Evolution of generalized couple-stress continuum theories: a critical analysis

Transcription

1 Evoluton of generalzed couple-stress contnuum theores: a crtcal analyss Al R. Hadjesfandar Gary F. Dargush Department of Mechancal and Aerospace Engneerng Unversty at Buffalo, State Unversty of New York, Buffalo, NY 460, USA ah@buffalo.edu, gdargush@buffalo.edu December 9, 04 Abstract In ths paper, we examne dfferent generalzed couple-stress contnuum mechancs theores, ncludng couple stress, stran gradent and mcropolar theores. Frst, we nvestgate the fundamental requrements n any consstent sze-dependent couple stress contnuum mechancs, for whch satsfyng basc rules of mathematcs and mechancs are crucal to establsh a consstent theory. As a result, we show that contnuum couple stress theory must be based on the dsplacement feld and ts correspondng macrorotaton feld as degrees of freedom, whle an extraneous artfcal mcrorotaton cannot be a true contnuum mechancal concept. Furthermore, the dea of generalzed force and ndependent generalzed degrees of freedom show that the normal component of the surface moment tracton vector must vansh. Then, wth these requrements n mnd, varous exstng couple stress theores are examned crtcally, and we fnd that certan devatorc curvature tensors create ndetermnacy n the sphercal part of the couple stress tensor. We also examne mcropolar and mcromorphc theores from ths same perspectve. Keywords Couple stresses Macrorotaton and mcrorotaton Curvature tensor Mcropolar theory Stran gradent theory Mcromechancs and nanomechancs

2 Introducton Classcal contnuum mechancs has provded a ratonal bass to analyze and understand the behavor of materals on human (or macro) scale for nearly two centures, snce the ntal work of Posson and Cauchy n the late 80s. Ths theory contans no length scale parameter n ts formulaton and, hence, produces sze ndependent solutons for all well-defned smooth problems. However, more recent experments show that the mechancal behavor of materals n smaller scales s dfferent from ther behavor at the more famlar macro-scales. Therefore, further progress n mcromechancs, nanomechancs and nanotechnology wll requre a consstent sze-dependent contnuum mechancs, whch can account for the length scale effect due to the mcrostructure of materals. Furthermore, ths sze-dependent contnuum mechancs can provde a more sutable connecton to atomstc models and the fundamental base for developng sze-dependent mult-physcs formulatons, such as those nvolvng electromechancal couplng. A revew of the early lterature reveals that classcal or Cauchy contnuum mechancs was based ntally upon an atomstc representaton of matter havng only central forces among partcles. As a result, the force-stresses j descrbe the nternal forces n the contnuum model []. However, n a more realstc representaton of matter the ntroducton of non-central forces n the underlyng atomstc model s nevtable. Ths led Vogt [] as a natural extenson to consder also the effect of couple-stresses j n the correspondng contnuum representaton, although he dd not develop a complete mathematcal theory. In the frst decade of the twenteth century, the Cosserat brothers [3] began to develop a mathematcal model to analyze materals wth couplestresses. In the contnuaton of ths development, the nteracton n bodes s generally represented by true (polar) force-stress j and pseudo (axal) couple-stress j tensors. The components of these force-stress and couple-stress tensors n the orgnal form as dentfed by the Cosserats are shown n Fg..

3 x 33 x x Fg. Components of force- and couple-stress tensors n the orgnal Cosserat theory As a result, the polar force-tracton vector t n and axal moment-tracton vector n m at a pont on surface element ds wth unt normal vector n are gven by These vector tractons are shown n Fg.. t n n, () j j n m n. () j j n m n n t x x ds x 3 Fg. Force-tracton t n n and moment-tracton m system 3

4 As shown n Fg., the force- and couple-stress tensors j and j have eghteen components altogether. Snce the tensors j and j are general non-symmetrc tensors, both can be decomposed nto symmetrc and skew-symmetrc parts j j j, (3) j j j. (4) Here we have ntroduced parentheses surroundng a par of ndces to denote the symmetrc part of a second order tensor, whereas square brackets are assocated wth the skew-symmetrc part. The lnear and angular equlbrum equatons n the orgnal quasstatc Cosserat theory n dfferental form are gven by where F and, (5) j, j F 0, (6) j, j jk jk C 0 C are the body force and the body couple per unt volume of the body, respectvely. Here jk s the permutaton tensor or Lev-Cvta symbol. We notce that the angular equlbrum can be wrtten as j jk lk, l jkck, (7) whch can be used to obtan the skew-symmetrcal part of the force-stress tensor j. Therefore, the sole role of the angular equlbrum Eq. (6) s to produce the skew-symmetrc part of the force-stress tensor. However, the appearance of body couple C n Eq. (7) s very dsturbng. After all, we expect the consttutve relaton for j be ndependent of the body couple C. Ths ssue s even more serous n some dynamcal models, n whch the spn nerta dstrbuton has also appeared. As a result, the total force-stress tensor can be wrtten as j j jk lk, l jkck. (8) 4

5 By usng ths expresson, the lnear equaton of equlbrum can be wrtten as [ j jk lk, l ], j F jkck, j 0. (9) Ths vector equlbrum equaton nvolves ffteen components of stresses. Therefore, t requres twelve extra equatons from consttutve relatons; a task that does not look optmstc n ths form. It s obvous that by neglectng the effect of couple stresses and body couples 0, C 0, (0a,b) j we obtan the classcal or Cauchy contnuum mechancs. In ths classcal theory, the angular equlbrum Eq. (6) or Eq. (7) show that the force-stress tensor s symmetrc j 0, j j j. (a,b) Ths means that the tensor j has sx ndependent components and we have three lnear equlbrum equatons n Eq. (5). In the classcal theory, the extra three equatons are obtaned by developng consttutve relatons, whch requres defnng a measure of deformaton n the materal. In classcal contnuum mechancs, the deformaton s specfed by the dsplacement feld u and the consstent measure of deformaton s the symmetrc stran tensor e j, defned by e j u, j u j,, () for nfntesmal deformatons. However, ths measure of deformaton s not suffcent n the more realstc sze-dependent contnuum representaton of matter, where there s no reason to neglect the effect of possble nternal couple-stresses j. We notce that n sze-dependent couple stress contnuum mechancs there are eghteen components of stresses, whereas we have sx lnear equlbrum equatons n Eqs. (5) and (6). Ths makes the number of ndependent components twelve, whch means the extra twelve equatons must be obtaned by developng consttutve relatons. Therefore, sze-dependent theores requre ntroducng new degrees of freedom and new measures of deformatons, n addton to the dsplacement vector u and stran 5

