ON INTERNAL STABILITY LOSS OF A ROW UNIDIRECTED PERIODICALLY LOCATED FIBERS IN THE VISCO-ELASTIC MATRIX. Resat KOSKER

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1 ON INTERNAL STABILITY LOSS OF A ROW UNIDIRECTED PERIODICALLY LOCATED FIBERS IN THE VISCO-ELASTIC MATRIX Resat KOSKER Yildi Techical Uiversity, Faculty of Chemistry ad Metallurgy, Dep. Mathematical. Egieerig., Davutpasa Campus, 34, Eseler, Istabul, Turey, oser@yildi.edu.tr I the preset paper, the microbuclig or iteral stability loss i the viscoelastic composites cotaiig uidirected fibers uder compressio alog the fibers is studied by use of piecewise homogeeous body model. I this model, it is used the Three-Dimesioal Geometrically Noliear Exact Euatios of Viscoelasticity Theory. The composite material was cosidered as a ifiite viscoelastic body with a row uidirected periodically located elastic fibers that have a iitial ifiitesimal imperfectio. Whe the iitial imperfectio starts to icrease ad becomes idefiitely, this is tae as a stability loss criterio ad co-phase microbuclig mode out of plae are tae ito accout. The umerical results about the ifluece of the iteractio betwee the fibers o the values of the critical time are abtaied ad preseted. Key words: Critical time, Iteral stability loss, Microbuclig, Uidirected fibrous composites, Row fibers, Viscoelastic composite. 1. Itroductio As we ca see some of them i [1-19], there are a lot of experimetal ad theoretical ivestigatios focus o composites icludig fibers ad aofibers. It is very importat to ow the mechaisms of the fracture of the composites uder uiaxial compressio alog the reiforcig elemets. For this purpose it is eeded to ivestigate the stability loss i the material structure (iteral istability or structural). So, the theoretical ivestigatios of the fracture o the uidirectioal composites uder uiaxial compressio alog the reiforcig elemets are mea to ivestigate the stability loss i the material structure, ad the value of the exteral critical forces is tae as the failure forces value i compressio (see: [14, 15, 19,, 1, ]). The review of ivestigatios carried out i this field is give i [4, 6, 1, 13, 17, 3]. It follows from these review that, with the use of piecewise homogeeous body model, two approaches are used to ivestigate the fracture ad the stability of the composite materials i compressio alog the reiforcig elemets. Oe of them is certai hypotheses applicatio related to deformatio of each compoets ad to the character of iteractig betwee them. The other is applicatio of the Three-Dimesioal Liearied Theory of Stability (TDLTS). The approaches related with the first oe used i [14, 19,, 1, 4, 5] ad others, but i the papers give i [5, 1, 13] ad others the secod approaches were prefered. It is clear that for cosidered problems, obtaied results with the use of the TDLTS are more reliable tha those obtaied by use of the approximate theories. However, i the studies doe by use of the TDLTS listed above were used the time-idepedet materials.

