UDK 539.42:539.61:536.7 ISSN 1318-0010 
Pregledni znanstveni ~lanek KZLTET 33(6)505(1999) 

PRIN CI PLES OF FRAC TURE ME CHANICS FOR SPACE 
AP PLI CA TIONS 

NA^ELA MEHANIKE LOMA ZA UPORABO V VESOLJU 
Mi chael P. Wnuk 
University of Wisconsin-Milwaukee, ASEE/NASA Summer Faculty at Caltech/JPL, USA 
Prejem rokopisa – received: 1999-11-13; sprejem za objavo – accepted for publication: 1999-11-22 
Eval u a tion of the ex ist ing and new ad he sives may in prin ci ple be re duced to the the o ret i cal and/or ex per i men tal de ter mi na tion of the ma te rial re sis tance to decohesion as mea sured by the spe cific "bond ing en ergy" which must be ex ceeded via an in crease of the ex ter nal loads and the re sult ing lo cally in duced state of stress in or der to break the bond be tween two ad he sively joined de form able ma te ri als. This en tity is not merely a ma te rial prop erty re flect ing sim ply t he strength of the ad he sive layer, but it also de pends on the elas tic moduli of the sub strate and the ma te rial bonded to it. It is, in f act, a mis match be tween the two sets of elas tic con stants that has an es sen tial in flu ence on the fi nal value of the spe cific en ergy of ad he sion. 
From the the ory pro vided by Non lin ear Me chanics of Frac ture it fol lows that in or der to dam age the struc tural in teg rity of an ad he sive bond, it suf fices to bring a min ute pre-existing crack-like de fect to a crit i cal lo cal stress level at which a sus tained prop a ga tion of frac ture be comes ther mo dy nam i cally fea si ble – as re quired by the clas s ic en ergy bal ance equa tion of Grif fith. For most load ings and geo met ri cal con fig u ra tions of the struc tural com po nent the ini ti a tion of crack ex ten sion is tan ta mount to the cat a strophic fail ure which in volves an un sta ble sep a ra tion and can not be stopped even when the ex ter nal loads are re duced to zero. 
In trin sic strength of the bond can also be al tered due to vari a tions in the ex ter nal con di tions such as tem per a ture, cy clic load ing, an in creased rate of load ing or the chem i cally ag gres sive en vi ron ment. The state of stress in duced in the neigh bor hood of the crack front con trib utes sub stan tially to the pro cess of decohesion and it can pose a for mi da ble math e mat i cal prob lem when frac ture prop a gates within the thin layer of the ad he sive placed be tween two de form able sol ids with dis sim i lar elas tic and ther mal prop er ties. Fre quently, the na ture of the prob lem re quires an ap pli ca tion of tech niques and con sti tu tive equa tions as so ci ated with highly de vel oped de for ma tion and frac ture pro cess. The nonlinearities en coun tered here are two-fold: (1) geo met ri cal and (2) phys i cal. The lat ter in volve time-dependent phe nom ena or plas tic ity de pend ing on the na ture and me chan i cal prop er ties of the sub stances in volved, the sub strate and the ad he sive layer. Thus, viscoelasticity so com mon for a num ber of com mer cial ad he sives and nonelastic de for ma tion dom i nated by the ir re vers ibl e plas tic com po nents of the strain ten sor re quires sig nif i cant mod i fi ca tions of the con sti tu tive equa tions. Both vis cous and inviscid de for ma tions have to be ac counted for by Non lin ear Viscoelasticity and the The ory of Plas tic ity. 
In de pend ently from these stud ies it is sug gested that the fractographic maps of the frac ture sur faces are re corded in the post-mortem in ves ti ga tion aimed at di rect ob ser va tion of the Wallner lines and river marks im printed on the frac ture sur face while the spec i men un der go ing frac ture is ir ra di ated with ul tra-sound waves of var i ous fre quen cies cor re lated with the speed of the shock wave which pre cedes the front of the prop a gat ing decohesion zone. 