6 tensor e j. However, the development of so many dfferent generalzed couple-stress contnuum theores n the last century shows that ths has been a dffcult task and, n partcular, there has been confuson n defnng knematcal degrees of freedom and measures of deformaton. In hndsght, ths ndcates that somethng fundamental has been mssed n the formulatons. We should notce that the man dffculty n developng a consstent couple stress theory from the begnnng has been the excessve number of components of force- and couple-stresses. As a result, we can suggest that there s a relaton between the number of stress components and the descrpton of deformaton wthn a true contnuum. Ths has been the man shortcomng untl recent tmes. Here we examne crtcally the evoluton of the Cosserats deas toward the development of a consstent theory. Ths ncludes the orgnal Cosserat theory [3], Mndln and Tersten [4], and Koter [5] couple stress theores, Yang et al. [6] modfed couple stress theory, stran gradent theores [7,8], as well as mcropolar, mcrostretch and mcromorphc theores [9-]. Manly based on the number of degrees of freedom and the addtonal measures of deformaton, we demonstrate that these theores suffer from varous nconsstences. As a result, t has been mpossble for any researcher to choose decsvely whch of these theores, f any, s selfconsstent and worthy of further study. Metaphorcally speakng, we mght say that all these dfferent theores have created a soup nto whch everybody adds some new ngredent based on hs or her taste. We organze the remander of ths paper as follows. In Sect., we consder the orgnal Cosserat theory and examne ts shortcomngs. Here we clarfy the knematcs of a contnuum by establshng the relaton between dsplacement and rotaton felds. Next, n Sect. 3, we consder the very mportant ndetermnate couple stress theory developed by Mndln and Tersten and Koter. By demonstratng the nconsstences n ths theory, we obtan the requrements for a consstent sze-dependent couple stress theory. In Sect. 4, we examne the modfed couple stress theory. Sect. 5 presents a revew of general stran gradent theores. Then, n Sect. 6, we examne mcropolar theory and ts more generalzed forms of mcrostretch and mcromorphc theores, whle Sect. 7 presents the recently developed consstent sze-dependent couple stress 6

7 theory. Sect. 8 contans a bref dscusson and an overall comparson of the theores for lnear elastc materals. Fnally, Sect. 9 provdes some general conclusons. Orgnal Cosserat theory The Cosserat brothers [3] formulated several theores for structural elements, such as beams and shells, whch represent one- and two-dmensonal objects embedded n three-dmensonal space. These theores nvolve dsplacements and ndependent rotatonal degrees-of-freedom n a natural way consstent wth a contnuum hypothess. Based upon these successes, the Cosserats then extended ths dea of ndependent rotatonal degrees-of-freedoms to the case of a full threedmensonal body n whch each partcle s outftted wth a trad of vectors, called drectors [3]. Such formulatons are today referred to as mcropolar theores (Erngen [9], Nowack [0]), whch attempt to capture the effect of dscontnuous mcrostructure by consderng a contnuous mcrorotaton n addton to the translatonal degrees-of-freedom u. However, ths extenson s problematc, because t requres embeddng a three-dmensonal contnuum, along wth addtonal ndependent rotatons, nto a three-dmensonal space. In a contnuum representaton of matter, t s assumed that matter s contnuously dstrbuted n space. As a result, the deformaton of the body s represented by the contnuous dsplacement feld u wthout consderng the dscontnuous mcrostructure of matter and moton of ndvdual partcles. Ths can be smply explaned by consderng an amount of gas n a closed contaner at thermodynamc equlbrum. In a contnuum mechancal vew, the velocty feld of the gas s zero. However, we notce that the ndvdual partcles (molecules) have random motons (translatons and rotatons). The Maxwell Boltzmann dstrbuton n statstcal mechancs descrbes partcle speed probablty densty as a functon of temperature. However, we notce that the average velocty or average momentum transfer s zero, whch s consstent wth a zero velocty feld n contnuum mechancs. When the gas flows, the contnuum mechancal velocty feld s actually the drft velocty at each pont, whch s the non-zero average velocty of the partcles. We notce that there s the same analogy n fluds and solds, where the atoms have ther vbratonal and rotatonal moton around ther equlbrum. However, these random 7

8 motons are not consdered n a contnuum mechancal model. Here t should be mentoned that these random motons contrbute to temperature and affect the materal propertes, such as heat capacty, vscosty and other materal propertes, whch are used n developng consttutve relatons n contnuum mechancs. Therefore, the contnuum mechancal velocty can be consdered as a drft velocty added to the random motons. The temperature related random fluctuatons (both translatons and rotatons) dsappear n the knematcs of the contnuum. We notce that the geometrcal ponts generally do not correspond to partcles. As a result, the rotaton feld n a contnuum s only defned based on relatve moton of these geometrcal ponts wthout consderng the ndvdual rotaton of partcles. Consequently, there s only one rotaton feld, derved from the dsplacement feld u, defned by jkuk, j. (3) Ths rotaton was later called constraned rotaton or macrorotaton by proponents of mcropolar theores, as opposed to mcrorotaton n the orgnal Cosserat theory. The concept of mcrorotaton was ntroduced to account for the rotaton of mcroelements or the drector trads, whch are dfferent from the contnuum mechancal rotaton. However, speakng of the rotaton of mcroelements or ndvdual partcles n a contnuum sense s meanngless, because we gnore the mcrostructure of matter after defnng the dsplacement feld. The moton of geometrcal ponts s only represented by the contnuous dsplacement feld u wthout consderng the dscontnuous mcrostructure of matter and moton of ndvdual partcles. Therefore, mcrorotaton, whch brngs extraneous degrees of freedom, s not a proper contnuum mechancal concept. How can the effect of the dscontnuous mcrostructure of matter be represented mathematcally by an artfcal contnuous mcrorotaton? Thus, a consstent szedependent couple stress contnuum mechancs theory should nvolve only true contnuum knematcal quanttes, the dsplacement u, and ts correspondng derved rotaton, wthout recourse to any addtonal artfcal degrees of freedom. Ths means that the rgd body moton of nfntesmal elements of matter at each pont of the contnuum s descrbed by sx degrees of freedom, nvolvng three translatonal u and three rotatonal degrees of freedom. 8

