Formulation of Deformation Stress Fields and Constitutive Equations. in Rational Mechanics

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1 Fomuaton of Defomaton Stess Feds and Consttutve Equatons n Ratona Mechancs Xao Janhua Measuement Insttute, Henan Poytechnc Unvesty, Jaozuo, Chna Abstact: In contnuum mechancs, stess concept pays an essenta oe. Fo compcated mateas, dffeent stess concepts ae used wth ambuty o dffeent undestandn. Geometcay, a matea eement s expessed by a cosed eon wth abta shape. The ntena eon s acted by dstance dependent foce (ntena body foce), whe the suface s acted by suface foce. Futhe moe, the eement as a whoe s n a physca bacound (eo eon) whch s detemned by the contnuum whee the eement s embedded (ena body foce). Physcay, the tota eney can be addtvey decomposed as thee pats: ntena eon eney, suface eney, and the bacound eney. Howeve, as foces, they cannot be added decty. Afte fomuatn the enea foms of physca feds, the defomaton tenso s ntoduced to fomuate the foce vaatons caused by defomaton. As the foce vaaton s expessed by the defomaton tenso, the defomaton stess concept s we fomuated. Futhemoe, as a natua esut, the addtve decomposton ves out the defnton of statc contnuum, whch detemnes the matea paametes n consttutve equatons. Thouh usn the eo dffeentas, the consttutve equatons ae fomuated n enea fom. Thouhout the pape, when t s sutabe, the eated esuts ae smpfed to cassca esuts fo ease undestandn. Key Wods: consttutve equaton, stess, defomaton tenso, atona mechancs, eo dffeenta Contents. Intoducton. Basc Notes on Reated Fomuaton 4. Moton Concepts n Contnuum Mechancs 4. Some notes about Tenso Theoy n Contnuum Mechancs 5. Notes on Exteo Dffeentas n Defomaton Geometca Fomuaton 6.4 Notes on Matea Invaance 8. Physca Feds Vaatons Caused by Defomaton 8 4. Geometca Fomuaton of Stess Feds Defomaton Descpton n Ratona Mechancs Inteo Lne Descpton 4.. Inteo Suface Descpton 4.. Exteo Suface Descpton 4. Suface Foce Deved Body Foce 4 4. Body Foce Deved Suface Foce Intnsc Stess Defnton Intena Reon Intnsc Body Foce and Sef-spn Extena Reon and Intnsc Stess 8

2 4.4. Exteo Suface Confuaton and Intnsc Vscous Stess Maco Statc Contnuum Defnton 5. Consttutve Equaton Fomuaton on Physca Bases 5. Isotopc Smpe Sod Matea 5. Poa Smpe Fud Matea 4 5. Lnea Consttutve Equaton n Genea Fom 5 6. Stess Fomuaton fo Chen S+R Addtve Fom Defomaton 6 6. Stess fo Chen Fom- Defomaton 6 6. Stess fo Chen Fom- Defomaton 7 7. Matea Paamete Invaance Featue 8 8. Concuson Refeences. Intoducton In cassca eastc defomaton of contnuum mechancs, the stess concept s we defned based on expements. Howeve, when the stess concept s ended to compcated mateas, the ambuty becomes cea. Athouh the stess can be defned by fomuatn the defomaton eney as the functon of stess tenso contact wth stan tenso n contnuum mechancs [-], what s the enneen pctue of stess st s a pobem [4-5]. In enneen mechancs, each stess component = ) s we defned as the suface ( foce actn on the suface on the decton. The symmety featue s defned as the statc baance condton of eement wthout otaton and s poved by the anua moment consevaton. If we accept both, then the suface foce actn on the suface on the decton s equa to the suface foce actn on the suface on the decton w become a natua concuson. Some oc ambuty may efect n hs easonn. If one examnes t by the tenso fomuaton ues [6-7] by above enneen defnton, the stess shoud be a mxtue tenso [8-9]. Why t s a covaant tenso n most conventona teatment? In most eseaches, the stess tenso s fomuated by foce baance ues. The peexstence of foce s assumed. The defomaton s vewed as a coodnato tansfomaton. Aon ths ne, the stess tenso concept s studed aan and aan by eseaches wth dffeent bacounds. In ths pape, the eometca fed fomuaton w be studed. In a esent pape [], Wate No suested a confuaton fomuaton composed by an ntena mappn and an ena mappn. By hs concept, the foce system n confuaton δ s a pa ( F nt δ, F δ ), both foce-baanced and toque-baanced. Hee, the ntena eon mappn and ena mappn ae defned ndependenty, but they ae addtve. So, the unt matea eement concept s epaced by a confuaton unt. The contnuum concept s ouhy defned as the mateay odeed sets. Fany spean, I cannot fuy undestand hs mathematca teatment, especay the enea teatment about defomaton: t s a mappn. Teatn the defomaton as a manfod mappn s vey common [-4]. In fact, t s accepted as a conventon. The ene n such a nd of teatment s that: tan defomaton eney as the functon of the defomaton tenso, then the physca pncpes ae apped to deduce the eated

3 equatons. Once the stess tenso s estabshed, the oca dffeenta coodnato tansfomaton concept s used to fomuate the equed tenso fom [-4]. Howeve, once the detas fo enneen appcatons ae equed, tte can be obtaned by ennees. So, an expct fomuaton [5-8] s equed. In ths eseach, the contnuum concept s ouhy defned as the mateay odeed sets (as Wate No defned). Geometcay, a matea eement s expessed by a cosed eon wth abta shape. The ntena eon s acted by dstance dependent foce (body foce), whe the suface s acted by suface foce. Futhe moe, the eement as a whoe s n a physca bacound whch s detemned by the contnuum whee the eement s embedded. Physcay, the tota eney can be addtvey decomposed as thee pats: ntena eon eney, suface eney, and the bacound eney. To fomuate the physca eney quanttes, the eo dffeenta toos ae used, combnn wth the tenso fomuaton of defomaton eomety n dan coodnato system [8, 9]. By ths fomuaton, the ntena eon eney s expessed by the mass cente moton eated eney. To expess ths eney (-fom), thee covaant base vectos ae equed. The suface eney s eated wth the unt matea suface confuaton. Usn the eo dffeenta toos, wede poduct of two base vectos ae used to descbe the confuaton suface (-fom). The bacound eney s eated wth the eney exchane between the unt matea eement wth the medum as a oba whoe and t s expessed as the wede poduct of thee base vectos (-fom). As an exampe [], the potenta foce s eated wth -fom (such as eectca fed), the suface foce s eated wth -fom (such as manetc fed), the eney densty s eated the -fom. Such a nd teatment s dffeent fom the teatment fom the methods based on Laane o Hamton quantty [-5], athouh the mped phosophy s dentca. Based on my pesona feen, athouh these eseaches ae ood enouh n mathematc fomuaton, the physca easonn s not so ston. It s tue that the defomaton s equvaent wth a oca dffeenta coodnato tansfomaton n mathematc teatment. Howeve, what ae the ea physca pctues? What ae the possbe methods to do the enneen measuement? O moe decty, fo ennees, how to undestand the eated fomuaton coecty? I do not thn that these papes can answe these pobems satsfactoy. It s easy to ntoduce the mappn concept o ts equvaent to fomuate the defomaton as a mappn n mathematc teatment. Even they ae coect n enea sense, ony when the ea physca aws ae apped to mappn, the ea physca pocess can be we expessed. In most eseaches, the physca aws ae pe-detemned. Hence, n essenta sense, these eseaches put the man ponts on endn the nown physca aws to enea cases. So, fo sovn mathematc equatons, they ae powe-fu. Yes, t s a way to mae scentfc fndns. Howeve, fo expann the physca aws n essenta sense, these methods ae not so effectve as they ae expected. Then, a detaed and moe enneen oentated teatment st s equed. Ths s the enea pupose of ths pape. Dffen fom above easonn, ths eseach w fomuate the defomaton concept by eometca fed n an expct way. Hee, the expct means t s the ea descpton of enneen opeaton way, athe than the mathematc opeato ways. Afte that, based on the enea eney fom (decomposed as thee tems of eo dffeentas), the stess concept w be ntoduced natuay thouh the defomaton tenso. Afte fomuatn the enea fom of eney by eometca feds, the defomaton tenso s

