C E R T A I N R E G U L A R I T I E S IN T H E S T R E N G T H OF S O F T - F I B E R

R E I N F O R C E D P L A S T I C S W I T H T W O - C O M P O N E N T W A R P S

~ . S. U m a n s k i i

Kiev Polytechnic Inst i tute Translated from Poroshkovaya Meta l lurgiya , No. 5 (23),

pp. 22-33, Sep tember -October , 1964

Original a r t i c le submit ted March 5, 1964

S t r e n g t h o f FRP wJEth O r i e n t e d T w o - C o m p o n e n t W a r p s From among the new synthet ic mater ia l s , f iber - re inforced plastics ( they wi l i be referred to as FRP), wh,.'ch are

produced by impregnat ing fibrous warps with shee t l ike or viscous- l iquid polymer compounds, are used to an eve r - increasing extent. The anizotropic ma te r i a l is obtained by impregnatL-,g fibrous warps of the o r i en ted- top type, while r e l a t i ve ly isotropic mater ia l s are obtained by impregnat ing a warp consisting of mutua l ly perpendicular tops or a warp consisting of random-or ien ted fibers. Most often, cotton tops serve as the fibrous warp, while mas t i ca ted polyvinyl chlor ide is used as the impregnat ing polymer.

The mechan i ca l and performance character is t ics of FRP can be improved by adding to the base h igh-s t rength e las t ic synthetic fibers, in par t icular , Kapron. Therefore, it is of interest to invest igate the phys icomechan ica l prop- erties of FRP with two-componen t warps. An a t tempt is made in the present a r t i c le to establish cer ta in theore t i ca l

dependences of the strength of FRP with two-componen t warps that are connected with changes in the composi t ion of the warp fibers and changes in the i r strength and deformat ion properties.

We shal l first consider uniaxia t extension of FRP specimens (Fig. ! ) with an oriented two-componen t warp

which are cut in the direct ion in which the fiber top is laid. Af te r impregnat ion, the strength of such a m a t e r i a l is de te rmined by the quantLty and the or ientat ion of the strongest e l e m e n t in its composi t ion - the fibers.

Obviously, one can only speak o f the predominant di rect ion of f ibers in an oriented ma te r i a l , s ince the fibers dev ia te from the assigned direct ion in the untreated top and als,- during the impregnat ion process. However, we shal l assume in the first approximat ion that the fibem in the c r i t i c a l cross sect ion dev ia te s y m m e t r i c a l l y from the di rect /on of lay by the same angle ~0 (which shall be referred to as the mean devia t ion angle). We shal l also assume

tb_at the transverse cross sections f l and f2 of fibers are tee same for each type of fiber.

The relationships expressing the deformat ion of fibers are known, and they can b e approximated by the equa- tions [i]

where s l and 6 a are the fiber strains, o i and OlI are the normal stresses in ~ e transverse cross sections of fibers, and A,_, A a, ki, and k a are the m a t e r i a l cons t an t , which are the same for a l l fibers in a cer ta in group under the given ex - tension conditions.

The u l t ima te strength of fibers is different for each fiber type. Considering that the number of fibers of each group in the spec imen is large, one can presume the exis tence of continuous distr ibution functions ~l and ~2 of the strength of both types of fiber.

We shal l further assume that nt and n 2 are the numbers of fibers in each group that pass through the c rms- sec t ional area F of the spec imen when the fibers are pa ra l l e l to the spec imen ' s axis; the corresponding numbers of incl ined fibers that pass through this area are equal to

(n,) =lz~ cos cO; (:'~2)~=n2 cos ~.

The relat ionships

F ' F

(2)

(3)

3"71

llj N

Y

N

Fig. I. Model of an FRP specimen with an oriented two-component warp.

5

0 --C

Fig. 2. Extension diagrams for group I and group II fibers.

F F /t

p(R)

b

Fig. 3. Approximation of the functions of strength distribution densities for group I (a) and group II (b) fibers.

can readily be determined by comparing the weights of the materi- al and of its components [2]. Generally, we have

(1 - x ) (1 - K ) m

(4)

x (I--K)) m 1 2

1 + 1 --), + Y3 m (,5)

where 7.~, ")% and 7s are the specific weights of type-I and -II fibers ~nd of the plasticized bonding film, respectively; k is the ratio of the weight of group-II fibers to the weight of the warp, and m = Qfib / Qfil is the ratio of the fibrous warp's weight to the weight of the film.

