STATISTIC MODELING OF THE CREEP BEHAVIOR OF METAL MATRIX COMPOSITES BASED ON FINITE ELEMENT ANALYSIS ~ YUE Zhu-feng (~-~t~) 1'2

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1 Applied Mamatics Mechanics (English Edition, Vol 23, No 4, Apr 2002) Published by Shanghai University, Shanghai, China Article ID: (2002) STATISTIC MODELING OF THE CREEP BEHAVIOR OF METAL MATRIX COMPOSITES BASED ON FINITE ELEMENT ANALYSIS ~ YUE Zhu-feng (~-~t~) 1'2 ( 1. Department Applied Mechanics, Northwestem Polytechnical University, Xi'an , P R China; 2. Institute Materials, Ruhr University, Bochum, Germany) (Communicated by ZHOU Cheng-ti) The one-dimensional problem motion a rigid flying plate under explosive attack has an analytic solution only when polytropic index detonation products equals to three. In general, : a numerical The analysis aim is required. paper is to In discover this paper, general however, creep by mechanisms utilizing for "weak" short shock behavior fiber reinforcement reflection matrix shock composites in explosive ( MMCs products, ) under uniaxial applying stress States small to parameter build a purterbation relationship method, an between analytic, macroscopic first-order approximate steady creep behavior solution is obtained material for micro problem geometric flying plate driven parameters. by various The high unit explosives cell models with were polytropic used to calculate indices or macroscopic than but nearly creep equal behavior to three. Final velocities with different flying micro plate geometric obtained parameters agree very fibers well with on different numerical loading results directions. by computers. The Thus an analytic influence formula with geometric two parameters parameters high fibers explosive loading (i.e. detonation directions on velocity macroscopic polytropic index) for creep estimation behavior had been velocity obtained, flying plate described is established. quantitatively. The matrix~fiber interface had been considered by a third layer, matrix~fiber inter.layer, in unit cells with different creep properties thickness. 1. Introduction Based on numerical results unit cell models, a statistic model had been presented for plane romly-distributed-fiber Explosive MMCs. The driven fiber flying-plate breakage had technique been taken ffmds into its account important in use statistic in model study for it behavior starts materials under intense impulsive loading, shock synsis diamonds, explosive welding experimentally early in creep life. With distribution geometric parameters cladding metals. The method estimation flyor velocity way raising it are questions fibers, results statistic model agree welt with experiments. With common interest. Under statistic model, assumptions influence one-dimensional geometric plane parameters detonation breakage rigid flying plate, fibers as normal approach well as solving properties problem thickness motion interlayer flyor is to on solve macroscopic following steady system creep equations rate governing have been flow discussed. field detonation products behind flyor (Fig. I): Key words: unit cell model; finite element method; MMCs; creep behavior; breakage fiber; statistic model; fiber parameters distribution CLC number: TGll3 Document code: A Introduction as a--t as The creep behavior short fiber reinforce Metal Matrix Composites (MMCs) depends on following factors, such as creep property matrix, elastic fractures properties where fiber, p, p, geometric S, u are pressure, parameters density, specific fibers, entropy arrangement particle velocity fibers detonation property products respectively, with trajectory R reflected shock detonation wave D as a boundary trajectory F flyor as anor boundary. Both are unknown; position R state parameters on Received it are governed date : by flow ; Revised field I date: central rarefaction wave behind detonation wave D by Foundation initial stage item: motion National flyor Natural also; Science position Foundation F China( ) state parameters products Biography: YUE Zhu-feng ( ), Pressor, Doctor hotmail, corn) 421

2 422 YUE Zhu-feng fiber/matrix interface fl-5] It should be considered simultaneously such mechanisms as load transferring from matrix to fiber, increased dislocation density around fiber, constrained flow matrix material around fiber, residual stresses arising from difference in coefficients rmal expansion between fiber matrix. The fiber breakage should also be taken into account for it starts early in creep life, when mechanical studies are carried out. So far se mechanisms in creep deformation for any given MMCs have not been adequately assessed. But generally, creep strain rate MMCs can be expressed as following qualitative equation = J"(ty, T,d m, Vf,/~p, 0, O'int, O'bf), (1) where a = external applied stress, T = testing temperature, ~ m : creep property matrix, Vf = fiber volume fraction, A p = geometric parameters fibers; 0 = fiber alignment angle (relative to applied loading), Crin t : fiber/matrix interface properties, abf = critical fracture stress The one-dimensional fiber. A complete problem description motion deformation a rigid flying in plate MMCs under should explosive incorporate attack has all an se analytic parameters. solution However only when it is not polytropic yet possible index with present detonation modeling products methods. equals In to fact, three. a In full general, a numerical analysis is required. In this paper, however, by utilizing "weak" shock description importance each geometric parameter such as reinforcement aspect ratio, behavior reflection shock in explosive products, applying small parameter purterbation volume fraction method, analytic, arrangement first-order is lacking approximate in literature, solution is obtained although for some problem investigators flying have plate acknowledged driven by various importance high explosives se with parameters. polytropic indices or than but nearly equal to three. Final velocities Previous works flying describing plate obtained deformation agree very mechanisms well with numerical macro results mechanical by computers. behavior Thus an MMCs analytic fall formula into three with categories. two parameters The shear-lag high approach explosive [63 (i.e. detonation its modified velocity methods polytropic concentrate index) for estimation velocity flying plate is established. on load transferred from matrix to fiber by shear loading at fiber/matrix interface, assuming that all matrix deformation takes place by shearing. This