Effect of specimen geometry on the toughness of fiber reinforced composites having bridged cracks

Scripta METALLURGICA et MATERIALIA

Vol. 25, pp. 2645-2650, 1991 Printed in the U.S.A.

Pergamon Press plc All rights reserved

I~I~CT OF SPECIMEN GEOMEFRY ON THE TOUGHNESS OF FIBER REINFORCED COMPOSITES H A V I N G

BRIDGED CRACKS

K. S. R a v i c h a n d r a n

WL/MLLN, Materials Directorate, Wright Laboratory Wright Patterson Air Force Base, Ohio 45433-6533, USA

(Received February 7, 1991) (Revised August 23, 1991) I Introduction Damage tolerance and stiffness of ceramics and brittle intermetallics can be improved by reinforcing these materials with high strength and high modulus fibers. A knowledge of the mechanics of fracture of such composites is vital for optimizing properties by controlling processing parameters and for life-prediction in service. Failure modes in composites, governed by mechanisms such as matrix cracking, fiber-matrix interface debonding, fiber fracture, fiber pullout and crack bridging have been extensively modelled [1-4]. The composite toughness has been suggested to be controlled by the fiber-matrix interfacial strength, fiber strength distribution, fiber stress-displacement behavior and matrix fracture toughness. Bridging of cracks by fibers is known [1-5] as a potential toughening mechanism in which the bridging tractions on the crack faces enhance the toughness of composite by partially shielding the (mack tip from applied stress. Toughness enhancement due to crack bridging tractions is generally modelled invoking assumptions of small scale bridging of a large crack in an infinite solid [4,5]. However, realistically, in addition to other factors, the size of the bridging zone under a given far field stress should be sensitive to the specimen geometry and crack length for a given fiber stressdisplacement behavior. Although the assumptions of infinite crack and specimen geometry simplify calculations, consideration of finite crack size and the geometry of the specimen or component is necessary to get reliable toughness levels. Experimental tests have often been done in small specimens and the measured toughness values are likely to be significantly different from the predictions based on infinite crack and geometry approximations. Therefore, a knowledge of the effects of specimen geometry on the toughness of composites is important. Estimations of toughness in finite specimens have been made by different approaches for cementitious composites [6,7] and also assuming the crack opening displacements being unaffected by bridging tractions, for SiC-glass composites [8]. In the present study, an approach to calculate toughness due to crack bridging in a specimen of any geometry, using appropriate Green's function is presented. As an illustrative example, the equilibrium crack opening displacement fields and the crack bridging stress distributions and stress intensity factors for bridged cracks are calculated using the Green's function for a single edge notch (SEN) specimen containing a mode-I crack. The calculations, made using an idealized stressdisplacement behavior for the fiber bridging the crack, illustrate the nature of opening of a bridged crack and the resulting evolution of toughness. It has been shown that toughness is strongly affected by the crack length and the fracture toughness of the matrix. By implication, the study suggests that the damage tolerance of composites containing bridged cracks should be assessed in the light of finiteness of crack length and the size and geometry of the specimen. H The Analysis To calculate the toughness enhancement due to crack bridging, the bridging stresses resulting from the self-consistent crack opening displacement field, determined iteratively [4], should be known for the crack and specimen configuration. In the present analysis, an approach to calculate the crack opening displacement fields and bridging stresses using the Green's function for a typical crack configuration is presented. The net stress intensity factor (SIF) due to the applied stress of a composite containing a crack bridged by fibers can be written as: K'~ ffiKbridge + Ktip

(1)

where K°° is the applied SIF and Kbridg e and Ktip are the SIF due to fiber bridging and the effective SIF at the crack tip, respectively. The criterion for crack extension in the composite can be specified by the relations: K°° = Kc or Ktip = Km (2)

2645 0036-9748/91 $3.00 + .00 Copyright (c) 1991 Pergamon Press plc

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where K c and K m are the fracture toughness of the composite and the intrinsic fracture toughness of the matrix respectively. It is necessary to calculate Kbridg e and Kti p from the bridging and the effective crack opening stresses across the crack faces respectively. This requires the determination of equilibrium opening profile of the bridged crack. The u.~e of Green's Function to Detprmin~ f'rack Op_enin~ Dis_vl~epment ~ield. For a finite specimen containing an edge crack, the SIF at the crack tip due to a mutually opposite force pair 'P' on the crack faces at a location 'x' from the specimen edge (fig. 1) can be written as K ffi P h(x,a) (3) where h (x,a) is the Green's function for the crack and specimen configuration. For a distribution of stress, c~(x), over the crack face, the SIF is given by K =

