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Dynamic Fragmentation of Brittle Solids: A Multi-Scale
Model
Christophe Denoual, François Hild
To cite this version:
A Multi{S ale Model
Christophe Denoual and Fran ois Hild
November 15, 2005
Abstra t
Modeling dynami fragmentation of brittle materials usually impliesto hoose be-tween a dis rete des riptionof the numberof fragments and a ontinuum approa h of damage variables. A damage model that an be used in the whole range of loadings (from quasi-stati to dynami ones) is developed. The deterministi or probabilisti natureof fragmentationisdis ussed. Qualitativeand quantitativevalidationsaregiven byusing a real-time visualization on guration for analyzing the degradation kineti s during impa t and a moire te hnique to measure the strains in a erami tile during impa t. Finally, a losed-form solution of the hange of the number of broken defe ts with the applied stress gives a way of optimizing the mi rostru ture of erami s for armorappli ations.
Bilayered armors with erami sas front plate and steels as ba king fa e have been used for several yearstoimprovetheeÆ ien yoflightormediumarmors(denReijer,1991). Thehigh hardnessof erami materialsfavorsproje tilebluntingand/orfailureandspreadsthe kineti energy on alarge surfa e of the du tile ba king. The weight of the armor is then redu ed in omparison to an armor made of steel only. The response of a erami impa ted by a steel rod isstrongly dependent upon the impa torvelo ity. Lowimpa t velo ities (approximately less than 1000m/s) lead to degradations su h as ra king prior to a signi ant penetration. It follows that ra king is the prevalent me hanism to predi t the residual properties of the erami before penetration and to assess its multi-hit apability. Higher impa t velo ities (rangingfrom about 1000 to 3000m/s)usually lead to degradations in ompressive and ten-silemodes (Espinosa et al., 1992).
Furthermore, the impa tor an penetrate the erami layer even though some parti u-lar on nement onditions may prevent penetration (Bless et al., 1992; Hauver et al., 1994). Ultra-highvelo ities(greater than3000m/s)leadtoafullyfragmented erami whose behav-ioris loser tothat of a uid rather than that of a solid material. Analyti al models an be used to des ribethe response of the material(Tate, 1967; 1969).
I.Fri tionatthe ra kfa ehas asigni antin uen e onthe damagedes ription(Espinosaet al.,1992; Halmand Dragon, 1998). Finally,the populationof aws that lead to ra k nu le-ationmaybedi erentintensionandin ompression. Toavoidanoverlappingof me hanisms that would make the modelvalidationdeli ate,only damagein tensionis onsidered herein.
In Se tion 2, a damage model des ribing the tensile fragmentation is derived. After the presentation of a simpli ed des ription for the initial (i.e., undamaged) material, the fragmentation is analyzed as an extension of the brittle fra ture regime observed in quasi-stati loadings by onsidering random arrays of ra ks. A multi-s ale approa h is proposed to model fragmentation with no onstrains on the stress rate. In parti ular, the transition between single and multiplefragmentationisanalyzed. Itfollows thatthe domainof validity of a ontinuum and lo alapproa h is obtained. A so- alled\Edge-On-Impa t" on guration isusedinSe tion3andallowsfortheobservation ofdamagepatterns andstrain eldsduring impa t. Results of multi-s ale simulations are dis ussed with respe t to experimental data. Finally, a materialoptimization, whi h makes use of a losed-form solutionof the hange of the numberof fragmentswith time or stress, is performed inSe tion 4to assess the ballisti performan e of four di erent SiC grades.
2 A Model for the Fragmentation of Brittle Materials
Forbrittlematerials,the analysisof failureduringquasi-stati loadings an beused tode ne the relevant features of the mi rostru ture in terms of aw density and failure stress distri-bution. The nu leation of a ra k in brittlematerialssubje ted to quasi-stati tensionis due to (point) defe ts de ned by a failure strength
f
(x). When an equivalent stress (x ), e.g., maximum prin ipal stress, is greater than
f
(x), a ra k emanating from the defe t leads to the failure of the whole stru ture. The failure strength is a random fun tion related to the defe t distribution and lo ation within the material. Therefore, the ultimatestrength of a erami spe imen is not deterministi and a failure probability P
F an be des ribed by a Weibulllaw(1939) P F =1 exp[ t ( F )Z e ℄ with t ( F )= 0 F 0 m (1) where t
is the defe t density, m the Weibull modulus, 0
the referen e stress relative to a referen e density
0 ,
F
the failurestress (i.e., the maximum equivalentstress inthe onsid-ered domain) and Z
e
the e e tivevolume, surfa eorlength (Davies,1973). The onstant
0 =
m 0
is the so- alled Weibull s ale parameter. In the following, when no spe ial mention is made, the development is valid for any spa e dimension n (i.e., 1, 2 or 3). Otherwise, it will be learly stated for whi h spa e dimension the results are valid. It an be noted that the previous formulation (i.e., Eqn. (1)) enters the framework of a Poisson point pro ess of intensity
t
(GulinoandPhoenix,1991; Jeulin,1991). Themi rostru ture ofthe undamaged materialis therefore approximated by defe ts of density
t
with random lo ations.
Moreover, the mean failurestress w
and the orresponding standard deviation sd are given by w = 0 (Z e 0 ) 1 m 1+ 1 m ; 2 sd = 2 0 (Z e 0 ) 2 m 1+ 2 m 2 w (2)
where istheEulerfun tionof these ondkind. TherelationshipsgiveninEqn.(2)are used to estimate the defe t density
t
[(t)℄ by using quasi-stati tests even in the dynami range (Denoual and Riou, 1995): up tostress rates of 10 MPa s
1
2.2 Simpli ation of the Damaged Material
In the bulk of an impa ted erami , damage in tension is observed when the hoop stress indu ed by the radial motionis suÆ iently large togenerate fra ture in mode I initiating on the mi rodefe ts already mentionedin Se tion2.1. It willbe assumed that the initial defe t populationleadingtodamageandfailureisidenti al whenthematerialissubje ted to quasi-stati anddynami loading onditions(DenoualandRiou,1995). Thisstatement orresponds totheassumptionthatasingle defe t (e.g.,avoid, ami ro ra k)breaksatastress levelthat is weakly dependent on the stress rate. However, the broken defe t population is strongly dependent on the stress history. For very low stress rates, only the dominant defe t of the wholepopulationbreaks (i.e., the weakest linkof astru ture) and the Weibull lawdes ribed in the previous se tion applies. When dynami loadings are onsidered, a part (in reasing with the stress rate) of the initial defe t population is broken. This point will be des ribed inSe tions 2.3and 2.4.