4 ntoduced to fomuate the eney vaatons caused by defomaton. As the eney vaaton s expessed by the defomaton tenso, the defomaton stess concept s we fomuated. Futhemoe, as a natua esut, the addtve decomposton ves out the defnton of statc contnuum by zeo defomaton eney. Thouh usn the eo dffeentas, the moton equatons ae fomuated n enea fom. Thouhout the pape, when t s sutabe, the eated esuts ae smpfed to cassca esuts.. Basc Notes on Reated Fomuaton To mae the pape be easy undestood, some concepts shoud be ceaed. Futhemoe, the mathematc fomuaton eated wth eo dffeentas shoud be ceaed.. Moton Concepts n Contnuum Mechancs The deveopment of mate moton concept n physcs s vey mpotant to undestand the dffcuty n undestandn the stess concept n contnuum mechancs. Thee ae two methods to descbe matte moton. One s n Newton sense, by poston dspacement, the veocty s an exampe. Anothe s n Ensten sense, by aue fed vaaton. Geneay spean, the moden eseaches put the effots on estabshn the poston dspacement ntepetaton, such as tace vaaton and moton path concept, o coodnato tansfomaton to epace the aue fed vaaton. Moe abstact foms n mathematcs ae based on above two moton concepts n essenta sense. If dspacement concept s used to descbe moton, the foce s defned to fom the basc concept of eney. Fo smpe moton, the eney vaaton s fomuated as: de = f ds () A vecto space concept s fomuated. The condton s that the envonment has no othe contbuton. Howeve, as Wate No pont-out, fo the moton n contnuum, as defned by A E Geen, the eney vaaton s n the fom: = () de S The basc descpton of moton descpton causes dffeent fomuaton system. Fo smpe moton n Newton patce mechancs, one has: d s ds m ds de = f ds = m d( ) = d dt dt dt () The veocty pays the essenta oe. When the tme paamete s mped, the potenta foce s defned as: f = u(s) (4) So, the esut s: de = u( s) ds (5) Hence, the potenta functon s named as eney functon. Combnn both toethe, the Laane mechancs and Hamtonan mechancs ae fomuated by defnn the enea eney functon: 4

5 E = K ± U (6) Its emty defnes the physca eaty. Summn above fomuaton, the poston dspacement domnate the moton concept. Howeve, the eectomanetc fed ves out anothe nd of moton. The basc equatons: A E = φ + µ, B = A (7) t They mae the Newton moton concept becomes a non-compete concept. The eney s n the fom: de = µ B + εe (8) In fact, t s fom the Maxwe equatons, the eatvty theoy s estabshed. The eseach on unfyn the avty foce and the eectomanetc foce onates fom ths basc fact. My undestandn s that: afte the estabshment of enea eatvty, the moton concept s domnated by dffeent fomuaton on aue vaatons. In ths eseach, the aue vaaton s fomuated n commovn dan coodnato system by ntoducn the basc aue vecto tansfomaton tenso. The Newton moton s aso fomuated coespondny.. Some notes about Tenso Theoy n Contnuum Mechancs In mathematcs, the physca tenso s taen as the nvaant obecto. So, fo dffeent coodnato system seecton, t s nvaant n fom. In most mathematc teatment, the dx and d ~ x meet Remannan eometca nvaant condton: ds ~ ~ ~ = dx dx = dx dx (9) Ths equaton means that when the coodnato system s tansfomed, the coespondn physca components ae tansfomed n the coodnato fowad tansfomaton fom (covaant component) o n the coodnato evese tansfomaton fom (ant-covaant component). In ths eseach, ths tenso fomuaton s apped to the nta dan coodnato seecton. In defomaton mechancs, the nvaant of matte unt unde dscusson s acheved by ven t a set of coodnatos. The aue tenso s used to descbe the eometca moton of the matte unde consdeaton. Fo nta confuaton, the dffeenta dstance vecto s expessed as: ds = dx. Fo cuent confuaton, the dffeenta dstance vecto s expessed as: ds = dx. In tadtona contnuum mechancs, the defomaton s expessed by the dffeenta dstance vaaton: δ s = ds ds = ( ) dx dx. () Many fomuatons of stan tenso ae based on ths equaton. Ths defnton omts the oca otaton, based on the fase beef that: the oca otaton s equvaent wth d otaton, so has no contbuton to dstance vaaton. Ths concept s eected n atona mechancs, whee the defomaton tenso s used to fomuate the defomaton moton. In ths eseach, the defomaton s expessed as the vecto dffeenta [8-9] : 5

6 δ s = ( ) dx () So, the ony way s to tae the dan coodnato system. In teatn fud moton, the oca otaton concept s used n contnuum mechancs. Hence, the defomaton tenso concept s used. In ths eseach, the defomaton tenso F s defned by: = F () Theefoe, the vecto vaaton n defomaton sense shoud be defned fo a fxed pont as: δ s = ( F δ ) dx () Whee, the dx s puey a scaa quantty. Unfotunatey, based by the coodnato tansfomaton concept, many eseaches defne the defomaton tenso as a coodnato tansfomaton [6]. Ths s not sound n physca easonn. One way to mae the defomaton concept cea s to et: F α up-ndex defned on nta confuaton, ts summn s aways about nta defned quantty; Low-ndex defned on cuent confuaton, ts summn s aways about cuent defned quantty. As an exampe: The man vewpont s that: Hence, the v = F α α, ~ α ~ α v = v α = ( v Fα ), so, v ~ α = v F. (4) dx ae fxed quantty fo the matea unt unde dscusson. ~ α ( ~ α = v = v F ) ony means an nvaant physca fed. As the α α α physca fed s chaned afte defomaton, the dffeentas ae: v = v ~ α α v = ( v ~ α F ) α v (5) Theefoe, the coodnato chane has no eaton wth the defomaton fomuaton. Geneay spean, fo physca consdeaton, f the matea has no ntnsc vaaton, the dentty shoud be: v = v α α v = α ( v Fα ) v A vaatons ae caused by eometca defomaton. Ths s named as: eometca defomaton to dstnush the physca ntnsc vaaton (physca defomaton). (6). Notes on Exteo Dffeentas n Defomaton Geometca Fomuaton Geneay spean, n mathematc fomuaton of eo dffeenta theoy, the vecto component s detemned by physca quantty (by dffeenta opeaton on functon defned on manfod). Hence, the coodnato dffeenta s sepaated as quantty. Smay, n defomaton tenso fomuaton, the base vecto s sepaated out aso n a sma fom. Even so, the eo dffeenta too [, 7-8] cannot be apped wthout detaed anayss. Some fom modfcatons ae equed. The deep easons ae that: n physcs, the physca fed s poduced by compan the 6

