Exposicion de la teoria de la plasticidas

8/17/2019 Exposicion de la teoria de la plasticidas

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Chapter 3

ELEMENTS OF

THE THEORY OF PLASTICITY

3-1. lntroduction

The theory of plasticity d


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AI

I

It1

I

IIr/

.I,I

/ec. 3-01 lements of the Theory of 2lasticity 55

wil .1 be considered iu 2art #our. H is very difficult. to describe% in arigorons aualyticnl '4.5% t he behavior of a metn.l m1der these conditions.Therefore% rertnin simplifying nssumptious are wmally ncccssary to obtain

a t.rnetahle rnat hemat ical solut.ion.6nother aspcct of pla"ir7ity is (8oncerned with ac*uiriog a hetter under st"1ndi11g of t hr nwd1"1nism of tlw pla"%,ti


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1 /ee hap. for t\ more 17omplctc 1lisousson of 1.hc matl%cmati


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/ec. 3-3L lements of the Theory of 2lasticity M

al ( bl (e )

#ig. 3-0. 71uealiIed flow curves. (a? nigid irlea.l plast.ic material, (b? ideal pl"u,ticmatcrinl with cln. lic rc%"tion, (e? picccwisc ! iuear (strain-h"m>ening? matrrial.

thc cla1%tc ami plast.ic rcgiom% (#ig. 3-0c?. This type of curve results insomewhat more com plicated mat.hematics.

3-3. True /train

*uat.io11 (1-1? describes t hc conventionul corn7ept. of unit. li11carstrain% namcly% t.1'1@ chaugc in lcngth rcfcrrcd to thc original unit lc11gth.

e =M, =1tL d L

Co Cu L$

This deti11ition of strain is satisfartor y for clast.ic strains whcre L is verysmnll. However% in plastic deformat.ion t.he strains n%r c frc*ucntly large%and during thc c$tcnsion thc gnge +cngth changc#. considcmbly. Cudwik 1

first proposcd t he defiuit.ion of trnc 1-1t rnin% or nat ural sCrain% % which obviates this difficulty. In this dofinitio11 of ;trnin thc chnugc in lcngth isreferred to the i nst ant.ancons gflgc lcngt.h% ralher than to t.he origi&+

g"1ge length.C1- Do N L2 - C1 N La - Ca N

E =

Í.¡ Lo L C0

t r, di, L

or E=

- =In ,., I, Lo

The relationshi p betwccn true strain and conve11tio1rnl linear strain

follows from *. ( l-1?.e = /j,?. L - C o = !::. 1

.Co l%o Co

e ! ! l.,Co

" = +n Co = ln (e # 1) (3-i?

t 2. Cutlwik% Olcmcu tc dcr t.cchnologischcn "+cchanik%O /pringer-Perlag4HE. :erlin% 1>?@Q.

L


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58 &echanical #undamentals Bhap. 3

Thc two measurcments of st.rain givc 11ca17ly ident.cal results np tostrains of about @.1.

:eca.use the volume rerrw.ns essentially constant duriug plastic dcformnt.ion% *. ("3-"3? can be writt.en in tel'ms of eit%hcr length or "irea.

L 6o" = l n -- = l n

-

Co (3-?

6lso% bccause of coustuncy of volume% thc summalion of thc three princi pal strains is c*ual to Iero.

(K-A?

This relationship is not valiUpcctcd tlmt yielding undcr ""t situat.o11 of com bned stresses isrelated lo sorne p"u't.icular combiuation of t.he princi pal str csses. 6yield critcrion can be c$prcsscd iu t%he general form

#$%,%2,%&'(,) , 8 8 8? 4% but wrc is at prmmn t no thcorctica.! way of calc11la.t%ing the relat.ionship between the stress componcnts to correlatc yielding in a three-dinwnsiona.! statc of stress with yielding in the uniu$ialtensio11 test. The yielding crit.eria are thercfore essentally cmpircalrelationships. 6t prcsent% there are two generally acce pted t%lworics for

pre


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/ec. 3.L lements of >he Theory of 2lasticity Q

ma$imum shear stress rcaches a critica> value H&1rn.! t.o thc shearing yieldstress in a nn ia $ial tension test. #rom ;*. (0-1?% tl1e ma$imum shcar

stress is given by \'

2¡

-

@'31m.-t:it (3-)

whero u1 is rn algchraically largest ancl u 3 is the ulgchrai('aly smallest princi pal stress.