Closed-form expressions for the e e tive properties of various ra k patterns are pro-posed in the literature (see forexample (Ka hanov, 1994)). These solutions are obtained for quasi-stati loadingsandwhen ra kintera tionsare onsidered,mostlyperiodi patterns are used. Inthepresentstudy,theapproximationsneededforananalyti alestimationofe e tive elasti properties of the ra ked solid no longer hold. The ra k velo ity has the same order of magnitude as the Rayleigh wave speed (Kanninen and Popelar, 1985; Freund, 1990) and stress equilibriumduring damage hange is never a hieved.
The onsidered ra k pattern is made of penny-shaped ra ks (instead of re tilinear ra ks) of random lo ations. It follows that periodi homogenization te hniques annot be used to des ribethis type of dynami ra king.
s ale 2 =:S D := 1 Z Z (x ):S :(x )dx (3) where S D
isthe omplian etensor of the damaged material (and S is that of anundamaged material), the mass density, a representative zone of measure Z, `:' the ontra tion with respe t to two indi es, (x ) the mi ros opi stress at point x , and the ma ros opi stress de ned by = 1 Z Z (x )dx : (4)
The aimof this se tion is tointrodu e allthe mi ros opi aspe ts of fra ture to des ribe the mi rostru ture of damaged material. A simpli ed stress eld is proposed in the following se tionfor a single ra k and then extended to a randompopulationof ra ks.
2.2.1 Single Cra k
Stresstensorsarenowexpressedasve torsbyusingVoigt'snotationsandfourthordertensors areredu ed tose ondorderones. Whenafra tureisinitiatedonadefe tk lo atedatx
k ,the stress state around the propagating ra k is a omplex fun tion of time, ra k velo ity and stress wave elerity. Fora ra k ofnormaln =x
1
submittedtoafar eld 0
,the mi ros opi stress eld (x )at pointx an bewritten as
(x)= h 1 R (x ) i 0 (5)
where R (x) isa se ond order tensor a ounting for stress modi ations (mainly stress relax-ation) arounda ra k.
In the appropriate oordinate system, the stress appliedto the ra k an be simpli ed as a uniform normal stress
0 n
in addition to a uniform tangential loading 0 t
normal aligned along dire tion x 1
, and a far eld stress n
, only the stress 11
(x) is relaxed over an important zone. For a pure tangential far eld stress
0 t in the (x 1 ;x 2 ) plane, only the omponent 12
(x ) is signi antly relaxed. Even though the whole relaxation tensor an be used, for the sake of simpli ity, only the rst and se ond most relaxed stress elds are onsidered inthe following. Two relaxation omponentsR
n and R t are de ned as 11 (x ) = [1 R n (x)℄ 0 n ; (6) 12 (x ) = [1 R t (x)℄ 0 t : (7)
Asimpli ationofthestress eldarounda ra kisproposedbyusingBooleanfun tions (JeulinandJeulin,1981). AnexampleofaBooleanrepresentationofstressrelaxationisgiven in Fig. 2{a and {b. Inside the Boolean fun tions
ij (x
k
) asso iated to a defe t of lo ation x
k
, the stress state is supposed to be ompletely relaxed whereas outside the far eld stress is applied ij (x )= 8 > > < > > : 0 if x2 ij (x k ); 0 ij otherwise : (8)
In the appropriate oordinate system, only two Boolean fun tions n (x k ) = 11 (x k ) and t (x k ) = 12 (x k
) are used to des ribe the stress relaxation for normal and tangential load-ings, respe tively.
It an be noted (see Eqns. (6), (7)and (8)) that the relaxation fun tions R ij
(x) orre-sponding to the `Boolean' stress eld are also simpli ed (i.e., R
ij (x) = 1 when x 2 ij (x k ) andR ij
(x )=0otherwise). Themeasure Z n
and Z t
ofthe relaxed (orobs ured)zones n (x k ) and t (x k
) are estimated for ea h stress omponents by assuming that the de nition of the ma ros opi stress (see Eqn. (4)) an be used for the real stress eld and the `Boolean' stress eld de ned inEqn. (8)
12 = 1 Z t Z 0 t (10)
whereZ isthe measure of. A spa es aling whi hmodi esalengthl intoitsdimensionless ounter-part l = l=a (where a the radius of the onsidered ra k) is used and allows one to derive anew expression for the measure of the obs ured zones
Z n = Z R n (x)dx=a n Z R n (x)dx=a n S n (11) and Z t = Z R t (x)dx=a n Z R t (x )dx=a n S t (12)
whereisadimensionlesszone,S n
andS t
aredimensionlessshape parametersoftheobs ured zone for normal and tangential far eld stresses, respe tively, and n is the spa e dimension (n=1for a line, n=2 for ashell, n =3 for avolume).
Forsake of simpli ity, itis assumed that ra ks in brittle solids rapidly propagate at a onstant velo ity kC 0 (C 0 = q
E= where E is the Young's modulus of the virgin material with k a onstant dependent on the material properties (Kanninen and Popelar, 1985) or (Freund,1990)). Therefore, Z n and Z t be ome Z n =S n [kC 0 (T t)℄ n ; Z t =S t [kC 0 (T t)℄ n (13)
where T is the present time and t the time to nu leation. It is worth mentioning that the shapeparameters S
n
and S t
may depend onthe Poisson's ratio but they are independent of time, i.e., the relaxed zones are self-similar. A numeri al estimation of S
n
and S t
is arried out by using Fabrikant's solution and Eqns. (11) and (12). It follows that S
n
= 3:74 and S
t
=1:15 for =0:15 and n=3.
2.2.2 Random Array of Cra ks
(seeareviewby JeulinandLaurenge(1997)). Forarandomarrayof ra ks duringadynami loading, stress elds an be approximated by the unions
[ ij
of the whole set of Boolean fun tions
ij (x
k
), ea h of them de ned as the relaxed stress state around a single ra k (Fig 2{ and {d) of random lo ationx
k [ ij = [ k ij (x k ): (14) The measure Z [ ij of [ ij
is expressed as (Jeulin and Jeulin, 1981; Serra, 1982; Denoual et al., 1997a) Z [ ij Z =1 expf t [(t)℄Z ij (t)g (15) whereZ ij
(t)denotesthemeanrelaxedzoneand t
[(t)℄theintensityofthePoissonpoint pro- ess (Eqn. (1)). The meanrelaxedzone Z
ij
(t)is al ulatedby averagingattimet these tion of the obs ured zonesZ
ij
(t )for anu leation at time andwith a density 1 t[(t)℄ d t dt [( )℄ Z ij (t) t [(t)℄= Z t 0 d t dt [( )℄Z ij (t )d (16) where Z 11 (t) = Z n (t) and Z 12 (t) = Z t
(t). A simple proof of these results is given in Ap-pendix 1.