Key words: ad he sive bond ing, frac ture me chan ics, con sti tu tive equa tions, cohezive crack model, ma te ri als prop er ties 
Oceno obstoje~ih in novih adhezivov lah ko izvr{imo s teoreti~no ali/in eksperimentalno dolo~itvijo odpornosti materiala proti dekoheziji, ki jo dolo~a specifi~na vezna energija. Ta mora biti prekora~ena z zunanjo obremenitvij o in lokalno induciranim napetostnim stanjem, ki je potrebno za prelom zveze med dvema adhezivno vezanima preoblikovalnima materialoma. Ta entiteta ni samo lastnost materiala, ki odra`a trdnost adhezivnega sloja, ampak je odvisna tudi od mod ula elasti~nosti povezanih materialov. Razlika med dvema vrstama elasti~nih konstant ima bistven vpliv na kon~no velikost spec ifi~ne adhezivne energije. 
Iz teorije nelinearne mehanike loma izhaja, da se lah ko po{koduje integriteta adhezivne zveze, ~e se majhna, `e obstoje~a razpoka, privede na lokalni kriti~ni nivo napetosti, pri katerem lah ko postane propagacija razpoke termodinami~no mogo~a, kot to zah te va klasi~na Grif fith-ova ena~ba o ravnote`ju energije. Za ve~ino obremenitev in geometri jskih ob li k strukturne komponente je iniciacija rasti razpoke predpogoj za katastrofi~ne po{kodbe, zaradi nestabilne propa gacije, ki jih ni mogo~e ustaviti tudi, ko se zunanje breme zmanj{a na ni~. 
Specifi~na trdnost zveze se lah ko spremeni zaradi spremembe zunanjih pogojev: temperatura, cikli~na obremenitev, pove~ana hitrost obremenitve ali kemi~no agresivno okolje. Stanje napetosti inducirano v okolici ~ela razpoke bistveno prispeva k procesu dekohezije in postane zelo te`ak matemati~ni prob lem, ko razpoka napreduje v tanki plasti adheziva med dvema trdnima materialoma z razli~nimi elasti~nimi in termi~nimi lastnostmi. ^esto narava problema zah te va uporabo tehnik in konstitutivnih ena~b povezanih z mo~no razvitimi procesi deformacije in preloma. Pri tem naletimo na dvoje vrst nelinearnosti: geometri~ne in fizikalne. Zadnje obsegajo tudi ~asovno odvisne fenomene plasti~nosti, ki so odvisne od narave in mehanskih lastnosti snovi, sub strata in plasti adheziva. Zato viskoelasti~nost, zna~ilna za mnoge komercialne adhezive in neelasti~ne deformacije, ki je odvisna od ireverzibilnih plasti~nih komponent tenzorja deformacije, zahteva pomembno spremembo konstitutivnih ena~b. Oboje, viskozno in neviskozne deformacije, je potrebno preveriti na nelinearno viskoelasti~nost in teorijo plasti~nosti. 
Neodvisno od teh ra zis ka v se priporo~a, da se zbirajo fraktografske mape prelomnih povr{in pri post-mortem preiskavah z namenom neposrednega opazovanja Wallnerjevih ~rt ter `il, ki nastanejo, ko razpoka napreduje zaradi obsevanja z UZ valovi z razli~no frekvenco odvisno od valovnega {oka, ki napreduje pred ~elom dekohezije. 
Klju~ne besede: adhezivna zveza, mehanika loma, konstitutivne ena~be, model kohezivne razpoke, last nosti materialov 
KOVINE, ZLITINE, TEHNOLOGIJE 33 (1999) 6 
M. P. WNUK: PRINCIPLES OF FRACTURE MECHANICS FOR SPACE APPLICATIONS 
One of the basic assumptions underlying all cohesive crack models used in the description of inelastic fracture has to do with the shape of the cohesive force distribution. The exact form of this distribution is unknown, but several very useful clues are provided by the experimental work on fracture at interfaces, cf. Hutchinson1. In principle it could be derived from considerations of the molecular forces exchanged between two adjacent planes of atoms which are subject to separation as the leading edge of the crack propagates along the interface. 
We shall return to this point after some mathematical preliminaries. The condition of finite stress at the tip of the extended crack, x < a (a visible crack stretches along x < c), valid for the stress boundary conditions 
..,0 < x < c 
px () =. 	(1)
Sx c < x < a.. - ( ), 

