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Deformation and fracture of a uniaxially reinforced composite material
Composites Science and Technology 59 (1999) 1847±1859
Deformation and fracture of a uniaxially reinforced composite material K.M. Ganey a, J Botsis b,* a Motorola, Semiconductor Products Sector, 2100 E. Elliot Road, MD EL 725, Phoenix, AZ 85284, USA Department of Mechanical Engineering, Swiss Federal Institute of Technology, CH - 1015 Lausanne, Switzerland
b
Received 26 June 1998; received in revised form 8 February 1999; accepted 12 February 1999
Abstract The eects of ®ber spacing on the mechanical response and fracture properties of a composite made of an epoxy resin and optical glass ®bers have been investigated. Two specimen types were tested: a notched specimen reinforced with one layer of equally spaced ®bers (monolayer) under remote tensile longitudinal loads and a specimen reinforced with several layers of equally spaced ®bers (multilayer) in a compact-tension (CT) con®guration. The experimental results indicate that in thepmonolayer specimens, the mac i roscopic stress at the onset of non-linearity, ci and ®ber l spacing, l are related in the form is a constant). The linear x 1 x 1 c p p portion of the stress/strain curves scale with the ratio lx = B (B is the specimen width). In the multilayer specimens, an eective i stress at the p onset of non-linearity, A , depends on the ®ber spacing along the ligament direction, lx , according to Ai Ao 2 = lx 2 and Ao are constants). Approximating the layers of ®bers with strips of eective material, two-dimensional simulations for an eective stress at crack initiation on the CT specimen are carried out by using a boundary-element linear-elastic model. The results of the simulations support the trend of the experimental data. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: B. Fracture; Composites; Fiber spacing; Scaling
1. Introduction The understanding of strength characteristics and fracture behavior of composites with long aligned ®bers has been the subject of intense investigation in the past several years. This is demonstrated by the many pieces of reported work that describe matrix fracture, crack bridging and interface properties in various composite systems. These analyses have played an important role in eorts to characterize the response of composite materials. Mechanistic investigations have shown that several processes accompany the deformation and fracture of composite materials. Some of these processes are related to micromechanics and others to macromechanics, thus involving a wide range of dimensional scales. Mechanical tests to determine the behavior of composites are usually performed on commercial or model composites and include the ®ber pull-out and push-out tests. It has been dicult, however, to investigate basic micromechanisms that contribute to strength and fracture * Corresponding author.
through testing of commercial composite materials and to relate interface behavior in pull-out or push-out tests to the overall behavior of these materials. Thus, it is a formidable task to determine the contributions of crack®ber interactions, tractions in the bridging ®bers, interface debonding and sliding, energy dissipation in the matrix material, etc., to the overall response of composite materials. The interaction of these mechanisms is the main obstacle to the characterization and modeling of fracture in a large class of modern composite materials. As an eort to further understand the role of the various elementary events leading to the fracture of composites and other heterogeneous materials, Botsis and co-workers have reported experimental and numerical research on composites with well-controlled ®ber spacing [1±4]. This approach oers certain advantages. By controlling the ®ber distribution, the properties of the constituent materials and the interface, various elementary events responsible for the non-linear response and fracture can be examined experimentally. Thus some of the pertinent parameters (®ber size, spacing, interface, etc.) aecting strength, crack growth and debond development can be identi®ed and thoroughly
0266-3538/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S0266 -3 538(99)00044 -5
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characterized. Such characterization is particularly important when a continuum approximation of the reinforcement is not realistic. These cases may arise when studying: (a) crack initiation, (b) driving forces of relatively short cracks, (c) the area around the tip of a long crack, (d) criteria for fracture, etc. Several studies have been reported in the literature that investigate the strength and fracture properties of model composites systems (e.g. polyester resin with glass ®bers [5]; epoxy resin with ductile metal wires [6]; PMMA with aluminum ®bers and other brittle systems [7]; metallic glass ribbon in epoxy resin [8]; polycarbonate or Nylon inclusions in epoxy resin [9], epoxy resin with aluminum [10]). Most of these studies, however, do not address the eects of ®ber spacing on deformation and fracture behaviors. In this paper, we report experimental and numerical results related to the mechanical and fracture properties of a composite with well-controlled ®ber spacing. In all studies the matrix material was an epoxy resin and the reinforcement consisted of mono®lament ®bers. Specimens reinforced with one layer of equally spaced ®bers (monolayer specimen) and also with several layers (multilayer specimen) were used in the experimental studies that were fractured under low, displacementcontrolled, tension ramp testing. The main dierences between the present studies and those reported previously [1±4] are in the relatively weak interface of the present composite and the geometries of the specimens. Eorts were concentrated on understanding the nonlinear deformation behavior and fracture response of two specimen geometries as a function of ®ber spacing with constant external specimen dimensions. To further elucidate the eects of ®ber spacing on the multilayer specimens on strength, a 2-dimensional boundary element method (BEM) analysis was used to simulate an eective crack-initiation stress. The paper is arranged so that in each section the results on the monolayer specimens are presented ®rst followed by those on multilayer specimens. Materials and methods are outlined in Section 2. Results and discussion of the mechanical and fracture behaviors of both specimen types are presented in Section 3, based on macro- and microscopic observations, and acoustic emission (AE). Analysis of the experimental data, including numerical simulations for the stress-intensity factor and crack-initiation stress on the CT specimens are presented in Section 4. The results are summarized in Section 5.