#or unia$ial tension @-1 = uo, u = u3 = 4% wherc o-@ is thc yield st.rengthin simple tension. Thcr efore% t he s0earin3 yicld stress for simple te11sio11ro is c*ual to one-half of tl1e tensile yicld "-"tress.

uo-ro 0

/11hHtit11ti11g thcse val ues into thc c*untion for the ma$imum shcu%ii'itressrcsults in

4' - O' s o

!nl:t" ro

0

(3-K?

or ("3-@?

This is i",omct.imcs writ.tcn as

(3-1@?

wliere u5 and u6 are thc devi"Ctor s of the principal slreH""-os nnd ) is t.heyicld stress for plll'e shear% i.c.% the stress at wh ich yiclding occurs in

f.on%ion% where u 1 = /ua.Thc mu$imum-shear-stres s thcory is in good agrcement with e$ perimental results% being slip,htly on thn safo Hide% aml is widcly 11sed hydesignen, for d11ctil e metals. t has rcplaced thc oidor "rnd far lcss accurate ma$irnum-stress thcory% nankine's theory.

2ragc17uud Hoclgc 1 havc poi nted out t.hat in certain plasticily problemsthe simple relatiomi o *. (" -Q? or ("-1@? ca.n not% he 11sed as thc yicldi ngconditions sincc it is not known which of thc t.l1ree principal stresses is the

la.rgest. In this case% tlw m uch more complicatcd g


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60 &echanical #undamenta ls Bhap. 3

7 on8 9ises, or :istortionener3 y, ;0eory

6 somewhat bet.ter fit with e$pcrimc1lt.al rcsulCs is provided by theyield cr)crion given in *. (3-10?.

(3-10?

6ccording to this critcrion% yielding will occur whcn t.he dilferences bet%ween thc principal Htrcsscs :$ pmssed by 1e l'ighl-ha11d side of thee*na.tion e$ceed 1e yiold stress in unia$ial tcnsion% uo. The devel(?prnen t of this yield criterion is associntcd with thc names of Pon "ilises%

Hencky% P+a$wcll% and Huher. Pon lP+ises proposed t.his criterion in theinvarfant form given by *. (3-13? primarily bn use it was rnat.hemat.ically simpler than the invariant fot*+ of t.he ma$imum-shear-stress theorygiven by *. (3-11?. /ubse*11ent. e$periments showed t%hat *. (3-13?

provides better over-all agreement with comhined stress-yielding datatha.n t.he ma$imum-she"u-stress theory.

(3-13?

J 0 is the second invariant of the stress devintor% and ) is the yield stressin pure shear.

6 number of attempts ha.ve been made to provide physical meaningto the Pon .i>lises yield criterion. 4ne commonly accepted conccpt isthat this yield criterion e$presses the strnin energy of distort.ion. 4n Che

basis of the distor tion-energy concepi% yielding will occur when thc straincnergy of distortion per unit vol ume c$ceeds the strain energy of distor

tion per unit. volume for a. specimen strained t.o the yield st%r esH in uuia$ialtcnsion or compression. The derivation of *. (3-10? on t.hc basis of distort.on cnergy is given helow. 6nother common physical intcrprcta. tionof *. (3-10? is tha.t it. represents the crit.icat value of the octahedral shcar stress (sec /ec. 3-M?.

The t.otal clastic strnin cllcrgy per unit volume (sce /ec. 0-1? can bedivided int.o two componeuts% t.he 1%lrai11 energy of dist%orCion% U, irndt.hc strain cnergy of volumc changc% U'. To illustratc the resolut.ion of total strain energy int.o its componen t.s% consider #ig. :-. This figure

illus t.rntes the poin t establish< d in /cc. 0-H that a general three-dimensiona.! state of stress can be e$presscd in tcrms of a spherimil or

hydrostatic component of stress% u",n.nd a stress deviat.or% u' . :ecauscc$perirocnt.s have shown1 that% u p to rather la.rge valucs of hydrost.atic

pressure n. hydrost.atic stalc of stress has no effect cm yielding% it. is validto assume that on ly the stress deviator cnn produce distortion.

Thereforc% t.he1 2. F. :ri


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llook ompany% +nc.% 9cw 'ork% 1Q0.


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/ec. 3-/ lements of the Theory of 2lasticity A1

strain energy of distortion will be bascd @11 the stress devi"!.tor. It reprcsents only thc strn.in encrgy associatcd with changing thc shapc of thespecimcn and ncglects thc strain cnergy associated wth chauges i nvolume.

