Toughened Plastics II - American Chemical Society

4 Influence of Adhesive Strength upon Crack Trapping in Brittle Heterogeneous Solids Todd M. Mower and Ali S. Argon 1

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Room D-450, Massachusetts Institute of Technology Lincoln Laboratory, P.O. Box 73, Lexington, M A 02173 R o o m 1-306, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, M A 02139

Downloaded by PURDUE UNIV on July 5, 2016 | http://pubs.acs.org Publication Date: May 5, 1996 | doi: 10.1021/ba-1996-0252.ch004

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An experimental technique was devised in which a crack could be propagated past macroscopic obstacles in a transparent brittle epoxy under stable conditions, to determine the effective toughening resulting from discrete, isolated, crack-trapping processes in the ab sence of all other toughening mechanisms. The results indicate that toughness may be enhanced by a factor of nearly 2 through the crack-trapping mechanism alone, with particle equivalent volume fractions of just 0.06. To produce this level of toughening, a high ad hesive strength was found to be required between the matrix and tough spherical particles. With long cylindrical trapping obstacles, the adhesive strength does not exert such a critical role.

T

O U G H E N I N G O F B R I T T L E S O L I D S by the inclusion of tough particles has

of-

ten been attributed to p i n n i n g o f the crack front. B y analogy w i t h dislocations b o w i n g a r o u n d impenetrable obstacles, it has been assumed that the local crack front bows out between such tough particles until a critical breakaway configuration is reached (J, 2). Recent numerical simulations o f crack-front interactions w i t h arrays o f tough particles predict enhancement o f global fracture toughnesses ranging from a factor o f approximately 2 (3, 4) to a factor o f nearly 3 (5) for volume fractions o f particles i n the range of 0.1 to 0.25. These n u m e r i c a l predictions and other analytical models (6, 7) assume that the i m p e d i n g particles are "perfectly b o n d e d " to the host matrix, a c o n d i tion seldom achieved i n practice. A precise understanding o f the adhesive strength necessary to promote crack trapping b y impenetrable obstacles i n brittle solids has still not been reached, despite numerous investigations. I n one o f the earliest studies, B r o u t 0-8412-3151-6

© 1996 American Chemical Society 45

Riew and Kinloch; Toughened Plastics II Advances in Chemistry; American Chemical Society: Washington, DC, 1996.


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T O U G H E N E D PLASTICS I I

m a n and Sahu used tapered double-cantilever beam ( T D C B ) specimens to measure the fracture energies o f epoxies containing glass spheres that h a d various surface treatments to modify their adhesive strengths (8). T h e y credited the crack-trapping mechanism for a maximum measured enhancement of fracture energy, by a factor o f approximately 3, i n specimens containing particles w i t h either u n m o d i f i e d or adhesion-enhanced surfaces. F u r t h e r increases i n fracture energy, to a level about five times greater than i n the neat resin, were measured w i t h composites prepared from spheres that h a d reduced adhesion. This augmented fracture energy was attributed to the greater fracture surface area resulting from rougher surfaces containing debonded hemispheres. W e suggest that other factors were responsible instead (such as crack-tip shielding via decohesion o f particles above and below the crack plane), because recent statistical analyses o f toughening due to increased surface areas suggest that, at best, such increases can be expected to account for only a 1 5 % rise i n fracture energy (9). O t h e r investigations of toughening by glass spheres i n brittle epoxies have c o n c l u d e d that little improvement o f toughness (K ) o f the composite results from increased strength of adhesion of the particles to the matrix (10-12). T h e p r i m a r y mechanisms suggested to contribute to overall toughness are crack trapping and crack-tip blunting, where the total toughness results from a c o m petition between these two mechanisms modulated by the degree o f adhesion (13-15). L a c k o f significant variation i n measured critical stress intensities w i t h apparent adhesive strengths (in particular composites) is not significant evidence that crack trapping is unaffected by adhesion levels. Rather, w i t h decreasing adhesion, other mechanisms may become operative while crack trapp i n g becomes less effective. T h e role that adhesion plays i n toughening by the process of crack trapp i n g has remained unclear for two distinct reasons. First, no p r i o r particulatetoughening research (that the authors are aware of) has i n c l u d e d a quantitative study o f adhesive strengths; rather, they have relied on fractographic information for qualitative rankings only. I n addition, previous studies have used measurements o f toughness i n composite materials w i t h microconstituents only. W i t h that methodology it is always difficult, i f not impossible, to separate the effects o f fracture mechanisms a n d deformation processes that may be operati n g i n unison. I n the study of crack trapping reported here, an experimental technique was devised i n w h i c h a crack c o u l d be propagated past tough macroscopic obstacles i n a transparent brittle epoxy under stable conditions. T h e shape a n d motion of the crack front were recorded cinematographically, w i t h increasing crack driving force, to determine the effective toughening resulting only from discrete, isolated, crack-trapping events in the absence of all other toughening mechanisms. A parametric approach was adopted to study the importance o f inclusion spacing, interfacial adhesion, and residual thermal stresses o n the crack-front behavior and enhanced stress intensity required to propagate the Ic


