Foundatons of consstent couple stress theory 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 July 9, 05 Abstract In ths paper, we examne the recently developed skew-symmetrc couple stress theory and demonstrate ts nner consstency, natural smplcty and fundamental connecton to classcal mechancs. Ths hopefully wll help the scentfc communty to overcome any ambguty and skeptcsm about ths theory, especally the valdty of the skew-symmetrc character of the couplestress tensor. We demonstrate that n a consstent contnuum mechancs, the response of nfntesmal elements of matter at each pont decomposes naturally nto a rgd body porton, plus the relatve translaton and rotaton of these elements at adjacent ponts of the contnuum. Ths relatve translaton and rotaton captures the deformaton n terms of stretches and curvatures, respectvely. As a result, the contnuous dsplacement feld and ts correspondng rotaton feld are the prmary varables, whch remarkably s n complete algnment wth rgd body mechancs, thus provdng a unfyng bass. For further clarfcaton, we also examne the devatorc symmetrc couple stress theory that, n turn, provdes more nsght on the fundamental aspects of consstent contnuum mechancs.. Introducton From the mddle of the twenteth century onwards, there has been a shft towards developng contnuum mechancs prmarly from a thermodynamcs perspectve. As a result, much progress has been made, especally n consttutve modelng. However, ths change n drecton also has led to a departure of the dscplne from the foundatons of mechancs n ts classcal form, n whch the fundamental enttes are forces and couples, along wth ther knematc conjugate
dsplacements and rotatons, respectvely. Of course, the former relate drectly to the basc conservaton laws of lnear and angular momentum, whle the latter descrbe the pure rgd body moton. In rgd body mechancs, the force and moment equatons are the governng equatons descrbng the translatonal and rotatonal moton of the body n space. Consequently, t seems n developng a consstent contnuum mechancs theory, we need to consder the rgd body porton of moton of nfntesmal elements of matter at each pont of the contnuum. Ths requres the ncluson of force- and couple-stresses n the formulaton. Snce the dsplacements and rotatons at each pont are the degrees of freedom of the nfntesmal body, the fundamental mechancal equatons are stll the force and moment equatons at each pont. However, to have a complete set of equatons, we need the consttutve equatons. Ths n turn requres consderaton of the deformaton or, more specfcally, the relatve rgd body moton of nfntesmal elements of matter at adjacent ponts of the contnuum. Cauchy elastcty, as the frst contnuum theory, focused on force-stresses and dsplacements. Couple-stresses were smply dsmssed from the very begnnng and, as a result, the moment equatons merely provde the symmetrc character of the force-stress tensor. Consequently, n ths theory, rotatons are left wth no essental role. Most formulatons untl recently have followed that drecton. However, wth the growng need to develop sze-dependent mechancs theory, there comes an opportunty not only to advance the dscplne, but also to reconnect wth some fundamental notons of mechancs. We beleve that, f possble, the four foundatonal quanttes (.e., force, dsplacement, couple, rotaton) should be at the very heart of such a theory and that ndvdual terms n vrtual work, as well as the essental and natural boundary condtons, should have a clear physcal meanng. Therefore, consstent contnuum mechancs must algn seamlessly wth rgd body mechancs. Beyond ths, there should always be an nner beauty and natural smplcty to mechancs, whch s what attracts many of us to ths feld. The formulatons presented n Neff et al. (05a), and n the other papers n ther recent seres, cannot possbly pont toward the future of mechancs. For example, the boundary condtons defned n equatons (4) and (5) of Neff et al. (05b) are far too complcated and non-physcal. Moreover, one cannot hope to prove a consstent theory wrong by patchng together several nconsstent theores, as those authors have attempted. There must
nstead be smple, elegant explanatons of sze-dependent response that wll lead to a meanngful, self-consstent descrpton of contnua at the fnest scales. Furthermore, we should note that the development of Neff et al. (05a) s lmted to lnear sotropc elastcty, rather than provdng generalty for contnuum mechancs as a whole. In ths paper, we wll not dwell on the detals of Neff et al. (05a), but nstead focus on presentng consstent couple stress theory (Hadjesfandar and Dargush, 0), as clearly and concsely as possble. However, we also wll examne the nconsstent devatorc symmetrc couple stress theory n ths paper, as ths helps to clarfy the requred consstency n a contnuum mechancs theory. It should be noted that elements of the consstent couple stress theory are based on the work of Mndln and Tersten (96) and Koter (964), whch use the four foundatonal contnuum mechancal quanttes (.e., force, dsplacement, couple, rotaton), wthout recourse to any addtonal degrees of freedom. Ths means the Mndln-Tersten-Koter theory s based mplcty on the rgd body porton of moton of nfntesmal elements of matter at each pont of the contnuum. In these mportant developments, Mndln, Tersten and Koter correctly establshed that fve geometrcal and fve mechancal boundary condtons can be specfed on a smooth surface. However, ther fnal theory suffers from some serous nconsstences and dffcultes wth the underlyng formulatons, whch may be summarzed as follows:. The presence of the body couple n the relaton for the force-stress tensor n the orgnal theory ;. The ndetermnacy n the sphercal part of the couple-stress tensor; 3. The nconsstency n boundary condtons, snce the normal component of the coupletracton vector appears n the formulaton. Ths nconsstent theory s called the ndetermnate couple stress theory n the lterature (Erngen, 968). Remarkably, consstent couple stress theory resolves all three of these nconsstences wth fundamental consequences. We notce that the major trumph n ths development s dscoverng In our prevous work on couple stress theory, we ncorrectly stated that the body-couple appeared n the consttutve relaton for the force-stress tensor n the Mndln-Tersten-Koter theory. We thank the authors of Neff et al. (05a) for pontng out ths error. 3