Thc strain cnergy of distortion will be dctermincd by first calculatngthe st.rnin cnergy of volumc changc a.nd thcn s1tbtracting thi"", t.crm from

O" i

/

t-

110' '

r-t 7

#iQ. 3-3. ;csohttion of stm,"", iuto hydrostatic strcs%, "md ",tr ss dcviator.

the total strain encrgy. ;efcrring again to #ig. 3-3% the strnn encrgy por unit volume associuted with a volumc chango will be

2 ' =


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(3-1M?

"#or u. unia$ial state of stress% u 1 = a@% a0 = u3 = 4.

(3-1K?


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#$ &echanical #undamentals Bhap. 3

'fhcrefore% the dist.ortion-energy yield criterion can he writt.en

At / 0o-u 0

=

(t : JI ((@-1 - Wr0? 0 # (


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Is''.;.:.. _ . 0.31---+ -+---i -+- +---t

/ec. 3-1 lements o the Theory of 2lastici ty A3

torsion% t%hc prindpal strcsscs from *. (0-M? "tre

rf7 - 0 # -7- # ';r,y

ises? theory.

6not.her type of combined >'itress test is t.o suhJect. 1i11-wall tubes totl$ial load "m


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(3-0%??

1 E. . 'l'aylor and H. [ui1111ey% 1ruc. Buy. Coc. $London, vol. 03@6% pp. 303-3A0%l>?"? !.

0 F. Code% . #$%si&, mi. 3E% pp. Q>""l-Qa>!% l>!0(i.

64 &echanical

#undamentals

Bhap. 3

This is the e*uaton of an ellipse whose maJor scmia$is is y0 uo andwhose minor semia$is is P\ uo.

6 convenicnt w:16 of comparing yielding critcria for a two-dimern%ional statc of slress is with a plot such as Pig. 3-. otc thatthc ma$imnm

shcar-strcss tlieory and thc distor l.ion-energy lheory predict thesamc yielcl stress for conditionsof uni

Distor tior1ener; y t$eor6

8-- +s$eor stresst$eor6

a$ial stress and ba.lanced bia$ialstress (u 1 = u0?. The greatest divergence betwccn thc two tbeoricsoccurs for a sialc of pure shear (@-1 = ->,). H has already been

shown t.hat. for this st")c of stressthc shear-strcss law prcdicts a yieldstress whi.ch is 1 per cent lower than the value given by t.hc distor tion-energy criterion.

A very sensitive method of diffo17ent.iating betwcen the two yield

#ig. 3-. ompnrison o yicifl criteriafor criteria is thc procedure adopted by plane str ess. Codc of determin ing thc effcct of

thc intermcdiat.c principal stress onyieldng. 6cconliug to Che ma$imum-shear-stress law% lhere should he noeffect of the value of thc intermedia.te stress%%a. Thus% (@-1 - u3?Ruo .l8or the distortion-cncrgy thcory% to account for the Jnftuence of theintermedia.te priucipal strnss% Codc introd uced t.he parameter D, (t>lcd

Lode's stress paraeter.

(a-0A?

/olving this c*uai.ion for u and eliminating @-0 frorn *. ("3-10? rcsults in

(3-2)


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$perimen tal data plot much better a.gainst grG. (3-0M? tha.n against thema$imum-shear-stress e*na.t.ion% indicati ng that the intermediatc

princi pal stress Xias au iufluence on yieldi11g.611othcr contributiou of Code was thc iutroductm1 of a. strnin

parameter v.

(3-0K?

wlierc Cl is a finite increment. of stmin. 6 p!o!., of D ngainst v should


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/E1c. 3-A! lements of the Theory of 2losticity #5

yicld a st.r")ght line at ] to lhe a$is if t,1e metal bclwvcs accor


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1 6. U"tdai% OThcory of #low ami #racture of /olids%O 0d cd.% vol. 1% pp. >?Q-1@%l+cEraw-Hi>l :ook ompany% !nc.% 9cw 5ork% CQ@.


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66 &echanical #undamentals hap. 3

shear stress corresponding to yielding in u nia.$al stress is given by

v0 %.T..t = *

3 ero = @.% lo-o (3-.'30?

4ct.ahedral strains are referred to the samc thl'ee-dimensional octahe dronas t.he octahcdral strcsscs. The ochlhcdral linear stra.in is given by

G N 0 # 3 oct =

3(3-33?

