proble n the orgnal MTK-CST whch he afterwards called ndeternate couple stress theory. A syetrc couple stress theory stll suffers fro the sae nconsstences and dffcultes wth the underlyng forulaton. In ths theory the syetrc part of the gradent of rotaton vector feld s taken as the curvature tensor. However careful exanaton shows that ths syetrc tensor s actually a torson tensor (Hadesfandar and Dargush 0 05a). Consequently the couple-stresses n ths theory create a cobnaton of torson and antclastc deforaton wth negatve Gaussan curvature for surface eleents of the contnuu. As n the orgnal MTK-CST the sphercal part of the couple-stress tensor also reans ndeternate n ths theory and cannot be deterned n a consstent systeatc way. Ths theory orgnates fro the work of Yang et al. (00) whch s coonly called the odfed couple stress theory (M-CST). In ther developent Yang et al. (00) consder an extra artfcal equlbru equaton for the oent of couples n addton to the two vectoral force and oent equlbru equatons of the classcal contnuu. Applcaton of ths unsubstantated equlbru equaton apparently leads to a syetrc couple-stress tensor. It sees the an otvaton for Yang et al. (00) n ther developent has been to reduce the nuber of couple-stress ateral paraeters for lnear sotropc elastc ateral fro two coeffcents and n the orgnal Mndln-Tersten-Koter theory to only one coeffcent. Recently Hadesfandar and Dargush (0 05a) have developed the consstent couple stress theory (C-CST) whch resolves all nconsstences n the orgnal MTK-CST. The truph of ths developent s dscoverng the subtle skew-syetrc character of the couple-stress tensor whch reduces the nuber of ndependent stress coponents to nne. As a result the curvature tensor n C-CST s the skew-syetrc part of the gradent of rotaton vector feld. The fundaental step n ths developent s satsfyng the requreent that the noral coponent of the couple- 3

4 tracton vector ust vansh on the boundary surface n a systeatc way. Ths s what Mndln Tersten and Koter ssed n ther portant developents although they correctly establshed the consstent boundary condtons. It s nterestng to notce that the skew-syetrc character of the couple-stress tensor edately resolves the ndeternacy proble by establshng that there s no sphercal coponent. As a result the couple-stress tensor s deternate n the skewsyetrc C-CST. It s portant to notce that the skew-syetrc couple-stresses create ellpsodal cap-lke deforaton wth postve Gaussan curvature for surface eleents of the contnuu. Although the reason for developng C-CST has not been to reduce the nuber of couple-stress ateral coeffcents t turns out that for lnear sotropc elastc ateral ths theory requres only one couple-stress ateral paraeter. Because of nconsstences t s nearly possble to fnd a true soluton for any probles n MTK-CST and M-CST that satsfes all boundary condtons. Ths can be observed n very eleentary practcal probles. For exaple there s no consstent soluton for pure torson of a crcular bar n these theores. We notce that the nconsstent approxate solutons for pure torson n these theores predct sgnfcant sze effect whch does not agree wth experents (Hadesfandar and Dargush 06). MTK-CST and M-CST also cannot descrbe pure bendng of a plate properly (Hadesfandar et al. 06). Partcularly M-CST predcts no couple-stresses and no sze effect for the pure bendng of the plate nto a sphercal shell. On the other hand C- CST predcts consstent results for pure torson of a crcular bar and pure bendng of a plate. In ths paper we take a fresh vew by exanng the valdty of MTK-CST M-CST and C-CST fro an energy perspectve. Energy ethods provde convenent and alternatve eans for forulatng the governng equatons of contnuu sold echancs. These ethods not only gve new nsght nto the forulatons but can also be used for forulatng effectve ethods to obtan approxate solutons. Therefore we concentrate on elastc bodes and derve the governng equlbru equatons and boundary condtons for the ndeternate MTK-CST and M-CST and deternate skew-syetrc C-CST. We nze the total potental energy functonal correspondng to these theores subect to the copatblty condton of the rotaton vector feld 0. Therefore we pose ths copatblty constrant by usng the Lagrange ultpler 4

5 ethod. As we shall see for MTK-CST and M-CST the Lagrange ultpler s the ndeternate sphercal part of the couple-stress tensor. Ths eans that the varatonal ethod results n a sphercal coponent for the couple-stress tensor n the orgnal Mndln-Tersten-Koter couple stress theory (MTK-CST) and syetrc odfed couple stress theory (M-CST). Therefore the couple-stress tensor does not becoe trace free as one ght conclude fro an ncorrect drect unconstraned nzaton. Ths result clearly deonstrates that the drect unconstraned nzaton of the total potental energy for MTK-CST and M-CST volates the dvergence free copatblty condton of rotaton vector feld such that 0. Thus another aspect of the nconsstency of ndeternate MTK-CST and M-CST s revealed based on the energy ethod. On the other hand for C-CST the correspondng Lagrange ultpler vanshes whch deonstrates the consstency of the skew-syetrc character of the couple-stress tensor n ths theory. Therefore the drect unconstraned nzaton of C-CST satsfes autoatcally the copatblty condton of rotaton vector feld 0. Ths result shows another nner consstency of C-CST. The reander of the paper s organzed as follows. In Secton we present a bref revew of the couple stress theory for lnear sotropc elastc aterals. Ths ncludes presentng MTK-CST M- CST and C-CST and exanng ther consstency fro a theoretcal and practcal vew. The an new results are then provded n Secton 3 where we derve the governng equatons for elastc bodes by nzng the total echancal potental energy for MTK-CST M-CST and C-CST subect to 0. Ths ncludes exanng the consstency of MTK-CST M-CST and C-CST fro an energy perspectve. Fnally Secton 4 contans a suary and soe general conclusons.. Couple stress theory Consder a ateral contnuu occupyng a volue V bounded by a surface S wth outer unt noral n as shown n Fg.. 5