the skew-symmetrc character of the couple-stress tensor. The mportant step n ths dscovery s to nvoke the fundamental contnuum mechancs hypothess that the theory must be vald not only for the actual doman, but n all arbtrary subdomans. (Ths, of course, s exactly the same hypothess that allows us to pass from global balance laws to the usual local dfferental forms.) Our nvolvement wth boundary ntegral equatons and the passon of the frst author wth the concept of rotaton throughout mechancs and physcs (Hadjesfandar, 03) have provded the necessary background. Furthermore, we should note that consstent couple stress theory offers a fundamental bass for the development of sze-dependent theores n many mult-physcs dscplnes that may govern the behavor of contnua at the smallest scales. The paper s organzed as follows. In Secton, we consder consstent couple stress theory n detal and clarfy some apparent ambgutes left n the orgnal presentaton. In ths secton, we demonstrate that a consstent contnuum mechancs theory should be based on the rgd body porton of moton for nfntesmal elements of matter at each pont n the contnuum and the relatve dsplacement and rotaton of these elements at adjacent ponts. Then, we establsh the skew-symmetrc character of the couple-stress tensor based on the requrements for havng consstent well-posed boundary condtons. After that n Secton 3, we examne the devatorc symmetrc couple stress theory, whch helps us to understand some nconsstences that have plagued dfferent sze-dependent contnuum mechancs theores. Fnally, Secton 4 contans a summary and some general conclusons.. Consstent couple stress theory Consder a materal contnuum occupyng a volume V bounded by a surface S wth outer unt normal n, as shown n Fg., under the nfluence of external loadng, such as surface tractons and body-forces. Let us begn wth the governng partal dfferental equatons for couple stress theory representng the force and moment balance equatons under quasstatc condtons, whch can be wrtten, respectvely, as: () j, j F 0 j, j jk jk 0 () 4
Fg.. The body confguraton. where j represents the true (polar) force-stress tensor, j s the pseudo (axal) couple-stress tensor, F s the specfed body-force densty and jk s the Lev-Cvta alternatng symbol. Any specfed body-couple densty can be rewrtten n terms of body-force densty and tangental forcetractons on the surface, and so does not appear explctly n the governng equatons. Here and throughout the remander of ths paper standard ndcal notaton s used wth summaton over repeated ndces and wth ndces appearng after a comma representng spatal dervatves. Please note that there s no need to complcate the presentaton wth concepts from Le algebra, orthogonal Cartan decompostons or generalzed coordnates. These are completely superfluous to the mportant arguments and only tend to dstract. We notce that the force and moment balance laws () and () are the governng equatons for translatonal and rotatonal equlbrum developed by consderng nfntesmal elements of matter. Therefore, we are concerned wth the rgd body porton of moton of nfntesmal elements of matter at each pont of the contnuum. However, the force and moment balance laws () and () do not by themselves have a unque soluton for dstrbuton of stresses n the contnuum. For ths purpose, we need to consder the deformaton n terms of relatve rgd body moton of nfntesmal elements of matter n the contnuum under the nfluence of nternal stresses. Ths provdes us wth the consttutve equatons, whch complete the set of equatons to permt a unque soluton of a well-posed boundary value problem. We consder next knematcs of a contnuum. 5
In a consstent contnuum representaton, t s assumed that matter s contnuously dstrbuted n space, whch requres the deformaton to be specfed completely by the contnuous dsplacement feld u. As a result, all knematcal quanttes and measures of deformaton must be derved from ths dsplacement feld. Fg. allows us to vsualze knematcs n the three-dmensonal case. At each pont, we defne a rgd trad, whch can be used to represent the rgd body porton of moton assocated wth nfntesmal elements at each pont of the contnuum. These rgd trads translate and rotate wth the medum to provde the underlyng rgd body porton of moton of each nfntesmal element, defned by the true (polar) dsplacement vector u and the pseudo (axal) rotaton vector. Thus, the rgd body porton of moton of nfntesmal elements of matter at each pont n three-dmensonal space s descrbed by sx degrees of freedom, nvolvng three translatonal u and three rotatonal wthn a contnuum descrpton restrans the rotaton dsplacement, whch of course shows that the rotaton feld degrees of freedom. However, the contnuty of matter to equal one-half the curl of the s not ndependent of the dsplacement feld u. Ths latter aspect was mssed by Cosserat and Cosserat (909) and by those advocatng for mcropolar and related theores. Nonetheless, the Cosserats should be credted wth the concept of the rgd trad and n elevatng the role of angular momentum balance n contnuum mechancs. Fg.. The knematcs of a contnuum. 6