The porosity of the material is determined by using the ex- pression

K=

),

(6)

where 7 and 74 are the specific weights of the material and of air, respectively.

Let the strain of the specimen be 6. Assuming that the hy- pothesis of plane cross sections hoNs, and neglecting the relative ~hift of fibers in the direction perpendicular to the extension axis, we readity find the strains and stresses in fibers of both groups:

r cos z ~; (7)

_% 1

( ( ) cos (p) ; = cosZ9 o,= ~- an ~ .k,

Consequently,

(8)

(9)

where 1

kA,] '

i---- k~ (io)

k, (:is.)

For the sake of determinacy, we shall further assume that the fibers of the first group have a larger elasticity rnodulns, a smaller ultimate strain, and a lower ultimate sEength. The second-group fibers are more elastic and stronger (Fig. 2). Since there is a scatter in the strength of fibers in both groups, a portion of fibers in eac}~ group will be ruptured at a certain stage of extension so that, at the equilibrium stage, we shall have

N = P = ~ s n l f I cos 2 tp[1 --c~i(al)l-kann,f2 cos ~ ~p [1 -- q>2 (%1)]" (12)

372

Considering (3), the nominal stress in the specimen's cross section is equal to

- N

As in the case of FRP with pure-cotton warps, we shall assume, for the sake of simplicity, that the density of the strength distr'ibution of fibers in the first and the second groups in the rma x, rmi n and Rma ~ Rmi n intervals is constant (Fig. 3).

1 1 0(5)= ; 0(%)=

r m a , - - r ~ i . Rmax-Rmi:" (14)

Then, the relative numbers of ruptured fibers in the first and the second groups are equal to

%(;[--/~min ~ l~rml n (;ii__Rmin i </), (~ ; q). 0 . ) =

r m a x - - rmin Rmax--~min Rmax-- Rmin (15)

By substituting expressions (18) and (9) in (13), we find

fmax--rmin ' ]~max--/~mi n " (16)

We shall find the stress in the unbroken fibers under which the specimen is capable of withstanding die maxi- mum load by setting equal to zero the derivative of N = ~F with respect to o I. We have

where

2 ix=A ' 62 'i , "A te2 " A 2 % Zx, ~I (17)

1 A I = -2- (rmax-- rmin);

1 Z~I=-~ -~ (Rmax--Rrnin). (18)

Hence, by substituting the actual vatue of the constant i, we determine (o i)u. It should be noted that the ex- tension diagrams for the fibers used in FRP warps can be considered as only slightly nonlinear so that, fo~ the sake of simplicity, we have k I ~ k 2. Then i = 1 and we find, from (17):

1 rmaxk~_~ @xRmax/~1~ ('I)d-- 2 ~ a -- , ,=* < , ~ T I U " " ~IT2

(%){i = ~ O~)u (19/

By introducing (19) in (17), we find ~ e mean ult imate strength of a specimen with_ an oriented two-component warp which is cut "along the fibers."

! I ,I 2

(7 0 , = cos" ~ ~,~= (%~=+x=%A~) (20)

For the extension of specimens cut in a direction perpendicular to the principal direction of fiber lay, the ex- pression for the ult imate strength will be obtained from Eq. (20) after the angle ~r/2 - ~o is substituted for ~p in thB equation. Thus,

(21)

The ratio of the ult imate strengths of FRP specimens with oriented two-component warps that are cut along and across the principal dJxection of lay is equal to

Gv) J (~v)~- - - ctg~ (22)

3?3

(_~,) ~ Z 2 f, ~,, (~ ,~ul , ' : - L ~ .~"

!

t,5 0,7

o,e

o.4

( j : /44 ':sO:4z) J=3

7 7

0 to 2030 40 50~0 7dO0 go/oo%f 10090 80 70 00 $o40 3~; 20 #0 O%LG

Fig. 4. Dependence of the u l t imate strength of FRP on the weight rat io of group-I and - I I fibers for different ratios

of fiber strengths, x = 0.5; in = 0.5; Yl

= 0.10; Y2 -- 0.25. 8

$

o,7 ~ / f -

0.6

I 1 "

o,, = T = 7 _ = , , : , o < ,

t 2 3 4 5 6 7 8 g~ [o r

Fig. 5. Dependence of the u l t ima te strength of FRP on the ratio of strengths of group-II and - I fibers in the warp.

