fracture mechanics of syntactic foam composites

~le experimental and theoretical tensile diagrams of the P~ for the partial cases of parallel and perpendicular packing are compared in Fig. 6.

~le proposed method of constructing the tensile diagrams can be used for strength calculations of components made of P~Is working in uniaxial strain state conditions, e.g., long thin-wall pipes (L>> D) loaded with internal pressure.

2o

3o

LITERATURE CITED

L. S. Shmelev, Yu. I. Sinel'nikov, N. I. Haturin, et al., "Experience with the rolling of porous steel sheets," Stal', No. i, 47 (1979). A. F. Tret'yakov, Yu. I. Sinel'nikov, A. G. Kolesnikov, and G. P. Polushkin, "A method of calculating the ultimate strength of porous net materials," Probl. Prochn., No. 12, 55-59 (1977). Yu~ I. Sinel'nikov, V. I. Hakarochkin, and A. G. Kolesnikov, "Experimental examination of stretching and porosity of a fibrous material in longitudinal rolling," Poroshk. Metall.~ No. ii, 6-9 (1974).

FRACTURE IfECHANICS OF SYNTACTIC FOAH CO~fPOSITES

P~ G. Krzhechkovskii UDC 539.4.015

l~le svntactic foam composite consists of polymer binders and hollow spherical fillers. As a result of its low density and comparatively high mechanical properties, the composite is used on an increasing scale in various areas of industry [i, 2].

In this report, the author proposes a variant of fracture mechanics of the syntactic foam composite based on the combined examination of the strength of the structural constit- uents of the composite in an arbitrary complicated stress state using the theory of scattered fracture. Previously, the problem was examined in a slightly different formulation for uniaxial tensile and compressive strains [3, 4].

We shall examine a composite material consisting of a polymer matrix with hollow thin spherical inclusions distributed at random in the matrix. The geometrical parameters of the spherical inclusions are identical~ A certain characteristic volume in which the number of inclusions is sufficiently large for averaging will be selected inside the composite. A system of forces characterized by the stress tensor o* is applied to the surface of this volume. Inside the volume, we select a single spherical inclusion whose center represents the origin of two coordinate svstems (Fig. i). The axes of the Cartesian coordinate system coincide with the direction of the main axes of the tensor 0*, and the spherical coordinates (R, e, ~) determined the position of the points at the interface between the matrix and the inclusion.

It is assumed that reinforcement of the composite is of random nature and, consequently, the stress state at the inclusion is determined by smoothed stresses characterized by tensor os which has the following form in the method of the self-congruent field:

~ = ~,o*. (i)

Here ~, is the tensor stress concentration coefficient determined from the equation [5]

cz, = L~ (L~, -f- L,) [L, (Ls --t-" L~,)] -~, (2)

where L,~ L s are the tensors of the elastic moduli of the composite and the inclusion; LI, is Hill's tensor [6]. The product of the tensors represents the convolution of the vectors in respect of tile internal indices.

After expanding the tensors included in Eq. (2) into volume and deviator components and carrying out the convolution operation, we can find the coefficients of stress concentrations at the inclusion for all-around uniform compression (tension) Gv and for the deviator stress state ~d:

Nikolaevsk Shipbuilding Institute. Translated from Problemy Prochnosti, No. ii, pp. iI0-i15, November, 19823 Original article submitted November 14, 1981.

1556 0039-2316/82/1411-1556507.50 �9 1983 Plenum Publishing Corporation

where

tta-' Z;'/' f /' f /~:7~..

Fig. 1

Fig. i.

Fig. 2.

G2

c 5t'-n..5~c E

Fig. 2

Calculation model of the syntactic foam composite.

Limiting curve of fracture of the filler (glass).

(I + %) .

~% -? K,

h (1 + ~ ) O~d--

gt =050c

(3)

(4)

m

3K,~ 290 -1- 3

K , , G, a r e t h e v o l u m e c o m p r e s s i o n and s h e a r m o d u l i o f t h e c o m p o s i t e ; Km, Gm a r e t h e same f o r the matrix material; Ks, Gs are the conventional effective characteristics of the hollow spherical shell [2].