6 Fg.. The body confguraton. In couple stress theory the nteracton n the body s represented by true (polar) force-stress and pseudo (axal) couple-stress tensors. The coponents of the force-stress and couplestress tensors n ths theory are shown n Fg x 33 x x Fg.. General coponents of force- and couple-stress tensors n the couple stress theory. 6

7 As n the Cosserat contnuu theory (Cosserat and Cosserat909) the force and oent balance governng equatons for an nfntesal eleent of atter under quasstatc condtons are wrtten respectvely as: F 0 () () k k 0 where F s the specfed body-force densty and k s the Lev-Cvta alternatng sybol. The force-tracton vector t and couple-tracton vector noral vector n are gven by t at a pont on surface eleent ds wth unt n (3) n (4) The force-stress tensor s generally non-syetrc and can be decoposed as (5) where and are the syetrc and skew-syetrc parts respectvely. The angular equlbru equaton () gves the skew-syetrc part of the force-stress tensor as εk lk l (6) Thus for the total force-stress tensor we have klk l (7) As a result the lnear equaton of equlbru () reduces to k lk l F 0 (8) 7

8 In nfntesal deforaton theory the dsplaceent vector feld u s suffcently sall that the nfntesal stran and rotaton tensors are defned as u u e u (9) u u u (0) respectvely. Snce the true (polar) tensor s skew-syetrcal one can ntroduce ts correspondng dual axal (pseudo) rotaton vector as We notce that the defnton () requres kk kuk () 0 () whch s the copatblty equaton for the pseudo rotaton vector. Ths condton constrans the for of a gven rotaton vector. It should be notced that the rgd body porton of oton assocated wth nfntesal eleents (or rgd trads) at each pont of the contnuu s represented by the dsplaceent vector u and the rotaton vector. As deonstrated n Hadesfandar and Dargush (05a) the sutable easures or etrcs of deforaton are defned based on the relatve rgd body oton of trads at adacent ponts of the contnuu. Ths eans the easures of deforaton are defned based u and. We should ephasze here that n classcal nfntesal theory only the on syetrc part of u defnes the deforaton. Should we expect that the entre tensor contrbute as a easure of deforaton n couple stress theory? Snce the tensor represents a cobnaton of bendng and torsonal deforaton of the ateral at each pont t can be properly called the bend-twst tensor (dewt 973). The nfntesal 8

9 pseudo (axal) torson and ean curvature tensors (Hadesfandar and Dargush 0) are defned respectvely as (3) (4) Snce the ean curvature tensor s also skew-syetrcal we can defne ts correspondng dual polar (true) ean curvature vector as Ths can also be expressed as kk (5) u u (6) 4 Mndln and Tersten (96) and Koter (964) have shown that the dsplaceent feld u specfed on a sooth part of the boundary surface S specfes the noral coponent of the rotaton nn n. Accordngly they have deonstrated that ateral n a consstent couple stress theory does not support ndependent dstrbutons of noral surface couple (or twstng) tracton nn n. Ths eans nn n nn 0 on S (7) Fro a atheatcal pont of vew these results show that we can specfy ether the dsplaceent vector u or the force-tracton vector t and the tangental coponent of the rotaton vector or the tangent couple-tracton vector. In the other words for three-densonal boundary value probles the nuber of kneatcal and echancal boundary condtons are each fve. However Mndln Tersten and Koter dd not realze the fundaental plcaton of equaton (7) as a 9

10 constrant on the for of the couple-stress tensor (Hadesfandar and Dargush 0 05a) whch postponed the defnton of a consstent couple stress theory for half a century. For boundary condtons n general couple stress theory one ay specfy dsplaceents u or force-tractons t u u on S (8a) u t t on S (8b) t and tangental rotaton or bendng couple-tracton Here on (9a) S on S (9b) S u and S are the portons of the surface at whch the essental boundary values for the dsplaceent vector u and the rotaton are prescrbed respectvely. Furtherore S t and S are the portons of the surface at whch the force-tracton vector t and the couple-tracton are specfed respectvely. In order to construct a well-posed boundary value proble we ust have S S S S S (0a) u t u t S S S S S (0b) Now we present specfc aspects of the orgnal ndeternate couple stress theory (MTK-CST) odfed syetrc couple stress theory (M-CST) and consstent skew-syetrc couple stress theory (C-CST)... Orgnal couple stress theory (MTK-CST) In the orgnal Mndln-Tersten-Koter couple stress theory (MTK-CST) there s no constrant on the couple-stress tensor. As a result there are 5 ndependent stress coponents. Ths ncludes sx coponents of and nne coponents of. 0

11 The correspondng easure of deforaton conugate to the couple-stress tensor s the bend-twst tensor k () where k () 0 For lnear sotropc elastc ateral the consttutve relatons are e e kk (3) Q 4k 4 k Q 4 4 (4) Here Q s the ndeternate sphercal part of the couple-stress tensor where Q s a pseudoscalar. We notce that the odul and are the Laé coeffcents for sotropc eda. These two coeffcents are related by (5) where s the Posson s rato. The paraeters and are the couple stress ateral coeffcents for sotropc eda. For ths ateral the elastc energy densty takes the for W e kk e e k k kk (6) where the postve-defnte elastc energy condton requres (7a-d)

12 We can defne the ratos l c (8ab) where c (9) Here l defnes a characterstc ateral length whch accounts for sze-dependency n ths theory. Therefore the relaton (4) can be wrtten as Q 4 l c (30) We notce that the ndeternacy of Q then carres nto the skew-syetrcal part of the forcestress tensor such that k lk l Q l k k k k (3) and the total force-stress tensor becoes Q e e l k k kk k k (3) whch also s ndeternate. Therefore we obtan the lnear equlbru equaton n ters of the dsplaceent as l u ( l ) u F 0 kk (33) In ndeternate couple stress theory (MTK-CST) the couple-tracton s Q 4 4 Qn 4 n n (34)

