Toughening in brittle systems by ductile bridging ligaments

Acta metall, mater. Vol. 40, No. 7, pp. 1531-1537, 1992 Printed in Great Britain. All rights reserved

0956-7151/92 $5.00+ 0.00 Copyright © 1992Pergamon Press Ltd

T O U G H E N I N G IN BRITTLE SYSTEMS BY DUCTILE BRIDGING LIGAMENTS M. B A N N I S T E R l, H. SHERCLIFF 1, G. BAO z, F. ZOK 2 and M. F. A S H B Y 1 1Engineering Department, University of Cambridge, Cambridge, England and 2Materials Department, University of California, Santa Barbara, CA 93106, U.S.A. (Received 13 December 1991)

Abstract--The characteristics of the toughening of brittle matrices by ductile inclusions are demonstrated by model experiments> and analysed using recently developed techniques. Design considerations, and the transition from single-crack mechanics to the regime of multiple cracking, or "continuum damage" mechanics are described. Rrsum~43n &udie les caractrristiques de l'augmentation de trnacit6 des matrices fragiles par des inclusions ductiles grfice fi des exprriences modrles; elles sont analysres en utilisant des techniques rrcemment drveloppres. On drcrit des considrrations de conception, et la transition de la mrcanique ~t une seule fissure au rrgime fi fissures multiples, ou m&anique du "drgfit continu". Zusammenfassung--Die Charakteristika der Z/ihigkeitsverbesserung spr6der Matrix-Materialien dutch duktile Einschliisse werden mit Modell-Experimenten gezeigt und mit kiirzlich entwickelten Methoden analysiert. Oberlegungen zur Auslegung und der l~bergang yon der Ein-RiB-Mechanik zum Bereich vieler Risse oder der "Kontinuums-Sch/idigungs"-Mechanik werden beschrieben.

1. INTRODUCTION Crack bridging increases the resistance to crack growth in a brittle solid. The bridging ligaments exert closing forces which reduce the stress intensity at the crack tip (Fig. 1). The form of the closing forces, described by a force--displacement ( F - u ) function, determine both the way in which crack resistance develops as the crack grows, and the final or "steady-state" toughness. The steady-state toughness is easy to calculate but seldom of real interest; almost always, it is the rising part of the resistance curve which is important in calculating the failure strength and this requires more elaborate analysis. This paper reports experiments on a model bridged system: glass, containing inclusions of lead. The lead particles, intersected by the growing crack, stretch, exerting closing forces which are sharply peaked at small displacements. Fibres, too, can give a bridging zone, with other, different force-Misplacement functions. A method for evaluating the crack resistance curve for any bridging function is described, and applied to the case of ductile particle toughening. The results are presented as design charts which allow the optimum microstructural features for the reinforcement to be determined when the desired resistance characteristics are known. A similar approach can be applied to the microstructural design of fibrous composites. When the resistance curve is steep, a transition takes place from failure by the propagation of a

single crack to failure by multiple cracking. The paper concludes with a brief analysis of this problem, identifying the conditions that must be fulfilled in order to develop multiple cracking. The paper builds on a considerable body of recent work on ductile toughening [1-14] and on the recently developed analyses of the bridging phenomenon [15-17]. The new contributions presented here are the direct observations of ductile bridging and its influence on crack shape, and the idea of design charts for optimum microstructure selection; in addition, the concept of a transition from single to multiple crack mechanics in this class of material is discussed. The last two are relevant to all bridging phenomena, not merely that due to ductile inclusions.

2. EXPERIMENTS: CRACK RESISTANCE IN GLASS/LEAD COMPOSITES

Specimens were prepared by mixing glass and lead powders, and hot-pressing the mixture to give a dense compact. The 20/~m glass powder was a soda glass containing a small amount of a borax flux to promote wetting. The 99.98% lead powder had a size range 30--50~m, and was of the quality used in earlier direct measurements of the force-displacement function [1, 2]. The mixed powders, with volume fractions of lead from 0 to 20%, were placed in a cylindrical graphite mould and hot pressed at 780°C under a small axial pressure (0.1 MPa).

1531

1532

BANNISTER et al.:

TOUGHENING IN BRITTLE SYSTEMS

Fmax (Zmox =

Umax 'STRETCH,u

O @

©

..0. @O

CLOSURE

FORCES ~.~.~

@

..

,

,

"

O

0 .

©

CRACK

O

°

O

-'--,o - . ]. o ...

Fig. 3. A bridging ligament approaching final failure. A single large void has begun to rupture the surrounding walls.