(~ (x) h (x,a) dx

(4) Alternatively, by the weight function approach [9,10], the SIF due to a(x) can also be calculated if the SIF, Kr, and the corresponding crack opening displacement field, u r (x,a) of a reference problem is known. Then, K can be expressed as a

K = H ~ (~ (x) our (x,a) clx Kr ]0 ~a (5) where H = E for plane stress and H = E/(1-1) 2) for plane strain. 'E' is the elastic modulus and '~O'is the Poisson's ratio. From equations (4) and (5), h (x,a) = H

/)Ur(x,a)

Kr 0a (6) If eqn. (3) is used as the reference case in eqn. (6), the displacement field 'Up' can be obtained by integrating eqn. (6), up(x,~,a) = ~l l c ' h (x,a) K(~,a) da (7) where K (~,a) = P (~) h(~,a) and '~' is the distance from the specimen edge (fig. 1). In eqn. (7), the lower integration limit is given by: c = '~' for a > x > ~ and c = 'x' for ~ > x > 0. Eqn. (7) gives the displacement field of a single edge crack, subjected to a point force P at x='~' as a function of position 'x' along the crack flank and the absolute crack length. Eqn. (7) can be used as a Green's function to calculate the crack opening displacement field for any arbitrary stress distribution a (x=~) as

u(x,a) =

I;

,~(~) up(x,~,a) d~ (8)

Combining equations (7) and (8) u(x,a)

= 1

c(~)

h(x,a) K(La) da d~

(9) When calculating the displacement field P(~) is taken as unity. Eqn. (9) is a general expression for the displacement profile of a crack of any len~,.h and for any specimen geometry. Using appropriate Green's functions, the displacement field due to any su-ess distribution can be calculated by numerical integration using Gauss-Chebychev Quadrature rule. Determination of Crack Bridcfina Stresses The equilibrium displacement profile of a crack subjected to bridging tractions can be arrived at iteratively [4]. To do this, the crack opening profile of an unbridged crack is first calculated as u"(x,a) = o"-[a Up(X,~,a) d~ .10 (10) The first set of bridging stresses due to the fibers is calculated by initially assuming that fibers along the crack wake withstand the crack opening displacements and finding the resulting stresses on the fiber from the fiber constitutive relation. The bridging stresses, a c (x), are related to the fiber stresses, ,~f(x), in terms of fiber volume fraction, vf, as Crc(X) = - vf (~i(x) = * vf f(u = u~(x,a) ) where f (u) is the fiber constitutive relation. The effective crack opening stresses then are cFt(x) = oo= - ~ ( x )

(11)

(12)

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The effective crack opening displacements, Ueff(x,a) with o °° at far field and bridging tractions across the crack can be found using ot(x) in eqn. (9). This displacement field is again used to calculate the new bridging and crack opening stresses from eqns. (11) & (12) and then the new displacement field from eqn. (9). The iteration, T, is repeated until

~(~,a)

-- ~(~,a)

(13)

In practice, convergence, with an accuracy of better than 1% is achieved in a few iterations. It was not always possible to obtain convergence, particularly when the crack transforms from a fully bridged to a partially bridged state, by back substitution. However, by choosing :

+

(14)

it was possible to obtain convergence in most cases by suitably adjusting the parameter '~'. This was not always successful at very low stresses where small changes in displacement cause large changes in crack bridging stresses. Additional problems arise when Ueff (x,a) is negative due to bridging stresses being larger than the crack opening stresses, implying a physically inconsistent situation of crack face interpenetration. Part of this problem of handling negative displacements was circumvented by adjusting the bridging stresses in between each iterative step in small amounts (< - 10%) successively, until crack face displacements are positive and subsequently resuming iteration. Experience with a number of fiber stress-displacement laws indicates that this procedure is quite effective in obtaining convergence although computationally very time consuming. Once the equilibrium bridging stresses are determined, it is straightforward to calculate Kbridg e and Ktip, consistent with eqn. (1), as [ - ac (x,a) h (x,a) dx

Kbridge

and

Iqip :

(o- - Oc (x,a)) h (x,a) dx

J0

(15)

The Fiber Stress-Disvlacement Relation

The specimen geometry effects are illustrated in this study by choosing the fiber stress-displacement behavior as shown in fig. 2. Several other forms [4,6,11] can also be used. The stress-displacement behavior can be represented by Ucrit - Um]

