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COMPUTATIONAL METHODS IN THE MECHANICS OF
FRACTURE
Vo lume 2 in Computational Methods in Mechanics
Edited by
Satya N. ATLURI
Center fo r the Advancement of Computational Mechanics Georgia Inslitllte of Ted lflo!ogy
At/alita , G eorgia . USA
\".
NORTH·HOLLAND AMSTERDAM . NEW YORK . OXFORD . TOKYO
CHAPTE R 5
Energetic Approaches and Path-Independent Integrals in Fracture Mechanics
Satya N. ATLU RI
Regents' Pr()fe::i.~or of MecJlllllics Cellter for the A(/I'(II/Cemellt of Compl/wliO/J(l( M{'("luJ/lics Schoof of O"il cngineeriflg Georgia IIISlilli/(' of Tt'c/Illology A llama. Gil 30332 U.S.A.
Computational MClhod~ In the r.kdl,jnlC'S of Fracture Edited b} S.N. Allun <D H-.c,ier SCience I'lLbll~hcrs IlV, 19H6
s_ .\ I llh",
I. Introduction
The success of modern fracture mechanics i ~ due. in a large measure. to the cclebnlled \\ork of IpA in in shov.ing Ihal. for ela~tic materials_ the crack-tip field ... arc gO\-erned by the ~)-called stress-intensity r .. ctor K. Likev.ise. in elaslic-pl;lslic material\. the well-known "ork of llutch inson. Rice. and Rosengren shov. ... th;1I for stat ionary cracks in qua\i-statically .. nd monotonically loaded bodies of pure power-law ha rdening materials. the st ress and strain fie lds in the vidnity of the crack-tip under yielding condi tions varying from small-scale to full yieldi ng arc con trolled by th e E\hclby-Cherapanov- Rice J-imegral.
This chaplc r i" concerned with a general dhcussion of crack-tip para mete r\ gO\ern ing quasi-static as v. ell as dynamic prupaga tion of cracks in clastic as well as clastic- plastic mate rials. These crack-tip pilrameleN are, in general. defined. ~a) for two-di mensional problems. as integra ls o\'er a circular path r;. with rad iu\ £
being "very smalr ·. The in tcgnmd. which invol\'e .. the crack- tip stress. strain, and displacemem field~. i ... in ge neral. such that it is of (l Ie) type at radius F' from the crack-tip. which renden the in tegral over I~ to be of a finite magnitude. This crack-tip integral p .. rameler is then sought to be represented equivalently as a far-field in tegral plu~ ,I " finite domain intcgnll". using the divergence theorelll . This alternative representation is convenient for computational analYM:s of fracture problems. Under some .. pecial circumst:mces. the aforementioned "finit e dOIll;!in integral" va nishes identic~llly - thus mak ing it possible to express the crack-l ip integral parameter solely as .. far-field con tour integral. These specia l circu msta nces arc cleilrly ~pcllcd ou t in this chapter. In Section 2 of this ch •• pter. we consider scl f-~i mil ar dynamic crack proplIga lion in an clastic solid sub ject to no n-uniform tempera tu re fields, and when the materi ;1 1 is considered to be non-ho mogeneous. In Section 3, we di scuss crack-tip paramete rs for quasi·sta tic as welt as dynamic crack propllgation in clastic-plastic solids. Finally. some remarh are also made concerning crack-tip parameters in creep crack-growth at elevated temperatures.
2. Elasto-dynamic crack propagation
2. 1. Prelimillllriel'
We consider the material to be non -linea rly clastic and finite ly deformed. We employ a fixed (global) Ca rtesian coordin;ltc system such that X, and Y, refe r.
re~pecti\'ely. to the coordinate~ of ,I given malerial particle before and afte r deformation. We introduce anolher "local" Cartesian .,y~tem _l , such that X l is locally normal to the crack border and in Ihe crack plane. x~ is norma l 10 the crack plane, and X _l i~ locally tangenllal to the crack border and in the crack plane. The deformation gradient i~ represented by F" = V"~, (aY,Ir1X, ) such Ihat dY, I~, dX,. Henceforth in Ihb ~eclion. we shall employ the "nominal" stress. denoted here by I" . as the mea~ure of .,tre!>~ in the deformed bod) . Note thai ' " = (TId" where T R is the first Piola- Kirchhoff stress [1].
The boundarY-"aluc problem in elasto·d) namic~ h. in ge neral. posed by the equation~ [2]:
(linear momentum balance):
I + ( - P" ' 1_' J I I
(2.1 )
(angular momentum bH l am~e):
(2 .2)
(constituti\e law):
f = a Wl aF iHVlae,. " I'
(2.3)
where
e" = 11/. ,; II = }' - X' / , / . F - + t5 /, - e
" "
(traction b.c):
lI , f ,/ =:0 ' / at S • (2A)
(displacement b.c):
II, = II, at S,. (2.5)
(initia l conditions):
II, = II:I(X.). . u(X ) /I /= V , k at t "" 0 . (2.6)
In (2.1 )-(2.6). all components refer 10 Ihe fixed (global) Cartesian system. In (2.1). f, arc body forces per unit initial volume. p is the mass de nsity of the undcformed body. and li, arc accele rations: where (.) denotes a material derivative.
LeI (T./ be the true (Cauchy) slress in the deformed body. Then il is well known
[II that:
(~X, I" = J y. (f~1
,) ,
S.N At/un
(2.7)
""here J i~ the determinant of F", If di~placements and their gradients arc infinite.,ima!. f'l "" 17,,: and Eq. (2.2) red uces to the condition of symmetry of U 'I.
Eq. (2.3) is valid, in gener:ll, for an inhomogeneou~ as \\ell a~ anisotropic body . The condition of material frame indifference imp<).,e~ certain reslriction .. [L Ion the structure of W: and hence. in general. it is a function anI} of C" ""' F~ , F41. \Vhen the Slrtu.:(Ure of IV is tllU~ prope rly defined. cond ition (2.2) becomes inherentl) embedded in thc st ructure of IV (sec. for instmlce. ReL121) . In (2A) and (2.5). S, and.\ arc parh of the e>:ternal surface of the undcformed bod), \\here traction ... and dll.placemcnh are re~pecti\'e ly .. pecified .
1.2. SI'If-similtlr £"rack propagmiofl
Con,ide r th e d) Il<lmic pro pagation of a crack in a self-slmi/ar fashion 'lich that the crack length increase., by (da) in time (dl). with a non-co nstant velocity of propagat ion. {/ (da df). The energy release 10 the cnu.:k-tip pcr unit of crad cxtelhion. denoted b} G. 1\ given from global energ} con\lderation\ ".,:
Gli DW, DI/ DU DT --'-+--------
01 01 DI DI
wherein. on the right-hand side of (2.8). the liN term represents the ratc of exte rnal work done. the second the ralC o f heat input to th e hody. the third the rate of change of internal energy in the bod~. and the fourth the rate of change of kinetic energy in the body . The heat -flux relation i~ given by:
~~ = - J " . rI ds = - J 'f' . " dv (2.9)
where Ii is the vector of heat-flux. and II i., the unit outward normal to the houndar~ S (If the bod). 'f' e,(iJ i(7 X , ). The componenh of /I are II , in the x , system and S. III the)(~ system. Aho. in an) Ihermu-mechanie:t l process III. wc ha\e:
D U DIY DII -- ~--+ --
DI DI DI
where (DIY/ Or) j, the "t re~~ powe r. and HI is the de n~ity of total sI res" worl\. NOle that OIVIDI t,/ , 121. U~ing (2.10) III (2.8). \\e obtain:
G "'" OWe. Da
DW DT Oil Oll
(2 . II )
f -Du , f Du , D f E I - ds+ f - du-- (W + T)du
, Dil ' DlI Oll (2 . 12)
v.here V is the tot.ll volullle of the cracked body, S, is the surfacc of V where
traction!. lIrc prc:>cribcd. f , arc bod~ forces, \V i!> the dcnsity of total stre!>!> work (or equivalently. the strain encrgy density. only in the e,lse of a non-linear chlstie material). and T ( ~ /Hi , li , ) .
Referring to Fig. I for nomenclature, we consider , for instance in a two~
dimemional problem. an arbitrarily small loop r. -.urrounding the erack~tip, such that the "volumc" (or ;Irca in;1 "plane" problem wi th unit thickness) imide I~ is V. (mcluding the crack-tip) . For inst.anec, in two-dimcn~ional problems r: rna} he
!
, X,
T· , a,
<Ul6/,I...:.S t.c / S (bolJ]dory of
bOdy V)
• r
r
s ,
V enllre vo.'lmc of the body
v( : v()/ume tYlClos~dby r: V r volume enclosed by r
S,:," S;r + S~r
Fig. 1 Norm:m:laum: fOl the erack~d-bod) <lnd ' :IrI"U ~ contours
T,
S,,"' IIrllm
considered to be a circle of radius E, while in three-dimensional problcm~ I; m:l) he considered to be a toroidal surface whose axis coincides with the crack-front and whosc cross-section i~ a circle of radius F . If we consider the volume (V V,) \\ hich docs not mclude the crack-tip. wc sec that the following equation of con .. ervation of energy hokb:
o I D" t , Oa' ds I D" I I, Oa' ds +
, . , , DII
[ '-D ' dv "
I)
I)" I (W + T)dv .
, , (2 . 13 )
If 1/ i~ the Ulllt "oul\\ard" normal in the c(lmentional sense, it i~ seen that the "cxternal" boundary of (V - Vr ) is (5, I~ ) if the "sen~" of I: is a .. <'ho" n in Fig. I. Using (2.13) in (2: . 12). it i<; seen that:
G = lim II '. ID)II, ds + I I , f}1)---.!~ dv - DD I (W + T) dV) . ,.,,\ II (/ a (2.14)
I . I . , "
Referring to Fig .. . 2(a) lind (b). P: and p~ represent the same material particle at times t and (t + dt). re<;pccti,-el). ""hen the crack propagates b) :In amount (du) 111
time (dt) . On the other hand. poinl<' PI and p, arc located at the same distance and orientation (i .e . r. 0 :I ... 111 Figs. 2a. b) from the crack·tipS at times t and 1+ d/. n:"pcctively . The fUlulamental idea. in se/f-~imil(/r. ('lasto-ciYllamic cmcJ.;. I"OIJlIK
mio". i~ Ihalthe crack-tip fields are self-similar OIl times I and (+ d/. Te~pcetively. execptthat their IIItt'lII'ilies m,IY differ. From thi., concept. we ~cc that Ihe changes in displacement. \elotity. ;,lIld slress al the laml' material parlie/I'. due to cnlck gro"th b) (da). arc given by:
(2 . 15a)
«(II1,/,la - (ill ,.iJ.\ I ) da (2 . 15b)
\\ilh ,imi luT relali{)n~ for changes in Ii, and f" , II i., important 10 note that.l , is :llong the direction of (~clf-~imi l ar) crack propagalion a~ in Figs. I and 2. Note I h ,l1lcrm~
such as (all ,l(ia) arise due 10 a change in the ~trength of the singul:lritics of Ihe crack-tip ficld~ cOTTe"ponding 10 an increa~e in crad. length of (du). while terms ... uch a~ (iJlI ,/(h l ) occur due to the tr3n~lation of the crack-tip field~ h) (dl/) . If the material particle i, at the external boundary of the specimen. it is cas) to ~cc from rciation, of the I)p.: (~ . 15) that:
rill , rIll , at S" : (ill ,hi
;it" _'il" (2.16)
<-It 5, . ,i (I (h i
/-.nergI'IIC approue/II'S am/ patll-mdt'peml<'lll lIIlt'grul:.
t I I I I I T , I
00
L~I [ .~ ~92
" , 2,1\" (0 ) ~
/ • !:,' ", ,
I So
I , , ,
I I ,I I I I , , I :t \: , , I T I
51 ,
,
L , , , ,
£"£,-00, £,£, 'e . , <:lL XI - 11- £.' 00 , ( b)
So
I I I I I I I I I T , \. \'
X2 \ .