S o,e J l I , o,~ I 1 i ~ - 2 Z

t ~ 1 7 I o,, ~ I i o , e - ~ ~ , 7 ' I / ~ " G '

o , - [ . . i 9,6 - ! . . / / ! i f . - i

I I1 / ,/_1,

o,s �9 i ! '),~ / / / - - o o,, 0,2 o,~ o,~ 45 4c o,~ o,8 o,~ ~ , _ s ' /

1 r a 0,3 i

~ z;2 o,~ a~ ~ 46 o,7 o , 8 o, gz:~, b

Fig. 6. Dependence of the u l t imate strength of FP, P on the rat io of e las t ic i ty modul i of the warp fibers, a) X = 34.4~ b) X : 60%.

and is, consequently, a constant quanti ty that does not depend on the quant i ta t ive composi t ion of fibers and of the

f i lm in the mater ia l ,

I t is obvious from the derived expressions that the strength of FRP with two-component warps is a function of many factors: the weight relationships between the fibers and the f i lm, the weight relationships between the fibers

of both groups, the strength of fibers, and the relationships between the e las t ic i ty moduli . By se lec t ing warp fibers with cer ta in specif ic properties and varying theis quanti ty, FRP with a strength assigned beforehand can be produced.

For convenience in analyzing the ef fec t of the above factors on the u l t imate strength of the mater ia l , we shall wri te expression (20) in the fol!owing form:

374

where

cos = qD (1 --K} 4(1 --v,)

1 +:ax I -- v----~ ~i) COS = (pr~ax I ~ ' q , 21--~I"- ~i=

4 ( r ~ a x ' r m i " ) l -~-~x i - ~ v 2

1 -t-btx 1 --v,~* 1 - -v2] %mr=ax

( 1 + ~ x 2 1 --~, (7~+'r + (72+~'3m) P 1 - - Vg (2a)

~ = R m a x . v t = r m i n " Y2=Rmin . k ~l

rma---7' rmax' Rm~----~' ~ = 1 - -X Ta (24)

For the sake of de te rminacy , we shal l assume that m = 0.5, Yi = 1.52, 72 = 1.14, 7a = 1.22 (which correspond to the specif ic weights of cotton and Kapron fibers, and the p las t ic ized polyvinyl chlor ide f ihn, respect ively) . More- over, v 1 = 0.1 and u 2 = 0.28. For these values, we shal l wri te expression (23) in the following form:

S = 5 , 9 (%) ~ _ 0,470 (1 + 1,2~•

COS 2 q) (1 - - K ) rmax ( t @ 0,821x ) (1 @ 1,2 lXx~ " ) ( 2 a %

The effect of the weight ratios of group-II and - I fibers can be es t imated from Fig. 4. The cu.'ves are plot ted for different ratios of the greatest strengths of fibers in groups II and I, while the figures in brackets ind ica te the ra t io s

j = g (1 + v a / 1 + v~) of the average u l t ima te strengths. The rat io of the e las t ic i ty modul i was assumed to be x = 0.5.

The above ca lcula t ions and Fig. 4 indica te that, even for average percentages of second-group fibers (X), the strength of the ma te r i a l changes l i t t l e with a sharp ir, erease in the rat io of the max imum or mean strength values for fibers of the second and the first groups. Thus, for X = 30%, when g changes from 1.5 to 10, the strength of the m a -

, t e r i a l increases by 10~ for X = 80~ it increases by only 22% Consequently, in the above range, the nse of

fibers with widely different strengths is not advisable .

Figure 5 provides a c lear idea of the effect of the ratio ~ of fiber strengths on the spee imen ' s strength for di f -

ferent f iber percentages in the warp. It is obvious that, with an increase in { = Rma x / r m a x , ~ e ra te of increase in the re la t ive strength (S) of the m a t e r i a l rap id ly diminishes. Therefore, the use of fibers whose strengths differ by a

factor greater than 3 in n, ro -componen t warps is not advisable .