7 physca quantty dffeence (vaaton) between two nea ponts. Howeve, n defomaton mechancs, the physca quantty dffeence (vaaton) at the same pont s equed. Fo manfod U, functons x on U (open set) ae named as coodnates. Any functons on U n can be wtten as: f ( x, x,..., x ). By eo dffeenta anuae, the oca coodnato s ntepeted as the coodnate -fom: dx. Fo smpe scaa potenta fed ϕ, the dffeenta between two ponts (coodnato dffeences dx pay the vecto oe) s epesented as a vecto: dϕ = ϕ dx. By defomaton eometca fomuaton adopted n ths eseach, t shoud be ϕ ( ) ϕ ( ) epesented as: dϕ = ( )( dx ) = ( dx ) = ~ ϕ. Compan both fomuatons, t s found that: n -fom d = ϕ ( ) ϕ dx, the vecto dx dx taes the oe of covaant base vecto mutped by the coodnato ncements; the ~ ϕ () ϕ = ( dx ) taes the oe of ϕ ant-covaant vecto component mutped by the scaa component (). To mae usefu smpfcaton n defomaton eometca fomuaton, the scaa potenta fed dffeenta shoud be undestood as a vecto n fom: ϕ ( ) dϕ = ( dx ) = ~ ϕ (7) Whee (above and hee afte), the bacets s used to show the ndex n t s not vewed as a summn conventon. That s to say, the physca vecto component s decomposed as the mutpcatve of quantty pat and base vecto pat. The man pont s that the dx s not a vecto n defomaton theoy (t s puey a scaa quantty). Hence, n defomaton fomuaton, the -fom (e eectca fed): ω ω = dx n eo dffeenta s efomuated as: ω ~ = ω( ) dx = ω (8) The eo dffeenta -fom (e manetc fed) s defned as: ω = ω dx dx. In defomaton fom adopted n ths eseach, -fom shoud be: Fo the -fom, ω = [ ω ~ ω ω = ω dx dx! ( ) dx dx ] = (9) dx, smay, t s efomuated as: ω = [ ω dx dx dx = ~ ( ) ] ω ()!! Usn above fomuatons s based on the consdeaton that: fo a fxed confuaton, the eo dffeenta s used to defne the physca fed and the moton aws. As the physca feds ( ω, ω, and ω ) ae we defned, by coodnatn the matea unt sze n ncementa 7

8 coodnato fom ( dx ), the feds ( ~ ω, ~ ω, and ~ ω ) ae we defned. In defomaton mechancs, the dx (fxed on odeed mateas whch fom contnuum and s nvaant) s vewed as ant-covaant coodnato seecton, so the physca components s vewed as covaant fom. Summn above expanaton, n defomaton eometca fomuaton of defomaton mechancs, the eo dffeenta fom s expessed as: -fom: -fom: f ω f ω -fom: -fom: ω ω ω ω! In Cuent Confuaton In Inta Confuaton! The mped physca meann s that: fo a manfod, accodn to a ven nta aue seecton, the eo dffeentas ae used to defne the physca fed. As the defomaton s ntoduced thouh defomaton equaton = F (o n dstance vecto fom ds dx =, and ds = dx ), so thee s no need to chane the eo dffeentas system as they poduce the equed physca fed, whe the defomaton the physca feds vaaton unde the defomaton..4 Notes on Matea Invaance In cassca defomaton mechancs, the matea featue s epesented by eastcty paametes. On physca undestandn, the matea s defned by the eastcty paametes. In attce dynamcs theoy, the eastcty paametes by physca fed (quantum mechancs). So, t s cea, the so-caed eastcty paametes ae detemned by physca feds. Based on ths undestandn, the physca feds ( ω, ω, and ω ) defne the matea featues unde defomaton. Ths vewpont s bued n contnuum mechancs. So, n ths eseach, the ona physca feds ( ω, ω, and ω ) ae supposed as nvaant. It s cea, fo ae defomaton, they ae not nvaant. Howeve, fo ncementa defomaton, they ae fomuated as nvaant n defomaton mechancs fomuaton. Ths assumpton s named as matea nvaance n ths eseach.. Physca Feds Vaatons Caused by Defomaton Based on above dscussons and fomuaton ues, the physca feds (defned by eo dffeenta fom) vaatons caused by defomaton ae fomuated n ths sub-secton. 8

9 u In fact, fo Gadent: u =, the defomaton caused vaaton s: u δ u = ( F δ ) = δu () In defomaton mechancs, the defomaton caused feds (defned n eo dffeenta foms) vaatons ae expessed as foown. -fom: δ f = f () = () -fom: δω ω ( F δ ) = δω -fom: = ω ( F F δ δ ) = δω δω (4)! m n m n -fom: δω = ω ( F F F δ δ δ ) m n = δω (5)! Hee (and afte), the nta confuaton s taen as efeence. Fo no defomaton, a above quanttes equa to zeos. By ths fomuaton, one fnds that: ω ( F δ ) ω dx (6-) ( F F δ δ ) ω dx dx ω m n m n ( F F F δ δ δ ) ω dx dx dx ω m n!! (6-) (6-) Whee, the eft sde quanttes ae the physca fed vaaton, the ht sde quanttes ae the ona physca fed defnton n eo dffeenta fom. Detaed eometca ntepetaton w be ven n the n secton. 4. Geometca Fomuaton of Stess Feds Geometcay, a matea eement s expessed by a cosed eon wth abta shape. The ntena eon s acted by dstance dependent foce (body foce), whe the suface s acted by suface foce. Futhe moe, the eement as a whoe s n a physca bacound whch s detemned by the contnuum whee the eement s embedded. 4.. Defomaton Descpton n Ratona Mechancs In cassca mechancs, the unt matea s unde dscusson. Its nta confuaton can be expessed by thee basc vectos. When the defomaton happens, the thee basc vectos ae tansfomed nto othe foms. Fo dffeent coodnato seectons, the basc vectos w be tansfomed n the sma way. So, they ae dentca n mathematcs sense. Howeve, fo a matea unt, the basc vectos ( dx, dx, dx ) o (,, ) ony defnes the aveae shape. O say, the enth decton shape. If the matea s not the cubc shape, howeve, t s abta shape, anothe set of basc vectos ae needed (,, ) (Refen Fue. Abta Shape of Matea Unt). By Remannan eomety tenso ntepetaton, they fom suface base 9

10 vectos. In eo dffeenta fomuaton, the 6 sufaces of the matea unt s defned by ( dx dx ) o ( ). As an exampe, fo a sphee shape, athouh the thee ectanua decton enth vectos foms the ntena confuaton, anothe thee suface vectos ae needed to foms the eo suface confuaton. In conventona eomety, the sphee suface s defned as: ( dx ) + ( dx ) + ( dx ) = d. Hence, n enea, the compete defomaton defnton [9] s: = F, = G (7) Whee, the suppe ndex o owe ndex s used fo nta confuaton. When the seecton = δ s made, the defomaton can be smpfed as: = F. It s cea that: G F δ =. In ths case, the voume nvaant defomaton s defned. The physca eaty of unt matea s a fnte eement wth fnte voume. Hence, at east thee basc vectos ae equed. In ths eseach, they ae (,, ). Fue. Abta Shape of Matea Unt Pue eometcay, a cubc eement can be constucted by 9 ndependent ponts. One cente, thee covaant vectos passn thouh t, wth 6 ponts; thee suface passn the cente pont, each equpped wth a noma vecto (ant-covaant base vectos), addtona ponts ae equed. The cente pont can be constucted by the othe 9 ponts, so t s not ndependent. Hence, f and ony f the 9 ponts ae tansfomed n such a way that the eatons ae ept nvaant, the coodnato tansfomatons ae acceptabe.