It is assumed that the components of the tensor o* have the form

0* i] = p u (do not add up), (5)

w h e r e ~ i j ~ [ - - 1 , 1 ] a r e some n u m e r i c a l c o e f f i c i e n t s ; p i s t h e p a r a m e t e r o f e x t e r n a l l o a d i n g ; 6ij is Kronecker's symbol.

Tensor o* will be expanded into volume and deviator components

o* = 3o(~ 2s(~

where

1 (0) _ o* -- o(~ (6) o~) = -5- o~h6~j; % ~/ ~: ,

V, D a r e t h e c o m p o n e n t s o f t h e u n i t t e t r a v a l e n t t e n s o r [ 6 ] .

T a k i n g i n t o a c c o u n t Eqs . ( 5 ) , t h e c o m p o n e n t s o f t e n s o r o* may b e w r i t t e n i n t h e f o r m

( ' ) 1 (o) = p6~j v i i -- -~ vhg

(do not add up with respect to i, j). (7)

~le stress tensor at the inclusion will also be represented in the form of the volume o(~) and deviator s(I) components

d = 3oCl)v + 2stuD.

The f o r c e s d e t e r m i n e d by t h e t e n s o r o s a t a n y p o i n t o f t h e s u r f a c e o f t h e s p h e r i c a l i n c l u s i o n a r e e q u a l t o

(8)

1557

0~}) : a~o!~)ni; s~} ) = ~dS~y)n7 (do not add up) (9)

whe re n j i s t h e c o m p o n e n t o f t h e u n i t v e c t o r o f t h e n o r m a l t o t h e s u r f a c e o f t h e i n c l u s i o n .

U s i n g t h e membrane t h e o r y o f t h i n s h e l l s , i t i s p o s s i b l e t o d e t e r m i n e t h e m a i n s t r e s s e s O , ( s ) and ~ a ( s ) i n any e l e m e n t o f t h e s h e l l w i t h t h e c o o r d i n a t e s 8 , 9 ( s e e F i g . l ) a s a f u n c - t i o n of the external loading parameter p. The main stresses in the element of the matrix at the interface with the inclusion are determined from the following conditions:

els) = e[=); e(]) = e~'~); %(,,,) - 9a~p, (i0)

where s l, ~2 are the relative strains of the matrix m and of the inclusion s at the interface

between them; $ =(I/3)Vkk.

Formulation of the Fracture Condition of a Composite Cell

The formulation of the fracture condition of the element as a whole on the basis of the known stress state of points of the element is the most complicated problem. Two main reasons

for fracture of the syntactic foam composite should be mentioned:

i) In compressive stress states, fracture takes place mainly as a result of the initial crushing of hollow glass microspheres~

2) In tensile stress states, fracture is caused mainly by a disruption of the adhesion

bond between the matrix and the filler.

It is evident that in cases in which both compressive and tensile stress states form in the points of the element of the composite, both factors leading to fracture of the ele-

ment are likely to operate.

The simplest method of solution is to carry out calculations for the critical point. However, this approach does not result in agreement between theory and experiment even in

calculations for uniaxial tensile loading [4] This indicates that the nature of fracture of the given material is more complicated.

Scattered fracture has often been taken into account in present-day calculations of

fracture and long-term strength of structures. This type of fracture is characterized bv the uniform accumulation of defects in the material which leads in the end to fracture of the entire element [7]. The degree of fracture is usually represented by a certain parameter of the state of damage ~ which varies within a specific range during fracture. The critical value of this parameter at a specific point or volume is regarded as the fracture criterion.

l]le scatter function of the state of damage ~(8, ~) will be related to the spherical surface of the interface between the matrix and the inclusion, taking into account the iso- tropy of the properties of the materials of the matrix and the inclusion and also the spheri- city of the cell. The invariant scalar characteristic of the state of damage will be represented by its mean value on the surface of the sphere. The critical mean value is equal to unity