13 As a result the noral surface couple (or twstng) tracton nn becoes nn n Q 4 nn (35) However ths coponent s not necessarly zero as requred n (7) even f we gnore the ndeternacy ter Q. Ths contradcton obvously shows that the MTK-CST s nconsstent. To resolve ths proble we ay apparently use the transforaton proposed by Koter (964) that a dstrbuton of noral surface twstng couple-tracton ( nn) on the actual surface S s replaced by an equvalent shear stress dstrbuton and a lne force syste. However ths transforaton s not consstent wth the dea of a contnuu echancs theory (Hadesfandar and Dargush 05a). A consstent couple stress theory ust satsfy ths condton drectly n ts forulaton that s nn n 0 on S (36) Therefore all troubles n the ndeternate couple stress theory (MTK-CST) are the result of not satsfyng ths condton n a systeatc way. Let us pose ths constrant at the present stage and nvestgate ts consequences on MTK-CST. We notce that by the fundaental contnuu echancs hypothess (Hadesfandar and Dargush 05a) the noral surface twstng coupletracton ( nn) ust not only vansh on the actual boundary surface S but on the boundary surface S a of any arbtrary subdoan wth volue V a as shown n Fg. 3 that s nn Q 4 nn 0 on S (37) a Fg. 3. The state of couple-tracton 3 n nsde the body.

14 However n ths relaton n s arbtrary at each pont and can be any arbtrary devatorc tensor; we ay construct subdoans wth any surface noral orentaton at a pont. Consequently the condton (37) requres that Q 0 and 0 (38) However the condton (7d) rewrtten equvalently as 0 requres Q 0 and 0 (39) whch eans that there are no couple-stresses n the body. Therefore the consstency condton has reduced MTK-CST to the classcal theory. We should enton that Koter s loadng transforaton ethod s ust an approxaton ethod whch conceals the nconsstency of ths contnuu theory n an unreasonable anner. Therefore fndng solutons n MTK-CST that satsfy all boundary condtons consstently s practcally possble for any probles. For exaple there s no consstent soluton for pure torson of a crcular bar n ths theory and the approxate soluton for pure torson n MTK-CST predcts sgnfcant sze effect whch does not agree wth experents (Hadesfandar and Dargush 06). In addton MTK-CST cannot descrbe the pure bendng of a plate properly (Hadesfandar et al. 06)... Modfed couple stress theory (M-CST) In ths couple stress theory orgnally proposed by Yang et al. (00) the pseudo couple-stress tensor s syetrcal that s (40) Therefore there are ndependent stress coponents n ths theory. Ths ncludes sx coponents of and sx coponents of. We notce that n ths theory the noral couplestress coponents 33 on the plane eleent surfaces create torson and tangental coponents 3 3 defor these plane eleents to antclastc surfaces wth negatve Gaussan curvature. Therefore the syetrc couple stress tensor creates a cobnaton of torson and antclastc deforaton for plane eleent surfaces of a contnuu. 4

15 The correspondng easure of deforaton conugate to the syetrc couple-stress tensor s the syetrc devatorc torson tensor where (4) (4) 0 For lnear sotropc elastc ateral the consttutve relatons are e e kk (43) Q 8 l 4 l Q (44) We notce that the couple stress constant can be expressed as l (45) The relatons n the odfed couple stress theory (M-CST) n orgnal for (Yang et al. 00) can be found by scalng 4 and l l n the present equatons. The elastc energy densty for ths ateral takes the for W e kk e e 4 (46) We notce that the ndeternacy of Q n (44) then carres nto the skew-syetrcal part of the force-stress tensor such that 5

16 k lk l Q l k k k k (47) and the total force-stress tensor becoes Q e e l k k kk k k (48) In ths theory for the lnear equlbru equaton n ters of the dsplaceent we obtan exactly the sae equaton as (33) n MTK-CST that s l u ( l ) u F 0 kk (49) In odfed couple stress theory (M-CST) the couple-tracton s Q 8l Qn 4l n n (50) As a result the noral surface couple (or twstng) tracton nn becoes nn n Q8 l nn (5) We notce that n M-CST ths coponent s not necessarly zero as requred n (7) even f we gnore the ndeternacy ter Q. Ths contradcton shows that the odfed couple stress theory s also nconsstent. As entoned above n Secton. the Koter transforaton s not consstent wth the dea of a contnuu echancs theory (Hadesfandar and Dargush 05a). A consstent couple stress theory ust satsfy ths condton drectly n the forulaton that s nn n 0 (5) Therefore all troubles n the M-CST as n MTK-CST are the result of not satsfyng ths zero noral couple tracton condton n a systeatc way. By posng ths constrant for the boundary surface S a of any arbtrary subdoan wth volue V a as shown n Fg. 3 we obtan 6