Double-torsion samples were cut from the sintered composites and tested in an instrumented doubletorsion rig mounted inside a scanning electron microscope. This arrangement allows direct correlation of the resistance curve and microscopic observation of crack advance and bridging [4]. A central groove is cut into the specimen to guide the crack. The stress intensity, K, at the crack tip is given by now-standard equations which can be found in Ref. [18]. As the crack advances and opens, the lead particles which bridge it are stretched. Figure 2 shows a bridging particle at an early stage in the process: it is starting to neck. Figures 3 and 4 show later stages in the stretching process: a single void has nucleated within the lead and has broken through the surface in the second of these pictures. Figure 5 shows a bridging particle on the fracture surface after final

failure. No debonding has occurred at this particle: at others, limited debonding was observed. Most particles failed by the nucleation of a single void within the lead; a lesser number failed by the multiple nucleation of voids within the lead, and by the formation of ring-cracks in the glass surrounding the lead--precisely the mechanisms observed and characterised in the earlier studies of constrained lead wires in glass [l]. As the bridging zone develops, the crack resistance grows. A typical increase in K R with crack extension is shown in Fig. 6. The crack starts to extend when the remote K reaches the value K 0 = 0.38 MPa ml/2; further growth requires an increase in K up to a final, steady value of Kss = 0.95 MPa m 1/2 at a crack extension of 10 mm, an increase of 21 times. The crack opening profile was measured from scanning electron-micrographs, taken under a load just less than that for propagation, in a sample with a fully-formed bridging zone, see Fig. 7. The solid lines are the parabolic profiles expected near the crack tip for an applied stress intensity equal to /Co and K~s respectively, applied to an unbridged crack. As expected, the observed profile lies between these limits, but is markedly different from either one.

Fig. 2. A bridging ligament in the lead-giass composite, in an early stage of failure,

Fig. 4. A bridging ligament at final failure. Only a thin section of lead still links the crack faces.

O

O

@

@ @

Fig. 1. A crack in a brittle matrix, intersected by ductile particles. The particles stretch and fail as the crack opens. The work of stretching contributes to the toughness of the composite. The force~listance curve for the stretching of one particle is shown inset.

BANNISTER et al.: TOUGHENING IN BRITTLE SYSTEMS

1533

G L A S S - L E A D PARTICLES

-/

8.0

7.0

~

K

:0.9

m 6.0

z m

d_ 5.0 c~

o_z 4.0

Fig. 5. A failed bridging ligament showing the central void and the strong bond between the lead and the glass. Damage to the glass has occurred in a roughly concentric ring around the lead particle.

3. CALCULATION OF THE RESISTANCE CURVE AND CRACK P R O F I L E 3.1. Small bridging zone

When the bridging zone length L (Fig. 1) is very small compared with both the crack length c and the sample dimensions, the crack resistance curve can be calculated from standard results. The initial and final values (K0 and K~) are easy. The resistance to advance of an initially unbridged crack is simply the fracture toughness K0 of the matrix and its profile is just that associated with a plane strain crack with this stress intensity applied

o

3.0i

<

zo

-."

/

.~"

=

8 ( 1 - v 2) K o r l , - - . '"

,:,~

2.0 4.0 6.0 B.O ~OO DISTANCE BEHIND CRACK TIP (mm)

Fig. 7. The crack opening displacement measured behind the crack tip during the R-curve determination. The two solid lines are the displacementspredicted using equation (1) with K, = 0.4 MPa.j'm and K, = 0.9 MPav~mm. where ~r(u) = F/rca 2 is the nominal stress required to stretch a ligament by u, and u* is its failure stretch. When a(u) is constant from u to u*, the result is AGes = Vr am,x u*

;

(3a)

when it is triangular (as in the inset of Fig. 1) it is A G•~ -- 5~Vf

am.~ u

*.

(3b)

The steady state resistance is given by K~ + ~

AG~,

E

u*

~r(u) du

(2)

[2512 AKin= •

GLASS- LEAD PARTICLES

7,_-T2 2 ~

rE g_

('t O-(X)

k'~) V,-Jov...~

d~"

o/Joo

(6)

o--

with

K,~ = K0 + AKs~.