The dependence of the strength of FRP specimens on the rat io of e las t i c i ty modul i of the warp f ibe~ is shown

in the diagrams of Fig. 6a and b, which were plot ted for X = 34.4 and 60% for g = 1, 1.5, and 2.

It is obvious that the strength increases a lmost l inear ly with an increase in x from 0.1 to 0 .5-0 .6 , while the ra te of increase and the slxength value depend only sl ightly on g. As z increases from 0.5-0.6 to t . 0 , the s~ength

kqcreases at a lower rate and begins to depend considerably on g. Thus, it is advisable to use fibers with w. > 0.5 (g > 1) in warps of two-componen t FRP.

S t r e n g t h o f FRP w i t h C r o s s e d T w o - C o m p o n e n t W a r p s

As before, we shall assume, in the fixst approximat ion, that the fibers in both crossed-top iayers s y m m e t r i c a l l y dev ia t e from the pr inc ipa l direct ion of lay by the same average angle ~. Let us eonsider the extension of a spec imen cut along one of the pr inc ipa l direct ions of lay. tn this case, one ha l f of fibers in each group forms the angle ~0 with

respect to the extension axis, while the other ha l f forms the angle r r / 2 - - ~o.

If the spec imen ' s strain is 5, the stresses in the unbroken fibers are equal to

1

(~)~= ~A, c~ I

(%)~= ~ c ~ ;

As in Russian O r i g i n a l - Publisher's Note.

!

1

(=,,).. = ,-( L ln' -i -<'>- l G . (27)*

375

Here, as in the preceding case, the hypothesis of plane cross secfiom remains in force, and the relative shift of fibers in the direction perpendicular to the extension axis is negIected.

From Eqs. (2q), it follows that 2

X--V

(%)~ = x (~1~ ( t g ~p)~'. (~8)

At the equilibrium stage of extension, we have

i , 1 N = P = ~ %)4hi, cos ~p t~-- qh (('h)~)l+ }- (%)~ n,h cos ~ ~p[1-- qh (o.~')1+

"1-- 1 ( 0 l ) ~ t 2 .~.__? f i l l 1S tn ' q) [ 1 - - f~)1((~ ~____ ~?)l-{" 1

+ ~ (%).~ _ J~2 s tn2 rp{l--q),[(oi,)~ _ l} , (~9)

where the relative numbers of ruptured fibers are equal to

~ l . rmin

Fmax~rmin ' ~max.--J~mi n ' 2

rmax--rmin g -- ~ Rmax-- Rmin (30)

After dividing Eq. (29) by the specimen's cross-sectional area F, substituting Eqs. (28) and (30), and assuming, as before, i = 1, k = k 1 = kz, we find

~ m a = F - - 4 (cos * tp+ sln ~ cp tg 2'k, q~) q4 ,

+(cos 2 tp+ sin~ qD tg~,'k~q~)~,: Rm~]~, - % , 4,k -x2 ' ,2 ? -- [(cos2 q~+sln~ cp tg4/k~ (P)A~ + (cos= qv-f sln2 q~ tg ~ ~ q~, -~-~-2 ] ~ }.

After setting the derivative of N with respect to o I equal to zero, we find

(al)

G), = a, a , x2~,]

2 [(COS~ q~q- sln~ q~ tg4i~, q~) ~ +(cos ' rpq-sin~ tp tg4/k, q~) - ~ ]

By substituting (32) in (81), we obtain the expressiort

(~v) ~ = 1 (cos 2 cpq-sln 2 tp tgv~ cp) (,hr~.xa2+xCd?~axaY 16A1A 2 (COS 2 qgq-sln 2 tp tg 4k tp ~)A2ff-x24,2Al)

S t r e n g t h of FRP w i t h O r i e n t e d and C r o s s e d C o t t o n - K a p r o n W a r p s

(a2)

(a3)

We shalluse the general equations derived above for estimating the strength of FRP with cotton-Kapron warps. Among materials with two.component warps, the above type is encountered most often.

