International Applied Mechanics, Vol. 45, No. 2, L. P. Khoroshun and L. V. Nazarenko

Transcription

1 ntenational Applied Mechanics Vol. 5 No. 009 LONG-TERM DAMAGE OF DSCRETE-FBER-RENFORCED COMPOSTES WTH TRANSVERSELY SOTROPC NCLUSONS AND STRESS-RUPTURE MCROSTRENGTH DESCRBED BY AN EXPONENTAL POWER FUNCTON L. P. Khooshun and L. V. Nazaenko The theoy of long-tem damage of homogeneous mateials which is based on the equations of the mechanics of stochastically inhomogeneous mateials is genealized to discete-fibe-einfoced composite mateials. The micodamage of the composite components is modeled by andomly dispesed micopoes. The failue citeion fo a micovolume is chaacteized by its stess-uptue stength. t is detemined by the dependence of the time to bittle failue on the diffeence between the equivalent stess and its limit. Given macostesses and macostains an equation of damage poosity balance in the composite components at an abitay time is fomulated. The time dependence of micodamage and macostesses o macostains is established in the case of stess-uptue micostength descibed by an exponential powe function Keywods: discete-fibe-einfoced composite stochastic stuctue long-tem damage effective chaacteistics poosity balance equation stess-uptue micostength ntoduction. The stuctual theoy of long-tem damage of homogeneous mateials based on models and methods of the mechanics of stochastically inhomogeneous media is outlined in [] and genealized to paticulate composites in [ ]. The theoy of long-tem damage with stess-uptue micostength descibed by an exponential powe function is genealized hee to composites einfoced with unidiectional spheoidal fibes. t is assumed that the matix is isotopic and inclusions ae tansvesely isotopic. The matix sustains damage. The stuctual theoy of long-tem damage of composites is based on the mechanics of stochastic micoinhomogeneous mateials. Damage is modeled by dispesed micovolumes destoyed to become micopoes andomly aanged in the matix [7]. The failue citeion fo a single micovolume is detemined by its stess-uptue stength descibed by an exponential powe dependence of the time to bittle failue on the diffeence between the equivalent stess and its limit which chaacteizes the tensile stength accoding to the Hube Mises citeion []. The tensile stength is assumed to be a andom function of coodinates whose one-point distibution is descibed by the Weibull function [7]. The effective elastic popeties and the stess stain state of a stochastic composite ae detemined using the stochastic equations of elasticity and the method of conditional moments []. We will develop an algoithm fo calculating the time dependence of micodamage macostess and macostain and plot elevant cuves in the case of stess-uptue micostength descibed by an exponential powe function.. Ñonside a composite with a matix einfoced by andomly aanged unidiectional discete fibes. t is assumed that the matix is isotopic and the inclusions ae tansvesely isotopic. As the composite is loaded the matix sustains micodamages which ae modeled by andomly dispesed empty quasispheical micopoes. The macostesses and macostains ae elated by kl i j k l. kl kl S. P. Timoshenko nstitute of Mechanics National Academy of Sciences of Ukaine Nesteov St. Kyiv Ukaine stochas@inmech.kiev.ua. Tanslated fom Pikladnaya Mekhanika Vol. 5 No. pp. 9 9 Febuay 009. Oiginal aticle submitted Octobe /09/ Spinge Science+Business Media nc. 5