The kineti s of damageis dis ussed in the following se tion. Both damagekineti s and des ription willbe de ned by using the simpli edstress eld des ription.
2.3 Damage Kineti Law
in several models (see a review by Meyers (1994)). Another fragmentation model was pro-posed by Grady andKipp (1980)andutilizedfor numeri al al ulations. The mi rostru ture was also des ribed through stress relaxation of spheri al shape around penny-shaped ra ks nu leated on initialdefe ts.
Tounderstandwhy a ra k nu leates, one has tomodelthe intera tion of nu leated de-fe ts and other defe ts that would nu leate. The spa e lo ation of the defe ts is represented in a simple abs issa of an x-y graph where the y-axis represents time (or stress) to failure of a given defe t. In this graph,a shaded ` one' represents the expansionof the obs uration zone with time due to nu leation and propagation of a ra k. A se tion of a one an be a volume,a surfa e oralength, depending onthe spa e dimension n (see Fig. 3-a). Inside this zone, the stress is de reasing and no new nu leation an o ur. An approximation of this zoneis given by [ n (and [ t
)i.e., thezone wherethestress normaltothe ra k isde reasing is assumed to be equal to the Boolean zone where the stress is relaxed. The defe ts lo ated outsidetheshaded ones an nu leateandprodu etheir own in reasingrelaxationzone(e.g., defe ts nos. 1, 2 and 3 of Fig. 3-a). Inside the ones, the defe ts that should have broken do not nu leate (e.g., defe ts nos. 4and 5 of Fig. 3-a) sin e they are shielded(orobs ured).
The total aw density t
an be split into two parts: b
(the broken aws) and obs (the obs ured aws). Furthermore, the distribution of total aws in a zone of measure Z is assumed to be modeled by a Poisson point pro ess of intensity
t
[(t)℄ in a ordan e with Se tion2.1(Eqn. (1)). New ra kswillinitiateonlyif thedefe t existsinthe onsideredzone and if it doesnot belong tothe relaxed zone
[ n d b dt [(t)℄= d t dt [(t)℄ " 1 Z [ n (t) Z # (17) with b (0)= t (0)=0.
Foravery high stressrate, most ofthe initialdefe ts nu leate ra ks beforeany signi - ant hangeof theobs ured zones,i.e.,d
averylowstress rateis applied,theobs ured zone o upiesthe wholevolume(Z n
(t)=Z 1) afterthe rst ra k nu leation, i.e., the nu leation is stopped afterthe failureof the weakest defe t. The initialdefe t populationdes ribed by
t
istherefore usedfor both dynami frag-mentation and quasi{stati failure but the number of nu leated defe ts varies with respe t to the stress rate. Another way of obtaining Eqn. (17) is given in Appendix 1 by using the horizon of a defe t. For a defe t P, the horizon is de ned as a spa e-time zone in whi h a defe t always obs ures P(Fig. 3-b).
The fra tion of relaxed zones Z [ n
(t)=Z is a good approximation for a damage variable D de ned in the framework of Continuum Damage Me hani s (Lemaitre, 1992),with D =0 for the virginmaterial and D=1 for the fully damaged one
D= Z [ n Z : (18)
It is worth noting that the damage des ription is not ne essarily isotropi even though it is hara terized through a volume ratio. Sin e the relaxation zones are relative to a ra king dire tion, an anisotropi damage des ription is needed. The ase of multiple superimposed ra kpatternsisstudiedinSe tion2.5wheredi erentvariablesD
i
areused forea hdire tion i of ra king. For any stress rate, the kineti s of D
i
is given in di erential form (a ording to lassi al results of Continuum Damage Me hani s (Lemaitre, 1992), the hange of D
i is stopped when d i =dt0) d n 1 dt n 1 1 1 D i dD i dt ! = t [ i (t)℄ n! S (kC 0 ) n when d i dt >0and i >0 (19) where i
is the eigen stress asso iated to penny-shaped ra ks of normal x i and t is the density of defe ts e e tively broken in the onsidered zone (n =1;2 or 3). The density
t is thereforeof probabilisti natureand may depend onagiven realization(i.e., one an have2, 0,1,5, et . defe t(s)broken for di erent niteelements
FE
ofvolumeV FE
submitted tothe same pres ribed loading).
When anin nitevolume is onsidered t isequalto t . For a nite size Z FE of agiven nite element FE
, the probabilisti density
t
break inadditionwith the density t
(see also(Benz and Asphaug,1994)). Thedensity t
is either equal to zero (no broken defe t), or equal to or greater than 1=Z
FE
, i.e., at least one defe t is broken in FE (see Fig. 4) Z FE t [ i (t)℄= 8 > > > < > > > : 0 if i (t) k ; max " Z FE 0 i (t) 0 ! m ;1 # otherwise. (20) The parameter k
is the failure stress of the rst defe t k able to break in FE
. The failure stress is obtained by randomsele tion of a failure probability P
F
2℄0;1[ with Z = Z FE
and isafun tion of the Weibullparameters (m;
0 =
m 0
)and the meshsize Z FE
(see Eqns. (1), (2) and (15)).
2.4 Continuum vs. Dis rete Approa hes
WhenContinuumDamageMe hani sisusedinnumeri alsimulations,themediumisassumed to be ontinuum on the s ale of a nite element in whi h numerous ra ks are expe ted to nu leate. However, ra k densities may strongly vary over the stru ture and the analysis of fragmentation through a ontinuum modeling may be deli ate. As an alternative, dis rete elementmodelinghas been proposed (Cama hoand Ortiz,1996; Mastilovi and Kraj inovi , 1999)whenthe fragmentsize isgreaterthanorequaltothe sizeof a niteelement. Espinosa et al. (1998) have developed a ontinuum/dis retemulti-s alemodelin whi h the ner s ale is dis rete and allows for the derivation of a ontinuum des ription on a higher s ale. In the present se tion, hara teristi s ales are introdu ed and enable one to hoose between ontinuum ordis rete approa hes.