As a result, the fracture condition of the element may be written in the form

I <II) ----- - ~ i" I] (0, q)) dS -- 1. (11)

Condition (ii) takes into account the local nature of damage at the points in the vicin-

ity of the point subjected to the highest stresses, and also the scale factor peculiar to the strength characteristics of brittle materials [8, 9]. To determine the function of the state of damage, it is necessary to determine an additional kinetic condition of fracture at the point. The invariant characteristics of the stress state of the composite at the interface will be represented by the stress intensities in the matrix and the inclusion

I

I

0 - 2 _ _ ~ = [~ q- 9(s) 0.1(s)0.:(s)] 2 . (12)

1558

It is assumed that the equations of the limiting curve of strength of the material of the inclusions in the plane stress state fs(O~(s), O2(s)) =0 and also the equation of the limiting strength surface of the material of the matrix fm(O~(m), O2(m), O3(m)) =0 which must take into account the possibility of fracture of adhesion bonds, are known. Conse- quently, for given ratios between the actual stresses in the matrix and the inclusion induced by external loading, the equations for the limiting fracture surfaces can be used to determine the corresponding limiting fracture stresses in the matrix ~j.(m) and the inclusion oj.(s) at any point of the interface. The function of the state of damage at this point is represented by the maximum value obtained from the following two equations for the state of damage of the matrix and the inclusion

where

FI (0, q)) = max {Ha, II,}, (13)

[ 4" l". [ 1".

ai.(m), ~i.(s) are the intensities of the limiting fracture stresses in the matrix and the inclusion; n and q are the exponents of the degree of nonlinearity.

As reported in [7], the accurate physical meaning of the parameter of the state of damage cannot always be determined unambiguously, it may vary depending on the method of indication. In the present case, function K, introduced by means of Eqs. (13), characterizes the intensity of the stress state at an arbitrary point of the external surface of the sphere and may assume values larger or smaller than unity, i.e., in the first case the given point is overstressed, in the second case it is insufficiently stressed.

s is assumed that in loading defined by Eq. (5), the main stresses in the matrix and the inclusion at a specific point with the coordinates 0, ~p are determined

-- 6 (]---- I, 2, 3); (14)

O/( s ) = c(,(/) PR �9 ~ q = 1, 2),

where aj (m) and ~j (s) are the numerical coefficients which depend on the running coordinates.

Let it be the case that qm=max{iaj(m)]} and ~s =max{iaj(s)]} �9

After substituting Eqs (14) into the equations of the limiting fracture surfaces of the materials of the matrix and the inclusion, we can determine the stresses oj. (m) and ~j.(s) for given values of 0 and

where ~o (m), Co (s) are the roots of the equations fm(oj(m)) =0 and fs(gj(s)) =0.

Substituting (14) and (15) into Eqs (13) and taking into account Eq. (12), we obtain

{ t3R]"{ ~im ~n. /pR kq( ~l, ]q (16) n., =t oo / _ t <P., ] , n, tw)

where ~m=oo(m) l~c, ~s =~o(S)l~c; Co is the ultimate strength of the material of the inclusions in unaxial compression. In subsequent considerations, it will be assumed that n and q are equal.

max! ~lm p(s) = / (17) (Pro ' (P8 "

We shall denote

Consequently, the strength condition of the cell of the composite (ii) may be written

in the form

T ) d S = 1. S

(18)

1559

~, I/1-_2 ~-u V~ rz

v

"~0 -I.35 -0.90 -a#5 0 ~o gg

Fig. 3. Theoretical diagram of limiting states for the syntactic foam composite: I) hydrostatic compression; II) biaxial compression; III) uniaxial compression; IV) shearing; V) uniaxial tension~ VI) biaxial tension; VII) hydrostatic tension i) ~o =--i; 2) ~ =+i~ �9 and A, experimental data from [Ii] and [12]).

