Crack Propagation and Damage Mechanics in Particle-Matrix Composites

A search of the literature has shown that a large number of empirical models exist for predicting the fracture toughness of particle-matrix composites based on their component material properties, but there is a poor understanding of the correlation of these models to the true physical basis for the models. The extent to which variations in fracture toughness can be attributed to factors such as crack deflection, crack-tip bridging, crack-front bowing, and interfacial adhesion, is an important area for new research . The objective of the present work has been to characterize the role of some of these factors by making dynamic observations of a propagating crack in a particle filled brittle matrix composite. High speed photography has been used to obtain photoelastic images of the state of stress at the leading edge of dynamic cracks as they intersect and pass disperse spherical particles in a birefringent polyester matrix. In order to further characterize the nature of the crack tip in the area of embedded spherical particles, cracks were also induced to arrest in close proximity to spherical particles in composite materials of the same type as used in the dynamic experiments. Experiments were performed using brittle polyester matrix materials with embedded disperse spherical particles. The particles were of three types including steel, glass, and rubber. Three series of experiments were performed. In the first series of experiments the physical characteristics of a polyester matrix material were evaluated including tensile strength, elastic modulus and birefringent properties. The second series of experiments utilized a CranzSchardin camera, and a circular polarizer to study the state of stress at the tip of dynamic cracks during progression of the cracks past particles in Modified Compact Tensile (MCT) test specimens. The third series of experiments was performed using the same component materials as the MCT experiments, in the form of single edge notch (SEN) specimens. In these experiments cracks were induced to arrest in close proximity to the spherical particles.

. Coefficients of Thermal Expansion Table 4. 458-C Polyester Fringe Coefficient Table 5. 458-C Polyester Matrix Critical Stress Intensity Kie Table 6. Matrix Material Dynamic Properties LIST OF FIGURES Figure 1 The crack tip stress field with Irwin ' s plastic zone correction                 37. The shape of a semi-infinite crack as it bypasses a single row of obstacles superimposed over bowed crack near a glass particle. Vlll

INTRODUCTION
The characterization of crack-particle interactions, and the study of the effect that crack-particle interactions have on a material's fracture toughness, are important considerations in many composite material systems. Examples of composite materials with properties that have been enhanced by the addition of particles include brittle ceramics with improved fracture toughness from embedded ductile particles, and polymers toughened by the addition of rubber particles.
The macroscopic properties of each of these particle-reinforced composites are dramatically influenced by the micromechanical deformations that occur in the area of the embedded particles.
The ability of two or more constituents to be effective as a composite depends on many factors including particle/matrix volume fraction, particle size, particle/matrix elastic modulus mismatch, matrix-particle adhesion, residual stresses, and also depends on the matrix and particle component properties, such as yield strength, plasticity/ductility and fracture toughness [ 1,2]. The interdependence and overall effect that each of these factors has on the toughness of a composite material is highly complex.
A considerable body of work exists that studies the behavior of particle-matrix composites.
Many of these works are directed towards establishing a fundamental understanding of composite material behavior based on the mechanics of crack propagation in the area of a particle. These include works on crack tip bridging [3][4][5][6], crack front bowing [7,8], crack deflection [9][10][11], and the effects from particle matrix interfacial adhesion [12][13]. The experimental research presented in this paper builds on these fundamental concepts of fracture mechanics of particle filled composites. Also, since only a limited number of previous works have studied crack-particle interactions under dynamic conditions [14][15][16] , the present work expands the knowledge of the physics of interactions of a dynamic crack with embedded particles.
By making both static and dynamic observations of cracks near disperse particles in polyester matrix materials it becomes possible to make a careful analysis of several of the factors that influence crack propagation in particle reinforced composites. These include observations of the variation in the dynamic stress intensity factor near particles with elastic modulus much higher than, and also with elastic modulus lower than, the modulus of the polyester matrix In the case of models of crack propagation in reinforced composite materials, such as with reinforcements bridging the crack tip, the mechanics of fracture at the crack tip are sometimes 3 distinguished from the cohesive zone models as having a near-tip component of matrix material toughness, added to a distinct material toughness component from the reinforcements . The combined model from these two processes might be described as a "small-scale-bridging" model.
The following review of the literature describes the origin of these models with particular emphasis on the bridging models. The review is then concluded with a presentation of recent work that investigates processes in the near-tip zone including crack deflection, crack bowing, crack tip trapping and crack tip bridging.