~"~. l ,p,,~ 1 '°2 -:--8 +/ ' /
XI ,; (e) \,
" " \, \, " . ,
hg. 2 Concept of ~ubtracl rng out 'i lllgulanHc\
Now consider Ihe Ihird lerm on the righ t-hand side of (2. 14):
2- J (IV + T)dv ' J « IV + T)( I',) - ( IV + T)(p ,) l/da dl' . Da
I ,
127
(2.17)
II is well know n 13j lhal in the ca<;e o f the propagating crack. IV and T m 3) posse!oo!'> si llgulariti e<; of Ihe order (I r) ncar the crack-tip . Further. «(]w 1r1u) and (arroa) mcrel) rc prc\e nt change in thc intensities of the singularitie ... "hile the order of their singul:lrit ie<; is slill (1 1,). lI o"cvcT. since (il WIJxt> and (aTi(h l ) mOl) lead 10
S \Irllm
" non-lIltcgrablc" singularities (which inva lidate the lt pplication of the dive rgence theorem to terms of the type L, (a lVl ax r) du, as done by Eshelby [41 and ot hers). it is more proper to use the concept of "subtracting out singularities" as illust rolled in Fig. l(e) and detailed in [3.51 . Thus. (2 . 17) may be wrillen as:
DD J(W + T)dtJ = -J ~ (W + T)du + J(\v + T)lI l dS. (2 .18)
(I (Ja , , , , , , Using Eqs. (2.15b) and (2.18), we rewrite (2.14) as:
J( au,)
G = (W + T)r1 I - I' ilx ds , '
-(J (-"- (IV + 1") - f rIa ' au
a /I , ,
limJ ( w + T)" I- I illI ' )dS , .~n ' ax , ,
aj, ) ax II, du , J
au ) I, -' ds au ,
(2.19)
(2 ,20)
Eq. (2 .20) fo llows from (2 .1,}) since the second term of (2.19) vanishes in the limit f'_O, due to the fact that cHVlaa and aT/i/a are still of order (IIr) ncar the crack-tip. I , is of OCr I ~ ). whi le (au ;f(1l1) h O(r' ~ ). The result for (tiC). analogous 10 that in (2 .20). has been derived earlier b) Atki nson ,lIld Eshelby [6J and Eshclby [71. even though not condusivcl) for;r crack propagating \\ith an arbitrary history of motion . Using ;trgulllents similar to those used above in deriving (2.19) from (2 .14) . one may likewise dcri\'e from (2 .12) tha t:
G= J (pv+ T)I1 , au )
I , -' ds ax 1 , (J( a ,)u aj, ) J au, ) - - (W + T) - f -'-- ,/ dv - 1 - <is
ilu ' ila ax 1 ' , all , , (2 .21 )
\\hcrc S. the external houndary of V. is such that 5 = 5, + Su' and use has been made of (2. 16). Note Ihat the second te rm in (2.21) does not vani~h: its evaluation in the practical problem of crack propagation in an arbitrary finite body involves. however. t\\O solutions for slightly different crack·length. (tl) and (a + da) . However . if onc considers the volume V, V, (where ris any path surrounding the crac\.;-ttp: see Fig. I) and thus excludes the crack-tip. it is a simple mailer to apply the dive rgence theorem and rewrite (2.20) as:
J ( au,) G "" (W + T)II [- I,- d .. ,]x I
I .~,
J ( d au , - - (W+T) - / -- I
aX l "" ax , " 1 i \
a'u ) -" -,,'. d v . "x,,,-, I
(2 .22)
C!9
We nOI\ restrict anention. without much l o~s of generality, to infinite~imaJ
deformations of cracked 1I01/·Jjllear clastic bodies 1\ hich arc subject to non-u niform temperature held .... In such a case, the infinitesimal strain tensor F,t may be decompo~cd a\:
(2 .23)
"here t' ~' arc "mechanical" 'i trains, and F~, arc " thermal" str:l ins (F :, = 0'08,, : "here a i~ the coefficient of thermal expansion, 8 is the temperature risc (1' - Til ): o 0(14): and r,, '\ the ambient temperature). The total strc,\s-I\ork ing den~ity is:
where
and
,
w f (T" dr"
" (2.24)
(2.25)
(2 .26)
In (2.25) [ 1/ i~ a tensor-valued func lion. Note Ihal in thc rl11oela~tic ity ~tn.'\ .. dcpcnd~ on the mechanical strain!. as well as explicitly on,\'4 , '\ incc the material may be non · homogeneous (either naturall) or due to temperature dependence of the material properties in a non-uniform temperature field). Thus
iJ U' (]r" iJ \V : u -+-
(1x, 'I (1.11 "x l
, iJt'" fl(]["1 a/'IIIf'~~
~ " - + -- + ----" tJx. ,}x I Hrl,",' ae m aO
II It
(2.27)
wherein the definit ion of (t1 U'l lIx l )e,pl'n1 i~ apparent. Usc of(2.27) in (2.22) re\ults in:
G l' = f ( U' + T)II) -I, iJ lI' ) d.\' ax. I • I.,
f ·( a,;, au, alV I ) lim pli , - + (I , - PII ,> -a + -a' dv .~II. iJx 1 XI .1 1 c·'r
I r \ ..
f( au ,) (W + 1' )" 1 - t, - ds iJ x 1 ,
(2,28a)
f [ a,l, all, II \\' I J lim PII, - + (f , - pii, ) -. +. du . (2.28b) , ." ax , a.t l (1'\ 1 ur , ,
lJO S \ All""
If the material I!> homogeneous in the x, direction, and (T - To) =:0 O. i.e. isothermal condition!> prevail in the ~olid. then (a W id X 1 )~'rhClI = O. On the o ther hand . consider for example an iSOiropic linear ela~ tic material with non-uniform tl!mpcr;lture distribution O(x~) and non-homogeneou~ material properties. Here.
(2 .29)
such that , .
" (2.30)
where () I = rJ( ) /(/Xi . Now. from (2 .29) one obtains:
(2.31)
Usc of (2.31) in (2.30) results in:
~.:.\: I~ 'I' "" ~ U".I F~ - ~ U'I C~,l - i (2,u. + 3A).1 0:20
2 - (2JL + 3A)aF.u 9.1
(2.32)
Note that Win (2.24) lind (2.26) is given for the present linear elastic isot ropic case as ' :
If the temperature dependence of,u. and A is ignored. the path-independent integral representation for energy-release rate. (2.28). becomes:
J (au,) J' ; (W+T)fl,-I,- ds ,7x ,
I •. 1.1
- f [pli,li,.,+(f,- pii,)II ,.I-(2,u.+3A)OEu O., Jdu \', \ .
where () 1 = a( ) /ax , . Nm\. note the identity :
f(2,u. +3 A)OFu all l ds = f (2,u. +3 A)OFu an , ds I. I - ~"
- f 1(2,u. + 3A)aOeu L du .
Vr " ,
(2 .33)
(2 .34 )
' Th,s 35sumc~ Ih31 a li:mperalure field is inItially preSCribed. and remains swtlonary during lhe mcchallicalloadlng of Ihe solid
Adding (2 .3ol) and (2 .33). \\c may define a parameter G. ~uch that :
(; J' + J (2~ + 3A)0f"Ha/! ! d\'= JIPV- + /')1/ 1 I,II '-l ld5 I. , :
, , J \(W ' + T)II I - ' ,u,-l lds
J [/lIiA .1 + (J , r, \
PI
(2 .35)
"hcre W ':-:: I1F" F" + (A / 2)t" i ~. On the other hand. the Na\ier equations of i~olropic cla~ticit)' in the presence of non-uniform tcmpcrature fields is gi\cn h} :
(2.36)
TllU~
[ (2~ + 3A) ,
2 8 II , (2.37)
If the temperature licld in the body is as!.umcd to obc) the harmonic equation
0.,,- 0. (2 .38)
then
(2.39)
I •. ~., I
Thu~. when the proce!!.!!. i!!. qll(lsi-Sl(llic (T "'" O. Ii , "'" 0) .. it i!!. pos!!.iblc to u!!.e (2.37) and
(2.3,) in (2 .35) and dcfine a modified pa rametcr (; such that
(2.40)
J J (2~ + 3A) 1W*" I- I , /lI1- A + ~ a
I . \. r
(2AI)
S \ Alilirl
Thu'l. (2 041) represen ts :1 pmh-il}depl.'/U11'II1 integr;11 exprc .. ~ion (without the prc .. c nce of a domain integral) for C. This re,u lt is due to Gurlin IR[. Note that J Q~ is the "qua 'Ii-static ,. \a lue 9 f the energy- rclca'le rate j' (i .c. when the Illlcrial inert ia is ignored). Thus. \\hile C represenb a m,l1 hematically convcnient integral in thc C,ISC: ( I) of linear isot ropic homogeneous elasticity; (2) 'Ahen J1 and A do not depend on tempera ture; and (J) 'A hen the tcmperature satisfies 0 " ~ 0; its ph )~ical
~ign ific;U1cc is ~ome\\ha[ oh,curc. The poim to be mitde here i~ that the situa tion when a meaningful crac~ -lip paramete r ca n he represe nled equivalenlly by a far-fie ld co ntour-integral alone (i .e. without the prese nce of a domain-inlegral) is rather fare in pracl ice.
Sometimes in IheTmocla'lticit} it i .. convenient to define the stress potentilll:
,~
v ~ f u" d,m
" (2.42)
" such that
V ~ V(F~ .rd (2,43a)
and
IT" ~ g,,(t:~. ' \ 1) . (V3h)
Note Ih;1I V is si mpl y a mathe matical potential ;tnd is not eq ual to stress-working de n~lIy, \~e have:
av ax ,
-" '" uF" a -- +
'I a.\ I
,~
f iJ!!",
ax, "
<I f' 'I iJ
I d,~ up
-,,"' . v I s u --+-
'I (h'l (l xl
~,p'
Thu~. o ne may define a crack-tip parameter:
C· =O ![(V+T)11 1 1,1I",ldS - ! [(V+T)I1 , - I,uLI I
'0
- ! [Pli,li" , I " I
I •. \,
+(/,- pii,) II ,I- UUo8 1+ aV I j. . ax , up
(V')
(2"5)
Ir the material is linear elastically isotropic . and A and J1 depend on temperature. we ha ve:
av (]x 1 I f':" [J E '/II
up rix I "
m d m 1J- II J1 'I (2 ,46a)
(2.46b)
1.'.'
From the relation.,:
(2.47)
one ma) \\nte (2A6h) a .. :
(2.48)
The ca .. c when (I) T ::= 0, (2) II, =0. and (3) the material constants arc independent of temperature , i.e. ("VI,]x,). w=O. has been reporled h) Ain<,worth et a1.191. Note, hO\\c\er, even thi .. ca .. c (or in gcncral) C- Ji- J '(iiiL G). It i ~ morc natura l to
deal with thc dcn~ity of st rclIlI-work. IV. in thc case of a non-lincar matc rial. When the mate rial ill homogencous in the x, direction . and when i.,othermal
cond itionll prevail. Eq. (2.28) reducell to
G J [PV+ T)II , -I, iJI~']d' a.l, _ , "
J [a,i , - (] II 1
pli - + (f - PII ) -' dv , ax, ' 'ax, r, \
where r i .. an) arbitrar} conlOur that encircles the crack-lip.
(2 .49)
The lIen.,e of path-ind\.'pcndence embodied in Eq. (2AY) implies thOlt for all}
cios(!(J l 'o/"me V·. "ilh a boundar) " -, IIU/ enc/using the crad..-tlp,;h in Fig . I, \\C
ha\\.' :
J ((II' + T)" , , . au ) J (,;'; ,;u ) I - ' ds - pfi -' + (f - plI ) -' dV = 0
, ax 1 ' ax , r 'r1x 1 , .