method has been used to describe stress state within a fiber 1. under Introduction applied load has been extended to allow for debonding at interface. But, limited by models mselves, predictability se Explosive driven flying-plate technique ffmds its important use in study behavior materials models is under questions. intense impulsive The Esheby's loading, "mean shock stress" synsis approach diamonds, has been used explosive to model welding work cladding hardening metals. composites The method reinforced estimation with elastic flyor ellipsoidal velocity inclusions [73 way. This raising method it are assumes questions that common plastic interest. strain is uniform within matrix se composites, although finite element modeling Under has indicated assumptions that one-dimensional strain is in fact plane highly detonation heterogeneous. rigid Models flying plate, using normal finite approach element continuum solving approach problem have recently motion been flyor used is to to model solve following stress-strain system characteristics equations governing flow field detonation products behind flyor (Fig. I): fiber plate containing composites have also been used to model process void formation at fiber/matrix interface by unit cell models [1'2'43. For an example, paper [ 1 ~ has carried out some basic calculation creep ap behavior +u_~_xp + by au plane elements axisymmetric elements. It is found, as pointed out by elastic-plastic au au analysis, 1 that strength creep behavior MMC with romly arranged fibers can not be described by modeling regular fiber arrangement, i.e., unit cell models. as These as papers have only carried out calculation on some special cases, a general result a--t has not been available. 1 Material Finite Element Model where 1.1 p, Material p, S, u are pressure, density, specific entropy particle velocity detonation products respectively, with trajectory R reflected shock detonation wave D as a boundary In order to assess predictability, an Al-matrix MMC will be used. The MMC contains trajectory F flyor as anor boundary. Both are unknown; position R state parameters 15% A1Mg on it are fibers. governed The details by flow field MMC I are central reported rarefaction in Ref. wave [8]. behind It is a detonation plane-romly- wave D distributed-fiber by initial stage composite. motion The stard flyor also; uniaxial position creep specimens F were state cut parameters out in such products a way that loading direction is within plane. Uniaxial constant load creep tests were carried out in

3 Creep Behavior Metal Matrix Composites 423 air in temperature 350~ at stresses ranging from 20 MPa to 40 MPa. Details uniaxiai creep testing are reported in Ref. [ 8 ]. In order to carry out finite element calculation, pure matrix is also crept at same temperature, steady creep behavior matrix is given out as following, which will be used in following calculation, by = 9.24 x 10-9o "3"1. (2) 1.2 Finite element model-unit cell In order to make problem tractable, unit cell with a quarter half fiber will be modeled. The unit cell means that, fiber is idealized as a regular, aligned period array. In this way, creep behavior MMCs as a whole may be represented by behavior unit cell containing half fiber surrounding matrix material. Since cells are identical, ir creep behavior will be identical under same loading conditions. Three-dimensional unit cell will be calculated as shown in Fig. 1. The center zone is a quarter The one-dimensional a half-fiber, which problem is surrounded motion by matrix a rigid material. flying plate The unit under cell explosive can be attack represented has an uniquely analytic by solution three only parameters when 1/L, polytropic d/d index did. The detonation uniaxial products loading directions equals to three. are within In general, plane a numerical xoz, which analysis is is plane required. In romly-distributed this paper, however, fibers. by utilizing The loading "weak" di.rection shock is behavior reflection shock in explosive products, applying small parameter purrepresented by angle 0, defined in figure. The effect adjacent reinforcements on terbation method, an analytic, first-order approximate solution is obtained for problem flying plate deformation driven by within various high unit explosives cell is accounted with polytropic for by applying indices or constraints than but to nearly outer equal surfaces to three. Final unit velocities celt so that flying it remains plate obtained geometrically agree very compatible well with with numerical or cells results deforming by computers. around Thus it. A an matrix/interface analytic formula interlayer with two is parameters induced to model high explosive interface (i.e. properties detonation velocity characteristics polytropic index) material for estimation manufacture. velocity To differentiate flying plate above is established. unit cell, model is named a matrix/ interface interlayer unit cell. The introduction interlayer is suitable for some cases, 1. Introduction as indicted by Ref. [91. The interlayer has different properties as Explosive driven flying-plate technique ffmds its important use in study behavior matrix fiber. But it is difficult to obtain thickness materials under intense impulsive loading, shock synsis diamonds, explosive welding property interface, So paper wil'l carry on,i -- --matrix cladding metals. The method estimation flyor velocity way raising it are questions general common calculations, interest. will give out influences thickness Under property assumptions interface one-dimensional plane overall detonation creep rigid flying plate, normal approach behavior. solving problem motion flyor is to solve following system equations governing In order flow to model field detonation cases products different behind angles between flyor (Fig. I): direction fiber loading, concept equivalent stress field is used here. It means that unit cell is loaded by same stress as applied stress on specimens. For example, to simulate constant stress creep test, normal tractions ( Tz ) are applied as to as end unit ceil in direction fiber such a--t that area integral traction in equal to total applied load / / Fig. 1 The unit cell computer model.['j TidA = ~Ai, where p, p, S, u are pressure, density, specific entropy (3) particle velocity detonation products respectively, with trajectory R reflected shock detonation wave D as a boundary where a is average applied stress in loading direction, A i is total area loaded surface trajectory F flyor as anor boundary. Both are unknown; position R state parameters on unit it cell. are governed Since by end flow field unit I cell central is constrained rarefaction to remain wave behind planar by detonation constraints, wave D tractions by are initial disturbed stage along motion end flyor surfaced also; in position order to F achieve this state displacement parameters constraint. products Normal tractions on outer side surfaces unit cell shear traction on all outer t ~ / interlayer D y