0.5 < £

O0

(s)

For a constant crack-bridging stress, a ( x ) = a m , ~ fromx=0to x = L , and AK,s = (8~L) 1'2 Vf O'max

oo

(4)

Everything depends on the stress-displacement function, ~r(u). It has been determined, for lead in glass, as a function of particle size and debond length in two earlier publications [1, 2], allowing Kss to be evaluated. The same result is obtained by considering the reduction in K caused by a distribution ~r(x) of closing stress, applied between the crack tip and a distance L (the steady-state bridging length) behind the tip. This reduction is given by [19]

)

15,

1.0

.

(I)

where E is Young's modulus of the composite and x is the distance measured from the crack tip. The fully bridged resistance is calculated from the work of fracture of one ligament and their concentration, measured by the volume fraction Vr of the ductile inclusions. The work of fracture, per unit area of the crack face, is [1] AG~ = Vt

/ K = O ~

I0

K~ =

u(x)

.~} ."

5 10 15 CRACK EXTENSION Ac (mrn)

20

Fig. 6. The resistance curve for glass containing 20% by volume of lead particles.

The steady-state bridging length is fixed by the failure stretch of the ligament. If the crack profile is unaffected by the closing forces, it can be calculated from equation (1). It can be shown that, for small AK~ (a necessary consequence of assuming that crack shape is unchanged by bridging) this result is identical to equation (4).

1534

BANNISTER

et al.:

TOUGHENING IN BRITTLE SYSTEMS

This alternative derivation has been introduced here because it can be extended to calculate the shape of the R-curve. Partial bridging over a length Ac < L is described by integrating equation (5) to an upper limit Ac rather than L. Evaluating the integral requires an explicit expression for a(x), which can be obtained from the a(u) function via equation (1), provided, once more, that the crack profile is unchanged by bridging. The procedure is straightforward; the results of applying it, using several different a(u) functions, are given elsewhere [4]. The arguments given above are valid when the assumptions of small-scale bridging and small crack size (relative to the specimen dimensions) are met. Further, they require the resistance increment, AK~s/Ko,to be small. When these assumptions are not met, a more elaborate treatment is needed. It is outlined in the next section.

3.2. Large-bridgingzone A glance at Fig. 7 shows immediately that the presence of a bridging zone of sufficient strength to double the crack resistance also changes the near-tip crack profile. The shape of the crack-resistance curve depends in a sensitive way on this profile. Setting it equal to equation (1), as we did in the last section, is not adequate. Mechanics solutions for the true crack profile are now available. They allow the crack surfaces to adopt an equilibrium shape [15, 16]; the distribution of closing forces [which depend on the crack shape and on the a(u) function] are used to compute the resistance curve. This is done by adding the components of crack opening displacement due to the applied loads and the bridging tractions, and requiring the sum of these to be consistent with the bridging traction law satisfied at all points on the crack surface [15, 16]. The result consists of an integral equation which, in most cases, must be solved numerically. A summary of the equations is presented in the Appendix. The results are used in the next section, to construct "microstructure design charts" which give guidance in selecting the optimum particle size, strength and debond length. 4. MICROSTRUCTURAL DESIGN CHARTS Toughening by crack bridging has the feature that the resistance to advance of the initial, unbridged crack is less than that when the bridging zone has developed. It might be thought that only the ultimate, steady-state, toughness should be of interest in design, but this is not true: the performance of any material which shows a rising resistance curve depends on the slope of this curve as well as its final value, and the dependence changes with different modes of loading. A way of quantifying this is outlined here. Consider, first, a sample containing an initial unbridged crack of length Co loaded in simple tension.

Unstable propagation (and failure) require that two conditions be satisfied simultaneously

K=KR }. and

(7) dK _ dK R dc dc

Almost always these conditions are met before the bridging zone is fully developed; sometimes they are met before it has formed at all. In practice, the fracture stress is usually determined by these conditions than by the steady-state value of the fracture toughness, Ks~. If the design criterion is energy absorption (rather than tensile strength), the slope of the resistance curve again enters. We show, in the next section, that appropriate combinations of slope and initial flaw distribution can lead to multiple cracking (rather than single crack propagation) even in tension. While this may not change the failure stress appreciably, it greatly increases the energy absorbed in causing the component to fail. But the microstructure which is best for energy absorption is not the one that is best for tensile strength. It would, clearly, be attractive to have some simple scheme for selecting the best microstructure to maximise performance in a given mode of loading. One, for optimising tensile strength of a sample containing an initial centre-crack of length 2c0, is illustrated here. The fundamental property of a bridging agent is its a(u) curve, that of the matrix is its unbridged toughness, K0. From this, all aspects of crack propagation (in a given mode of loading) follow. For tensile strength we are interested in the fracture stress, o'f; we measure it in units of the unbridged fracture strength