matrix. FiberGuide Industries Anhydroguide G optical ®bers, referred to as `®bers', were used as the reinforcing material. The ®bers consisted of a fused silica core, ¯uoride doped silica cladding and polyimide jacket with a total diameter of 465 mm [Fig. 1a]. Manufacturer data for the Young's modulus of the core and strain to failure were 66.67 GPa and 5±10%, respectively. There were two reasons that lead to the selection of the particular matrix and ®bers. The ®rst one was that the resulting interface (i.e. epoxy/polyimide) was relatively weaker as compared to the previous studies of glass/ epoxy and Kevlar2/epoxy systems [2,11]. The second was that the present ®bers have a precise circular crosssection as compared to the tows used previously. Thus, ®ber pull-out and post-failure analysis could be easily observed. The matrix material was a three-part epoxy system, supplied by the Dow Chemical Company. It consisted of a `hard' epoxy resin (D.E.R. 383), `soft' epoxy resin (D.E.R. 732) and an epoxy curing agent (D.E.H. 24). A unit mixture of the epoxy consisted of a hard resin, soft resin and curing agent mixture in a weight ratio of 3.5:1.5:1. Curing of all specimens was carried out at laboratory atmosphere conditions (temperature 22 C and relative humidity 50±74%) for a minimum of 24 h. The ascured specimens were transparent with negligible void formation and small shrinkage. Using standard ASTM methods, the fracture toughness and Young's modulus of
2. Materials and methods 2.1. Materials The composite system investigated consisted of continuous uniaxially aligned optical ®bers in an epoxy
Fig. 1. (a) Cross-section of the ®ber, (b) schematics of molds for the monlolayer and (c) multilayer specimens.
K.M. Ganey, J. Botsis / Composites Science and Technology 59 (1999) 1847±1859
p the matrix material were measured to be 3.3 MPa m and 2.2 GPa, respectively. 2.2. Experimental methods In this work two specimen-loading con®gurations were examined. The ®rst was a specimen with one layer of equally spaced ®bers under a remote load. The second one was a compact tension (CT) specimen reinforced with several layers of equally spaced ®bers along the ligament and thickness directions. These testpieces are hereafter called monolayer and multilayer specimens, respectively. 2.2.1. Monolayer specimens Monolayer specimens were prepared using the mold shown in Fig. 1(b). Grooves 0.25 mm deep, 0.5 mm wide and 0.75 mm center-to-center were machined into each half of the mold ends. This allowed the 0.465 mm diameter ®bers to be well aligned and held ®rmly in place in center-to-center increments of 0.75 mm. Commercially available mold release for epoxy (Crown 3070) was sprayed on the clean mold surface before placing the ®bers. Afterwards, the ®bers, 50 mm longer than the length of the mold, were aligned in the grooves at the desired center-to-center spacing. The mold halves were then screwed together and all joints and grooves sealed with silicone gel (General Electric) to prevent leakage of the epoxy. To eliminate sagging of the ®bers across the length of the mold, their ends were gripped and put under tension. Rubber gasket strips, 2.8 mm thick, were glued to the inside of two 25 mm binding clips. The clips were placed on the ®ber ends in such a manner that the rubber gasket strips ®rmly gripped the ®bers. One clip was then anchored while a dead load of approximately 2 N was applied to the other clip. The dead load resulted in an average pre-stress of 2 MPa on the ®bers. This stress was neglected in subsequent analysis. The epoxy mixture was then prepared and slowly poured into the mold. Two specimen types were machined from the as-cured composite bars, single-edged notched and smooth. For the notched specimens three ®ber spacings were investigated, 0.75, 1.5 and 2.25 mm, corresponding to ®ber volume fractions of 7.3, 3.8 and 2.5%, respectively. The
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machined notch was 1 mm deep, angled at 60 and about 1 mm from the ®rst ®ber. For the smooth specimens two ®ber spacings were investigated, 1.5 and 2.25 mm, corresponding to ®ber-volume fractions of 4.1 and 2.5%, respectively. The overall specimen dimensions are shown in Fig. 2(a) and (b). Using experimental and manufacturer's data for the matrix and ®ber, the rule of mixtures (ROM) was used to calculate the Young's moduli of the composite specimens. Pertinent data for the monolayer specimens are shown in Table 1. Preliminary studies indicated that for the range of 1± 50 mm/min tension tests, the mechanical and fracture response of the monolayer specimens were practically the same (Table 1). Therefore, in the present investigations, displacement controlled experiments with 1 mm/ min for the smooth specimens and 30 mm/min for the notched specimens were performed. The lower rate for the smooth specimens was intentionally chosen to facilitate use of AE sensors in the study of the micromechanics of deformation prior to specimen fracture. Similar studies on the notched specimens were not carried out because all experiments showed that fracture initiated at the notch and subsequently arrested at the interface of the ®ber closest to the notch tip. 2.2.2. Multilayer specimens To prepare the multilayer testpieces, the mold shown in Fig. 1(c) was designed and used. Brass screens with 0.5 mm diameter holes and 0.75 mm center-to-center hole spacing were used to align the ®bers (screens with smaller or larger ®ber diameter can be used). The screens were attached to frames, one which was anchored to the mold and one which was part of a mobile screw mechanism. The frames were positioned apart to approximately the desired length of the composite bar. Prior to ®ber placement, a mold release for epoxy was sprayed on the bottom and side plates of the mold. Fibers approximately 50 mm longer than the desired length of the bar were aligned through the screens at the desired spacing. Silicone gel was used to seal completely the screens and encase the extra length of the ®bers, 25 mm at each end, in one large mass of gel. The gel was allowed to cure for a minimum of 6 h. Afterwards the mobile frame was moved outwards for a few millimeters. The silicone gel massed on the extra length of
Table 1 Testing conditions and results for uniaxial-loaded monolayer composite geometries. Specimen Test rate Fiber spacing type mm/min lx (mm)
Fiber content vol %
Young's modulus (GPa) (experiment)
Young's modulus (GPa) (ROM)
Stress at onset of nonlinearity (MPa)
Fracture strength (MPa)
Notched Notched Notched Smooth Smooth
7.3 3.8 2.5 4.1 2.5
5.8 4.2 3.7 4.6 3.7
6.8 4.6 3.7 4.8 3.7
100 75 53 67 52
2485 1638 1215 16110 1165
30 30 30 1 1
0.75 1.50 2.25 1.50 2.25
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the ®bers and also prevented them from slipping through the screen, thus, putting them under slight tension. Next, the epoxy was slowly poured into the mold producing a composite bar which was machined into
Fig. 2. (a) Geometries of notched monlolayer, (b) smooth monolayer and (c) multilayer CT specimens.