These arguments ndcate that rgd body moton s so fundamental n understandng contnuum mechancs that the quanttes u and must drectly appear as prmary varables. Furthermore, ths all suggests that consstent contnuum mechancs theory should be developed as an extenson of rgd body mechancs, whch then s recovered n the absence of deformaton. To complete the deformaton analyss, we need to defne sutable measures or metrcs of deformaton based on the relatve rgd body moton of trads at adjacent ponts of the contnuum. For ths purpose, consder two nfntesmal elements of matter at arbtrary ponts P and P, as shown n Fg.. The dsplacements and rotatons of these elements (or trads) are denoted by u P P P and at pont P, and u and at pont P. Therefore, the relatve translaton rotaton of the element P relatve to the element P can be expressed as P P P, j j P P u and u u u u dx (3) and P P P, jdx j P (4) respectvely. These equatons show that the relatve rgd body moton of nfntesmal elements of matter s descrbed by the gradent of the translaton tensor u, j and the gradent of the rotaton tensor, j. Ths result suggests that the tensors u, j and, j are of prme mportance n deformaton analyss and should appear n defnng the measures of deformaton. It should be mentoned that n some tme-dependent phenomena, such as vscoelastcty and flud mechancs, ths relatve moton s descrbed nstead by the velocty and angular velocty or vortcty vectors. We recall that n classcal contnuum mechancs, the symmetrc part of u, j, the stran tensor e j, accounts for the deformaton by measurng stretch of straght element lnes. Ths means we only consder the translatng relatve moton from (3) of nfntesmal elements of matter wthn the contnuum n the classcal theory. On the other hand, n sze-dependent contnuum mechancs, we also need to consder the relatve rotaton of nfntesmal elements (.e., the relatve rotaton of the 7
rgd trads), as n (4). Ths necesstates the contrbuton of the gradent of rotaton tensor, jn the defnton of the bendng metrc or measure of deformaton, whch ultmately wll reduce to curvatures, as we shall see. From ths knematcal analyss, other gradents of deformatons, such as j, k e and, jk, do not appear as measures of deformaton n a consstent contnuum mechancs. Although the gradent of deformaton tensor u, js mportant n the analyss of deformaton, even n the classcal case, t s not n tself a sutable measure of deformaton. In small deformaton theory, ths tensor can be decomposed nto the true (polar) symmetrc stran tensor e j and the true (polar) skew-symmetrc rotaton tensor j, where e u u u (5) j (, j), j j, j u[, j] u, j uj, (6) Notce that parentheses around a par of ndces denote the symmetrc part of the second order tensor, whereas square brackets ndcate the skew-symmetrc part. Then, the pseudo (axal) rotaton vector dscussed above, dual to the true skew-symmetrc rotaton tensor j, s defned as where we also have the relaton jkkj jkuk, j (7) j jkk (8) Now the prncple of vrtual work can be developed by frst multplyng () and () by energy conjugate vrtual quanttes and then ntegratng over the volume V. In the case of couple stress theory, these energy conjugates must be the true (polar) vrtual dsplacement u and the pseudo (axal) vrtual rotaton for equatons () and (), respectvely. Here we should note that () s a true (polar) vector equaton, whle () s n the form of a pseudo (axal) vector relaton. 8
Multplcaton by the conjugate vrtual felds defned above produces n both cases a true scalar, whch represents a vrtual work densty that s then ntegrated over the doman. In ths manner, the development of the prncple of vrtual work begns by wrtng: j j j j jk k, F u, j dv 0 (9) V Note that ths approach wll provde a formulaton wth the correspondng real knematc felds as the essental varables. Thus, the dsplacements and rotatons wll become the prmary degrees of freedom and we wll have a contnuum formulaton based upon the fundamental enttes of mechancs. For sutably dfferentable felds, we may rewrte (9) by ntroducng the relatons j, j j, j j, j u u u (0) j, j j, j j, j () whch after nvokng the dvergence theorem provdes the followng: t u m ds u F u dv 0 () S ( n) ( n) j, j j, j jk jk V where the force-tracton true (polar) vector and couple-tracton pseudo (axal) vector are defned as t n (3) ( n) j j m n (4) ( n) j j respectvely, wth n j representng the unt outward normal to the surface S. Fg. 3 llustrates force-tracton and couple-tracton vectors at an arbtrary locaton on the surface. 9
Fg. 3. Arbtrary force-tracton and couple-tracton vectors on surface. However, j s dual to the axal vector k, such that Then, () reduces to the followng: j jkk (5) V e j j j, j dv u ds u dv ( n) ( n) t m F S V (6) If one places a restrcton now to knematcally compatble vrtual felds on the boundary, then (6) would represent the prncple of vrtual work from the Mndln and Tersten (96) ndetermnate couple stress theory. We notce that the left hand sde of (6) shows that the stran tensor e j s energetcally conjugate to the symmetrc part of force-stress tensor j, whch s consstent wth our noton n classcal contnuum mechancs. Addtonally, ths relaton shows that j and, j are energy conjugate tensors. Ths confrms our predcton that, j should contrbute n the defnton of the bendng measure of deformaton. Mndln and Tersten (96), and Koter (964) consdered the devatorc tensor, j as the bendng measure of deformaton. However, ths creates some nconsstences n the formulaton, such as the ndetermnacy n the sphercal part of the couple-stress tensor. Most mportantly, the vrtual work prncple (6) shows 0