4 424 YUE Zhu-feng surfaces unit cell as zero. For those unit cells with loading directions or than longitudinal transverse ones, above boundary conditions may be too strict. On such cases, two kinds boundary conditions, one is above keeping plane constraint, or is free out surface, will be calculated simultaneously, compared with each or. The fiber is modeled as elastic while matrix deforms both elastically by Norton power law creep, shown in Eq. (2). The material elastic parameters for fiber matrix are given out in Table 1. Table 1 Elastic parameters fiber matrix elastic modulus E/GPa Poisson ratio v fiber matrix The one-dimensional problem motion a rigid flying plate under explosive attack has an analytic solution only when polytropic index detonation products equals to three. In general, The a following numerical triaxial analysis constitutive is required. law In is this used paper, to relate however, components by utilizing creep strain "weak" rate shock to behavior effective stress reflection deviatoric shock in stress explosive tensor products, applying small parameter purterbation method, an analytic, first-order approximate ~0 = cae n-, solution is obtained for problem flying Sij, (4) plate driven by various high explosives with polytropic indices or than but nearly equal to three. Final velocities flying plate obtained agree very well with numerical results by computers. Thus an where analytic a e = formula N/If ~ 3 S~flj, with two Mises parameters effective high stress, explosive S~ (i.e. = cr~j detonation - ~-8ij 1 akk, velocity deviatoric polytropic stress index) tensor, for c estimation is parameter velocity Norton flying power plate creep is established. law, n is exponent law. The constitutive law thus formulated can be reduced to familiar Eq. (2), Norton power law, under uniaxial conditions. Perfect bonding is 1. assumed Introduction to exist between fiber matrix as well as interlayer. Explosive driven flying-plate technique ffmds its important use in study behavior The unit cell described above is solved using ABAQUS, version 6. 1, a general finite materials under intense impulsive loading, shock synsis diamonds, explosive welding cladding element code. metals. For The each method case studied, estimation unit flyor cell velocity is divided into way a mesh raising it 510 are questions 20-noded isoparametrical common interest. three-dimensional elements. Convergence studies show this number elements is sufficient Under to describe assumptions creep one-dimensional behavior accurately, plane compared detonation with rigid cases flying plate, element normal number approach 800, solving problem A Newton-Rhapson motion flyor technique is to solve with adaptive following time system stepping equations is used. governing Calculations flow are carried field out detonation for 500 products or more behind increments flyor (Fig. at I): stop time, steady creep behavior has already been obtained. 2 Axial Stress Fiber 2.1 Under longitudinal loading The development axial stress as is illustrated as in Fig. 2, showing magnitude axial stress along fiber direction, i.e., a--t loading direction, at different creep time. Initially, due to its large elastic modulus, fiber bears a large part load on composite, compared to applied stress. As matrix creeps around fiber, it forces fiber to extend because where prefect p, bond p, S, at u are pressure, fiber/matrix density, interface. specific This entropy elastic straining particle velocity in fiber detonation increases products stress respectively, with trajectory R reflected shock detonation wave D as a boundary furr, especially near fiber center where a peak value several times : applied stress is trajectory F flyor as anor boundary. Both are unknown; position R state parameters attained. on The it are stress governed prile by along flow centerline field I central rarefaction fiber changes wave significantly behind during detonation course wave D creep by initial deformation. stage Comparison motion flyor with also; shear-lag position model F prediction state for parameters an elastic fiber products in a creeping matrix shows that stress state within fiber is more complex than can be modeled

5 Creep Behavior Metal Matrix Composites 425 within ir ory. Furrmore, from calculation, it is found that shear-lag model can not describe high stress in fiber near fiber end due to intense shearing at side interface normal traction at fiber end, as pointed out by Ref. [ 1 ] i~.. l:t =400 s 300" ~ - - m ~ 2:t =300 s 400~ '~J,.._ 2-~ 3 : t = 150 s 250" 3oo. -~-...~. ~ =, = sos 200- l = lo,l =6,d = l.2,d =O.308,d~ = l Sl -10 L-6 d-1 D d~-i ; =o ;0s, ",,-2,~ g l = 10,L =6,d =0.5,D = 0.308, dy = t/s The one-dimensional problem motion a rigid flying plate under explosive attack has an analytic Fig. 2 solution Development only when axial stress polytropic or'= in index Fig. detonation 3 The development products equals peak to three. axial In general, a numerical fiber (from analysis middle is required. to end) In this paper, however, stress by a'= utilizing fiber under "weak" axial shock behavior reflection shock in explosive products, applying small parameter purgeometric relative values: l = 10, loading (cr = 20 MPa) terbation method, an analytic, first-order approximate solution is obtained for problem flying L = 6, d = 1, D = 0.308, different geometric plate driven by various high explosives with polytropic indices or than but nearly equal to three. Final velocities dy = flying l, cr = plate 20 MPa obtained agree very well with numerical fibers results by computers. Thus an analytic formula with two parameters high explosive (i.e. detonation velocity polytropic index) One for estimation important phenomenon velocity is that flying re plate is a is steady established. value for axial stress state in fiber for each case. Fig. 3 shows peak axial stress varying with creep time for some cases, as an example. At this time, macro creep 1. has Introduction already gotten into creep steady stage. It means that