(2.50)
which may he verified easi ly under thc assum pt ion of (2.!). ma terial h01l1oge1ll.::ity along Xl' and when \V is a llingle valued fu nction of /'~I'
Because of the usc of J' as defi ned fo r any path I' as in (2A9) in\'o ! \'e~ a l'olllme-il1tegral, the above notion of pmh-il1l/l'IJemJellce has been pronou nced by 1ll;IOY to be ulleless. Thi:. vie\\ point. howevcr, is somewhat ort hodox . True. the evaluation of (2A9) invo!vc$ taking the limit o f the vol ume in tegr;llto the cracl--tip: and thus, on the surface. it ilppcars to involve .. "knowledge of the cr;lCk-tip ticld,", which the so ·called J -i ntegral of e/aSIO-srlllics [when Ii = II = II in (2.-19)1 doell not invoh c. FirM of all, it is clear from (2A9) that its usc dOCll not rcquire a knowledge of the crack-tip strcss-strain fieldll, hut on ly of displ;lCemcnl. \elocit~ . and accelcration. Furthe rmore. a comparison of (2.-19) ("hen evaluated (l\er the extern;l! surface 5) .md (2.21) reveals th;1I
I." S \' Alfu"
lim f ,~II
( a,i, all,) pli -+ (f-pii) - dV
, thl
' 'i1x1
f [ ,) .. au, af ,] f au, ~ - (IV + 7) - f - - - 1/ <Iv - t - ds
iJa ' ax I ilx I ' , (ill , s
(2.51 )
and Illu .;" the left-hand Side of (2.51) remains finite in Ihe limit f ---10 O. This is intcrc,ting if one note .. Ih.l\. in known 'l1l:.llytical asymptotic solutions [101. Ii, - OCr I ~) and 'I, - OCr \ ~ ); and hence. on first glance. the Icft·hand side of (2.51) appears to contai n non-integrable !>ingularitics. It has also been verified direcl!y [lOllhal for known analYlical asymptotic solutions for IIlfinlc bodies. the
\olume integral in (2.49) docs have a finite limll . due to the faci that the angular variat ion of the integrand I'> '>lIch thaI:
" ,
!i'!l •. f r J (plI,II,~)r dr J dO ...... 0 . (2.52) . "
E'en though finding the :,0l U1ion of " " Ii ,. II , ncar Ihe crack-tip in a finite body is a difficuh problem (I/1(1/Ylically, it i:. a relatively si mpl e task in comlJllrmiol1(1/ mec/ulllin', Tbis has been demonstrated conclusively 111 , 121 in a variety of crac/.. -prop(lgmioll problem'> in finite bodie!>, even while u:.ing the simple:.t of cracl. -tip finite e lement~ which do 1/01 model {my of the singularities in stn,in , veloci ty, o r acceleration.
If one considers the energy-re lease ralC per ullit lime in self-similar elastodynamic crack propagation, one sees thai Ihis quantilY is represented by:
= J ((\V + T)C,N, - C,l, (:;' ) ds I; k
(2 .530)
where C hi the tlOfl-('OllSttllll velocity of crack propagat ion along the XI direction, " I
is the component of a unit normal to r: along X I' while Ck and Nk are components of the instantaneous velocity vector and the unit normal to f .. , respectively, along the X, directions (see Fig , I) . (Note that the ve locity vector C with ICI = C. along the x I direction in se lf-simila r propagation. may be considered to have components Cl along the Xl direction,,) It is now a simple task (1) to apply the divergence theorem, (2) 10 use the coordin~lIe-i nvarianl fo rms of the linear momentum bal ance laws of (2.1). under the aSl>ull1ption:
(2 .54 )
hrt'rJ:t'lir lIf1pmllrlU.1 unil pllllr-Iflrif'pfmlf'nl mlf'8'ul.\· 1J5
i.e . W docs not dcpend explicit ly on ,111 thc X k (or the materi:11 is homogencou~ in all the X, dircctions). to derive from (2.53b):
.. ., [ J ( ,'" ) ((~ "" (,JA "" hm (W+T)Nl - 1 - X' ds . - 11 ' a k r .\
J ( a,i, . a,,) 1
- pli , ax, + (I, - pU , ) ax' dV C, . r, \ '
(2.55 )
The sen<;c of path-independence embod ied in (2.55) i<; simi lar to that in (2.49). (2.50). In the above. S" which is eq ual to (S: I + Sd) (+ anti - referring. arbitrarily. to thc crack faces) is the crack surface enclosed within r . whilc S, h, the total crack ~urface. Thus. an eva luation of J~ not onl y in volve<; a vol ullle integral. but •• 1<;0 an integral along the crack faccs. The infinitc!>imal Mrain counterparb of the J~ integrab have been fir~t <; Iated in [10]. based on a ~implc modification to the J, integrals for dynamic cf:lck propagation given in 13].
It b importan t to note the meaning of (2.55) - it still govern~ the energy release per unit timc. due to self-similar propagation (along the x ,-axis) . 1; would simply char .. cteriLc the total cne rg) change due to a unit lraflslm/oll of Ihe crae/.; as (l
whole. rigidly. in thc Xl direction. T hus. J~ lioes flOI characteri..:e the ellergy rell'tlSe
due to a unit motioll of the crack-tip in the Xl direction (.mcl thus kinkillg the origill<ll cr;lck) . In facl there arc no "imple integnlb that characterize the energy release due to kinklllg of a crack. as is often erroneously implied in the literature 113.14]. This is tlue to the fact that in deriving (2 . 18). "'hich forms tbe hash of all the en~uing p;lIh-integral .. thereof. use has been made of the sclf-~imilari ty of solutions at time I and t + dt. which is valid only in self·similar elastic crack propagation hut not in the case. in general. of a kinked crack .
A'>wming for the moment that the global and the crack- tip coordinate!, coincide. one m;.y tlefine:
J: -= lim . .-11 J ( W + T)N! - I, ~X", ) cis " ! / \. ,
J ( . ",i,
pll - . , (iX, f au , ) + ( , - pii,) (")X
2 dV (2.56)
which would characterize the total energy change for a unit rigid tr'IIl~ lation of the crad.. a~ a "hole (anti 1101 a III/il growlh oflhe crack-lip (I/olle) in the X z direction. A~~lIIlling tern hod) force. traction-free crack faces. and cla!>tO-Matic deform.lIion~. one ma) reduce (2.56) to:
J.= J ( WN.- t ~~.!... )dS+limJ(w · -W )dS . .' a,, ! ,·_ 11 , . .. (2.57)
S_N AII,m
wherem, for a Oat crack face, N; = - N~ = - I . The definition of) , of Budiansky and Ricc 1151. on the othcr hand. does not im oh e the crack-face integral. which ,tccount\ for tliM:ontinulties of \V along the crack face. Thus. a~ 1Iiso noted hy 116. 17).): a~ gi\cn b} liS ) is 1101 pmh-ilUlepemlent. Evc n though (2.57) appca r~ to lI1\"olve a knowledge of crack-tip W for its successful application as a p.llhindependent HlIegra!. the use of (2.57) has bee n concl usive ly demonstrated 112. 18\ in compllI(I{iolla/ approaches using simple (non-singuill r) crack-tip finite clements.
From thc .toove di<;cu<;<;iol1<;, it should be clear that neither the integrals)~ nor an} ot her ~imi lar l y "pa th-i ndependent" integrals prO\ide any information as to
kinking o f a crack or of the direction of propllgmioll o/Ihe crack-tip in anything other Ih;m a collincar fa~hion. contr,try to speculations often made in the literature [14. 19.20[.
U<;ing the a<;ymplOtic solu tions in self-similar crack propagation . even under arbitrary time history of motion of the crack-t ip . n<uncJy Ii, - C,JII, lax,. it is seen that Ihe energy- rclea<;c ratc expressio n in (2.5301) reduce .. to thaI of Freund 1211. It i~ worth noting that (2.53a) as well as Freund 's result arc valid for an arbitrary shal)(' of th e loop I ; ncar lhe crack-tip . On the o the r hand, if consider.llion is restricted to fully SII.'(U/y-s/(J{I! (i.e. the fi eld evcrywhe re is invariant with respect to
an ob!loervcr moving with the crack-tip ) .vei/-simiitlr prop:tgation at a COIISWI/t
crack-tip velodty, it is seen that el'('rywhere in V. one hll<;: Ii ,"'" - Crill / iJX I :
li ,. 1 == - Cf1 l ll ,inx~ : II, == C ~a ~ ll ,IiJx~. INo tc, ho\\e\cr. even at con<;tant \cloci l}. un'ltcad) cond ition~ in ge neral imply that : Ii , = (dIl 1dt) - qall ,lax l ) and Ii, =
(n ~ tI ,Iat~ ) + C~(n ~ lI ,i fJx~ ) - 2C(a l ll, l iJx 1 iJl)l. Thus. when bod) force,/ , = (). for slc:tdY-!low te. com.tant velocity propagation. the volume integral in (2.-19) disappea rs: and the re.'!ulting expression. wi th 2"1 = pC~(all ,/ a), I ) ~. becomcs ide ntical to that given by Sih 122). evc n though the far-field con tour conside red in 1221 movcs along \\ ith thc crack-t ip al the same velocity . It ma) be noted, however. thai such stcady-statc comtant-velocil) propagation seldom occu~ in practical problem<; of fa~t fracture in finite bodies: see [231 for fu rt her dcw ils.
Ina<;m uch a .. J'( ' G) as defi ned in (2.20) has a well-defi ned physical mcaning as the crack- tip energy release rate and can be convenicntly computed from sim ple numerical proced ures from far -field quantities through (2.49). it can be used as a parametc r governing elaslo-dynam ic cnICk propagalion and arre~t. The rel ations betwee n j~ ,md the dynamic su ess-intensit y factors arc given in 1101. j ' is. in general . a function of Ihe crack-tip velocity 1101. In a dyn<tmic fracture problem. init iat ion of propagation occurs at j' = ):. and during crack prop;tgation. j' = ) l~(C) where )I~ and )I~ are mlt terilll properties. Exa mplcs of IJredictioll of crackpropaga tion histories and crack-arrest using thcse crit erill and compa rison with experimen tal resu lts may be found in Refs. Ill . 12. 18, 23. 241·
As nOled, the far-lield pat h r in (2.49) is fixed in sp'lce. On the o ther hand. con .. idering a far-fie ld contour r to be a rigid path surrou nding the crack-tip and in tranS\;lIion at the samc \clocity C along Ihe Xl-axis. a p,lIh-i ndcpcndelll integral. denoted here by ) K' was given by Bui 125.26\ and Erlacher \271 for infi n itesim.t l
deformation:
n~a* , " " - Jil l - Pli, II ,.I C l /1 11ds + ~I J PII , /I ,.1 dV ,
137
(2.58)
where ( ) ,1 = ri ( ) inx l' When the material derivative for a Illo\ing control \olumc containing singularities is properly treated. it may be sho .... n (sec Refs. (10. 231) that (2 .5S) is equivalen t to (2.49) . I lowcvcr. experience has shown that (2A9) with a fixed path is ca~icr to usc directly in a computational scheme [11. 12. 18.23.241.
For linear elastic II1fl1eriafs undergoing infinitcsim:.1 deformations. In\in 1281 and Erdogan 1291 gave Ihe cxprcs'iion for cnerg),+Telcasc ratc in dynamic crack propagation. as:
aIr I
~( :J ,. f 1,(X1·/u)II ,/(X , ( a(t) - a(lu») ·lu]dx1 · (2 .59) ~ (',,1
Thus. it is the work of tractiom al I II in moving through the displacement<. lit the corresponding poims al lime tu + dt. The validity of (2.59) ha~ been eSlabli~hcd for line" r cI .. ~to-dynamics by Gunin and Yatomi 1301. On the other hand , Ache nbach (311 give", . for finite dcformation~ as well a~ non-linear ela~tic behavior. the expression for G. ;IS:
"" (2 .60)
" . where II , (X , .0 ' , /) denote the particle velocities at the cnlck surface~. x 2 = 0 ' . The tip of the crack i~ denoted by x, = (I. and E a sma ll number . Now. consider the pa th r. in (2 .20) to be a rectangle of height 28 (in the x~ direction) and width 2E (in the x. direction) and cemered at the crack-tip. Thus. we may write from (2 .20) that:
G = tim lim J (cw + -'-)/1 , - I , all ,) d, .~o 6 . () ilx .