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Influence of Adhesive Strength

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cracks past obstacles. This chapter briefly details the experimental procedures, presents typical crack-front images, and discusses the trends observed i n the measured toughening due to crack trapping. T h e influence o f adhesion o n these results is emphasized.


Experimental Details Specimen Fabrication. T h e crack-trapping experiments reported here were performed with double cantilever-beam ( D C B ) fracture specimens having a square cross section o f 38 χ 38 m m and a height o f 89 mm. The specimens were cast from a transparent epoxy bearing tough, 3.17-mm-diameter rods or spheres to act as obstacles to crack propagation. Epoxy was chosen for the matrix material be cause it lacks other distracting damage and energy-absorption mechanisms, such as crazing and microcracking, and can be formulated to a brittle state. A standard D G E B A epoxy (Shell 815), cured with a bifunctional polyamide (V-40), was employed so that partial curing at room temperature would occur, in order to position obstacles i n the desired locations within the specimens. A resinrich composition o f 3 parts resin to 1 part curing agent was employed to create an epoxy solid that was as brittle as possible using this particular curing agent. A n ele vated postcuring temperature (135 °C) activated the secondary amines and com pleted the cross-hnking of epoxy groups, creating a matrix material with a glasstransition temperature, T , of 80 °C and a room-tenrperature fracture-initiation toughness as low as K - 1 . 0 M P a λ/τη, G ~ 350 J/m , as measured with tapered D C B specimens at displacement rates of 0.1 to 10 mm/min. The materials chosen for obstacles, polycarbonate (PC) and nylon-6, had elas tic properties very similar to those of the epoxy matrix, so that stress concentrations would not be present and the analysis and interpretation of data would be simpli fied. A l l rods and spheres were solvent-cleaned and then thoroughly dried prior to their inclusion i n the specimens to prevent plasticization of adjacent epoxy by dif fusion o f water from the rods during curing o f the epoxy. Interparticle spacings (jR/L, as defined i n Figure 1) of 0.125, 0.187, and 0.250 were used, corresponding to equivalent volume fractions of approximately 0.06, 0.14, and 0.27. Specimens were milled to the dimensions just indicated, and a semicircular groove was machined across the bottom of each specimen, parallel to the crack plane, to accommodate a pin that provided vertical support during testing. To pro mote planarity o f crack growth, side grooves were cut with a 0.5-mm-thick slitting saw to a depth of 6.3 m m , and a reverse-chevron-tipped starter crack was cut to a g

I c

I c

Figure 1. Geometry of particle spacing and local crack advance. Reproduced with permission from reference 26.


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T O U G H E N E D PLASTICS II


length of 43 m m . The specimens were then finish-sanded and polished with alumi na paste. Photographs of a typical specimen are shown in Figure 2. Constituent Material Properties. Fracture of most epoxies at room tem perature occurs i n an unstable, jerky manner, which results from blunting of sta tionary cracks, followed by their re-initiation as sharp cracks and energetically "su percharged" extensions, and then blunting again upon arrest (16,17). This mode of fracture gives way to stable, continuous fracture as the yield stress is increased, as a result of either increased loading rates or reduced testing temperatures (18). In the epoxy used here, stable fracture was consistently found to occur at a tempera ture of - 6 0 °C. Because stable fracture was required both to photograph the crack front as it interacted with the obstacles and to accurately measure the resulting i n creases i n toughness, the experiments were performed at - 6 0 °C. Relevant mechanical properties at this temperature, such as Youngs modu lus, E, yield strength, σ^, and fracture toughness, K , were determined for the i n dividual constituent materials and are listed in Table I. Glass-transition temperaI c

Fignre 2. Crack-trapping

specimen containing PC spheres.