three to four identical specimens. This mold can be used to fabricate composite bars, not only with a regular ®ber distribution, but also with a 3-dimensional random distribution, particularly with mono®lament ®bers. The as-cured composite bar was machined into CT specimens with characteristic external dimensions as speci®ed by the ASTM standards for fracture testing of isotropic materials. A two step notching procedure was adopted for the CT con®guration: ®rst, a straight notch was cut using a straight head disk cutter 1.5 mm in thickness and, second, a razor blade was used to give a symmetrical and relatively sharp tip. The specimen surfaces were hand-polished to allow for in situ observations of crack growth. In the present study, three specimen types were tested: (a) unreinforced matrix specimens with W=36 mm for determining the fracture toughness of the matrix, (b) specimens with W=32 mm for the deformation characteristics and strength of the composite under single ramp testing, and (c) specimens with W=24 mm for AE studies of the deformation and fracture mechanisms, under quasi-static loading/unloading conditions. In all multilayer specimens the reinforcement in the thickness direction had the same center-to-center distance, ly =1.5 mm, while the spacing, lx , was varied along the ligament direction [Fig. 2(c)]. For the deformation and strength characteristics (single ramp testing), three ®ber spacings were investigated, lx 0:75, 2.25 and 3.0 mm. For the AE observations, specimens with two ®ber spacings, lx 1:5 and 2.25 mm were tested. Parameters pertaining to the multilayer specimens are summarized in Table 2. All experiments on the multilayer specimens were displacement controlled with a rate of 1 mm/min. During single ramp testing the actuator was moved at the aforementioned rate until specimen failure. For the quasi-static loading/unloading experiments, the specimens were stretched up to a given level of grip displacement. The level of the grip displacement at the end of the unloading phase was not brought exactly to zero. This was intentionally chosen to avoid compression of the specimens. Corresponding readings of load levels were about 50 N for all specimens. The initial loading phase was programmed for a displacement of 1 mm with subsequent increments of 0.5 mm until the specimen fractured.
Table 2 Average values of parameters for the multilayer CT specimens (test rate 1 mm/min) Fiber spacing lx mm
Specimen length, W (mm)
Crack length, a (mm)
Crack tip to ®ber, (mm)
Stress at onset of nonlinearity
Eective Fracture p toughness, (MPa m)
Fracture strength (MPa)
Matrix 0.75 2.25 3.00 1.50 2.25
36 32 32 32 24 24
16.5 18.5 17.1 16.2 ±a 15.5
n/a 0.50 0.65 0.55 ±a 0.50
33.52 68.07 48.55 57.06 n/a n/a
3.30 6.90 4.55 5.80 n/a n/a
33.52 1368 1095 1055 n/a n/a
a
Fracture morphology prevented post-mortem measurements.
K.M. Ganey, J. Botsis / Composites Science and Technology 59 (1999) 1847±1859
All experiments were performed on an Instron 8500 with Instron Series IX software to collect load/displacement data. A Physical Acoustics Corporation (PAC) Spartan 2000 and PAC SA-DAQ software were used to record AE signals during dual ramp testing. Two 6 mm diameter PAC S9220 sensors were attached to the specimens with silicone gel as the couplant and connected to a PAC 1220A preampli®er with 40 dB gain. Acoustic emission recording parameters were set at 40 dB threshold and peak de®nition, hit de®nition and hit lock-out times of 50, 200 and 300 ms, respectively [12]. Fracture events at the crack tip and the extent of debonding along the ®bers were monitored using a traveling optical microscope with magni®cations up to 200. A video camera was attached to it for real time recording. The extent of debonding along the ®ber was observed with the aid of a light beam directed at an angle of about 30 to the specimen's plane. Optical and SEM microscopy techniques were combined with mechanical and AE data to explain the deformation and failure mechanisms. The reproducibility of the results was examined by performing three to four experiments for each ®ber spacing. A variation of the experimental data less than 5% was seen in the monolayer specimens and less than 10% on the data of the multilayer specimens tested under the same conditions. 3. Results and discussion 3.1. Mechanical and fracture behavior 3.1.1. Monolayer specimens Experimental stress/strain curves for the notched monolayer specimen including the matrix material are shown in Fig. 3. At approximately 1% strain the behavior of the matrix became non-linear with fracture occurring at about 3.6% strain. The data in Fig. 3 show that the addition of reinforcing ®bers not only resulted in stier specimens but also increased the range of linear elastic behavior. The addition of the ®bers increased the strain to failure of the composite specimens by about 20±25%, as compared to that of the pure epoxy, but, without a de®nite trend as a function of ®ber spacing. As shown in Table 1, the elastic moduli as well as the strength characteristics of notched and smooth specimens are practically the same. In all notched monolayer specimens, crack growth started from the notch at a very early stage of the loading sequence. As the crack approached the ®rst ®ber, interface separation was observed along that ®ber. Growth of the interface separation and/or sliding for a few ®ber diameters took place before the crack front reached the ®ber. Upon continuing loading, the crack and the interface separation did not grow any longer and the specimen fractured at some other place of the
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Fig. 3. Typical stress/strain curves for the monolayer notched specimens.