(2.6 \)
Jail ,
= lim lim - t - ds T~U~_U ' ax. (2.62)
since " , i-~ zero on ~egments pa rallcl to the x , -ax is. and the integral of ( \V + -'-)11, vanishe~ ,I long segments parallel to the x ~ -axb in the limit 8 _ O. Also. ncar the crack -tip. Ii , = - Call ,lax •. Thus. it appears on first glance th,lt (2.60) is the correct limit of (2.62). 1100·\ever. this has conclusivel) been disproved by Yatomi 1321 who shows that the limit 8_0 muq be taken a/fer the integral in (2.62) is evaluated: and in any eve nt. (2.20) always leads to the correct result. eve n for fi nite-
DS S V IIllu"
deformation non-linear elas tic problems. irrespecti\c of the !>hape of r. . For the ~peci:.l p,nh described above. thc gcncral \aJidit) of(2.61). (2.62) is establi shed h) Guni n and Yatomi POI.
Stnfor<:. [191 and Ca rl~son (20]. on the other hand. apply the "princlple o f'irtual v.ork·· to an arbitrary part VI of the body containing a crad (a~ in Fig. I). which thcy considcr to havc a " finit e cohesive zonc". Their [19.201 definitio n of "an appa rent crack-ex tension force". ",riuen below. for in-.lancc. in thc~, direction. is arrived at by thcm [19.201 by considering ,irtual di')placements of th c form 811, - (ill ,la., I ove r VI as well as over a cohesi\'l! zone of "ite F. a~:
F :::: J [/ '/ (rlll l) ax1 ) - (f, - pl',)(a ll ,laX 1)] ds J 1,(all ,laX I ) d~ I ', I . \ t
. - limJI ; [(all ,lax l ) - (iJ lI ,lihl ) Idx l . (2.63)
• ·11
" From th e preccding argume llls. it may be seen that the ext reme right-hand sidc of Eq. (2.63) docs not havc thc mean ing of an cnc rgy-relca"e nile eve n in thc limited ~i tua tion -. when (2.60) I1la ) be valid . becausc of the onl ) one-<'idcd limit o f integration appeari ng in (2 .63). Furthcr. sincc I ,t (all t,,!a., . ) and (1/ ,(111,/(1.\' 1) rn a) ha\c singu l ari lie~ of ordcr grealer than (r I ), the limit of the integral ove r l', Illu!.t be considcred scparate l). Thus. e,'en though Fin (2 .63) i~ pat h independent. ils meanmg 1<; not clear. Kishimoto et aL 113] define a para meier such Ihal :
j = - J 1,(,]u ,lfJxl)dS ,
(2.6-1 )
where 1; i\ ;I non·di!>torting "small" ' con to ur \\ hich move" at the <;amc "peed a~ the wick- tip . Even though j as in (2.M) i!> defined in 113] a~ one o f the cornponcn h needed in analyzi ng crack growth a t an angle to the initial dircction. this concept i~ q uc~tionable for reasons discussed earlier. Further. for arbitrary ( ,j a!o> in (2.M) is th e nile o f work done on th e process zone of sile r. . by the ~urrou nding medium and i<; not the energy releasc to the crack-tip. From (2.64). and the diverge nce theore m, Ihey [13] derive the "far-field" expression :
. J ( ,)" ) J J :: Wil l - I, - ' ciS + ax I
1 ,-1., II
f ) il l/ , dV , (IX I J IV" l ds . ,
(2.65)
Note the presence o fa near-field integral on r;. in Ihc "far-field " expre~~ion (2 .65). In 1131 this inlegral ovc r ( on the righi-hand side of (2 .65) i\ dropped (sec Eqs . (24 ). (25) of [13)) by consideri ng a special c:l~e of r; to be a rectangle of size (2£ x 25) ce nte red at the crack-tip. Howevcr. it should be noted th;lt the intcgral
o f \\''' , over r; d()e~ not \ani~h for arbitrary r.. Also if the integral over I~ is dropped from (2.65) and the resulting integral is con~idered in the limit .... hen G is ~hrunl.. 10 I;. one obtains a nea r-field definition of j from (2.65) that i~ different ~rom the o riginal definition. (2.M)! Kishiomoto et al. [141 in a later pilpcr . redefi ne
j J 1 \\''', ',(JII,Ilh dl dS , = J IWII , ',(illl,iil.l.,>i dS+ f (pli ,-f,)((ill ,la.l.,) dV
, ,
(2.66)
(2.67)
and !;On~ider [141 j in (2.60), (2.67) as th e "energy relea~e rate per unit o f crack trarhilltion in the x, direction". Comparing (2.66) with (2.20). it is seen thllt ~tleh i~ not the case for arbitrary shapes of r;. si nce (2.66) docs not contain the rate of change of kinetic energy in the e ne rgy balance for dynamic crack growth .
It is ea~y to see th,l1:
J pII ,(a ll ,lilx l ) dV = J [J(p'-i ,II ,)/iI.t·1 - pII ,«(ill ,l .. h,)l dV \ I. I
= f (pll,Il,II, ds)ds- f PII ,II " ' l ds I ~ \.. I
- f pII ,(iI ,"i,lJx , )dV . (2.6R) , ,
Using (2.68) and the rather ex traordina ry case when I, arc constants li .e. I.'" I,(·l dl. one rna) derive from (2.67) what Ou)ang ]331 defi nc~ a~ a parameter Y1 for ela!>to-dynamic crack propagation (and an associated parameter Y~. slightly different from Y\, to account for plasticity). as:
y ) = j + J PU,II ,", ds J [(W + p'-i,II, )1I 1 - ' ,(all ,!il.t·,)1 cis , . , .
(2.69) , ,
Comparing (2.69) \\lth (2.20) it is see n that Y, is not in general an energy rele ase rate for e l a~to·d) n amic crack propagation. ,md hence ib usc as a fracture parameler is questionable . Likewise. the parameter Y1 of [33\. which is the integral in time of Y, lsimilar to Ihe time integral of G of (2.17) which would give the tOlal fraclUre energy up to the current time 11 is not a me'lIlingful fracture parameter.
,., S,V Ar/u,;
More recently . however. Auki CI :11. [34) define a parameter .... hich. for cla~lo-dynall1ic crack propagation. may be defined as;
j= J [(W+ T)II) ' ,(iJlI ,IiJxll]ds (2.70) I,
(2.71)
Note the prese nce of a ·'nea r-tip·· integral (over I~) in the suppo~cdly far-Held expression (2.7 1) for I They [34\ go on 10 consider the limit of (2.7 1) for two
differelll shapes o f I~. In any event. the presence of the integral o\'er r. makes it inconvenient to u~c (2.71) in <I meaningful computational sense (i.e . without an accura te ncar-tip modelling) . [I is a simple mailer to usc the dive rgence theorem and elimi nate the integral over r. from (2 .7 1); in which case. the resulting far-fie ld expression for the energy release rate is none other than J' of Eq . (2.49).
Finally we mention the following path-independent integrals of Ni lsson (35] and Gunin [36]. respectivcly, for :l sUltionary crack in a /ineM clasto-dynamic field:
I( p) = J H ~V + ~ Pol/ii/i, )111 - i,(ali,!ax d] ds (2.72)
and
(2.73)
In (2.72). IUl) dcnotes a Laplace transform of let) and n denotes a Laplace Iransform of ( ). Likew):!>c . in (2 .73). [(f)*(g)] denote:!> a convolution integral in the timc domain. of Iwo functionsf(l) and g(l ). Thus. both (2.72) and (2.73) do not easily give the illS/(JnUlllfOIlS valu e of the crack-t ip parameter. which is use rul in analyzi ng dynamic crack propagation and :!rrest in a finite body . Further, in the case of a sUl/iollary crack in a dynami c field, the energy release due to incipien t crack growth (I{ allY illSUml of time is given from (2.20) and (2.21) as:
G .u""nm = J [Will - t,(au,lax.)1 ds (2.74) , ,
" (all ,lax.)] ds . J all, Itm (f, - PIt,) - dV (2.75) ,-II. ax.
\, . ,
(2.74) follows from (2.19) since T is no longer si ngul ar at the slatiollary crack-tip; (2.75) follows from (2.74) due 10 the divergence theorem.
We now turn to a class of "path-independent integrals" derivable from the
'41
'.pplic.Hioll of Noclhcr's theorem 1371 in Ihc form of "con~crv"liol\ 1<I\\'s" (3. 3H- ·[!J.
2.3. " Comerl'(l/iOIl 1(l1I'~" (It"/ their relewlI/ce 10 /mclIIre mechal/ics
The density of the "Lagrangian" for a (linear or non-linear) ellisio-dynamic prohlcm i, defined as L (\V - T - P) v. here W is the main cncr~D den!>!I}. T the kinetic c nerg) densil}. and P Ihe polential of external forces . In Lagrange 's descnption of motion (\\llh material coordinates X, as independent variables). L may be c()ll\idcrcd. in general. to be II function of the variables Y,,/(=(]Y,liJX/ ) (or cquh •• kntl) of 11,). Ii,. II , as well as that of the independen t variables X, (for a non-homogeneous "yslcm) and r (for a non-holanomic sY'., \cm) . Thu~.
, L "::: f J L(X,. 11 , . II,. U ',I ' /) du dl. (2.76)
I" I
Nocthe(~ theorem [371 concerning the invariance of L · with rc~pc(t to (enain trall!.forrnations of the <lrguments of L leads to corresponding conditions which may he labelled a., conser-ation laws. Eshelby [4. 7. HI \\a~ the first to intuitivc1~ recognize the importance of these in con nection with ··forces·· on point defect!. and crad.~ . Gunther [381 was apparently the first to appl) the formalism of the Noethcr\, theorem to obtain general conser-'ation la\\~ in c1a~to<;tatics . Knowle .. and Sternberg 1391 provided. independently. a thorough treatment ,llso in the case of finit e elastostatics: and this work was later extended by Fletcher (401 to linear c1asto-dynamics. although the claim in (.wI that Eqs. (3.1 )-(3A) :lnd (3 .6) therein can ca~il}' he extended to finite elasticity should be viewed with some caution . More recently, Go1cbiew~ka- 1-Ierrmllnn 142. ~31 prc~cntcd studies of con~crv;lIion law!. in finite elasto-dynamics using both Lagrange as well as Eu1cre'lIl descriptions of motion .
Ilcre we brieny discuss the case of Lagrange's description of motion and conside r the COllscrv;H ion law~ that ari~c from (2.76) when it is required to be invariant under va rious tr:msformations, when body forcc~ I, arc present:
( 1) invil ri,mce under tillle translation U = t + ,):
d _ a aL I -d (W+ r) + I ,li, + ax (tt 1i }=- - . t l I 1 at .. pl .. ,1
(2.77")
when L docs not depend on I explicitly. this leads to a "conservation law·· for a closed volume V· (with a surface r·. see Fig. 1) that does 1/0/ contain the crack-lip:
(2 .77b) , . . . , .
.\. V AI/un
Eq. (2.77b) is analogous to the energ}-ba lancc rel;llion (2.R) except for a subtle difference: (2.77b) docs not impl~ any crack growth. whcrca~ (2.8) is wrinen 'pccifically for crad, gro\\Ih.
(2) imariance under translation of Y,(Y, = Y, + t', ):
t" + f, = fm . (2.783)
I lenceforth in IIIi5 seclion wc u~c the notalion. ( ) ,' S fi( ) '(1X,. Eq. (2.78a) mol} be wrillen in in tegral form a!>:
J N,t" cis + J f, dv - J pi;, dv = n . (2.78b) I • I · • I -
Eq~. (2.7Ka. b) arc. re<;pectivcly. local and global equation~ of balance of linear momentum.