17 nn Q 8l nn 0 on S (53) a However n ths relaton n s arbtrary at each pont and can be any arbtrary devatorc tensor because we ay construct subdoans wth any surface noral orentaton at a pont. Consequently the condton (53) requres that Q 0 and l 0 (54) whch eans that there s no couple-stresses n the body. Therefore the consstency condton has reduced the odfed couple stress theory to the classcal theory. Fro a practcal pont of vew t s also generally possble to satsfy all boundary condtons correctly n the soluton of ost probles usng M-CST. Ths can be observed for the pure torson of a crcular bar where t s possble to obtan an exact contnuu M-CST soluton. Furtherore the apparent approxate pure torson soluton predcts a sgnfcant sze-effect whch contradcts recent experents for pure torson of cro-daeter copper wres (Hadesfandar and Dargush 06). In addton M-CST cannot descrbe the pure bendng of a plate properly (Hadesfandar et al. 06). Surprsngly M-CST predcts no couple-stresses and no sze effect for the pure bendng of the plate nto a sphercal shell. These characterstcs also ake M-CST of questonable value to serve as a bass for sze-dependent structural odels such as beas plates and shells. Perhaps ths character of M-CST has not been fully understood but n any case t s unfortunate that ths theory has been used so extensvely n structural echancs..3. Consstent couple stress theory (C-CST) In ths couple stress theory the pseudo couple-stress tensor s skew-syetrcal (55) Therefore there are nne ndependent stress coponents n ths theory. coponents of and three coponents of. Snce the couple-stress tensor s skewsyetrc the couple-tracton 7 Ths ncludes sx gven by (4) becoes tangent to the surface. As a result the

18 couple-stress tensor creates only bendng couple-tracton on any arbtrary surface n atter. The coponents of the force-stress and couple-stress tensors n ths theory are shown n Fg x 33 x 3 x Fg. 4. Coponents of force- and couple-stress tensors n consstent couple stress theory. We notce that n ths theory the couple-stress coponents defor the plane eleent surfaces to ellpsod cap-lke surfaces wth postve Gaussan curvature. It should be ephaszed that the skew-syetrc character of the couple-stress tensor s a fundaental contnuu echancs property whch has nothng to do wth any consttutve relaton. As a consequence ths result s n no way lted to lnear elastc aterals or to sotropc response. In C-CST we can defne the true (polar) couple-stress vector dual to the tensor as εkk (56) 8

19 Ths relaton can also be wrtten n the for (57) k k Consequently the surface couple-tracton vector whch obvously s tangent to the surface. reduces to n n (58) k k The correspondng easure of deforaton conugate to the skew-syetrc couple-stress tensor s the skew-syetrc ean curvature tensor. We notce that the skew-syetrc part of the force-stress tensor fro equaton (6) becoes (59) Thus for the total force-stress tensor we have klk l (60) As a result the lnear equaton of equlbru () reduces to F 0 (6) For lnear sotropc elastc ateral the consttutve relatons are e e kk (6) 8l 4l (63) 9

20 Here we have l as the characterstc ateral length n the consstent couple stress theory where the couple stress constant can be expressed as The elastc energy densty n ths theory takes the for whch can also be wrtten as Then by usng (59) we obtan l (64) W e kk ee 4 (65) W ekk ee 8l (66) (67) l k k for the skew-syetrc part of the force-stress tensor. Therefore the total force-stress tensor becoes e e l (68) kk k k whch s fully deternate. Interestngly for the lnear equlbru equaton for sotropc elastc aterals n ters of the dsplaceent we obtan exactly the sae equaton as (33) and (49) n MTK-CST and M-CST that s l u ( l ) u F 0 kk (69) In the consstent couple stress theory (C-CST) the couple tracton s n 4l n (70) We notce that the noral surface couple-tracton nn vanshes that s nn n nn 4l 0 nn (7) 0

21 Therefore there s no ndeternacy and no noral surface couple-tracton nn. Ths s the an reason that ths theory s consstent. As a result probles wthn C-CST can be well-posed and boundary condtons can be satsfed precsely. For exaple for the pure torson of an elastc crcular bar the soluton n C-CST reduces to that n classcal theory where there s no sze effect. Interestngly ths predcton copletely agrees wth recent experents for pure torson of crodaeter copper wres (Hadesfandar and Dargush 06). In addton C-CST s the only couple stress theory whch descrbes the pure bendng of a plate properly (Hadesfandar et al. 06). It should be entoned that dscoverng the skew-syetrc character of the pseudo couple-stress tensor results n defnng the skew-syetrc pseudo ean curvature tensor. Snce the couple-stress and ean curvature have true vectoral character we can defne the true couplestress vector and ean curvature vector. Interestngly ths result gves the clue how to defne the ean curvature vector n dfferent space and hgher densons. It turns out that ths has a fundaental pact n understandng soe physcal phenoena. For exaple Hadesfandar (03) has developed the geoetrcal vortex theory of electroagnets n fourdensonal space-te where the electroagnetc four-vector potental and strength felds are the four-densonal velocty and vortcty felds respectvely. Ths theory shows that the agnetc and the electrc felds are the crcular and hyperbolc vortcty-lke felds respectvely. Therefore the hoogeneous Maxwell s equatons are the necessary copatblty equatons for the electroagnetc vortcty vectors whereas the nhoogeneous Maxwell s equatons govern the oton of these vortctes. Geoetrcally the nhoogeneous Maxwell s equatons are the relaton for the ean curvature four-vector of the electroagnetc velocty feld. It turns out that these equatons sply show that the four-vector electrc current densty s proportonal to the fourdensonal ean curvature of the four-vector potental feld..4. Dscusson In ths secton we have presented dfferent versons of couple stress theores (MTK-CST M-CST and C-CST) and exaned soe of ther theoretcal and practcal aspects. We have notced that MTK-CST and M-CST not only suffer fro dfferent nconsstences such as ndeternacy and ll-posed boundary condtons these theores also cannot descrbe properly several eleentary

