Finite Element Determination of Critical Zones in Composite Structures

Transcription

1 Finite Element Determintion of Criticl Zones in Composite Structures Alexey I. Borovkov Dmitriy V. Klimshin Denis V. Shevchenko Computtionl Mechnics Lb., St. Petersburg Stte Polytechnicl University, Russi Abstrct Modeling nd structurl nlysis of composite mterils (CM) with complicted microstructure is very lborconsuming job. In the current pper new method of composite structures nlysis is suggested. It enbles to determine with high degree of ccurcy criticl zones of potentil filure where criticl stressed stte is chieved nd only then nlyze in detils micro stress fields. Min steps of the method re:. Homogeniztion of composite structure - computtion of effective thermo-mechnicl chrcteristics nd filure surfce tensors.. Mcro nlysis of the construction stressed stte nd determintion of criticl zones. 3. Sequentil heterogeniztion in criticl zones nd determintion of micro stresses. Suggested pproch is bsed on new form of Tsi-Wu tensoril-polynomil filure criterion construction nd ppliction of up-to-dte computtionl methods implemented into ANSYS. This pproch differs from the one considered in the pper of Tsi nd Wu [] by numericl but not experimentl determintion of filure surfce tensors components for the periodicl cell. After tht, during mcro nlysis of the homogenized structure tensoril-polynomil criterion is pplied to determine criticl zones. The criterion is stisfied when the point tht chrcterizes stressed stte of the whole periodicl cell in the spce of verged stresses is locted outside effective filure surfce. To construct effective filure surfce the mcro in ANSYS Prmetric Design Lnguge (APDL) ws developed tht enbles to clculte bout 300 points of the filure surfce. Every point is result of solution of problem for the periodicl cell with boundry conditions corresponding to the loding type. The number of points clculted in the present work llows creting 4-th order surfces. It corresponds to keeping items contining 8-th order filure surfce tensors in the Tsi-Wu criterion. Verifiction of the method developed is done by solving plne stress problems nd determintion of zones where criticl stressed stte is chieved: compression of fibrous periodicl composite plte; complex loding of fibrous periodicl composite plte with centrl hole. Etlon solution is lso obtined by direct finite element modeling of the construction contining ll microstructure in order to estimte "qulity" of the results obtined with use of new method. It is determined tht method developed enbles to define efficiently ll criticl zones in the construction nd with high degree of ccurcy compute micro stresses. Introduction Modeling nd structurl nlysis of composite mterils with considertion of complex microstructure is very lborconsuming job tht includes mny problems. One of them is determintion of zones where criticl stress stte is observed. Even despite recent chievements in computer techniques nd science-intensive nlysis softwre the solution of this problem involves difficulties. Tht s why the necessity is rising of the new methods development tht could solve such problems with minimum time nd lbor. In the current pper the new method of stressed stte of composite mterils is suggested. This method enbles one to determine with high degree of ccurcy criticl zones of potentil filure nd then nlyze in detils micro stresses fields. Min steps of the method re:

2 . Homogeniztion of composite structure - computtion of effective thermo-mechnicl chrcteristics nd filure surfce tensors.. Mcro nlysis of the construction stressed stte nd determintion of criticl zones. 3. Sequentil heterogeniztion in criticl zones nd determintion of micro stresses. Suggested pproch is bsed on new form of Tsi-Wu tensoril-polynomil filure criterion construction nd ppliction of up-to-dte computtionl methods implemented into ANSYS. Verifiction of the developed method of potentil filure zones where criticl stressed stte is reched is performed for number of plne stress problems:. Plte mde of periodicl fiber composite under compression;. Plte mde of periodicl fiber composite with centrl hole under complex loding. In order to verify the obtined results by direct FE modeling the reference solutions were obtined with full ccount of rel microstructure of the composite mteril. It ws found tht the developed method enbles to find ll criticl zones of potentil filure in the construction nd with high degree of ccurcy clculte micro stresses. To illustrte the suggested method unidirectionl fiber composite ws considered. Mteril of the mtrix is brittle polymer, fiber steel. Periodicl cell is presented in Figure, where = 5 µm, b = 0 µm, с = 5 µm. Volume concentrtion of the fiber is V f = Elstic chrcteristics of the mterils re listed in Tble. Figure. Periodicl cell Tble E(GP) р (MP) c (MP) τ (MP) Y (MP) mtrix fiber Sttement of the problem During the development of criterion by Tsi-Wu [] computer techniques didn t llow to crry out complex numericl simultions nd composite filure could be nlyzed only during experiments. At present time computers nd softwre give possibility to solve complex problems with ccount of physicl nd geometricl non-linerity. Nevertheless, determintion of criticl zones in composite mterils with complex microstructure is still very consuming tsk. Tht s why it is suggested in the present work to turn to homogeneous mteril nd nlyze filure process for it. To nlyze filure it is suggested to use tensoril-polynomil criterion tht hs the following form (Tsi-Wu, []): i F F + F + [ F ] + [ F ] +... = () where filure surfce tensor of i rnk. In the coordinte form the criterion is looking s the following: F + F + F + F +... () = ij ij ijkl ji lk ijklmn ij lk nm ijklmnps ij lk nm sp