Irwin's Energy Balance and Energy Release Rate/Stress Intensity Factor
Modem methods for analysis of crack tip stresses originated in 1920 with Griffith ' s [17] concept of energy balance, which was later refined by Irwin's [ 18] addition of an incremental measure of plastic work per unit of crack surface created. An outcome of these works was the concept of stress intensity at the crack tip and an intrinsic fracture toughness of a material.
Irwin and others developed closed-form solutions for the stresses in the area of a crack tip.
These solutions for the state of stress near the crack tip in linear elastic materials describe the stresses as varying with the inverse of the distance from the crack tip (1/-.Jr), and are asymptotic to the crack tip at radius equal to zero (stress field O'ij = kl'1r fij(8) + other terms) [1]. The asymptotic region of stress at the crack tip is said to be the singular zone and is depicted below in figure 1. In this region the proportionality constant K, the stress intensity factor, defines the state of stress.
In 1961 Irwin [ 19] proposed that in the case ofreal materials with non-infinite stresses at the crack tip, the non-infinite stresses can be accommodated by defining a plastic zone at the tip of the crack. This "correction" by Irwin assigned an effective crack length which is the sum of the actual crack size and a plastic zone, represented by an "effective" stress intensity factor Keff = C (aeff) CJ -In aeff, where aeff is a geometry correction factor.
The Irwin plastic zone correction. In 1959 Barenblatt [20,21] [26,27] of the perturbation of a stress field in the presence of a crack. Rose's solution to the perturbed crack with springs/ligaments is a dislocation density along the length of the crack that satisfies the Bilby/Eshelby analysis, and also has an inverse square root singularity at the crack tips. As part of Rose's analysis, he derived solutions of the 6 crack dislocation density that were dependent on the nondimensional crack length parameter of c/a and nondimensionalized spring-crack parameter kl. Rose applied this concept of bridging ligaments to a range of cracks, from the case of small scale bridging ( c/a ~ 1 ), up to reinforcement from a fully bridged crack (c/a = 0). He developed models which employed hard springs (kl>> 1 ), and soft springs (kl<< 1 ), as well as linear and nonlinear springs, and used interpolating functions and numerical solutions for springs of intermediate stiffness. Rose's model is characteristic of a second class of models of crack tip behavior (distinguished from the cohesive models of Dugdale and Barenblatt) known as bridged crack models. While the bridged crack models, like the cohesive models, have tractions between the crack faces, in the bridged crack models the non-linear separation processes allow for a singularity-dominated region at the crack tip. As a result, the bridged crack models can be used to represent non-7 linear regions with more than one physical phenomena such as fracture events with distinct tip and wake phenomena. For example, a bridged crack model of a fiber-matrix or particle-matrix composite has brittle fracture of the matrix (a singularity in the non-linear region), in combination with tractions from crack bridging phenomena such as fiber or particle pullout processes that also occur in the non-linear region, but which might be behind the brittle fracture event. (A)

7 Crack tip bowing
Lange [28] studied the interactions of a crack front with second phase dispersions in a composite, brittle material. He hypothesized, based on the Griffith Model of energy absorption, that the major energy-absorbing process during fracture is the production of surface energy.
Lange speculated (based on experimental observations) that fracture energy increases when inhomogeneities in a brittle material act as obstacles to progression of a crack front. Increased fracture toughness results from "crack bowing" when the crack front increases its length as it is simultaneously pinned by two or more inhomogeneities. Lange's model supposes that the crack front bows out between the second-phase dispersion while still remaining pinned at all the 9 positions where it encounters the dispersion . During crack propagation, both new fracture surface is formed and the length of the crack front is increased due to its change in shape between the pinning positions. Using this model a functional relation between the fracture energy and the dispersion spacing is derived for the case of a Griffith crack under an applied tensile stress. Lange states that " ... the increment of energy absorbed (~U) for an increment of crack extension (~C) can be divided into two parts, i.e. one part (~Us) associated with the energy to form new surface area and one part (~UL) associated with the energy to form the increased length of the crack front." Evans [29] concluded that toughening cannot only be due to particle spacing, and added to Lange's model for crack front bowing by also relating toughening to the inclusion-size-tospacing ratio, rather than just the spacing.
Green et.al. [30,31] studied crack shape changes in the vicinity of un-bonded spherical nickel inclusions in a glass matrix. Green postulated that observed increases in fracture energy resulted from crack tip blunting although increased toughness was less than predicted by theory.

Crack plane deflections
In 1980 Cotterel and Rice [32] solved for the increased driving force necessary to propagate a crack subjected to a local planar tilting misalignment (crack deflection). In their work the tilted stress intensity factors K\ and Kt 11 and Ktm are solved for and stress intensity is plotted as a function of angle of inclination of the crack.
Faber and Evans [9,10] by a geometrical analysis extended Cotterel's and Rice's planar solution to three dimensions (crack deflection including tilting and twisting). They deduced that tilted 10 cracks have mode I opening, and II sliding components, while twisted cracks have mode I and mode III tearing components. Toughening, and crack advance are presumed governed by the strain energy release rate G from the segment of the mis-aligned crack. The effects from aspect ratio of the particles, and particle spacing, were studied for both rods, discs and spheres. This geometrical analysis predicts that tilt is non-contributing to fracture toughening, except in the case of discs, where tilt provides a small benefit. In all cases crack deflection by twisting is the predominant toughening mechanism (refuted later by Pezzotti 1993 [ 11 ]), and rods of high aspect ratio are the most effective for deflecting cracks with increased toughness as high as 4 orders of magnitude for rods with aspect ratio of 12 or higher.