2 I the papers [6, 7], a approach is proposed to ivestigate stability loss i the time-depedet layered composite material usig the TDLTS. I the paper [8], the approach [6] is developed for the uidirected fibrous composite material. However i [8], a ifiite legth fiber embedded ito ifiite viscoelastic is cosidered. The filler cocetratio i the composite is small ad the iteractios betwee the fibers is igored I [9], the approach give i [8] developed to tae ito accout the iteractio two fibers ad it is assumed that the ifiite viscoelestic media cotais two eighborig fibers. I [4], the approaches [8, 9] are developed to aalyse the stability loss i the ifiite viscoelastic matrix cotaiig a row uidirected periodically located elastic fibers. It is assumed that the midlies of the fibers are located i a plae ad they have co-phase curvig relative to each other. I this ivestigatio, the approaches [4] is developed to study the stability loss i the case where the midlies of the fibers are i parallel plaes ad the fibers have co-phase curvig accordig to each other. This case will be called co-phase out of plae. The stability loss criterio is tae as the imperfectio starts to icrease ad becomes idefiitely. By this way, i the cosidered problems, it is estimated the values of critical time which occurs as a result of the iteractio betwee the fibers. Below, we will deal with determiatios of these critical times values ad the co-phase out of plae stability loss mode of row fibers will be ivestigated. The ivestigatios are made by the use of the piecewise-homogeeous body model i the framewor exact three dimesioal geometrical o-liear euatios of the liear viscoelasticity theory. Throughout the studies repeated idices are summed over their rages; but uderlied repeated idices are ot summed. Furthermore, the tesor otatio will be used to simplify the cosideratio.. Formulatio of the problem Periodically located row fibers embedded ito a ifiite body is cosidered. The fibers have isigificat iitial imperfectios. We associate O x1 x x 3 cartesia ad cylidrical O r cylidrical coordiates system with the midlie of each fibre (Fig. 1).,...,, 1,,1,,..., deote the fibres umber. As ca be see i Fig. 1 betwee these coordiates we have the followig relatios: i i x x, x3 x3 x3, x1 R1 x1, r e R1 r e, It is assumed that the legth of the period of the iitial ifiitesimal curvig of the fibers is the same ad the middle lies of the fibers are located i the parallel plaes with respect to each other, we suppose that the middle lies of the fibers are i the plae x1. The euatios of these lies are give as folloes x Lsi x () 3 By this iitial imperfectios, we will ivestigate the co-phase stability loss out of plae. Note that the correspodig results about stability loss mode of two fibers i a pure elastic matrix have bee obtaied i [13, 16]. It is assumed that, the cross-sectio perpedicular to the middle lie of each fiber is a circle with costat radius R ad this is ivariat alog the etire legth of the fiber. It is itroduced a small parameter L, ( 1).where, L is iitial curvig amplitude of the fiber ad is the legth period of the iitial curvig. (L is smaller tha ). The degree of fiber s iitial imperfectio is characteried by this parameter.

3 Figure 1: The cosidered material structure ad coordiates. Below, the values related to the fibers will be deoted by upper idices (), the values related to the ifiite matrix will be deoted by upper idex. We sssume that matrix ad the fibers materials are homogeeous, liear viscoelastic ad isotropic. The developmet of the ifiitesimal iitial imperfectio of the fibers is ivestigated whe the body is compressed at ifiity by uiformly distributed ormal forces with a itesity p actig i the directio of fibers. For this purpose, i the cylidrical system of coordiates ad i the geometrical oliear statemet ad we write the goverig field euatios withi the ifiite matrix ad fibers: ( ) i j ( ) j i g u ( ) ( ) ( ) ( ) ( ), jm jum mu j ju mu, e, ( ) *( ) ( ) *( ) ( ) ( i) i ( i) *( ) *( ) where ad are the followig operators. (Fig. 1): t *( ) ( ) ( ) (.) (.) ( t )(.) d, e (3) ( ) ( ) ( ) ( ) rr t *( ) ( ) ( ) (.) (.) ( t )(.) d (4) We assume that, the completely cohesio coditios are satisfied o the iter-medium surface S g u g u, u ( ) i j ( ) j i j j j j S S ( ) j uj S S (5) where j are the the uit ormal vector compoets of the surfaces S. I additio, the coditios ( ) ( ) ( ij), u() i, p, r ( ij) r ( ij) are valid i the cosidered case. Covetioal tesor otatio is used ad physical compoets of the correspodig tesors are showed by subscripts i paretheses i Es. (3)-(5).