3 where Fijklmnsp components of 8-th rnk filure surfce tensor (i,j,k,l,n,m,p,s =,). Geometriclly the filure criterion cn be interpreted s some surfce in the spce of stresses, i.e. condition of filure is stisfied when defined stress vector crosses this filure surfce in point A (Figure ). Figure. Geometric interprettion of filure condition The suggested wy of filure surfce construction differs from the considered in Tsi-Wu s work by the method of filure surfce tensor components determintion. In Tsi-Wu s pper experimentl pproch ws utilized, while in the present work filure surfce tensor components re clculted bsed on numericl nlysis of periodicl cell tht is crried out with use of FE softwre ANSYS nd APDL. In generl filure surfce is being built in 6D nd it is closed. The bsence of closure mens infinite ultimte stress in some direction. In the current work it is suggested to consider D stressed stte so tht filure surfce would be built is 3D spce < >< >< >. Algorithm of composite structure nlysis hs the following steps:. Switch from heterogeneous model of composite mteril to homogeneous (determintion of C effective elstic moduli tensor);. Construction of closed filure surfce (in stresses spce) ccording to tensoril-polynomil criterion for 4 homogeneous mteril bsed on heterogeneous periodicl cell (estimtion of F ( < > ), F ( < > ),... effective filure surfce tensors); 3. Usge of the constructed filure surfce for the detiled nlysis of heterogeniztion zones (micro stress concentrtion zones, ); 4. Sequentil heterogeniztion in the zones of criterion stisfction to obtin more exct solution. Switch from heterogeneous to homogeneous model of the composite mteril Switch from heterogeneous model of the composite mteril to homogeneous one ws crried out by direct homogeniztion method. This method includes:. Computtion of effective Young s modulus Е 3 bsed on volume concentrtion of fibers V f f f m f E = E V + E ( V ) Two problems of trnsverse tension of the periodicl cell (plne strin) to clculte effective Young s moduli Е nd Е, nd lso Poisson s coefficients. 3. The problem of periodicl cell trnsverse sher (plne strin) to clculte effective sher modulus G. 4. Two problems of periodicl cell longitudinl sher (nti-plne strin) to clculte effective sher moduli G 3 nd G 3... Boundry conditions for the first problem of trnsverse tension: 0 x = 0.5 : u = u, x = 0.5 b : u = 0, = 0; x = 0 : u = 0, = 0 = 0; x = 0 : u = 0, = 0

4 .. Boundry conditions for the second problem of trnsverse tension: x = 0.5 : u = 0, 0 x = 0.5 b : u = u, = 0; x = 0 : u = 0, = 0; x = 0 : u = 0, = 0 = 0 Using effective determinte reltions for mcro-orthotrophic heterogeneous medi nd connections between effective Young s moduli nd Poisson s coefficient the following system of equtions is formed: E E ε 0 = 0 = 0 = 0 = ε 0 = 0 = E 3 = E ( ) 3 3 ( ) ( ) 33 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 ( ) ( ), E3 3 = E ( ) 33 ( ) 33 ( ) ( ) 33 ( ) 33 ( ) (3) The solution of the current system of equtions: = E E 3 ε E ( ) E + E ( ) E3 ( ) ( ) =, E E () E3 () () =, where, = E () E E3 ε E E3 3 3 =, E E3 3 3 =, () ( ) () ( ) ij = ii jj jj ii, ij = ji + ; 3 3 = ; (4) 3 3 =, 3. Boundry conditions for the problem of periodicl cell trnsverse sher (computtion of G ): x = 0.5 : u = 0, 0 x = 0.5 b : u = u, Then: = γ G, = 0; x = 0 : u = 0, = 0; x = 0 : u = 0, u 0 γ = b ( ) = 0 = Boundry conditions for the first problem of the periodicl cell longitudinl sher (computtion of G 3 ): Then: x = 0.5 : = 0; x = 0 : = 0 0 x = 0.5 b : u = u ; x = 0 : u = 0 = γ 3 G 3, u 0 3 γ 3 = b ( ) Boundry conditions for the second problem of the periodicl cell longitudinl sher (computtion of G 3 ): (5) (6)