that there s no room for stran gradents as fundamental measures of deformaton n a consstent couple stress theory, as was concluded above n our knematcal analyss. The rght hand sde of the vrtual work prncple (6) shows that the boundary condtons on the surface of the body can be ether vectors u and as essental (geometrcal) boundary condtons, or t and ( n ) m as natural (mechancal) boundary condtons. Ths apparently makes a total ( n ) number of sx boundary values for ether case. However, ths s n contrast to the number of ndependent geometrc boundary condtons that can be mposed (Mndln and Tersten, 96, Koter, 964). In partcular, f components of u are specfed on the boundary surface, then the normal component of the rotaton correspondng to twstng where n nn n n n (7) k k nn knk (8) nn cannot be prescrbed ndependently. Therefore, the normal component 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 nn (9) k k 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 on a smooth surface s fve. Next, we let nn m and ( ns) m represent the normal and tangental components of the surface coupletracton vector m ( n), respectvely. The normal component nn nn m m n (0) where
nn ( n) m m n nn () k k j j causes twstng, whle ( ns) ( n) ( nn) kj k j j m m m n n n n () s responsble for bendng. Therefore, the boundary couple-tracton vrtual work n (6) can be wrtten as n nn n ns ns m ds m ds m ds S S S S nn nn ns ns m ds m ds S (3) 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, ts correspondng generalzed force couple-tracton) must be zero, that s ( nn) m (.e., the torsonal component of the nn ( n) m m n nn 0 on S (4) k k j j As a result, the boundary moment surface vrtual work n (3) becomes n ns ns ns m ds m ds m ds (5) S S S Ths shows that a materal n couple stress theory does not support ndependent dstrbutons of normal surface twstng couple-tracton ( nn) m, and the number of mechancal boundary condtons also s fve. Consequently, whle the force-tracton may be n an arbtrary drecton, the coupletracton must le n the tangent plane, as shown n Fg. 4. Ths means a consstent couple stress theory must satsfy the boundary condton (4) automatcally n ts formulaton.
Fg. 4. Force-tracton and tangental couple-tracton vectors on surface. Ths fundamental result was frst establshed by Mndln and Tersten (96) and more fully by Koter (964). However, the non-symmetrc form of the couple-stress tensor j n ther theory does not satsfy ths requrement drectly n the formulaton, where a generally non-zero dstrbuton of ( nn) m seemngly can be appled on the boundary surface S. In fact, the fundamental mplcaton of (4) as a constrant on the form of j was not understood fully untl recently. To resolve ths problem, Koter (964) proposed that a dstrbuton of normal surface twstng couple-tracton ( nn) m on the actual surface S be replaced by an equvalent shear stress dstrbuton and a lne force system. Ths s analogous to the transformaton of twstng shear dstrbuton to an equvalent vertcal transverse shear force and end corner concentrated forces n Krchhoff bendng theory of plates. However, we notce that there s a fundamental dfference between couple stress theory and the Krchhoff bendng theory of plates. The Krchhoff plate theory s a structural mechancs approxmaton to a contnuum mechancs theory obtaned by enforcng a constraned deformaton. Therefore, 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. On the other hand, couple stress theory s a contnuum mechancs theory tself and should be vald everywhere, ncludng near to and on the boundary, wthout any approxmaton. After all, we expect that the sze-dependency and effect of couple stresses are 3
more mportant near boundary surfaces, holes and cracks. Therefore, a consstent couple stress 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 condton (4) and ts mpact on the formulaton was not apprecated at the tme. It turns out that satsfyng the condton (4) n a systematc way yelds the consstent couple stress theory by revealng the fundamental character of the couple-stress pseudo tensor as follows. We notce that by the fundamental contnuum mechancs hypothess, the prncple of vrtual work and ts consequences are vald not only for the actual doman V, but for any arbtrary subdoman wth volume V a havng surface S a, as shown n Fg. 5. Therefore, the normal surface twstng ( nn) couple-tracton m on the artfcal surface S a must vansh, that s nn m nn 0 on S (6) j j a Fg. 5. The state of couple-tracton n m nsde the body. In the orgnal Mndln-Tersten-Koter theory, a generally non-zero dstrbuton of on the boundary surface ( nn) m appears S a. However, we notce that the Koter loadng transformaton method for ths possble dstrbuton of ( nn) m on the artfcal surface S a s ncompatble wth the arbtrarness of the surface S a. Ths means that the couple stress dstrbuton n the doman has to 4