The experimental results obtained in investigating the strength and deformability of FRP with an oriented and crossed warp containing 67% medium-fiber cotton and 33% K apron under conditions of uniaxial extension are given

3'16

o, d a N / c m z 45t71

3 s 0 i / _

3aO r / i

250

200 ....... ~ 2

ida ] "~/

looi rn2fi__ b ~3 o,4 o,5 o,~ o,7 o,80fi!

Fig. 7. Theoretical and experimental dependences of the u l t ima te strengths

of FRP with an oriented cot ton--K apron warp on the weight rat ios of the fibers

and the f ihn in the mater ia l . 1) Ex- tension "along fibers"; 2) extension

"across fibers. ~

? ' : ' ~ 4 - ' - - . . . . '

"'i - 7 - T F

t 1 u,.~, i i ~ [ ~ - - - - + -

! ! [ ! t

[ ! ! I i Q., .~.-a--.--..U .--.--k.___. ...... r r - 11D

~;3 (d,,v ~15 ~5 0,,7 US ~fzl

Fig. 8. Ratio of the u l t imate strengths of FRP with a crossed and an oriented

c o t t o n - K a p r o n warp (the specimens were cut along the pr inc ipa l d i rec t ion of f iber lay). Sol id l i n e - theore t ica l dependence; dotted l i n e - expe r imen t - a l values.

in [2,3]. We shal l give a theore t i ca l e s t ima te of the strength and d e -

formabi l i ty in dependence on the rat io of fibers to the bonding f i lm for

this m a t e r i a l We shal l first consider FP, P with an oriented warp. Theo-

r e t i ca l ly , as follows from (22), the rat io of the u l t ima te strengths of spe-

c imens cut along and across the fibers is a constant quant i ty regardless of

the composi t ion of the mater ia l .

Experiments performed on five batches of the oriented c o t t o n -

Kapron ma te r i a l (m = Q f i b / Q f i l = 0.4, 0.5, 0.57, 0.67, 0.8) have shown that the above rat io is in fact constant and equal to 2.1 [2,3]. Conse-

quently, cotan2~o = 2.1; r = 34~ '.

The m e a n u l t ima te strength of specimens cut along the p r inc ipa l d i rect ion of top lay can be de termined by using Eq. (20) or (23). The specif ic weights of cot ton and Kapron fibers and of the p las t ic ized po ly - v inyl chlor ide f i lm are given above. From the comple t ing condi t ion X

= 0.33, by compar ing the weight of the m a t e r i a l and of its components ,

we readi ly find the ra t io of the air vo lume to the volume of the mater ia l s

(K)

As was shown in [2], the dependence of this ra t io on the f ibe r s - to - f ihn weight ratios is c lose to a l inear dependence and can be writ ten:

K=--0 ,0606+0,25 m; ( 0 , 4 4 m 4 0 , 8 ) (34)

Consequently, from Eqs. ('4) and (34), we find

(0,57292--0, ! 350m) m % = 9co~ 1 +-0,891m (35)

',52 ~,~'~ 0,33. !,52 ~ 0 , 6 o . 9~ (1--)) y~. 0,67.1,14 (36)

In de termining the distr ibution parameters , one _.must bear in mind the considerable scat ter of the strength of fibers, the dun! character of

their fai lure in FgP (d i rec t rupture and breakdown of adhesion), and also

the chosen form of the curve of strength distr ibution density. In a n a l y z - ing the strength of FRP with pure-co t ton warps, we used u, = r m i n / r m a x = 0.1 for a constant distr ibution density. Kapron fibers are more h o m o -

geneous, and, therefore, one can use u 2 = Rmi n / R m a x = 0.25-0.30.

For the ratios of the e las t ic i ty modul i to the max imum strength values of Kapron and cot ton fibers, we used x = 0.60 and ~ = 1.45, r e -

spect ively . After substituthug the above values in (20) or (23), we obtained the following dependence of the m a t e r i - a l ' s mean strength on the f i be r s - t o - f i lm weight ratios:

a) for extension "along the fibers,"

(~v)ll = i 105,7m -- 260,5m z 1 + 0 , 8 9 1 m

b) for extension "across the fibers,"

(37)

526,5m-- 124,0m' 1 + 0 , 8 9 1 m (38)