Crack pinning
As noted above, in 1970 Lange speculated (based on experimental observations) that fracture energy increases when inhomogeneities in a brittle material act as obstacles to progression of a crack front. Increased fracture toughness results from "crack bowing" when the crack front increases its length as it is simultaneously pinned by two or more inhomogeneities.
In 1985 Rice [33] developed a three dimensional weighted function theory to study stress intensity factors along a nonlinear crack front. The numerical methods allowed first-order elastic field variations due to varying crack front geometry to be analyzed.
Rose [34] proposed that a two-phase stress intensity factor can be used to model the restraining effect of inclusions. In the case of a bowed crack front between inclusions, Rose characterizes unbroken ligaments stemming from the inclusion and behind the furthest extent of the crack front as "stretched" ligaments which toughen the matrix. The two-phase parameter which accounts for crack front bowing and "crack bridging" is dependent on inclusion size, spacing and volume fraction and describes a property of the two-phase composite which is not necessarily related to the properties of the separate phases. Rose's parameter correctly characterizes the volume fraction dependence of the fracture toughening seen in Lange's experiments.
Evans, Budiansky and Amazigo [4] performed theoretical analysis of elastic-perfectly plastic particles bridging crack faces, spaced to simulate a brittle ceramic bridged by ductile metal particles, and compared their analysis to experimental results. They employed Budiansky's and Rose's distributed spring model for the partially pinned crack. Their model utilizes an elastic spring constant which they derive based on a randomly distributed concentration of "smooth punch" bridging elements. They studied the toughening effects of particle spacing (volume concentration), and from ductility of the bridging elements. In their model they conclude that "Confrontation of experimental data on metal-reinforced ceramics with the predictions of the present theory seems to imply suspiciously high particle strengths. If such high strengths are not confirmed, toughening mechanisms in addition to crack bridging may be operative in particulate-reinforced ceramics." [ 4 p 181] Rice and Gao [8] used Rice's technique to study variations in elastic fields as a crack front penetrates (is trapped by) an array of obstacles. They assumed that progressive states of equilibrium occurred as the applied load increased and the crack front moved forward to achieve a geometry where the stress intensity factor is matched to the local fracture toughness.
Their first-order model extends only to cases of particle toughness close to matrix toughness and also only in cases of crack bowing aspect ratios up to approximately 1.2 Fares [35] used Boundary Element Methods to discretize crack growth profiles. The BEM method can be applied to wider bounds of particle/matrix toughness ratios than Rice's, and also, the BEM method models observed crack behavior more closely including cases when the crack fronts circumvent obstacles leaving bridged particles behind the crack front.
Bower and Ortiz [36] extend Rice's first-order perturbation analysis to include large perturbations by beginning with a known initial geometry (such as a circular or half-plane crack) and using a succession of perturbations to calculate stress intensity factors. As examples of the method they present solutions to the problems of a semi-infinite crack trapped by a periodic array of tough particles, and unstable growth of a semi-infinite crack through material of decreasing toughness. A difficulty associated with the method is the presence of singularities in the integral calculations. The singularities are regularized using the known behavior of cracks (generally known only for planar cracks). Continuity is preserved by using weighted interpolation in the area of the singularities. 13 Bower and Ortiz [6] went on to apply perturbation analysis of crack propagation to include particles which remain intact after a crack front has bowed and then coalesced, after propagating past the particle. This is the case of crack bridging. As the crack faces separate the Mower and Argon [14) studied the process of crack trapping by high toughness inclusions that had elastic modulus near the matrix modulus. The result is crack bridging by the tough particles. The effects of particle size, spacing, adhesion and residual stresses on the crack front 14 behavior and toughness were observed. Bowed crack front shapes were seen to match Bower's and Ortiz's predictions very well.
While working with particle reinforced ceramics Pezzotti [11] differentiated between toughening achieved by microfracture mechanisms at the crack tip, from those in the wake zone following behind the crack tip. His small scale bridging model describes these two toughening effects that can give rise to R-curve behavior because of the potentially cumulative nature of the wake zone effects. In 1996 Pizzotti et al. [3] studied the possible existence of an intrinsic fracture toughness value in ceramic composites. They described both the Dugdale and the distributed spring models as involving shielding-stress distribution functions which are continuous over the entire range of crack length. They contrasted a continuous stress distribution for fiber-reinforced materials, to the observed R-curve ("pop-in") behavior in particle reinforced ceramics experiments, and concluded that a discrete stress function is more physically realistic when the inter-particle distance is of the same order of magnitude as the bridging zone length . When only one or few particles act as the bridges in the wake of the popin crack, Pezzotti et al. modeled the particles behavior as stiff nails held between the crack faces and locally exerting point forces that pin the crack faces. They performed FEM calculations assuming infinite strength of the interface, and with bridging particles that were considered to behave elastically. They generally found that, in the range of Young's modulus and strength reported for ceramic single-crystals, no more than one single particle could operate elastic bridging whatever the volume fraction, which was consistent with the particle morphology in their experiments. 15