11 Howeve, n defomaton mechancs, fo abta shape eement, anothe way shoud be found. Ths s descbed beow. 4.. Inteo Lne Descpton Fo a matea eement cente pont ( x, x, x ), any ne vecto ds passn thouh the cente pont and tae the ntesecton ponts wth suface as ts two ends s expessed by the covaant base vectos constucton as: ds = dx. By ths fomuaton, the dstance vecto between two nehbo eements aso s ds = dx. Refen Fue. (Cente Ponts Dstance Desced as Inteo Reon Lne). Fue. Cente Ponts Dstance Desced as Inteo Reon Lne By ths defnton, the matea cente pont coodnato ncements ae ( dx, dx, dx ), and at the same tme, they ae the coodnato ncements fo any ne passn thouh the cente and connectn the two suface ponts. Afte the whoe medum s coodnated, the defomaton s consdeed as the base vecto vaaton. Fo nta efen confuaton, ds = dx. Afte defomaton, fo cuent confuaton, ds = dx. Hence, the confuaton vaatons ae descbed. The Remannan eomety eques that: ds s nvaant to mae the dffeenta oca coodnato tansfomaton possbe. Howeve, fo defomaton, ds s not nvaant. So, fo defomaton, the eement confuaton s defned by the numeca coodnato

12 vaaton and the ndependent base vectos whch ae defomed. Ths s the essenta meann of defomaton n atona mechancs. Physcay, ths s the enson of pont moton concept. Body foce s based on ths pont of vew. The dspacement fed u (numeca) s based on ths fomuaton system. Chen s eseach [8-9] shows that: u F = δ + (8) Ths equaton shows that the defomaton tenso can be detemned by measun the dspacement feds. Fo detas, pease see [8-9]. 4.. Inteo Suface Descpton On the othe hand, the eement eo suface can be defned. The suface can be fomuated as ( dx dx, dx dx, dx dx ) n numeca sense, and n base vecto sense as: (,, ). (Refen Fue. Inteo Suface Base Vectos). Fue. Inteo Suface Base Vectos eement, The nteo sufaces tae the eement cente pont as ts eometca cente. Fo unt voume = δ. Ths defnton s used n contnuum eomety to cove the whoe medum. So, t s a oba defnton. Hence, t s named as nteo sufaces. In eometca theoy, any suface passn thouh the eement cente can be decomposed as the nea combnaton of thee basc sufaces. Fo ths eason, the ant-covaant base vectos w not be used as the basc vectos fo defomaton. In fact, each nteo suface can be thouht as a squae whch taes the fou eement cente ponts on the squae pane as the fou cone ponts. Then, the suface s moved to tae a matea

13 cente as ts eometca cente. The stess concept n cassca defomaton mechancs s defned on these basc sufaces. 4.. Exteo Suface Descpton The above nteo suface defnton foms the eement as a cubc shape and ts cente s the matea cente. So, the ea eement suface s not descbed. In ths eseach, the eon not cosed n (by the ea suface of eement) s defned as eo eon. So, hee, the suface on eement s defned as eo suface. In ths eseach, the eo sufaces ae expessed as: dx ( ) dx ( ). The suface spanned out fom the pane suface dx ( ) dx ( ) (whch centeed at eement cente) s defned as eo suface. Its eometca meann s expaned n Fue 4. (Exteo Suface Defnton). Fue 4. Exteo Suface Defnton Theefoe, the nteo suface passn matea cente dx dx foms the eo suface spanned out on decton based on ht-hand chaty ues. The nteo suface passn matea cente dx dx foms the eo suface spanned out on decton. The two eo suface foms a cosed suface. Smay, othe two foms can be expaned. Hence, the enea fom dx ( ) dx ( ) defnes 6 spanned out sufaces. They fom thee nds of confuaton sufaces. As a eometca conventon, the spanned out suface shoud be the appoxmaton of ea suface shape of matea

14 eement. In defomaton mechancs, the ea aea of spanned out suface s not a pobem. Ths s because the aea vaaton of spanned out suface s popotona wth the aea vaaton of nteo suface. Fo defomaton = F, the eo suface defomaton s expessed as: = F F (9) Theefoe, the abta shape unt matea defomaton ae fuy descbed by the defomaton tenso F. Fo smpcty, the eo suface may be smpy caed as suface. 4. Suface Foce Deved Body Foce In ths eseach, the body foce and suface foce ae actn on the matea eement. The Body foce s defned as: actn on eveywhee. Fo defomaton mechancs, the mass cente o eometca cente pont can be used to epesents ts acton pont. The Suface foce s defned as: actn on eement suface. Fo defomaton mechancs, thee ae 6 sufaces to fom a cosed eon. Such a nd of cosed eon s defned as eement unt. In mechancs, the ea pobems ae the body foce w poduce suface foce and the vse vesa. In conventona eomety, body foces ae eated wth covaant base vectos eated wth pont pa. Suface foces ae eated wth ant-covaant base vectos eated wth suface. Fue. 5. -Fom Suface Foce It s a fact that the suface foce s expaned by the ena apped foce. But, t cannot ue out the ndependent exstence of suface foce. Fo the ndependent exstence of suface foce, some papes ae avaabe. [9-5] 4

