Fracture mechanics of unidirectional fibrous composites with metal matrix under compression

Theoretical and Applied Fracture Mechanics 3 (1985) 151-155 North-Holland

151

FRACTURE MECHANICS OF UNIDIRECTIONAL FIBROUS C O M P O S I T E S W I T H METAL MATRIX UNDER C O M P R E S S I O N A.N. GUZ and M.A. C H E R E V K O ln~tltute , f Mechanws oJ the Academy of Scwnces of the Ul~ramian SSR. Kwc. USSR

l w o different approaches are used to evaluate the critical loads of the unidirectional fiber composites. The 5 are based on the three-dimensional linearized elasticity theory. The constituents of the composite are assumed to have elastoplastic behavior. In the first approach, the composite is assumed to be homogeneous and orthotropic at the continuum level while the second approach assumes piecewise homogeneity where the fiber and matrix interaction at the interfaces are accounted for. For different ratios of the fiber and matrix moduli, critical loads and deformations are obtained and compared with experimental values.

!. Introduction When unidirectional composites are subjected to compressive loading along the direction of reinforcement, one of the failure mechanisms is the stability loss of the structure elements, namely the fibers. This is characteristic of composites with a high compliance matrix which behaves in a elastic-plastic fashion such as metals. Theoretical solution of unidirectional composites in compression has been obtained using the bar model with lateral constraint. [1.2]. Results based on energy methods can also be found in [3]. Subsequent studies used the linear elasticity theory for the matrix and the fibers were modelled as bars [4]. In [5,6], linearized three-dimensional theory of elastic stability were used. Plasticity of the composite was accounted for in [7]. The present study considers two approaches. The first assumes the composite is globally homogeneous and orthotropic. The second approach distinguishes the difference in moduli between the fiber and matrix which are individually homogeneous. These approaches have been successfully applied to composites by assuming both elastic and elastic-plastic behavior of the constituents [8-14].

coincide with 0x 3 axis. The three-dimensional linearized equations of stability will be used [7.15] when the subcritical state is taken to be geometrically linear. The theory of small elastic-plastic deformations [16] is applied to the fiber and matrix prevails only in the critical state and not in the analysis of stability loss. Since the compressive loading is unidirectional, the fibers and the matrix are assumed to deform only in the 0x~ direction, i.e., e = E , , , . As was mentioned earlier, in this model global homogeneity and orthotropy are invoked. Before fracture initiation, stress state is uniaxial: ot I> = -q8,~8,3, l

(1)

where q is the load and 8,i the Kronecker delta. The stress state will be disturbed as the fracture process proceeds. Under static loading, the system of governing equations into [7] are: L,,,lu ~ = O, a ~L,,,, :

a'+ (1 -

)<<,,,>

+(1 - a , , , ) < o , , , , > a , , , - z T - a , , , , q ax7

2. Homogeneous and Orthotropic Model Consider a composite that consists of a matrix and parallel fibers with radius R. A cartesian coordinate system is introduced such that the fibers

a.v,,,ax ,

(2) ax 3

in which equation (1) has been used. At the initial stages of fracture, microvolume changes result in changes in the macrovolume. These changes affect the properties locally and globally independent of boundary conditions. The

0167-8442/85/$3.30 '~ 1985. Elsevier Science Publishers B.V. (North-Holland)

A.N. Guz. M.A, Cherevko / l')'acture mechamc.s ol untdtrectumal hhrous cornposttes wtth metal matr*a un&>r ~ompres.~#on

152

disturbances of the system are governed bv equations (2). Fracture initiation is associated with the nontrivial solutions of the system of equations (2) while the solution for the homogeneous stress state coincides with the initial state given by equation (1). From these considerations, only solutions of the hyperbolic type of equation (2) are of interest: o,,., = (G,,>,

(3t

where GL~ is the shear modulus and %., the fracture load. Due to structure defects, the local shear modulus is lower than the mean value, and hence equation (3) can be written as o.,,,<~ ( G L ~ )

orolm,=min{(Gl~)}.