satsfy the condton (6) drectly wthout recourse to any loadng transformaton. Thus, for any pont on the arbtrary surface S a wth unt normal n, ( nn) m must vansh. Ths requres nn m nn 0 n V (7) j j However, n ths relaton, n s arbtrary at each pont; we may construct subdomans wth any surface normal orentaton at a pont. Consequently, n (7), nn s an arbtrary symmetrc second j order tensor of rank one at each pont. Therefore, for (7) to hold n general, the couple stress pseudo tensor j must be skew-symmetrc, that s (8) j j Ths s the fundamental dscovery of consstent couple stress theory, whch shows that the coupletracton vector n m n (4) s tangent to the surface, thus creatng purely a bendng effect. We should emphasze that there s no menton of consttutve relatons n any of ths development, so that these results are n no way lmted to lnear elastc materals or to sotropc response. 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. The skew-symmetrc character mmedately resolves the ndetermnacy problem. Snce the dagonal components of the couple-stress tensor vansh, we notce that the couple-stress tensor automatcally s determnate n ths consstent couple-stress theory. Interestngly, ths result ndcates that there s an nterrelatonshp between the consstent mechancal boundary condton (4) and the determnacy of the couple-stress tensor; resolvng one, resolves the other. Ths s the amazng result of the fundamental hypothess of contnuum mechancs that the theory must be vald not only for the actual doman, but n all arbtrary subdomans. Ths realzaton s what was mssed by Mndln, Tersten and Koter n ther quest for a consstent couple stress theory. The components of the force-stress j and couple-stress j tensors n ths consstent theory are n shown n Fg. 6. Snce j s skew-symmetrc, the couple-tracton m gven by (4) s tangent 5
to the surface. As a result, the couple-stress tensor j creates only bendng couple-tractons on any arbtrary surface. The force-tracton n t and the consstent bendng couple-tracton n m actng on an arbtrary surface wth unt normal vector n are shown agan n Fg. 7. 3 3 3 3 x 33 x 3 x 3 3 3 3 3 3 3 3 Fg. 6. Components of force- and couple-stress tensors n consstent couple stress theory. n n t m n ds n Fg. 7. Force-tracton t and the consstent bendng couple-tracton n m. 6
It should be notced that n ths consstent contnuum theory, the shear force-stresses,.e. the tangental components of t on any surface, completely account for the torsonal loadng n the ( n ) materal, a character smlar to classcal contnuum mechancs. The true (polar) couple-stress vector dual to the pseudo-tensor where we also have the relaton j s defned as εjkkj (9) j jkk (30) Consequently, the surface couple-tracton vector tangent to the surface n m reduces to n ns m m n n (3) j j jk j k Here, t should be emphaszed that the couple-tracton vector n m s a pseudo vector, whereas the couple-stress vector s a true vector. Snce the couple-stress tensor s skew-symmetrc, we can obtan the skew-symmetrc part of the force-stress tensor from (), as j jk lk, l, j (3) Thus, for the total force-stress tensor, we have j j jklk, l j, j (33) Therefore, there are nne ndependent stress components n consstent couple stress theory or general sze-dependent contnuum mechancs. Ths ncludes sx components of j and three components of. 7
Interestngly, the relaton (3) can be elaborated further f we consder the pseudo (axal) vector s dual to the skew-symmetrc part of the force-stress tensor j, where Then, by usng (3) n (34), we obtan s jk[ kj] (34) s jkk, j (35) It s amazng to notce that the apparently complcated moment equlbrum equaton () reduces to the smple curl relaton (35). Ths s the result of the skew-symmetrc character of the couplestress tensor. Consequently, the lnear equaton of equlbrum reduces to [ ] 0 j j,, j F (36) whch shows that there are only three ndependent equlbrum equatons. Therefore, we must obtan the necessary extra sx remanng equatons from consttutve relatons. Now by returnng to the vrtual work prncple (6), we notce that the skew-symmetrc part of the tensor, j, namely, j, j, j j, (37) s the consstent curvature pseudo tensor. Further nspecton shows that the pseudo tensor j s the mean curvature tensor, whch represents the pure bendng of materal (Hadjesfandar and Dargush, 0). Moreover, the true (polar) mean curvature vector dual to the pseudo-tensor j s defned as where we also have the relaton jkkj (38) j jkk (39) 8
After some manpulaton, (38) can be wrtten as j, j uj, j u (40) 4 Interestngly, the mean curvature vector also can be expressed n terms of stran gradents as ekk, ek, k (4) Here, we should emphasze that ths relaton shows the curvature vector cannot be expressed n terms of the arbtrary gradents of stran e j, k, but rather a very specfc set of dervatves. On the other hand, we notce that the symmetrc part of the tensor, j, that s, j, j, j j, (4) s the torson pseudo tensor (Hadjesfandar and Dargush, 0). The skew-symmetrc character of the couple-stress tensor necesstates that the symmetrc torson tensor j does not contrbute as a fundamental measure of deformaton n a consstent couple stress theory. Now by assumng knematcally compatble vrtual felds n (6), the prncple of vrtual work balancng nternal and external contrbutons s wrtten: W nt W (43) ext e dv t u ds m ds F u dv (44) V ( n) ( ns) ( ns) j j j j St Sm V where t and ( n ) m represent the prescrbed force-tractons on S t and tangental couple-tractons ( ns ) on S m, respectvely, whle are the tangental components of vrtual rotaton. Note that snce ( ns ) e j s symmetrc, only the symmetrc part of the force-stress tensor j contrbutes n (44). Interestngly, the followng observatons can be made from our development, whch demonstrate the nner beauty and natural smplcty of consstent contnuum mechancs: 9