Crack path selection
Erdogan et. al. [3 7] discuss crack propagation of an arbitrarily oriented crack near an inclusion and the tendency of a crack to propagate towards or away from a perfectly bonded inclusion depending on relative elastic properties. They use the "single-valuedness condition of displacement" [37 pl009] and the superposition of the problems of 1) stress distribution near a circular inclusion without the crack, and the problem of 2) stress disturbance of the normal and tangential stresses resulting from the presence of a crack, to calculate stress-intensity factors.
Their result predicts that a crack will propagate toward an inclusion if the inclusion modulus is less than the matrix and away if the inclusion has higher modulus.
Dekkers and Heikens [38] found that the mechanisms for craze formation near glass beads in a matrix are fundamentally different for adhering and non-adhering beads. propagate along the interface, or deviate into the adjoining material and extend parallel to the interface. They found that for a given combination of opening and shear loading at the tip of an interfacial crack, as characterized by the phase angle l.jl, there is a critical ratio of interfacial and matrix toughness, R1 !Rm, which must not be exceeded if the crack is to remain in the interface. They showed that the measured interface fracture energy may be strongly influenced by the crack trajectory, as governed by l.jl, due to crack shielding and plasticity effects.
Lefebvre, Gerard et al. [ 40] studied the influence of an elastomeric interface layer on the fracture properties of glass beads in an epoxy matrix. The critical stress intensity is seen to increase fracture toughness and is attributed to the promotion of shear yielding in a localized zone near the crack tip.
Lee, et al. [ 41] considered whether a driving force for a crack to grow along a remote interface can exceed the fracture energy of the interface, before the driving force for the primary crack approaching the interface exceeds the fracture energy for the matrix. They found that an interface may deflect primary cracks once it contains defects and there is enough driving force for its growth. However, this is true only when the interface debond crack does not kink out of the interface and when the primary crack stops with no further crack growth. They studied these behaviors by making static observations of the morphology of cracks which they had caused to arrest in Single-Edge-Notch specimen fracture experiments.

CHAPTER ID THEORETICAL ASPECTS OF BIREFRINGENCE AND STRESS STATE AT THE TIP OF A DYNAMIC CRACK
Particle-filled composites generally exhibit isotropic behavior. This is because the particles are uniformly distributed, and the physical behavior of the composite is symmetric in three dimensions. In some cases, such as injection-molded components where particle alignment might occur due to the molding process, particle filled composites can exhibit a higher degree of anisotropy. However, the experiments reported here study the behavior of symmetric particles embedded in a matrix.
In order to isolate the effects of interaction between a dynamic crack and a single embedded particle, experiments were performed on materials comprised of disperse spherical particles in a brittle matrix. By having spacing of approximately 5 diameters between particles and a total matrix thickness of approximately 4 diameters, the effects from particle interactions were minimized. In this way, observations were made of dynamic cracks propagating through an orthotropic medium, and of the dynamic crack encountering disperse embedded particles in the medium. Khanna [13], in dynamic photoelastic and strain gauge experimental work, calculated that the correction factor for dynamic stress intensity factor due to the effect of materials bridging the crack tip in an orthotropic medium is less than 2%. The experimental work presented in this paper utilizes photoelastic methods for orthotropic media in making determinations of stress intensity. The theoretical basis for determining the stress intensity factor and stress field around a dynamic crack in orthotropic materials is presented here. 18 Sanford's and Dally's [42]over-deterministic solution relating stress intensity factors at a crack tip to isochromatic fringes using Irwin's series representation of the stress field around constant speed cracks (43] Irwin's series representation of the stress field around a constant speed crack expresses the stresses at the crack tip as a power series containing r(n-1 )/2 terms. The birefringent behavior of polyester matrix materials allows stresses to be observed as dynamic isochromatic fringes at Here Co "'-l2rr is Ki , the stress intensity factor, and 2 Do is the remote stress ax Remembering that the maximum in-plane shear stress 'Tm is related to the Cartesian components of stress (thus to the isochromatic fringes) by: 20 Substitution oflrwin's dynamic stress field equations into the stress optic relationship gives expressions for ( crx -cry) and (•xy) in terms of the series constants and the coordinates of the point x , y in the region of the crack tip, and relates them to the fringe order around the crack tip. This is a system of non-linear equations to be solved for the unknown coefficients Cn and Dm . The over-deterministic method of Sanford and Dally, which is described below, can be used to solve for the unknowns.
Sanford's and Dally's method, as it applies to static stresses at a crack tip is as follows : The local stresses in the neighborhood of crack tip are: with r and 8 being polar coordinates (r/a < < 1 ), and with the origin at the crack tip. The term cr 0 x is a correction term for specimen geometry (from Irwin, Proc. of SESA 16, 93-96 1958) 21 The maximum in-plane shear stress 'tm is related to the Cartesian components of stress (and to the isochromatic fringes) by: Combining these equations gives the relation for the isochromatic fringe pattern in the area of the crack tip:   23 Then, by iteration, the solution for [~k] can be used to correct the previous estimates for K 1 , Kn and CT 0 x by: This is repeated for several iterations until the solutions converge.
In the case of the dynamic stress field equations above, and using the over-deterministic method