When dynami (and proportional) loadings are onsidered with a onstant stress rate d
i
=dt = ,_ one an de ne a dimensionless aw density ( = = ), time (t = t=t ), spa e measure (Z =Z=Z )and stress ( i = i =
) fromthe ondition (Denoual and Hild, 1998)
Z =1 with = t (t_ ) and Z =Z n (t ) (21)
where the subs ript ` ' denotes hara teristi quantities. A hara teristi stress is de ned by
=t_
. Equation(21) expresses thefa tthatthe hara teristi zoneofmeasure Z
(21), the hara teristi parameters are given by
t = " m 0 0 S(kC 0 ) n _ m # 1 m+n ; Z = " ( 0 kC 0 ) m S m=n 0 _ m # n m+n ; = " m 0 _ n 0 S(kC 0 ) n # 1 m+n : (22)
This s aling is useful, in parti ular, when losed-form expressions an be given for the nu- leated defe t density, damage kineti s and ultimatestrength (Denoual and Hild, 1998). By usingEqns. (1), (15),and (16)a losed-formsolution anbederived forthedi erential equa-tion(17)inthe aseofa onstantstressrate_ (seeSe tion4). ByusingEqns. (15),(16),(18) andbyassumingthat
t =
t
,the hangeofanyofthedamageparametersD i
isdeterministi (Denoual and Hild,2000)
D i =1 exp " m!n! m+n i (m+n)! # : (23)
The applied stress i
is relatedto the lo al(ore e tive) stress i by i = i =(1 D i ). The ultimatestrength (d i =d i =0),denoted by max , is thereforeexpressed as max = " 1 e (m+n 1)! m!n! # 1 m+n : (24)
These losed-formsolutionsforquasi{stati (Eqn. (2))anddynami loadings(Eqn.(24)) an be validated by using Monte{Carlo simulations. In a ubi volume of 1.7mm
3
, a set of aws of density
t
[(t)℄is randomlylo ated. Whenthe stress rate in reases(with a onstant stressrate ),_ obs urationzones followingthe pro ess des ribed inSe tion2.2.1 aremodeled. The ma ros opi stress is obtained by averaging the mi ros opi stress in the non-relaxed zones. The behavior of this ` nite volume' is not deterministi and numerous al ulations have to be performed when average values are sought (e.g., average ma ros opi ultimate strength and standard deviation). Su h al ulations are shown in Fig. 5 where the ma ro-s opi ultimatestrength is plottedagainst the stress rate ._ It an be noted that the results obtained with the multi-s alemodel (Eqn. (19)) are very omparable (in terms of mean and standard deviation) to those given by Monte-Carlo simulations (Denoual and Hild, 2000), withaCPUtime dividedby 3000. Forastress ratewithin[0,500 MPas
1
Duringthesingle/multiplefragmentationtransition,thedi eren ebetweenthedashed lines (Eqns. (2)and (24)) and the Monte-Carlo simulations does not ex eed 10%. The stan-dard deviation signi antly de reases in the ase of multiple fragmentation when the stress ratein reases. Furthermore,for S-SiC erami s, astress rateup to10MPas
1
has shown nostressratee e t onthemeanfailurestrength (DenoualandRiou, 1995). This observation is ingoodagreementwith the resultshown inFig. 5.
The losed-formsolutionsforquasi{stati (Eqn.(2))ordynami regimes(Eqn. (24))are now used to determine when dis rete or ontinuum approa hes an be used. The transition between single and multiplefragmentation an beestimated as the interse tion between the weakest link and the multiple fragmentation solutions(see Fig. 5)
max
()_ = w
: (25)
The transitionde ned by Eqn. (25) leads tothe following inequalities 8 > > > > < > > > > : _ Z m+n mn <f Single fragmentation _ Z m+n mn f Multiple fragmentation (26) with f = 0 1=m 0 S 1=n n kC 0 " e m! n! (m+n 1)! m+1 m m+n # 1 n : (27)
8 > > > > > < > > > > > : Z Z ()_ <g(m) Singlefragmentation Z Z ()_ g(m) Multiplefragmentation (28) with g(m)= " e m!n! (m+n 1)! m+1 m m+n # m m+n : (29) ThesizeZ
anthereforebe onsideredasthe hara teristi s aleforwhi hasingle/multiple fragmentation transition is observed. Furthermore, Fig. 5 shows that, when Z=Z
1, the ultimate strength s atter is very smalli.e., when the stress rate in reases, the hara teristi s ale of the fragmented erami de reases and the stress estimated over Z be omes a good approximation of the average stress.
Furthermore, a hypothesis of uniformity of the damage variables over the horizon (see Fig. 3) is needed in a lo al (and ontinuum) approa h. When the mesh size Z
ZE
is smaller thanthe horizon,twoneighboringintegrationpointshavetheir horizonsoverlapping: aspa e lo ation may be in uen ed by two a priori independent sets of variables. To avoid su h a situation,theminimummeshsizemustbegreaterthanorequaltothehorizon. Equation(23) showsthatD i ( i =1) = 0andD i ( i =2) =
1(i.e.,mostofthe damage hangeo ursduring a time intervalequal tot
). During t
, the measure of the horizonis limited by Z n (t )=Z . Therefore the minimum mesh size is Z
. The size Z
is dependent on the loading rate: the higher the stress rate, the smaller the mesh size. This is onsistent with the general pra ti e ofmeshre ningwhensho kwavesaresuspe tedtoo ur. The hara teristi size anbeused inFE omputations inwhi hthe meshsize Z =Z
FE
has to begreater than orequal toZ
to use a ontinuum (and deterministi )des ription of damage(Eqn. (28)).
Continuum (and lo al) Me hani s approa h an be used. In the transition regime, dis rete approa hes may no longer be needed while Continuum (Damage) Me hani s hypotheses are not yetreasonable.
2.5 Damage Des ription
Theaimofthisse tionistoestimateofthe omplian etensorS D
ofadamagedbody. The ase of ra ks in one dire tionis analyzed in Se tion 2.5.1 and generalized thereafter to multiple ra k patterns in Se tion2.5.2.
2.5.1 Cra ks in One Dire tion
The damage state an be des ribed by using only one s alar variable D 1
. Let the shape parameter S t be written asa fra tion of S n S t = S n (30)
with 0:31 for = 0:15 and n = 3, see the numeri al evaluation of S t
and S n
in Se -tion 2.2.1.
