METHODS OF TESTING THE STRENGTH OF FILAMENT-WOUND COMPOSITES V. V. Partsevskii

UDC 620.1:539.3:678.067

To produce structures made of composite materials, it was necessary to develop the mechanics of these materials [1-3] and to develop and substantiate special methods of mechanical tests [4]. The most widely used and simplest methods of determining the ultimate strength in the circumferential direction of composites produced by winding are the loading of circular specimens with stiff sectors on the internal (tension) and external (compression) surfaces. The shortcoming of these methods is the considerable heterogeneity of the stress field in the zone of failure [5-7]. This heterogeneity decreases with increasing number of loading sectors, but the design of devices becomes more complicated. A t t e m p t s to reduce the stress concentration by the use of a special ring specimen with straight sections [8] have not been successful: the stress concentration is not eliminated; moreover, it becomes difficult to control, and the structure of the specimen differs from that of the component. To eliminate this shortcoming, the field stress at loads in the vicinity of the fracture load should be axisymmetric and the circumferential stresses should change only slightly through the thickness of the ring. The latter condition is satisfied in not too thick rings (r~/r2 > 0.8). Axisymmetry can be achieved if the radius of curvature of the loading surfaces of the sectors and the dimensions of the ring specimen satisfy the following conditions: ra < ro in tensile loading of the ring and r2 > ro in compressive loading (Figs. la-c), and the aperture angle of the loading surface of the sector ~ ~/N (N is the number of sectors). We shall determine the required value of ro.

The circumferential

strain in the specimen

c~ = ( o ~ -- V ~ r 0 r ) / E ~ 2 o ~ / E ~ , s i n c e ~ r << a~, and v ~ r = 0 " 05-0 " 1+ for filament-wound - composites. Assuming that ~,~ is equal tO the ultimate strength in tensile g~ and compressive ~ loading,

and taking into account that on the internal surface of the ring in tensile loading E:~ = (ro-r~)/ro, and on the external surface in compressive loading ~ = (r2 -- ro)/r2, we obtain, respectively, ro/r2 ~ 1 - - U ~ - / E ~

In the tensile loading of the ring, at N > 2, whereas in compressive loading cases, the edges of the loading surfaces grooves for penetrating into each other

.

(1)

this modification of the method can be realized only the number of sectors may be arbitrary. In all the of the sectors should be rounded (Fig. Id), contain [9], and measures should be taken to reduce friction~

To substantiate the method and interpret accurately the test results, it is essential to examine the dependence of the stress field in the ring on the radial displacements of the sectors A in all the loading stages. The minimum value A at which contact occurs on the entire surface of the sector with radius ro will be denoted by • We shall solve a planar periodic contact proble_m of loading a polar-orthotropic circular ring with a system of stiff smooth dies at S > 4. In loading the ring on the external surface r = r2, the solution of the differential equations of the theory of elasticity with respect to the displacements u r and u,~ [6] should satisfy the boundary conditions ~=~=0;

~=0

{ (A,-A)cos ~ + a l - r 2

(r=rl);

( ~ [ 0 , ~])

(r=~),

(2)

Ut

~= 0 Here

(~[~o,

z/N]).

al=[ro 2 - (A,--iA)2sin 2 ~]'/'; a~= (ro--f)cos(~--~p) + (A, --A) cos qo; aa2-- (ro+f)2+ (A,-,A)2+2(A,-A) 9

(r+f)cos~--f2;&, is the displacement

of the sector A at which the solution is axisymmetric;

Moscow Energy Institute. Translated from Mekhanika Kompozitnykh Materialov, 336-340, March-April, 1983. Original article submitted June 17, 1982.

0191-5665/83/1902~0259507.50

9 1983 Plenum Publishing

Corporation

No. 2, pp.