on steady creep stage, axial stress will keep constant. In fact, at this time, stress Explosive driven flying-plate technique ffmds its important use in study behavior stste within overall unit cell has little change. The transfer time from elastic state to materials under intense impulsive loading, shock synsis diamonds, explosive welding creep steady state depends on applied stress geometric parameters. Generally, ' case cladding metals. The method estimation flyor velocity way raising it are questions with common bigger interest. external applied stress needs shorter transfer time. This characteristic axial stress is also different Under from assumptions prediction one-dimensional shear-lag plane model detonation that calculation rigid flying plate, Ref. [6]. normal approach The steady solving value problem peak axial motion stress within flyor is fiber to solve is always following center system in equations middle governing section, flow is an field important detonation parameter. products The behind dependence flyor (Fig. steady I): value peak axial stress on external applied stress micro geometric parameters is shown in Fig. 4: For each unit cell model (keeping micro geometric parameter fibers), steady value axial stress is proportional to external applied stress, can be expressed simply by following equation as a= as = k~, (5) where a... is steady value peak a--t axial stress in fiber, k is slop, depending on geometric parameters loading p directions. =p(p, s), On longitudinal loading, k depends on parameters l/l, d/d dy/d. Increasing where l/l, p, d/d p, S, u are dy/d pressure, Will result density, in decreasing specific entropy slop k. particle But velocity parameter detonation 1/L has products biggest respectively, with trajectory R reflected shock detonation wave D as a boundary influence on slop k. It also means that decreasing volume faction fiber will trajectory F flyor as anor boundary. Both are unknown; position R state parameters increase on k, it are governed peak by axial stress flow will field depend I central more rarefaction extemal wave behind stress. detonation wave D 2.2 Transverse by initial stage loading motion flyor also; position F state parameters products A different stress state will exist within fiber when unit cell is applied by

6 - lo0 426 YUE Zhu-feng transverse loading. At this time, fiber is loaded by compress stress, for compatibility deformation. The peak value compress axial stress also exists in middle section. As creep deformation increases, difference axial stress between in center middle section at end fiber increases. It is found, as that case longitudinal loading, that re is a steady value. The steady value for peak axial stress depends on external applied, also on geometric parameters. Eq. (5) can also relate steady value peak axial stress to external applied stress. The parameter k is also depended on geometric parameters fiber. l:l=10,l=6,d=o.308,d=1.2,dy=1 2:1 12,L =6,D = 0.308,d 1,dy= 1 3: l = lo,l =6, D=O.308,d = l,dr= l 4:l=8,L=6 D=O.308,d=l,dr=l 300- I~176 ls:t-- 10,L =6,D-0.308,d =0.8,d r = 1 S~ I ' 6:l=lO'L=6'D=O'308'd=O'5'dr=l,r ~ ~ 4 J The one-dimensional problem motion a rigid flying plate under explosive attack has 10oan analytic 7O0- solution only when polytropic 5 index detonation products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing "weak" shock 5OObehavior reflection shock in explosive products, applying small parameter pur- 1: constrained plane boundary terbation 300" method, an analytic, first-order approximate solution is obtained 2: free-surt~ce for problem bound,~ry flying plate driven " by various high explosives with polytropic indices or than but nearly equal to three. 100" Final velocities flying plate obtained agree very well with numerical results by computers. Thus 0 5 " 1'0 " 1'5 " 2'0 " 2'5 " 30 " 3'5 " 4~ '~5 an analytic formula with two parameters high explosive 0 (i.e. detonation 2'0 4b velocity 6b polytropic 8b a/mpa O index) for estimation velocity flying plate is established. Fig.4 The dependence steady values peak axial stress a= on external Explosive driven applied flying-plate stress a technique ffmds its important D = 0.308, use in d = study d r = 1 behavior materials under intense impulsive loading, shock synsis diamonds, explosive welding 2.3 Loading in or directions cladding metals. The method estimation flyor velocity way raising it are questions If loading direction is not along nor transverse to fiber direction, axial stress common interest. varies Under with loading assumptions directions one-dimensional between two plane extreme detonation cases. It rigid is also flying found plate, that re normal is a approach steady state for solving each case. problem At steady motion creep flyor stages, is to solve stress following state within system unit equations cell will governing almost be unchanged. flow field The detonation steady products value behind axial flyor stress (Fig. will I): change tensile on Fig.5 1. Introduction The steady peak axial stress or= varies with loading direction a = 10 MPa, l = 10, L = 6, longitudinal loading to compressive on transverse loading, as shown in Fig. 5. The critical angle for change is about 25 ~ to transverse direction (refer to Fig. 1 ) on geometric parameters fibers studied here. Two kinds boundary conditions are analyzed y simultaneously, =0, results are shown in Fig. 5 for comparison influence boundary conditions on peak axial stress at as as creep steady stages. The boundary a--t condition remaining plane constraints gives higher axial stress than free boundary. The difference is about 10%. It is believed that real result for MMCs is between two results. Since difference is not too big, in following where discussion p, p, S, u are following pressure, statistic density, model, specific entropy average peak particle axial velocity stress will detonation be used. It products is found respectively, with trajectory R reflected shock detonation wave D as a boundary that peak axial stress localizes also in center middle section fiber for each trajectory F flyor as anor boundary. Both are unknown; position R state parameters case. Eq. on it (5) are can governed describe by steady flow field value I central peak rarefaction axial stress. wave It behind is shown that detonation as increasing wave D l/l, by d/d initial stage dy/d motion will result flyor in increasing also; position slop F steady state value parameters steady products value peak axial stress to external applied stress.