specimen with an audible sound in two pieces and multiple matrix cracking (Fig. 4). Limited fragmentation and secondary matrix cracking followed an inverse trend to the ®ber spacing, i.e. lower ®ber spacing yielded more secondary matrix cracking. The mechanisms of deformation and fracture are discussed in Section 3.2. 3.1.2. Multilayer specimens Load/displacement curves for single ramp loading of the matrix and the multilayer CT composite specimens are shown in Fig. 5. Linear behavior up to fracture prevailed in the specimens of the matrix material since the specimen dimensions were chosen according to ASTM standards for fracture toughness. As for the composite specimens, crack initiation from the notch was observed in a range of 400±500 N. Subsequently, non-linear load/displacement behavior was recorded. Before the crack tip reached the ®rst row of ®ber, the clear epoxy matrix started to become opaque ahead of the crack tip at the location of the ®rst ®ber row. With continued loading, the opaqueness propagated vertically along the row of ®bers. As the crack approached the ®rst ®ber row, similar opaqueness started to grow along the second row of ®bers and propagated parallel to the ®bers. The observed opaqueness indicated a form of interface separation and/or plastic deformation of the matrix material growing along the ®bers. The process of fracture described above, schematically shown in Fig. 6(a) suggests the presence of a crack arrest mechanism, similar to one described earlier for the monolayer specimens. Some details on the underlying mechanisms are given in the next section. Crack growth in the ligament direction was arrested at, or within fractions of a millimeter after the ®rst ®ber
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Fig. 6. Schematics of (a) crack growth, (b) fracture path and (c) a photograph indicating the ®ber surface after fracture on a typical multilayer specimen.
Fig. 4. Schematic of fracture sequence on a monolayer specimen. Phases (2) to (4) were not directly observed and ®ber fractures were seen during post-failure observations.
Fig. 5. Typical load/displacement curves for the multilayer CT specimens.
row. As the grip displacement was increased past the point of crack arrest, fracture began to propagate along the ®ber direction. The load decreased slightly before specimen fracture in the ®ber direction. Fast fracture followed a curved path and the specimen separated into two or three pieces [Fig. 6(b)]. The experimental data in Fig. 5 also show that the addition of the ®bers increased the overall ductility of the specimens. Although a small dierence in the crack lengths between the specimens may have contributed to the observed behavior, note that a weak ®ber/matrix interface existed in these composite specimens. Moreover, as the ®ber spacing was decreased, the amount of ®ber-matrix interface was increased. Therefore, the interface area also has contributed to an increase in ductility and toughness of the composite CT specimens. Other sources contributing to the stiness reduction may be irreversible deformation of the matrix material between the ®bers as well as crack growth. Recall that the external specimen dimensions are the same in all cases with small dierences in the initial crack length (Table 2). Thus, a decrease in the ®ber spacing lx [Fig. 2(c)] resulted in a decrease of the thickness of the matrix material between the vertical layers of ®bers. As a result the extent of non-elastic deformation of the epoxy matrix was more pronounced on the specimens
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with smaller lx . The extend to which the amount of interface, matrix deformation and crack growth contributed to the load/displacement behavior shown in Fig. 5, was not determined in the present studies. It is also interesting to note that the load to fracture was practically independent of the ®ber spacing (Fig. 5). This behavior was attributed to the weak interface and the mechanisms preceding global fracture. It is well known that the sensitivity of crack initiation and growth to local notch geometry decreases with a decrease in interface strength and damage around the vicinity of the notch tip [13]. For the present material, even though the number of ®bers was increased, it did not aect the load to fracture since their eects were diminished or even eliminated by the weak interface and possibly plastic deformation of the epoxy around the crack tip towards the end of the loading course. 3.2. Microscopic observations 3.2.1. Monolayer specimens One of the most interesting features on both specimen types was the onset of non-linear deformation behavior and its dependence on ®ber spacing. To shed light onto the mechanisms controlling the observed behavior, AE methods were coupled with mechanical testing. The number of hits and the associated energies superimposed on the load/displacement curve for a
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typical monolayer smooth specimen are shown in Fig. 7. Note that acoustic activity began as the stress/strain behavior started to deviate from linearity. Both the number of hits and energy were observed to increase proportionally, up to a displacement of about 5 mm, suggesting that only one damage process was dominant. A sharper increase in energy was seen at about 6 mm which continued until specimen fracture. The exact damage processes which result in acoustic activity are usually unknown, especially in new materials. Correspondence of an acoustic signal with a particular event may be achieved by specifying the event with the pertinent acoustic signal through detailed experimentation. While attempts have been made in the literature to correlate acoustic activity with speci®c damage processes no absolute relationships have been found. On the other hand, it is commonly reported that matrix damage, such as plastic deformation, is a source of low acoustic energy while ®ber/matrix separation, debonding, delamination and friction are sources of medium acoustic energy, and ®ber and matrix fracture are sources of high acoustic energy [12]. Yet, without identifying the AE behavior of each mechanism separately as well as in composite form, one cannot make speci®c statements as to the source of acoustic activity. Although an identi®cation was not performed in the present studies, based on the optical observations it is possible that acoustic activity that started at the onset of
Fig. 7. Acoustic emission signals [(a) Hits and (b) Energy versus displacement] of smooth monolayer specimen superimposed on load displacement curve. `n' indicates the onset of nonlinearity (®ber spacing lx 2:5 mm).