The parameter of external loading is determined from Eq. (18)

I l

--~- S" Isr" [3--- (19)

= I p~ where Isr (S)dS. integration is carried out on the surface of the hemisphere. s

Knowing the value of p, the limiting values of the components of tensor o* are determined from Eq. (7).

~ms, the determination of the limiting load for the syntactic foam composites for an arbitrary stress state and for q =n was reduced to the computation of the surface integral Isr of the function of the state of damage determined on the surface of the interface between the matrix and the inclusion. The value of Isr can be calculated by numerical methods using a computer~

Construction of the Limiting Curve of Fracture of

Glass in the Plane Stress State

To construct the function of the state of damage of a hollow spherical inclusion, it is essential to have the limiting curve of fracture of the material of the inclusion in the plane stress state. The filler used in the syntactic foam composite is represented by thin, hollow glass microspheres with a diameter of 3-150 ~m and a thickness of 1-2 ~m. Regardless of the fact that glass is used extensively in industry and its properties have been studied sufficiently, no exhaustive universal theory which would make it possible to clarify all the special features of fracture of glass is available. Glass is characterized by a very wide range of strength values, a large scatter of experimental data, and, most importantly, the existence of a scale factor [8, 9]. The experiments aimed at determining the strength of high- and low-strength glasses in biaxial compression carried out at the Institute of Prob- lems of Strength of the Academy of Sciences of the Ukrainian SSR have shown that the theory of the highest normal stresses can be applied to compressive stress states. The applicabil- ity of this theory for low-strength glasses in the region of tensile-compressive stresses has been confirmed experimentally in [13, 14], and the suitability of the theory for high- strength glass fibers has been established in [15].

The direct experimental determination of the ultimate compressive strength of silicate glass used for the production of microspheres is very difficult. However, using the results published in [i0], the value of ~c can be determined for a specific experimental value of the ultimate hydrostatic strength of the syntactic foam composite. Taking into account the possibility of formation of initial thermal and shrinkage stresses in the inclusion [16], the Oc value equal to (10-12).102 MPa is in complete agreement with the experimental data obtained for silicate glasses [8].

1560

\I ___--J i

t I 0

y 02 ~/~u -0,5 -0,2 02 R5 ~]P

Fig. 5

0,~ ~ ~ _

-1,0

~7 ~ o U. , I

021 ~ i 7 - ~

-0.2

-O.e

ft. 1 -~.o -~8 -0.~ -04 -a2

Fig. 4

Fig. 4. Theoretical fracture curve of the syntactic foam composite in the plane stress state: - and A) experimental data taken from [ii] and [12].

Fig. 5. Theoretical dependence of the effect of lateral pressure on the strength of the syntactic foam composite in axial compression (I) and axial tension (2).

The ultimate strength of glass in uniaxial tensile loading a t can be determined approxi- mately using the diagram of the dependence of the theoretical strength of glass on the crack dimensions [17]. Taking into account the fact that the thickness of the glass microsphere used for the manufacture of syntactic foam composites is 1-2 ~m and assuming that the microspheres contain initial microcracks, the tensile strength of glass estimated using this diagram can be taken as equal to (5-6).10 2 }~a The diagram of the limiting stresses for glass used in subsequent calculations is shown in Fig. 2.

Selection of the Criterion of Adhesion Strength

The construction of the limiting surface of fracture of the material of the matrix which is in contact with the inclusion is an independent complicated problem because of the vague physicochemical nature of the adhesion forces. In addition to this, the adhesion strength of a modelling system measured by any method cannot reflect the actual value of the strength of bonding of the filler with the binder since the adhesion strength always reflects other factors whose effects in the model and in the real material differ. Consequently, adhesion strength varies within a very wide range depending on specimen geometry and the test conditions, irrespective of the identical nature of adhesion interaction [18].