259

~(A) is the coordinate of a "jump" in the curvature of the loading surface is determined from the given value of A. Figure Ic-e yields

of the sector, w h i c h

A, = [ (r~- ? ) z - (ro+ ~)~ sin ~ (p] '/~- (ro + ?) cos ~; @=arctg The coordinate of the v_ance. The right-hand r2(~) -- r2, where the ment by & follow from

ro sin~ A,-A+rocos~

(3)

"

boundary of the contact area ~o(&) (see Fig. le) is not known in ad, parts in the conditions (2) were determined from the equations u r = equations of the loading surface of the sector r=(~) after its displacethe geometrical considerations (Fig, le).

In loading the ring on the internal surface r = r~ with the stiff smooth sectors, • t h e conditions (2) and subse_quent equations it is necessary to change r~ and r~ in certain places, to change the signs at r and (A...-- A), and to replace s in Eq. (3) by A, = ( r o - ~) cos ~--[(rl -- ~)2 _ (ro --p)2 sin s ~]172" The algorithm of solution of the given contact problem may be described as follows~ The ro and s are calculated from the geometrical, elastic, and strength characteristics of the ring; i is specified, and ~ is determined from Eq. (3). Subsequently, the pFoblem is solved by the iteration method because ~o is unknown. The first approximation ~o(~)is specified, K and the final line of the equations (2) is replaced by the condition Ur = Z Ck(~/N-~) h (r=r2, k=0

~ [ ~ 0 (1),~/N])

with the unknown coefficients

Ck.

The solution of the equations with the resultant single-type boundary conditions has the form of an expansion with respect to trigonometric functions ~ [6]. The coefficients C k of the first approximation, included in the linear form in the solution, are determined bY the method of boundary collocation from < ~o (1) in relation to the sign of the difference of the derivat'ves i ~ U r / ~ at the points ~ = ~ o (~) • 0 (r = r2) [i0]. Iterations continue until the smoothness of u r with the given accuracy is ensured. The next s is then defined and calculations are repeated. Some of the results of solving the problem of loading the ring with two sectors (half yokes; N = 2) at the external contour, obtained in a computer, are given in Figs. 2-5, The following values were used in the calculations: E ~ = 60 GPa; E r = 15 GPa; Gr~ = 6 GPa" Vr~ = + = 1.4 GPa; ~ = 0.95 GPa; r~/r2 = 0.95 ;r/r2 = 0,i. 0. 3 ; E~ Equations (i) a~d (3) yield ro/ r2 = 0.984, A~./r2 = 0.187. Figure 2 shows the curves of the contact stress o for various loading levels. At A < r ~, the solution of the problem gives an alternating function for Or, i.e., s = ~ is the condition of contact on the entire surface of the sector. In this example, s = 0.882A ,. The stresses Or~(~) for r = (r~ + r2)/2 are given in Fig. 3, and the stresses o~(r) for ~ = 7/2 are given in Fig. 4. It may be seen that the steep_hess of the stress gradients at the edge of the sector rapidly decreases during loading at 5 < s < s At s > 5, the gradients again become steeper. It is important to note that at the values of 5 near the ~ the tangential * ~ stresses Or~ are low and the dependences of the mean ~ ( A ) for ~ = =/2 and [max ~ ( )I rapidly increase (Fig. 5). Therefore, even if the values of the ultimate strength'in the circumferential direction are greatly scattered, failure should take place in a narrow range of in the vicinity of s i.e., practically in a uniform field of the stresses, This }s the advantageous feature of the modified methods of testing in comparison with the conventional methods in which the stress gradients are minimum in early stages of loading and very steep at the fracture loads. All the above-mentioned special features of solving the problem of loading the external contour of the ring at ro < r2 are also typical of the solution of the problem of loading on the internal contour for ro > r~ and also for N > 2. In order to determine the boundaries of application of the given modifications of the strength test methods, the maximum stresses should not exceed the ultimate strength in the initial stages of loading in which contact does not yet take place on the entire Surface of

260

[l m-4

___rJ J d

~

"

ro

:/';

e,

Fig. i. Modifications tems of ring specimens bers.