4ctahedral shear strain is given by

'5oct \B(1 -

0? 0 N (0

-

_3? 0 N (3

-

G? 02' (3-3?

3-M. lnvarants of /tress and /train

His fre*uently useful to simplify the representaton of a comple$ stateof stress or strain by means of invariant funct.ions of stress and strain.If the plastic stl'ess-strain curve (the flow curve? is plotted in terms of invaria.nts of stress and strain% appro$imately the same curve will beobt.ained regardless of the state of st%ress. #or e$ample% the fiow cur-vesobtained in t unia$ial-ten sion test and a bia$ial-torsion test of ti thin tubewi th interna> pressl'e will coincide when the curves are plotted in termsof inva.riant stress and strain f unctions.

9adai Ihas shown that t.he octahedral sbear stress and shear strain areinvariant functions which describe the flow curve inclependcnt of the type

of test. 4ther fre*uently uscd invariant functions are the effecti ve, or Gi3nificant, stress and strain. Thcsc *uant.itics are dcfincd by thc following e*uations for lhe case where the coordinaCe a$es correspond to the

principal dircctions"

ffective or signiftcant stress

(" -3?

ffecti ve @17significnnt st.rain

(3-KE?

9ote that both dfoctive stress an


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?e:. 3-@1 "e+ents of the + $or6 of Basti:it6 67

arn also related to thc octahcdral shcaring stress and strain% as cau he

sccn by compari ug *s. ("-3@? and ("-3-l? wilh the above e*ualions.

1'@%i . lr( -C -*" ("3-3M?

Drucker 1 has pointed out that% t.herc are a largc number of diffcrcntfunctions of stress a1Hl stra%i n which migh t serve as i nvarian t stress andstrnin para.meters. #or e$ample% X1e shows th"it combincd stress data for aluminum-a.lloy tuhes show hettcr areement when the e*uivalent sh9tr ing stress r.[% de`ned helow% is plotte


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1 D. . rucker% .R. A!. M*+$., vol. lll% pp. 3%H!-3M% 1Q-1Q.


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68 &echanical #undamentals Bhap. 3

&n thematical t.!ieories of plasticity can be dividcd roughly into twotypcs. :e.foration t0eorie.%6 relate the stress to t he strain% while fiowt0eories relate t.he stress to the st.rau rat.c% ot7 thc vdocity of stntin.Deformation t.heories u t.iliM.e n n "wern.gi n process ovcr thc ent%irc defotmation history "tlld rehte t.he total plast.ic st.rain to thc fina.>stress. This t7 peof theory is valid when the material is s11bJectcd to proporf)mal loading%

but% it is not generally considered to be reliable 1 when the dil'ec t ion of loading is cbanged dudng the t est. #low t.heorics consider a. suc cesson of infinitesimal increments of distort.on in which the im,ta.n t.aneous stress isrela.ted to t.he incremen t of the sti-ain rate. :ecause a fiow theor.yconsi


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/ec. 3-Q1 lemen>s of the Theory of 2lasticity #+

D = v. 6lt hough devintions fmm 1%odc's rdat.ionship h"we been shown

by c$perimcnt % it appenrs t hat t hc proportionalit.y Ccl%wten stress dcvi")or and strain incremen t ii"" u reasonably good appro$imat.ion.

To provide additioual .simpli tic"tt.ion to t he analysis% it is oftcn assurncdt hnt th' bo

3.+. Flow Theories

Bi3id +deal 1la stic 9 aterial

6 flow thcory for a rigid ideal plastic material bnsed on thc propor tionalit.y betwceii strc",s deviator ami strain rato is thc outgrowt.h of work

by /t. Penant% Cevy% and Pon i!iscs. Thc Cevy-Pon i.Plises e*uations aregivcn below for a gcnernl coorc.linate systcm. /. is a. proportionality constant%and OO' is t.ho hydrostat ic componcn t of stress. otc that a dot over t hesymhol for strnin indicat.es the time derivn.t.ve of stran% i.e.% tl1e strainrate.

;,.y = A)0,E

T'', = 6--y%%%T: = 6O(%,I

(3--10?

+n terms of t%he principal stresses% the Cfffy-.5on l+i"",es e*uations can be written

(3-3?