g0_ K0

,/;Z0 It depends, as we have said, on the slope of the resistance curve: it captures one aspect of the a(u) function. Another is captured by the steady state toughness, K~, which we measure in units of the unbridged toughness, K 0. The tensile-strength design charts (of which Fig. 8 is an example) have axes of O'f/O"0 and Kss/Ko.The grid of lines shows the range of these which are accessible for a given volume fraction of a given reinforcement, and initial crack size. One set of lines shows the particle size (in microns), the other, the debond length in units of the particle size; these are the two microstructural variables which determine performance. The three charts shown here are for 20% lead reinforcement in glass, for initial crack lengths co of 1, 10 and 100mm. In the calculations, the bridging traction law is assumed to be linear, with the peak stress, a . . . . occurring at u = 0 and the stress decreasing to zero at u = Umax.This law agrees

BANNISTER et al.: TOUGHENING IN BRITTLE SYSTEMS

1.4

(a) I

I

i

I

I

one of the dashed lines corresponding to a fixed t. The optimal debond length increases with the initial crack length, and decreases with increasing particle size. An additional noteworthy feature pertains to the particle size: both Kss and ar increase with t. However, for combinations of short cracks and long debonds, there is only little benefit in increasing t beyond a certain value if we are mainly interested in at, because of the way resistance curve slope influences ~r.

I

c o = 1 mm

~

1.3

o~

1.2

o

~ \~

~

~

1535

Ld/t

2

"2 1.C

"s

";o

I 2

t (~m) ~ 4

I 3

"'%

......... "'5o4 I 5

I 6

I 7

K s J Ko

(b)

[

I0

2.4 - CO = 1(3 turn

0.21

I

2.2

I

1

" ~ 0.5

2.0

•

-

xxx ~x 1

~

Ld/I

"'-... 4 14

5

10

t(gm) 20

50

12 ~0

1 2

/ 4

[ 3

I 5

I 6

I 7

Kss/Ko

(c) :50 I co=lO01mm .

.

.

. Ld/t 1 .

0,5 _ - -

-~

"'-.

0.2/"

"'".

35

x~ .

0

30

_-

4

5O ---~_

~

2O

I~ 2.5 2.0 15 1.0

I

2

I

3

I

4

I

5

I

6

I

7

Kss/Ko

Fig. 8. The design chart for centre-cracked plate, with (a) Co= 1 mm, (b) co= 10 ram, and (c) co = 100 mm.

reasonably well with the a(u) curves obtained in previous studies on Pb/glass monofilament specimens [1, 2]. The effects of debond length on Crm~and Um~ are obtained from the earlier work [2]. The relevant equations are contained in the Appendix. Calculations of the R-curves (from which the fracture strength is derived) incorporate the effects of the bridging tractions on the crack profile, as described in Section 3.2. The design charts show that, in general, conditions which maximise ~f do not maximise Ks,, and vice versa. For example, for a given particle size, K,, increases monotonically with debond length. However, there is an optimal debond length at which ~f is maximised. This can be illustrated by following

AM40/7~

5. SINGLE VERSUS MULTIPLE CRACKING: THE TRANSITION TO DAMAGE MECHANICS A brittle material containing an initial distribution of small crack-like flaws may fail, when loaded, by the unstable propagation of a single crack (usually, the largest one); or it may fail by the simultaneous, initially stable, propagation of many cracks which ultimately link to give a final failure. At the macroscopic level (the design of a component, for instance) the first of these is treated by fracture-mechanics; the second by continuum-damage mechanics. Single-crack vs multiple-crack response is determined by three things: the initial flaw distribution in the materials, the stress state applied to it and the form of the crack resistance curve of the material. A material containing an inital flaw distribution and with a constant Kc (that is, no change of crack resistance with crack length) shows single-crack behaviour in tension but multiple cracking ("crushing") in compression. But if the material has a steeply rising resistance curve and the right sort of flaw distribution (defined in a moment) then it will show multiple cracking in simple tension, and even in extreme tensile states like those which can appear at the root of a notch. Our aim here is to draw attention to this transition and the conditions which cause it to occur. Figures 9 and 10 illustrate the key points. The first illustrates the importance of stress state. A brittle body with a constant Kc (no change of crack resistance with length) is immediately unstable, in tension, when the first crack propagates because the stress intensity at its tips, roughly a ~ , ~ 0 , increases as the crack grows. In compression things are different: the first crack to propagate is the longest one at an angle (roughly 45 °) to the principal compression (in the way shown in the lower part of Fig. 9), but it does so stably--meaning that the stress must be increased to drive it further (see, for example, Refs [20 22]). The increase in stress causes other, slightly shorter, cracks to start extending, again stably, requiring further increase in stress and the activation of a further segment of the initial flaw population. Instability occurs only when the growing population start to interact strongly [21, 22]. The second figure illustrates the second circumstance: the stress state is tensile, but the material has a rising crack-resistance curve. The population of small cracks described by the inset crack length