9 Consequently, all mcropolar formulatons, whch nclude an nconsstent ndependent contnuous artfcal mcrorotaton, suffer from ths basc flaw. The concept of ndependent rotatonal degrees of freedom may also orgnate from the dscrete model of matter n molecular dynamcs. In molecular dynamcs, the lumped part of matter can be modeled as rgd bodes. As a result, the moton of each part can be descrbed by moton of ts center of mass and ts rotaton. In rgd body dynamcs, ths rotaton s ndependent of the moton of the center of mass. Therefore, descrbng the translaton and rotaton of ndvdual partcles, such as atoms, molecules and grans, requres dscrete pont functons. On the other hand, the ndependent contnuous artfcal mcrorotaton cannot represent these dscrete pont functons. 3 Indetermnate couple stress theory of Mndln, Tersten and Koter (MTK theory) Some researchers, such as Mndln and Tersten [4], and Koter [5], speculated that n a consstent contnuum theory, the deformaton s completely specfed by the contnuous dsplacement feld u. They consdered that the knematcal quanttes and measures of deformaton are derved from ths dsplacement feld. Hence, n Mndln-Tersten-Koter (MTK) theory, the rgd body moton of the nfntesmal element of matter at each pont of the contnuum s descrbed by sx degrees of freedom (.e., three translatonal u and three rotatonal ). As a result, energy consderatons show that hgher order measures of deformaton must be related to the rotaton feld. We notce that for a materal contnuum occupyng a volume V bounded by a surface S, the prncple of vrtual work or weak formulaton for equlbrum Eqs. (5) and Eq. (6) can be wrtten as [3] n n edv j j dv t uds m ds F udv C dv j, j. (4) V V S S V V 9

10 Ths relaton shows that j and, j are energy conjugate tensors. Mndln and Tersten [4] and Koter [5] consdered the tensor j, as the nfntesmal curvature tensor, that s. (5) j j, However, ths curvature tensor creates some dffcultes n the correspondng couple stress theory. Frst, we notce from Eqs. (3) and (5) that, (6), 0 whch shows that the tensor j s devatorc, and smlarly for the varatons, j, and thus s specfed by eght ndependent components. Ths character creates ndetermnacy n the couplestress tensor. Ths can be seen by decomposng the general tensor j nto sphercal S j and devatorc parts n the followng manner: D j S D, (7) j j j where S. (8) 3 j kk j By denotng Q, (9) 3 kk we have S Therefore, the couple-stress tensor can be wrtten as j kkj Qj. (0) 3 D j j j Q. () Accordngly, we notce that 0

11 D j, j j j, j j. () Ths shows that t s actually the devatorc part of the couple-stress tensor D that s energetcally conjugate to the devatorc curvature tensor j. As a result, we can only specfy the devatorc part D n ths theory. In other words, the sphercal part of the couple-stress j j tensor S j Q s ndetermnate. j The ndetermnacy of Q then carres nto the skew-symmetrcal part of the force-stress tensor, such that C j jk lk, l jk k Q C. D jk, k jk lk, l jk k (3) We should menton that the appearance of body couple C n ths equaton s also a major ssue. However, the ndetermnacy of the couple stress tensor does not affect the force equlbrum Eq. (5), snce Q C j, j D C. jk lk, lj jk k, j D jk, kj jk lk, lj jk k, j (4) The other major dffculty n ths development s the nconsstency of the boundary condton for the normal component of the moment tracton. The rght hand sde of Eq. (4) shows that the boundary condtons on the surface of the body can be ether vectors u and as essental (geometrcal) boundary condtons, or t n n and m as natural (mechancal) boundary condtons. Ths apparently makes a total number of sx boundary values for ether case. Consequently, there s no other possble type of boundary condton n sze-dependent couple stress contnuum mechancs. However, ths s n contrast to the number of geometrc boundary condtons that can

12 be mposed [5]. In partcular, f components of u are specfed on the boundary surface, then the normal component of the rotaton correspondng to twstng n nn n n n. (5) k k where nn n, (6) cannot be prescrbed ndependently. Therefore, the normal component nn s not an ndependent degree of freedom, no matter whether the dsplacement vector u s specfed or not. However, the tangental component of rotaton correspondng to bendng, that s, ns n k k k k nn, (7) represents two ndependent degrees of freedom n the global coordnate system, and may be specfed n addton to u. As a result, the total number of geometrc or essental boundary condtons that can be specfed s fve [5]. Next, we let m nn and m represent the normal and tangental components of the surface moment-tracton vector m ( ns) n, respectvely. The normal component where nn nn m m n, (8) nn ( n) m m n nn, (9) k k j j causes twstng, whle m ns n nn m m n, (30) s responsble for bendng. Therefore, the boundary moment surface vrtual work n Eq. (4) can be wrtten as n nn n ns ns m ds m ds m ds S S S S S nn nn ns ns. (3) m ds m ds