One of the initial investigations into this problem was carried out by Skudra and Kirulis [19]. They determined the strength of the contact layer using a strength criterion in which the relationship between the octahedral stresses was assumed to be linear. The constants included in the equation of the limiting surface were determined in pulling and shearing tests. The case in which both contacting surfaces are planar and the contact layer can fracture under the effect of both tensile and tangential stresses was examined. In the syntactic foam composite, the stress state in the contact layer is three-dimensional, even in the case of the simplest uniaxial loading. In addition to this, in selecting the strength criterion it is necessary to take into account the fact that the criterion can be used for compressive stress states to construct the function of the state of damage of the matrix, and also the experimentally determined fact according to which the adhesion strength in pulling is lower than in shearing. Consequently, the adhesion strength criterion is represented by Balandin's criterion [20] which leads to the following strength condition:

~ + ~2 + ~ __ l(m) 2(m) 3(m) ~l(m)O~(m) -- ~2(m)~3(m)

where ac(m), at(m) are the ultimate strengths of the adhesion layer in compression and pull-

ing.

1561

To describe more accurately the limiting surface of fracture in the region of the combined action of the tensile and compressive stresses, one of the experimental constants will be represented by the ultimate adhesion strength in shearing for the epoxy binder--glass pair. According to [21, 22], this strength is equal to 26 MPa. The ultimate compression strength Oc(m) is assumed to be equal to the limit of induced elasticity of the binder (89 MPa) [4]. The ultimate strength in uniaxial tensile loading Ot(m) is linked with the values of Tc(m) and Oc(m) by the relationship resulting from criterion (20)

2 3"~c(rn) ( 21 ) O't(m) -- Oe(m )

Construction of the Limiting Surface of Fracture of

the Syntactic Foam Composite and Analysis of Results

The fracture loads for the syntactic foam composite with a complex stress state were determined by the above-mentioned method in an EC-1020 computer. The integration step in respect of width and meridian was selected equal to 7/80. Thus, the surface of the hemisphere was divided into 3200 elements. Consequently, the error of the results in comparison with the accurate values (e.g., for triaxial tension or compression) was not greater than 2%. The following initial data were used in the calculations: K m =4.15.10 3 )4Pa; ~m =0.37, K s =2.].0 3 MPa; E c =0.65-105 MPa; Vc =0.21, R/6 =26.5; K, =2.8"10 3 }fPa, G, =0.95-10 3 MPa; G s =0.75-I0 s MPa; the nonlinearity parameters n =i.

The results of the numerical calculations of the theoretical surface of fracture of the syntactic foam composite are given in Fig~ 3. The abscissa gives the normal octahedral fracture stresses related to the ultimate strength of the composite in uniform compression ~u, whereas the ordinate gives the corresponding comparative intensities of the normal stresses. Consequently, the diagram shows the meridional sections of the fracture surface combined on the same plane.

As sho~.~n by the results of the calculations, the entire fracture surface of the syntactic foam composite can be regarded as consisting of three zones. The first zone corresponds to the stress states from triaxial to biaxial uniform compression, i.e., 0u~<~o<~_(2/3)O2c. In this zone, fracture of the syntactic foam composite is caused by the loss of strength by the filler in which compressive stresses only form, their limiting values are determined by the straight line BF (Fig~ 2). The limiting fracture surface in this zone has the form of a cone of revolution whose tip is positioned at the point corresponding to uniform compression.

The second zone of the fracture surface extends from biaxial uniform compression to pure shear (2/3 O2c<~o~_0). At these loads the syntactic foam composite fractures as a result of the combined action of compressive and tensile stresses in the filler and also owing to possible disruptions of adhesion strength. The limiting surface in this zone proved to be irreg- ular and depended on the type of stress deviator Zo [20], although this dependence is not strong as a result of the initial strength characteristics of the materials of the inclusions and the matrix considered in the calculations.

Finally, the third zone of the fracture surface includes the stress states with the positive values of the normal octahedral stresses. The loss of strength of the syntactic foam composite in this zone is caused by disruption of adhesion bonding between the matrix and the inclusion. Owing to the fact that the initial fracture surface of the matrix is regular, the calculated fracture surface of the composite is also regular and has the form of a surface of rotation similar to a paraboloid.