4 3. Method of Solutio It is used a versio of the boudary shape perturbatio method [7] to ivestigate the correspodig problem. Usig the fiber cross-sectio coditio ad E. (), we easily derive the S iterfaces euatios as follows. 1 3 r ( ( ( t )) si 1) ( ( t )( ( t )) ( t ))si R ( ( t3)) ( ( t3)) ( ( t3)) (1 ( ( t3)) )si d ( t3) t3 ( t3) r ( t3)si ( t3) ( t3), ( t3) (6) dt The euatio of the middle lie of the -th fiber is deoted by ( t 3) where t3 (, ) is a parameter. the boudary shape perturbatio method, the uows are preseted i series form i : ( m ) ( m ) ( m ) ( ), ( ), ( ), ( ) ( ) ( ) m m m ij ij u i ( ij) ( ij) u( i) ; ; ; ;, (7) 1 After some calculatios, the followig expressios are obtaied from E. (6) for the compoets of the uit ormal vector to S : a (, 3), t3 b t3 1 1 r R t 1 (, ), r 1 c (, t3) , d (, t ), g (, t3) (8) a (, t 3),, g (, t 3) fuctios i E. (8) ca easily be caculated from E. (6). For each approximatio i E. (7), a set of euatios ca be obtaied by substitutig E. (7) i E. (3). We expad each approximatio (7) values i the series form i the viciity of ( r R, t3 ) by usig E. (8). Cotact coditios satisfied i r R, t3 for each approach i E. (7) is obtaied by usig r, ad give i E. (8) ad substitutig last expressios i E. (5). It is clear that, the E. (3) are valid ad the coditios (5) are replaced by the same oes satisfied i r R, ( ), t3 for the eroth approximatio. It is assumed that u j 1 is satisfied ad j ( ) j, j j therefore gm mu ca be replaced by where are Kroecer symbols. From that, we obtai the followig system of euatios ad ad cotact coditios, respectively for the eroth approximatio ( ) ij, ( ), ( ), ( ), i, ij jui iuj, (9) 1, u ( ),, ( ij) r ( ij) R r R ( ),, ( i) r ( i) R rr, u ; ( ij) rr, r, r, ( i) r,, If the last assumptio is tae ito accout, the followig system of euatios ca be obtaied for the first approximatio ( ) ij,1 ( ) i, u ( ) j,1 i, ( ),1 ( ),1 ( ),1 ij jui iuj. (11) Additioally, the costitutive relatios are writte as follows: ( ),1 *( ) ( ),1 *( ) ( ),1 e, (1) ( i) i ( i) The cotact coditios ca be writte as follows usig physical compoets of the displacemet vector ad stress tesor for the first approximatio.,,,1 ( i) r ( i) r,,, ( i) r f 1 1 r ( i) r 1, ( i) 1, ( i) r 1, 1, 1,

5 ,,,1 u( i) u( i) ( i) f 1 1 r 1, 1, u where ( i) r,,. The rest of the cotact coditios are obtaied from (13) by meas of cyclic permutatio of the idices r, ad oly i the compoets stress tesor (the first idex is permuted) ad displacemet vector. I (13) the followig otatio is used. s, ( ), s, s ( t3) ; f 1, s 1 ( t3)cos ; 1 R ( t3)cos, r ( t3) Rcos ; R ( ) t3 si ; ( t3)cos ; ( 3) si( 3 ) si( 3) R (14) We ca obtai similar cotact coditios for the subseuet approximatios. Let s determie the uow values of these approximatios. Assume that the materials of each fiber () are the same ad the material of the fibers is pure elastic with elastic costats E (Youg s moduli), () (Poisso coefficiet), material of the matrix is viscoelastic with operators (6). We will suppose (), where (t is time) ad the stresses arisig i the eroth approximatio as a t () () result of for t will be igored, because these stresses have a order O( ) ad accordig to [1], do ot have oly cosiderable effect o umerical results. Thus, taig the above stated ito accout cosider the determiatio of each approximatio separately. Let us determie the eroth approximatio. This approximatio has a exact aalytical solutio i the pure elastic case. Here, we obtai a solutio to the correspodig uasistatic problem by replacig the elastic costats by correspodig operators. The followigs are obtaied: (), ( ), E ( ),, p ( p; p; * ; ),,,,, u * u ; ur r; E E ( ), () ( ),, ( ), ( ),, () ( ) ur r u u, ( ij) ( ij) ; 1 ; E E,,...,, 1,,1,,..., ; ( ij) rr,, r,, r (15) where * E ad * are the followig operators t * t * (13) E E E ( t ) d : ( t ) d (16) It is time to get the first approximatio. Usig the solutio of eroth approximatios, we derive the followig for E. (11). ( ),1 ( ),1 ( ),1 ( ),1 rr 1 r r 1 ( ),1 ( ),1 ( ), ur rr, r r r ( ),1 ( ),1 ( ),1 ( ),1 r 1 ( ),1 ( ), u r, r r r ( ),1 ( ),1 ( ),1 ( ),1 r 1 1 ( ),1 ( ), u r. (17) r r r E. (1) will ot chage, but, because of pure elasticity of the fiber material, E. (1) for the fibers are replaced by folowig e. (18) ( ),1 () ( ),1 () ( ),1 ( i) i ( i) Moreover, the geometrical relatios for both the matrix ad fibers have the followig form. ( ),1 ( ),1 ( ),1 ( ),1 ( ),1 ( ),1 ur ( ),1 u ur ( ),1 u ( ),1 1 ur u u rr,,, r r r r r r r ( ),1 ( ),1,

6 ( ),1 1 u u, r r ( ),1 ( ),1 u r u ( ),1 ( ),1 ( ),1 1 r From E. (15), the cotact coditios of the first approximatio become as follows:.,1,,1 rr,1, (),, t 3 r r ( ) cos, u,1, u,1,,1 r. (19) u. () t (time) is a parameter i the all euatios writte for the fiber but a idepedet variable i the euatios writte for the matrix. Here, Laplace trasform is applied st ( s) ( t) e dt, (1) with s, to all relatios ad euatios (except the cotact relatios ()) belogig to the matrix material. From the procedure, Es. (17), (19) are valid for the Laplace trasforms of the correspodig sought-for uatities ad the costitutive relatios (1) are trasformed to the followig oes:,1,1,1 E ( i) e i ( i), (1 )(1 ), E. () (1 ) As it has bee oted above, the Es. (17)-(19) coicide with the correspodig euatios of the TDLTS, therefore to solve the obtaied euatio systems we ca use the followig represetatios i the cylidrical system of coordiates (see: [16]). 1 1 u r, u, r r r r u3 ( ) ( ) 1 ( ), 1. (3) r r r r The fuctios ad are determied from the euatios 1 1, where 1,, 3 (4). (5),1 Es. (3)-(5), are used for fibers ad matrix. The uatities u () i,, ad are replaced by u() i, ( ),1 () () (),,,, respectively for the matrix ad by u () i,,, respectively for the fibers. If we tae the cotact coditios () ito accout, the Es. (4) ca be solved as follows: ( ),1 ( ) () A t I 1 t r i si ( ) ( ( ) )exp( ), ( ),1 ( ) () ( ) () B t I t r C t I 3 t r i cos ( ) ( ( ) ) ( ) ( ( ) ) exp( ) (6),1 A s K 1 s r i si ( ) ( ( ) )exp( ), B s K s r C s K 3 s r i cos ( ) ( ( ) ) ( ) ( ( ) ) exp( ), (7) where ad I( x), K( x ) are Bessel fuctios of a purely imagiary argumet ad ( Macdoald fuctios, i tur. Moreover, the uows ) ( A () t,, ) C () t, A () s,, C () s are

7 the complex umbers ad satisfy the relatios A ( t) A ( t), B ( t) B ( t), C ( t) C ( t), Im A Im B Im C, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) A ( s) A ( s), B ( s) B ( s), C ( s) C ( s), Im A ( s) Im B ( s) Im C ( s). (8) We wat to obtai the expressios of the the first approximatio values of by satisfyig the cotact coditio (). To mae these operatios, the expressios (6) ad (7) must be represeted i the -th (,...,, 1,,1,,..., ) cylidrical coordiate system. We already preset the expressios (6) i the -th cylidrical system of coordiates ad It be used the summatio theorem (Watso ([8]) for the K() x fuctio i expressios (7). The theorem ca be writte for the case at had as follows r exp i ( m) R exp i r exp i m m 1 m m 1 m m m for m, m for m, c cost. r R1 K ( cr )exp i ( 1) I ( cr ) K ( c m R )exp i( ) expi ; (9) Usig Es. (3)-(5), (7)-(9) ad E. () the expressios Laplace trasform of the values of the first approximatio related to the matrix are obtaied. Now, we cosider the determiatio of the iverse