(16)

where Om, Um and Ucrit are respectively the maximum stress, displacement corresponding to maximum stress and the displacement at fiber rupture. The exponent 'n' can be varied to obtain different forms of stress-displacement behavior. The typical values selected are: Om = 200 MPa, u m = 1 X 10-6 m, Ucrit = 50 X 10-6 m and n=0.2. Toughness Calculations for the SEN Svecimen Geometry v The Green's function for the SEN geometry, with an accuracy better than 1% for any a/w, is given as [12] --2-

h(x,a)--

G(~,~) (l"~w)l ~

(17) ax+

where

ax+

a

x

g l ( b ) = 0.46 + 3.06(b ) + 0~4(1-~wy+ 0.66(a ~(1--~-~ W/~ W J, :

g4 ( a )

6.17-

÷

W

W

and

g2(w ~) 2.

= -352(w~~ t

~W/~-

-etl-,-e

W~

= _ 6A3 + 25.16 (w~-) - 31.04 ( a ) 2 + 14.41~wla--~ + 2 ,[1-a-a ~w, + 5.04 ( 1 - a y + 1.98 ~w-, [a~2 (1 ~-a~ 2w.

where x, a and w are respectively the co-ordinate along the crack, crack length and specimen width respectively. m Results and Discussion

The crack opening profile of an unbridged crack is a function of specimen geometry, specimen size and crack length. Hence for a giver fiber stress-displacement behavior, the bridging stress distributions and accordingly the composite fracture toughness, Kc, would be sensitive to these geometrical factors. Further, the matrix fracture toughness, Km, decides extent of crack opening and hence the degree of bridging thereby limiting the toughness by the onset of crack extension in matrix. This is illustrated in fig. 3(a) and (b) by mapping the variation of K°° and Kbridge with Kti p for several a / w values. K~° and Kti p are synonymous with Kc and Km respectively. The calculations were done for a 100 mm wide SEN composite (with 0.45 volume fraction (vf) of fiber of 400 GPa modulus in a matrix of 100 GPa modulus), having a modulus of 235 GPa and a Poisson's ratio of 0.3. Calculations for a / w > 0.5 were not made due to numerical instability and difficulty in