22 practcal probles such as pure torson of a crcular bar and pure bendng of a plate. On the other hand C-CST s consstent wth well-posed boundary condtons and can descrbe the pure torson of a crcular bar and pure bendng of a plate properly. Although the fnal governng equatons for sotropc lnear elastc aterals n ters of the dsplaceent vector are the sae n these theores the dstrbuton of nternal stresses and boundary condtons are dfferent. Because of the ndeternacy t s practcally possble to fnd solutons to probles n MTK-CST and M-CST whch satsfy all boundary condtons consstently. We notce that the nconsstent approxate solutons for pure torson n these theores predct a sgnfcant sze effect whch does not see to agree wth experents (Hadesfandar and Dargush 06). The recent physcal experents by Song and Lu (05) Lu et al. (03) and Lu and Song (0) entrely agree wth the predcton of skew-syetrc consstent couple stress theory (C-CST) that there s no sze effect n the elastc range for pure torson of cro-daeter copper wres. Furtherore MTK-CST and M-CST also cannot descrbe pure bendng of a plate properly (Hadesfandar et al. 06). Partcularly M-CST predcts no couple-stresses and no sze effect for the pure bendng of the plate nto a sphercal shell. Interestngly Tang (983) has appled MTK-CST based bea bendng of Kao et al. (979) and Tzung et al. (98) to exane four-pont bendng and unaxal tensle data of varous sze cylndrcal and square specens for three grades of graphte: H-37 H-45 and AGOT. The evaluatons ndcate that the data can be nterpreted by lnear couple stress theory (MTK-CST) wth 0.85 for H-45 graphte. Furtherore the results are proved by consderng a non- lnear effect whch yelds the new value As we can see these result n hndsght show that the experental data actually approach the consstent couple stress theory (C-CST) correspondng to. On the other hand one would need for verfcaton of M- CST whch clearly s not consstent wth ths experental data for graphte.

23 In lght of all of the above we realze that the confuson n the developent of couple stress theory for ore than a half a century has gven soe status to MTK-CST and M-CST. Had Mndln Tersten and Koter dscovered the skew-syetrc character of the couple-stress tensor n the 960s there would not have been the present confuson n couple stress theory. We should notce that the consstent couple stress theory (C-CST) systeatcally lnks efforts of Cosserats Mndln Tersten and Koter and others n a span of a century. Although the fnal lnear equlbru equatons n ters of the dsplaceent n all three dfferent versons of couple stress theores (MTK-CST M-CST and C-CST) are the sae the odfed couple stress theory (M-CST) and consstent couple stress theory (C-CST) are not specal cases of the orgnal Mndln-Tersten-Koter theory (MTK-CST). Ths s because the dfferent curvature tensors n M-CST and C-CST are the syetrc and skew-syetrc parts of the bend-twst tensor respectvely. It apparently sees that for sotropc lnear elastc aterals the forulatons n M-CST and C-CST can be obtaned by lettng and respectvely n the orgnal MTK-CST. However we ust notce that these cases are excluded by condton (7d) ( < ) for the ndeternate MTK-CST. In addton ths pecularty s only vald for sotropc ateral. There s no sple analogy for general ansotropc or non-lnear cases. Furtherore a consstent theory such as C-CST cannot be consdered as a specal case of an nconsstent theory such as MTK-CST. However we should reeber that MTK-CST stands as a fundaental pllar n the developent of the consstent couple stress theory (C-CST). Ths s obvous fro the fact that eleents of the C-CST are based on the orgnal MTK-CST. Although the work of Mndln Tersten and Koter s fundaental n developent of couple stress theory nn they dd not realze that satsfyng the boundary condton 0 n a systeatc anner reveals the deternate skew-syetrc nature of the couple-stress tensor. To perceve better the status of MTK-CST M-CST and C-CST n contnuu echancs we can use the followng sple llustratve analogy to classcal contnuu echancs:. MTK-CST wth k as the curvature tensor s analogous to a classcal theory wth u as the stran tensor; 3

24 . M-CST wth as the curvature tensor s analogous to a classcal theory wth stran tensor; as the 3. C-CST wth as the curvature tensor s analogous to the correct classcal theory wth e as the stran tensor. However we notce that the classcal theores based on u and are not correct and are never consdered as possble theores. It s also obvous that the consstent classcal theory wth e as the stran tensor s not consdered as a specal case of the general theory wth u as the stran tensor. Interestngly t sees possble to solve soe sple probles n the classcal theores wth u and as stran easures of deforaton. For exaple the sple tenson or bendng of a slender bar can be solved reasonably n all of these theores. However these eleentary solutons do not ustfy the theores based on u and as vable theores. As dscussed there s a slar stuaton n couple stress theores. In recent papers (Neff et al. 06; Ghba et al. 06; Madeo et al. 06; Münch et al. 05) the authors have claed that the consstent skew-syetrc couple stress theory s not the only possble theory to represent the contnuu consstently. They advocate that the other theores are also vald and can be arbtrarly used. For exaple they suggest that the couple-stress tensor ay be chosen syetrc and trace free (Münch et al. 05). Although these papers use a labyrnth of atheatcal forulae and are nearly penetrable the work s stll lted to lnear sotropc elastcty rather than provdng generalty for contnuu echancs as a whole. Interestngly these authors have also claed dscoverng the correct tracton boundary condtons n the ndeternate couple stress odel (Neff et al. 05). However the newly defned boundary condtons n the ndeternate odel are far too coplcated and non-physcal. It s also not known why the couple-stress tensor would stll be ndeternate n a consstent odel. These authors do not realze that the ndeternacy eans there s stll a trouble or nconsstency n ths odel whch s why soe researchers gave up on MTK-CST and revved the dea of crorotaton concept (Erngen 968). 4