15 Hee, as the eo dffeenta toos ae used, the covaant base vectos ae used and the ant-covaant base vectos ae not used decty. Fo a statc unt eement, n defomaton mechancs, the pue -fom suface foce can be expessed as: = () Ths suface foce s actn on the suface (ht-hand chaty s seected). That means the foce ne s on the suface and taes the ht-hand chaty as postve decton. Fo the suface eometca meann of suface foce component =, efen Fue 5. (-Fom Suface Foce). I s cea that, fo the uppe-haf suface and owe-haf suface fomed cosed suface (tops sepaated by dx ), the bounday of both s the ede of pane dx dx. The foce actn on the ede aon the pane noma decton can be vewed as the actn on the pane (o a body foce dstbuted on the pane). Ths concuson can be nfeed fom Stoes theoem. Hee, a smpe eometca expanaton s fomuated. = suface, up-haf suface (-) suface =, ow-haf suface (-) Hence, fo the pane dx dx = dx dx, the body foce deved fom the cosed suface foce s: On the nteo suface noma decton ~ = ( ) (-) = e (ths opeaton s caused sta opeaton n eo dffeentas), the pane suface foce can be fomuated as: = ( e ) ~ = () whee, ~ = ( ), ~ = ( ), ~ = ( ) ae eement suface foce deved body foce components. It s cea that, the symmety stess = w cause zeo noma decton body foce ~. In eo dffeenta fomuaton, the symmety pats ae omtted. Howeve, fo defomaton mechancs, fo eneaty, the symmety pats ae eseved. Ths deved body foce s actn on the ntena eon of the unt matea eement. Heeafte, t s named as nteo body foce. Thouh above fomuaton, t s concuded that: the symmety stess can ve on a cosed suface ndependenty wthout poducn net nteo body foce on noma decton. Theefoe, the suface foces ae cassfed nto two types: one nd ( ) s actn on the eo sufaces dx ( ) dx ( ), that s they ae defned on the 6 sufaces of unt eement; 5

16 ( ) ( ) anothe nd s the net suface foce actn on the nteo suface (wth aea dx dx ) noma decton. In essenta sense, t s a body foce dstbuted on the suface. In cassca defomaton mechancs, ony the ate s defned. In tadtona eastcty, the stess s defned as the tota contact foce actn on the suface aea. The Equaton () shows that ony the ant-symmetca eo suface stess components can be chaned nto nteo body foce. It exposes that: the stess symmety denes the eo suface foce w poduce ntena body foce. It s the statc baance condton n tadtona mechancs theoy. The we-nown ntepetaton s that: f the stess s not symmetca, the eement w otate. ( ) In fact, n tadtona stess defnton, the pue body foce actn n unque decton (hee s taen as ) can be expessed as: ( ) = (no summaton) (-) One way s to defne: = α, then one has: ( ) ( ) = = ( α ) = (-) By ths way, the body foce s tansfomed nto (suface foce decomposton) cassca stess. ( ) Ths way s wdey accepted n cassca mechancs. The quantty ( α ) s defned as stess. In fact, ntepetn the ndex as the body foce decton and the ndex as the suface decton, the stess s defned. Ths stess expanaton (defnton) s wdey used n tboos. The above fomuaton aso shows that, by sutabe seectn the coodnato system, the cassca stess aways can be expessed as the pue noma foce fom: ( ) ( ) =. Ths = fact s we expaned n mechancs tboo. Reasonn fom above anayss, the vtua suface foce tadtona stess concept. = s ued out n 4. Body Foce Deved Suface Foce The tadtona stess defnton can be fomuated by the -fom fomuaton stcty. In cassca contnuum mechancs, the body foce s enay apped on the suface of matea eement, et = e (sta opeaton n eo dffeentas) be the suface noma decton, one has: = (4-) By eo dffeenta ues, one has: d = (4-) Whee, = (4-) 6

17 Hence, the body foce spata vaaton w poduce eo suface foces. In tenso eometca fomuaton, one has: = (5) Hence, n tadtona defomaton mechancs, fo cubc unt voume eement, the cassca stess tenso components ca ae defned as: cs ( ) ( = + = + ) (6) It shows that, the pue enay apped body foces vaaton can be decomposed as the aveae stess (nteo suface foce) on the nteo sufaces of matea eement and the dea -fom eo suface foce. Theefoe, ena body foce w poduce aveae stess concept actn on the suface of matea eement, whch w eque that the stess s symmety. Heeafte, the symmety pats of -fom eo suface stess w be named as cassca stess, the ant-symmetca pats w be named as otatona stess. 4.4 Intnsc Stess Defnton Based on the anayss about suface deved body foce and the body foce deved suface foce, t s shows that: ) ony the ant-symmetca eo suface foce w poduce nteo body foce, whe the symmety eo suface foce w not poduce nteo body foces; ) the pue eo body foces vaaton can be decomposed as the aveae suface foce actn on the suface of matea eement (named as cassca stess) and the dffeence suface foce actn on the suface of matea eement (named as otatona stess). Hence, the cassca stess has no effects on the nteo moton of matea eement. The nteo eon s sepaated out by physca featues. Ths esuts suppot the two eons sepaated by cosed suface adopted n ths eseach about matea eement. Afte sepaatn the suface foce deved nteo body foce fom the eo body foce and sepaatn the ena body deved suface eo stess fom the vtua suface foce, the unt matea eement foce system s descbed by thee sets: ) the nteo body foce, whch s actn on the ntena eon of the matea eement and tan the eometca cente of confuaton as ts cente; ) the eo body foce, whch s apped to the matea eement fom ts suoundns; ) the nteo suface foce, whch s nteactn wth the ntena eon; 4) the eo suface foce, whch s nteactn wth ena eon. Now, we ae eady to adopt the cassca stess concept, whch s defned as the body foce actn on unt suface aea on the suface noma decton. The ntnsc stess s the stess not poduced by the maco defomaton. Ths s to dstnush the defomaton stess puey caused by defomaton Intena Reon Intnsc Body Foce and Sef-spn Fo physca eaty, the nteo eon can exst a vtua body foce f nt, they w act on the ntena sde of the confuaton suface. The enea foce n the ntena eon w be: f = f nt (7) Hee, the symbo s used to show that: the two vectos have dffeent featues, they cannot be 7

18 added decty []. Note that, as =, athouh fo standad ectanua system (o nta confuaton) f =, then =. But, fo abta shape confuaton, the symmety may be boen. So, n enea sense,. Hee, ths pont s dffeent fom pue eo dffeentas. By pevous anayss (Equaton ()), on the matea eement nteo suface, the eo suface foce w poduce an ntena suface foce. If the nteo body foce components ae thouht as actn on the matea cente pont, then, as the thee ntena sufaces passn thouh the eement cente, thee pncpe ntnsc body foce components can be defned as: nt = ( ) + f nt (8-) nt = ( ) + f nt (8-) nt = ( ) + f nt (8-) Ths body foce s defned on the cente pont of matea eement. It shows that the eo suface foce deved nteo body foce can affect the moton of nteo eon cente pont. They fom a compete foce system fo cente moton descpton. Ths foce w mae the cente pont moton towad the foce vecto decton. Based on ths pont, the ntnsc defnton about maco confuaton statc contnuum s:. Fo maco confuaton statc contnuum, the condton s expessed as: nt = ( ) + f nt = (9-) ( ) + f nt = (9-) ( ) + f nt = (9-) Consden an atom mode, the f nt s Couomb eectca foce towads the cente pont (so, they ae neatve). To baance ths foce, a otatona stess must exst on the atom shes. If the ntena body foce s neatve (postve), the otatona stess s ht-hand chaty (eft-hand chaty). So, eneay, fo maco confuaton statc contnuum, the matea eement has sef-spn. As the eated detaed dscusson may be to boad, hee ths topc w not be dscussed futhe. Smpy, the above Equaton (9) can be vewed as the defnton of eement unt sef-spn Extena Reon and Intnsc Stess The eo eon can exst a vtua body foce f, they w act on the eo eon of the confuaton suface. The enea foce n the ena eon w be: f = f (4) 8