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non-linear stress/strain behavior was due to ®ber/matrix debonding and/or sliding. Shortly before specimen fracture, a relatively high energy signal was recorded (displacement 6.85 mm, Fig. 7). This signal may be due to crack initiation around a ®ber as shown in Fig. 8. The specimen did not fracture at this point because the ®bers around the crack reduced the stresses at its tip. Upon continued loading, the recorded signals may be due to either ®ber fracture and sliding or crack growth. The largest peak was seen at specimen failure shortly after the presumed crack initiation and with a very small additional load and audible sound. The scenario of deformation and fracture outlined above suggests that matrix-crack initiation occurred well after the proportional limit. This observation is dierent from what is seen in brittle materials. For example, it has been
Fig. 8. Photographs of fracture surfaces of monolayer specimens: (a) and (b) indicate morphologies of fracture initiation and (c) indicates a typical morphology of the surface at fast fracture (arrows point at ®bers).
reported by Kim and Pagano [14] that matrix cracking in a brittle matrix composite may occur, not at the proportional limit as commonly assumed, but before this limit is reached. In every monolayer specimen, catastrophic failure occurred at the peak load. Composite failure initiated with matrix cracking which propagated along the entire width of the specimen. The ®bers were unable to support the excess load thrust upon them in the moments after matrix cracking (Fig. 8), therefore composite failure occurred in a very short time after matrix cracking. Interestingly, in all notched specimens fracture of the specimens into two pieces occurred away from the notch and shortly after matrix cracking. Failure of the specimens at a location dierent from the notch can be explained by the weak interface present in the specimens. The observed debond growth along the ®rst ®ber caused crack blunting and, thus, a reduction of the stresses at that point. Other defects in the specimen became more severe leading to crack initiation and specimen fracture. The phenomenon of interface failure in the path of a growing crack is typical in composites with `weak' interfaces and has been described by Cook and Gordon [15]. Inspection of the fracture surfaces revealed that the fracture mechanism was identical in all specimens. Fig. 8(a), (b) and (c) show the fracture surfaces of two typical specimens around the location of fracture initiation. The photographs in Fig. 8(a) and (b) indicate that a radial crack initiated at the ®ber/matrix interface, i.e. interface between the ®ber polyimide coating and the epoxy. Note here the smooth fracture surface associated with slow growth of the radial crack and the relatively rough surface indicative of fast fracture [Fig. 8(c)]. Combining the in-situ observations, AE signals and post failure analysis, the fracture process sketched in Fig. 4 was presumed for the monolayer specimens. 3.2.2. Multilayer specimens Fig. 9 shows a sequence of load/time curves superimposed on the AE hits for a specimen with ®ber spacing 1.5 mm obtained during the loading/unloading experiments. A similar behavior was seen on a specimen with ®ber spacing 2.25 mm. Thus, only the results from the specimen with a 1.5 mm ®ber spacing are discussed herein. In the ®rst phase `a', the specimen was loaded up to 800 N. During this phase, low activity was observed at approximately 100, 450 and 800 N. The activity at 100 N was attributed to mechanical noise associated with the alignment of the specimen and grips. Based on the in±situ observations described in Section 3.1, the activity at 450 and 800 N was presumed to be due to crack initiation and growth from the notch. In the second phase `b', no activity was seen until 800 N, the maximum load of phase `a'. The absence of acoustic activity until some previous threshold load has
K.M. Ganey, J. Botsis / Composites Science and Technology 59 (1999) 1847±1859
Fig. 9. Acoustic emission signals of CT multilayer specimen superimposed on load displacement curve (®ber spacing lx 1:5 mm).
been reached is known as the `Kaiser' eect and is an important tool in identi®cation [12]. After this point, the rate of activity was fairly constant until shortly after unloading and was attributed to crack growth and interface sliding. In phase `c', the activity began at 800 N, 200 N less than the maximum load of phase `b'. This observation coupled with the events in phase `b', i.e. activity after unloading, suggested that friction and separation of the interface were taking place after 800 N. Yet the activity rate did not greatly increase until about 1000 N, the maximum load of the previous loop. The rapid increase in activity afterwards suggested that crack growth and interface fracture were taking place (recall the opaqueness growing parallel to the ®ber direction). Similar to phase `b', activity continued until just after unloading began. A similar scenario was repeated in loop `d'. Activity attributed to interface friction and separation began at approximately 150±200 N and continued at a low rate until 1200 N. At this point crack growth began. Interestingly the activity ceased just prior to the maximum
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load. This may have been a sign of crack arrest at the ®rst ®ber row. Note that at all loading steps, crack growth did not resume until the maximum load of the previous loading loop was reached. The acoustic behavior of loop `e', in which specimen fracture occurred, was quite unique. At a load of about 200 N the greatest amount of activity was recorded. Based on the behavior of previous loading loops it was concluded that the intensity of this burst of activity occurred at a load too low to be due to crack growth. Combined with optical observations it was assumed that this activity was due to the interface sliding, most of it already in place from the loading and subsequent unloading of the previous phases. The activity spike at about 1300 N was due to specimen fracture along planes parallel to the ®ber row (Fig. 6). Interestingly, a decrease in the load was recorded prior to reaching the programmed maximum displacement. This correlated well to the observation that the peeling behavior just prior to fracture was accompanied by a slight decrease in load (Fig. 5). The experimental evidence so far in this composite system points to a weak interface. Further evidence of a weak interface was found during SEM observations of the multilayer specimens. The photograph in Fig. 6(c) shows evidence of the adhesive bond between the ®ber and matrix. Therefore, it was assumed that the change in optical properties along the ®bers during loading was due to ®ber/matrix separation (Section 3.1). According to these observations the following scenario of events was assumed. As the crack tip approached the ®rst row of ®bers, the crack induced higher normal and shear stresses at the ®ber/matrix interface causing debonding and interface separation [Fig. 6(a)]. This process is similar to the one well explained in by Cook and Gordon [15]. As the loading continued, interface separation and crack growth occurred prior to fracture along the ®ber direction leading to specimen fracture [Fig. 6(b)]. 4. Analysis 4.1. Monolayer specimens In previous works [1,3] the fracture strength, c , of similar composite geometries was correlated with ®ber spacing, lx , on the basis of dimensional analysis arguments. The behavior of those specimens was linear elastic up to fracture and had the same mechanisms of deformation throughout the entire loading phase. A dierence between the present results and those reported earlier [1,3] is that the composite specimens used herein displayed a nonlinear response after a certain load level. These eects were attributed to a relatively weak interface and the nonlinear behavior of the epoxy matrix. Using dimensional analysis for the fracture
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strength of the data in Fig. 3 may not be appropriate since the mechanisms of deformation and their extend, especially in the non linear part of the curve, depend on lx . Note that for lx 0:75 mm, the entire range of the stress/strain curve is linear. Therefore, in this work the same expression, as in earlier studies [1,3], was used to correlate the stress at the onset of the non linear behavior, ci , with ®ber spacing, lx p ci l = lx 1 p where 1 (Pa m) is a parameter. The data in Fig. 10(a) show that the experimental values for i c follow expression (1).