25 Furtherore t s ncorrect to thnk that the energy ethod could ustfy gnorng the ndeternate sphercal part of the couple-stress tensor. As wll be deonstrated n the followng secton the drect nzaton of total potental energy for MTK-CST and M-CST volates the dvergence free copatblty constrant 0 whch has led Neff and colleagues to erroneous conclusons. 3. Varatonal ethod and ts consequences In ths secton we derve the governng equatons for an elastc body by usng a varatonal ethod nzng the total echancal potental energy. We consder the orgnal odfed and consstent couple stress theores (MTK-CST M-CST and C-CST respectvely) and develop the energy ethod for the general non-lnear ansotropc elastc case under nfntesal deforaton theory. It should be notced that we gnore the aforeentoned nconsstences of MTK-CST and M-CST n our varatonal ethod n ths secton. The total potental energy for the elastc body s (7) WdV Fu dv t u ds ds V V St S where S t and S are the portons of the surface on whch t and Here W represents the general elastc energy densty functon where are prescrbed respectvely. W W e k n the orgnal couple stress theory (MTK-CST) (73a) W W e n odfed couple stress theory (M-CST) (73b) W W e n consstent couple stress theory (C-CST) (73c) The equlbru condton corresponds to the nu of the total potental energy. It should be notced that n a consstent theory we do not need to pose the dvergence free copatblty constrant 5

26 0 (74) n the nzaton. We expect that ths constrant s satsfed n all steps of the nzaton process. However for assurance we pose ths constrant by usng the Lagrange ultpler ethod and nvestgate f t vanshes n the fnal results. If the Lagrange ultpler perssts wthn the forulaton then ths ndcates that the drect nzaton of cannot satsfy the dvergence free copatblty constrant 0 at least n one of the steps of the process. Thus there s an nconsstency n the correspondng couple stress theory forulaton. Therefore by usng the Lagrange ultpler ethod to enforce the constrant (74) we defne the Lagrangan functonal WdV Fu dv t u ds ds q dv (75) V V St S V where q s the Lagrange ultpler that can be a functon of space. The equlbru condton corresponds to 0 (76) where s the frst varaton of the functonal. If q 0 n the fnal results then the constrant (74) s satsfed autoatcally n the nzaton of n (7). Ths would show that the correspondng couple stress theory s consstent. On the other hand f q does not vansh n the fnal result then t ndcates that the drect nzaton of n (7) volates the constrant (74). Ths eans that there s soe nconsstency n the correspondng couple stress theory. Interestngly the physcal eanng of the Lagrange ultpler q wll be revealed after obtanng the governng equatons and the boundary condtons. 3.. Varatonal ethod for orgnal couple stress theory (MTK-CST) For the orgnal Mndln-Tersten-Koter couple stress theory (MTK-CST) W We k we have V V St S V and u e k WdV Fu dv t u ds ds q dv (77) 6

27 Therefore the frst varaton of s W W e k dv F uds t uds ds q dv e V k (78) V St S V where k (79) We should notce that the varaton of k ust be consstent wth the varaton of constrant (80) k 0 whch has been posed n (78) by usng the Lagrange ultpler ethod. Note that n t n and F are specfed quanttes not subect to varaton n (78). By consderng the condtons u 0 on S u and 0 on S (78) can be wrtten as W W e k dv F uds t uds ds q dv e V k (8) V S S V By soe anpulaton we obtan W W W u dv q dv e V e k V Fu ds t u ds ds S V S (8) By usng ntegraton by part on the second ter ths becoes V W W u dv e e W W q q dv k V k Fu ds t u ds ds S V S (83) 7

28 At ths stage we notce that kuk (84) Therefore the varaton (83) can be wrtten as W W W q n u dv e e k V n n V W q dv k FudS tuds ds S S S (85) Agan by usng ntegraton by parts ths becoes W W W q n u dv e e k V n n W W W q udv n V e e kn n V S W q dv k S FudS tuds ds S (86) Now we apply the dvergence theore to the frst and thrd ters n the volue ntegral n (86) and by notcng that q 0 (87) we obtan the relaton 8

29 W W W F udv e e k V n n W W W q n n t uds e e k S n n W q n ds k S (88) We should recall the condtons u 0 on S u and 0 on S to obtan the relaton W W W F udv e e k V n n W W W q n t uds e St e kn n W q n ds k S (89) The varaton u s arbtrary n the doan V n (89). The varatons of u and are also arbtrary on the boundary surfaces S t and S respectvely. Therefore the ndvdual ters n the ntegrals ust vansh separately and we have W W W F 0 e e k n n n V (90) W W W t q n e e kn n on S t (9) W q n k on S (9) 9

30 For lnear sotropc ateral where W s gven by (6) these equatons reduce to l u ( l ) u F 0 kk n V (93) t e e l q n kk on S t (94) q n 4 l c on S (95) By coparng the governng equaton (93) and boundary condtons (94) and (95) wth ther correspondng equatons n secton. for MTK-CST we recognze the Lagrange ultpler q as the sphercal part of the couple-stress tensor Q that s and obtan the general consttutve relatons q Q (96) e e kk (97) Q 4 l c (98) kq k l k k (99) As we can see the varatonal ethod shows that the couple-stress tensor s stll ndeternate. Here we have deonstrated that the constraned nzaton of produces the fundaental governng equatons n the ndeternate couple stress theory (MTK-CST) for lnear elastc sotropc aterals. Interestngly the Lagrange ultpler correspondng to the constrant (74) 0 s the ndeternate sphercal part of the couple-stress tensor. Ths developent shows that f we do not pose the constrant (74) n our varatonal ethod the drect nzaton process of volates ths constrant n the frst step of varaton n (78) where k s not necessarly zero. Therefore the ncorrect drect nzaton of results n a trace free couple- 30

31 stress tensor n the orgnal couple stress theory. Rearkably the fact that the constrant (74) s not satsfed autoatcally n the nzaton of n (7) deonstrates that there s soe nconsstency n the Mndln-Tersten-Koter couple stress theory (M-CST). 3.. Varatonal ethod for odfed syetrc couple stress theory (M-CST) For the odfed couple stress theory (M-CST) W We and we have u e WdV FudV tuds ds q dv (00) Therefore the frst varaton of s V V St S V W W e dv FudS tuds ds q dv e V (0) V St S V where (0) We should notce that the varaton of ust be consstent wth the varaton of constrant (03) 0 whch has been posed n (0) by usng the Lagrange ultpler ethod. Agan by consderng the condtons u 0 on S u and 0 on S we can wrte (0) as W W e dv F uds tuds ds q dv e V (04) V S S V Therefore ths can be wrtten as 3