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These e*uat.ion"%, are similar to the e*uations of viscosity for nu incorn pre""%si ble fluid. Thc import"rn t% differcnce is t hat for tite case of the fl uidthe proport .iona.lity constn.nt U is a% true material const"1nt% thc coefficicnt


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70 &echanical #undamental> lhap. 3

of viscosity. #or the ca.se of the phistic body% the value of 6 depends onthe valucs of stress and strnin. 6 can be cvaluated when thc yield critcrion is cstablished.

Thc 5on "&ises yield criterion is given by

& 2 = c

@1' (3-?

/ubstituting *s. (3-3? into *. (3-? 1esults in

J %0 0(7l0 # *08G#

*a0? - 5(.

FG (3-%?

'l'he *uan tit,y " 1 #0

0

N"a0

is an invariant of strain rntc. /ubstituting

*. ("3-? ba.ck into *s. (3-,rn) givcs

% P0 uo1U = G3(10 N 00 N a 0 )B 1

omplctely anttlogous e*uations follow for u auda6.

*uations (3-3? can be wl'i tten

(3-A?

A=.. Jo -


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(3-K?

Thc "bove two e*uations% plus the consta.ncy-of -volume relationship

1 # 0 # 3 4% constit.ut.e a system of differen tial e*uations that must be integratcd over a particular stress path or strain path for the solutionof a particular problem.

Elustic1last fr J! aterial

The e$tension of the Cevy-Pon &ises c*uations to thc


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/ec. 3-QL lemnts of the Theory of 2lasticity M1

t.he rate of change of plastic strain is proportional to thc stress deviatorrcsults in

(3-->Q?

The time derin1tivc of Hookc's law e$prcssed i n t.erms of stress ft> strainde%7iators B*. (0-@?L gi%7c"-, t he c7o1Te%-ponding e*untion"-, for clast.ic st%rain.

(3-@?

ombiuing *s. (3-Q? and (3-@? rcsults m e$pressions for the timederivativc of total slr-i.

If it. JG", aiisumcd t hat the Pou "+i1%es critcrion of yiclding f lpplies and Chatthere is no strain hardening%

#rom *. (3-?!. .i2 J

'( '?0 # N 1 = .= 8 # ( '?0 = K4'G 911 ( 0 00 L43 03 (3-0?

'l'his e.$pression can he usc


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(3-A?

1 This derivation folows the proce


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,$ &echan ical #undamentals Bhap. 3

Phe11 t.he stress is in t.he cLastic ra.nge% or in unloading from the plasticregion % *s. (3-? do not. apply. The proGXer c*uat.io1ts are given by theelasticity e*uftt.ions like .*. (0-@?.

3-1 '. Defor!"#o$ Theor#e%

Hei1cky proposed t.hat. for small st.rains the stress devator could beconsidered proportioria.l to the strain dcvia tor.

u' = 2Ort' (3-M?

lastic strains are ncglcctcd in *. (3-M?. Or is a plastic slrnar modulus

which varics depcnding upon the values of stress and straiu. :ecauseof the assumptfon of co11stancy of volmne% l' 4% and " t. There forc%*. (a-M? can be e$panded in t.erms of principal stresseG%, arnl .strains logivf i

'l'hc analogy is apparent between t.he right.-hftnd side o *s. (H-K? andthe familiar e*uations of elasticity e$pressing strain in terms of the

principal stresses B*s. (0-0".?L. !t'or the plastic case% 2oisson's ratio has been taken e*ual to - E1 can beconsidercd to be a plastic oddus that

&-I /I /

/ .i, R Ep'" /

1ra----- (: - --O1.

/ntf ico t str c>

#iQ. 3-A. Den11itiou of EKK

is adually a variabl< depencling uponthe stress ancl strain. The evaluationof 6l' from an invariat1+% strcss-sCrainc"urve is shown in #ig. "-A.

(3-Q?

9adai 1 has dcvelopcd rclatonshpssimila.r to *s. (-K? based on t.hee*ual ity of Code's stress and si.rai npn.rr>n

ctors. The fact that D. = v leads to the co1tclusion tlmt the rntios of

thc principal shearng stresscs to lhe pl'^ucipal shearing strains are e*ual%1 6. Uadai% O2lasl.iciCy%O pp. MM-M>=% ".++cr"i.w-Hill :ook orupuny% lm.".% "'.'l'cw 5ork%

l>?"31.

t


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1Punch

-)

€y =X

/

/ec. 3-11) lements of the Theory of 2lasticity ,3

an


31/35

74 &echanical #undamentals (hap. 3

Ier o for plane strain% it docs not follow that t.here is Iero stress in thisdirection . !t can be shown 1 that for phme st.l'ain u, = (cr.% # u".%?R0 or u( = (u1 # u0?R0. If this value is substituted into the e$pression for thePon &ises crit.crion of yielding% the yicld criterion for plaue strain bccomcs

value of &. *uatiou ("-A1? is e*ually valid when writ.leu .interms of the st%ress dcvif ttor.