1536

BANNISTER et al.: TOUGHENING IN BRITTLE SYSTEMS (7 C~

ttt

SINGLE CRACK

/

TENSION (CONSTANT Kc)

(7 (5"

MULTIPLECRACKING COMPRESSION

to the toughness-resistance curve. The figure illustrates two extremes. The first has small initial cracks and a low dGR/dC: the material fails (in tension) by the propagation of the single largest crack. The second has a wider distribution of initial cracks and a steeper dGR/dC: now a large fraction of the cracks in the sample propagate (absorbing energy) before instability. A simple geometric argument illustrated by the figure gives one of the conditions for multiple cracking in tension: it is, roughly put, that the peak of the initial crack distribution lies outside the initial tangent to the GR(C) curve, assumed convex upwards. This requires that 9--.

\ 6c /~=~o

E Fig. 9. Two samples with the same initial flaw distribution fails in tension by the unstable propagation of a single flaw; in compression it fails by the stable propagation of many flaws.

distribution. The rest of the figure shows the toughness-resistance curve, GR(C), used here in place of KR(c) because the loading curve is linear in crack length, c t7 2 7CC

G(c) = --U-

(1

- v2).

(8)

The instability condition [equation (7)] is that a straight line drawn from the origin (c = 0) is tangent

GR

/ ~

)

N(Co) / / /

R

~ G~o G

/ A

/

//

C

K

HIGH LOW bOGR/(~ C

SMALLFLAWS

I L::'

AC Co~ GR

, '///

N(Co)

/

.... ~ G o

When this condition is satisfied the energy absorbed in fracturing the material, though brittle, becomes large because many cracks propagate instead of just one. It is worth observing that the condition is most easily achieved by making c0 large: doing so reduces O'f but increases the fracture energy dramatically. 6. CONCLUSIONS Crack bridging increases the crack resistance of brittle solids. The mechanism of crack bridging studied here--that created by an array of ductile particles--can give a substantial increase in fracture toughness, but this increase develops only when the crack advances. The rate of increase is at least as important as the final, steady-state toughness in determining component response. This suggests the idea of microstructure-design-charts, from which the best combination of ductile particle characteristics for a given component application can be read. A rising crack-resistance curve can lead to a transition from single-crack to multiple-crack ("damage") mechanics. The conditions for multiple crack propagation are outlined and the consequences - - a n increase in energy absorption, at the possible penalty of a loss of strength--are explored. Acknowledgements--The authors wish to acknowledge the support of the Defence Advanced Research Projects Agency for financial support under ONR Contract N00014-86K-0753, the stimulating collaboration with Professor A. G. Evans and his colleagues at the University of California at Santa Barbara, and the considerable technical assistance of Mr Alan Heaver of the Materials Group, Cambridge University Engineering Department.

/ / ~

LOW o LOWO~"G./a c LA.OEFL

Co

S

&c

Co-.,~..-

Fig. 10. A brittle solid containing an initial flaw population N(co) fails by single crack propagation when the flaws are small and OGR/Oc is low. Under the opposite conditions it fails by multiple cracking even in tension. In the lower figure all the flaws of size between (c0)~ and (Co)2 are activated.

REFERENCES

1. M. F. Ashby, F. J. Blunt and M. Bannister, Acta metall. 37, 1847 (1989). 2. M. Bannister and M. F. Ashby, Acta metall, mater. 39, 2575 (1991). 3. L. S. Sigl, P. A. Mataga, B. J. Dalgleish, R. M. McMeeking and A. G. Evans, Acta metall. 36, 945 (1988).