13 As we know from theoretcal mechancs, the generalzed forces are assocated only wth ndependent generalzed degrees of freedom, thus formng energetcally dual or conjugate pars. From the knematc dscusson above, ( nn) s not an ndependent generalzed degree of freedom. Consequently, the correspondng generalzed force must be zero and, for the normal component of the surface moment-tracton vector ( n) m, we must enforce the condton nn ( n) m m n nn 0 on S. (3) k k j j Furthermore, the boundary moment surface vrtual work n Eq. (3) becomes n ns ns ns m ds m ds m ds. (33) S S S Ths shows that a materal n couple stress theory does not support ndependent dstrbutons of normal surface moment (or twstng) tracton m nn, and the number of mechancal boundary condtons also s fve. Ths result was frst establshed by Koter [5], although hs couple stress theory does not satsfy ths requrement. To resolve ths problem, Koter [5] proposed, based on the Sant-Venant s prncple, the possblty that a gven nn m has to be replaced by an equvalent shear stress dstrbuton and a lne force system. He gave the detal analogous to the Krchhoff bendng theory of plates. However, there s a dfference between couple stress theory and the Krchhoff bendng theory of plates, as we explan. It should be realzed that Krchhoff plate theory s once agan a structural mechancs approxmaton to a contnuum mechancs theory obtaned by enforcng a constraned deformaton. Consequently, results from ths plate theory are not vald on and around the boundary surface, and near concentrated pont and lne loads. It s a fact that the plate theory usually gves better results n the nternal bulk of the plate far enough from boundary and concentrated loads. However, couple stress theory s a contnuum mechancs theory tself and should be vald everywhere, ncludng near to and on the boundary, wthout any approxmaton. Ths means that a contnuum theory should treat all parts of a materal body wth the same mathematcal rgor and should not be consdered as a structural mechancs formulaton. Nevertheless, ths fundamental dffculty wth boundary condtons and ts mpact on the formulaton was not apprecated at the tme. 3

14 For further nsght, we examne ths nconsstent theory for small deformaton elastcty. In an elastc materal, there s an elastc energy densty functon W, where for arbtrary vrtual deformatons about the equlbrum poston, we have W e. (34) j j j j Therefore W W e,. (35) j j Because of the devatorc character of the curvature tensor, we notce that j D j j j W e, (36) whch shows that W j, (37) e j D j W. (38) j For general lnear b-ansotropc elastc materal, the energy densty functon W takes the form W Ajklejekl B jkl jkl Cjkle j. (39) kl Here b-ansotropc means that there s a cross-lnk relatonshp between e j and j through the tensor C. However, when C 0, the materal becomes ansotropc. The tensors A jkl jkl jkl, Bjkl and Cjkl contan the elastc consttutve coeffcents and are such that the elastc energy s postve defnte. As a result, tensors A jkl and B jkl are postve defnte. We notce that the tensor A jkl s actually equvalent to ts correspondng tensor n Cauchy elastcty. Snce the stran tensor e j s symmetrc and the curvature tensor j s devatorc, we have the symmetry relatons A A A jkl klj jkl, (40) 4

15 B C jkl jkl B, (4) klj C, (4) jkl wth constrants B kl 0, B 0. (43) jkk C 0. (44) jkk These show that for the most general case, the number of dstnct components for A jkl, B jkl and C jkl are, 36, and 48, respectvely. Therefore, the most general lnear elastc b-ansotropc materal s descrbed by 05 ndependent consttutve coeffcents. Snce ths theory requres very many materal coeffcents, t s less attractve for practcal and expermental applcatons. By usng the energy densty Eq. (39) n the general relatons for stresses, Eqs. (37) and (38), we obtan the followng consttutve relatons As a result A e C j, (45) jkl kl jkl kl B C e. (46) ( D) j jkl kl klj kl Q B C e (47) j j jkl kl klj kl where agan Q s ndetermnate. For lnear sotropc elastc materal, the symmetry relatons requre A jkl j kl k jl l jk, (48) B 4 4, (49) jkl k jl l jk C 0. (50) jkl The modul and have the same meanng as the Lamé constants for an sotropc materal n Cauchy elastcty. The materal constants and account for the couple-stresses n the sotropc materal. As a result, the energy densty takes the form 5

16 W e e ee jj kk j j j j j j. (5) The followng restrctons are necessary for postve defnte energy densty W 30, 0, 0,. (5) The frst two are dentcal to those from classcal theory. As a result, we have the followng consttutve relatons for the symmetrc part of the force-stress tensor and couple-stress tensor, respectvely, e e j kk j j, (53) Q 4 4 j j j j Q 4 4. j j,, j (54) Then, by usng Eq. (54) n Eq. (3), we obtan Q C j jk, k jk k jk k, (55) for the skew-symmetrc part of the force-stress tensor. Therefore, the total force-stress tensor becomes Q e e C j jk, k kk j j jk k jk k, (56) Notce the ndetermnacy due to Q and the presence of the body couple C k. Then, for the equlbrum equaton n terms of the dsplacement, we obtan ukk, ( ) uf jkck, j 0. (57) The dsappearance of the elastc constant n the force-stress tensor j n Eq. (56) and equlbrum Eq. (57) can be seen as the ndcaton of nconsstency n ths theory. Interestngly, the rato 6

17 l, (58) specfes a characterstc materal length l, whch accounts for sze-dependency. Thus, the fnal equatons governng the sotropc lnear sold n the small deformaton couple stress elastcty theory under consderaton can be wrtten as e j kkj ej, (59) j Qj 4l j, 4 l, j, (60) j jkq, k ekkj ej l jk k jkck, (6) l uk, k ( l ) u F jkck, j 0. (6) Subsequently, Stokes [4] brought ths formulaton nto flud mechancs to model the szedependency effect n fluds. It turns out that ths s an nterestng concdence. George Gabrel Stokes generalzed Naver equatons for fluds, whle much later Vjay Kumar Stokes brought MTK theory nto flud mechancs. However, as mentoned above, MTK couple-stress theory suffers from some serous nconsstences and dffcultes wth the underlyng formulatons, whch are summarzed as follows:. The body-couple s present n the consttutve relatons for the force-stress tensor n the MTK theory.. The sphercal part of the couple-stress tensor s ndetermnate, because the curvature tensor, s devatorc. j j 3. The boundary condtons are nconsstent, because the normal component of moment nn tracton m appears n the formulaton. 7