7; 7 ,

of loading syswith stiff mem-

0 0

-0,025

oy\/

0,96"/.~'%~

-0,02

o,92-l

-0,050 - 0,04 -0,075 -%~oo or

I

-0,06 ,

0,45 Fig.

0,50

- 0,0s

OfiO

f

2

r

0,45 Fig.

Fig. 2. Contact stresses in compressing at the curves give the values of b/A,.

I

3

the ring with two half yokes.

Fig. 3. Tangential stresses in the center of the thickness at the curves give the values of b/b,.

Vqotq ._1 0,50

of the ring.

The figures The figures

the sector (4 < A). Despite the fact that the external loads are not high in this case, the stress field is highly nonuniform. To evaluate the stress maxima, we shall examine the most unfavorable case: compression of a ring with two half yokes with a load at which the gap between the ring and the half yoke in the section ~ = 0 is overlapped. We shall use the rod model of the ring compressed with two radial concentrated in the sections ~ = • The radial displacements of the points at the horizontal cal diameters are

forces Q and verti-

u~(o)= zJQ~ ( ~2 21) ," u~(n/2)=- E~---Tq---~ (f=lA(q+r2);

I=ha[12; h=r~-rl).

261

GPa

1'501 1.25l_

/]

/

i1~%o '10z

1,00!

0,88 0,92,.,.,.~ i /

I

3 2 1

/

0,75'

1 , 0 0 ~

0,50 9

0,75

\

025 j/

f-

I /

-'(~cp

.

,

5. rz - r~

0.50I/" o 025 0.50 0,75 ~p0

0

I

I

A/A. "~

J

0,85 0.90 0,95 ~,00 1,05

Fig. 4

Fig. 5

Fig. 4. Distribution of the circumferential normal stresses in the section (P = ~/2. Tile figures at the curves are the values of A/A,. Fig. 5. Dependences of the stresses in the ring on displacement i) o~(~/2); 2) max o~; 3) max ~r~"

of the half yoke:

For the above values of the ring parameters, contact in the section ~ = 0 takes Dlace at a value Ur(~/2 ) near (ro -- r2). Actually, at Ur(~/2) = ro -- r2, Ur(0) = 0.149r2 = 0.796A,. This corresponds to the position of the edges of the die at ~ : -+87.9~ . Thus, the use of the system of the ring compressed with two concentrated forces in the sections -+~/2 for evaluation of the stresses is justified. Assuming that Ur(~/2) ~ ro -- r2, the following equation for the magnitude of the force is obtained from the above equations: 9

4~

(r2-ro) EJ

~- 8

fa

"

The maximum values of the bending moment and the lateral force are respectively equal to 2Q~/~ and Q/2. Assuming on the basis of Eqs. (i) that (r2 -- ro)/r = ~$/Em, the conditions of the strength with respect to the circumferential normal and tangential stresses give the estimate

--~min

f

.

4

'

,.

4H~-

;

~<,

4H~-

For example, for a glass-fiber-reinforced plastic, the strength condition with resoect to ~gives h/r ~ 0 . 4 7 0 , and with respect to ~r~: h/r <~0.213. Thus, the modified methods can even be used for N = 2 for very thick rings. It should be mentioned that the dependences of the load exerted on t h e s e c t o r on its displacement in the given methods are highly nonlinear: they are flat and "soft" in the stage A < A, and steep, almost linear in the stage & > A (coincide with the curve 1 in Fig. 5). CONCLUSIONS The modification of the methods of testing strength in the circumferential direction in the tensile and compressive loading of the filament-wound composite has been proposed and theoretically substantiated. LITERATURE CITED i.

V. V. Boiotin and Yu. N. Novichkov, Mechanics

of Multi-ply Structures [in Russian], Moscow

(1980). 2. 3. 4.

A. K. Malmeister, V. P. Tamuzh, and g. A. Teters, Strength of Stiff Polymer Materials Russian], Riga (1972). Composite Materials [in Russian], Vols. 1-8, Moscow (1978). Yu. M. Tarnopol'skii and T. Ya. Kintsis, Methods of Static Tests [in Russian], Moscow

(1981).