7 _ ' Matrix/fiber interlayer model Creep Behavior Metal Matrix Composites 427 The introduction matrix/fiber interlayer will decrease axial stress, compared to unit cell model with same geometric parameters creep properties, with assumption that interlayer between matrix fiber (none creep). Like above unit cell models, re is a steady state axial stress for each case. And peak value exists also in center middle section fiber. It is found that steady value axial stress is also proportional to external applied stress, can be expressed by Eq. (5). The parameter k in Eq. (5) depends not only on micro creep law No. thickness interface ~ =9.410-~%st =1 3 0,075 r = I~ /~" ~ =9"410-1%7t / "2 6ooi ~ 4 9 / ~3 geometric The one-dimensional parameters problem loading conditions, motion a rigid flying plate under a/mpa explosive attack has an which analytic have solution been discussed only when above, but polytropic also on index Fig. detonation 6 The dependence products equals to steady three. peak In general, creep properties a numerical analysis is thickness required. In this paper, however, axial by stress utilizing a= on "weak" applied shock behavior interlayer. Increasing reflection shock creep stress in exponent explosive products, stressa applying (l = small 10, L parameter = 6, purterbation method, an analytic, first-order approximate solution is obtained for problem flying interlayer or increasing thickness D = 0.308, d = d r = 1) plate driven by various high explosives with polytropic indices or than but nearly equal to three. Final interlayer velocities will decrease flying slop plate k obtained matrix/fiber agree very well interlayer with numerical model, as results shown by in computers. Fig. 6. Thus an analytic formula with two parameters high explosive (i.e. detonation velocity polytropic 3 Creep Behavior From Unit Cell Model Matrix/Fiber Interlayer index) for estimation velocity flying plate is established. Model 2002 " lb " 1'5 " N 2'5" 3'0 ' 3~ 4b d5 3.1 Unit cell model 1. Introduction In presenting results analysis, macroscopic creep true strain creep rates are defined Explosive respectively driven as flying-plate technique ffmds its important use in study behavior materials under intense impulsive loading, shock synsis diamonds, explosive welding m, li cladding metals. The method estimation r = In flyor lio velocity way raising it are questions (6) common interest. Under assumptions one-dimensional ~i - d(ln( plane li/l'~ detonation ) rigid flying plate, normal (7) approach solving problem motion flyor dt is to solve following system equations governing where li0 is flow initial field length detonation unit products cell in behind loading flyor direction, (Fig. I): li is length unit cell in long direction in different creep time. All simulated creep curves from analysis show an initial decreasing creep strain rate transient followed by a steady state deformation. The initial transient behavior does not result from primary creep matrix which is not modeled in this study, but rar results directly from transfer load from matrix to fiber. Fig. 7 show a set creep behavior. It is clear that re are steady creep stages for all cases. The as as steady creep strain rate depends on external applied stress geometric parameters. For a a--t given unit cell ( geometric parameter fibers are fixed), a power law Can be related steady creep strain rate to external applied stress, following equation can be used to where describe p, p, S, steady u are creep pressure, strain density, rate as specific entropy particle velocity detonation products respectively, with trajectory R reflected shock detonation wave D as a boundary ~s = a a u. (8) trajectory F flyor as anor boundary. Both are unknown; position R state parameters For all on it unit are cells, governed by stress exponent flow field N I central steady rarefaction creep strain wave behind rate equals detonation 3.1, with wave less D than 10% by initial deviation. stage The motion value equals flyor also; stress position creep exponent F matrix, state parameters given out in products Eq. (2), different to experimental creep stress exponent MMC, which equals 9.2. It is

8 428 YUE Zhu-feng reasonable for unit cells to have same stress component as matrix. As on steady creep stage, stress states in unit cell including fiber matrix will almost keep constant change little. So creep strain rate is equal to that matrix, can be proved simply by following illustration. Because isotropic material matrix, from Eq. (4), one can get creep starin rate in loading direction l ~a = c a~ -1SIt, (9) extending rate unit cell in loading direction l On small deformation assumption, creep strain rate along load direction l can be obtained as i, ft ltdz fl ca2-~s~tdl The one-dimensional problem motion a rigid flying plate under explosive attack has an analytic solution only when su - polytropic - index - detonation products equals to three. ( 1In 1 ) general, a numerical analysis is required. In this paper, however, by utilizing "weak" shock behavior reflection shock in l explosive l products, l applying small parameter purterbation In above method, equation, an analytic, [ dl first-order is a constant approximate on above solution assumption, is obtained for ae, problem St~ are constants flying plate driven by various high d l explosives with polytropic indices or than but nearly equal to three. Final respect velocities to creep flying time, plate so s a obtained is also a agree constant very respect well with to numerical creep time. results The by computers. dependence Thus ~ it an on analytic stress formula is exponents with two parameters ae Sit. high The explosive stress exponent (i.e. detonation N velocity unit cell equals polytropic n, index) stress for exponent estimation matrix. velocity flying plate is established. The calculation Ref. [ 1 ] on MMCs by applying longitudinal loading shows stress exponent unit cell equals stress 1. Introduction exponent matrix, although paper did not indicate out it as a conclusion. The present conclusion furr indicates that MMCs with Explosive driven flying-plate technique ffmds its important use in study behavior plane-radomly-arranged can not be modeled by