5 Then: 0 x = 0.5 : u = u ; x = 0 : u = 0 x = 0.5 b : = γ 3 G 3, = 0; x = 0 : u 0 3 γ 3 = ( ) 3 3 = 0 In Tble effective chrcteristics obtined by this lgorithm re listed. Tble Е (GP) Е (GP) Е 3 (GP) 3 3 G (GP) G 3 (GP) G 3 (GP) (7) Construction of the filure surfce Construction of the closed filure surfce for homogeneous mteril contins the following steps:. Cretion of representtive points.. Approximtion of the creted representtive points with closed surfce in the stress spce with use if lest squres method. Cretion of representtive points To clculte representtive points coordintes it is necessry to solve number of problems for the periodicl cell with vrious boundry conditions. Averged micro stressed stte will be clculted nd representtive points coordinted will be clculted bsed on it. In every problem periodicl mteril with vried Hshin Rosen boundry conditions [8] is considered. Generl sttement of these problems is presented in Figure 3, where l chrcteristic vlue of the boundry displcement; u, v, w integer numbers. Figure 3. Hshin-Rosen boundry conditions In the current work only plne stressed stte is considered. Infinite strins tensor hs the following structure: where c fiber dimeter, ~ ε ε = ε xx yx ε ε xy yy = l u c l w c l w c l v c (8)

6 l chrcteristic vlue of the boundry displcement, u, v, w integer numbers. Every problem is chrcterized by its own combintion of boundry conditions nd, consequently, its own combintion of vlues of u, v, w. Such boundry conditions enble to relize ll potentil stressed sttes of periodicl cell. Chrcteristic vlue of boundry displcement l is chosen smll enough not to rech ultimte stress in ny periodicl cell. Representtive point is creted by verging of micro stress field fter the solution of problem with corresponding boundry conditions. Averging of the stress fields gives representtive point А (< >, < >): < ( A ) ij > = S 0 ijds S0 where < >, < > - verged field of stress tensor components,, S 0 - centrl cell surfce. To rech criticl stress it is required to find out correction fctor (multiplier) K tht turns stresses into criticl. Switch from point А into point В is relized by mens of correction fctor: (9) ( B) ( A ) < ij >= K < ij > (0) where i, j =,. Scheme of representtive point cretion is presented in Figure 4. Figure 4. Scheme of representtive point cretion Multiplied by fctor K mximum principl stresses in the mteril become equl to criticl: p mx c min τ mx K = inf(,,,y ), () where infimum is clculted over ll nodes of the mtrix,, - principl stresses, mx = mx{ }, int =. Consider cretion of one of representtive points (for plne stress) with prmeters w =, u = k =. FE mesh is presented in Figure 5. In this cse infinite strins tensor hs the following structure: int

7 ~ ε = Figure 5. FE mesh for cretion of one representtive point In Figure 6 stress tensor component field is presented for the whole model nd centrl cell. Kinemtic boundry conditions of Hshin Rosen doesn t stisfy the condition of sttic comptibility of the deformed periodicl cells. To stisfy these conditions in the centrl periodicl cell it is suggested to dd lyers of periodicl cells surrounding the centrl cell. At this the necessity of the following reserches is rising:. Prcticl convergence depending on the number of degrees of freedom (NDF).. Prcticl convergence depending on the number of periodicl cells lyers surrounding the centrl cell. Figure 6. Stress field of one representtive point. The loction of point F, for which the convergence ws investigted, is shown in Figure 7, nd the convergence plot for mtrix stress ( F) depending on the number of degrees of freedom is shown in Figure 8. For Figure 7 point F is situted in the middle between points G nd H. For Figure 8 FE mesh corresponding to the point mrked with red circle is tken for the following nlysis. Averging of stress fields in the centrl cell gives the following results (point А): < >= 0.7 MP < >= 0.3 MP < >= 0.04 MP

