5.2 Plasticity I: Hypoelastic-Plastic Models

Transcription

1 5. Platicity Hyolatic-Platic Modl h two main ty o claical laticity modl or larg train ar th hyolaticlatic modl and th hyrlatic-latic modl. h irt o th i dicud in thi ction. 5.. Hyolaticity n th hyolatic-latic modl o laticity th latic ron i aumd to b hyolatic. A dicud in 4..4 hyolatic matrial ar charactrid by contitutiv rlation o th orm & ( d) (5..) and in mutiaxial roblm thi imli that th ron cannot b xrd in trm o an latic train nrgy unction. hu th ron i ath-dndnt and diiation may occur vn though th matrial i uod to b latic. Howvr th ida with th hyolatic-latic modl i that th latic train ar aumd to b rlativly mall o that any rror in th conrvation o nrgy i vry mall and o can b nglctd. h tr-rat & in th hyolatic quation 5.. mut b objctiv. hu th matrial drivativ o th auchy tr can not b ud but any o th many objctiv rat or xaml th aumann Grn-Naghdi rudll tc. can b ud. A larg cla o hyolatic matrial i ncomad in th linar rlation btwn objctiv tr-rat and th rat o dormation d (5..) whr i an objctiv tr rat and i th corronding ourth ordr tnor o latic moduli which may itl dnd on th tr in which ca it mut b an objctiv unction o th tr. For a givn initly dormd tat th (mall) incrt in tr and train ar linarly rlatd and ar rcovrd uon unloading. Howvr or init dormation th work don in a clod ath may not b zro. h latic modulu tnor i alo calld th tangnt modulu. t o th minor ymmtri du to th ymmtry o d and. t i uually aumd to o alo th major ymmtri. Examl onidr th ollowing hyolatic contitutiv quation d (5..) whr i th aumann tr-rat Eqn..5.0 & w w. h rudll trrat dind by Eqn..5. & l l tr( d) i thn 60

2 w w l l d d tr ( ˆ ) d ( d) tr ( d) (5..4) whr ˆ d d d. hu 5.. can b xrd a d tangnt modulu givn by ˆ. with th rudll t i intrting to not that i i contant thn i not. Furthr i ha th major ymmtri thn do not (inc do not). For thi raon on otn u th Kirchho tr rathr than th auchy tr. For xaml th aumann rat o th Kirchho tr i & w w. hn with & tr( d) ( d d ) ( ˆ ) d d (5..5) Now i ha th major ymmtri thn o do. 5.. Hyolatic Platic Modl Additiv Dcomoition o th Rat o dormation h rat o dormation i now dcomod additivly into latic and latic art according to d d d (5..6) h latic ron i thn h yild condition i ( d d ) (5..7) ( ) 0 (5..8) whr rrnt any othr variabl() bid uon which dnd. h low rul i ( ) d & λ G (5..9) 6

3 whr g i th latic otntial; λ i th latic multilir. For aociativ low-rul g. h variabl ar aumd to ollow an volution law o th orm h tnor G i uually xrd in th orm G g ( ) / ( ) & & λ A (5..0) h loading and unloading condition may b xrd a & λ 0 0 & λ 0 (5..) h ar known a th Kuhn-uckr condition. h irt o th tat that th latic multilir rat i non-ngativ; or latic loading & λ > 0 othrwi & λ 0. h cond tat that th tr mut li on or inid th yild urac. h lat condition aur that th tr tat rmain on th yild urac during latic low. hi lat condition can alo b xrd in rat orm & 0. hi i known a th conitncy condition & & & 0 (5..) otroic Matrial From th objctivity rquirt th yild unction ( ) in Eqn mut b an objctiv calar unction o. hi imli that mut b a unction only o th invariant o. hu th orm ncarily rrnt th yild unction o an iotroic matrial. For aniotroic matrial on mut u a dirnt tr maur or S. xaml th yild unction could b xrd in trm o th PK tr 0 onidr thn an iotroic matrial with ( ). hn it can b hown that { Problm } whr i ar th invariant o (5..) that i th tnor and / ar coaxial. onidr now a ormulation in trm o th aumann tr-rat. From 5.. it ollow that { Problm } & (5..4) h conitncy condition 5.. can now b xrd in trm o th aumann tr rat 6