As a matter of a qualitative comparison, the fracture p stress was also plotted against 1= lx [Fig. 10(a)]. Interestingly, when a linear relationship was forced through the origin, the deviation from the linearity diminished with increased ®ber spacing. Based on the later observation, the linear region of each stress/strain curve was extrapolated until the experimental failure strain of the composite specimens [approximately 4.3% in every case, pFig. 3(a)]. This stress was then plotted against 1= lx . As can be seen in Fig. 10(a), the extrapolated fracture stress displayed the same trend as ci , thus suggesting that non-linear eects may be responsible for the deviation of the fracture stress from relation (1). Fiber spacing was also expected to have an in¯uence on the stress/strain behaviorpof p composite speci the mens. The scaling parameter lx = B was chosen over the entire stress range. It is worth mentioning that a similar ratio is often used in fracture mechanics, i.e. crack length versus specimen width. Scaling or `collapse' of the stress/strain curves in the linear region was observed for the range of ®ber spacing used here [Fig. 10(b)]. The deviation of the scaled curves after the proportional limit is attributed to the dierent extent of non linear deformation in each specimen. 4.2. Multilayer specimens The stress intensity factor (SIF) for a CT specimen can be expressed as [16] p 2 KA A W ÿ aF a=W where A is a nominal stress that accounts for both tension and bending of the ligament of the specimen and, for a specimen of unit thickness, is given by A Atension Abending
PA 6PA a 0:5 W ÿ a W ÿ a W ÿ a2 3
p Fig. 10. (a) Various characteristic stresses plotted against 1/ lx for the monolayer specimen (see text for details.) (b) Scaled stress/strain curves of the notched monolayer specimens.
where PA is the applied pin load; a and W are the crack length and specimen width, respectively [Fig. 2(c)]; F a=W is a dimensionless parameter that accounts for the specimen geometry. To analyze the mechanical and fracture responses of the CT specimens as a function of ®ber spacing, the stress given by Eq. (3) was used. Recall that the only variable ®ber spacing in the multilayer CT specimens was lx [Fig. 2(c)]. From the load/displacement curves (Fig. 5) the load at the onset of non-linearity was measured and subsequently used to determine the corresponding stresses i A from Eq. (3). The results of these calculations as a function of lx are shown in Fig. 11. Each datum represents the average value of stress from specimens with a characteristic length of 32 mm with a particular ®ber spacing while the bars represent a 10%
K.M. Ganey, J. Botsis / Composites Science and Technology 59 (1999) 1847±1859
error. The data in Fig. 11 can be approximated by the following equation p Ai Ao 2 = lx 4 p m) and Ao (Pa) are empirical parawhere 2 (Pa meters. Interestingly, a non zero intercept [as compared to relation (1)] was found for the multilayer specimens and the value of the slope 2 was about 3 times smaller than 1 . Note also that the fracture stress of the CT specimens, calculated from Eq. (3), can be described by a similar equation (Fig. 11). In this work, we do not attempt to give a physical interpretation of the intercept. However, for the data shown in Fig. 11, relation (4) gives Ao =37 MPa. This value is about 4 MPa higher than the specimen fracture stress found experimentally using Eq. (3) for the unreinforced matrix CT specimens. Moreover, it is interesting to point out that in previous studies on multilayer CT specimens of epoxy matrix/E-glass ®bers, the crack initiation stress as a function of ®ber spacing was described by relation (4), or for that matter relation (1), with Ao 0. A key dierence between the present studies and those reported in previous works was that the external geometry varied with the ®ber spacing [3] while here the external geometry was constant. Investigations of strength as a function of a characteristic length in polycrystalline materials (ceramic and metals) have also yielded relationships similar to Eqs. (1) or (4). Two types of mechanisms have been discussed in attempts to explain the strength of polycrystalline solids. When Ao 0, it is assumed that
Fig. 11. Stress at the onset of nonlinearity and fracture stress of the CT multilayer specimens.