8 Figure 7. The loction of point F Figure 8. The convergence of stress ( F) Field of principl stresses in the centrl cell is presented in Figure 9. For the current sttement of the problem correcting fctor К is clculted in the mrked points, becuse there reches its mximum. So, the filure occurs t tension. Figure 9. Field of principle stress in the centrl cell Thus, the correcting fctor for the current representtive point is equl to К=50.4 nd representtive point coordintes re the following:

9 < >=.7 MP < >= 7.63 MP < >= MP. Convergence of < > depending of the number of periodicl cells levels (n) is presented in Figure 0. Figure 0. Convergence of < > Tke 3 periodicl cells lyers for following nlysis. Cretion of effective filure surfce With use of ANSYS APDL mcro ws written tht enbles one to crete representtive points in < >< >< > 3D spce (plne stress) by mens of the describe bove method. Generl view of representtive points in shown in Figure. nd their projection on < >< > plne is shown in Figure.. Figure.. Generl view or representtive points

10 Figure.. The projection of representtive points on < >< > plne Algorithm of composite mterils nlysis includes shift from heterogeneous model to homogeneous. Tensorilpolynomil criterion for homogeneous mterils cll effective tensoril-polynomil criterion. It hs the following form: F ij < + F ij ijklmnps > + F ijkl < ji >< < ji >< lk >< nm > + F >< sp > lk < ji >< +... = where i, j, k, l, n, m, p, s =,. The geometricl interprettion of this criterion is closed surfce in < >< >< > spce. Cretion of this surfce ws done by method of lest squres. At this not ll points turn out to be locted inside this surfce. To estimte better order of pproximtion it is suggested to introduce the vlue of so-clled Averge Reltive Devition (ARD) Θ of ll points of the creted surfce. The eqution of the sought surfce hs the following form: Ω + F ( ) = Fij < ij > + Fijkl < ji >< lk > + F < ji >< lk >< nm >< sp > ijklmnps ijklmn ijklmn < ji >< lk lk >< where i, j, k, l, n, m, p, s =,. Reltive devition of the representtive point (χ) reltive to the surfce cn be evluted by substitution of representtive point coordintes into eqution of surfce. Sign of χ will be negtive in cse of point hit inside the surfce. The closer point will be to the surfce, the less χ vlue will be (Figure ). Θ is rithmeticl verge of χ: Θ = N χ(i) i=.. N where N - number of representtive points ( in this cse). >< nm nm > + > + () (3) (4)

11 Figure. Definition of x Sign Effective surfce of the second order nd its cross-sections projections on < >< > plne re shown in Figure 3, where number mrk cross-sections projection t < > = 0 MP, t < > = 0 MP, 3 t < > = 0 MP, 4 t < > = 30 MP, 5 t < > = 40 MP. For the current surfce ARD = ; 3 points re locted inside the surfce nd 90 points - outside. Figure 3. Effective surfce of the second order Account of third order items in the eqution of filure surfce doesn t mke significnt chnges. Effective filure surfce of the fourth order nd its cross-sections projections (t < > = 0, 0, 0, 30 MP) on < >< > re shown in Figure 4, where number mrk cross-sections projection t < > = 0 MP, t < > = 0 MP, 3 t < > = 0 MP, 4 t < > = 30 MP. For the current surfce ARD = -0.07, 49 points re locted inside the surfce nd 73 points - outside. Figure 4. Effective surfce of the fourth order