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Table I. Physical Properties of Constituent Materials at -60 °C Material

T (°C)

Ε (GPa)

80 145 50

3.5 3.0 3.6

g

Epoxy PC Nylon

a (MPa) y

145 90 (20) 140 (22)

K

(MPaVm)

lc

0.5 1.9 (20, 21) 3.3 (22, 23)

ture, T , was determined both with differential scanning calorimetry and thermomechanical analysis. The values of T are also listed i n the table. The uniaxial modulus and yield strength of the epoxy were detennined by compression exper iments. Fracture toughness of the epoxy was measured with T D C B specimens at displacement rates ranging from 0.1 to 10 mm/min. Tensile moduli or the P C and nylon rods were measured with the aid of a strain-gage extensometer. More de tails on the conduct of these tests can be found elsewhere (19). Included i n Table I are yield-stress and fracture-toughness data for P C and nylon, derived from the g


g

Adhesive Strengths. The strengths of the adhesive bonds between the i n clusion materials and the epoxy were determined with an experimental technique described i n detail elsewhere (24). The technique involves embedding a sphere of the candidate material i n the contoured-neck portion of a cylindrical epoxy bar. The decohesion of the sample from the epoxy is then observed as the bar is strained i n tension. The radial stress at decohesion is then determined by means of a numerical solution of the deformation problem. Because no stress singularities exist at the interface prior to debonding, as is often the case with the usual popular tests, the technique used here provides a measure of the "true" adhesive strength for the bimaterial systems employed. The specimens tested contained either nylon or P C spheres and were made using the same epoxy formulation as was employed i n the crack-trapping experi ments. A l l spheres were solvent-cleaned and dried prior to inclusion i n test speci mens. Analysis of the test results indicates that the nylon-epoxy system exhibits an adhesive strength ( σ ) of approximately 31 ± 5 M P a . Various attempts to enhance the adhesion through acid etching or use of primers did not result i n adhesive strengths outside the indicated range. Polycarbonate-epoxy systems consistently exhibited higher adhesive strengths, such that σ = 53 ± 2 M P a . Modification of the P C surface with one coat of release agent lowered the strength to 33 ± 4 M P a ; a second coat resulted i n σ = 25 ± 3 M P a . β

α

α

Residual Thermal Stresses. Residual thermal stresses i n the model spec imens at the surface of inclusions (i.e., maximum matrix stresses) have been cal culated (25, 26) using appropriate equations from the literature and thermal strains measured with a tnermomechameal analyzer (TA 2940). The actual ther mal stresses present i n the model specimens were equal to the sum of the stress es generated during the initial cooling from the T of the epoxy to room temper ature, reduced by some unknown amount of inelastic relaxation, plus the stresses generated during cooling from room temperature to - 6 0 °C. Because experiments with a similar epoxy have demonstrated a relaxation time of several years at room temperature (27), the residual stresses during the crack-trapping experiments are taken to be the sum of those generated during the two cooling stages, as indicatg


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Table II. Computed Matrix Thermal Stress (MPa) at Inclusion Surfaces Inclusion

80 to 20 °C

Spheres

Final

0"r

Nylon PC

1.8 -6.2

Rods

σ

Nylon PC


20 to -60 °C

Γ

-1.0 -3.7

-0.9 3.1 σ

ζ

-0.1 0.55

2.2 -2.2 σ

Γ

-0.4 -1.0

-1.1 1.1 σ

ζ

-0.3 0.15

4.0 -8.0 σ

Γ

-1.4 -4.7

-2.0 4.0 σ

ζ

-0.4 0.7

Testing Procedures. The crack-trapping experiments were performed at - 6 0 °C in a temperature-controlled chamber. Cooling was accomplished with forced convection of liquid nitrogen boil-off vapor, and was maintained to within ±2 °C with a controller, utilizing a thermocouple placed adjacent to the specimens. Testing was executed with a screw-driven (displacement-controlled) machine op erated at a constant crosshead velocity of 2.5 mm/min. Loading of the specimens was performed with a polished, stainless steel, 30° wedge, prying apart the D C B specimens. Specimens were supported with a cylindrical steel pin positioned paral lel to the fracture plane. The load was directly recorded by a load cell, while the wedge displacement was measured with a linear potentiometer sensing the relative motion between the wedge and the load-cell platen. To prevent sticking of the actuator, a resistive film heater was mounted on the potentiometer housing. Both load and displacement data were digitally recorded and stored. Images of the crack front and its interaction with obstacles were recorded on 16-mm film at a frame rate of 24 frames per second. Two cameras were used: one with a normal view (looking along the axis of the rods) to image the crack front, and one looking down on the specimens at an angle of approximately 25° i n order to record possible debonding along the rod-epoxy interfaces. Lighting for the crackfront images was provided by light from a tungsten lamp reflected by a heated mir ror (to increase the spot size) and passed through a diffuser plate. Lighting for the debonding images was provided by a fiber-optic light source aimed directly at the surface of the obstacles where crack intersection was anticipated. The elapsed time from crack initiation to specimen failure was typically about 40 s, so approximately 1000 photographic images were obtained during crack growth i n each specimen.