(3) invariancc unde r rotations of Y, (Y, = )', + e'l l. WI Y. ): Il ere. e 'l l. i~ Ihe allernati ng tensor. The h" lance law i!>:
When (2 .78;1);'1 u~cd. it i<; seC Il that (2.79a) i!> but a disguiscd form of the angular momcntum balance. (2 .2) . The corrc!oponding "conscnation 101\\" i!>:
J 1" 11. YI( II. PI'.) du + J 1" 11. YINptpl. d.f = (J . (2.79b) I • T '
Note that V· is the volumc in undcformed configuration. (4) ill\ariancc under translation of X,(X, = X, + e,):
Note Ihal tran .. lation of the coordinates in the llIu/elormc(/ body arc con<;idercd (or equivalently the translation of th e clastic field rcfcrred to the undefonned geomctry). The ba lance law is:
(2.80a)
When wc consider: (I) a volume V · that docs not con tai n any s i ngular i tie~ .. uth that each of thc terms in (2.80a) is il1fl'grable in V·. and (2) L doc~ not explicit ly depcnd on X,. i.e. the ma~eria l is homogeneou", in alllhc X, directions. \\e obtain from (2.80<1) the conscrvat ion law:
J (:t (pli l ll l _, ) - I~ 111.' ) dv + J [( \V - T)N, '1111_' \ ds ,-; 0 . (2 .80h) I . ,..
The ahove con<;cna tion la\\ . and its alternative representation". were di'cu"<.,cd In
13\. Note that E!.helhy [4. 7, 43[ names (he te rms in square brackets in (2.80a) the "energy- momentum te nsor".
(5) imariancc under rotations of X,(X, = X, + e" lll',X.): Thi~ is po~sihle only when the (linear or non-linear cI .. stic) m;lIerial is isotropic. The bahUlce law is :
(2.8Ia)
When the global angular momentum conservation law (2 .79b) h u~ed. the conservation law corre':lponding to (2.8Ia) can be written as:
o e".{f [(pII", [,,,)U,,I.IXl +Plim,im,X, plI,lIddu , .
+ f [(W T)X,N, + N",tm,ll, Nl' il,mUm,X,[d.\.} (2.8Ib) , .
where V· is the volume tha t is void of <Ill} singula rities. (6) invari:U1ce under scalc changes of X ,. tIX,: (1 + F)X, . ,-".." (I + F)/j:
This i .. po .. sible onl) "hen the material is IiI/ear. The corre~ponding conservation I:m III lil/I'ar l'la.'ilO-dnwm;cs is [40[:
f :, (pli ,(II , + 1I" X l + Ii ,') + Lt) du , . + J {tN/XI N/"(U,+ U, ,X, +li,r)}ds = O. (2.R2) , .
Now \\e c()n~idcr the applica tion of the conserva tion law~ (2.80b) and (2.8Ib) to non- linear da~to·d) !lamic crack propagation. We consider a volume VI V, wh ich docs nOI contain the crack-tip. \, here r is an) path enclosing the crack-t ip, V, i~ a small vol ume wit h the boundary f ~ also enclosing the crack- tip; thu~ r + S" - r. is the bound:lry of V V.. Note that Ih~ divergence theorcm. in the prc ... encc of possible llon· in tcgrable singularities. may be applieu on ly in V - V, in the limi t (I:'
F_ O. Ba .. cd on the<;,c arguments. further elaborated upon in [3 1. \\c obtain from (:U';Ob) thc pmh-ilU/ep('1u/£'1I1 integrals:
lim f [( \V - T)N_ - t,l/ , ll d.'i , .11
(2.K]a) ,
lim , . 11
f I(IV , , (2.83b)
, .... S.N AI/urI
Comparing (2.83a. b) wilh (2.20) and (2.49), it should be evident that i l of (2.83a) arc 110/ associllf('(J 11/111 ,I/{! ("(matJi of all energy relellse ratc. hut rather (as the root:. of thei r derivation would indicate) are associated wil b the rate of change of LagrangiclO L · of the system, due to unit translation of the crack in Ihe XI. direction [31. ThaI the eq uivalent "energy-momen tum tensor" in c la),to-dyn:Hllic~
doc)' not lead to an energy release rate was also noted by Eshclby [71. Thus. the relevance of (2.8.1a) a~ a "fracture parameter" i~ vacuous. Likewise. letting global Xl coincide "ltb the (;rac"-tip local Xl' the integral of (2.83b) in time. say.
, ,
Y1 = J J] dt "- !j~l.J ( f f(W 1')/1 1 - IIft/,l ldS/ dt t" I • . ~.,
,
-f (J (f1.1l4.])dV} lI t+ f Vli/II /.l d UI:" (2.K') " r, \ , \', \',
has little relevance to fracture - it is the total change of Lagrangian from 10 to r. Eq. (2.H4). in a sligh t I) less general fo rm . for the case of infinitesimal deformation. along with the assumption of a nlther special ~e t of COl/suml body forces. i.e . Il #- It (.r,) (which rende rs the volume integral of flllu to be a surface integral of ItlllflJ)' appea rs in a paper by Ouyang 133).
If one reali/e~ the identi ty:
(HS)
and adds (2.85) to (2.S3b). one recove rs the in tegrah.J~ of (2.20) and (2.49) which arc associated with the ene rgy release rate. as was done o riginally in 1101.
Analogously to the \\a) in which (2.83) is derived from (2.80b). we mily derive the follov.ing path-indepe ndent in tegrals from (2.8Ib):
L, = ('"k f [(W - T)X!N, + Nm l""II. NI,I,,,,,lIm .,X!1 dJ , ,
., e 'l l { f (W - T)X*N, + N", t""I1* - Npt,,,,,u,,, ., Xtl ds r· \r
+ f [{pu", - fm)Il", ., X k + pli",li", ,X! - PII ,lIddV } (2.86) r, \',
which would have the meaning of the rate o f change of Lagrangian L pcr un it rotation of the cnlck. In o rder to obtain an equivalen t "cnergy release" interpret ation , we may add the iden tity
f J~ (2Te,!, X A ) du - f 2e l /. TX,N, ds= f 2e l !lTXl N, us r, I , /_\, I,
to (2.86) and obwm:
L ; e"l f I(W + T}x~,! , + 1I ", l nol ll, - 1I ,.1,.", II"", x , 1 ds ,
= e"l { f ](\" + T)X, II , + " n,tml lll " ,.I,.m ll ", IX, 1 ds / .1.1
+ f [(PII ,-fm) IIII/ "Xk pli m,i,,"Xk Pli,/l ( ]dU} . , ,
IJ5
(2.87)
(2.88)
Finally . we not e that the 'io-called i\.f-integral for lim'{/{ cI;t~\() -dynamic~ call be ocrivcd from (2.82).
1 . .J . Complemelll(lfv repre,~ellt(l/;Oll of pml!-illdependellf illff'gm/s ill (lloll-finellr)
e/(l\"fo·tlYI1f1mic fracilire
I-I ere we define the complementary energy densit) (per unit initial volume) of the mate rial. denoted herc by IV, . through the con tact (Legendre) tran;,fonnation ,
(2.89)
EYalu;ltion (If (2)~Q). however. invol\'es finding the invcr'\c of lhe mCl;s-'Ilrain relation.
(2.90)
Tha I the in\'er~e of (2.90) i ~ nol unique is now well c<;t abJ ishcd {I . 21 . 1-I 00\ cver. by defining th e <,o-c<tll ed !'Iecond Piala- Kirch hoff st ress te11"or,
(2.91 )
we Illay definc a "valid " cO(llplemcmary energy:
11'(" )- r C' \\'(C') , "" - "'I 'I - 'I
(1.91)
\\here C" = (YA. , }',., ), E\aJuation of (2.92) ill\ol\cs finding the in\ er<,e of
(2 .93)
146 !} \ AI/un
\\ hkh i.., known 10 be uniquc [1.21 ... lIeh Ihal :
c,' (~ IV (1S . " (2.94)
We nov. define a Lagrangian in terms of the com plementa ry energy. IV~. as:
,
L(S", II , . Ii ,. 11,- .X,.I)= J fIS'I C" IV.(S,, ) T - P ]dudf, (2.95) r" I •
B~ appJ)i ng Nocther' .. theurem. II I l> 110\\ possihlc to dcri\c a va ric IY of conse rvation laws. and complement,lTY path-i ndepende nt integrals. fro m (2.95 ). We omit further detaih. but refe r for cxamp lcl> o f Ihese \0 15. 441.
We conclude this ,celion h~ noting tha i of Ihe !teemingly infi nit e \ariet ic~ o f " pa th- independen t integrals" and att enda nt "conservation l aw~" possible in nonlinear (or linear) cl<lslo-dyn;Ullic crack propag.lt ion. on I} J' of (2.28) and (2.49). and Ihe eq ui\alent ) 1I o f (2.58) have the propcnies: ( I) they characterize an e nergy relc;!!>e rate due to crack propagation. (2) they are mca!>urable. and (3) they a re
meCl!>ure" of d) namic cnlck-ti p fields. In linea r elas to-d~ namic crack propagation. even though ~ome of the olhe r " path-i ndependen t indepe nden t in l egT<L I ~'·. such a .. j of (2.M) and (2.66). and J~ u f (2.S3a). do not have the same physic;11 meani ng as (J' l\fHJ JII ). (hey ma} he related (Q the dynam il.~ stress-i nten!>iIY factors 11<.(/)1. Such rel a tiofh, wh ich are, o f course, d iffe rent from tho!>c betwee n J' and k(t), are given
in 1101.
2.5. E.\·1Jerimelllal metlsllrtlbility of J' for elastic mll/nitl/s
We co nside r here thl.' q uestion o f the mea, ura bi lity. o n laboratory tcst ~pec i
mc n~, of the m.Herial propert y, namel). the rate of energy release per uni t crack gro\\lh . G of (2.8), which ha~ been defi ned ;l lso as ,I pHth-indc pcndcllt intcgral J' th rough Eqs. (2.20). (2.28) a nd (2A9), fo r eta.wic mate rials.
We consider two test speci mens o f ident ical geometry except fo r the lenglh~ of
crack!> which arc (a ) and (a + d u) , respectively. in the two bodi e!>. We conside r th e two hodies to be suhJetl to idl.!n tical load histories. Le t this " loading" be through pre~ rihcd di~p lacemcnts II, on a portion 5" and th rough prescribed tractions t, on a
portion 5, of the boundaries of each cracked body; and le t the body fo rces /, be zero. Prior and up to the Ollsel o f c rack propagation. the equilibrium process in each body implies bee [.l4]):
J (W + T)ln l dv == - J J II: ") <I I, d.f + J i,II:~) <Is + J J 1:"1 dli, ds. \ ,\11 I '.11
(1.96)
I',/u"grl/c l'ppT(HJchn Imd parlHmfrprlldl'fu ml/:gflll~ 147
In (2 .96) the \upc r\cri pt «(r) ..... hich varics from I to 2. denotes the cracked body in quc-s tion. The fact that the crack length in the second specime n i'> larger than that in the fir!>t hy (da) b reflected in : (I) the strain and kinetic cne rgy densitie!> Wand r. rc !>pcctively , being different in the two specimens, (2) the dbplacemcnts Il , at the traction-prescribed houndary .II, bei ng different for the two specimens, and (3) the tr:lctions ;11 the displacement-specified boundar} 5" being differe nt for the 1\\0
specimen'> , Thu,> , from (2.96) one may dcri,e :
, "
d J J J du , - J - du , J J do (W + T)du= ~ da d /,ds+ r, dll ds + dt, _ da dll, ds .
I 5.0 I, .1.0
(2.97)
Lei (dA) reprc~cnt the ~um of: (I) the difference in area under the load (i,) ve rsus di-splacclTlcnt (II,) diagram for surface 5, and (2) the difference in area under the load (I,) ve r\u~ dbplacemcnt (ii, ) diagram for 5
11, Note that the sign conven tion
employed here implies thaI:
> 0 '.
at .II,;
Thu,>
,
J J du, - J J dA - - dl ds-da ' , " S. 11
dr I d~ u < 0
dr, _ -dduds.
u ' (2.98)
Using (2.97) III (2 .98). one has:
J - du d J dA = t -' ds - - [W + T] du . , dll da
(2 .99) ,
A~ long as the cracks in both the bodies remain stationary, the stress-strain fi elds in both the bodie!> may be a~sumed to be of similar mathematical form, at ;,11 times. Thus. using procedures si mi lar to those in con nect ion with Eqs. (2.15)- (2.20), one may write :
dA = J ( au ) J - au J a( IV + T) (W +T)1l 1 -t,-' du+ t,-' ds - du o ax 1 aa i/o
s, I (2. 100) I .. "
Recall that Eq , (2 .5 1) holds only for purely elastic materials. whose properties arc independent of loading- unloading histories. For such materials. uo;ing (2 .5 1) (in the absence of body forces), one obtains from (2 .100) that:
'" S.N AI/IITI
dA "'" J (( W + T)Il, - I, "a:~ , ) ds - J (PIt _"'_i, - pii au..! ) dv "" ' ax,' ilX ,
I , , \ I ,
E f ((\V + T)II, - I, ~:) us . ,.