4 & & 0 (5..5) h latic multilir can now b olvd or uing th hyolatic rlation 5..7 th low rul 5..9 th volution quation 5..0 and 5..5 { Problm } d & λ (5..6) A G Subtituting thi xrion into th low rul 5..9 and uing th hyolatic rlation 5..7 thn lad to { Problm 4} d ( G) A G (5..7) h tnor i th lato-latic tangnt modulu. t conit o an latic comonnt and a comonnt which rult rom latic low. Whn th low i aociativ o that G / th lato-latic modulu o th major ymmtri. h laticity quation can alo b bad uon othr tr-rat; or xaml th rudll tr-rat. For th am raon a dicud arlir although ha th major ymmtri or aociativ low rul th rudll modulu will not. howvr th quation ar ormulatd in trm o th Kirchho tr thn th rudll modulu will o th major ymmtri (in act all th rlvant rlation abov ar valid or th Kirchho tr with imly bing rlacd by ). Not that i th latic train ar mall and latic dormation ar iochoric (volum rrving) thn and. Small Strain n th ca o mall train in th abov rlation on rlac d with ε& and dcomo th mall train rat according to ε & ε& ε&. h tr rat i th tim drivativ o th auchy tr inc objctiv tr-rat ar not now a conidration. h hyolatic rlation i & ( ε& ε& ) and th rmaining rlation ollow or xaml th latolatic tangnt modulu i givn by 5..7b with rlacd with th mall-train latic modulu tnor. h mall-train ormulation i valid or aniotroic latic moduli. 5.. Flow hory n th Flow hory on aum th matrial i iotroic latic low i indndnt o th hydrotatic rur latic low i incomribl and th yild urac ud i th 6

5 Von Mi yild urac. An ctiv tr i ud to gnrali th uniaxial bhaviour to multiaxial tr tat. Firt to b xamind i th iotroic hardning modl. otroic Hardning t i uul to xr th quation now in trm o th Kirchho tr and th aumann tr-rat. n that ca on again ha Eqn d d d with th hyolatic Eqn now rading h Von Mi yild unction can b xrd a ( d d ) ( ) Y Y 0 (5..8) (5..9) whr i th cond invariant o th dviatoric Kirchho tr and trm act a an ctiv tr ˆ. dv. h h (aociativ) low rul 5..9 can b xrd a ( th Andix to thi ction or dtail o th dirntiation involvd hr) d & λ G G (5..0) h only variabl ncary in th modl i th calar accumulatd ctiv latic train ˆ ˆ ˆ ˆ ˆ ε ε dε & ε dt dε dε dε (5..) h rood volution law or i imly (Eqn with th unction A ) ( & ˆ ε ) With th dinition o th latic modulu bing & & λ (5..) dy ( ˆ ε ) ε th conitncy condition 5.. i rom 5..9 H ˆ (5..) d ˆ ε & ( ˆ ε ) ˆ& ε 0 & H (5..4) 64

6 65 h latic multilir rat i now givn by Eqn d H ˆ ε λ & (5..5) Finally th lato-latic tangnt modulu Eqn i d H ˆ ε (5..6) t i uul to dcomo th ron into volumtric and dviatoric comonnt. Firt xr th latic moduli a in Eqn κ (5..7) whr n m j i jn im. Not thn that { Problm 5} (5..8) o that th tangnt modulu 5..6b can b xrd a H κ / 4 / (5..9) Kinmatic Hardning o accommodat kinmatic hardning on introduc a anothr hardning aramtr th back tr. h yild condition 5..8 i now (in trm o th Kirchho tr) 0 ˆ ˆ Y ε ε (5..0) h low rul i G G d λ & (5..) h volution quation or th back-tr i or xaml uing a linar hardning rul and th aumann tr-rat

7 cd ( c & λ G) (5..) Not that th u o th aumann rat in th back-tr volution law can lad to hyically unraonabl tr ocillation in iml har or larg dormation (Nagtgaal and Dong 98). Howvr thi bhaviour i not igniicant rovidd th latic train ar not too larg. A ormulation bad uon th Grn-Naghdi train can liminat uch tr ocillation (ohnon and Bammann 984) Druckr Pragr Modl h yild critrion or a Druckr-Pragr matrial i a modiication o th Von Mi unction o a to incororat a latic ron du to a hydrotatic rur ( ) Y 0 (5..) whr tr. h aociatd low rul i thn d & λ G G (5..4) whr dv. A uitabl non-aociativ low-rul might b g d & λ G G g (5..5) 5..5 Elatic Moduli and Objctivity t wa tiond abov that objctivity rquir that a yild unction o th orm ncarily rrnt an iotroic matrial. Objctivity alo lac rtriction on th latic moduli. Firt conidr a contant modulu tnor. Objctivity o a contitutiv quation rquir that * ( d ) * (5..6) or Q Q ( Qd ) Q (5..7) o uing th indx notation ( ) [ Q Q Q Q ( ) ]( ( d ) ) i qj rk l qr kl (5..8) n ordr that objctivity b atiid on mut hav 66