Results L o a d - D i s p l a c e m e n t H i s t o r y . A typical load versus displacement (Ρ-δ) trace obtained from testing a specimen containing no obstacles is shown i n F i g u r e 3. As the crack extends through the chevron starter section, its w i d t h increases, resulting i n a nonlinear rise i n load. T h e m a x i m u m load is reached w h e n the crack reaches the full w i d t h o f the specimen (end o f the chevron), and is followed b y a steady drop i n load d u r i n g further crack extension. A pre cipitous drop i n load results as the crack approaches the e n d o f the D C B a n d final specimen cleavage occurs. I n crack-trapping specimens, the obstacles



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Wedge Displacement (mm) Figure 3. Typical Ρ-δ plot obtained from testing a neat specimen. Reproduced with permission from reference 26.

were positioned such that the crack front encountered t h e m d u r i n g stable ex tension. Typical Ρ-δ traces from specimens containing two P C spheres are shown i n F i g u r e 4. T h e slight reduction i n load at the initiation o f fracture (pop-in) can be seen i n the data from the specimen w i t h full adhesive strength (> 54 M P a ) . W h e n the crack front reaches the obstacles a n d becomes "trapped/' the a p p l i e d l o a d (and hence the stress intensity) rises as the crack front begins t o advance past the obstacles and assumes locally increasingly b o w e d configura tions. A further load increase causes the local crack front to continue to b o w between the obstacles u n t i l the breakaway configuration is reached. A t this point, one or more o f the obstacles are left b e h i n d , and the previously inde pendent crack fronts coalesce into one and continue to grow. I f the i m p e d i n g obstacle was tough enough to be left intact (as was the case here w i t h P C ) , a n d i f it is sufficiendy well-bonded to the matrix, t h e n bridging o f the crack flanks takes place. I n specimens containing P C spheres w i t h adhesive strength re-



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Wedge Displacement (mm) Figure 4. Effect of adhesion on load-displacement data obtained from testing two specimens containing PC spheres: one with full (> 54 MPa) and one with reduced (22 MPa) adhesive strength. R/L = 0.125. Reproduced with permission from reference 26.

d u c e d to about 22 M P a , crack trapping was still exhibited, but the cracks were able to bypass the obstacles b y fracturing along the interfaces. As indicated by the data i n F i g u r e 4, such behavior resulted i n a greatly reduced m a x i m u m app l i e d loading d u r i n g the crack-trapping process. Analysis of Crack-Front S h a p e s . T h e cinematographically recorded images provide clear histories of the crack-front interactions w i t h the obstacles i n each o f the m o d e l specimens. D i g i t i z e d and enhanced reproductions o f frames recorded d u r i n g the testing o f a typical specimen w i t h P C spheres are shown i n F i g u r e 5. These sequential images give direct evidence for the complete evolution o f the crack-trapping mechanism, from the initial " p i n n i n g " o f the crack faces to the final breakaway configuration and transition to bridging, once the crack front has gone around the obstacles and has left



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Figure 5. Digitized images from 16-mmFilm, showing the evolution of local bowing of a brittle matrix crack past tough PC spheres, R/L = 0.125.

them b e h i n d . Registration of the movie films w i t h the applied loading provides the magnitude o f applied loading (and corresponding toughness enhancement) that results from forcing the cracks into the configurations recorded i n each image. D e t e r m i n a t i o n o f Stress Intensity. T h e accurate determination of the elevation o f stress intensity due to the crack-path obstacles i n the specific D C B specimens requires taking note of the special features o f the specimen and its mode of separation. T h e simple beam theory solution does not account for shear deformations or the compliance of the " b u i l t - i n " cantilever ends. A