(2.101 )
It i~ importiUlt to nOle that Eq. (2.101): (I) b not valid during dynamic crack
propagation even for purc:l~ elastic materials . ~incc (2.96) ill not valid for propagating cracks: and (2) is not valid for e/aslo-IJ/aslic materials.
On the OIher hand. during dynamic crack propag;lIion. J' can be determmed cxpcrimclllally. from Its definition as an integral mer r.. if the rclc\ 'anl ncar-lip data loan be measured experimentally. Bcincrt and Kalthoff 1451 have been successful in applying the method of catl~tic~ in measuring the dynamil.: ~lrc!>!>
intensity factor ncar the tip of a propag:H ing crack. From the thus-mCH~urcd dynamic K-faclor. J' a~ a function of crack velocity fo r a propagating crack can he determined using the crack velocity-dependent rcl:ltion between J ' and the K-factor given in 110[ . On the other hand. if pertinent data 10 evaluate the integral on the extern .. 1 boundary S of the specimen as in (2.-l9) can be measured. one rna) usc a hybrid experimenwl-numerical procedure to evaluate the integral on V - V, and determine j' from (2.49). wherein n s considered to be the external boundary .
3. Inelastic (llIId dynalll ic) crack propagat io n
We fir~t con~iuer crack-growth il1iti(lliol1 under qll(lJi-stfltic condit ions. in clasticplastic material,>. The most widely u~ed parameter so far. and the one Ihat has made possible certain impressive advances in elaslo- plastic fracture has been the l-integral [~61. In the context of incipient self-~imilar growth. under quasi-static conditions. of a crack in an clastic material. 1 [which is equal to l ' \\hen Ii , and II , are sel to zero III (2.-l9)1 has the meaning of energy release per unit of crack extension. A~ in the case of j ' of (2.49). Ihe palh-indepe ndence of 1. e\aluated now only as a contour integral. C;1Il be established when the strain energy density of the material is a single-villued function of !>train ;lI1d the material is appropri:l tcly homogeneous and the bouy forces arc ze ro. In a deformation theory o f pla!>ticiIY. which i~ valid for radial monotonic loading but precludes un loading (and t h u~ i~
essentially and mathematically equivalent to a non· linear theory of elasticity). J still characterizes the crack-tip fields. However. in this case 1 does lIot have the mealHng of an energy release rate; it is simply the total potential-energy difference between identical and identic'llly (monotonically) loaded cracked bodies which diner in crack length~ by a differential amount . It should be emphasized that even thi~ interpretation of 1 under a deformation theory of plasticity is "alid only up to the point of crack growt h 1I1Itiation 144]. as disclissed in Chapter 3. Moreover. in a flow theory of plastidly. under arbitrary load hbtories. the path-independence of J.
evaluated ..- .. a contour integral. is no longer valid: and further. under the~c
circum <; tances. J docs not have any physical mean ing . 1100\ e\er. ~ign ificant advances have been made. in the past decadc. in the
problem of crack grow th initiation in monotonicall y IO<lded structures, using the conccpt of J-integral. The principal contributions that made these possiblc may pcrh<lps be identified. as : ( I) the \\ork of Hutchinson 147] and Rice and Rosengre n 1481, .... ho ~how thalthc stresses and strains near the crack · tip in a monotoll1cally loaded body of a pure pov.er-Iaw hardening material. under yield ing condition:. \arying from small -scale to fully plastic. are controlled by J; (2) the work o f Beglcy and Landc .. ]49] and Rice et al. [50[ on the measurcmen t of J frOIll ,>malllahuratory teM ~pecimem : and (3) si mple procedures for estimation of J . by interpolating be tween fully plastic solutions and clastic solution~, based on the work~ of Bucci e l al. [51 [. Shih and I lutc hin~on [52]. and Rice et al. [50]. On th e other hand . a large amou nt of crack growth in a ductile ma terial is necessa rily accompanied by a significa nt nOll-propo niOlwl plastic de formation which invalidate .. the deformation theory of pl:lsticity. Thm . the valid it y of i. as a contour in tegral as defined by Eshelby [4] and Rice [461. is questionable under these circurnst'lI1ces. Fo r limited amount.. of cracK growth. howeve r. Hutchinson and Paris [53[ argue that the far-field J , denoted a., i f in Chapter 3. is sti ll a controlling parameter. For such ~i tuation !. of if-controlled growth. Paris et al. [54] introduced the concepts of a " tearing mod ulu~" and"J resistance curve·' to analyze the ~Whility of such growt h. Using th e aho\c concepts and the related concepts of CTOA. engineering approache, to cla .. tic- plastic fracture analyses .... ere elaborated upon by Kum;lr e t OIL [55\ and Kanninen et al. [56]_
The mechan ics of crack growth initiation, and substa nti .. 1 amounts of stable growth. in clastic-plastic material s subject to arbitrary load histories is not yet umlcr!ltood . Thi!. ~ ta te of affairs is due, in pan, to the rea~on cited hy Rice 1571 in 196R that " . .. no succe~s has bee n met in attempb to fo rmulate similar general results for incremen tal plasticit y".
Among the first attcm rm to find a suitable parameter. that is theoretically val id in ela~tic-pla .. tic frac ture mechan ics, were those by Bi lby 15H ] and Mi ya moto and Kageyama 159[ who defined an integral:
i "," :-:: f ( W"N! IJ:l ~. )ds , (3. I )
where W " i .. the clastic ~train energy density , and /3,. is the ··clastic di stortio n tensor"' !luch Ihat the incremen ts of clastic displacements .Ire given by: dfl ~ = (3 ~ d \ ,. The integral (3.1) is path-independen t only for paths in the region of th e hody that remains ela~tic, but is path-dependent for contour; pas~ing through the plastic region. Some st udies on J,, ~ , wcre presented b) Miyamo to and Kagcya ma [59.60[ .
1<;(1 .\ \ All""
Alw. from time to time. ideas of "encrg) balance" and "energy release rale,>". similar to those in the prcvious scetion (Scl;lion 2). arc pre!>cnled in the literature for clastil;- pla!>tie materials. Il owevcr. such ideas of "cnergy relca,>c rate·' are well
known 161.621 to he unworkable for el:l~tic-pla~ tic materi'll~ \\ herein stres~
loalurale~ to a finite value at large \alues of st rain. In such matcri .. ls. under
qua!:!i·!>!alic eondilion~, 1\ has been lohown 161 - 63] Ihal the energy release rale vanbhe,> [i.e. the value (.lU ~/.l (l) lend~ to zero when .lu -O, where .lV· b the
totul change in global cncrgy due to crack growth by amount .1(/1 . Of cour::.e. the tOlal energy release for a finite growth !>tep ..1(/, denoted a::. C" .1 , remains finite and depends on .la 162.631 . It is this dependence on the size .1(/ that precludc::. a mtional utilization of the "ene rgy releasc" concept or the generalization of thc original Griffith energy balance concept. in ela!:!tic-plaloic fracture mechanics. Abo. the derivation of inlcgr:ll<;, that may characterize "energy rc1c:I~e·' even in finite
growth ::.teps, along the lines of tho!>c in Section 2. arc no longer pos<;ih1c in clastopla~ticity, since the ~olu lions near the crack-ti p at time I ;md lit time (I + ..1 1) (during which the crack grow,> by ..1 (/) arc. in general non-steady crack propagation cases, no longer self-similar - due to the claMic unloading that aecompanic~ crack
growth . First conside r a cnlcked clasto- plastic bod) that is !>uhJect to quasi-!.tatic.
monOlOnic. and proportional (n.dial) loading ~uch thai a deformation theory of plasticity may be valid. Further, we consider (I) the material \() be homogc llcou,>,
at lc .. ~ t in the Xl direction; (2) the loading to be only through !>urf .. ce tractions. i.e. the hody forces <Ire zero. and restrict our attention to crack gro," th initiation onl) . For this case of .. stationary crack. one rna) define a crack-tip pluameter:
f ( ')". ) , )1 = Will - I , ,1x dl , , ,
(3 .2)
For .. s1:ltionary crack. the integral in (3.2) renMins finite. for all values of F '>uch
Ihat 30 < F < R" (see Chapter 3). For the clasto-plastie body. U' is the total stre~~·,"ork at a material point (per unit volume) and is defined as:
•
U' "" J a'i df·'1 '
" (3.3)
Under arbitrary history of ~tnlining. U' in (3 .3) i::. not a single-valued function or f ·'I' Ho,"ever. under conditions of validity of the deform ation theory of plasticit) as
delincatcu above, W may be considered to be a si ngle-val ued function of £'1" Thu~.
under the restrictive :1,>:-,urnptions stated above,
(3A)
'" 111l1\, J, in (3.2) i:. a path-independent intcgml in itse lf. \\ilhout the presence of a domain integral: and hence i , = J. where j is the integral over an} arbitr,lf) contour ,. of the irllcgrand \\hich is identical to that in (3.2).
On the other h:md . consider a stationary crack in a homogcncou::. (atlcast in the x 1 direction) claslo- plastic bod) that is subject to dynamic !ourf,lCC traction~. such that materialme rtia plaY''' a dominant role . In thi::. (a::.c. one may define;1 crack-tip
parameter [which remains finite for any f similar to i , of (3.2)1. a\:
l' J (au,) (W + T)Il , - I,- du o ax ! ,
(3 .5 )
Under a general lran::.ic nt dynamic loading. a deformation th eory of pl;t::'licit y clocs not. in general. hold. i.e. W is not a single-va lu ed funclion of 1'", Thus.
J ["(\~~ T) ,
II ( " "')] - <T - du ax, 'I ax!
illI,) _ (; I/'j - + f - d,. ;Ix ! ' (1x 1
(3.6)
In (3.6) it is implied that (ri \vl ax ! ) is evaluated dircctly b) fiN calculating \V as per (3.3) and then differentiating it with re~pcct tox!. Thu~. Eq. (3.6) implicsthatj ' in (3.5) i~ not a path-independent integral by itself. and the domain-illlegral of(3.6) is prc!>cllI in any path-independent integral definition for J'.
Thc physical meaning of J, of (3_2) under the restrictive assumptions mentioned earlier. hllS been discussed in Chapter 3 of this book. while J' of (3.5). in the context of an ela\tic-plaSlic hod). has no ph)~ical i11lcrprelation other than that it is a parameter thaI quantifies the crack-tip fields .
Now con~ ide r the case of quasi sta tic stable crack gro\\lh in an e l ;btic-pla~li c
body. If a two-dimen sional situat ion is considered. any integral over an arbitrarily small circl/hlr path I~ near the crack-t ip (with radius F being small and tending to zero), with the integrand being !ouch that: (I) it depends oillhe strc!>s. slr;lin . and di'placcment state ncar th e crack-ti p. lmd (2) it has a (1 / r) variation ncar the crack-tip. would se rve as a valid crack-tip parameter. Since the integrand has a (Ih) variation at J~. i\ is seen that the integral crack-tip parameter remains fi nite. This crack-tip integral parameter is then sought 10 be represen ted equivalen t I) a\ a far-field integral plus a "finite domain illlcgral", using the divergence theorem. This .. lt e rnalive representation is convenie nt for computational :lIlalyses of fracture problems. Under certain idealized and special circumstances. hov.cver. Ihe aforementioned " finile domain integral" vanishes identically - Ihus making it poso;ible to e \press the crack-lip integral parameter solely as 1I far-field contour integral. To define a crack-tip integral parameter of the aforementioned type. a knowledge of the .. teady- as v.ell 'IS non-steady-state lIsymplOtic .. olutions ncar a
1~2 S 'V At/Ilr1
grm .. ing crad. III a hardening material i ~ nccc~~<l ry. While some progress has been made in reccnt years. ~uch a complete asymptotic solution ye t remains elusive for the practicall) important problem of mode -I crack gro .... th in plane strllin / pillne .. trcss. For an ideal!} plastic material. the asymptotic crack- tip fields in a mode- I problem ha\l,! been recently "tudied b} Slepyan 164]. Gao \65]. and Rice et al. [66]. Later. Rice refined thc~e !.olu tions [671. While the solutions in [671 arc valid during non-.. lcady a .. \\cll a .... ,cady-slale growth . the other SOlutions 164-66\ arc \'alid only for .,tc,uJy-\t;l\c gro\" tho Finall). Gao 168] has developed lin a~ymptotic !.olution for .,teady-<,tatc gro\\ th in a power-law hardening material; ho\\e\er, it was later noted by Gao \C19] that there is a deficiency in these solutions. namely that the pla~tic part of the ~train rate doc" not vanish as 0 ( the angular coordina te centered at Ihe crack-tip) approaches the boundary between the plastic loading and elastic un loading ~eclor!>. Tim." the i ~.,ue ofasymplotic fields ncar the crack -tip in mode- I growth in a il train-hardening material is in need of further .. tud y.