Laplace trasform. For this purpose, we will use the Schapery [9] method. This techiue is described i [3]. Obtaied ifiite system must be replaced with the correspodig fiite system of euatios to get umerical results.. It ca be prove the validity of these replacemets. Note that such a proof was also performed i [7]. Coseuetly, we ca replace algebraic euatios ifiite system by the followig oe for umerical ivestigatios: N N () () 3, (), Y Yv Fv F Y ( ), 1,,..., N (3) v where,1,,...,, 3 1 ad, if 3. The values of N ad N i E. (3) are determied from the covergece reuiremet of umerical results. I this way we determie the uows for ay selected t ad withi the framewor of the first approximatio fid the critical time from the criterio max u ( ),1 r (, );, (31) The critical time (or critical compressive force for pure elastic problem) we deote as t ( p ). 4. Numerical Results ad Discussios We assume that the costitutive relatios for the matrix material are described by the operators * * E E 1 R '( ), * 1 * 1 R '( ), * 1 * 3 1 R, (1 ) (1 ) * 3 * 3 1 R, (3) (1 ) (1 ) where E, are the istataeous values of Youg s modulus ad of the Poisso coefficiet,

8 respectively;, are the istataeous values of Lamé s costats,,, are the * rheological parameters of the matrix material, R is the fractioal-expoetial Rabotov operator [4]. The Rabotov operators be allowed to describe with reuired accuracy, the iitial parts of the experimetally ad theoretically costructed creep ad relaxatio graphs ad to determie the asymptotic values of these graphs with very high accuracy. Moreover, these operators have may simple rules for various complicated mathematical trasformatios. So, these operators are employed successfully to describe various polymer materials ad epoxy-based composites with cotiuous fibers or layers.. We itroduce the dimesioless rheological parameter ad the dimesioless time 1 (1 ') t' ( ) ( ) () t ad assume that.3, E E. Moreover, we itroduce the parameters p E, R ad ivestigate the iteral stability loss of the ifiite viscoelastic matrix cotaiig a row fibers. Uder iteral stability loss we will uderstad the stability loss i the material structure, which arises for certai relatios of the stiffess ad geometrical parameters of the matrix ad fibers ad does ot deped o boudary surfaces, sies ad forms of members of costructios. It is well ow that uder ivestigatio of stability loss problems for viscoelastic materials the exteral compressive force p must satisfy the followig ieualities cr. p E p E p E (33) where is a obtaied at t, is a obtaied at t. A lot of ivestigatios reviewed i [4] show that, the iteral stability loss pheomeo for uidirected fibrous composites () occurs uder E E. Thus, taig the above-stated cosideratio ito accout we aalye the umerical results related to the iteral stability loss of the cosidered viscoelastic matrix cotaiig a row of fibers. First, we cosider umerical results related to the iteral stability loss i the above described sese for the pure elastic deformatio state uder t ad t. Cosider the case where () () E E 5 (for all umerical ivestigatio we will assume that E E 5, R.3 ) ad itroduce the parameter R1 R through which we will characterie the iteractio betwee the fibers uder their stability loss. The correspodig results obtaied for ad for various values of ad are give i Tab. 1. Note that these results relay to pure elastic problem ad coicide with correspodig obtaied i [1]. Tab. 1 shows that the depedece betwee, ad is mootoic, i.e. the values of (or ) icrease mootoically with ad approach asymptotically the correspodig values of (or ) obtaied for a sigle fiber [11].