8 ( ) Q iqqjqrkql ( ) kl qr (5..9) which how that i th modulu i contant it mut b iotroic. on rquir an aniotroic latic modulu on mut u a ormulation bad on a coniguration othr than th atial coniguration a dicud in th nxt ub-ction orotational Str Formulation h modl dicud thu ar hav two drawback. th yild unction mut b an iotroic unction o th tr. i th latic moduli ar contant thn th moduli mut b iotroic o ovrcom th rtriction to iotroic matrial on can ormulat a laticity modl in trm o or xaml th corotational tr. h corotational tr i dind by Eqn..5. R R (5..40) ing a ormulation bad on th Kirchho tr th corotational Kirchho tr i R R (5..4) Rcall rom.5. that th PK tr S i th ull-back o th Kirchho tr # S χ * F F ; th corotational tr i th ull-back o but with rct # F R to R and not F R R χ. * Din now th corotational rat o dormation with th dcomoition d R dr (5..4) d d d (5..4) Not that th corotational tr and rat o dormation ar innitiv to rigid body rotation Q to th currnt coniguration * * ( ) ( R R) ( QR) ( QQ )( QR) * * ( d ) ( R dr) ( QR) ( QdQ )( QR) d (5..44) h corotational tr-rat i 67

9 & R R R& R R R & R R& (5..45) A rigid body rotation Q to th currnt coniguration alo rult in { Problm 6} * ( ) & & (5..46) With th corotational tr-rat objctiv th latic ron givn by & d (5..47) Not that rigid body rotation to th currnt coniguration lav & and d unchangd and o rmain unchangd alo. hu unlik th iotroic rtriction 5..9 thr i no objctivity rtriction hr on th orm o and o can rrnt in gnral an aniotroic ron. Anothr way o looking at thi i a ollow or an aniotroic matrial and uing a ixd coordinat ytm th comonnt o an latic modulu tnor will in gnral chang a th matrial rotat. With th corotational ormulation howvr th ax rotat with th matrial and o matrial rotation ha no ct on. h rmaindr o th ormulation ollow a bor or xaml th yild condition would b ( ) 0 and th low rul d & λ G( ). h volution law or th variabl() i & & λ A( ). Again th calar yild unction may now rrnt in gnral aniotroic matrial bhaviour. h loading and unloading condition ar again givn by 5... h latic multilir rat i now ( 5..6) and d & λ (5..48) A G d ( G) A G (5..49) 5..7 Problm. h drivativ o th invariant o th tr tnor ar ( Eqn...) 68

10 th rlation to how that or th yild unction ( ) and / ar coaxial.. th act that and / ar coaxial and Eqn..9.h ( B) ( B A) ( A ) B th tnor A to how that & whr & w w.. Driv Eqn th indx notation to vriy that ( A d) G ( G) ( A ) d whr A d G ar cond ordr tnor and i a ourth-ordr tnor. Hnc driv th lato-latic modulu 5..7b. 5. th rlation.9.6 and.9.64 to how that or an arbitrary tnor A A κ( tra) dva A A κ( tra) dva dva with givn by Hnc uing.9. how that or a dviatoric tnor A Finally u 5..0 to driv Eqn A A A A A A 6. h alication o a rigid body rotation Q to th currnt coniguration rult in a chang to th corotational tr rat * * * * * * * * * * & R& R R & R R R& QR QQ QR QR QQ QR QR QQ QR th rlation Q Q Q& Q Q Q& 0 * & to how that ( ) R R & Andix to 5. Dirntiation o th Von Mi Yild Function h Von Mi yild critrion i k 0 whr. ing th roduct rul o dirntiation in indx notation 69

11 70 q q q q mm nj mi q q kj ki nj mi q q kk q q q q q q (5..50) n tnor notation tr tr (5..5)