a special unit ceil, which can only model materials under intense impulsive loading, shock synsis diamonds, explosive welding regular-arranged-fiber MMCs. But unit cell models can provide basic information creep cladding metals. The method estimation flyor velocity way raising it are questions deformation common interest. l The geometric parameters loading directions have influences on steady creep behavior Under unit assumptions cells, which will one-dimensional be discussed plane on detonation next section. rigid flying plate, normal approach solving problem motion flyor is to No. solve thickness following creep law system equations governing flow field detonation products behind flyor interface 1 (Fig. 0 I): ~, = ast /~,4: N = = 10,L =6,D =0,308,d = 1.2,d r = ~ = t0o-st v/ <~e3:n = 3.91 /./ l = lo,l=6,d=o.308,d=l,dr=l 1E , = :~/~/S:II : ~ 331? l=8,l76,d=o.308,d=i,dr=l ~ IE-5 / /J ' ~ I E - 6' as as 0:005- l = lo, L=6,D=O.308,d=O.5,dr=l a--t... 1Elo~ 3~ ' 560d' 76o0' 9dod p =p(p, ' i1'000 s), "1'0 t/s a/mpa where p, p, S, u are pressure, density, specific entropy particle velocity detonation products respectively, with trajectory R reflected shock detonation wave D as a boundary Fig. 7 The development macroscopic Fig. 8 The influence interface on trajectory F flyor as anor boundary. Both are unknown; position R state parameters on it strain are governed with creep by time flow field I central rarefaction steady wave creep behind ratel = 10, detonation L = 6, wave D by initial (a = stage 10 MPa) motion flyor also; position D F = 0.308, state d parameters = dy = 1 products

9 Creep Behavior Metal Matrix Composites The matrix/fiber interlayer model The general conclusions obtained from above calculation on unit cell models can also be obtained from matrix/fiber interlayer model, except creep stress exponent, creep stress exponent N matrix/fiber interlayer model in Eq. (8) is no longer equal to value n matrix, The creep stress exponent matrix/fiber interlayer model depends not only on creep properties matrix, but also on creep properties thickness interlayer, as shown in Fig. 8. Fig. 8 shows that increasing creep exponent interlayer increasing thickness interlayer will increase creep stress exponent N composites. It is also found that loading direction micro geometric parameters have also influences on value. 4 Statistic Model 4.1 The Statistic one-dimensional model problem motion a rigid flying plate under explosive attack has an analytic The calculations solution only when unit cell polytropic models have index been shown detonation that products a special equals unit cell to model three. In can general, a numerical analysis is required. In this paper, however, by utilizing "weak" shock not describe creep behavior MMC with romly fiber arrangement. But dependence behavior reflection shock in explosive products, applying small parameter purterbation steady creep method, behavior an analytic, on first-order geometrical approximate parameters, solution arrangement is obtained for fibers problem external flying plate loading driven stress by various has been high obtained. explosives with Here polytropic according indices to or measurement than but nearly equal geometrical to three. Final parameters, velocities a statistic flying model plate obtained is presented agree for very well creep with behavior numerical results MMCs. by computers. Since Thus fiber an breakage analytic is formula open observed with two during parameters early creep high in explosive MMCs [l~(i.e., detonation in model, velocity breakage polytropic index) for estimation velocity flying plate is established. fiber will be taken into consideration. In order to study quantitatively, it is an attempt to define dependence geometric parameters defined above on steady state creep rate, a parametric study was highlighted Refer 1. to Introduction Fig. 1 for unit cell model, fiber geometric parameters arrangement relative to loading direction can be adequately described by Explosive driven flying-plate technique ffmds its important use in study behavior materials following under three intense parameters, impulsive 1/L, loading, d/d, shock dy/d synsis 0. diamonds, The statistic explosive distribution welding se cladding parameters metals. will be The presented method in Section estimation 4.2. Mamatically, flyor velocity this procedure way raising separates it are questions effect common one interest. three particular variables on creep rate loading direction by following Under assumptions one-dimensional plane detonation rigid flying plate, normal equation approach solving problem motion flyor is to solve following system equations governing flow field e~ detonation = fl( l/l)f2( products d/d)f3( behind dr/d)f4( flyor (Fig. O)f(m). I): (12) In above equation, Jl(1/L), f2(d/d), f3(dz/d) f4(0) represent influences l/l, d/d, dz/d 0 on steady creep strain rate, respectively, f(m) is creep behavior matrix. Here, it is taken as Eq. (2) for unit cell model. The separation might not be physically realistic, but it is simplest in mamatical expression, which can describe adequately macroscopic creep behavior. Consequently, as as influence three parameters a--t loading direction on steady creep rate was also conducted. The results are reported below. Keeping constants or geometric parameters fibers, influence parameter I/L on steady creep strain rates will depend on where loading p, directions. p, S, u are For pressure, example, density, on longitudinal, specific entropy transverse particle loadirlg velocity 0 = detonation 45 ~ loading, products Fig. 9 respectively, shows parameter with trajectory a Eq. R (8) reflected varies with shock parameter detonation l/l. wave It D as is a shown boundary that on trajectory F flyor as anor boundary. Both are unknown; position R state parameters longitudinal on it are loading, governed decreasing by flow 1/L will field decrease I central most rarefaction steady wave creep behind strain rate. detonation It means wave that D on by condition initial stage decreasing motion l/l, flyor also; fibers position will produce F a greater state constraint parameters on products matrix material within cell, making creep deformation more difficult.