19 The eo foce defned ntnsc stess components ae: ca =, ca =, ca = x x x (4-) cs ( ) ( = ), (4-) Fo maco confuaton statc contnuum, a cassca ntnsc stess components ae zeo. So, the maco statc condtons ae: ( ) =, eo eon (4-) and the statc eo suface s defned by equaton: ( One smpe souton s: + ) + ( + ) =,, on eo suface (4-) f = Const, = (on suface) (4) That s: ena body foce s constant fo eveywhee, hence w not poduce cassca stess (so no maco defomaton). At the same tme, the eo suface ony has pue otatona stess. Hence, the matea eement can have sef-spn. Refen Fue 6. (Smpe Statc Contnuum Defnton). Compan wth the eod suface defned on the Eath, one may et bette undestandn about ths concept. Fue 6. Smpe Statc Contnuum Defnton 4.4. Exteo Suface Confuaton and Intnsc Vscous Stess In enea cases, fo a ven eo suface foce, the statc ntnsc vscous 9

20 stess on suface can be defned as: vs η ( ) ( = + = + ), (44) The non zeo vscous stess w cause the toson of the eo suface of matea eement wthout poducn cente ponts moton. Ge s a typca matea n ths statc state, whe the ntefaca mco fows do exst. The equaton aso shows that, f the suface ntnsc vscous stess s zeo, the eo body vs foce must be a vecto potenta fed (e manetc fed). That s: f η =, then + = (45) As n ths case =, the eo suface foce s puey otatona Maco Statc Contnuum Defnton It s we nown that the maco statc contnuum has mco dynamc moton. Based on above fomuatons, the maco statc contnuum s defned as: ( ) + e f nt =, nteo eon (46-) ( ) =, eo eon (46-) ( + ) + ( + ) =,, on suface (46-) Hence, fo ven body foces (ntena and ena) and suface foces, the suface confuaton s defned by equaton: = e f nt ( + ), (47) Fo a matea eement, a foce components ae the functon of poston efen to the eement cente. So, ts souton ves out a cosed suface. In theoetc sense, mutpe soutons may exst. In ths case, the eement can has mut-possbe confuatons. In some cases, ths phenomenon can be expaned as the ntena mut-scae featues of mateas. It means that, fo statc state, the cosed suface of matea eement has fxed confuaton. The suface foce on the cosed suface s ntnsc. As a maco mechancs theoy, the matea eement s vewed as a pue suface confuaton as a whoe. Geometcay, the nteo body foce and eo body foce w foce the matea eement poduce sef-otaton about ts cente. It s named as sef-spn o spn of matea eement. The matea eement suface has sheddn functon thouh chaty seecton. Ths s vey mpotant to undestand the compcated featues of modem mateas, especay atfca mateas. On physca sense, the body foce s actn on eveywhee and s eated wth the mass cente moton, hence, the cassca mechancs ony consde the body foce eated wth mass cente moton. Its success on dspacement concept s based on ths ntnsc featue. It s cea, when =, the matea eement has no chaty. Reasonn fom ths pont, the cassca mechancs ony consdes the eectonc statc potenta fed. The attce dynamcs taes ths as ts basc featue.

21 5. Consttutve Equaton Fomuaton on Physca Bases In eectomanetc fed, the eectc fed s -fom vecto and the manetc fed s -fom vecto. Athouh they ae dffeent, the can act on the matea eement at the same tme. In fact, n physcs theoy, t s we accepted that an abta foce can be decomposed as the addtve of two pats: one s cu ess (that s -fom hee), anothe one s dveent ess (that s the -fom). Futhe moe, the dea eement s the epesentatve of aveae matea featues n contnuum. Geneay spean, the defomaton (o moe exacty, the ncementa defomaton) s many poduced by the eo foces actn on the matea eement as a whoe. So, physcay, the stess shoud be defned thouh the enea eo foce vaaton. Ths s the man tas fo ths subsecton. Summn up above esuts, fo a matea unt, the enea foce s composed by ona (vtua) body foce and ona (vtua) suface foce n nta confuaton: f = f (48) Afte defomaton, n cuent confuaton, f the ntnsc featues of mateas have no vaaton (as t s supposed n nfntesma defomaton), the foce s: By ths way, the defomaton f = f f = f (49) = F w poduce the defomaton foce concept defned as: F δ ( F F δ δ ) (5) ( ) Hence, efen to the nta confuaton, wthout ntnsc vaaton of matea featues, (that s fo nvaant f and ), the foce poduced by defomaton s expessed as: ~ f = f ( F δ ), spn-e tem (5-) ~ = ( F F δ δ ),,, shea tem (5-) They ve out the ntnsc defnton of defomaton foce feds on physca bases. Hee, the up-ndex s used fo ntnsc foce component to show the nta confuaton s taen as the efeence. Fo eo eon, efen the ntnsc stess defnton Equaton (4), the eo defomaton stess components ae defned as: = ( F ), = ( F ), = ( F ) (5-) f f f ( f ) ( ~ ~ = f F + + ), (5-) The above equaton s the consttutve equaton fo enea mateas n defomaton mechancs. (Heeafte, the powe ndex w be dopt out). If =, then ~ = ~, the eo suface defomaton contbuton s zeo. The stess components n eo eon ae puey poduced by ena body foce. The eement

22 sef-spn has no contbuton to defomaton stess. Fo nteo eon, the nteo defomaton stess pncpe components ae defned as: ~ ~ ~ mn nt = e ( ) = e ( Fm Fn Fm Fn ) (5) mn nm mn If = = δ, the suface defomaton contbuton s zeo. Ths stess w be sensed by the matea cente pont. If t s not zeo, the cente pont w move ts poston. As a esut, the suface confuaton o eo suface stess w be chaned coespondny. Ths topc s too boad to be ncuded n ths pape. As the eo defomaton stess s taen as a conventon n cassca stess, the foown fomuaton s named as the Cassca Fomuaton of Stess on Physca Bases. 5. Isotopc Smpe Sod Matea In cassca nfntesma defomaton theoy, the cassca stan s defned as: ε = ( F + F ) δ (54) Omttn hhe ode nfntesmas, one has: ( ε (55) δ F F δ δ ) Fo sotopc smpe sod mateas, defned as f = =,, = f = f f, = (efen Fue 7. Isotopc Smpe Sod Mateas), the defomaton foce fed s: ~ f = f F f = f ( ε + ε + ε ) (56-) ~ f = f F f = f ( ε + ε + ε ) (56-) ~ f = f F f = f ( ε + ε + ε ) (56-) ~ = ( F F δ δ ) ε, (56-4) Hence, the eo defomaton stess tenso s defned as: = f ε, = f ε, = f ε (57-) =, (57-) f ε ε As t s we-nown, fo sotopc smpe mateas, the cassca stess-stan equaton (consttutve equaton) s: ( = = + (58) ) λ( ε ) δ µε By compan the shea components, t s cea that: As the fst nvaant of defomaton stess s: µ = ( f ). (59) + + = f = ( λ + µ ), = ε + ε + ε (6) So, we have: f λ + µ = λ + = f. Then, we have:

23 λ =. (6) Howeve, fom above fomuaton, t s found that, the foown eometca condton s mped: F = F (6) Ths bued eometca condton n cassca consttutve equatons s named as defomaton confoma condton. Many eseaches have been done on ths topc n dffeent fomuaton foms. Fue 7. Isotopc Smpe Sod Mateas Summen above anayss, the sotopc smpe matea s defned by ntnsc fed: f = =, (6-) = f = f f, = The Lame constants ( λ, µ ) ae detemned by the ntnsc fed as: λ =, µ = ( f ) (6-) Physcay, the sotopc smpe matea has sotopc body foce and sotopc eo ntnsc stess both tae the matea eement eometca cente as the stat pont of oentaton decton. By Equaton (44), the sotopc smpe matea has an ntnsc vscous stess: vs η = ( + ) = (64) Hence, the Lame constant λ s adhesve foce on matea eement. Ths ntepetaton s we nown and expaned n t boos. Ths smpe case shows that the stess fomuaton n ths eseach expans the matea

24 featues n ntnsc sense. As two speca cases: ) Incompessbe Fuds: fo λ =, that s when the eo suface adhesve (vscous) stess s nea zeo, the smpe sotopc contnuum s dea fud. Snce on the matea eement suface the eo ntnsc stess s sotopc, the mco fud eement can be thouht as a sphee ba. The zeo eo suface foce means that any cosed suface can be teated as the matea eement suface, theefoe, the fud has no fxed confuaton. By f µ, the fud vscous stess s poduced by the eo body foce. ) Dscete Medum: fo f, = µ, the dscete medum s defned. Fo ths nd of medum, =, means that matea eements ony have contact eaton such as sands (puey body foce system, such as the dea as n cassca themodynamcs). 5. Poa Smpe Fud Matea In cassca nfntesma defomaton theoy, the cassca Stoes asymmetca stan s defned as: ω = ( F F ), ε =, (65) Fo poa smpe fud mateas, defned as: = f = f, = = ~, ~ f = f. (Fo ts statc defnton, efen Fue 7, but the sotopc stess n eo suface s epaced by ht-hand chaty otatona stess). Unde ths condton, fo defomaton = F eo suface vaaton s: ~ = ( F F δ δ ) ω,, the (66) The eo defomaton stess fed s: = f ε, = f ε, = f ε (67-) ~ = ω f, (67-) When the aveae sotopc stess The consttutve equaton s fomuated as: ~ p (fud pessue) s defned as: p = ( f ), = ε + ε + ε (68) = + ~ (69) pδ µ ω Whee, ~ µ = f and the ndex s oweed to foow conventons. Ths equaton s vey common n fud dynamcs eseach. Compan wth Equaton (58) =, a moe enea fom of consttutve λ( ε ) δ + µε equaton fo smpe mateas s: ( = ) = λ( ε ) δ + µε + ~ µ ω (7) 4

25 ~ Whee, λ =, µ = ( f ), ~ µ = f. In ths case, the sod-featued shea and fud-featued shea ae dffeent. The physca bacounds ae dffeent. Fo eastc fud concept, ths equaton s appcabe. Ths equaton expans a on-standn queston n my ban: why Stoes asymmetca stan s dscaded n sod defomaton mechancs theoy, whe n fud dynamcs the Stoes asymmetca stan pays the man oe. (In cassca sod mechancs, µ = ~ µ s used). 5. Lnea Consttutve Equaton n Genea Fom Stcty spean, the cassca stan defnton omts the asymmetca components of defomaton tenso. Then, the defomaton stess tenso s symmetca. Ths shotae s we exposed n many eseaches. Hee, a moe exact nea fomuaton w be pesent. In fact, the actua way s to defne: ( F F δ δ ) = ( F δ )( F + δ ) + ( F + δ )( F δ ) (7) Hence, δ δ ) = ( F δ ) + ( F ~ = ( F )( )( F + δ F + δ F δ ) ( F δ ) + ( F δ ) (7) As, ( F = Intoducn paametes: mn δ ) + ( F mn = ( δ nδ mδ + ~ = C ( F δ ) ( F δ ) δ ) δ δ δ + n m nm n m nm ( F δ δ δ )( F δ ) δ δ δ δ ) n m (7) ~ C n m n m mn nm = ( δ δ δ + δ δ δ ) (74) Then, ntechane the ndex,, the eo suface foce vaaton s: ~ ~ = C ( F δ ), (75) ~ n m n m mn nm Whee, C = ( δ δ δ + δ δ δ ). Defnn the matea featues paamete as: C ~ ~ mn nm = ( C ) ( )( + C = + δ n δ mδ + δ nδ mδ ) (76) It shows that ony symmetca pats of eo suface have contbuton to defomaton stess. The eo defomaton stess feds ae: = f ( F ), = f ( F ), = f ( F ) (77-) ( ) = f F + C ( F δ ), (77-) Ths s the eneazed nea consttutve equaton. The wea non-nea matea paametes can be defned as: 5

26 C mn nm ( ε ) = [ ( Fn + δ n ) δ δ m + ( Fn + δ n ) δ mδ ] (78) In ths case, the matea paametes have a wea dependence on defomaton tenso. The wea nea dependence can be expessed as the nea functon of stan: C mn m nm m ε ) = C + [ δ δ ε n + δ δ ε ] (79) ( n The wea nea dependence eatons ae wdey used n enneen mechancs fo dffeent easonn and fomuaton. Hee, the fomuaton can be vewed as a enea fom. 6. Stess Fomuaton fo Chen S+R Addtve Fom Defomaton Howeve, the soated consdeaton of stan defnton and defomaton stess defnton ae not the best way. Hee, a new way s expaned. Fom eometca consdeaton, fo enea defomaton = F eomety, the defomaton tenso can be decomposed as the Chen Fom-:, based on defomaton F = S + R (Θ) (8) Whee, S s symmetca tenso, R (Θ) s an unt othoona otaton tenso wth otaton ane Θ. O Chen Fom-: ~ ~ F = S + R ( θ ) (8) Whee, S ~ ~ s symmetca tenso, R ( θ ) s an unt othoona otaton tenso wth otaton ane θ. The defomaton stess tensos w be dscussed beow, espectvey. 6. Stess fo Chen Fom- Defomaton Fo Chen Fom-defomaton, F = S + R. tenso s: Case : If thee s no oca otaton, the defomaton s: F = S + δ, the defomaton stess, = f S, = f S (8-) = f S ( ) ( ( f + f ) S + = ) C S, (8-) Case : If thee s no ntnsc stetchn, the defomaton s: F = R, as the eo suface vaaton s: ~ = ( F F δ δ ) = (ths s because that the othoona otaton w not chane the symmety featue of, whe the ant-symmetca pats have no contbuton to defomaton stess), and the unt othoona otaton tenso can be expessed as: R = δ + sn Θ L + ( cos Θ) L L = δ + sn Θ e L + ( cos Θ)( L L δ ) (8) 6