32 W W W W u dv q dv e e V V Fu ds t u ds ds S V S (05) By usng ntegraton by part on the second ter ths becoes V W W u dv e e W W W W q q dv V Fu ds t u ds ds S V S (06) where we have kuk (07) Therefore the varaton can be wrtten as W W W W qn u dv e e 4 n n V n W W q dv V Fu ds t u ds ds S S S (08) Agan by usng ntegraton by parts ths becoes 3

33 W W W W q u dv e e 4 n n V n W W W W q udv n 4 V e e n n n V W W q dv Fu ds t u ds ds S S S (09) Now we apply the dvergence theore to the frst and thrd ters n the volue ntegral n (09) and by notcng that q 0 (0) we obtan the relaton W W W W F udv e e 4 V n n n W W W W qn n t uds e e 4 n n S n S W W q n ds () By applyng the condtons u 0 on S u and 0 on S we obtan the relaton 33

34 W W W W F udv e e 4 V n n n W W W W q n t uds e 4 St e n n n S W W n ds () We notce that the varaton u s arbtrary n the doan V n (); and the varatons of u and are also arbtrary on the boundary surfaces S t and ndvdual ters n the ntegrals vansh separately and we have S respectvely. Therefore the W W W W F 0 e e 4 n n n n V (3) W W W W t q n e e 4n n n on S t (4) W W q n on S (5) For lnear sotropc ateral where W s gven by (46) these equatons reduce to l u ( l ) u F 0 kk n V (6) t e e l q n kk on S t (7) 8 q n on S (8) 34

35 By coparng the governng equaton (6) and boundary condtons (7) and (8) wth ther correspondng equaton n secton. for M-CST we recognze the Lagrange ultpler q as the sphercal part of the couple-stress tensor Q that s and obtan the general consttutve relatons q Q (9) e e kk (0) l Q 8 Q l 4 () kq k l () k k Therefore we have deonstrated that the constraned nzaton of produces the fundaental governng equatons n the ndeternate odfed couple stress theory (M-CST) for the lnear elastc sotropc aterals. We notce that for ths case the Lagrange ultpler correspondng to the constrant (74) ( 0 ) s also the ndeternate sphercal part of the couple stress tensor. Ths developent shows that f we do not pose the constrant (74) n our varatonal ethod the drect nzaton process of volates ths constrant n the frst step of varaton n (0) where s not necessarly zero. Therefore the ncorrectly developed drect nzaton of results n a trace free couple-stress tensor n the odfed syetrc couple stress theory. Snce the constrant 0 s not satsfed autoatcally n the nzaton of n (7) one should realze that there s soe nconsstency n the odfed couple stress theory (M-CST) Varatonal ethod for consstent couple stress theory (C-CST) One ght expect that consderng the copatblty constrant (74) ( 0 ) s not necessary n nzaton of for the skew-syetrc consstent couple stress theory (C-CST). We wll see f the fnal result confrs ths speculaton. 35

36 For ths case the energy densty s W We The frst varaton of s. Thus we have u e WdV FudV tuds ds q dv (3) V V St S V W W e dv F uds t uds ds q dv e V (4) V St S V where Ths relaton shows that we always have whch does not depend on the varaton of the constrant (5) 0 (6) 0 (7) posed by usng the Lagrange ultpler ethod n (4). Ths character sees to ply that we do not need to pose the dvergence free copatblty constrant (3) n the nzaton process fro begnnng for ths case. However we go forward and deonstrate ths nterestng character by dervng the governng equlbru equatons and the correspondng boundary condtons. By consderng the condtons u 0 on S u and 0 on S (4) can be wrtten as W W e dv F uds tuds ds q dv e V (8) V S S V By a slar ethod as before we obtan the relaton 36

37 W W W W F udv e e 4 V n n n W W W W qn n t uds e 4 St e n n n S W W q n ds (9) We notce that n (9) the varaton u s arbtrary n the doan V whle the varatons of u and are also arbtrary on the boundary surfaces S t and ndvdual ters n the ntegrals vansh separately and we have S respectvely. Therefore the W W W W F 0 e e 4 n n n n V (30) W W W W t q n e e 4n n n on S t (3) W W q n on S (3) However we notce that the condton requres that nn n 0 on S (33) W W n q nn W W nn qnn 0 on S (34) 37

38 W W Snce the expresson ultpler dsappears that s s skew syetrc the relaton (34) shows that the Lagrange Therefore the boundary condtons (3) and (3) reduce to q 0 (35) W W W W t n e e 4n n n on S t (36) W W n on S (37) For lnear sotropc ateral where W s gven by (65) these equatons reduce to l u ( l ) u F 0 kk n V (38) t e e l n kk on S t (39) 8 n on S (40) By coparng the governng equaton (38) and boundary condtons (39) and (40) wth ther correspondng equatons n secton.3 for C-CST we obtan the general consttutve relatons e e kk (4) 8l (4) Therefore the total force-stress tensor becoes (43) l k k e e l (44) kk k k 38