18 4. For lnear b-ansotropc elastc materal, ths theory requres 05 materal constants, whch makes the theory less attractve from both practcal and expermental standponts. Interestngly, the ndetermnacy of the sphercal part of the couple stress tensor Q j n ths nconsstent theory has been smply gnored wthout any reasonable justfcaton n some work [5-]. Erngen realzed ths ndetermnacy as a major mathematcal problem. As a result, he was the frst to call ths ndetermnate couple stress theory [9]. In response to the appearance of ths ndetermnacy, Erngen and some other researchers returned to the orgnal Cosserat theory and revved the dea of an ndependent artfcal mcrorotaton n developng many dfferent mcropolar theores. We wll consder mcropolar theory n Sect. 6. As mentoned prevously, for sotropc lnear elastc materal, the second elastc couple-stress constant does not appear n the fnal governng equatons. It turns out for the twodmensonal case, the stress boundary condtons are ndependent of ; thus, the correspondng boundary value problem only depends on the frst elastc couple-stress constant. As a result, there have been many applcatons n the lterature for two-dmensonal sotropc elastc problems. However, for three-dmensonal problems, ths theory requres both couple stress materal constants and. For example, n the torson of a cylnder, ths theory predcts appearance of couple-stresses [5], whch depend on both constants and. It should be noted that MTK theory s very nfluental n the hstory of couple-stress related theores. As wll be seen, ths theory has a drect mpact n formulatng the consstent couple stress theory. The most mportant advancement ntroduced n MTK theory was takng the contnuous dsplacement vector u as the fundamental varable to represent the deformaton of the contnuum doman. However, after developng the ndetermnate couple stress theory [4], Mndln hmself was not entrely pleased wth hs formulaton. Ths s obvous from the other formulatons he developed, such as stran gradent theores [7,8] and mcromorphc theory []. All of these developments suggest that perhaps Mndln was not certan about the valdty of any of hs theores. 8

19 In a sharply crtcal passage n ther defntve text on contnuum mechancs, Truesdell and Noll [3] provded the followng summary of the collectve understandng, just after the shortcomngs of MTK theory were realzed: The Cosserats masterpece stands as a tower n the feld. Even the recent recreators of contnuum mechancs, whle they knew of t, dd not know ts contents n detal. Had they mastered t, not only would tme and effort of redscovery have been spared, but also a paragon of method would have lan n ther hands. As we shall soon see, ths judgment n favor of the Cosserat approach also was made n haste and the struggle to defne a consstent couple stress theory remaned for another half century. In retrospect, the nconsstences n MTK theory make one suspect that perhaps the number of ndependent stress components n a consstent couple stress sze-dependent contnuum mechancs theory should be less than eghteen. Ths s one reason why the theory wth a symmetrc couple-stress tensor was developed, whch we consder n the followng secton. 4 Modfed couple stress theory of Yang, Chong, Lam and Tong (YCLT theory) Yang et al. [6] developed a model of couple stress,.e., the modfed couple stress theory, that consders an addtonal equlbrum equaton for the moment of couple, n addton to the two equlbrum equatons of the classcal contnuum. Applcaton of ths equlbrum equaton, apparently leads to a symmetrc couple-stress tensor, that s, j j. (63a,b) 0 j As a result, the vrtual work prncple Eq. (4) shows that the symmetrc part of, j s the correspondng curvature tensor n ths theory. We notce that j, j j,, (64), (65) 0, whch shows that the tensor j s devatorc, and thus s specfed only by fve ndependent components. As a consequence, all the nconsstences n MTK theory, such as the 9

20 ndetermnacy n the couple-stress tensor and the appearance of reman ntact n ths theory. We explore the detals as follows. nn m on the boundng surface, Frst, we notce that the devatorc character of j requres that D j j j j, (66) D whch shows that the devatorc part of the couple stress tensor s energetcally conjugate to the devatorc curvature tensor j. As a result, we can only specfy the devatorc part j of the D j couple-stress tensor n ths theory and the sphercal part of the couple-stress tensor S j Q s j ndetermnate. Ths ndetermnacy appears n the skew-symmetrcal part of the force-stress tensor, that s D Q C j jk lk, l jk, k jk lk, l jk k. (67) However, as n MTK theory, ths ndetermnacy does not affect the force equlbrum Eq. (5). Snce the couple-stress tensor j s symmetrc here, ths couple stress theory also does not satsfy the requred boundary condton nn m nn =0 on S. (68) j j One mght fnd recourse to Koter s method to replace a gven m nn by an equvalent shear stress and force system based on Sant-Venant's prncple. However, agan ths s ncompatble wth the fact that the couple stress theory s a contnuum theory, whch should be vald everywhere, ncludng the surface wthout any approxmaton. For an elastc materal n ths theory, the elastc energy densty functon W s defned, such that W W e,. (69) j j 0

21 Therefore j D, j j j W e, (70) whch shows that j W, (7) e j D j W. (7) j For lnear b-ansotropc elastc materal, the energy densty functon W takes the form W A e e B C e jkl j kl jkl j kl jkl j kl. (73) The tensors A jkl, Bjkl and Cjkl contan the elastc consttutve coeffcents and are such that the elastc energy s postve defnte. As a result, tensors A jkl and B jkl are postve defnte. Snce the stran and curvature tensors are symmetrc, we have the symmetry relatons wth constrants Ajkl Aklj Ajkl, (74) B B B jkl klj jkl, (75) C C C jkl jkl jlk, (76) B kl 0, B 0, (77) jkk C 0. (78) jkk These show that for the most general case, the number of dstnct components for A jkl, B jkl and C jkl are, 5, and 30, respectvely. Therefore, the most general lnear elastc b-ansotropc materal s descrbed by 66 ndependent consttutve coeffcents.