(3-R2)

3-&2. SI#'-f#el( Theor)

onsidct7a volume elemeut in plirne st.rain within a plastic region of a body. #igure 3-Ka represents the two-dmensional statc of stress withrespcct to arbitrnry cartesian coordi natcs. .It is possihlc lo determine

t.hc principal planHs snch that t.he shea.r st.l'esses vauish (#ig. 3-Kb?. The pri ncipal st%r csscs u.re simply f unction",, of thc s phcrical component of

stress% ,/ ', and the shcaring stress &. & is 1. constant throughout the plastic rcgion if .sl.rain hardcning is ueglccted% but u" varies from poin t to

poin t. The ma$imum shcar stress will occm7on planes %1 to the direction of the principal strc.sses. TCus% thc cril.ical shea17stress & wll f)st

r each its val ue ou these planes. This condit.ion is shown in #ig. "-Kc%whc.r e it. is seen tbat thc ma$imum shea.r stress occurs i n t.wo ort.hogonal

dircctions%


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f / r!!

/ec. 3-10L lements of the Theory of 2lasticity 75

are not the slip lines% or slip ban1ls% ohserved 1111ck"r the microscope @11 thesurfocc oGXlastically dcformcd metal. Ths lattcl' t.ype of slip lincs willhe discusscd more f ully in thc 11c$t ehapt%cr.

(a% (b) (el

f ig. 3-:. Two-dimensional ist"ltc of str cisli in pl"me strnin.

Dy comparing #ig. "3-Kb and e, it is sceu t!u) th


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u"- 0kefG constan t along {3lino

("3-A?

The slir% lnes at a free surface nrnst% make au auglc of -1] witli thesurfo.ce (#ig. "-?% sincc there a11 be no rcsultant tangent.al force at a


34/35

MA &echanical #undamenta ls Bhap. 3

free surface. /ince there is no resultant normal stress at a free suriacc%o- 1 4 a11d by g*s. (3-A3? u" /). Thcrcfor c%u2 = -0k%"wd the transverse principnl stress is compressive witll a value of 0k.

6s a fu r ther e$ample of the use of slip lincs% consider the deformationof an ideal plastc metal by a fiat punch. 1 Thc friction betwecn the face

#is. 3-1@. /lip-line lield prochwcd by inden tu%t.ion of tt punch.'

of the punch and the metal is considered to be ncgligiblc. 2lastic defor mation will first start% at thc corners of the punch and wll result in asli p-line field such as is shown in #'ig. 3-1@. onsider the point M.

/ince this is at a free slll'face% the normal stress is Icro and u" &. +naccordance with *s. (3-(H?% the e*uaCion of Chis slip linc may bewrittenu" # 9+:/ = &. There is no change in the value of u" until we reach poi nt ;, whcre thc slip line devia.tes from a straight% lino. +n going from * to , the slip linc turns through an angle F, = /8n/S2 so that its e*ua t%ionat point is u"- 0k(1fOR0? &. /incc no furthcr change takes place in


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"!e+ents oft$e4$eor6of2laslicit

y

U ma$ = *= 1 # - % EOo7 v3 0

Thns% t he t heory prcdid"s that. full-sc"i le plastic flow% wi t h thc rcsult ingi11dcntation % wll occnr when the stress aeross the fa('e of thc pu11c>t rcachest hrrc t.imes t hr yicl'.E. Hodgc% %!r." OTlu..ory o 2crl'. Hi>!% pp. 1@0l-l@3@% 1>?.3

P. 2rnger% ;rans. Uoy. + ns f. '

l'cc0ol., Ctod8&w/, no. Eri% !H ,:-.V . ?. Tholli.,,c11% J. H p p/. &l tc0., vol. 0% pp. K1-K-1% l>li .8 :. :. Hnndy% 9 eiulluryia, voC.1>?% no. 0@3% pp. 1@Q-11K% 1Q.

0uo ( 11'? %

Exposicion de la teoria de la plasticidas

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