10 430 YUE Zhu-feng 101 ~ l 1-1 g l/l d/d 5 Fig.9 The influence l/l on ralative Fig. 10 The influence d/d on ralative creep strain rate ~s on different loading direction ( D = , creep strain rate on different loading direction ( l = 10, The one-dimensional d= 1, dr= 1) problem motion a rigid flying L=6, plate dy= under 1) explosive attack has an analytic solution only when polytropic index detonation products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing "weak" shock behavior The parameter reflection l/l has shock much in more explosive influence products, on creep applying on longitudinal small parameter loading pur- than terbation transverse method, loading. an analytic, Thus, first-order expression approximate f1 (l/l) solution should is be obtained taken for loading problem direction flying into plate consideration, driven by various can high be deduced explosives by with polytropic least square indices method or as follows: than but nearly equal to three. Final velocities flying plate obtained agree very well with numerical results by computers. Thus fl (l/l) = ( ) ( l/l)~ an analytic formula with two parameters high explosive (i.e. detonation velocity polytropic index) It is shown for estimation that as loading velocity direction flying shifting plate is established. from transverse to longitudinal ones, importance l/l on creep strain rate will increase. Fig. 10 shows influence parameter 1. Introduction d/d on steady creep strain rate. The influence is also concerned with loading directions. Taking loading direction parameter 0 into consideration, Explosive f2(d/d) driven flying-plate can be expressed technique as ffmds follows: its important use in study behavior materials under intense impulsive loading, shock synsis diamonds, explosive welding Jz(d/D) = ( )(d/O) 0" cladding metals. The method estimation flyor velocity way raising it are questions As common same interest. process, expression f3(dy/d) can be obtained as Under J3(dy/D) assumptions = (0.472 one-dimensional plane 229 detonation 0)(dy/D) 0" " rigid flying plate, o, normal approach solving problem motion flyor is to solve following system equations governing 40- flow t/laxial field loading detonation products behind expression flyor (Fig. I): f4(0) can be J' d/daxial loading f4(0) = e -0" O / ~o,.. ran loading / I/L ~ 20- tran ioatl Since breakage should be taken into ng/o: / D / dr/d ax a loading Y au au consideration 1 in statistic model, axial ~ ~ D - ~ t r a n loading stress expression must be obtained. The axial as 0. a--t " " 315 " 410 " l/l, d/d, dr/d as stress, like steady creep strain rate, depends on geometric parameters fibers as well as loading directions applied stress. It should be mentioned that axial stress varies creep time, where Fig. p, 11 p, The S, u influence are pressure, density, geometric specific para- entropy but re is particle also steady velocity value for detonation axial stress products in respectively, meters with on trajectory relative steady R reflected peak shock detonation wave D as a boundary steady creep stage. The peak axial stress exists trajectory F axial flyor stress as a= anor boundary. Both are unknown; position R state parameters on it are governed by flow field I central always rarefaction in center wave behind middle detonation section. wave Here D by initial fiber(a stage = 10 motion MPa) flyor also; only position steady F peak state axial parameters stress products will be

11 Creep Behavior Metal Matrix Composites 431 considered. The influence parameters l/l, d/d, dy/d 0 on parameter k Eq. (5) will be analyzed. Fig. 11 shows influence l/l, d/d, dy/d 0 on k. In order to obtain quantitatively expression k, a similar procedure like steady creep rate will be applied. The finial expression can be deduced by following equations a.. =.s a, (13) where J~ = d/d , J~ = ( O)dJD + ( ), J~. =, (l/L) -~ , The one-dimensional f0 = problem motion a rigid flying plate under explosive attack has an analytic solution only when polytropic index detonation products equals to three. In general, 4.2 Statistical a numerical distribution analysis required. geometrical In this paper, parameters however, by utilizing "weak" shock behavior According reflection to expression shock in steady explosive creep products, strain rate applying expression small parameter steady pur- peak terbation axial stress, method, only an analytic, distributions first-order l/l, approximate d/d, dy/d solution is 0 obtained will be presented. for problem First, 0 flying can be plate driven by various high explosives with polytropic indices or than but nearly equal to three. assumed as a uniform distribution from 0 = 0, corresponding to transverse loading, to 0 = 90, Final velocities flying plate obtained agree very well with numerical results by computers. Thus an corresponding analytic formula to with longitudinal two parameters loading, high can explosive be expressed (i.e. detonation as following velocity probability polytropic density index) function for estimation velocity flying plate is established. h(o) = 8/90, (14) where h (O) is probability density 1. function, Introduction 0 is variable. Lognormal distributions are found to describe parameters l/l, d/d djd best [~2]. Explosive driven flying-plate technique ffmds its important use in study behavior Lognormal probability density function is materials under intense impulsive loading, shock synsis diamonds, explosive welding cladding metals. The method estimation flyor velocity way raising it are questions h(x) _ ~l-r=--exp[-2-'l ~/27t (lnx-/,ax )z], (15) common interest. O" x X Under assumptions one-dimensional plane detonation rigid flying plate, normal approach where h (O) solving is lognormal problem probability motion density flyor function, is to solve ax is following stard system deviation, equations /z is governing mean. flow field detonation products behind flyor (Fig. I): 5 The Results Statistic Model A stard Monte Carlo method is applied to obtain macroscopic steady creep strain rate by volume coverage 'N as as 1 ~si (16) a--t where N is number simulation. p =p(p, It is found s), that N > 10 6 can lead a steady value steady creep Strain rate. Fig. 12 shows comparison between statistic model experimental where results. p, It p, is S, shown u are pressure, that density, statistic specific model entropy can describe particle steady velocity creep detonation strain rate products well. In respectively, Fig. 12, with parameters trajectory are input R as reflected values shock in Table detonation 2. The volume wave faction D as a boundary fiber can be trajectory F flyor as anor boundary. Both are unknown; position R state parameters calculated on it as are 15 governed %, which by equals flow field experimental I central value. rarefaction wave behind detonation wave D The by initial influence stage motion distribution flyor on also; steady position creep F strains can state be parameters analyzed by products statistic model.