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m o r e exact analysis considering these problems was developed b y K a n n i n e n using the m o d e l o f a beam resting on an elastic foundation (28). K a n n i n e n s derivation was for crack extension under conditions of constant opening dis placement, a n d yields the M o d e I stress intensity as:

w h e r e the correction term, Ψ, is a function o f the ratio o f ligament length to beam height a n d is not reproduced here; nonetheless, its m i n o r influence i n these experiments was recognized. F o l l o w i n g convention, Ρ represents the ap p l i e d load, Β a n d h are the beam w i d t h and height, a is the crack length, a n d ν is Poissons ratio. A l t h o u g h the application o f equation 1 to data obtained from fracture i n the neat portions o f the D C B specimens results i n K values that are i n exact agreement w i t h the value obtained from our T D C B fracture tests (0.5 MPa>/m, at - 6 0 °C), a procedure was adopted to eliminate the effects of m i n o r speci men-to-specimen differences. T h e toughness enhancements (effective stress intensity (Kf), remotely measured, normalized by matrix toughness) due to trapping o f the crack front b y tough particles were d e t e r m i n e d using the ratio given b y I c

w h e r e the numerator is evaluated throughout the crack-growth experiment, a n d the denominator is evaluated at the point where the crack just reaches the obstacles. T h e effect o f residual stresses was accounted for b y a superposition procedure, using remote loadings measured from testing o f specimens c o n taining no particles.

Discussion I n f l u e n c e o f R e s i d u a l Stresses. T h o u g h the magnitude o f residual t h e r m a l stresses i n the vicinity o f the obstacles i n the m o d e l specimens was small i n comparison to crack-tip stresses, their effect on crack growth was no ticeable and qualitatively predictable. T h e compressive longitudinal stress (-0.4 M P a ) i n the epoxy adjacent to the nylon rods caused a slight retardation i n the rate of crack growth as the rods were approached. Conversely, the longi tudinal tensile stress (0.7 M P a ) near P C rods caused slight accelerations i n crack growth. I n the case o f spheres, the effect o f residual stresses was more dramatic. W i t h P C spheres, the tensile, tangential matrix stress (4 M P a ) at tracted approaching cracks to the sphere equators. In the matrix near the sur-


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face o f nylon spheres, the combination o f compressive tangential (-2 M P a ) and tensile radial (4 M P a ) stresses caused approaching cracks to spall away from the obstacles, so that no crack trapping was i n d u c e d i n these specimens. I n f l u e n c e o f A d h e s i v e S t r e n g t h s . T h e effective toughening due to crack trapping i n specimens containing either nylon or P C rods is plotted i n F i g u r e 6 as a function o f the local bow-out amplitude of the crack, along w i t h a numerical prediction provided by B o w e r and O r t i z (5, 29). T h e particle spacing o f these data (R/L = 0.125) corresponds to an equivalent particle volume fraction o f - 0 . 0 6 . T h e data indicate that, at a given applied stress intensity, the local crack fronts i n specimens w i t h nylon rods were able to advance slightly farther than the n u m e r i c a l m o d e l predicts. This advance was accompanied b y l i m i t e d , albeit steady, debonding (fracture) o f the nylon-epoxy interface,

2.0

.5

1.0

1.5

2.0

Figure 6. Effective toughening (Kf, normalized by matrix toughness, K"lf ) resulting from crack trapping as a function of the local crack advance, normalized by particle diameter (dp). R/L = 0.125. Numerical results, shown by the curve, are from Bower and Ortiz (5, 29). r