For the sake of comparison, Fig. 3 gives the experimental data obtained in the examination of the strength of the syntactic foam composite in a wide range of complicated stress states. The data were taken from [Ii]. Attention should be given to the completely satis- factory agreement between the above-proposed theory and the experimental data. The largest difference is observed for the stress states indicated by the points on the diagram to the right of the point corresponding to uniaxial compression (see Fig. 3). It is evident that the difference is caused by the fact that the strength and adhesion characteristics of the epoxy compound used as the matrix in the syntactic foam composite in [ii] were higher than those considered in the present work.

Figure 4 shows the theoretical curve for the plane stress state of the syntactic foam composite~ For the sake of comparison, the experimental data taken from [ii, 12] are also given o

1562

Figure 5 shows the theoretical dependence of the lateral pressure (compressive or tensile) on the strength of the syntactic foam composite in axial compression and tension. The abscissa gives the ratio of lateral pressure po to axial stress o; p, is the axial fracture stress. As expected, application of the compressive lateral pressure increases the strength of the syntactic foam composite in axial loading whereas loading with the tensile lateral pressure reduces its strength.

In conclusion, it should be mentioned that in the present work the strength of the syntactic foam composite was determined for simple loading because the stresses in the matrix and the inclusion increased in proportion with the parameter of external loading p. The proposed calculation method can also be used to determine the effect of the loading path on the limiting state of the composite.

The author is grateful to A. A. Lebedev for his help and comments.

LITERATURE CITED

i. A. A. Berlin and F. A. Shutov, Hardened Gas-Filled Plastics [in Russian], Khimiya, Moscow (1980).

2. J. A. De Runtz, Jr., "Micromechanics and macromechanics of syntactic foams," in: Pro- ceedings of Conference on Civil Engineering Materials, Vol. i, Southampton, pp. 405-419.

3. C. K. Ko, "Elastic stresses in two-phase composites," in: Proceedings of International Conference on the Mechanical Behavior of Materials, Vol. 5, Kyoto (1971), pp. 19-27.

4. V. A. Telegin, E. M. Filyanov, and E. B Petrilenkova, "Examination of the strength of syntactic foam composites," Mekh. Kompozitn. Mater., No. i, 73-78 (1979).

5. P. G. Krzhechkovskii, "Determination of the effective moduli of composite materials," Mekh. Kompozitn. Mater., No. 6, 995-999 (1981).

6. T. D. Shermerger, Elasticity Theory of Microheterogeneous Media [in Russian], Nauka, Moscow (1977).

7. V. P. Tamuzh and V. S. Kuksenko, Micromechanics of Fracture of Composites [in Russian], Zinatne, Riga (1978).

8. G. S. Pisarenko, K. K. Amel'yanovich, Yu. I. Kozub, et al., Structural Strength of Glasses and Glass Ceramics [in Russian], Naukova Dumka, Kiev (1979).

9. G. M. Bartenev and L. P. Tsepkov, "Scale factor and the strength of glass," Dokl. Akad. Nauk SSSR, 121, No. 2, 260-263 (1958)~

i0. P. G. Krzhechkovskii, "Determination of the elastic and strength properties of composites based on hollow spherical inclusions," Probl. Prochn., No. 3, 37-40 (1979).

ii. J. A. De Runtz, Jr. and O. Hoffman, "Static strength of syntactic foams," Prikl. Mekh., Ser. E., No. 3, 181-186 (1969).

12. P. G. Krzhechkovskii, "A criterion of the strength of composites based on hollow spherical inclusions," Probl. Prochn., No, 2, 68-71 (1978).

13. N. N. Davidenkov and A. N. Stavrogin, "A strength criterion in brittle fracture and the plane stress state," Izv. Akad. Nauk SSSR, Otd. Tekh. Nauk, No. 8, 101-109 (1954).

14. D. E. Vyazun, E. M. Mikhailovskii, and L. M. Sedokov, "Strength of cast iron and glass in the plane stress state," in: Strength and Ductility of Materials [in Russian], Tomsk Polytechnical Institute (1970), pp~ 26-31.