27 whee L s the components of unt otaton axe decton vecto), the defomaton stess tenso s: = f cos Θ)( L L ), (84-) ( = f cos Θ)( L L ), (84-) ( = f cos Θ)( L L ) (84-4) ( ( ) = f ( R δ ), (84-5) Any Chen Fom- defomaton can be vewed as the stac of the two defomatons. Hence, based on the eneazed nea consttutve equaton, stess fo Chen Fom- Defomaton s: = f S + ( cos Θ)( L L )] (85-) [ = f S + ( cos Θ)( L L )] (85-) [ = f S + ( cos Θ)( L L )] (85-) [ ( ) ( ) ( ( f + f ) S + f ) ( R δ + = ) C S, (85-4) Ths consttutve equaton can be used fo compcated mateas. Hee, t s the fst tme to ve ts ntnsc fom. 6. Stess fo Chen Fom- Defomaton ~ ~ Fo Chen Fom- defomaton, F = S + R ( θ ), the othoona otaton s: ~ snθ ~ ~ ~ R = δ + L + ( )( L L + δ ) (86) snθ ~ ~ ~ = δ + e L + ( ) L L ~ Fo pue otatona defomaton, F = R ( θ ), the suface foces vaatons ae: ~ = ( ) (87) cos θ Hence, fo pue otatona defomaton, the stess fed s: ~ ~ = f ( ) L L (88-) ~ ~ = f ( ) L L (88-) ~ ~ = f ( ) LL (88-) ( ) = f ( R δ ) + ( )( + ), (88-4) cos θ By Equaton (44), the tem ( )( + ) s the adhesve (vscous) stess on cos θ eo suface of matea eement. Hence, ths Chen Fom- defomaton s many eated wth 7

28 tubuence o fatue-cacn. Foown the sma way as the ast sub-secton, based on the eneazed nea consttutve equaton, stess fo Chen Fom- Defomaton s: ~ ~ ~ = f [ S + ( ) L L ] (88-) ~ ~ ~ = f [ S + ( ) L L ] (88-) ~ ~ ~ = f [ S + ( ) LL ] (88-) ( ) ( ) ~ ( ) ~ ~ = ( f + f ) S + f ( R δ ) + ( )( + ) + C S, (88-4) cos θ Ths consttutve equaton s dffeent fom Equaton (85). Fo the same defomaton, t seems to be that two dffeent decompostons w poduce dffeent fom of consttutve equaton. Ths concept s won. In fact, both equatons ae deved fom the enea nea consttutve equatons. But, these two consttutve equatons do cea the dffeent contbuton of defeent defomaton mechansms. Ths s vey mpotant when the ea enneen mateas ae concened. By Equaton (44), the ast Equaton (88-4) can be wtten as: ( ) ( ) ~ ( ) ~ vs ~ = ( f + f ) S + f ( R δ ) ( ) η + C S, (89) cos θ Summn up above esuts, the consttutve equatons based on Chen fom decomposton of defomaton tenso can be used to expess the stess caused by oca otaton n a much smpe way. 7. Matea Paamete Invaance Featue Obsevn the statc condton Equaton (46) (fo easy eadn, they ae ewtten at beow) fo a maco statc contnuum, t s found that the eo ona suface s detemned by the nteo eon body foce and the eo eon body foces. ( ) + e f nt =, nteo eon (46-) ( ) =, eo eon (46-) ( + ) + ( + ) =,, on suface (46-) Obsevn the enea stess defnton Equaton (5), (fo easy eadn, they ae ewtten at beow), t s found that when the defomaton s emoved ( F the stess becomes to zeo. ~ δ ) afte an abta defomaton, = ( F ), = ( F ), = ( F ) (5-) f Whee, = ( F F δ δ ),,. f f ( f ) ( ~ ~ = f F + + ), (5-) Howeve, by the stess defnton, afte the defomaton pocess, the zeo stess ony ves out 8

29 vs a concuson: the vscous stess ~ η ( ~ = + ) s etuned to ts ona vaue. Hence, possbe vaatons on ( ) can be poduced dun the defomaton pocess. Ths tem w affect the ntena eon of matea eement. If the nteo eon body foce s chaned, n enea sense, the matea has chaned (moe o ess). Ths means the evouton pocess of mateas. To descbe the matea evouton pocess, the nteo eon s studed beow smpy. The nteo defomaton stess pncpe components ae defned by Equaton (5) (fo easy eadn, they ae ewtten at beow) as: ~ ~ ~ mn nt = e ( ) = e ( Fm Fn Fm Fn ) (5) Defnn the -fom of t as: ~ ~ nt = (9) Then, one has the -fom defned by the suface foce vaaton (dveence): ~ ~ d nt = e (9) 6 On physca sense, the -fom s the eney passn thouh the eo suface of matea eement. Theefoe, an ntena eney tem U = ω can be ntoduced. In nta statc state, defned by maco statc contnuum, the nta statc ntena eney s defned by: U = ω (9) Whee, by defnton of maco statc contnuum, t s not zeo: nt nt nt ω = + + (9) Afte defomaton, the cuent ntena eney s defned by: U = ω (94) Whee, the S + S + S. Theefoe, the ntena eney vaaton s: d ω = ( ) ω ( S + S + S ) ω (95) On the othe hand, n defomed state, the nteo eon new statc condton s: ~ ( ~ e ) + f nt =, nteo eon (96) ~ Whee, f = f f ). Hence, by Stoes equaton, one has: nt ( nt nt 9

30 ~ ~ ~ ~ ( ) ( ) ( ~ ~ ) dω = [ + + ] (97) Ths ntena eney vaaton s not ecoveabe by emove defomatons. In cassca eastc-pastc mechancs theoy, ths tem s named as pastc eney. Based on above fomuaton, the matea evouton equaton can be defned as: t ( ~ ~ ) ( ~ ~ ) ( ~ ~ ) t dω = [ + + ] dt = ( S + S + S ) ωdt (98) Whee, the tme paamete shows that the ntena eney vaaton s detemned by defomaton hstoy. In atona mechancs theoy, ths s named as memoy effects. By ths fomuaton, the eo suface foce evouton equaton s obtaned as beow: ( t) ( t) t = S dt ω (99-) ( t) ( t) t = S dt ( t) ( t) t = S dt (99-) (99-) Whee, ( t) = ( t) (). Futhe moe, by the ecovey pats, afte emove the defomaton, one has: ( t) + ( t) = () + () () The fna eo suface foce evouton equaton s obtaned as: t ( t ) = () + e ( S ωdt) dx, no summaton () unt It ony suppes the ant-symmetca pats. Hence, the matea paametes ae nvaant. mn nm C = ( + )( δ n δ mδ + δ nδ mδ ) () That means that, the consttutve equatons ae nvaant. It s cea that the pastcty of defomaton s we expaned by ths fact. 8. Concuson In ths eseach, the physca fed on maco statc contnuum s estabshed fsty wth the eo dffeenta tos, then the defomaton s ntoduced to et the vaaton of the physca fed. Statn fom ths teatment, the stess concept s studed by the physca vaaton. The pape ves out the eated stess defnton exacty. As a oc consequence, the enea consttutve equatons ae obtaned, whee the nta physca feds ae the matea paametes. The advantaes of such a nd of consttutve equaton ae that: t s decty usabe fo detaed anayss of vaous mateas based on physca undestandn about the mateas [6-4]. Futhe, the consttutve equatons fo Chen Fom- and Chen Fom- defomatons ae obtaned. By ths way, the Chen atona mechancs theoy s compcated as a unted system. It s sue thee ae many pobems ae watn to be studed futhe, that w be expessed by othe papes n nea futues.