1537

BANNISTER et al.: TOUGHENING IN BRITTLE SYSTEMS 4. M. Bannister, Ph.D. thesis, Cambridge Univ. (1990). 5. B. D. Flinn, M. Riihle and A. G. Evans, Acta metall. 37, 3001 (1989). 6. B. D. Flinn, F. Zok, F. F. Lange and A. G. Evans, in Proc. Syrup. Innovative Composite Processing. TMS (1990). 7. H. C. Cao, B. J. Dalgleish, H. E. Ddve, C. Elliott, A. G. Evans, R. Mehrabian and G. R. Odette, Acta metall. 37, 2969 (1989). 8. T. C. Lu, A. G. Evans, R. J. Hecht and R. Mehrabian, Acta metall, mater. 39, 1853 (1991). 9. H. E. Ddve, A. G. Evans, G. R. Odette, R. Mehrabian, M. L. Emiliani and R. J. Hecht, Acta metall, mater. 38, 1491 (1990). 10. M. C. Shaw, D. B. Marshall and A. G. Evans, Proc. Mater. Res. Soc. Symp. 170, 25 (1990). 11. F. Zok and C. L. Hom, Acta metall, mater. 38, 1895 (1990). 12. J. Besson, M. De Graef, J. L6fvander and S. M. Spearing, J. Mater. Sci. In press. 13. K. T. Venkateswara Rao, G. R. Odette and R. O. Ritchie, Acta metall, mater. 40, 353 (1992). 14. H. E. Ddve and M. J. Maloney, Acta metall, mater. 39, 2275 (1991). 15. R. Odette, B. L. Chao and G. Lucas, to be published. 16. B. N. Cox, Acta metall, mater. 39, 1189 (1991). 17. G. Bao and F. Zok, to be published. 18. B. J. Pletka, E. R. Fuller Jr and B. G. Koepke, Fracture Mechanics applied to Brittle Materials, ASTM STP 678 (edited by S. W. Freiman), p. 19. ASTM (1979). 19. H. Tada, P. C. Paris and G. R. Irwin, The Stress Analysis of Cracks Handbook, Del Res. Corp., St Louis, Mo (1985). 20. S. Nemat-Nasser and H. Horii, J. Geophys. Res. 87, 6805 (1982). 21. M. F. Ashby and S. D. Hallam, Aeta metall. 34, 497 (1986). 22. M. F. Ashby and C. G. Sammis, PAGEOPH 133, 481 (1990). APPENDIX Here we present the equations used to compute both the R-curves and the fracture strengths of the lead/glass composites. The composite body is assumed to contain an initial (unbridged) centre-crack of length 2c0, and the body loaded in tension, as shown in Fig. A1. The dimensions of the body are assumed to be large compared with the crack length. Following equation (5) in the text, the variation in fracture resistance with crack length can be written as

KR=Ko+2V f

0x/r~

dz

(A1)

where 2c is the total crack length and z is the distance measured from the center of the crack along the crack plane. Computing a(z) requires satisfying the condition that the crack opening displacement be consistent with the bridging traction law a(u) at all points along the crack surface. To do this, the total crack opening displacement is expressed as the sum of two components u = uA + ua

(A2)

where UA is the component due to the remote stress and uB is the component due to the bridging tractions. These are given by [19] UA--

4 ( 1 - v 2) K R ~ E x/~

(A3)

and

l

T

l

UB--

E

f;

o

a(~H(z, [, e) d[

(A4)

l

l

l

2c

l

l

l

Fig. A1. The crack with closure forces. where H is the Green's function and is given by [19]

1 i ~/~ws~_~:+~/c2_ ;2. n=;log

(AS)

Assuming a(u) exhibits linear softening (as described in the text), equations (A2)-(A4) can be combined to give

u ~ l _ a ( z ) ] _ a ( 1 - - v2) KR L amax/ E x/~

a([)H(z, [, c)d[.

4(1--v2)Vr E

~ 2 (A6)

,d Co

Equation (A1) is substituted into (A6), and the resulting equation is solved for a(z) as a function of crack length. Combining this result with equation (A1) gives the variation in K R with c. This result is then used along with equation (7) in the text to find the strength af. The effects of debond length on the a(u) function for the lead reinforcement have been obtained from an earlier study by Bannister and Ashby [2]. Their experimental results are adequately described by the linear softening model, with Om ' a0Xo"= 6'511 +i+Ld/t0"23 Ld/t lj and Umax

t

- 0.2 + 0.77 Ld/t.

Here % is the initial yield stress of the lead. Table A1 summarizes the values of the material properties used in the calculations. Calculations which incorporate more general relations between debond length and amax/a0 and Ur~x/t will be presented in a future publication [17]. Table AI. Properties of the lead/glass composites E = 40 GPa v =0.2 K0 = 0.4 M P a , / ~

-4(1 - v 2)

l

a0 = 5.2 MPa Vr= 0.2