Coefficients of thermal expansion
The thermal expansion coefficients of each of the component materials are as follows: Low-Carbon Steel 11 x 10-6 Borosilicate Glass 12 x 10-7

Experimental Determination of Fringe Coefficient
Two     Results for the three samples tested were as follows:  The SEN specimens had the geometry shown in Figure 5, and the MCT specimens had the geometry shown in Figure 6. Test specimens of both types were prepared in an oversized, 300 mm long, by 315 mm wide, by 12 mm deep mQld. The mold (lined with a Mylar film and coated with mold-release to ease sample removal) was first leveled on its side within+/-0.4 mm of horizontal, after which a 6 mm deep layer of polyester was poured into the mold. At the beginning of gelation of this first layer of polyester, an arrangement of particles was made on the surface. While the first layer was still not fully cured, the mold was covered over (except one comer was left open to allow filling), and the mold was inclined to approximately 45°. The mold was then filled by pouring polyester in from the uncovered end. After initial curing of the polyester, the specimens were removed from the mold, and the desired geometry's were achieved by a series of machining operations, and thermal cycles to shape, and stress-relieve the test specimens.

Dynamic Photoelastic Experiments
Dynamic photoelastic experiments were performed using the MCT test specimens. The

4.11
Test apparatus -dynamic experiments In each dynamic experiment the MCT specimen was statically loaded to 150 pounds, using a hydraulically driven transverse wedge in a split-D fixture . This method of loading resulted in a predetermined stress intensity, Kin, at the initially blunt crack tip prior to initiation of fracture.
A solenoid-actuated knife was than used to initiate a dynamic crack in the specimen. The moving crack operated the Crantz-Chardin Spark-Gap camera by cutting a conductive film on the surface of the specimen, thereby triggering the camera just prior to progression of the crack past the first embedded particle.
Two groups of ten consecutive photographic images were taken in each experiment. By carefully measuring the elapsed time between each photo (approximately 5 microseconds), and the delay between the two groups of photos (approximately 20 microseconds), sequential photos of the dynamic crack in the area of 2 widely spaced particles were obtained during each MCT specimen experiment. The result was ten photos depicting the photoelastic state of stress at the tip of the dynamic crack as it approached, intersected, and exited the region of the embedded particles. Similar experiments were performed with specimens made with each of the three particle types.

Photographs of isochromatic fringes as a dynamic crack intersects a Buna-N particle
Photographs of the dynamic interaction of the crack with an embedded particle, in an experiment with Buna-N particles, can be seen below. The trigger input that initiated the camera was also used as an input trigger to a digital oscilloscope to begin a trace on the oscilloscope. Using a light detector at input "A" to the oscilloscope, the elapsed time between sparks (and time between photographs) of the Crantz-Chardin Spark-Gap camera was measured. By using the photographs to make dimensional measurements of the crack advance from one photo to the next, and knowing the time elapsed between photos, the crack velocity was determined. Velocity profiles are shown below in  The dynamic stress intensity factor The dynamic behavior of spherical particle reinforced brittle matrix composite materials was studied using the birefringent behavior of the 458-C polyester matrix. The twenty high speed still photographs of the advancing crack, obtained using the Crantz-Chardin Spark-Gap camera,   ... As can be seen in figures 1 7, 18 and 19 the apparent change in the stress intensity factor at the crack tip as it passes the embedded particles is small, except for the case of the glass particle which shows an increase of approximately 30 percent as the crack passes the particle. Also, the velocity of propagation of the crack tip by the particles is constant, except for near the glass particle where the velocity is reduced. In the case of the Buna-N rubber spheres, the stress intensity factor dips as the crack tip exits the area of the particle. In the case of the steel spherical particles the stress intensity factor either remains the same or increases slightly as it exits the particle, with higher scatter in the data.