4- ~ ( / ) ; 4- ~ ( m ) ~ ( , n l

13 ~.__Z'i_°__ L_'_ ~.--17--C-° _ _ ~ , , ,

"~'(_~f),¢', m U]3

J_/1 T [i

d

<°"m<

a_ , ¢ ~ ( / ) ~ / ' 2 ( m ) T ,J 1"113

~13

S(,,,,)

c(r)t~

G'~ ~,

w ' 1 3 ~,* q-

+

"~{h.,

.4' .... g( .... .

In ~

in which S(/~ and S ~'''> stand, respectively, for the volume fraction of fiber and matrix. Equation (7) gives the deformation limits for zero approximation and equation (8) corresponds to the first approximation. In both cases the load is determined from the relation ' oj ....

-

A ¢'''~ 5' ....

~l ....

f;

~-S:7 7

(4)

By including the elastic-plastic behavior of the constituents, the normalized shear modulus ( G ~ ; is calculated at the moment of stability loss. Making use of the bilateral Hashin-Strickmann relation, the theoretical strength limit are obtained: G(f)(I

,I

(5)

13

3. Piecewise homogenous model I-or composites with a low fiber volume fraction. the interaction between fibers has an insignificant effect on the overal composite strength and can be neglected. In this study, the stability of one circular fiber in a matrix is analyzed. The fiber axis is located in a cylindrical coordinate system (r. % .v3) such that the 0x~ axis coincides with the fiber. An asterisk will be used to denote fibre quantities in the deformed state. The matrix is assumed to obey the theory of small elastic-plastic deformations and the hardening law given by equation (6) and the subcritical state:

where S denotes volume content and the subscripts ( f ) and (m) refer to the fiber and matrix, respectively. Assume that the matrix material behaves according to the hardening power law

{ r z -k ( q 9

%0=A( %0)~

The displacement field is governed b\

(6)

in the initial state with the stress intensity o,I) and 0 Thus, for elastic fiber deformation intensity %. with modulus of elasticity E >> 1, the following approximations are obtained: 1 + S (f~ A (''~ 3S~.,)S(r) [

1 A ( ' ' t S ~'''~

E

2 1

73S(1)

~< (o

E

E

E

S(,),

10)

)

3u, +~-=0

11)

Both the matrix and fiber should satisfy the equations [15]: . . . . 0,

(12)

such that

1 A ('') S ( ' ' i 2

') -"' ~. = 0.

3u, l(0zq 3t--7 + ~+u~

W,t

S~

A ("')

S ' ....

lim~

"

(7)

t " = ~' ,,,l~W,,Ul~+ ~ ' " g / I ~

(i3)

and applies to the matrix and _(o) + k ( tlim

m)

( 1 + s(/' 33 t.))e

((o) } lira ,

1 +2di°2, A ,) S ( , , ) l.(o) tirn

o)) + k•( . , ) ( 7 ~1 S ( " ) : _ ( (lim

t " = oy " " W,,Ul~

(14)

to the fiber. Complete adhesion is assumed between the fiber and matrix. Therefore, continuity of displacements and stresses in the initial and deformed state is

A.N. Gu:, M.A. Chereuko / Fracture mechanics of unidirectional fibrous composites with metal matrtx under compression

in which

invoked:

."=, ,,'."

(15)

,,2 = 4 7 ; t ~' = t ~'*,

u' = u "

1 Ott,I = ~Oq,q. .

((,r r : ( qI)q "

(16)

The govering equations can be related to two potentials q, and X [15]: J - sf-~

O.v~

de&. dq03 t _P(r)=

~l; el2" ~3" P;

dr

dr

I'"

for r = R.

For a hollow fiber with an inner radius R~ whose surface is free of tractions, it is assumed that the initial stress o)~ and stress arising from loading vanish at large distances. Two methods of solution will be used. The initial state in the first method is taken to be homogeneous while nonhomogeneity is accounted for in the second by application of a numerical procedure. If the initial state of the fiber and tnatrix is homogeneous, then

,

153

.i~*(r)={~b," ~b2" ~3; d ~ t d~ d~ I dr ' dr ' dr ]

The quantities A ( r ) and A * ( r ) stand for coefficients in the matrices given by equations (21). Numerical evaluation of equation (21) results in a system of homogeneous algebraic equations. For a non-trivial solution, the determinant of this system equations must vanish. This leads to a characteristic equation that determines c the minimum value of which gives the critical strain ~,~,. For the case of incompressible elastic fiber, k~/~ = 1. The same relations may be used if the fiber were elastic-plastic.