22 By usng the energy densty Eq. (73) n the general relatons, Eqs. (7) and (7), we obtan the followng consttutve relatons A e C j, (79) jkl kl jkl kl B C e. (80) D j jkl kl klj kl Thus, we have Q B C e, (8) j j jkl kl klj kl For lnear sotropc elastc materal, the symmetry relatons requre A jkl j kl k jl l jk, (8) B 4 4, (83) jkl k jl l jk C 0. (84) jk The sngle materal coeffcent accounts for the couple-stresses n the sotropc materal. As a result, the energy densty takes the form W e e ee 4 jj kk j j j j. (85) The followng restrctons are necessary for postve defnte energy densty W 3 0, 0, 0. (86) Therefore, for lnear sotropc elastc materal, the consttutve relatons reduce to e e j kk j j, (87) Q 8 j j j Q 4, j, j j, (88) and the total force-stress tensor becomes Q e e C j jk, k kk j j jk k jk k. (89)

23 In ths theory, for the equlbrum equaton n terms of the dsplacement, we obtan u ( ) u F C 0 k, k jk k, j. (90) By usng the characterstc materal length l defned by the rato l. (9) we can rewrte the consttutve and governng equatons as e e j kk j j, (9) Q 8l, (93) j j j Q e e l C j jk, k kk j j jk k jk k, (94) l u ( l ) u F C 0 k, k jk k, j. (95) Interestngly, the force-stress tensor and the fnal governng equlbrum equatons are smlar to those n MTK theory for sotropc materal. Therefore, we notce that the modfed couple stress theory nherts all nconsstences from ndetermnate MTK theory. Nevertheless, the appearance of only one length scale parameter for sotropc materal makes modfed couple stress theory more desrable from an expermental and analytcal vew. As a result, ths theory has been extensvely used n many problems, such as bendng, bucklng and post-bucklng, and vbraton n recent years to nvestgate the mechancal behavor of the structures at small scale. We notce that the results for two-dmensonal sotropc problems are smlar to those n ndetermnate MTK theory. However, ths theory s dfferent for three-dmensonal and ansotropc cases. Interestngly, for torson of cylnder, ths theory also predcts the appearance of couple stresses [6]. 3

24 It should be emphaszed that the modfed couple stress theory cannot be taken as a specal case of ndetermnate MTK theory obtaned by lettng. (96) Ths s obvous by notcng that ths case s excluded by condton Eq. (5) for the ndetermnate MTK couple stress theory. In addton, ths smlarty s only vald for sotropc materal, and there s no smple analogy for general ansotropc and b-ansotropc cases. There have been some doubts about the valdty of the modfed couple stress theory from a dfferent fundamental aspect. As mentoned, the symmetry character of the couple-stress tensor s the consequence of the pecular equlbrum equaton for the moment of couple, besdes the two conventonal equlbrum equatons for force and couple. However, ths requrement s an addtonal condton, whch s not derved by any prncple of classcal mechancs, as mentoned by Lazopoulos [4]. Therefore, the modfed couple stress theory not only nherts all nconsstences from ndetermnate MTK theory, but also s based on an unsubstantated addtonal artfcal equlbrum of moment of couples n the set of fundamental equatons. Ths new equlbrum equaton has no physcal explanaton, and has been nvented to make the couple-stress tensor symmetrc. Let us examne ths nconsstency n more detal. As s known, a couple of forces s a free vector n conventonal mechancs of rgd bodes. The moton of a rgd body s not affected by changng the poston of a concentrated couple at pont A to any arbtrary pont B n the body. However, the same couple s not a free vector, when we analyze the dstrbuton of the nternal stresses and deformaton of the body n a deformable contnuum mechancs theory. In ther development, Yang et al. [6] clam that the effect of a couple at a pont A s equvalent to the effect of ths couple at pont B plus the moment of ths couple about pont B. However, ths clam can be refuted very easly by realzng that the stresses and deformatons of these two loadngs cases are not the same. Interestngly, we expect that the concentrated couple creates sngularty n stresses at pont A for the frst case, whle t creates sngularty at pont B for the second case. These two dfferent sngulartes ndcate that the effects of these two couple systems are not the same. Furthermore, the governng equatons 4

25 n rgd body dynamcs are based on the equatons for the rate of change of lnear momentum P and angular momentum L of the system of partcles,.e. dp F, (97) dt dl M, (98) dt where F and M represent the sum of the external forces and the moment of external forces, respectvely. Based on Newton s thrd law, the nternal forces and ther correspondng moments dsappear n these relatons. As a result, these fundamental equatons establsh that forces are sldng vectors and couples are free vectors, as long as the moton of rgd bodes are concerned. However, there s no addtonal analogous equaton for the moment of angular momentum for system of partcles n mechancs. Ths s because n ths equaton the effect of nternal forces wll not dsappear. Therefore, Yang et al. [6] have volated fundamental laws of mechancs to make the couple stress tensor symmetrc. Smplfyng a theory mght be acceptable, but only as long as t does not create fundamental nconsstences. We summarze the nconsstences of the YCLT modfed couple stress theory as follows:. The symmetrc character of the couple-stress tensor j s based on an artfcal fundamental law for equlbrum of moment of couples, whch has no physcal realty.. The sphercal part of the couple-stress tensor s ndetermnate, because the curvature tensor j s devatorc. 3. The body-couple s present n the consttutve relatons for the force-stress tensor. 4. The boundary condtons are nconsstent, because the normal component of moment nn tracton m appears n the formulaton. 5

26 5 General stran gradent theores General stran gradent theores were also ntroduced n the 960s by some researchers, ncludng Mndln [7], and Mndln and Eshel [8]. In these theores, varous forms of gradent of stran tensor have been taken as a fundamental measure of deformaton. Interestngly, some forms of these theores have been also used n developng sze-dependent multphyscs dscplnes, such as flexoelectrcty [5,6]. These theores descrbe the knematcs of the contnuum by the dsplacement feld u, as the fundamental varable. However, n these theores, the second gradent of deformatons, such as u, jk and j, k e appear n the formulatons explctly. In some versons of these theores [7,7], the thrd gradent of dsplacement and hgher order stresses are also ntroduced. However, we notce that although these theores utlze the contnuous dsplacement vector u as the fundamental varable to represent the deformaton of the contnuum, the second gradent of deformatons u, jk and j, k e are not drectly related to the rotaton gradent,. j As mentoned prevously, the left hand sde of the vrtual work prncple Eq. (4) n n edv j j dv t uds m ds F udv C dv j, j, (99) V V S S V V shows that the general stran gradent e j, k s not a fundamental measure of deformaton n a consstent couple stress theory. Ths relaton also shows that there s no room for the thrd and hgher gradents of deformaton n the formulaton, because t also would requre addtonal mproper essental boundary condtons. Therefore, stran gradent theores are nconsstent contnuum theores. 6 Mcropolar, mcrostretch and mcromorphc theores Soon after realzng the nconsstency of ndetermnate MTK couple stress theory, researchers also began to develop sze-dependent theores that more closely resembled the Cosserats 6