MCT specimen fracture surface morphology
The surface morphology of the MCT specimens was studied after each experiment.   The samples with glass and steel spheres had cracks with surface morphologies that were alike, and that were more complex than that of the Buna-N rubber spheres. Like the samples with Buna-N spheres, the advancing crack first intersected the glass or steel spherical particles at or near the equator of the particle (see figures 24 and 28), and then began to travel by the particle as a planar crack in the matrix. But, unlike the Buna-N crack trajectory, the leading edge of the crack in the matrix advanced at a slower velocity near the sphere than away from the sphere (crack tip bowing). Also, upon intersecting the glass or steel spheres, the crack progressed around the hemispherical particle-matrix interface, at the same time that it progressed through the matrix at the equator of the sphere. And, unlike the planar crack trajectory in the matrix around the Buna-N particles, the cracks in the matrix near the glass and steel spheres was not planar, and was instead deflected slightly towards the pole. In the present work the interactions between an advancing crack in a brittle matrix, and spherical second phase particles embedded in the matrix, has been studied in order to learn the factors influencing the intrinsic fracture toughness of particle-filled brittle matrix composite systems. In particular, the present paper is focussed on the evolution of the crack front as it passes a single particle. Several fundamental aspects of crack-particle interactions are observed during the evolution of crack advance in these experiments. Crack advance is seen to include processes of crack deflection, crack-front bowing, crack face bridging, and crack path dependence on particle-matrix interface de-bonding. The role of matrix-particle elastic modulus mismatch, and the role of matrix-particle interfacial adhesion are seen to be parameters that largely define the path and shape of the advancing crack in the area of the embedded particles.
The crack front progression, and particle-matrix-crack tip interactions have been studied by a variety of techniques. These include static observations of arrested cracks that were induced in single edge notch specimens (SEN). In these experiments, the energy available at crack initiation was varied, with the result that cracks were made to arrest at varying stages of interaction between the crack front and a spherical particle.
A series of dynamic experiments were also performed, using photoelastic techniques to make observations of a dynamic crack-front passing spherical particles in modified compact tensile specimens (MCT). The particle spacing and matrix material thickness was kept the same in both the SEN and MCT specimens, which allowed inter-comparison of crack propagation results. And finally, by observation of crack surface morphology in the area of the spherical particles (including an MCT specimen with a crack arrest event due to a bridged crack at a particle), it has been possible make several conclusions about crack-particle interactions. The mechanisms that contribute to a material's fracture toughness that have been investigated in this work are: crack deflection, crack tip bowing, crack arrest, crack bridging, and the relationship between interface strength and crack path selection. The purpose of the work has been to assess experimentally the size of the process-zone, the toughening contribution by the various mechanisms, and the role of the mechanical properties of each individual constituent phase.

Mismatch of particle/matrix elastic modulus
An elastic modulus mismatch between particles and the matrix is known to result in uneven stress distribution around the circumference of the embedded particles. Eshelby [44] developed a closed form solution for the state of stress inside a particle in an isotropic material under a remote stress as follows: where K, , µ, ,~and f..lo are the bulk and shear modulii of the particle and the matrix respectively, and rpo 00 = 2(4-5v 0 ) 15 (1-v 0 ) with V 0 being Poisson's ratio of the matrix.
By fabricating composites using Buna-N rubber particles with elastic modulus less than the polyester matrix material (Ebuna 1Ematrix < 1), and using Glass and steel particles with elastic modulus much higher than the polyester matrix (Esteel/glass 1Ematrix >> 1), the effect of elastic modulus mis-match on the path of a dynamic cracks in a particle-matrix composite was studied. Depicted below in figures 30a and 30b are the locations of stress concentrations near embedded particles for the case of elastic moduli of the particle higher than the composite matrix material (Eparticle 1Ematrix >> 1 such as with Glass or steel), and lower than the matrix material (Eparticle 1Ematrix < I such as with Buna-N rubber), under conditions ofremotely applied loading, as was the case in these experiments Stress concentrations around and within the particle Tangential stress concentration around the particle As can be seen in figures 26 and 29 above, the crack propagation paths near the steel and glass particles were similar, while, as can be seen in figure 23, the crack propagation path near the composite with Buna-N rubber particles was quite different. The path of cracks in the polyester matrix near the low modulus rubber particles was along the mid-plane of the particles, with crack fronts that were planar and linear. These low modulus rubber particles remained intact, and bridged the crack faces, by remaining bonded to the matrix along the embedded hemispheres on both crack faces. The two hemispherical matrix-particle interfaces continued to remain bonded to the opposing crack faces long after the crack had progressed by the particle.
In the case of the high modulus glass and steel particles, the crack front exhibited a nonlinear progression as it advanced both through the matrix on each side of the particle (nearly planar, nonlinear matrix crack front), while at the same time progressing along the hemispherical particle-matrix interface (highly nonlinear hemispherical interface crack). The crack paths reformed as a planar linear crack approximately one diameter past the particles. As can be seen in figures 20 and 21 above, crack tip bowing and out of plane crack deflection were observed for both steel and glass particles, and in the case of glass particles, figure 22 above, crack arrest was also observed.

5.3
Effects from crack-to-particle spacing (crack path deflection near a particle) In the present work the specimen thickness was 4 particle diameters (12mm/3mm). The space between particles in the plane of the specimens was four particle diameters or greater in all directions. During these experiments cracks intersected particles on the particle centerline, offcenter, and cracks also passed at varying distances "near particles". That is, cracks passed by particles at distances greater than 2 diameters (minimal effect on crack trajectory) and at all distances less than two diameters, with varying effects on crack path trajectory. Cracks that passed particles at a distance of 1 diameter or less from the particle, but without intersecting the 50 particle, showed evidence of a significant non-uniform strain field around the particle. This was observed as non-planar crack faces as shown in figure 31 below. In the case of cracks that passed at a distance of 2 diameters away from particles or greater there was no observable deflection of the crack face . Moschovidis and Mura [45) showed  Crack surface morphology near a rubber particle showing a non-planar crack path. Spacing is approximately 2/3 particle diameter from the particle. 51