. In classcal contnuum mechancs, there are no couple-stresses, such that j 0. As a result, the force-stress tensor j s symmetrc.. In couple stress contnuum mechancs, the force-stress tensor j s not symmetrc, whereas the couple-stress tensor j s skew-symmetrc. In addton, the skew-symmetrc part of force-stress tensor j s expressed n terms of the couple-stress tensor curl relaton (35). j va the elegant Ths result shows that both classcal and couple stress contnuum mechancs enjoy some level of symmetry n ther nner structures. We have demonstrated that n consstent contnuum mechancs, we must consder the rgd body porton of moton of nfntesmal elements of matter (or rgd trads) at each pont of the contnuum. Therefore, n ths consstent couple stress theory, the dsplacements and rotatons provde the prmary degrees of freedom. Ths s entrely compatble wth the fundamental knematc varables n classcal mechancs, whch defne drectly all of the basc rgd body moton. We also notce that the number of basc conservaton laws of lnear () and angular () momentum at each pont s consstent wth those for a rgd body. The essental boundary condtons on a smooth surface n ths couple stress theory for threedmensonal problems become the three dsplacements and two tangental rotatons to form a set of fve ndependent quanttes. Meanwhle, natural boundary condtons consst of the forcetracton vector wth three ndependent components and the tangental couple-tracton vectors to apply bendng. As mentoned prevously, ths result was actually establshed by Mndln, Tersten and Koter. Unfortunately, they dd not realze that satsfyng these boundary condtons n a systematc manner reveals the determnate skew-symmetrc nature of the couple-stress tensor. Instead, by consderng a general non-symmetrc character for the couple-stress tensor, Koter approxmately enforced the requred boundary condtons by usng the loadng transformaton method from structural mechancs. However, the resultng couple stress theory was ndetermnate 0
and nconsstent. We notce that n the classcal contnuum mechancs theory, we only consder the moton of ponts or the relatve translatonal rgd body porton of moton of nfntesmal elements of the contnuum. As a result, the rotatons are left wth no essental role and the dsplacements become the prmary degrees of freedom n ths theory. What could be more beautfully-consstent and physcally-motvatng for the defnton of contnuum boundary value problems than to base the theory on the four central quanttes of mechancs? These are exactly the quanttes, whch descrbe the rgd body porton of moton of nfntesmal elements of matter at each pont of the contnuum. Fundamental solutons, varatonal prncples, boundary ntegral representatons, fnte element methods, boundary element methods, fnte dfference methods, and solutons to a sgnfcant number of boundary value problems already have been developed for ths consstent couple stress theory, wthn the context of both sold and flud mechancs. Addtonal work s underway, as are physcal experments, to assess crtcally these formulatons. Tme wll tell to what extent ths self-consstent theory algns wth nature. 3. Devatorc symmetrc couple stress theory Perhaps we should emphasze a further pont. In consstent couple stress theory, the dagonal components of the couple-stress tensor always vansh due to the skew-symmetrc character. Consequently, the determnate couple-stress tensor s also devatorc. Therefore, we may conclude that the devatorc skew-symmetrc couple stress theory s the fully consstent and determnate theory. Ths s n contrast to the devatorc symmetrc couple stress theory, whch suffers from many nconsstences. We examne ths theory n detal n the followng, as ths mght be helpful n apprecatng more deeply the consstency and beauty of skew-symmetrc couple stress theory. Neff et al. (009) support a theory based on the devatorc (trace free) symmetrc couple-stress tensor. Ths theory s also related to the work of Yang et al. (00), whch s commonly called the modfed couple stress theory. In ther development, Yang et al. (00) consder an extra equlbrum equaton for the moment of couples, n addton to the two equlbrum equatons of the classcal contnuum. Of course, ths addtonal law has no support n physcal realty.
However, applcaton of ths unsubstantated equlbrum equaton, apparently leads to a symmetrc couple-stress tensor, that s (45) j j The man motvaton for Yang et al. (00) n ther development has been to reduce the number of couple-stress materal constants for lnear sotropc elastc materal from two n the orgnal Mndln-Tersten-Koter theory to only one constant. For ths theory, the vrtual work prncple (6) shows that the symmetrc tensor j s the correspondng curvature tensor n ths theory. However, we notce that (46) 0, whch shows that the tensor j s devatorc, and thus s specfed by only fve ndependent components. As a consequence, all the nconsstences n Mndln-Tersten-Koter theory, such as the ndetermnacy n the couple-stress tensor and the appearance of nn m on the boundng surface S, unfortunately reman ntact n ths theory. Although, Yang et al. (00) do not offer any reason for the dsappearance of the ndetermnate sphercal part of the couple-stress tensor, many proponents of ths theory assume the couple-stress tensor s also devatorc, that s, (47) 33 0 There have been some doubts about the valdty of the fundamental aspects of the devatorc symmetrc couple stress theory. As mentoned, the symmetry character of the couple-stress tensor n ths theory s the consequence of the pecular equlbrum equaton for the moment of couple, besdes the two conventonal force and moment balance laws. However, ths requrement s an addtonal condton, whch s not derved by any prncple of classcal mechancs, as mentoned by Lazopoulos (009). Ths smply shows that modfed couple stress theory s not consstent wth basc rgd body mechancs. For more explanaton about ths fundamental nconsstency, see Hadjesfandar and Dargush (04). However, there are some other ssues wth ths theory, whch we examne next.