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strength is controlled by a Grith-type ¯aw with size proportional to the grain size. When Ao > 0 it is implied that plastic deformation takes place prior to fracture. Expression (4) is known as the Hall±Petch relationship for metals [17] or the Orowan±Petch relationship for ceramic materials [18]. To investigate further the strength behavior of the CT specimen and its dependence on ®ber spacing, numerical analysis of crack initiation in a 2-D model of the actual CT composite specimen, as shown in Fig. 12(a) developed previously [4] was used herein. The remainder of the paper outlines the essential elements of the numerical scheme and presents results pertaining to the eects of the ®ber spacing on the fracture initiation stress on a CT specimen with constant external dimensions. The BEM was used to obtain the strain energies U1 and U2 of a specimen at two crack lengths, a1 and a2 a1 0:1 mm. From these values the energy release rate for mode I loading was calculated as U2 ÿ U1 5 a2 ÿ a1 Assuming linear elasticity, the SIF for mode I loading was found from G1
Fig. 12. Schematics of approximations for the reinforcement. (a) discrete model; (b) eective model; (c) results of numerical analysis using the two 2-D approximations.
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p K1 G1 E0
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6 0
Plane-strain conditions were considered, i.e. E Em / ÿ 1 ÿ 2 . In the present simulations it was assumed that (a) the crack initiated from the pre-crack at the matrix material, (b) the crack front was suciently far from the interface of the ®rst layer of ®bers so that an inverse square root singularity for the stresses can be assumed, (c) the interface behaves as a perfect one, (d) the Young's modulus, Em , was taken as that of the matrix and the Poisson's ratio, was equal to 0.3 for both matrix and ®bers. The stress at crack initiation was determined as follows. Relation (2) can be written as KA P KO P1
7
where KO P1 is the SIF due to the unit force. Usually the applied force, PA , is measured experimentally. However, it can be calculated numerically by assuming that the crack initiates at the matrix and imposing a Grith fracture criterion. Namely, crack initiation begins when the total SIF, Kr, of the composite specimen equals the fracture toughness of the matrix material, Kmc. If the SIF in the composite is proportional to the applied force at crack initiation one ®nds PA Kmc = Kr P1
8
Here Kr P1 is the stress-intensity factor due to a unit applied force in the composite specimen which is determined numerically from Eqs. (5) and (6). It has been reported that when using an experimental value for the Kmc the numerical scheme correlates very well with experimental data of crack initiation stress of a CT epoxy/glass specimen [4]. In view of the weak interface present in the composite specimen examined here, and the perfect interface assumed in the numerical model, it would be unrealistic to correlate the experimental data shown in Fig. 11 with results from numerical analysis. However, using the numerical studies we can examine if expression Eq. (4) is the result of the spacing variation or the result of the specimen geometry. Approximating the actual 3-D ®ber architecture with the 2-D geometries shown in Fig. 12(a) and (b), crack initiation stress [Eq. (3)] as a function of reinforcement for each approximation can be easily simulated numerically as a function of the various geometrical parameters of the CT specimen. In the present studies three discrete approximations for the reinforcement, with spacing 1.5, 2.25 and 3.0 mm, were examined. A layer thickness of 0.5 mm, approximately equal to the ®ber diameter, D, was used as the reinforcement. The modulus of a layer was calculated
using the rule of mixtures, with the elastic properties of the constituent materials and the ®ber spacing taken from the actual specimen geometry. For the second geometry, the discrete reinforcing layers were assumed to behave as a single eective material equal in dimension to the ligament of the specimen minus the distance of the crack tip to the ®rst layer of ®bers. The rule of mixtures also was used here to calculate this `eective' modulus of the reinforcement from the actual discrete ®ber distribution. Another important length parameter in the present simulations was the distance of the crack tip to the ®rst layer of ®bers, . Although care was taken during specimen preparation to assure that this distance was the same in all specimens, small variations from specimen to specimen were not easy to avoid. This distance was between 0.5 and 0.65 mm (Table 2). Since the present simulations were aimed at examining the trend of the initiation stress on a CT composite specimen as a function of ®ber spacing, and not modeling the experimental data, was taken equal to 0.8 mm. The results of the simulations [Fig. 12(c)] show that the dierences between the two approximations are small. As expected, for large values of ®ber spacing, there are some dierences between the two solutions, with the eective model giving lower critical stress. The trend is reversed for small ®ber spacing. Some of these dierences may be attributed to the numerical approximation in the BEM scheme. Note also in Fig. 12(c) the presence of the intercept and the relatively weak dependence of the strength on the spacing of the reinforcement, as in fact was shown from the experimental results (Fig. 11). For a composite specimen with W=32 mm and matrix fracture toughness equal to 3.3 p MPa m, the numerical results shown in Fig. 12(c), extrapolate to a stress of 35 MPa, a value very close to the experimentally measured strength of an unreinforced specimen. The experimental results presented in this paper and in Refs. [3,4], as well as the numerical analysis, demonstrate that not only the spacing of the reinforcement but also the external geometry is important to the strength behavior of the CT composite specimens. Further work is needed to establish the limits of relation (4) and the physical meaning of the intercept. 5. Summary The deformation behavior and strength characteristics of a composite with long aligned ®bers as a function of ®ber spacing were investigated. Two types of specimens were used in the experimental studies: monolayer specimens under remote load and multilayer specimens in a compact tension loading con®guration. For the range of parameters used, the following conclusions can be drawn:
K.M. Ganey, J. Botsis / Composites Science and Technology 59 (1999) 1847±1859