can be set up as follows 
a 	a pxdx () 
0 = KTOT (., S) - 2 = 
..022
a 	- x 
c
. 	().
a .dx a [. - Sx ]
= 2 .+ . = (2)
. 
22 .c 22
..
.0 	a - x 
a 	- x . 
a .. a - sx dx () . 
= 2 
.. ... 2 .c 
a 2 - x 2 ..

If the stress distribution S(x) is normalized by the reference cohesive stress S0, say S(x) = S0G(x), then Eq. 
(2) reduces to 
aG() x dx .. 
Q = , Q= (3) 
c 22
. 	2S
a 	- x 0 
When the variable x is replaced by x1, x = x1 + c, Eq. 

(3) reads 
Gx 1)dx 1R (
Q = 	(4)
.
02 2
a 	- (x1 + c) 

or, better yet 
1 	G( )( 1- md .
. )
Q = 
(5)
. 
0 
1 -[(1 - m).+ m]2 

Here, . = x1/a while m is a parameter related to the crack length c and the length of the extended crack, a = c 
+ R, namely, m = c/a. In what follows we shall limit the considerations to the case of R << c, i.e., for m~1, which is pertinent for "small scale yield condition" met in all cases of practical importance in the context of Materials Science. For this limiting case the integral in Eq. (5) can be simplified as follows: 
[Q m ]m( ) ~1 = = 0 1 .  1 1 2 - - m m  1 - G( )d. . .  =  1 2 0 1- m .  1- G( )d. . .  (6)  
506  

Valuable clues regarding the distribution G( .) are gained from studies of fracture occurring at the interface between two dissimilar materials joined together either by direct adhesion or by a thin bonding film. In order to account for the experimental data, two main features are expected. First, the stress S should reach a maximum at a certain distance . from the crack front. This maximum stress Smax may in some cases become substantially larger than the reference stress S0. It is assumed that Smax is attained somewhere within the process zone, most likely at its outer edge, x1 = .. To the left of this point S drops off rapidly to zero to match the boundary condition of stress-free crack at x1 = 0, while to the right of this point S falls down again and levels out at the value S0, toward the end of the cohesive zone, x1= R. 
In order to account for such behavior we propose a strongly nonlinear function composed of a power function and an exponential. We submit, therefore, a two-parameter distribution function of this form 
. x1 . n ..- x1 ..
Sx 1) = S 0 .. exp ..1 . , 1
( 	x = R.
..
·R .·R .
..G() . =.n exp [.(1 - .], 0 .. .1 (7) 

in where . and n are yet undetermined parameters. This function is now substituted into Eq. (6), yielding 
n 

R 1 . exp [.(1-.)]
Qm (	) = 
(8)
.
2c 	0 

1 -. 

Note that for m~1, the expression (m - 1) can be replaced by R/c, while the integral in Eq. (8) can be cast into a closed form, cf. 2 
W (.	, n) = 1 . 
. 3 ..
. exp( ) ( .. n +1) +1 F1 .1 + n, + n,-.. . (9)
.
. 3 ..	
·2 ..

.. 	+ n .
·2 .