It follows (see Eqns. (15), (16) and (18)) that
1 Z t Z =(1 D 1 ) : (31)
1 1 ra k in dire tion d 1 is surrounded by a zone Z t
that relaxes the shear stresses 12
(x) and
13 (x).
2.5.2 Cra ks in Multiple Dire tions
When three orthogonal ra k patterns are superimposed (i.e., the Boolean fun tions an be superimposed), the omplian e tensor S
D
is obtained by using Eqn. (3)
S D = 1 E 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 1 1 D 1 0 0 0 1 1 D 2 0 0 0 1 1 D 3 0 0 0 0 0 0 1+ (1 D 2 ) (1 D 3 ) 0 0 0 0 0 0 1+ (1 D 3 ) (1 D 1 ) 0 0 0 0 0 0 1+ (1 D 1 ) (1 D 2 ) 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (d 1 ;d 2 ;d 3 ) : (33)
The omplian etensorS D
isde nedinthedire tionsof ra king(d 1 ,d 2 ,d 3 ). Thesedire tions asso iatedtoD 1 ,D 2 and D 3
may hangeatea htimestepuntilD 1
rea hes athresholdvalue D
th
= 0:01 (the e e t of the threshold value was found to be negligible in the simulations). Then, only the dire tion d
1
is lo ked, the other dire tions follow the eigen dire tions of , with the onstraint to be perpendi ular to d
1
. When D 2
rea hes the threshold value, the whole dire tions d
i
are lo ked. It an benoted that the same type of result an be obtained by using mathemati alarguments on ase ond order damage tensor. The only hange is the value of the power : =
1 2
(Cordebois and Sidoro , 1982). Lastly, another des ription of the stress eld (e.g., relaxation fun tion R (x) obtained by a numeri al analysis instead of using Fabrikant'ssolution)would probably lead to yet anothervalue of the onstant .
straight ra ks. The inability to deal with rapidly rotating prin ipal dire tions of stress is however a limitationof the model.
3 Comparison with Experiments on SiC Cerami s
On e the elasti properties and the Weibull parameters are known, the model has no other parameters totune. A spe ial emphasis willbe put onsili on arbide erami s. Sin e sili on arbide erami s an be obtained by di erent pro essing routes, the present study mainly fo uses ontwo SiC grades whose properties are listed in Table I. The rst grade, referred to asS-SiC,wasprovided by CeramiqueetComposites(Fran e). The se ondgrade(SiC-B) has been manufa tured by CERCOM (USA). The S-SiC erami is naturally sintered (sintering temperature: 2000
Æ
C). The end produ tis an -SiC(6H hexagonalstru ture). The material isnot fullydense. No se ondaryphase anbeobserved but B
4
Cin lusionsare present(Riou, 1996) be ause boron was added to enhan e di usion during sintering. Transgranular failure is the dominant me hanism. On the other hand, SiC-B erami s are obtained by pressure assisted densi ation. Aluminum is used to eliminate porosities (pro essing temperature: 2000
Æ
C, pressure: 15MPa). An alumina-ri h se ondary (glassy) phase is present (Forquin, 2000). Be ause of the lower strength of the se ondary phase, the failure mode is predomi-nantly intergranular.
i s (Denoual et al., 1997b) and the EOI on guration an therefore be used to validate the damagekineti laws for numeri alsimulations of the behavior of lightarmors.
Figure6{a (top)shows astress rate map4s afterimpa t withthe orresponding dam-aged zone (bottom). When damage is generated, the stress rate is about 10
3
MPa/s. One ansee inFig. 6that thisloading annotbemodeleda urately byusingeither ontinuumor dis reteapproa hes, i.e., more than one defe t breaks but the material annotbe onsidered as ontinuumin aFE ell. It followsthat the multi-s alemodelisused. It an benoted that forea hnumeri alsimulationthe setof randomnumbersis hara terizedbyaninteger alled the `seed' of the random generator (Press et al., 1992). A given probabilisti simulation is then de ned by this integer and an always bereprodu ed by using the same `seed'.
3.1 Real Time Visualization
FEvolumeof1mm 3
. Forhighstressrates(i.e., infrontof theproje tileandintheHertz-like one ra k), many defe ts nu leate in a FE ell. For a velo ity of 185m/s, failure of an ele-ment set, whi h an be ompared to ma ros opi ra ks, an be observed in addition to the ontinuousdegradationgenerated attheedge ofthe proje tile(see Fig.7{b). However, there are some diÆ ulties in handling ma ros opi ra ks. The failure of a FE ell is not always followed by a ra k generation and propagation,and when su ha ra k is reated, there isa tenden y to follow the dire tionof the FE mesh. The des ription of ra k propagation may be improved by onsidering the failureof interfa es between nite elements instead of bulk failure(Cama ho and Ortiz,1996; Espinosa et al.,1998; Mastilovi and Kraj inovi , 1999).
3.2 Moire Te hnique
A se ond EOI on guration provides quantitative strain measurements over a eld of 32 32mm
2
during impa t. Details on the moire photography set-up an be found in (Bertin-Mourot et al., 1997). The advantage of the moire measurement is that a quantitative rather thanqualitativeanalysis anbeperformedbetween experimentsand simulations. Figure8{a is the fringepatternapproximately2s after impa t.
the nite element pa kage PamSho k (1998)). The average and standard deviation of the hoop and radial strains are plotted in Fig. 8. The multi-s ale modelyields good predi tions of thestrain levels. Allthe experimentalmeasurementsfall inthe greyshadedzone, i.e., the experiment may be ompared to one realization of the 500 numeri al simulations. The use of an anisotropi modelis ne essary if one wants to a urately predi t the strain levels. An elasti omputationunderestimatesbothradialand hoopstrains. Anisotropi damagemodel would have given even lowerstrain levels(Denoual et al., 1996).