Thermal expansion coefficient mismatch
Another material property mismatch that had a bearing on crack propagation paths is the thermal expansion coefficient mismatch which caused residual stresses due to cooling from the elevated curing temperatures during specimen fabrication. For the case where a spherical particle is embedded in an infinite medium, the stresses in the matrix at a distance r from the center of the particle are given by Miyata  EP where crrr and, cr 88 are radial and tangential stresses, respectively, a is the thermal expansion coefficient (see table 3) tff is the cooling range and R is the particle radius. Figure   Residual stresses, and possible fracture paths, near embedded particles for the case of thermal expansion coefficient a of the particle higher than, and lower than the composite matrix material ( aparticle /amatrix >I such as with rubber, and aparticle /amatrix < I such as with Glass or steel -0- Im > Ip

5.5
Crack path as a function of mixed mode loading at the particle matrix interface In these experiments the high modulus particles became unbonded as the crack moved by the particles. The mode of crack advance was by progressive failure at the particle matrix interface. The degree of adhesion of the matrix to the high modulus particles was such that the through-thickness crack front advanced with varying velocity, and in preferential directions, along the particle-matrix interface. Evans, He, Hutchinson and Dalgleish [39] have postulated that the crack trajectory is dictated by the mixed mode loading phase (\jf =function of the ratio shear mode versus tensile mode loading), and by the relative fracture toughnesses of the materials at the interface. Evans et al. define the loading phase angle \jf as: where r is the distance from the crack tip, and u and u are the relative shear and opening displacements, and with where Dundurs' parameters a and 13 are given by: Depicted in figures 33a and 33b from Evans' et al. ' s [39] work are predictions of interface crack path directions for materials with varying elastic modulus mismatches. In the case of particles with higher modulus (such as steel and glass, in polyester matrix), negative phase angles exist around the equator of the particles (\II from high shear component, and negative sign from particles with modulus higher than matrix). Phase Angl e of Load ing. \jl ; In the case of cracks advancing past spherical particles with higher modulus than the matrix, the hemispherical geometry of the crack-particle interface results in variations in the degree of mixed mode loading along the crack front. This is seen in the interface cracks of the steel and glass particles in the experiments reported here. As the crack tip passes over the pole of the particle (on the centerline of the steel and glass particle/MCT specimens), the crack front accelerates into a region of tensile loading. At the same time, in the region of the matrix on either side if the particle (away from the particle, towards the outside surfaces of the MCT specimen), the loading is also tensile. But, in the region between the equator of the particle and the pole of the particle, there is a sharp transition from the remote tensile loading in the matrix, to highly mixed mode loading at the equator, and back to tensile loading over the pole of the particle. It is this variation in relative shear mode u, and opening mode µloading (\Jf = f (KJ  As can be seen in figure 34, as the crack progresses towards the back edge of the particle, a Vshaped stepped surface is formed as the crack exits the particle-matrix interface. This is due to a transition from tensile loading over the pole of the particle, to mixed mode loading behind the pole. At the same time the two arms of the crack front close to the equator (in mixed mode loading) coalesce with the out of plane interface crack exiting the pole, forming a non-planar crack purely in the matrix. Approximately one diameter past the particle the crack front has reformed as a linear and planar crack.   These curves are modeled after crack front bridging where the fracture toughness of the bridging element is greater than that of the matrix, and is a constant across its cross-section. As a result, the bridging particle either remains intact as the crack front passes, or the r:esulting fracture of the bridging element is behind the more advanced (bowed) crack front on either side of the bridging element.
It can be seen in figure 3 7 below that when the crack front bowing profiles described above are superimposed on the real crack front bowing of these experiments, that the shape of the crack front in the matrix has evolution similar to the case of the puturbation analysis. This is in spite of a much more complex crack front evolution inside the area of the particle-matrix interface than that of the purturbation analysis model. Figure 37. The shape of a semi-infinite crack as it bypasses a single row of obstacles superimposed over bowed crack near a glass particle.