The theore t i ca l dependence of the u l t ima te strength of FRP with c o t t o n - K a p r o n warps oi1 the f i be r s - t o - f i lm weight ratios is given by the solid lines in Fig. 7; the points correspond to the exper imen ta l values obtained in s tand- ard tests in machines at a deformat ion ra te .c lose tO 3, 7% per sec. The theore t ica l and expe r imen ta l strength values are also given in Table 1.,

377

The strength of FRP with crossed cot ton.Kapron warps can be estimated by using Eq. (33) which, for the above deviation angte of fibers in each layer, ~o = 34~ ' and p = 0.66 (33% Kapron and 67% cotton), assumes the following form:

1 --vl'~ t 1 + 0 , 6 6 x 1 - - ~ /

(~v) = 0,459 r m ~ ,5:

(i+0,66x~ 1--~I ! \ 4 (1 - -v~)

1--~ 2 r . ) (39)

In calculating the r value by means of Eq. (4), one must take into account the porosity of the type of FRP in In [2], for FRP win a crossed cot ton-Kapron warp, we found

K,~=--0,0650+O,22m (0,4~<m~<0,8).

question.

(40)

Then,

'~'1 = 0 , 5 7 5 1 m - - 0 , 1 1 8 8

1 + 0 , 8 9 1 m (41)

It should be mentioned that the ~i and r values for the crossed and oriented cot ton-Kapron materials are close to each other and that their ratio in the m = 0.4-0.8 range is not greater than

t

q01 = 1,03. q)l (42)

By comparing Eqs. (39) and (23), and taking into account (42), we find the ratio of the ultimate strengths of FRP with crossed and oriented cotton--Kapron warps, tt is equal to

o,7o (or)it (43)

and is virtually independent of the fibers-to-fi lm weight ratios in the mate r ia l It is obvious from Fig. 8 that the experimental values of this ratio are close to the theoretical value.

After the substitution of all the parameters, the final expression for the ultimate strength of this type of FRP as- sumes the following form

-- 7 5 3 , 4 m - 155,6rn 2

('~ v)~* = 1 + 0 . 8 9 1 m i44) Table 2 provides a comparison between the experimental and the theoretical values of (Ov) #. The above data indi- ca te that there is good agreement between the theoretical and experimental results. Consequently, the above sire- plified model of a composite specimen is entirely suitable for estimating the strength of reinforced plastics with two- component warps.

We shall now determine the number of ruptured fibers at the given stage of extension for the mean stress ~ in the specimen's transverse cross section. For an oriented material and extension "along the fibers," we have, from Eqs. (18) and (16):

4A 1 (~lhz+x2'~2h I) q)~- -2 [~lhz (rmax--2rmin)--x~sAl (/~max--2Xrmin)]

r ~[2`,51A2-t-x~z (Rm.~--zr~i.)]+ 2A2 o = 0 . COS ~ q0

Hence, we find

~ 1 ~ [~1A2 (rmax--2rmin)nt-xA1 '~ (Raax'--2xrmin)] __+

(45)

3'/8

TABLE 1 TABLE 2

~fib In = Qfil

0,80 0,67 0,50 0,40

Mean u k i m a t e strength, d aN/cm z

"'along~fAbers" ! 2_o,r o~_iibers ~

theor, expt l .

419 419 390 401 348 348 295 293

theor.

199 186 166 141

exptl .

In

198 0,80 195 0,67 163 0,50 141 0,40

Ul t ima te strength, d a N / c m z

theor, exptl .

294 312 273 261 234 238 204 226

+---V'[t~,a~ (rmax" 2rmin)-F xA~_d?z (Rmax-- 2xrmin)] z - - 4& ('h&+x~,%a 0

COS ~ [p

4Az (9~h~.+x~tp~h~) ' (46)

1 qb~ = ~ [2zh I qb 1 _ (R~i __xr~in)] .