2 whee kl is the effective stiffness tenso which is defined [6] as a function of the elastic moduli of the damaged components [ ] kl [ ] and kl the volume faction of inclusions c and the shape paamete of inclusions s s / s i.e. [ ] [ ] kl kl kl kl c s s s / s. whee the supescipts efe to the inclusions and matix espectively; s and s ae the tansvese and longitudinal semiaxes of spheoidal inclusions espectively. [ ] [ ] The tensos kl and kl ae defined [] in tems of the stiffness tensos kl and kl of the skeletons of the components and thei poosities p and p chaacteizing damage: [ ] [ ] [ ] [ ].. kl kl kl p Given kl o we can use fomulas. the elations kl kl kl p [ ] kl kl. and the equations whee c c c c.5 kl kl kl and ae the mean-stess and mean-stain tensos of the damaged th component to find kl [ ] kl.6 kl [ ] kl klmn mn hee [ ] [ ] mn kl klmn klpq klpq [ ] pqmn [ ] pqmn c.7 whee klmn is a unit tenso. The mean stesses in the skeleton of the th component ae elated to the mean stesses component as in the damaged th p..8 f it is the matix that sustains damage we will use the Hube Mises failue citeion fo a micovolume of the undamaged potion of the matix []: k /.9 whee is the mean deviatoic stess in the undamaged potion of the matix; k is the limiting value of the invaiant which is a andom function of coodinates. f the invaiant fo some micovolume of the matix does not each the limiting value k then accoding to the stess-uptue citeion damage occus afte a lapse of time k which depends on the diffeence between and k.nthe geneal case this dependence can be epesented as some function and k k and k accoding to k k.0

3 The one-point distibution function F k fo k may be eithe a powe function on a finite inteval [7]: 0 k k0 n k k0 F k k0 k k k k 0 k k. o the Weibull function on a semi-infinite inteval: F k 0 k k0 mk k 0. exp k k 0 whee k 0 is the minimum value of k fom which failue begins in some micovolumes of the matix; k n m and ae the constants chaacteizing micostength scatte. Assume that the andom field of the tensile micostength k is statistically homogeneous which is typical of eal mateials and micodamages and the distances between them ae negligible compaed with the micovolume. Then the distibution function F k is egodic because it defines the undamaged potion of the matix in which the tensile micostength is less than k. Theefoe if the stesses ae nonzeo the function F defines accoding to.9. and. the content of instantaneously destoyed micovolumes in the matix. Since destoyed micovolumes ae modeled by poes then we can wite a balance equation fo destoyed micovolumes o poosities of the matix subject to shot-tem damage [7]: p p0 p0 F. whee p 0 is the initial poosity of the matix. f the stesses jk act fo some time t then accoding to the stess-uptue citeion.0 those micovolumes will be destoyed that have k such that whee the invaiant is defined by.9. t k k. f the time to bittle failue k fo eal mateials is finite fo abitay values of which may be obseved at high tempeatues k can be epesented by an exponential powe function []: k expl 0 k n n.5 which has enough constants 0 l n n to fit expeimental data. Substituting. into. we aive at the inequality k / n t ln l / n t t 0..6 Consideing the definition of the distibution function F k we conclude that the function F [ t ] whee / n / n t ln t l.7 7

4 defines the elative content of the destoyed micovolumes in the undamaged potion of the matix at the time t. Then in view of.8 the poosity balance equation fo the matix subject to long-tem damage can be epesented as p p p F 0 0 t.8 p whee the poosity p is a function of the dimensionless time t and the invaiant defined by.8 is a function of the macostains o macostesses accoding to.6. At t 0 the poosity balance equation.8 with.8.9 and.7 defines the shot-tem instantaneous damage of the mateial. As time elapses Eq..8 with.8.9 and.7 defines its long-tem damage which consists of shot-tem damage and additional time-dependent damage.. We will use the above model to study the defomation of a discete-fibe composite with an isotopic matix and tansvesely isotopic inclusions. We assume that only the matix sustains micodamage i.e. p 0. The effective elastic constants and of the composite ae defined by the following fomulas [6]: and cc [ ] [ ] [ ] [ ] K z [ ] [ ] [ ] [ ] [ ] K z K z cc [ ] [ ] [ ] K [ ] [ ] [ ] z K z K z. cc [ ] [ ] [ ] K z K z [ ] K z [ ] K 5 K 5 cc cc K 66 K K K K z z K K K [ ] [ ] [ ] [ ] [ c ] [ ] [ ] c c. [ ] [ ] [ ] [ ] c. [ ] [ ]. whee c c The paametes K n n 5 ae given by c c K c A c c 8 A K A A c c c c c 5 K A A A A A c c c c c K A A A 8