CONCLUSIONS AND RECOMMENDATIONS
The only studies known to this author, involving dynamic fracture mechanics in the area of embedded particles, include a series of experiments that employed the method of caustics to study stress intensity factors during crack-particle interactions [15,16], and a series of experiments that employed low temperatures to obtain slow, stable crack progression past particles [14]. A new series of experiments have now been performed using photoelastic techniques and the birefringence of a brittle polyester matrix with embedded spherical particles, to study the interactions between dynamic cracks and embedded spherical particles. The study using caustics to measure stress level employed single edge notch specimens having particle diameters equal to the thickness of the specimen. With particle diameters equal to the thickness of the specimen the fracture experiments are more representative of a through thickness defect in a flat plate, than that of a spherical particle embedded in a matrix. In the present study, difficulties associated with fabrication of a specimen with embedding particles inside a matrix with material thickness greater than the particle diameter, have been overcome. The present study is of dynamic crack propagation in the area of particles with diameters of 1/4 to 1/3 the overall thickness of the matrix.
The stress induced birefringence of the polyester matrix allowed photoelastic observation of dynamic stresses at the crack tip during crack propagation. Modified compact tension specimens (MCT), 12mm thick, with 3.97mm diameter spherical particles spaced approximately 6 diameters apart on the mid-plane of the specimen were used in the dynamic fracture experiments. 59 By fabricating composites using Buna-N rubber particles with elastic modulus less than the polyester matrix material, and using Glass and steel particles with elastic modulus much higher than the polyester matrix, the effect of elastic modulus mis-match on the path of a dynamic crack in a particle-matrix composite has been studied.
It was found in fracture experiments with low modulus rubber particles that the path of cracks were through the mid-plane of the particles, with crack fronts that were planar and linear. Also, the Buna-N rubber particles bridged the crack faces by remaining bonded to the matrix on hemispheres on each crack face. These hemispheres remained bonded to the opposing crack faces long after the crack had progressed by the particle. The net stress intensity factor, observed as birefrigent fringes of the through thickness crack tip, was briefly reduced as the crack exited the vicinity of the Buna-N rubber particle. The average velocity of crack propagation was constant, although fringe tilt, which is an instantaneous indication of crack tip acceleration was observed as the crack exited the particle.
The high modulus steel particles became unbonded along one hemisphere as the crack progressed by the particles, by progressive failure at the particle matrix interface. The degree of adhesion of the matrix to the high modulus steel particles was such that the crack advanced with varying velocity, and in preferential directions, along the particle-matrix interface according to the mixed mode loading phase(\!!= Ki !Ku). A small degree of crack tip bowing was observed, as well as out of plane crack deflection. The net through thickness crack tip stress intensity factor showed small variations, due to the compounded effect of retarded crack advance at the periphery of the particle, and accelerated crack advance across the pole of the particle. As with the Buna-N rubber particles, the average velocity of crack propagation past the steel particles was nearly constant, although fringe tilt, which is an instantaneous indication of crack tip acceleration was observed as the crack exited the particle.
Like the steel particles, the high modulus glass particles became unbonded along one hemisphere as the crack progressed by the particles, by progressive failure at the particle matrix interface. Again, like the steel particles, the degree of adhesion of the matrix to the high modulus particles was such that the crack advanced with varying velocity, and in preferential directions, along the particle-matrix interface according to the mixed mode loading phase (\jl = Ki !Ku). Crack tip bowing was more pronounced in the case of the glass particles and crack arrest were observed, as well as out of plane crack deflection. Unlike the steel particles, the glass particles showed a more distinct, approximately 30% increase in the net through thickness crack tip stress intensity factor, as well as a reduction in crack tip velocity. Like the steel particles, there was retarded crack advance at the periphery of the particle, although this effect was more pronounced in the glass particles. Also, there was accelerated crack advance across the pole of the particle.
In order to better understand the nature of the crack-particle interactions observed as fringe patterns in the dynamic experiments, additional studies were performed using Single Edge Notch specimens. Cracks were induced to arrest near the embedded particles. This allowed an approximate determination of the shape of the crack front, which was then correlated to the stress intensity factor and crack velocity.
One of the outcomes of the work reported above, is that it makes it possible to recommend several avenues for continued study of the interactions of crack fronts near embedded spherical particles in brittle polyester matrix materials. First, the particles studied in this investigation have elastic modulus either much lower than the matrix material (Buna-N particles), or many 61 orders of magnitude higher than the matrix materials (glass and steel). And, due to either the particles high elongation (Buna-N), or due to toughness that was higher than the matrix materials (glass and steel), none of these particles fractured as a result of progression of the crack by the particles. A possible means to achieve higher overall composite toughness in particle filled composites is to use particle materials with elastic modulus nearly equal to the matrix materials, but with higher toughness, such as with Lexan spherical particles. In future experiments, by using particles with elastic modulus nearly equal to the matrix materials, the path of the crack would not be deflected by unequal strain, and the associated stress concentrations at the matrix-particle boundaries would be reduced prior to cracking. Also, the tougher materials would tend to resist cracking, possibly resulting in crack trapping behavior that could be observed in the MCT photoelastic experiments.
Second, the effect from interfacial adhesion between the polyester matrix and the glass particles is believed to have been an important factor affecting the interactions of the cracks and particles. In these experiments there was moderately increased fracture toughness near the glass particles. This slightly increased toughness is consistent with predictions in the literature for enhanced toughness from crack tip deflection around particles. Additional work could be performed to study the effect from varying interfacial adhesion.
And third, the effect on fracture toughness from the interaction of one Buna-N particle with the advancing crack, which was seen in these experiments as a reduction in fracture toughness as the crack exited the area of the particle, could be studied and compared to predictions in the literature. The mitigating influence from bulk modulus effects as the crack first intersects the Buna-N particles, coupled with the near perfect adhesion and 300% or greater elongation of the Buna-N materials, have a role in the resulting fracture toughness. These influences, as well as the effect from increased particle density, could also be studied as part of continuing work.