27 orgnal theory. The dea of mcrorotaton, a feld ndependent of dsplacement feld u, was agan consdered to be a fundamental knematc quantty n an attempt to remedy the aforementoned ssues wth nconsstent ndetermnate couple-stress theory. Erngen [9], Nowack [0], Mndln [] and Erngen and Suhub [] were the frst to revve varous forms of the orgnal Cosserat theory that now s more commonly referred to as mcropolar, mcrostretch and mcromorphc theores. In mcropolar theores, to each pont sx degrees of freedom are attrbuted, whch are represented by the dsplacement feld u for translatonal degrees-of-freedom and the mcrorotaton feld for the rotatonal degrees-of-freedom. However, as we explaned before, the artfcal mcrorotaton as an ndependent varable s not compatble wth the dea of a contnuous medum and cannot descrbe the dscontnuous mcrostructure of matter. Consequently, we expect that mcropolar theores also create some nconsstences n ther formulaton. For more clarfcaton, we examne the mcropolar theory of Erngen [9] deeply and fnd varous new nconsstences. Mcropolar theores apparently seem to cure the ndetermnacy problem, boundary condton problem and accommodate a place for body couple dstrbuton [9]. However, we dscover that these nconsstences are transformed to new nconsstences, as we now explan n more detal. In mcropolar theores, t s customary to assume that the moment-tracton vector n m s energetcally conjugate to the mcrorotaton vector wthout any reasonng. As a result, the vrtual work theorem n mcropolar theores s apparantly wrtten as n n j u, j jk k dv j, jdv t uds m ds F udv C dv. (00) V V S S V V However, careful examnaton shows that there s no reason why the macrorotaton does not have any contrbuton to the work of the moment-tracton n m on the surface n Eq. (00). We notce that the left hand sde of Eq. (00) shows that the tensors u, (0) j, j jk k k j, (0), j 7

28 are energy conjugate to the force-stress tensor j and couple-stress tensor j, respectvely. As a result, these tensors are the measures of deformaton n mcropolar theory [9], where j s the mcropolar stran tensor and k j s the mcropolar curvature tensor. It s obvous that these tensors are generally non-symmetrc and each s specfed by nne ndependent components. For an elastc materal, the elastc energy densty functon W becomes W W, k. (03) j j Therefore W k, (04) j j j j whch apparently shows that j j W, (05) j W. (06) k j For lnear b-ansotropc elastc mcropolar materal, the energy densty functon W takes the form The tensors A jkl, Bjkl and Cjkl W A B k k C k jkl j kl jkl j kl jkl j kl. (07) contan the elastc consttutve coeffcents and are such that the elastc energy s postve defnte. As a result, tensors A jkl and B jkl are postve defnte. Snce the deformaton tensors are general non-symmetrc tensors, we have only the symmetry relatons A B jkl jkl A, (08) klj B. (09) klj 8

29 These show that for the most general case, the number of dstnct components for A jkl, B jkl and C jkl are 45, 45, and 8, respectvely. Therefore, the most general lnear elastc b-ansotropc mcropolar materal s descrbed by 7 ndependent consttutve coeffcents. Thus, mcropolar theores requre more materal parameters n consttutve relatons than other theores we have examned thus far. By usng the energy densty Eq. (07) n the general relatons for stresses, Eqs. (05) and (06), we obtan the followng consttutve relatons A C k, (0) j jkl kl jkl kl B k C. () j jkl kl klj kl For lnear sotropc elastc mcropolar materal, the symmetry relatons requre A () jkl j kl k jl l jk and the energy densty becomes Bjkl jkl k jl l jk, (3) C 0. (4) jkl W jjkk jj j j kjjkkk kjkj kjkj. (5) The followng restrctons are necessary for postve defnte energy densty W 3 0, 0, 0, (6) 3 0, <, 0. (7) As a result, we have the followng consttutve relatons for the force-stress and couple-stress tensors, respectvely:, (8) j kk j j j k k k. (9) j kk j j j 9

30 We notce that for lnear sotropc mcropolar elastc solds four addtonal constants are requred, whch presents a more dffcult task from an expermental standpont. Now we realze that the dffcultes of ndetermnate theory of MTK have been transformed to new troubles, such as the ncreased number of materal propertes. By usng these consttutve relatons n the equlbrum Eqs. (5) and (6), we obtan the governng equatons n terms of the dsplacement u and mcrorotaton. However, the specfcaton of boundary condtons s more problematc n ths formulaton. For example, as stated by Sadd [8], t s not clear how to specfy the mcrorotaton and/or moment-tracton boundares. n m on the As mentoned, the ndetermnacy of the couple-stress tensor manly drected Mndln and others to return to the Cosserats orgnal theory and develop new mcropolar theores. What they dd not realze s that the couple-stress tensor s stll ndetermnate, whch we demonstrate n what follows. Snce u, j s a true (polar) second order tensor, Eq. (0) shows that the mcropolar stran tensor j s a true (polar) second order tensor. Ths requres the mcrorotaton to be a pseudo (axal) vector. As a result, the dvergence of ths vector,.e.,,, s a pseudo-scalar. Ths means, changes sgn under an nverson of the coordnate system. However, the appearance of a pseudoscalar n the knematcs of a contnuum s not possble, whch means only scalar quanttes exst. Ths requres that the dvergence, vansh, that s. (0), k 0 As a result, based on the Helmholtz decomposton theorem, the mcrorotaton vector can be represented smply by the curl of a true (polar) vector feld, where jkk, j. () 30