12 432 YUE Zhu-feng Table 2 Lognormal distribution parameters l/l, d/d dy/d l/ L d/ D dr/ D mean (,u) stard deviation ( a ) 0.05 The influence means l/l, d/d dr/d on creep stress exponent steady creep are shown in Fig. 13. It is shown that means d/d dy/d have little influence on exponent in parameter range considered. But increasing mean l/l will increase exponent. From expression steady value peak axial stress fiber, one can know that decreasing mean l/l will increase steady value peak axial The stress, one-dimensional which increases problem probability motion a breakage rigid flying plate fibers. under explosive The concept attack has an probability analytic solution breakage only when fibers polytropic can also index be used detonation explain products little equals influence to three. In general, means a d/d numerical dr/d analysis on is required. exponent. In For this paper, two however, cases, by deviations utilizing d/d "weak" shock dy/d behavior have also little influence reflection on shock exponent. in explosive For products, case l/l, applying increasing small (elative parameter deviation, purterbation method, an analytic, first-order approximate solution is obtained for problem flying defined as a//.z, will increase exponent, for increasing breakage probability plate driven by various high explosives with polytropic indices or than but nearly equal to three. Final fibers. velocities flying plate obtained agree very well with numerical results by computers. Thus an analytic formula with two parameters high explosive (i.e. detonation velocity polytropic index) 1E for - 4! estimation. _ t /. velocity / * flying plate is established. 10.2: 1 /-ul/l / a 9.8: 1E - 6~ 9 matrix 1. _ ~ Introduction / 9.4: 7 9 statistic model results J 9 ~.~ IEto Explosive 9 edriven x p flying-plate e r i n technique l e n ffmds its 8.629'0i important use in study behavior materials under intense impulsive loading, shock synsis diamonds,,( explosive welding 1E / cladding metals. The method estimation flyor velocity way raising it are questions common interest. 8.2./ 1E- Under 12 assumptions one-dimensional plane detonation 1' rigid 2.5 flying 3.0 plate, 3.5 normal 4.0 approach solving problem a/mpa motion flyor is to solve following l/l, d/d, dr/d system equations governing flow field detonation products behind flyor (Fig. I): Fig. 13 The influence geometric para- Fig. 12 Comparison between statistic meters on creep stress exponents model experimental ap +u_~_xp results + au N composites Fig. 14 shows influence fracture stress for breakage fiber on exponent steady creep. It is interesting as that as assumed fracture stress for breakage fibers a--t has no influence on creep stress exponent on studied range. It can be explained as follows: increasing decreasing fracture stress fibers will change number breakage fibers, but also change times breakage fibers. where p, p, S, u are pressure, density, specific entropy particle velocity detonation products The existence matrix/fiber interlayer will have influence on creep stress exponent respectively, with trajectory R reflected shock detonation wave D as a boundary trajectory composite F flyor from as anor interlayer boundary. model Both calculation. are unknown; The creep position stress R exponent state from para- meters interlayer on it model are governed will influence by flow creep field stress I central exponent rarefaction from wave statistic behind model, detonation as shown wave in D Fig by Comparing initial stage motion two statistic flyor models, also; one position can find F that difference state parameters creep products stress exponents between two statistic models is equal to difference creep stress exponents

13 Creep Behavior Metal Matrix Composites 433 between unit cell model matrix/fiber interlayer //jr z" ~: J cr dmpa , 3.0 " 3:5 " 410 ' 415 " 5'.0 " 515 ~.0 Fig. 14 The influence fracture Fig. 15 The influence exponent nln t The one-dimensional problem stress o'f fiber on motion a rigid flying plate interlayer under explosive on exponent attack has an analytic solution only when polytropic index detonation products equals to three. In creep stress exponent N N statistic model general, a numerical analysis is required. In this paper, however, by utilizing "weak" shock behavior reflection shock in explosive products, applying small parameter pur- 6 Conclusions terbation method, an analytic, first-order approximate solution is obtained for problem flying plate driven The aim by various high paper explosives is to discover with polytropic general indices creep or mechanisms than but nearly for equal short to three. fiber Final velocities flying plate obtained agree very well with numerical results by computers. Thus reinforcement matrix composites (MMCs) to build relationship between macro steady an analytic formula with two parameters high explosive (i.e. detonation velocity polytropic index) creep for behavior estimation micro velocity material flying structural plate is parameters. established. The unit cell models are used to calculate macroscopic creep behavior with different micro geometric parameters fibers on different loading directions. The influence 1. Introduction geometric parameters fibers loading directions has been obtained, described quantitatively. The matrix/fiber interface properties have Explosive been considered driven flying-plate by a third layer, technique matrix/fiber ffmds its important interlayer, use in in unit study cells with behavior different materials hardening under material intense properties impulsive loading, thickness. shock Although synsis calculation diamonds, shows explosive that a special welding unit cell cladding metals. The method estimation flyor velocity way raising it are questions model can not describe adequately creep behavior MMCs with romly-distributed common interest. fibers, unit cell model can provide basic information creep deformation. Based on Under assumptions one-dimensional plane detonation rigid flying plate, normal approach numerical solving results problem unit cell motion models, flyor a statistic to solve model has following been presented. system equations The fiber governing breakage has flow been field taken detonation into account products in behind statistic model flyor (Fig. for it I): starts early in creep life. With distribution geometric parameters fibers, results statistic model agree well with experiments. With statistic model, influence geometric parameters breakage fibers as well as properties thickness interlayer on macroscopic steady creep rate have been discussed. Acknowledgment The author is as grateful as to Alexer von Humboldt Foundation for awarding a--t chance to study research in Germany. The host pressor is Pr. Dr. Gunr Eggeler. References : where p, p, S, u are pressure, density, specific entropy particle velocity detonation products [ 1 ] Dragone T L, Nix W D. Geometric affecting internal stress distribution high temperature respectively, with trajectory R reflected shock detonation wave D as a boundary trajectory creep F rate flyor discontinuous as anor boundary. fiber reinforced Both methods are unknown; [ J ]. Acta Metall position Mater, R 1990,38 (4) state : 1941 para- - meters on it are governed by flow field I central rarefaction wave behind detonation wave D [ 2 I by Dong initial M, stage Schmer motion S. Transverse flyor also; mechanical position behavior F fiber reinforced state parameters composites-fe products model- ing with embedded cell models [ J]. Computational Materials Science, 1996, 5(l) : /Zint