4. D i s c u s s i o n

~=0:

where ~ is the Laplace operator. On the other hand. if the initial state is nonhomogeneous, the initial state is first solved. Then the matrix displacements are u,. = q'l( r ) cos ~ COS o~.v:~,

(22)

of results

Equations (7) to (9) give the limiting values of the strains and stresses. Results corresponding to the upper limits are presented in Fig. 1. The matrix is assumed to be elastic-plastic and fibers to be elastic. Dots refer to the limiting values of the stresses and open circles to the strains. These values are taken from [17] whose accuracies were improved in [10]. These are shown by the solid lines in Fig. 1 for the limiting stresses; and dotted

uq = q>:(r) sin ~ cos c~.x~. J

u~' = ~ ( r ) cos q~ sin a x , .

I

I

I

J

(18)

The quantity p as in equation (13) is cos q~ cos a x 3.

p = p(r)

(19)

The corresponding fibre displacement components in the deformed state are u,* = ~]( r ) cos eg cos a.x 3 , u~ = 4 : ( r )

sin ~ cos o<~:~,

u,* = ~ 3 ( r ) cos ~ sin a.v3.

(20)

F r o m equations (12) a system of ordinary differential equations is obtained: d.P(r) dr

O

A(r)~(r);

r> R,

-A*(r)P*(r);

R > r>.> R],

s

•

.~

ID

i5

Sq~

dP*(r) dr

i z

(21}

20

25

! 30

,35

o..

Fig. 1. Values of olin. and el,m as a function of fiber volume fraction.

154

A . N . Guz. M . A . Cherevko / Fracture mechantcs o/ unidtrectional fibrous composites with m e t a l m a t r i x u~td~r ~ompre;sh:m

line in Fig. 2 have been obtained ior R~_ and 10R~ It is shown that when R~ increases to 30R, the results change no more than 5% 1121. I,imit loads may be calculated and compared to the experimental values. For io,x fiber volume fraction, results of the critical strains are shown in Figure 3 for the matrix and fiber with properties mentioned above. Determined critical strains are 5.7~, 5.1% and 4.8% and the}' ~tre similar l() the experimental values in [17] for S " = 4.1~i: Figure 2 shows that the effect of inhomogeneit} has insignificant effect on the critical loads. For small hole radii with R 1 < 0,2R, the fiber may be assumed to be solid. This resutt~ in an error not exceeding 5%. For higher value> ;:,f R~, the values of critical strain can differ greativ from that for a solid fiber. When the influence ~)f plasticity of the fiber is accounted for, fracture occurred at considerably lower strain in contrast )(~ the case of an elastic fiber.

Table 1 Value of parameters taken from [17-19] Curve No.

Parameters A ( x 107 Pa)

/,

to 10 6.8

o.i 0.25 0.25

~[]81 2 [17] 3 119]

lines for the strains plotted as a function of the fiber volume fraction S (/). For the fiber, it was assumed that ~,=0.3 and E = 2 × 1 0 1 1 Pa. The three curves labelled 1, 2 and 3 correspond to the values of the parameters shown in Table 1. Numerical results have been obtained for the piecewise homogeneous model in the initial state. The body was approximated by a cylinder of finite radius R~. The center of the fiber was excluded from the region of integration, except for the cylinder with radius 0.02R. Increase of the fiber radius to 0.05R did not substantially affect the results. Values of (lira corresponding to the solid

0,07

J

I

5. Conclusion Use ts made of the three-dimensional linearized stability theory for estimating fracture of unidirec-

t

~=0,t

0,0'}2_22 __

ojee

................................

/

!

o~oL

I, I= i

:L

0,9

,

I

}

]

o,oJ L-

//,,o

gTE~

/

I

o,oe

8, ?5

ct~

o,,o/

o

o2

qe

o,~

o,~

a,5

Fig. 2, Piecewise-homogeneous model: Results exhibiting the influence of inhomogeneity of the initial state ( E / A = 2,000).

3t

Fig. 3. Piecewise-homogeneous model: Strain versus rigidity ratio for different power hardening coefficients.