12 Averge Reltive Devition for effective filure surfce of the fourth order is smller, then one for second order. As it cn be seen, fourth order surfce is better fitting ll representtive points. Account of the fifth order items in the eqution of filure surfce doesn t mke significnt chnges. As the result the fourth order surfce describes better the creted representtive points. Fourth order of the surfce is equivlent to the keeping of filure surfce tensor up to 8-th rnk in the tensoril-polynomil criterion. At construction of effective surfce with use of lest squres method some representtive points re locted inside filure surfce. When using this criterion it cn led to the fct tht not ll criticl zones will be determined. To eliminte this wek point the surfce ws modified by introducing correction reduction coefficient k (i), where i the number of representtive points tken to construct the surfce. In this cse i = : Ψ + F (8) () = Fij < ij > + Fijkl < ji >< lk > + F < ji >< lk >< nm >< sp >= + k ijklmnps ijklmn () < ji >< lk >< To clculte effective reduction coefficient it is necessry to define internl point tht is locted frthest from the surfce: inf( (8) ( ) ) nm > + Ψ (6) where infimum is clculted through ll representtive points. Effective reduction coefficient cn be found in the following wy: k where Ψ ~ ( 8) ) = ~ Ψ (8) ( ) ( ) (7) ( - function of Ψ ( 8) ) with coordintes of the internl point locted frthest from the surfce. The obtined in this wy coefficient k () = 0.. Correction of the filure surfce is shown grphiclly in Figure 5. Initil surfce (45 points inside) is shown with green, nd modified surfce (0 points inside) with blue. (5) Figure 5. Correction of the filure surfce The obtined effective filure surfce fits only for the internl periodicl cells. Quite often criticl zones in composite structure re locted ner the boundry. Thus the necessity is rising of the cretion of filure surfce for the boundry cells. At this it is necessry to consider two types of boundry conditions: kinemticl nd hybrid boundry conditions, wht will give two filure surfces. First of ll consider kinemticl boundry conditions. Sttement of the problem nd FE models re shown in Figure 6. Periodicl cells where stress verging is done re mrked with red. In our cse periodicl cell hs rectngulr shpe, so there re only two vilble borders types.

13 Figure 6. Boundry conditions nd FE models for two cell loctions Filure surfce for boundry cells t free boundry reduces to two representtive points on corresponding xes: A nd B for the periodicl cells ner free horizontl border (Figure 7.), C nd D for the periodicl cells ner free verticl border (Figure 7.). These points ly outside the corrected filure surfce for inner cells. So it is possible to use this surfce for both (inner nd boundry) cells types. Figure 7.. Representtive points for periodicl cell, locted ner horizontl border

14 Figure 7.. Representtive points for periodicl cell, locted ner verticl border Determintion of criticl zones in composite structures To illustrte the suggested method unidirectionl fiber composite ws considered. Mteril of the mtrix brittle isotropic polymer, fiber steel. Periodicl cell of the composite is shown in Figure 8, where = 5 µm, b = 0 µm, с = 5 µm. Volume concentrtion of the fiber V f = Mteril properties re listed in Tble 3. Figure 8. Periodicl cell Tble 3 E(GP) р (MP) c (MP) τ (MP) Т (MP) mtrix fiber In Tble 4 effective chrcteristics obtined with use of direct homogeniztion method re presented: Tble 4 Е (GP) Е (GP) Е 3 (GP) 3 3 G (GP) G 3 (GP) G 3 (GP)

15 To nlyze working cpcity of the developed criterion the reference problem ws solved. I.e. problem, where FE model of the composite with ccount of ll microstructure is represented. Sttement of the problem nd its solution re shown in Figure 9. Comprison of solutions is done is the mrked point A. Figure 9. Sttement of the problem nd its solution To determine criterion work efficiency it is suggested to introduce co-clled 5% rnge, which includes stresses forming 85% of ultimte stresses. Criticl zones where ultimte stress is exceeded defined for micro heterogeneous medi (reference solution) nd 5% rnge re presented in Figures 0 nd. Figure 0. Criticl zones of reference solution Figure. Criticl zones of 5% rnge in reference solution Finite element effective tensoril-polynomil criterion (FE ETPC) hs the following form: Ψ + F (8) () = F ijklmnps ij < < ji ij > + F >< lk ijkl < >< ji nm >< >< lk sp > + F > k ijklmn () < ji >< lk >< Criticl zones obtined with use of FE ETPC with reference problem FE mesh re shown in Figure., nd with rbitrry FE mesh in Figure.. It should be noted tht ppliction of this criterion in some cse gives more criticl zones then in reference problem. Stresses in the lower zone in Figures. nd. re close to criticl but doesn t exceed limits. Criterion stisfction in this zone cn be explined by filure surfce correction with effective nm > + (8)