9 Table 1 : The values of ( fort ' ) ad of (for t ' ) obtaied for.3,.5with various R1 R R1 R () E E =5, Figure : The graphs of the depedecies betwee t ad R1 R for various values of uder.5,.5. I Fig. the graphs of the depedecies betwee t ad are give for.5 uder.5 ad for various. It follows from these graphs that the values of t approach the correspodig values of t obtaied i [8] for a sigle fiber i a viscoelastic matrix. These graphs show that depedecies betwee t ad are mootoic. Moreover, these graphs show that t ( t ) as ( ) ad as a result of the iteractio betwee the fibers the values of t decrease sigificatly. It follows from the above-discussed umerical results that iteractio betwee the fibers i the viscoelastic matrix is more cosiderable tha that i the pure elastic matrix. It follows from the compariso of the graphs costructed i the various figures that, for the same the values of t icrease with. I Tab. the values of t are give for various values of which shows the order of the sigularity of the operator (33). These results are obtaied for the case where.5 with various

10 ad. The Tab. shows that uder t.5 ( t.5 ) the values of t icrease (decrease) mootoically with. Moreover, this table shows that the iteractio betwee the fibers becomes more sigificatly with. As it has bee oted above, the umerical results aalyed here are obtaied withi the framewor of the first approximatio. Uder obtaiig these results the ifiite system of euatios are replaced by the correspodig fiite oe. For the illustratio of the umerical results with respect to the umber of the euatios i this fiite system i Tab. 3 the values of t obtaied for various umber of the euatios are give uder.5,.1,.5 for various. It follows from these table that the covergece of the solutio method employed is highly effective. Table : The values of ' t obtaied for.5 with various R R, ad ' Table 3 : The values of t obtaied for various values of N ad N i euatio (3) uder.1,.5,.5 with various N ( N N ( N 17 ) 5. Coclusios I the preset paper, the iteral stability loss (microbuclig) i the structure of the viscoelastic uidirected fibrous composites uder compressio alog the fibers is studied withi the framewor of the piecewise homogeeous body model with the use of the Three-Dimesioal Geometrically Noliear Exact Euatios of the Theory of Viscoelasticity. This study cocers maily the cases where the iteractio betwee the fibers is tae ito accout ad all ivestigatios are carried out for the ifiite viscoelastic matrix cotaiig a row of fibers. As a microbuclig criterio the iitial imperfectio criterio is used ad two cases of the locatio of the iitial imperfected fibers (co-phase periodical curvig of the fibers out of plae) with respect to each other are cosidered. For the stability of the risig of the iitial imperfectio with the time uder fixed exteral compressed forces the Three-Dimesioal Geometrically Noliear Exact Euatios of

11 the Theory of Viscoelasticity is employed. Itroducig the dimesioless small parameter characteriig the degree of the isigificat iitial imperfectio for the solutio to the correspodig oliear boudary value problem, the perturbatio of the boudary-shape method is employed. It is prove that the euatios ad relatios related to the first approximatio are the correspodig euatios ad relatios of the TDLTS. For first approximatio the correspodig closed system of liearied euatios ad cotact coditios are obtaied ad for the solutio of these euatios the Laplace trasformatio with respect to time ad method of separatio of variables are employed. For determiatio of iverse Laplace trasform the Schapery method is used. It is prove that the values of the critical parameters ca be determied i the framewor of the eroth ad first approximatios oly (see: [9]). So, these parameters have bee determied i the framewor of the eroth ad first approximatios. The umerical results related to the critical time are also aalyed. Accordig to these results it is established that, for the cosidered microbuclig mode of the fibers as a result of the iteractio betwee the fibers the values of the critical time decrease. Further, this iteractio is more sigificat tha that i the pure elastic buclig. The umerical results obtaied i the preset wor i the particular cases coicide with the correspodig oes obtaied i the other ivestigatios. This situatio guaraties the correctess of the developed approach. Refereces [1] Liu, Y.