Frst, we notce that a theory based on the constraned devatorc (trace free) symmetrc couplestress tensor cannot be physcally acceptable. We demonstrate ths by usng physcal contradcton. If we assume the couple-stress tensor j s devatorc and symmetrc, t can also be dagonalzed by choosng the coordnate system xxx, 3 such that the coordnate axes x, x and x 3 are along ts orthogonal egenvectors or prncpal drectons. Therefore, n ths coordnate system, the couple-stress tensor j s represented by 0 0 j 0 0 (48) 0 0 33 where the dagonal components, and 33 are the torsonal couple-stress components around the coordnate axes x, x and x 3, respectvely. However, from a practcal vew, we notce that the loadng along these drectons are ndependent. Ths means we are allowed to exert torson couple-stress n any drecton; ts amount s arbtrary. Therefore, f we can exert torsonal couple-stresses, and 33 on some element of the matter, these three components must be ndependent of each other. Ths physcal fact contradcts the mathematcal devatorc condton expressed by (47). Therefore, couple stress theory wth a devatorc symmetrc couple-stress tensor s nconsstent and cannot be accepted on physcal grounds. We also notce that the symmetrc tensor j s the torson pseudo-tensor representng the pure twst of materal (Hadjesfandar and Dargush, 0). Snce ths tensor j s symmetrc, t can also be dagonalzed by choosng the coordnate system xxx, 3 such that the coordnate axes x, x and x 3 are along ts orthogonal egenvectors or prncpal drectons. Therefore, n ths coordnate system the torson tensor j s represented by 0 0 j 0 0 (49) 0 0 33 3
where the dagonal components, and 33 are the torsons around the coordnate axes x, x and x 3, respectvely. Therefore, the torson tensor (49) does not represent the bendng deformaton of the materal at all. Ths fact also suggests that ths tensor should not be chosen as the sole bendng measure of deformaton. Therefore, the devatorc symmetrc or the modfed couple stress theory not only nherts all nconsstences from ndetermnate Mndln-Tersten-Koter theory, but also suffers from new nconsstences, whch are summarzed as follows:. The unsubstantated addtonal artfcal equlbrum of moment of couples n the set of fundamental equatons;. The physcal nconsstency of the constraned devatorc symmetrc couple-stress tensor j ; 3. The devatorc symmetrc torson tensor j does not descrbe the bendng deformaton. As a fnal ssue, one mght thnk that the ndetermnacy of the sphercal part of the couple-stress tensor s analogous to the behavor of an ncompressble materal under pressure. For an ncompressble materal, the ncompressblty condton s u, 0 (50) Assume the dstrbuton of the constant pressure p, where j p (5) j Consequently, the normal force-tracton on the surface s t n pn (5) 4
We notce that the pressure stress dstrbuton (5) does not contrbute to the nternal work, because we have for the nternal compatble vrtual work j ej pu, 0 (53) As a result, ths loadng does not create any deformaton n the body. However, we notce that an ncompressble materal s a mathematcal concept, and physcally does not exst. Ths means that the stran tensor e j never becomes devatorc n realty. The ncompressblty condton (50) s just an artfcal assumpton to smplfy cases of near-ncompressblty. Interestngly, for the lnear sotropc elastc materals, the ncompressblty corresponds to Posson rato s excluded based on energy consderatons (Malvern, 969)., whch On the other hand, we notce that the devatorc character of the couple-stress tensor n Mndln- Tersten-Koter and modfed couple-stress theory s the drect result of devatorc tensors j and j, respectvely, ndependent of the materal behavor. It s ths devatorc character, whch makes these tensors unsutable as measures of bendng deformaton. Nevertheless, we have already establshed that the skew-symmetrc mean curvature tensor bendng deformaton, whch of course has no sphercal part. j s the consstent measure of 4. Conclusons The recent papers by Neff et al. (05a-c) have motvated us to reexamne contnuum mechancs from a fundamental perspectve. However, what s most fundamental n developng a contnuum mechancs theory? Is t the defnton of thermodynamc potentals? Balance laws? Boundary condtons? Vrtual work? Of course, all of these are mportant, but we beleve that frst and foremost the development should be founded on concepts emanatng from the classcal mechancs of partcles and rgd bodes, n whch all varables have clear physcal meanng. Thus, the fundamental objects of nvestgaton n mechancs should be forces and couples, or ther ntensve contnuum counterparts, namely, force-stresses and couple-stresses. Furthermore, the knematc varables must be dsplacement and rotaton, whch are needed to descrbe rgd moton of entre 5