5 5 c c c c c c K A A A A. whee A 8 c c s f f q f s fs q A 8 c c s f s fs q A 8 c c s fs q.5 whee and ae expessed diffeently depending on the sign of the paamete Q defined by Q f q..6 f Q 0 then D D s Q D D.7 s whee D i actan zi zi 0 / zi z i ln zi 0 / zi z i i = z f s Q z s f s Q..8 s f Q 0 then s z Q z z a accot z a z aln z a z a accot z a z aln s z.9 whee z z z z z a z z a z a z a f s s..0 f Q 0 then s fd fz D z f s. z whee 9

6 s z actan z 0 z s D s s z ln z 0. z f. The paametes A and A 5 in. ae defined by the fomulas A 8 c 66 s E E A s 5. c 8 s 66 whee s s actan s 0 s s E s s s ln s 0. s. The paametes f q and in Eqs..5. ae expessed as c c c c c c f q c c c c c The tanscendental equation..8 can be solved using the following iteative pocedue. At the nth iteation the poosity of the matix p n is elated to the limiting intensity of mean tangential stess in the skeleton of the mateial k n n i.e. to the intensity of mean tangential stess which in tun is elated to the poosity of the matix at the n th iteation p n n the effective elastic moduli of the composite at the n th iteation and the macostains kl o macostesses accoding to.6.9. The effective elastic moduli ae also elated to the poosity p n accoding to.. Thus expessions.6.8 yield whee k n is given We assume the Weibull distibution: n n 0 0 p p p F k.6 n n / n k t t ln t l..7 / n n k k n 0 0 F k n n exp m k k k k Accoding to.6.9 we have n n n n p kl.9 0

7 p = p = t t Fig. Fig. o n n n n p.0 t follows fom.. that n n c p s.. Thus given macostains we can use Eqs... to detemine the effective elastic chaacteistics of a poous tansvesely isotopic mateial accoding to the equality n lim n.. Given macostains and having effective elastic chaacteistics we can use fomulas.. to calculate the macostesses.. We will now use the above pocedue and poosity balance equations to plot as an example nonlinea stess stain cuves and to study the behavio of a discete-fibe-einfoced composite with an isotopic matix and tansvesely isotopic inclusions. Fomulas allow us to detemine the volume faction of micodamage in the matix and to analyze the stess stain state of the composite fo the function t defined by.7 when eithe macostesses o macostains kl ae given. The inclusions and matix ae made of quatz and epoxy esin espectively with the following chaacteistics: 8. GPa 07 GPa GPa 9 GPa 5.8 GPa E GPa The volume faction of the inclusions the initial poosity of the matix and the shape paamete of the inclusions ae the following: c p 0 0 s.. The paametes of the Weibull distibution and stess-uptue micostength ae the following: k GPa m 000 l n n.. Figues and show the poosity p of the matix as a function of time t fo c 0.5 and c 0.50 espectively and diffeent values of. Figues and show the macostain as a function of time t fo c 0.5 and c 0.50

8 = = t t Fig. Fig. p = GPa 0.0 = t t Fig. 5 Fig. 6 espectively and diffeent values of. As can be seen fo all values of the macostain and poosity each at some values of t the citical levels at which failue begins. Compaing the esults obtained with the exponential powe function.5 and the factional powe function [] eveals that given the same macopaametes the macostain and the poosity p depend diffeently on time t : they have a hoizontal asymptote fo macostesses smalle than a cetain level in the case of factional powe function and each a citical level fom which failue begins fo all macostesses in the case of exponential powe function. Figue 5 shows the poosity p as a function of time t fo = and c = 0.5 solid line c = 0.50 dashed line and c = 0.75 dotted line. The notation is the same in Fig. 6. t can be seen that the micodamage p inceases with the macostain fo all volume factions of inclusions and abitay time t. Hee damage builds up with time wheeas the expeiments on polymes [] indicate that damage does not change noticeably at constant stain. This disageement may be attibuted to eithe stess elaxation in polymes due to ceep which is neglected hee o the appoximateness of the finite-time damage model. Figue 6 shows the macostess as a function of time t fo = and diffeent values of c.as is seen the cuves ae descending fo all values of c. Compaing the esults obtained with the exponential powe function.5 and the factional powe function [] shows that given the same macostains the macostess and the poosity p show simila dependences on time t.

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