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recorded by the upper camera as a bright area growing from the point of crack contact, along the interface to a distance o f approximately one r o d diameter. T h o u g h the crack faces remain " p i n n e d " at the trailing edges o f the nylon rods, the action o f the partial debonding is to reduce the effective R/L and thus d i minish the toughening from the levels that w o u l d be achieved w i t h "perfect" bonding. Apparently perfect b o n d i n g was exhibited by the specimens containing P C rods. N o debonding was ever observed i n these specimens, not even i n those w i t h adhesive strengths reduced by a surface coating o f release agent. T h e effective toughening levels corresponding to local crack advances i n all the specimens containing P C rods or spheres compare w e l l with n u m e r i c a l predictions (which assume perfect bonding) u p to the stages at w h i c h the cracks approach the final breakaway configuration. A t this point, the tougheni n g observed i n the m o d e l specimens begins to exceed the predictions b y as m u c h as 1 5 % . T h e probable cause o f this discrepancy is not clear, but it is likely related to some finite size effect manifested i n the experiments at the h i g h est load levels. T h e behavior o f the specimens containing nylon rods, w h e n compared w i t h the behavior o f those containing P C rods, is consistent w i t h the measured adhesive strengths o f 31 and >54 M P a , respectively. A n unexpected result was that reduction of the P C - e p o x y adhesive strength to about 28 M P a , accomplished by treating the r o d surfaces w i t h a release agent, d i d not result i n any debonding. F u r t h e r reduction, w i t h a second coat, to about 22 M P a still d i d not result i n any d e b o n d i n g or decrease i n trapping-induced toughness. Evidently, the toughness o f the P C - e p o x y interfaces remained high enough to prevent the probing cracks from prying o p e n the interface. I n contrast, the nylon-epoxy interfacial toughness must be sufficiently low to enable the l i m i t e d debonding that was observed i n these specimens. This observation is supported by determinations o f interfacial toughness (30) obtained from analysis o f adhesive-strength tests performed o n specimens fabricated from these material pairs (24). I n those experiments, the toughness o f release-agent-coated P C - e p o x y interfaces was found to be about 2 0 % greater than the toughness of nylon-epoxy interfaces, even though the two systems exhibited similar adhesive strengths. R e d u c t i o n o f the adhesive strength of spherical particles had a m u c h m o r e dramatic effect on their ability to trap cracks, as illustrated by F i g u r e 4. W h e n the adhesive strength o f P C spheres was reduced from above 54 M P a to about 22 M P a , crack trapping was still exhibited, but the cracks were able to bypass the obstacles by fracturing along the interfaces. A s a result, the maxim u m levels o f toughness enhancement achieved i n such specimens was only o n the order o f 1 0 - 1 5 % o f the toughening possible with "perfect" adhesion. M a x i m u m Toughness E n h a n c e m e n t Achievable by C r a c k Trapping. In specimens containing well-bonded obstacles at an equivalent v o l u m e fraction of approximately 0.06 (R/L = 0.125), the m a x i m u m measured


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toughness enhancement consistently approached 8 0 - 9 0 % p r i o r to crack-front breakaway. Increased toughening was measured as a function o f decreased particle spacing, such that at an equivalent particle volume fraction o f about 0.27 (R/L = 0.25), the measured crack-trapping toughening was approximately 270% relative to the neat epoxy. It is emphasized here that the m o d e l matrix material does not craze or produce microcracks. It was completely brittle at the test temperature of - 6 0 °C: the calculated plane-strain plastic-zone size (radius, r , approximated by [K /a ] /6ir) is less than one μηι. Consequently, toughening by local plastic flow or any mechanism other than crack trapping was inconsequential i n the m o d e l specimens.


p

Ic

y

2

Summary Experiments were performed to definitively determine the magnitude o f toughening generated b y crack trapping i n the absence of all other toughening mechanisms. M o d e l composite specimens consisting of a transparent brittle matrix containing two or more tough inclusions i n a row perpendicular to the crack front were fractured under conditions o f stable, controlled crack growth. C r a c k - p a r t i c l e interactions were recorded cinematographically both to d o c u ment the evolution of the crack-trapping mechanism and to enable identifica tion of various stages of crack growth with the applied stress intensity. T h e sample images presented here reveal the evolution of crack-front trapping from the initial p i n n i n g to the fully bowed, breakaway configuration and finally the transition to crack-flank bridging by inclusions i n m o d e l frac ture specimens. T h e quantitative results indicate that toughness enhancement by nearly a factor o f 2, relative to neat matrix values, may be achieved through the crack-trapping mechanism alone, w i t h particle equivalent volume fractions of just 0.06. To produce this level o f toughening, it was found that a high adhe sive strength (of the order of one-third of the matrix flow stress) is r e q u i r e d between the matrix a n d tough spherical particles. W i t h cylindrical trapping ob stacles, the requirement o n the adhesive strength is not as critical because the crack front cannot readily escape from the p i n n i n g configuration.

Acknowledgments Support for this w o r k was provided by the U . S . Department of the A i r F o r c e and is gratefully acknowledged. Opinions, interpretations, conclusions, a n d recommendations are those o f the authors and are not necessarily endorsed b y the U . S . A i r Force. W e thank Professors A . F. B o w e r and M . O r t i z of B r o w n University for providing results o f specific n u m e r i c a l simulations relating to our experimental conditions, and acknowledge several helpful discussions w i t h t h e m . T h e skilled assistance o f M i c h a e l Imbeault (photography) and Chester Beals (computer graphics), both at the Massachusetts Institute o f Technology


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L i n c o l n Laboratory, was instrumental i n this w o r k a n d is gratefully appreciated. T h e generous provision o f epoxy by the Shell C h e m i c a l C o . is acknowledged.


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10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

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