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getting convergence with acceptable accuracy in self-consistent crack opening displacements. This is mainly due to the increased sensitivity of specimen compliance for a / w > 0.5 resulting in large changes in crack surface displacements for small changes in crack surface stresses. In fig. 3(a), Kbridg e increases steeply with Ktip followed by a region over which it is constant and subsequently decreases at large Ktip. The range of Ktip corresponding to constant Kbridge decreases as a / w is increased. The broken line is the locus of Kbridge and Ktip combinations at which the crack transforms from a fully bridged to partially bridged configuration. The transition is accompanied by fracture of several fibers causing a loss in bridging potential. This behavior is also reflected in the shape of K** versus Kti p curves in fig. 3(b). For low a/w, there is a proportional increase in K** with Ktip while at a/w=0.5, K** is constant over a range of Ktip implying a sudden change from fully bridged to partially bridged configuration. If the matrix fracture toughness is in this regime, unstable crack extension would occur at constant load. This tendency increases with an increase in a/w. For Ktip > 70 MPa~/m, the effect of a / w on Kbridge and K** is insignificant, since the size of the bridging zone being small relative to crack size. The nature of crack opening profiles and the distribution of bridging stresses at different applied stresses are presented in figs. 4 and 5 for a choice of a / w of 0.1 and 0.5 respectively. The applied stresses were appropriately chosen to yield representative Ktip values of 5 and 35 MPa~/m to illustrate the nature of fully bridged and partially bridged cracks. The degree of crack closure due to bridging can be realized by comparing the crack opening profiles of unbridged and bridged cracks. In fig. 4(a), the crack opening profiles for a/w=0.1 exhibiting the same degree of bridging at Ktip = 5 MPaqm and Ktip = 35 MPa~/m are presented. The cracks are fully bridged for both conditions since they fall in the fully bridged region as in fig. 3. In fig. 4(b), the corresponding bridging stresses (Oc) are equivalent giving same Kbridge while the crack opening stresses (or) differ producing different K** levels. For Ktip = 5 MPa~/m, Oc is close to the applied stress (o** = 104 MPa), over most of the crack wake except near the crack tip, thus promoting strong bridging (K°° = 22 MPa'/m, Kbridge = 17 MPa'/m)). The rise in o t close to the crack tip is due to the initial elastic portion of the fiber stress-displacement law (fig. 2), and this significantly contributes to Ktip. It is to be noted that the bridging potential of the fibers vanishes for applied stresses beyond OmVf = 90 MPa. Hence, for Ktip = 35 MPa~/m, the additional increase in applied stress (o*° = 241 MPa) does not cause any further increase in Kbridge (fig. 3(a)) but serves only to increase Kti p. Hence, the increase in toughness (K** = 52 MPaqm, Kbridg e = 17 MPa'/m) mainly comes from the larger value of Kti p. The data for a/w=0.5 (fig. 5) provide an interesting example of the transition from fully bridged to partially bridged crack configuration. The unbridged crack profiles are identical (fig. 5(a)) for both examples due to the applied stresses being almost the same (o ~ = 59 MPa for case (1); fro. = 60 MPa for case (2)). However, with fibers in place, the crack is fully bridged (K*" = 68 MPa'/m and Kbridg e = 63 MPa~/m) for Kti p = 5 MPa~/m and with a slight change in applied stress, the crack transforms to a partially bridged condition (K** = 69 MPa'/m and Kbridge = 34 MPa~/m) thereby increasing Ktip to 35 MPa~/m, consistent with fig. 3. This causes significant changes in the crack opening profiles (fig. 5(a)) and the crack surface stresses (fig. 5(b)). In the fully bridged condition, the crack opening displacements are low (< Um= 1 X 10-6 m) due to the applied stress being lower than OmVf. The crack bridging stresses in the partially bridged condition can be seen to follow the shape of the fiber constitutive behavior and are balanced by negative crack opening stresses in the regime 0.3 < a / w < 0.5. The variation of fracture toughness of the composite, Kc (K**), with a / w at different Ktip (Km) levels are presented in fig. 6. These R-curves show that the rate of increase in Kc with a / w is a strong function of the matrix fracture toughness. For Km varying from 2 to 5 MPa~lm, entire crack extension occurs in fully bridged state and would be expected to continue until unstable fracture intervenes at large a/w. However, since o~* required to maintain constant Ktip would continuously decrease, the fully bridged condition will be maintained and Kc would continue to increase exponentially at large a/w, consistent with experiments [8]. On the other hand, for Km values of 20 and 35 MPa~/m, crack extension beyond a/w=0.3 approaches unstable fracture due to the crack attaining partially bridged state (fig. 3) and hence toughness is saturated. Thus, Km appears to control the extent of stable crack growth in the composite and the shape of R-curve. For a given fiber constitutive behavior, prolonging the extent of fully bridged crack growth by suitable choice of Km, toughness can be enhanced. These trends should exist irrespective of the choice of specimen size and geometry, although the absolute toughness and the extent of fully bridged crack growth would be strongly dependent on specimen size and geometry. IV Condusio~ns An approach to explicitly calculate the fracture toughness and the extent of crack bridging in fiber reinforced composites having bridged cracks in finite specimens has been presented. This should be useful for the calculations of toughness and R-curves in several geometries for which the Green's functions are known [12]. The calculations made for a composite of single edge notch geometry indicate that the bridged crack opening profiles and the resulting bridging stresses are strongly sensitive to crack length and matrix fracture toughness. The results also imply that toughness would be very sensitive to the size and the geometry of the specimen and the crack length suggesting that comparisons of experimental

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toughness values generated from specimens of varying geometry and size are not straightforward. The rate of toughness development or the R-curve, including its shape, is also strongly sensitive to the matrix fracture toughness. Acknowledgement The support of the US National Research Council in the form of a research associateship and the computing facility at WL/MLLN, Wright Patterson Air Force Base, for numerical calculations is gratefully acknowledged. It is a pleasure to acknowledge helpful discussions with Mr. J. L. Kroupa, Dr. T.A. Parthasarathy at various stages of this work. References