-Q., et al., Air Permeability of Naofiber Membrae With Hierarchical Structure, Thermal Sciece,, 18, 4, pp [] Wag, P., et al., Eergy Absorptio i Frictio-Based Stab-Proof Fabrics ad the Pucture Resis tace of Naofiber Membrae, Thermal Sciece,, 18, 1A, pp [3] Liu, P. ad He, J., Geometric Potetial A Explaatio of Naofiber s Wettability, Thermal Sciece,, 18, 1A, pp [4] Abarov, S.D., Stability Loss ad Buclig Delamiatio: Three-Dimesioal Liearied Approach for Elastic ad Viscoelastic Composites, Spriger, Berli, Germay, 1: [5] 5 Abarov, S. D., Gu, A. N., Stability of Two Fibers i a Elastic Matrix with Small Strais, Soviet Appl. Mech., 1, 1985, 1, pp [6] Abarov, S. D. Ad Gu, A.N., Statics of Lamiated ad Fibrous Composites With Curved Structures. Appl. Mech. Rev., 45, 199,, pp [7] Abarov, S. D., Gu, A. N., Mechaics of curved composites, Kluwer Academic Pubishers, Dortrecht/ Bosto/ Lodo,. [8] Abarov, S. D., Koser, R., Fiber Buclig i a Viscoelastic Matrix, Mechaics of Composite Materials, 37, 1, 4, pp [9] Abarov, S. D., Koser, R., Iteral Stability Loss of Two Neighbourig Fibers i a Viscoelastic Matrix, It. J. Eg. Scie.,.4, 4, 17/18, pp [1] Babaev, M.S., et al., Stability of the Row of Fibers i the Elastic Matrix at Small Precritical Deformatios., Soviet Appl. Mech., 1, 1985, 5, pp [11] Babich, I., Stab. of Fiber i a Matrx with Small Str., Soviet Appl. Mech., 9, 1973, 4, pp

12 [1] Babich, I.Yu., Gu, A.N., Stability of Fibrous Com., Appl. Mech. Rev., 45, 199,, pp [13] Babich, I.Yu., et al., The Three-Dimesioal Theory of Stability of Fibrous ad Lamiated Materials, It. Appl. Mech., 37, 1, 9, pp [14] Budiasi, B. ad Flec, N.A., Compressive Failure of Fibre Composites, J. Mech. Phys. Solids, 4993, 1, pp [15] Grescu, I.B., Fracture of Composite Reiforced by Circular Fibers From Loss of Stability Of Fibers, J. AIAA, 13, 1975, 1, pp [16] Gu, A.N., Fudametals of the Three-Dimesioal Theory of Stability of Deformable Bodies, Spriger-Verlag, Berli, Germay, [17] Gu, A. N.; Lapusta, Yu. N., Three-Dimesioal Problems of the Near-Surface Istability of Fiber Composites i Compressio,. Iter. Appl. Mech., 35, 1999, 7, pp [18] Gu, A. N., Rushchitsii, Ya. Ya., Naomaterials: o the Mechaics of Naomaterials, It. App. Mech., 39, 3, 11, pp [19] Rose, B.W., Fiber composite materials. Amer. Soc. For. Metals, Met. P., Ohio, USA, [] Dow, N.F., Grufest, I.J. Determiatio of Most Needed Potetially Possible Improvemets i Materials for Ballistic ad Space Vehicles, Geeral Electric Co. Space Sci Lab., USA, 196. [1] Drapier S., et al., A structural approach of plastic microbuclig i log fibre composites: compariso with theoretical ad exp.results. It. J. Solids ad Str., 38, 1, 8, pp [] Chamis, C.C. Micromechaics stregth theories. [for uidirectioal composites], NASA Lewis Research Ceter, Clevelad, Ohio, USA, [3] Abarov, S.D., Three-dimesioal stability loss problems of the viscoelastic composite materials ad structural members. It. Appl. Mech,.43, 7, 1, pp [4] Sadovsy, M.A., et al., Buclig of Microfibers, Tras ASME, 34, 1967, 4, pp [5] Schuerch, H., Predictio of Compressive Stregth i Uiaxial Boro Fibermetal Matrix Composite Materials, J. AIAA 4, 1966, 1, pp [6] Abarov, S.D. et al., O The Fracture of the Uidirectioal Composites i Compressio. It.J.Eg. Sci. 35, 1997, 1/13, pp [7] Abarov, S.D. et al., The Theoritical Streght Limit i Compressio of Viscoelastic Layered Composite Materials. Composites Part B: Egieerig, 3, 1999, 5, pp [8] Watso, G.N., A Treatise o the Theory of Bessel fuctios, Cambridge Uiversity Press, Eglad., [9] Schapery, R.A., Approximate Methods of Trasform Iversio for Viscoelastic Stress Aalysis, Proceedigs 4th US Nat. Cogress o App. Mech., Berely, USA, 196, pp [3] Gu, A.N.. et al., Plae Problems of Stability of Composite Materials with a Fiite Sie Filler. Mecha. Comp. Materials., 36,, 1, pp