(4q)

If we know ~ and ~z, we can read i ly de te rmine ~he ra te of the fai lure process, reduced to unit area of the spe- c imen ' s transverse cross sect ion [4]:

~ dG , d ~ + d~. /, dO, d.O,~

By di f ferent ia t ing Eqs. (46) and (47) with respect to 8 and substi tutNg these values in (48), we find

COS 2 v,, / [2 'h~&+x~2A~ ( R , . ~ - 2 ~ ' r . q . ) l z -

- -4a 1 (~i/~ 4-~.2',52&) [cos ~ q~ r.~,,:~ [2',51~-F• 2 (Rm.~--Xrmi~) �9 (49)

The fai lure process becomes unstable when Vp --> ~o . Since d 8 / d r is a f ini te quanti ty, Vp ~ o~ when ~ e d e - nominator in (49) tends to zero. This occurs when

- 1 (',51r~.axh 2 @ x'~2Rm~xhl) ~ ~_~ _ _ C O S 2 q)

8 @% (91G+x~'%h0 which corresponds to the mean stress in ~ e spec imen under tahe greatest load (20), i .e . , it corresponds to the u l t ima te s t r eng~ of the specimen. Here, the s t r~s in unbroken fibers is de termined by using Eqs. (19). Thus, hhe fa i lure of FRP specimens consti tates an equi l ibr ium process uc_,til the te~tsite force attains its ma x imum va lue also in ~he case

of two-componen t warps. Let ,us compare tthe m a x i m u m strains of specimens cut along and across the fibers. We shall def ine the re la t ive strain as the mean s~ain of the fibers of both groups at the moment of transit ion to the non- equi l ibr ium stage of fai luie . For specimens cut a!ong the pr inc ipa l d i rec t ion of fiber lay,

1

(g),, = (n G cos ~ 1

l=1 i--1 (5o)

379

The total strains of the fibers of both groups, consisting of the strains of ruptured and unbroken fibers at the moment of transition to the nonequilibrium failure stage, are equal to

(n,}~ (~ } v

2 (e,0= (n,)~ ff A,~[.p,d~+(n,)~il--ch,(a,v~lA,d[(. , rmin

(ns}~ (~II ~x"

2 (%,)=(n,), ff A,,J[tp,da, t+(n~),p[l--cl)z (~11 ')] AgcIkt" l=l

Rmln

By substituting (51) in (50), we find

(el)~., (~'II) x"

cos 2 r rmin Rmin

+A~~ 1 - q)~ (~u,)1 }"

After substituting the Pl and Pz values from Eq. (14) in Eq. (52) and integrating, we obtain

(51)

(52)

Cos2(p t 2A1(kl_4_l)X i,,, m,n.J--

+ G,,)- (~ �9 ca3) In the extension of specimens cut in a direction perpendicular to the principal direction of fiber lay, we simi-

larly have

-}- Alaf(, [ ~ " (~1 (ai ~--I~.('I1 v)] } "

The ratio of relat ive strains

~._~ = c tg ~ (p ~H (54)

is a constant quantity which does not depend on the f ibers- to-f i lm weight ratio.

For FRP with cotton--Kapron warps, r = 34*3o'; consequently,

~-3-x = 2 ,1 . ~11

For the investigated five batches of FRP [2], the experimental value of the above quantity is actually independ-

ent of In = Qfib/Qfil ; it is equal to 1.96.

S U M M A R Y

Formulas are derived for the Mtimate strength of fiber-reinforced plastics on an oriented two-component warp, on a crossed two-component warp, and on an oriented co t ton-Kapron warp,

The formulas make possible an analyt ical calculat ion of the strength of these materials, proceeding from the properties of the fibers, and thus selection of the optianal composition of the material ensuring the given properties,

380

L I T E R A T U R E C I T E D

1. A . A . I I 'yushin , P las t i c i ty [in Russian] (GITTL, OGIZ, 1948).

2. ~ . S. Umansk i i , DAN UkrSSR, No. 2 (1962).

3. ~ . S. Umansk i i and L. I. Rusakovich, TsNIIKP S c i e n t i f i c Research Papers, Co l l e c t i on No. 15 (1964).

4. S . D . Volkov, Zavod. lab . , No. 3 (1960).

All abbreviat ions of per iodicals in the above bibliography are let ter-by-let ter transli ter- ations of the abbreviat ions as given in the original I,~ussian journal. Some or all o f t h i s per i -

od i ca l l i t e r a t u r e m ay we l l be a v a i l a b l e in E n g l i s h t rans la t ion . A complete l ist of the cover- to- cover Engl ish t ranslat ions appears at the back of this issue.

381