4 Towards Material Optimization
A ne fragmentation ofthe erami leadsto alo alizedstrain aroundthe proje tiletip. The energy needed for the penetration into an armor is thus redu ed in omparison to a oarse fragmentationwherelargefragmentsspreadthestrainwithinthevolumeand onsumeenergy (see (Woodward et al.,1994)and Fig.9). Therefore, anoptimization riterionassumed tobe relevantforarmoristhatanin reaseoffragmentsize(i.e.,ade reaseofbrokendefe tdensity) leads to an in rease of stru tural strength. A losed-form solution for
b
(t) (Eqn. (17)) is used with a onstantstress rate of 5MPans
1
and a maximum tensile stress max of 1GPa max ( max )= m m+n m!n! (m+n)! !m+n m " m m+n ; (m+n)! m+n max m!n! # (34) where [p;x℄ = R x 0 t p 1
exp ( t) dt is the in omplete gamma fun tion. When the maximum tensile stress is rea hed, the kineti s of broken aw density is stopped (see Fig. 10{a). The material parameters hosen to be optimized are the mean failure stress
w
and the Weibull modulus m. For ea h ouple (
w
;m), the Weibull parameter 0
= m 0
is omputed by using Eqn. (2). The results are shown in Fig. 10{b where the broken defe t density is plotted as a fun tion of
w
low porosity of SiC-B erami s leading to a high average failure stress with a redu ed s at-ter (i.e., a high Weibull modulus). However, the S-SiC grade has the oarser fragmentation leading to a better ballisti performan e (Beylat and Cottenot, 1996). Finally, two other grades of sili on arbide alled SiC{HIP (Riou, 1996) and SiC{150 (Leroy, 1999) are plotted inFig.10{b. The SiC{HIPgradeexhibitsbetter ballisti performan esthan theS{SiCgrade (BeylatandCottenot,1996),asshown inFig.10{b. The SiC{150grade,whi hhasaporosity of10-14%,shows that agoodmaterialmusthavea lowWeibull modulus,i.e., alarges atter of failurestresses and a low porosity ontent, i.e., ahigh average failure stress.
5 Summary
A fragmentation model based on a me hanism of nu leation of aws and stress relaxation around propagating ra ks is derived. By onstru tion, this approa h is non-lo al and the horizon of a defe t onstitutes the key ingredient. When a onstant stress rate is applied, a losed-form solution for the number of nu leated defe ts is given. A damage kineti law is derivedfromthefragmentationmodel. Theanalysisofstressrelaxationaroundthe propagat-ing ra ks leads naturally to ananisotropi des ription of damage. A di erentialequation is obtainedforthe kineti sofdamagevariablesinordertobeimplementedintoaFE ode. The probabilisti nature of this model willhelp in understanding the non-deterministi behavior ofstru turesmadeofbrittlematerialsandsubmittedtoawiderangeof loadings(from quasi-stati to dynami ones). This model is able to des ribe a high density of ra ks of random lo ation. It is therefore well suited for des ribing degradations from the very early stages (i.e., nu leation of few ra ks) up tothe onset of ra k oales en e.
lo al-failurestrength (e.g., the SiC{B grade) is well reprodu ed by the model. The strain history duringimpa tisalsopredi ted,inparti ular whenthe materialseemstobe intensively dam-aged (e.g., the S-SiC grade).
The set of hypotheses shows that this model an only be used for damage in tension. Damage in ompression should lead toa very di erentmodeleven if the same kind of me h-anisms (i.e., aw nu leation, obs uration zones) are used. Moreover, when rapid rotating stressesare onsidered,the resultingdamageisobtained throughthe superpositionof orthog-onaldamagepatternsinsteadof hangingthedire tionofthe ra kpropagation. Thismaybe seen asalimitationof the model, aslong asthe di eren es between superposition ofdamage patternand rotating ra ks interms of overall stru tural response is proven.
Thetransitionzoneforwhi hthenumberofnu leated awsisgreaterthanbut neverthe-less lose toone ina FE elliswell reprodu ed by the multi-s alemodel. The orresponding behavior, neither ontinuum (deterministi and lo al) nor dis rete (probabilisti and non-lo al) is one of the major features of this model. Lastly, it is expe ted that these models are appli able to other brittle materials(su has ro k, glass or on rete). Sin e the numbers of parameters to identify is very limited and an be arried out under quasi-stati loading onditions, the model an betested ona large lass of brittle materials.
6 A knowledgements
Benz W., Asphaug E., 1994. Impa t Simulationswith Fra ture. I. Method and Tests. I arus 107, 98{116.
Bless S.J., Ben-Yami M., Apgar L., Eylon D., 1992. Impenetrable Targets Stru k by High Velo ity Tungsten Long Rod. In: Pro . 2
nd
Int. Conf. onStru tures under Sho k and Impa t, Portsmouth (UK).
Bertin-Mourot T., Denoual C., Deshors G., Louvigne P.-F., Thomas T., 1997. High Speed Photography of Moire Fringes: Appli ation to Cerami s under Impa t. J. Phys. IV oll. III-C3,311{316.
Beylat L., Cottenot C.E., 1996. Post Mortem Mi rostru tural Chara terizationof SiC MaterialsAfterIntera tionwith aKineti Energy Proje tile. In: Pro .SUSI'96symposium, 459{468.
Cama hoG.T., Ortiz M.,1996. ComputationalModeling of Impa t Damage in Brittle Materials. Int. J.Solids Stru t. 33,2899{2938.
Cho K., Katz N., Bar-On I., 1995. Strength and Fra ture Toughness of Hot Pressed SiC Materials. Ceram.S i. and Eng. Pro ., 16.
Cordebois J.-P., Sidoro F., 1982. Endommagement anisotrope en elasti ite et plas-ti ite. J.Me . Th. Appl., spe ial issue, 45{60.
Davies D.G.S., 1973. The Statisti al Approa h to Engineering Design in Cerami s. Pro .Brit. Ceram. So .22,429{452.
Denoual C., Riou P., 1995. Comportement a l'impa t de eramiques te hniques pour blindages legers. CREA, Report95 R 005 (inFren h).
Denoual C., Cottenot C., Hild F., 1996. On the Identi ation of Damage During Im-pa t ofaCerami by a HardSteelProje tile. In: Pro .16
th
Int. Symp.onBallisti s,ADPA, 541{550.
an Impa ted Cerami . In: Pro . Sho k Compression of Condensed Matter, S hmidt S.C., DandekarD.P. and ForbesJ.W. (Edts.), AIP Press, New York (USA), 427{430.
Denoual C., Hild F., 1998. On the Chara teristi S ales Involved in a Fragmentation Pro ess. J. Phys. IV Fran e 8, 119{126.
Denoual C., HildF., 2000. A DamageModelfor the Dynami Fragmentationof Brittle Solids. Comp.Meth. Appl. Me h.Eng. 183, 247{258.
Espinosa H.D., Raiser G., Clifton R.J., Ortiz M., 1992. Experimental Observations and Numeri al Modeling of Inelasti ity in Dynami ally Loaded Cerami s. J. Hard Mat. 3, 285{313.