39 As expected the result q 0 shows that we do not need to pose the copatblty condton constrant 0 n the nzaton process of the total potental energy n the consstent skewsyetrc couple stress theory (C-CST) fro the very begnnng. Therefore the varatonal ethods used n the forulatons developed n Darrall et al. (04 05) and Salter and Rchardson (04) are atheatcally consstent. We can see that the drect nzaton process of can produce the fundaental governng equatons n the consstent couple stress theory (C-CST) for lnear elastc sotropc aterals wthout volatng the dvergence free constrant Dscusson By usng an energy ethod n ths secton we have deonstrated another aspect of the nconsstency of the ndeternate MTK-CST and M-CST for elastc solds. We have shown that the drect unconstraned nzaton of the total potental energy for these theores volates the dvergence free copatblty condton 0. On the other hand the drect unconstraned nzaton of total potental energy for C-CST satsfes ths copatblty condton autoatcally. Ths result once agan deonstrates the nner consstency of C-CST. In ther varatonal forulaton for odfed couple stress theory (M-CST) Park and Gao (008) gnored the ndeternacy of the couple-stress tensor copletely. As a result they consdered the unconstraned nzaton of total potental energy for M-CST whch volates the dvergence free copatblty constrant 0. Forgettng to pose the dvergence free constrant 0 n ther varatonal ethod Neff and hs colleagues also stakenly concluded that a varatonal ethod results n a trace free couple-stress tensor n M-CST (Neff et al. 06; Ghba et al. 06 Münch et al. 05). How can the couple-stress tensor be trace free n M-CST when the unconstraned nzaton of volates the dvergence free copatblty constrant 0? 39

40 Neff et al. (06) have also argued that the ndeternacy of the sphercal part of the couple-stress tensor s analogous to the behavor of an ncopressble ateral under pressure. For an ncopressble ateral the ncopressblty condton s u 0 (45) Because we can apply any pressure to an ncopressble sold wthout changng ts shape the stress cannot be unquely deterned fro the strans. For a lnear sotropc ncopressble elastc ateral n classcal theory the consttutve relaton for force stresses are p e (46) where p specfes the negatve of the sphercal part of the force-stress tensor. As a result the lnear equlbru equaton n ters of the dsplaceent n classcal elastcty becoes u F 0 p (47) Interestngly we notce that the elastc energy densty for ths ncopressble case takes the for W e e (48) As we know the pressure stress dstrbuton does not contrbute to the nternal work because e de pe pu 0 (49) 0 Interestngly we notce that the pressure p n (46) becoes the Lagrange ultpler correspondng to the ncopressblty condton (45) n a varatonal energy ethod (Spencer 980; Fosdck and Royer-Carfagn 999). However t sees Neff and hs colleagues dd not realze the consequence of ths analogy n ther varatonal ethod n couple stress theory for the copatblty condton of the rotaton vector feld 0. As entoned they have forgotten to enforce ths constrant n ther varatonal energy ethod by usng the Lagrange ultpler ethod for couple stress theory. 40

41 We should also notce that an ncopressble ateral s a atheatcal concept and physcally does not exst. The ncopressblty condton (45) s ust an artfcal assupton to splfy cases of near-ncopressblty. We notce that ths relaton s not the result of the general character of u or defnton of the stran tensor as e u u (50) Ths eans that the stran tensor e never becoes devatorc n realty. Interestngly for the lnear sotropc elastc aterals the ncopressblty condton (45) corresponds to Posson s rato whch s excluded based on the energy consderatons (7a) that requres (5) On the other hand the devatorc characters such that k 0 (5) are the result of the atheatcal defnton of k and for any arbtrary ateral. As a result the sphercal part of the couple-stress tensor n MTK-CST and M-CST becoes ndeternate for all aterals ndependent of the ateral behavor. It s ths devatorc character whch akes the tensors k and unsutable easures of bendng deforaton. We should notce that the ncopressblty of ateral u 0 s ust an approxaton for soe specal cases. However the dvergence free constrant of the rotaton vector 0 s a atheatcal constrant for all bodes based on defnton whch has been the source of all knds of confuson n the evoluton of sze-dependent contnuu echancs at least for half a century (Hadesfandar and Dargush 05b). 4. Conclusons In ths paper we have exaned soe theoretcal and practcal aspects of the three prary couple stress theores naely MTK-CST M-CST and C-CST. It has been shown that MTK-CST and 4

42 M-CST not only suffer fro dfferent nconsstences such as ndeternacy and ll-posed boundary condtons these theores also cannot descrbe eleentary practcal deforatons such as pure torson of a crcular bar and pure bendng of a plate. On the other hand C-CST s consstent wth well-posed boundary condtons and can descrbe pure torson of a crcular bar and pure bendng of a plate. Furtherore by usng an energy ethod we have also derved the governng equlbru equatons for elastc bodes n MTK-CST M-CST and C-CST. Ths developent shows that the drect nzaton of the total potental energy for MTK-CST and M-CST volates the dvergence free copatblty condton of the rotaton vector feld 0. Ths deonstrates another aspect of the nconsstency of the ndeternate MTK-CST and M-CST for elastc bodes based on the energy ethod. Therefore fro a atheatcal standpont the total potental energy functonal correspondng to MTK-CST and M-CST ust be nzed subect to the copatblty condton 0. Therefore one way to pose ths copatblty condton constrant s by usng the Lagrange ultpler ethod. Through ths approach we fnd that n MTK-CST and M-CST the Lagrange ultpler s the ndeternate sphercal part of the couplestress tensor. Ths eans the varatonal ethod results n a sphercal coponent for the couplestress tensor n MTK-CST and M-CST. On the other hand the drect unconstraned nzaton of total potental energy for C-CST satsfes ths copatblty condton autoatcally whch once ore deonstrates the nner consstency of C-CST. References Cosserat E. Cosserat F Théore des corps déforables (Theory of Deforable Bodes). A. Herann et Fls Pars. Darrall B. T. Dargush G. F. Hadesfandar A. R. 04. Fnte eleent Lagrange ultpler forulaton for sze-dependent skew-syetrc couple-stress planar elastcty. Acta Mech