16 reduction coefficient. Thus FE ETPC determines under-criticl stress zones. Differences between under-criticl zones nd criticl zones depends on effective reduction coefficient k (). Figure.. Criticl zones obtined with use of FE ETPC Figure.. Criticl zones obtined with use of FE ETPC with rbitrry FE mesh For more ccurte solution definition in the zones of criterion stisfction it is suggested to use sequentil heterogeniztion, i.e. entering the heterogeneous periodicl cell lyers in the homogeneous model. The cell, which were touched upon the zones of criterion stisfction, re tken s 0 lyer. Other cell re situted by the concentric lyers round 0 lyer. Sequentil heterogeniztion nd convergence of principl stress in the mrked point A depending on the number of heterogeniztion lyers (n) re presented in Figures 3 nd 4. Figure 3. Lyers of sequentil heterogeniztion

17 Figure 4. Convergence of principl stress depending on the number of heterogeniztion lyers The developed criterion must be stisfied in ny cse of loding. To verify this it ws suggested to solve number of cse studies with vrious boundry conditions (sher, compression, tension) nd nlyze working cpcity of the criterion in every cse. Cse study Let us consider the micro heterogeneous medi, presented in Figure 5. Specific chrcter of this tsk is represented by centrl hole, which is the stress concentrtor. Criticl zones of ultimte stress exceeding defined for micro heterogeneous medi (reference solution) nd 0% rnge (rnge includes stresses forming 90% of ultimte stress) re presented in Figure 6. Figure 5. Sttement of the problem Figure 6. Criticl zones of reference solution for problem Criticl zones obtined with use of FE ETPC with rbitrry problem FE mesh re shown in Figure 7.

18 Figure 7. Criticl zones of homogenous model for problem Cse study Then it is suggested to consider the micro heterogeneous medi with non symmetric lod, which is presented in Figure 8. Criticl zones of ultimte stress exceeding defined for micro heterogeneous medi (reference solution) nd 0% rnge (rnge includes stresses forming 90% of ultimte stress) re presented in Figure 9. Figure 8. Sttement of problem Figure 9. Criticl zones of reference solution for problem Criticl zones obtined with use of FE ETPC with rbitrry problem FE mesh re shown in Figure 30.

19 Figure 30. Criticl zones of homogenous model for problem Cse study 3 Finite element sttement of the third cse study is presented in Figure 3. Criticl zones of ultimte stress exceeding defined for micro heterogeneous medi (reference solution) nd 0% rnge (rnge includes stresses forming 90% of ultimte stress) re presented in Figure 3. Figure 3. Sttement of problem 3 Figure 3. Criticl zones of reference solution for problem 3 Criticl zones obtined with use of FE ETPC with rbitrry problem FE mesh re shown in Figure 33. Figure 33. Criticl zones of homogenous model for problem 3

20 Conclusions The new method of criticl zones estimtion in composite mterils ws suggested. Min steps of this method re:. Homogeniztion of composite mteril computtion of effective mechnicl chrcteristics nd filure surfce tensors;. Representtive points were constructed to build filure surfce. Ground for filure surfce order choosing nd necessity of its modifiction re presented. 3. Mcro nlysis in the suggested construction of determintion of criticl zones; 4. Sequentil heterogeniztion in criticl zones nd evlution of micro stresses. The developed method ws pplied to the composite structure (plne stress) nd detiled development of the method lgorithm ws done. The method ws lso verified in cse studies of complex loding of composite structure. As further development of the methods it cn be expnded to the nlysis of 3D constructions. References ) Tsi S.W., Wu E.M. A Generl Theory of Strength for Anisotropic Mterils. J. Composite Mterils, V. 5, 97, pp ) Tsi S.W. Strength Chrcteristics of Composite Mterils. NASA CR-4,965 3) Vnin G.A., Micromechnics of composite mterils, Nukov dumk, 985, Kiev 4) Mechnics of composite mterils, Ed. Sendeckyj G.P., Vol., Acdemic Press, NY, 974 5) Pobedri B.E. Mechnics of composite mterils, Moscow, 984 6) Azzi V.D., Tsi S.W. Anisotropic Strength of Composites. Experimentl Mechnics, September 965, pp ) Hoffmn O. The Brittle Strength of Orthotropic Mterils. J. Composite Mterils, V., 967, pp ) Hshin Z., Rosen W. The elstic moduli of fiber reinforced mterils. J. Appl. Mech., 964. V. 3, pp. 3 3