bodes or, n the contnuum case, of nfntesmal elements. Snce the force and moment balance laws for these nfntesmal elements of matter are not suffcent to determne unquely the dstrbuton of stresses n the contnuum, we need to consder deformaton. To gan a better understandng of the knematcs of deformaton, we may envson a rgd trad assocated wth each nfntesmal element. However, the contnuty of matter restrans the relatve moton of these rgd trads, such that here, unlke n Cosserat theory, the trad translates and also rotates wth each nfntesmal element. There s no ndependent rotaton; rather the rotaton of each nfntesmal element, and ts attached rgd trad, s defned by one-half the curl of the dsplacement feld. In classcal contnuum mechancs, the deformaton then s attrbuted solely to the symmetrc part of the relatve translaton of adjacent nfntesmal elements (or rgd trads). However, ths s an ncomplete pcture, whch assgns a mnor ancllary role to rotatons and ndcates that classcal Cauchy contnuum mechancs s not fully algned wth partcle and rgd body mechancs. We must extend ths classcal vew to accommodate the relatve rotaton of these adjacent nfntesmal elements (or rgd trads) as well, and elevate rotatons to the level of knematc degrees of freedom, along wth dsplacements. Thus, relatve trad translaton provdes dsplacement gradents, whch lead to the dentfcaton of strans, or stretches n prncpal drectons, as the sze-ndependent measure of deformaton, exactly as n the classcal theory. On the other hand, relatve trad rotaton offers rotaton gradents as the canddate from whch a szedependent deformaton measure can be derved. Next, by gvng careful consderaton to the ssue of ndependent boundary condtons on both the real surfaces and any arbtrary nternal surface, we fnd that the normal twstng couple-tracton must vansh on all surfaces. Satsfyng ths requrement n a systematc way restrcts the form of the couple-stress tensor to be skew-symmetrc. Ths s what was mssed by Mndln, Tersten and Koter n ther quest for a consstent couple stress theory. Because of ts skew-symmetrc nature, the couple stress tensor also s automatcally devatorc wthout mposng non-physcal constrants on the components. At once, ths resolves all of the ssues of nconsstency and ndetermnacy that have plagued pror couple stress theores, ncludng the orgnal Mndln-Tersten-Koter and modfed couple stress theores. Furthermore, the deformaton measure that s energy conjugate to the skew-symmetrc couple stress tensor becomes the skew-symmetrc part of the rotaton 6
gradent tensor, that s, the mean curvature tensor, whch captures sze-dependent bendng deformaton. Fnally, we may menton the nterestng symmetres present n the two man contnuum theores. In classcal contnuum mechancs, there are no couple-stresses, and the force-stress tensor s symmetrc. On the other hand, n consstent couple stress contnuum mechancs, the force-stress tensor s not symmetrc, but the couple-stress tensor s skew-symmetrc. Ths suggests once agan that the mathematcal descrpton of nature may favor a certan level of symmetry and beauty n ts nner structure. References Cosserat, E., Cosserat, F., 909. Théore des corps déformables (Theory of deformable bodes). A. Hermann et Fls, Pars. Erngen, A. C., 968. Theory of mcropolar elastcty, Fracture, vol, ed. H. Lebowtz, Academc Press, New York, 66-79. Hadjesfandar, A. R., 03. Vortex theory of electromagnetsm. vxra: 009.35. Hadjesfandar, A. R., Dargush, G. F., 04. Evoluton of generalzed couple-stress contnuum theores: a crtcal analyss. arxv: 50.03. Hadjesfandar, A. R., Dargush, G. F., 0. Couple stress theory for solds. Int. J. Solds Struct. 48 (8), 496-50. Koter, W. T., 964. Couple stresses n the theory of elastcty, I and II. Proc. Ned. Akad. Wet. (B) 67, 7-44. Lazopoulos, K.A., 009. On bendng of stran gradent elastc mcro-plates. Mech. Res. Commun. 36 (7), 777 783. Malvern, L. E., 969. Introducton to the Mechancs of a Contnuous Medum. Prentce-Hall Inc., Englewood Clffs. 7
Mndln, R. D., Tersten, H. F., 96. Effects of couple-stresses n lnear elastcty, Arch. Ratonal Mech. Anal., 45 488. Neff, P., Münch, I., Ghba, I-D., Madeo, A., 05a. On some fundamental msunderstandngs n the ndetermnate couple stress model. A comment on recent papers of A.R. Hadjesfandar and G.F. Dargush. arxv: 504.0305. Neff, P., Ghba, I-D., Madeo, A., Münch, I., 05b. Correct tracton boundary condtons n the ndetermnate couple stress model. arxv: 504.00448. Neff, P., Jeong, J., Ramézan, H. 009. Subgrd nteracton and mcro-randomness Novel nvarance requrements n nfntesmal gradent elastcty Int. J. Solds Struct. 46 (5-6), 46 476. Yang, F., Chong, A. C. M., Lam, D. C. C., Tong P., 00. Couple stress based stran gradent theory for elastcty, Int. J. Solds Struct. 39 (0), 73 743. 8