If one con,icler ... Ihe problem of initia tion of growth from a stalionllrY crack in an cla~lic-p la~tic .. oli<.l under .Irhitrary load history, it is clear Ihal J, as defl ned in (3.2) remain .. flnite for any value of F. including E _ U. Mo reover. under the restrictivc :I~llumplions which validate ,I deform.llion theory of plasticity, for a stll tionary crack. J is path-independent as discussed earlier. As the crack begins 10 grow, there is no rellson to expeclthat the crack-tip integral as deflncd in Eq. (3.2), wh ich had tlCen flnile until gro\\lh initiation. would not continue to remain flnite. On the other hand ..... ith elll!-.tic unloading accompanying large amounts of gro\\ th and Ihe con~cquent in\alidation of a deformation theory of plasticity, one \\ould expect that crack- tip integral as defined in (3.2) would nol remain path-independent by itself. without the presence of a domain integral. With these in mind. one may defl ne a crack- tip paramete r for quasistaticllily growing cracks in clastic-plastic
solids under arbitra ry histories. as: 2
T' J( au,) Wil l - I, ax. (3.7)
'. where the de nsity of stress-work. W. ill ail defined in Eq. (3.3). The pathindependent integral representation of T*, which now involves a domain-integral term, may he written. using the divergence theorem. as:
.,.*= (3.8)
ror reasolls rullydlscu~sed b)' Brusl el al.l71. 72].lhe path /'mthe dcfilllUOIl of T' of(3.7) IS as rollow~, In mode· ' !,""rack gro"'th. al an) cr<lci. length. t: Illelude~ a Sf'mlCi,d~ centered at the currellt end·up <llld of radiu) F. alld tra,,:rse) along the "'ake or the alll,m~'ns crack-up alld paralle1lO the enel. aXI) 31 a dlst:mee -F. Thus. r: ill the defimtioll of T' of(3.7) I) no lunger a n,d~or r:ldlu) F 3S III the delinilioll nr J, of (3.2) for a St3110naT) crack
r.nrrl!"!w (IPf1TO(lrlu' \ WId fllllh-mdrf'f'fu/rlll mlegm!1 L~]
wherein ( (1\Viu.\ I) i ~ e\al uated by fir..t computing \V at each mat erial point a~ per the definition given in (3.3) and then computing the partial of \V \\ Ith re~pec t to x I '
The 'olume Integral in Eq. (3.8) docs not. in genera l. \ "ni\h for prohlel1l~ of cr.lck gro ... .th III linite hodie ... If o ne defines a far ·field i , through the ('quation:
J, f ( au,) W" I - I, -. dl'. a.t I
(>.9)
it ;~ clear that the crack· tip para meter T' differs from the f,lr-field i f through the term:
. f (a \V t1c" - all ,) f .-: J - --a-+[ - du I . iJx 1 'I ax I 'a, , ,
(3. IU)
where V i, the total volume, While the volume integrals in (3.8) ;lIld (3.10) do not vanb·h in the general cases of crack growth . there Illay be !'pecial ci fl; Ul1lstance~
when they do . Consider. for intance, the hypothetical c:lse of a IOwl/y \ tea dY'~ l ate
crack growth . Le. a sit uation in wh ich the stress /strtlin ficld~ not only asym ptoti cally clo~e \0 the crack-ti p. but also e"erywhe re in the solid, remain invariant wilh respect to an observer movi ng with the crack-tip . To this end. conside r a t"o-dimensional problem for inslance: and let ({l>X~ ) be a coordinate system cente red al the moving crack- tip . such that:
(3. " )
where A i~ a monotonicall) increasing time-li ke parameter. ;iI1d c(A) is the coordina le of the crack-tip in the space· fixed coordinate s)slem r,. It is e:l")' to sec from (3. 11 ) Ihal :
(3. 12,,)
amI
(7/j I ilA (3. 12b)
In the lotall y steadY>Male problem. one has. eve rywhere in the solid .
(3. 13)
and. hence,
a() ~_ a() 1 (h'l (1A. , «(ic faA) .
(3 . 14)
154 S.N AI/url
Thu'!
a lV I OWl at"1 I = = (T,/ (lA., ox 1 «(]claA) aA , «(]CloA)
(3. 15)
and
OF,/ OE,) 1 (Jx, (ac/aA) aA.,
(3.16)
From (3. 15) and (3. 16). it follows that. when! , =(). th e vol ume integral'! in (3.8) <md (3.10) vanbh identically in the totally steady-state casco l lowevcr, in pract icc, whcn ~table crack growth occurs in a finite body. such as a "compact tension" type Illboratory test specimen . the st ress- strai n field asymptotically clo~e to th e crack-ti p may att ain OJ steady slate (thus resulting in a constant value of "'* during sustained crack growth). while the far -fi e ld stress-strain fields do not remain invariant with re~pcct to ;m observer moving with the crack-tip (thus rcsultin g in an eve rincreasing v:!luc of far-field ' r)' This situation is illustrated in Fi g. 3_ which shows re~ulh of numerical simulation of an experi ment on a compact tension test specimen [70.711. The difference between the far-field 'f and the ncar-tip T o.. i.e. the volume integral in (3.10). can be accounted for. mainly. by even t'! immed iately in the plastic zone nellr the crack-lip and elastic unload ing which accompanies crack growth . Exte n,!jve studies of the use of T* to .malyze st..,ble crack growth under monotonically risi ng load. as well as the predictive cap" bi lity of T* in si tuatio ns of cr;lI.:k growth after :. cycle of loading. unloading to ze ro load. followcd by
p
J,
9
T' '_-----------'------ 0_·0 5 . .
Fig ., T\'p,(";,l T':)l,llt) for J, "lid T' (hlTlllg ~t:,hle era.:J.; gro""th In 3 oompa("t t en~l(m ,pc(",mcn ( Based on numen~3t ",mulat,oll of c \pcnmelltal data. See Ref PI1 for furthef deta,ls)
t ... <;
reloading. have been presented by Brust et ,II. [70-721 rece nt ly. The re~ulb of a careful. combined numerical /experimental study showed 1721 that the T" parameter accuratel) predicted the e.>.:perimentally observed bchavior (crack growth to begin o nly after 50':'c loading in the reloading phase). while the o the r paramete rs 11, (i.e. the far-field J as is widely used) and CTOA (cnlck-tip ol>c ning angle)! .... ere seriou~l) anti-conservative (Le. they predict crack growth to recommence only al the lOOq. load level in the reloading phase).
Likewi~e. for the ca~e of dynamic crack propagation in II dynllmically loaded inelastic hody (such a~ a rate-dependent or visco-plastic body). o ne may define a crack-tip parameter -P ' . and its eq ui valent path-i ndependen t integral (including a domain -integral term) representation. as:~
TO' = f ( W + 'f )". - I, :_:~) dr , .
f ( au,) (W + T)" . -I, - . dr
j).\ • / \.,
f [( " IV "" ' ) (. a,i, ,'u,) f ,)u,] -- - u -- + P /I - - U - + - d u nXI 'I ax. ' "XI ' ax . ' (Ix . " ,
(3.170)
.... herc IV i~ the t01;l1 st ress work at a material point. a.!. defined in (3 .3). and (a IVlax.) i .. e\a luat ed directly from the computed valuc~ of W at two infinitesimal ly close materia l particles . Once again. the domain integral in (3. 17h) docs not vani <; h cxcept in rather very !opecial ci rcumstances . Conside r the ca~e of mode- I inc!:l\ tic dynamic crack propagation along the x .-axis. As in 0. 11) . we introduce a coordinate sy!otem ceflt ered at the propagating crack-tip . such that
~. :::: X I - e(1) (3.1~)
where I i\ the Newton ian time. We now consider the ma te rial derivative of II , (i.e. material velocity Ii,). mate rial
derivat ive of Ii, (Le. m,Herial acce le ration II,). and Ihe mate rial deriv,lIive of ~trcss work \V (i.e. stress power ~V) . as follows:
lIere Ihe dchmhon of Ihe palh t: is ~,"l1tar 10 that gl\cn In conneellon With Ihe ql.l~sl-~I(me easc, as In
(.H).
S. v A ilim
illi , I a li, ", a li , u, =
,11 " at " at ,7{1
(] ~ I~ , I _ 2 (7c iI ~ ll . + a !I~, ("k f (j " , (7 ~ c
",. " il t iJ la{, a{ j iJ l (){I (7 t ! (3. 19b)
IV (1 W i " IV iJ W ae
= -- --,), J/ {, a"~ a, (3. 1ge)
i (k"1 _ ae' l I ae'l ae -- ---
" ", . al c, J{ l al (J. 19d)
If onc rCM rict s ,lIIe ntion 10 conslanl~velocity crack propagation : ( I) with zero body fl)rcc':>.f. 0 : and (2) \\ith full) s teady·slaiccondition~. i. e. the ~ trc~~ . ~ train .
di\ placcllIenl. vclocity. and accd eration fields eve rywhe re in the solid (a ~ymptoli
call y cio .. e to thc crack -tip a .. wcll a~ in the far-fi c ld ) remain in v;lriant with respect 10 an olhcfver moving with the cnlck-tip . we ha ve from (3. 19):
(1c a ll , ,le fi ll , (3.20a) Ii , -= - -
ii' a"~ a, ax, ,)C ali ,
( J.lOb) " = --, (71 ax,
~F!I. = ' " (J.2Oc)
ax , (ac /at)
il lV ,] W aw - I 1 = W =- (7,/" . r1.,. 1 a{, (ac l il t) ", (ac/iJ l ) ( ilc lat)
(J.lOd)
ThU'~. UlH.lcr full) steady-sl:lte conditions. Eqs. (3.20a- d ) Impl y that the volume 1I11cgrai in (3. 17b) i ~ l ero . and the contour-integral T" as defined in (3. l7a) is path -independent in itself. Howcver. for thc case o f accelerating / decelerating crad, motion\ in finit e bodies . the attainm ent of full y stcadY-':>tat c conditions c \'erywhere in th e solid is an unlikely event. as in the case o f quasi-stati c crack ~rowth described carlie r (and illustrat cd in Fig . 3) .
It i .. thu~ see n that. in general. the path-independent int egral represe nt a tio ns for thc crar k-lip parameter. T " . in situations of qua~ i -static and d)rHlmic crack propagation in inelastic solids. invol ve a domain integ rallCr1l1 a~ in Eqs. (3.8) and
(3. 17b). rc~pcc l i \c1y.