A.N. Gu-. M.A. Cheret'ko / Fracture mechanics of unidireetional fibrous composttev wtth metal matrix under eompresston

tional fibrous composites assumption

under compression.

of the constituents

elastic-plastic

manner

has

to behave

led to more

The in a n

accurate

evaluation of the fracture load. Solutions obtained neous

from the piecewise-homoge-

model, revealed that inhomogeneity

of the

initial state may be neglected. The hollowness the

f i b e r is i m p o r t a n t

only

for large

radii

of and

plasticity effect has a negligible influence on the limit load.

References [1] B.W. Rosen, "Mechanics of composites strengthening, fiber composite materials", (Ohio, Amer. Soc. Metals, 1965). [21 H. Schuerch, "'Prediction of compressive strength in uniaxial boron fiber-metal matrix composite materials". AIAA Journal, 4, Jan. (1966). 102-106. [31 L.B. Greszczuk, "'Microbuckling failure of circular fiberreinforced composites", AIAA Journal. 13, Dec. (1975) 1311-1318. [41 M.A. Sadowsky, S.L. Pu and M.A. Hussain, "Buckling of microfibers", J. of Appl. Mech., 34. Set. E. N 12 (1967) 1011-. 1016. [5] A.N. Guz, "Constructing a stability theory for monodirectional fibrous materials", Prikl. Mech., v. 5, N 2 (1969) 62 70. (in Russian: English Transl.: So,,'. appl. mech., v. 5, N 2 (1969)). [6] A.N. Guz, "'On the determination of the theoretical limit in compression for reinforced materials", Dop. Akad Nauk Ukr. RSR, Ser. A, N 3, (1969) 236-238 (In Ukrainian). [7] A.N. Guz, "Stability of three-dimensional deformable bodies", (Naukova Dumka, Kiev, I971, in Russian).

155

[8] l.Yu. Babich, "'On the stability of a fiber in a matrix under small deformations", Prikl. Mech. v.9. N 4 (1973) 29-35. (in Russian; English Transl.: Sov. appl. mech., v. 9, N 4 (1973) 370-375). [9] A.N. Guz, '" Mechanics of composites-material failure under axial compression (plastic failure)", Prikl. Mech.. `,'. 18, N 11 (1982) 21-29 (in Russian: English Transl.: So',. appl. mech., v. 18, N 11 (1982) 970-977). [10] A.N. Guz, "'Continuum theory of fracture in the compression of composite materials with a metallic matrix", Prikl. Mech.. v. 18, N 12 (1982) 3-11 (in Russian: English Transl.: Soy. appl. mech., v. 18, N 12 (1982) 1045-1053). [11] A.N. Guz, ""Fracture of unidirectional composite materials in axial compression". In: Strength and Fracture o/ ComposHe Mater,als (Riga (1983) 284-292. in Russianl. [12] A.N. Guz, M.A. Cherevko, " O n fracture of unidirectional fibrous composite with elastic-plastic matrix in compression", Mech. compos, mater. N ,5 (1982) 987-994 (in Russian). [13] M.A. Cherevko, "'Stability of fibre in elastic-plastic matrix", Dokl. Akad. nauk Ukr. SSR, Set. A,, N 9 (1982) 43-46 (in Russian). [14] M.A. Cherevko, "'Stability of hollow fibre in elastic-plastic matrix", Dokl. Akad. Nauk Ukr. SSR, Ser. A, N 11 (1982) 35-38 (in Russian). [15] A.N. Guz, "Principles of the stabilit,, theor,, of mine workings", (Naukova Dumka, Kiev. 1977, in Russian). [16] A.A. llyushin, "'Plasticity, part. 1.", Elastic-plastic deformations, (Moscow-Leningrad, 1948, in Russian). [17] M.P. Pinnel, A. Lowley, "'Correlation of uniaxial yielding and substructure in aluminum-stainless steel composite". Met. trans, v. I, N 5 (1970) 1337-1348. [18] A.P. Smeryagin, N.A. Smeryagina. A.V. Belowa. Industrial Nonferrous Metals" and Alloys Refereme Book, (Metallurgiya, Moscow, 1974). [19] R.W.K. Honeycombe, The Plastte Dqformatton o[ Metal,~. (Edward Arnold, 1968).