15. W. J. Kroenke, "Mechanical test for anisotropy failure of aluminoborosilicate glass fiber under combined loadings of tension and torsion," J. Am. Ceram. Soc., 49, 508-513 (1966).

16. P. G. Krzhechkovskii, V. A. Naumenko, B. Ya. Artem'ev, and V. A. Nikitin, "Thermal and shrinkage stresses in syntactic foam composites," in: New Methods of Producing Gas- Filled Polymers and Applications [in Russian], Proceedings of the Second All-Union Con- ference, Vladimir (1978), pp. 177-178.

17. C. J. Phillips, "Fracture of glass," in: Fracture, Vol. 7, Academic Press, New York-- London (1972), pp. 2-33.

18. A. A. Berlin and V. E. Basin, Fundamentals of Adhesion of Polymers [in Russian], Khimiya, Moscow (1974).

19. A. M. Skudra and B. A. Kirulis, "A criterion of adhesion strength under the effect of normal and tangential stresses," Mekh. Polim., No. 2, 246-251 (1974).

20. G. S. Pisarenko and A. A. Lebedev, Deformation and Strength of Materials in Comp!icated Stress States [in Russian], Naukova Dumka, Kiev (1976).

21. L. M. Vinogradova, Yu~ V. Zherdev, R. V. Simonenkova, et al., "Measurement of the adhesion of polymers at constant internal stresses in the adhesive--substrate system," Mekh. Polim., No. 2, 270-276 (1974).

1563

22~ Go Do Andreevskaya, High-Strength Oriented Plastics [in Russian], Nauka, Moscow (1966).

EXAMINATION OF THE STRESS STATE OF A HELICALLY

REINFORCED COMPOSITE IN SHEARING

Go Eo Freger UDC 678.067.5:539.3

~ie helically reinforced composite is made of individual preshaped structural units which are produced by winding fibers or threads of the auxiliary reinforcement on the threads or bundles of the main reinforcement. The thickness, physicomechanical properties, and structural parameters of the auxiliary reinforcement generally vary within a wide range. In the manufacture of, for instance, winding components based on such a material, the cross section contains easily visible individual structural components which are circular and consist of the core (main reinforcement fiber) and the layer (auxiliary reinforcement fiber). The latter may be combined and consist of several sections with different properties. The stress state of the model of such a helically reinforced composite for the case in which the layer is quite thin and has isotropic properties was examined in [i].

In the present work, attention is given to the stress state of a helically reinforced composite with an auxiliary layer consisting of two sections with different physicomechanical properties.

We shall examine the model of a unidirectional reinforced material with a small amount of filling subjected to transverse shearing (Fig. i). It is assumed that the cross section of the main reinforcement is circular, the properties of the main reinforcement in the examined section are isotropic in the transverse direction, and the main reinforcement is encir- cled by the auxiliary layer consisting of two parts whose material is assumed to be homo- geneous and isotropic. When a shear load is applied to the matrix, a plane strain state will form in the system. The problem is solved using Kolosol--Muskhelishvili complex poten- tialso

It is assumed that complex potentials are given for the regions 0, I, 2, and 3; at the contours Lo, L~, and L2 these potentials satisfy the boundary conditions for the first and second main problems for the circle [2]

(t) + t~' (t) + ~ (t) = [; (1)

Jt

Fig. 1

\5111 // //,// // // ~ A / / , / / / H / / / / /

Fig. 2

~ 0 . 48

2

/ / / /

/ /

0.20

Fig. I. Calculation model of a helically reinforced composite.

Fig. 2. Dependence of equivalent stress H~o at the boundarv of the first section of the layer with the core on relative stiffness C~ and D.

V0roshilovgrad Engineering Institute. Translated from Problemy Prochnosti, No. ii, pp. i16-i19~ November~ 1982. Original article submitted October 13, 1981.

1564 0039-2316/82/1411-1564507.50 �9 1983 Plenum Publishing Corporation

fracture mechanics of syntactic foam composites

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