Since mo .. t lIle lasti c crack propagation analyses ha ve to rely on computational approachc\. and ;,, _nce almost a ll computation,ll approaches fo r incla<;tic analyses lIl\ ol\e rate (or lIlcrcmental ) formulations. it is COTl\ cnie ll1 to define rale (or increment al) crack-tip parameters. ab-initio 13 . 44. 73. 741 .~ This approach will also
' \ or ,\ (()mprehensilc dlscu~Slon of _he rclarion uf Ihew nile 1m I11Cfemenlal) I'aramctcr~ 10 rh ... c .. n~·epl tlf ··Ctl!l\cflal,on tails" In rate l heuric~ of Incl •• -ric 'tllld, l'UI;h a' rhe genera t Jlo .. Ilu'on of el •• 'tullla'helh). \cc Rd,. [3 . +t. 73. 7~ J
157
facilitate a eomputa tion of the derivative in the x, direction of the total slress-\\orl.. In a more simple manner. To thi" end. we consider the more general dynamic case and define an increment,11 crack ·tip parameter [with the p;lIh r. as defined in connection wilh (3.7)1. as:
JT ·= f ( JW +JT)II I ,
nJ/I, (II I (I II ,) , -- - J, - ' - JI - dr.
, ~ , .1 , ., oX I oA t f XI
(3.21 )
Det.uled discll!>!>ioO'. of the relev.mce of J T· in the cont ext of the flO'\ theory of plasticit) ma) be fOllnd in 1441. The last term mthe integrand ollihe right-hand side of (3.21) is of second order but is retained for beller accllraq in an incremenlal anal),>i ... The incremental ~Ire),,>-work. JW. is defined <I),:
(3.22)
where JE'I is the incremental strain. and Jif,/ the incremental st ress. Eq. (3.21) i') valid for any materi;11 model. Consideri ng. for example. Ihe case of elasticpla!>licity. one rna) decompose Ihe incremental str;tin as
(3.23)
The incremental st ress-st rain relation may be written. in general. as:
(3.24 )
where E:/4I is the tangent consti tutive matrix. For classical clasto-pla'ilicity. for instance.
(3.25)
where r = 0 in an ela!> tic process (no thange in plastic strain) and r = I in a plastic process (change in plastic strai n): g is a parameter re lated to isotropic and /or kinem:nic hardening of the material: and N,/ is a unit normal to the yield su rface. Even though the materi;,l may be el astically homogeneous. Eqs. (3.24) and (3.25) make it evident that E:/4I is an explicit function of the location of the material particle. i.e. it depend!> on \\hcther the material particlc in question is undergoing an clastic process or a plastic .proccs. It is thus seen that:
(3.26)
From (3.24) it is see n that:
From (3.27) it foIIO\\~ that:
S \ Alillr'
I a..1u'1 - ..1t' --
2 'I ax ]
U<;i ng (3 .27) in (3 .26). one finds that:
" ..1 W ( I ) (}..1F'1 ( aU'1 I --= u + - Ju -- + - + -"l I, ., 'I d" "' ,. 2 . I - ., ] 0 ... !
(3.27a)
(3 .27b)
(3.28)
U!>c of (3.2S) ,.Iong with the divergence theorem. re'-)ulh in the following path·independent integral representation for ..1 T' of (3.21) (as::.uming that 110ninertial hody force::. arc lero):
..1T J. «..1W+JT)IIJ
- (I + ..1 1)(1..111 ' _ ..11 BII' )dr , I ax ' ax
I • \ ••
f [ (a,,, I a~" , ) (au" I ,UU,, ) + J (T --+--- - ~, -+---"'x >a '". >, " J - X l r.\ ] _ ( X .
I ", I
a..1l1, a..1 ,1 all , a,i 1 + p(ll , + JIl,) - il- - p(li , + J li,) __ , + pJII, - - pJli -' du.
X l a .l l aXI ' ax ]
(3.29)
The ahove incremental parameter is such that: (I) it im-ol\e<; only the incremental ~trc .. ..,-work which can be defined for any m;l terial coml itutive model; (2) the f;, r-field defini tion b inherently path-independen t. but involve.., a domain integral: (3) its defi nition can easily be modified for non·i~othcrrn:,l conditions: (4) it is a path-independent type crack-tip parameter valid for l:lrgc :unou nt ~ of crack growt h and general non-steady-state cond itions: (5) it is a valid cnlCk-ti p parameter for arbitrary hi\torie:; o f loading and unloading: and (6) \~hen specialil"ed to the case of fully steadY-Slate crack propagation. th e domain integr:tl in the far-field repre~entation of ..1 r- \anishe~ identically.
Com.ider . for instance. the case when material inertia. kinetic encrg~. and body force~ arc negligible as In SillJalions of creep crack gro\\ Ih in structures operating at elevated telTlpcralure~ . Ignoring ,,>econd-order term.., in rale,. a rate parameter go\erning the crack-tip conditioni> in !ouch situation, ma) be \\ nlten a, :
· f ( au,. T · = W" - I -- t 1 , U.l I ' ,
In the above.
ali t - ' , i)x I
i all') dJ" , ii x I
(3 .3 1 )
is the "stress power" and hence is defined for any m:ltcri al irrespecti \'c of the postu lalcd constitut ive re lation. Specifically. the ralC par.llllc tc r t * of (3.30) rC l1lain~ valid fo r c rack growth in silualion~ wherein c lastic. plastic. and creep !o tTlJins may be simultaneously prese nt. When the stress state in the body saturat es and pure !.Icady-:. tatc No rton's powe r-law type c reep occurs, i. e. E - u~ . the situa tion is oft en refe rred to as "steady-stat e c reep" , In thi<; se nse, the paramete r t· of (3.30) remains va lid in "arbitrary no n-steady cree p" conditions and , of course, in the limit a~ the so-called "steady-state creep" condit ions atl ain . In the li mit as "stcady-statc creep" condit ions develop , &,,- 0, i , - 0; and t · of (J.30) becomes:
- f (l¥" - I ali,) d r - f i au" d u I ' ax " a
, _ \ I , ',_ I , X I
(3 .l2)
where in the subscripts (SSe) on t · ind icalC "steady-st:lt e creep",
Ea rl ier , Landes and Begley 175] and Goldman and H utchinson [76] considered the special case o f "steady-sta le creep" and proposed a crack-tip paramete r C·. A~
mentioned ea rlie r. in the so-called "steady-state creep", only creep strains arc prese nt ; and, furt her , the creep strain-rate is propo rtional to the 11th power of stress
( Norton's power law) which sa turates to (Tat any mate rial particle, i.e, i - (T ~ . This b enl ire ly ana logous 10 the case of deformation theo ry of plasticity, wherein E - (I' '', Based on this ana logy, La ndes and Begley ]75] and Goldman and
H utch inson [76 ] de fi ne C· , .malogous to the deformation theory j , as:
c·= f( w.u, (l.ll) , .
(l.l' )
I~I
\\here ,
\Y" :-:: J IT", n di",,, .
"
5 .,\ A/11m
(3 .35)
Note that \V* in (3 .35) i .. nOllhe st ress-power: il i!o. !.imply a pseu dopolcnli .11 for tT"
in te rms of t"" ,md thus W· has no physical mea ning. Note that W · is a single-val ued funct ion of £". just as \V in the definition of deformation theory o f plasticity J b a single-valued function of Err Thus. in "steady-slate creep"
,1 W fT" = - ,-.-:
( Fit (3 .36'.0)
It is because of (3.36b) that Ihe domain integral. which arises from the applica tion of the divergence theorem \0 th e integral over r; in (3.33). vanishes identically; and hence C* is p;lIh -indcpcndcn t as .. contour integral alone. On the other hand , th e "stress-power" \V, which is defined for "non-steady creep" , in general. such that
\Y - if ' . - I, ' ,, '
Note that c\cn under the so-ca lled steady-state creep conditions.
(t~~( C·) :-: J [lV~),c - W · ]m j <.II' ~O , .
(3 .37)
(3 . .18)
where the subscript ::. (SSe) on ~V denote "steady-state creep". Fo r instancc. for Norton's powcr-law type "stcndy-slatc creep" (i - u" ). we have:
. ' ( )" '. £" =~ )'ue ,, a " (.1 .39)
where), is the fluidity par.ameter. u.~ the eq uivalent stress. and a; , th e stress devilllor. (a ~" = l tr;, IT;, ): II the hardcning parameter. For the matcrialmodel of (3.39), we have:
(HO)
where the subscripts (SSe) dcnote steady-state creep. On the other hand,
, ,
W ' = J (JAI)
10'
ThUl>.
IV _-,1--;-;:(~)'"(" )1~'I.It:::: )' (~)~"I . (W~,("- ) =-;(,, +1) " y c~ (1/+1) "4 (3.42)
Also earlier. Stonesifer and Alluri [77.78] considered a crack-tip p.tramcter fur general " non-steady creep" conditions. T" defined (I":
~ J (IV" - I "u,) - J ' ') ,. '-, I I, ( \ " \
(. "a,,) I" -, dv.
J.\ I (3.43)
Comparing (3.30) and (3.·B). it i .. see n th:H for general non-l.lcady conditions.
(,-"4)
whercal. . under the so-called ··~lcadY·l.lall· creep" conditions,
(J.<5)
wherein the subscripb. (SSC) indicate .. ~ t c<ldy-.,ta lc cree p", Recently BTU'>! C\ a l. 170- 7'1. 79J ha \'c prc.,clllcd a number o f l>lud ic\, tl l> ing T
(or j T - o r t - ). pertaining to stahl e crack gruwth under monotonic as \\ell ,,\ cycl ic loading in cla,tic-plastic bodies :In d creep rr:Kk growth at elevated temperatures . Unde r monoton ic lmlding. T · increase" monoton ically ;lIld is equal to i f for ~rna ll
amounts of growth : while for modera te to large amounh of growth . i f conti nucr" to increa~e .. uh~tantial1y whi le T' le\els off to a con!>tant valuc . Thus . .,. . has the (eaturc~ of ,I combined (ir- CTOA) criterion . A detailed account o f the'i;c resu1t ~ is give n in [70.71[ . Ref. [7'2 1 prese nts a ~tudy of Ihe use of P in sit uation .. of cyclic loadi ng. In turren t practice . the i i- integral (far-field value) h common l) u~ed 10
determinc the re ... istance of pla~tlcall} deformed structures to cont inued crack grO\\I h. This approach ha .. been ~hO\\n [71. 72[ to be valid on ly for vcry small amounts o f growth under monotonic loading: yct in the engi neerin g community the i(- rc"'i~tancc cunc OIpproachjs believcd to bc eonsen'alivc for all gene ral loadings. On the o thc r hand . the T ' para mcte r is formulated 10 he valid under a wide range of loadi ng/ unloadi ng condition '" The~e two criteria ('I'. and i f)' along \\ ith CTOA (crack-tip opening angle) \\cre examined for their validit), and predictive capabilit }' in situations of cr;lck grO\\ th after a cycle of loading. unloading to zero load. followed by iI reloading [72[ . The re"' lLlt ~ of a careful. combi ned numeric'll experi mental .. wdy shO\\ed [72[ th,l1 the r para meter 'Iecuratel}' pred icted the
Ih2 S. \' At/II"
behavior (crack. growth to begin only afte r SOq. 10,ldin g in the reloading phase), while the o lher paramcters (1 and CTOA) were se riously anti-conse n 'alivc. In another shldy (791. the relevance of the parameters "t'., C·, and 1~ in characterizing creep cr:lc" gTO" th wa!. exa mined. Experimental data on creep crack gro" th in a 316 stamlc,,!. Mec] .. ingle-edge- no tched specimen was numerically simulated . and the variation!. of various parameters during crack gro" lh were asce rtained . These reSul1') \\cre found to be in fil\or of the t · parameter in charaClcrizing creep crack gn)l,\ th under non-steady creep (not pure po"cr-];I" cree p) as well as in situations " herein time- independent plastic strains :Ire significanl. in addition to creep ~trains.
Ilowe\'er, the present sta te of knowledge is toO premafllre to concl ude that fracture-charac teri/ing parameters are rationally establi~hed for inelastic and dynamic fracture .
Acknuv. ledgellu.' lI ts
The rc~ulh he rein were obtained during the course of investigations !.upPorled by the US National Science Foundation under grant MEA-8306359 and the US Of/icc of Naval Research under con tract NOOOI4-78-C-0636. The encourage ment of Drs. Clifford Astill and Y. Rajapakse is gralcfully .. ck nowledged. The assistance of r>.ls. J . Webb in preparing this manuscript is Ihankfully acknowledged .
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