31 Here the factor s arbtrarly added. Remarkably, ths expreson for shows that the vector feld can be taken as a dsplacement feld, whch can be convenently called the mcrodsplacement feld. Ths s the complement of our prevous dscusson that the ntroducton of mcropolar rotaton actually nvolves consderng the contnuty of matter for a second tme, whch creates nconsstency. On the other hand, the compatblty Eq. (0) shows that the mcropolar curvature tensor k j s also devatorc, and thus s specfed only by eght ndependent components. As a result, the couple-stress tensor n mcropolar theory s also ndetermnate. The appearance of the mcrodsplacement feld creates other speculatons, such as ts contrbuton as a vrtual dsplacement n the expresson of vrtual work. Interestngly, for general b-ansotropc mcropolar materal, the condton Eq. (0) requres that B kl 0, B 0, () jkk C 0. (3) jkk These show that for the most general case, the number of dstnct components for A jkl, B jkl and C jkl are 45, 36, and 7, respectvely. Therefore, the most general lnear elastc b-ansotropc mcropolar materal, n what should be called ndetermnate mcropolar theory, s descrbed by 53 ndependent consttutve coeffcents. As a result, the consttutve relatons become A C k, (4) j jkl kl jkl kl B k C. (5) D j jkl kl klj kl Therefore, mcropolar theores have created new nconsstences wthout resolvng any of the former nconsstences n MTK theory. We summarze nconsstences of mcropolar theores as follows: 3

32 . The artfcal mcrorotaton s not a contnuum mechancs concept. A contnuous mcrorotaton functon cannot descrbe the rotaton of ndvdual pont partcles.. There s no reasonng for takng the degrees of freedom u and as the energy conjugates of generalzed tractons t n and m n, respectvely. 3. Snce the mcrorotaton s a pseudovector, the mcropolar curvature tensor kj, j s devatorc. Ths makes the couple stress tensor ndetermnate. 4. For lnear b-ansotropc elastc mcropolar materal, ths theory requres 53 materal constants. Ths makes the theory less attractve from both practcal and expermental standponts. Interestngly, mcropolar theores are consdered as specal cases of general mcrostretch and mcromorphc theores [, ]. In ths latter model, there are addtonal degrees of freedom: three mcrodsplacement vector components and nne addtonal ndependent mcrodeformaton tensor components. It s obvous these theores are also nconsstent. Some mcromorphc and mcropolar theores also typcally nclude n the dynamcal case a spn nerta dstrbuton, whch cannot exst n a consstent contnuum mechancal theory. Instead, we must notce that n consstent contnuum mechancs the nerta of matter s only descrbed by mass densty. There s no such rotary nerta per unt volume n consstent theores of contnuum mechancs. Therefore, the concept of mcrorotaton and spn nerta dstrbuton are the result of confuson caused by mxng dscrete molecular dynamcs wth Cosserat contnuum mechancs theory. 7 Consstent sze-dependent couple stress theory In ths secton, we present the consstent sze-dependent couple stress theory, whch ends the quest for a consstent Cosserat theory. Ths development nvolves only true contnuum 3

33 knematcal quanttes wthout recourse to any addtonal artfcal degrees of freedom. We notce that elements of developng ths theory are based on MTK theory. Interestngly, we realze that the ndetermnate MTK theory s an ntal ncomplete verson of ths consstent theory. Based on our arguments n Sect. and 3, we postulate that n a consstent contnuum theory the deformaton s only specfed by the contnuous dsplacement feld u. Therefore, the rgd body moton of each nfntesmal element of matter at any pont of the contnuum s descrbed by sx degrees of freedom, nvolvng three translatonal moton u, and three rotatonal components. As mentoned, the prncple of vrtual work or weak formulaton for equlbrum Eqs. (5) and (6) can be wrtten as [3] n n ejdv j j, jdv t uds m ds F udv C dv. (6) V V S S V V Ths relaton shows that a compatble curvature tensor must be related to, j. However, n Sect. 3 and 4, we demonstrated that snce ths tensor and ts symmetrc part j are devatorc, these cannot be consdered as proper curvature tensors. Therefore, one mght suggest that the skew-symmetrc part of, j j, j j,, (7) s the consstent and proper curvature tensor, because ts sphercal part vanshes. Interestngly, n recent work [3], the present authors have demonstrated that ths s actually the case by developng a consstent couple stress theory, whch resolves the ndetermnacy of the couplestress tensor j and the ssue wth boundary condtons for the normal component theory. Let us next examne ths development n more detal. nn m n MTK In Sect. 3, by usng arguments from Koter [5], we demonstrated that the consstent couple stress nn theory requres that the normal surface moment-tracton m on the boundary vansh, that s 33

34 nn ( n) m m n nn 0 on S. (8) k k j j However, as we dscussed, ths requrement was volated by former couple stress theores. It s mportant to notce that ths fundamental dffculty wth boundary condton and ts mpact on formulatons was not understood before. Interestngly, ths requrement has been fulflled wth remarkable consequences n Reference [3]. By usng the concept of an arbtrary subdoman, we have shown that at any pont n the body on any plane wth normal drecton n, vansh; that s nn m must nn m nn 0 n V. (9) j j Now snce nn j s symmetrc and arbtrary n Eq. (9), j must be skew-symmetrc. Thus, j 0, j j, (30) whch means j j n V. (3) Ths s the subtle character of the couple stress tensor n contnuum mechancs, whch has not been recognzed by early nvestgators. We should notce that the skew-symmetrc property of the couple-stress tensor s the result of ts fundamental character, and has nothng to do wth any consttutve relaton. Ths property s vald for any sold, sotropc or ansotropc, elastc or nelastc, lnear or non-lnear. In ths development, there are no addtonal assumptons beyond that of the contnuum as a doman-based concept havng no specal characterstcs assocated wth the actual boundng surface over any arbtrary nternal surface. Interestngly, the skew-symmetrc character of the couple-stress tensor resolves the ndetermnacy problem. We notce 0 n V, (3) whch shows that Q vanshes, and thus the couple-stress tensor j does not have any sphercal part, that s S j Q 0. (33) j 34