Espinosa H.D., Zavattieri P.D., Dwivedi S.K., 1998. A Finite Deformation Contin-uum/Dis rete Model for the Des ription of Fragmentation and Damage in BrittleMaterials. J. Me h.Phys. Solids46, 1909{1942.
Fabrikant V.I., 1990. Complete Solution toSome Mixed Boundary Value Problemsin Elasti ity. Adv. Appl. Me h.27, 153{223.
Forquin P., Denoual C., Cottenot C.E., Rota L., Hild F., (2000). Experimental Ap-proa h and Modeling of the Compressive Behaviour of Two SiC Grades. J. Phys. IV 10, Pr9-735{740.
Freund L.B., 1990. Dynami Fra ture Me hani s, Cambridge University Press, Cam-bridge (UK).
Grady D.E.,KippM.E., 1980. ContinuumModelingofExplosiveFra tureinOilShale. Int. J. Ro k Min.S i. Geome h. 17,147{157.
Gulino R., Phoenix S.L., 1991. Weibull Strength Statisti s for Graphite Fibres Mea-sured from the Break Progression in a Model Graphite/Glass/Epoxy mi ro omposite. J. Mat.S i. 26,3107{3118.
HalmD., DragonA., 1998. AnAnisotropi ModelofDamage andFri tionalSlidingfor BrittleMaterials. Eur. J. Me h.A/Solids 17,439{460.
Ce-Hornemann U., Kaltho J.F., Rothenhausler H., Senf H., Winkler S., 1984. Exper-imental Investigation of Wave and Fra ture Propagation in Glass - Slabs Loaded by Steel Cylindersat HighImpa t Velo ities. EMI reportE 4/84, Weil amRhein (Germany).
Jeulin D., 1991. Modeles morphologiques de stru tures aleatoires et de hangement d'e helle, These d'
Etat, University ofCaen (in Fren h).
Jeulin D., Jeulin P., 1981. Synthesis of Rough Surfa es by Random Morphologi al Models. Stereol. Iugosl. 3/Suppl. 1, 239{246.
Jeulin D., Laurenge P.,1997. SimulationofRough Surfa es by Morphologi alRandom Fun tions. J. Ele troni Im. 6,16{30.
Ka hanov M., 1994. Elasti Solids with Many Cra ks and Related Problems. Adv. Appl. Me h. 30,259{445.
Kanninen M.F., Popelar C.H., 1985. Advan ed Fra tureMe hani s, OxfordUniversity Press, New York (USA).
LemaitreJ.,1992. ACourseonDamageMe hani s, Springer-Verlag,Berlin(Germany). Leroy Y., 1999. Private ommuni ation.
Mastilovi S., Kraj inovi D., 1999. High-Velo ityExpansionof aCavitywithina Brit-tleMaterial, J. Me h.Phys. Solids 47,577{600.
Meyers M.A., 1994. Dynami Behavior of Materials, Wiley Inters ien e, New York (USA).
MottN.F.,1947. FragmentationofShellCases. Pro .Roy.So .LondonA189, 300{308. PalikaR., 1995. Private ommuni ation.
PamSho k, 1998. User's manual, PamSystem International,ESI.
Press W.H.,Teukolsky S.A., VetterlingW.T., FlanneryB.P.,1992. Numeri al Re ipes inC, Cambridge University Press, Cambridge (UK).
den Reijer P.C., 1991. Impa t on Cerami Fa ed Armor, PhD thesis, Delft Te hni al University.
lors d'un impa t de basse energie : appli ation aux blindages, PhD thesis, E ole Nationale Superieure des Mines de Paris (inFren h).
Riou P., Denoual C., Cottenot C.E., 1998. Visualization of the Damage Evolution in Impa tedSili on Carbide Cerami s. Int.J. Impa t Eng. 21, 225{235.
SerraJ.,1982. ImageAnalysisandMathemati alMorphology, A ademi Press,London (UK).
Straburger E.,Senf H.,1994. ExperimentalInvestigations of Waveand Fra ture Phe-nomenain Impa tedCerami s. EMI report 3/94, Weilam Rhein (Germany).
Tate A., 1967. A Theory for the De eleration of Long Rods after Impa t. J. Me h. Phys. Solids 15,387{399.
Tate A., 1969. Further Results in the Theory of Long Rods Penetration. J. Me h. Phys. Solids 17,141{150.
Weibull W., 1939. A Statisti al Theory of the Strength of Materials. Roy. Swed. Inst. Eng. Res. 151.
WinklerS., SenfH., Rothenhausler, 1989. Waveand Fra turePhenomena inImpa ted Cerami s. EMI reportV 5/89, WeilamRhein (Germany).
The kineti s of nu leated aw density or damage variable an be obtained by using the onditions ofnon-relaxationfor agiven defe t by examiningthe inverseproblem(Denoualet al., 1997). It onsists in onsidering the past history of a defe t that would break at a time T. The defe t willbreak if no defe ts exist in its horizon. For a given defe t D, its horizon is de ned as aspa e-time zone in whi h a defe t willalways obs ure P (Fig. 3). Outside the horizona defe t willneverobs ure P.Equation (17) be omes
d b dt (T)= d t dt (T)[1 P o (T)℄ with b (0)=0 and t (0)=0 (35) where 1 P o
is the probability that no defe t exist in the horizon (P o = Zn Z ). The variable P o
an be split into an in nity of events de ned by the probability P(t) of nding at t a new defe t during a time step dt in a zone Z
n
(T t). This probability in rement is written by using a Poisson point pro ess of intensity d
t
=dt. Those independent events an be used toderive the followingexpression for P
o 1 P o (T) = T t (1 P(t)) = T t exp " d t dt (t)TZ n (T t) # exp " Z T 0 d t dt (t)Z n (T t)dt # (36) where Z n
Parameters S{SiC ℄ SiC{B [ SIC{HIP ℄ SiC{150 \
Young's modulusE (GPa) 410 455 465 350
Poisson'sratio 0.15 0.16 0.15 0.25
Density 3.15 3.20 3.18 2.76-2.89
Porosity 1.8% 0 NA 10-14 %
Weibull modulusm 9.3 27 8.6 15
Mean failure strength w (MPa) 370 560 590 225 E e tive volume Z e (mm 3 ) 1.7 1.5 1.2 1.4 Numberof samples 65 30 26 NA
Type of exuraltest 3-point 4-point 3-point 3-point
℄
: (Denoualand Riou, 1995) [
: (Palika,1995; Cho et al., 1994) \
1 Exampleofstressrelaxationfun tionsfornormaland tangentialloadings. The normal stress
11
and shear stress 12
have the greatest relaxation zones for normaland tangential loadings,respe tively. . . 33 2 Exampleofsimpli ationfora omplexmi rostru ture ontainingpenny-shaped
ra ks submitted to a normal far eld stress. The stress eld around a single ra k (a) is transformed into a Boolean fun tion (b) on whi h Boolean op-erations (i.e., union) an be performed (d). This simpli ation is used as a representation of omplex mi rostru tures ( ) for whi h e e tive elasti prop-ertiesare deli atetoobtain. Forthe sakeof simpli ity,onlythe stress eld
11 isrepresented even if the method isapplied ontensor elds. . . 34 3 a-Depi tionof obs uration phenomena.