262

[in

5. 6. 7. 8. 9.

i0.

V. P. Nikolaev, "Testing rings of glass-fiber-reinforced plastics using stiff sectors~ ~ Mekh. Polim., No. 6, 1132-1134 (1973). V, V. Partsevskii, "Stresses in an anisotropic ring in tensile loading with stiff sectors," Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, No. i, 90-92 (1971). V. V. Partsevskii and A. Ya. Gol'dman, "Mechanical tests of ring specimens made of glass-fiber-reinforced plastics," Tr. Mosk. Energ. Inst., No. 74, 125-128 (1970)~ B. W. Rosen and N. F. Dow, "Mechanics of failure of fibrous composites," in: Fracture (H. Liebowitz, ed.), Academic Press, London (1971), pp. 612-672. V. D. Popov, V. N. Nikolaev, N. F. Savel'eva, A. V. Lavrov, and A. K. Sborovskii, Methods of Evaluating the Strength of Glass-Fiber-Reinforced Plastic Produced by Winding [in Russian], Leningrad (1977). V. V. Partsevskii, "Substantiation of the method of testing anisotropic ring specimens, '~ Mekh. Polim., No. i, 171-174 (1972).

EXAMINATION OF ANISOTROPIC PROBLEMS OF THE MECHANICS OF DEFORMED BODIES BY THE HOLOGRAPHIC MOIRE METHOD V. A. Zhilkin, V. B. Zinov'ev, and T. V. Gorbunova

UDC 620.17;678.067

Regardless of the rapid development of numerical methods, especially the finite-element method, there is still a sufficiently large group of the problems of mechanics of deformed media where the experimental methods represent the only possible or simplest solution: determination of the physical characteristics of real materials, problems of fracture mechanics, nonlinear problems, etc. The experimental solution of anisotropic problems by the photoelasticity method on optically transparent models is restricted by the unavailability of transparent anisotropic or crystalline materials with the required combination of the mechanical Droperties [i], and the method of photoelastic coatings can be used only at specific ratios between the minimum stiffness of the specimen and the stiffness of the coating [2]. The construction of highly coherent light sources (lasers) facilitated the development of highly sensitive methods based on the holographic methods of recording information: the method of holographic interferometry, speckle interferometry, the holographic moir@ method, which do not impose any special requirements on the properties of the material of examined objects~ and the process of conversion of the mechanical quantities (displacements) into the optical signals in these methods is not associated with the distortion of the measured parameter. In this article we shall discuss the simplest method (as far as the conditions of technical realization are concerned), i.e., the holographic moir@ method, whose principle may be described as follows. A high-density (of the order of I000 lines/mm) metallized grid is deposited in the examined area of the specimen [3]. (The amount of labor required to produce the metallfzed grids and deposit them on the surface of the component is not greater than that required for pres ing and bonding photoelastic coatings.) After placing a high-resolution medium in front of this region of the component, the system described in [4] is use@ to record the holographic interferogram in the opposite beams; the interferogram makes it possible to compare the copies of the undeformed and deformed grids in diffraction orders with a high degree of accuracy. These interferograms are reproduced not only in reflection but also in transmission [5] and make it possible to obtain, from a single interferogram, interference images characteristic of the above-mentioned methods. The simultaneous registration and decoding of these images increase the reliability of the investigations. All the subsequent considerations will be made on the assumption that the unit vectors of the normals to the wave fronts of the illuminating wave and the wave reflected from the surface of the examined object are distributed in the yOz plane. (The relationships presented in the article can be easily extended to the general case.)

.... Novosibirsk Institute of Railroad Transport Engineers. Translated from Mekhanika Kompozitnykh Materialov, No. 2, pp. 341-347, March-April, 1983. Original article submitted August 13, 1982. 0191-5665/83/1902-0263507.50

9 1983 Plenum Publishing Corporation

263