(1) A.G. Evans and D. B. Marshall, Acta Metall., 37 (1989) 2567 (2) M.D. Thouless and A. G. Evans, Acta Metall., 36 (1988) 517 (3) B. Budiansky and J. Amazigo, J. Mech. Phys. Solids, 37 (1989) 93 (4) B.N. Cox, D. B. Marshall and M. D. Thouless, Acta Metall., 37 (1989) 1933 (5) L.R.F. Rose, J. Mech. Phys. Solids, 35 (1987) 383 (6) R.M.L. Foote, Y. W. Mai and B. Cotterell, J. Mech. Phys. Solids, 34 (1986) 593 (7) J. LLorca and M. Elices, Acta Metall. Mater., 38 (1990) 2485 (8) F. Zok and C. L. Hom, Acta Metall. Mater., 37 (1990) 2143 (9) H.J. Petroski and J. D. Achenbach, Engg. Fract. Mech., 10 (1978) 257 (10) X.R. Wu and J. Carlsson, J. Mech. Phys. Solids, 31 (1983) 485 (11) C. H. Hsueh and P. F. Be(her, J. Am. Ceram. SOc., 71 (1988) C-234 (12) H. Tada, P. C. Paris and G. R. Irwin, The Stress Analysis of Cracks Handbook, 2nd Edition, Paris Productions Inc., St. Louis, Missouri 6310@,1985

A i 0. 0

2oo

150

:... J~

50

W

o

;I

10.5

I

0

|

410.5

Displacement,

100 m

i

i

!

tO0

!

i.

80

J 6O

Partially Bridged

~

,o

•

2o

m

Fully

o

I

0

~

/

I

~

r

(.,w) o::'

~

" .2

,o

partllily Bridged | ~

r'..L ~ r ..

0.5 0.4 3 •

610 "s

(2U) (m)

Fig. 2. The fiber stress-displacement behavior.

Fig. 1. Point loaded crack in SEN specimen.

~

I;

I

1 104

o.1

o.1

%,

•

-

8ddged I

20 40 Crack Tip SIF,

I

~-I

I

60 80 KIn, (MPI~/m)

Fig. 3(a) Kbrid~e versus Ktiv for different a / w values.

I O0

I

20 40 Crack TIp SIF,

I

I

60 80 K p IMPadm)

Fig. 3(b) K°° versus Ktiv for different a / w values.

100

2650

TOUGHNESS OF COMPOSITES

i

Ev3.5 lO"6

i

:

250

:

i

Unbddged Ccack (1): K" .22;(2): K - - 5 2 MPm/m 3.OLO~

...... Bridged Crock

--

(1): Ktlp. 5; (2): K.p. 35 MPm/m

i

25, No.

i

i

CrackOpenlng Stress, o t

...... Crack Bridging Stress, o©

~ 30o

c

1

i 2.510"8

15q

_ 2.010 '6 O 1.$10'6

Vol.

~ ...i '--..." ' ~

Crack

10,

(1)

::::::::::::::::::::::::::::::::::::::::::::::::::::::::::.)

Tip

..........

~ 1"010"6 ~ 5,0104 o

"~- . . . . . . . . . . . . . 1. . . . . . . . . . . . . . .

0

0.02 Position

r ............

0.04 0.06 0.0e a l o n g t h e Ccaok, x / w

o~,

002

0,1

oo;1

oo;,1

OlO1

Position along the Crack, x/w

Fig. 4: (a) Half crack opening profile and (b) Crack surface stresses for a crack of a/w:0.1 for typical Ktip conditions. 100

7.510"6 A i

Unbrldged Ccack (1): I C . 6 8 ; (2): K - . 6 9 MPm/m :~. 8.0 10"6 ~

Crack

(1): Kip- 5; (2): Ktlp- 35 MPm/m

,

,

~ Crack Opening Strees, a t ...... Crack Bridging Stress, o©

70

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

,

l .....-"~)"'"

,-. ............. ;; .........

40

~

(2) °"°'°.o...°,

~.30

~ 1.510~

O o

0

0.1 0.2 0.3 P o s i t i o n a l o n g t h e Crlm.,k,

0.4

I " '~" ~ n -50

0.5

I

Crack TIp

0.1101

I I I 0.202 0.303 0.404 Posltlon along the Crack, x/w

Fig. 5: (a) Half crack opening profile and (b) Crack surface stresses for a crack of a/w--0.5 for typical Ktip conditions. ~ 11111;

E

a

i

i

a

~p - 2 MPaqm ~

K~ - 20 MPeqm

--e-- Kip. 3 MPaqm ~ Kip- 5 MPm/m

Kip.36 MPm/m

i

,o

!2o .

o

I 0.1

I 0~

I 0.3

I 0,4

I 0.5

0.6

Normallz~l Crack Length, a/w

Fig. 6. The composite fracture toughness, Kc = K°° as a function of a / w for typical Ktip conditions.

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