b-Horizonfor a defe t P. . . 35 4 Change of broken aw density with time predi ted by the multi{s ale model
(solidline) and the fragmentation theory (dashed line). . . 36 5 Ultimate ma ros opi strength vs. stress rate predi ted by the multi-s ale
modeland Monte{Carlosimulations(500realizationsforea hpoints(Denoual and Hild,2000)) for anS-SiC erami . . . 37 6 Requirements tobeful lled touse a ontinuum damagemodel an be he ked
by using a maximum stress rate map.
elementsof 111mm 3
).
a{Exampleofrandomfailurestress k
forthe rst defe tabletobreakinea h FE ell.
b{ and { Tile upper part: simulations (multi-s ale model), tile lower part: experiments. Damaged zones (D
1
> 0:5 in the dark zones) for two impa t velo ities. . . 39 8 a{Typi al exampleof moirefringes.
b{ Strain hange given by a moire te hnique (dots) and by the multi-s ale model(plain urve: average, grey bandwidth: standard deviation). . . 40 9 Strainsaroundapenetratingproje tilewhenthe fragmentationis oarse (left)
and ne (right). . . 41 10 a{ Stress history for material optimization. The dashed urve is any positive
monotoni allyde reasing fun tion.
b{ Broken aw density map as fun tion of Weibull modulus m and average failurestress
w
Normal Loading
σ
11
0
Tangential Loading
σ
12
0
σ
12
0
σ
11
(x)/
σ
11
0
σ
11
(x)/
σ
11
0
σ
12
(x)/
σ
12
(x)/
σ
12
0
1
2
3
a
σ
=
0
σ σ
=
0
ij
σ
= 2
σ
0
ij
R
ij
=1
R
ij
=0
R
ij
=-1
1
2
1
2
−
−
−
−
Figure 1: Denoual and Hild
Figure 1: Example of stress relaxation fun tions for normal and tangential loadings. The normal stress
11
and shear stress 12
-a-
σ
11
around a
single crack
(closed-form
solution according
to Fabrikant (1990))
-b-Simplified σ
11
field around a
single crack
(Boolean
function)
-d-Simplified σ
11
field around
many cracks
(union of Boolean
functions)
-c-
σ
11
around a
random array
of cracks
∼
∼
Figure 2: Denoual and Hild
Figure 2: Example of simpli ation for a omplex mi rostru ture ontaining penny-shaped ra ks submitted to a normal far eld stress. The stress eld around a single ra k (a) is transformed into a Boolean fun tion (b) on whi h Boolean operations (i.e., union) an be performed (d). Thissimpli ation isused asa representationof omplex mi rostru tures( ) for whi h e e tive elasti properties are deli ate to obtain. For the sake of simpli ity, only the stress eld
11
b
-Space
Time
Stress
A defect in this
zone never
obscures P
Horizon of P
(a defect in this
zone always
obscures P)
a
-Space
Obscured flaws
Broken flaws
Relaxed
(or obscured)
zone
Stress
1
2
3
4
5
Time
Figure 3: Denoual and Hild
P
Brok
en def
ect density
,
λ
(T) b
Time
Failure of the first defect
Figure 4: Denoual and Hild
Monte-Carlo simulations
Weibull law
10
1
10
2
10
3
10
4
10
5
300
400
500
600
700
800
Multi-scale
model
Stress rate (MPa/µs)
Ultimate tensile strength (MP
a)
Figure 5: Denoual and Hild
Single Fragmentation
Multiple Fragmentation
10
-5
10
-4
10
-3
10
-2
10
-1
1
10
10
2
10
3
Dimensionless volume, Z/Z
c
σ
w
− σ
sd
σ
w
+ σ
sd
σ
w
Σ
max
V = 330m/s, T = 4
µs
Ultimate strength
Single
fragmen-tation
Multiple
fragmen-tation
Stress rate
(MPa/µs)
10
1
10
2
10
3
10
4
-a-
-b-Figure 6: Denoual and Hild
Figure 6: Requirements to be ful lled to use a ontinuum damagemodel an be he ked by using amaximum stress rate map.
a{ Example of maximum stress rate for the rst eigen stress 4safter impa t (top) and the orrespondingdamaged zone(bottom). Thelinedensity isafun tion ofbroken aws density and the line dire tionisthat of ra king.
550 MPa
450 MPa
-a-
-b-V=185m/s
-c-V=513m/s
Figure 7: Denoual and Hild
Figure 7: Numeri alsimulations of a SiC-B erami in anEOI on guration (5010010 elements of 111mm
3 ).
a{Example of random failurestress k
for the rst defe t able tobreak inea h FE ell. b{ and { Tile upper part: simulations (multi-s ale model), tile lower part: experiments. Damaged zones (D
1
0.0
0.004
0.008
0.0
–0.004
–0.008
–0.012
Hoop
str
ain
Radial strain
Moiré
technique
Multi-scale
model
Elastic
model
M
-a-
-b-Figure 8: Denoual and Hild
10 mm
Figure8: a{Typi al exampleof moirefringes.
Projectile
Strain localization
(fine fragments)
Fragments
Figure 9: Denoual and Hild
5
10
15
20
25
30
1
10
9
10
10
10
11
10
12
10
13
10
14
A
ver
age f
ailure strength,
σ
W
(MP
a)
Weibull modulus, m
1100
1000
900
800
700
600
500
400
300
200
SiC-B
SSiC
Fine
fragmentation
Coarse
fragmentation
SiC-HIP
SiC-150
σ(T)
T
σ
max
dσ
dT
-a-
-b-Figure 10: Denoual and Hild
Figure 10: a{ Stress history for material optimization. The dashed urve is any positive monotoni ally de reasing fun tion.
b{Broken aw density map as fun tionof Weibullmodulus m and average failurestress w