Here the stan dar d notation for the gamma function (.) and the hypergeometric function ( 1F1) is used, cf. 3. Physical interpretation of the integral (9) leads to the energy dissipated within the cohesive zone, hence the symbol W. Finally, combining Eqs. (8) and (9) allows us to define the length of the cohesive zone: 
.. K1 . 2 
R = .. (10) 
2W 2 ·S 0 .

When KI attains its critical level KIc, the Eq. (10) predicts the characteristic microstructural length parameter, Rmax = ( ./2W2)(KIc /S0)2. The primary conclusions of this contribution can be summarized as follows 
1. 	A generalization has been proposed that encompasses all previous cohesive crack models and provides a platform for novel investigations of the influence of the structured nature of the nonlinear zone on the early stages of fracture; 
KOVINE, ZLITINE, TEHNOLOGIJE 33 (1999) 6 

M. P. WNUK: PRINCIPLES OF FRACTURE MECHANICS FOR SPACE APPLICATIONS 
2. 	
By proper choice of parameters . and n we are able to quantify the inner structure of the cohesive zone, the so-called "fine structure", which accounts for the existence of the small process zone of size . embedded within the larger R-zone; 

3. 	
Microstructure of material is now represented by properties such as the overstress factor, k = Smax /S0 and the ductility parameter, . = Rini/., in which Rini denotes the threshold value of R associated with the onset of fracture; 


For a given k and ., the parameters that determine the shape of the S-distribution, . and n, can be evaluated explicitly by matching the ratio Smax /S0 = (n/ .)nexp( . ­n) with the given overstress factor, k. Solving the 
equation 
. n . n 
.	. exp [. - n]= k (11)
·..

for the coefficient ., we obtain ..= ln)( k.n ) (12)
.-1 

Since ./n represents the reciprocal of the coordinate . at which the maximum in S occurs, we have 
. 1 Rini 
= ==. (13) n . max . 

Combining it with Eq. (12) results in the transcendental equation 
. n 	(14)
ln( k. ) -.= 0 .. - )
(	1 

For any given input set of data, such as specified . and k, the other two variables, . and n, can be solved for (numerically, of course). Since the input parameters are deduced from the microstructural data, and can be measured experimentally, the fine structure characteristics . and n are not accessible to an experiment, we have provided a link between the two sets of parameters pertaining to micro-level of fracture. The next step, of course, is to evaluate the macro–level entities such as W and R. Our model makes these calculations possible, too. And thus, we have indeed 



Fig ure 1: Dis tri bu tion of the co he sive force S( .)/So within the R-zone for the fol low ing meso-structural pa ram e ters: 
-duc til ity in dex, . = 10, and 
- over stress fac tor, k = 5 Slika 1: Porazdelitev kohezivne sile S( .)/So v R zoni za naslednje mezo – strukturne parametre: 
-in dex duktilnosti . = 10 in 
- faktor prenapetosti k = 5 
constructed a bridge between the micro-and macro-scales of fracture representation. To illustrate this statement, we set . = 10 and k = 5, 
and then using the equations written above, we obtain n = 0.2403 and . = .n = 2.4031, while the nondimensional dissipation of energy for those microstructural input data 
is W ( ., n) = 4.4805, and the length of the nonlinear zone is Rmax = 0.3506(KIc/S0)2. 
Finally, Fig. 1 shows the predicted shape of the G-function, which represents a nondimensional cohesive force distribution within the R-zone for the choice of micro-parameters used in our sample calculation. 

REF ER ENCES 
1 J. W. Hutchinson, "The Role of Plasticity in Toughening of Ductile Metals and Interfaces", seminar at Northwestern University in the series "Colloquia on Mo der n Topics in Mechanics", March 1997, Evanston, IL 
2 Gradshtein and I. M. Ryzhik, "Tables of Integrals, Series and Products", Academic Press, 1980 (translated from Russian) 3 "Encyclopedia of Mathematics", Kluwer Academic Press, 1997, The Netherlands 
KOVINE, ZLITINE, TEHNOLOGIJE 33 (1999) 6