1. In the monolayer specimens, the proportional were limit, ci , and the ®ber pspacing, lx , p prelated according to ci 1 = lx . The ratio lx = B was found to scale the linear stress/stain behavior of the composite. 2. In the multilayer specimens the proportional limit (an eective stress accounting for tension and bending) and the ®ber spacing along the ligament p direction, lx , were related by Ai Ao 2 = lx , where Ao is a stress parameter close to the stress for crack initiation of a CT-matrix specimen. It may be argued that the experimental data, presented in Fig. 11, are limited and relation (4) must be further examined with more experiments. Although this should be the objective of a thorough experimental program, relation (4) is also obtained using a 2-D numerical scheme for the crack initiation stress on a CT specimen with a constant external geometry and variable lx . 3. Real-time observations, combined with acoustic emission data and fracture-surface observations, suggested that weak ®ber/matrix interface damage may have contributed to the nonlinear deformation behavior. 4. The results of the present studies and those reported earlier [3,4] suggest that the intercept Ao depends primarily on the external specimen geometry. 5. The results of the present studies may be very useful in characterizing the in¯uence of the various length parameters on deformation and strength of various heterogeneous materials. They are also directly applicable to deformation and fracture of layered composite materials. Acknowledgements Part of this work was carried out at the Department of Civil and Materials Engineering, University of Illinois at Chicago, IL. The authors wish to acknowledge the ®nancial support from the AFOSR under grant
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93-1-0319 at the University of Illinois. Dr. W. F. Jones was the program monitor. Thanks are also due to DOW Chemicals for providing the epoxy. References [1] Botsis J, Beldica C, Zhao D. On strength scaling of composites with long aligned ®bers. Int J Fract 1995;68:375±84. [2] Botsis J, Beldica C. Strength characteristics and fatigue crack growth in a composite with long aligned ®bers. Int J Fract 1994;69:27±50. [3] Zhao D, Botsis J. Experimental and numerical studies in model composites, part IÐexperimental results. Int J Fract 1996; 82:153±74. [4] Beldica C, Botsis J. Experimental and numerical studies in model composites, part IÐnumerical results. Int J Fract 1996;82:175±92. [5] Harris B, Ankara AO. Cracking in composites of glass ®bres and resin. Proc Royal Soc London 1978;A.359:229. [6] Bowling J, Groves GW. J Mater Sci 1979;14:443. [7] Zok F, Hom CL. Large scale bridging in brittle matrix composites. Acta Metall Mater 1985;38. [8] Friedrich K, Fels A, Hornbogen E. Fatigue and fracture of metallic glass ribbon/epoxy matrix composites. Compos Sci Technol 1985;23:79. [9] Mower TM, Argon AS. Experimental investigations of crack trapping in brittle heterogeneous solids. Mechanics of Materials 1995;19:343. [10] Walter ME, Ravichandran G. Experimental simulation of matrix cracking and debonding in a model brittle matrix composite. Experimental Mechanics 1997;37:126. [11] Beldica C, Ganey K, Botsis J. unpublished data. [12] MISTRAS 2000, users manual. Princeton (NJ): Physical Acoustics Corporation, 1995. [13] Hull D. An introduction to composite materials. Cambridge Solid State Science Series. New York: Cambridge University Press, 1981. [14] Kim RY, Pagano NJ. Crack initiation in unidirectional brittlematrix composites. J Am Ceram Soc 1991;74:1082±90. [15] Cook J, Gordon E. A mechanism for the control of crack propagation in all brittle systems. Proc Royal Soc London 1964;A-282:508±20. [16] Tada H. The stress analysis of cracks handbook. Hellertown, (PA): Del Research Corp., 1973. [17] Hirth JP, Lothe J. Theory of dislocations. New York: McGraw± Hill, 1968. [18] Lawn B. Fracture of brittle solids. 2nd ed. Cambridge Solid State Series. New York: Cambridge University Press, 1993.
Deformation and fracture of a uniaxially reinforced composite material K.M. Ganey a, J Botsis b,* a Motorola, Semiconductor Products Sector, 2100 E. Elliot Road, MD EL 725, Phoenix, AZ 85284, USA Department of Mechanical Engineering, Swiss Federal Institute of Technology, CH - 1015 Lausanne, Switzerland
b
Received 26 June 1998; received in revised form 8 February 1999; accepted 12 February 1999
Abstract The eects of ®ber spacing on the mechanical response and fracture properties of a composite made of an epoxy resin and optical glass ®bers have been investigated. Two specimen types were tested: a notched specimen reinforced with one layer of equally spaced ®bers (monolayer) under remote tensile longitudinal loads and a specimen reinforced with several layers of equally spaced ®bers (multilayer) in a compact-tension (CT) con®guration. The experimental results indicate that in thepmonolayer specimens, the mac i roscopic stress at the onset of non-linearity, ci and ®ber l spacing, l are related in the form is a constant). The linear x 1 x 1 c p p portion of the stress/strain curves scale with the ratio lx = B (B is the specimen width). In the multilayer specimens, an eective i stress at the p onset of non-linearity, A , depends on the ®ber spacing along the ligament direction, lx , according to Ai Ao 2 = lx 2 and Ao are constants). Approximating the layers of ®bers with strips of eective material, two-dimensional simulations for an eective stress at crack initiation on the CT specimen are carried out by using a boundary-element linear-elastic model. The results of the simulations support the trend of the experimental data. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: B. Fracture; Composites; Fiber spacing; Scaling
1. Introduction The understanding of strength characteristics and fracture behavior of composites with long aligned ®bers has been the subject of intense investigation in the past several years. This is demonstrated by the many pieces of reported work that describe matrix fracture, crack bridging and interface properties in various composite systems. These analyses have played an important role in eorts to characterize the response of composite materials. Mechanistic investigations have shown that several processes accompany the deformation and fracture of composite materials. Some of these processes are related to micromechanics and others to macromechanics, thus involving a wide range of dimensional scales. Mechanical tests to determine the behavior of composites are usually performed on commercial or model composites and include the ®ber pull-out and push-out tests. It has been dicult, however, to investigate basic micromechanisms that contribute to strength and fracture * Corresponding author.
through testing of commercial composite materials and to relate interface behavior in pull-out or push-out tests to the overall behavior of these materials. Thus, it is a formidable task to determine the contributions of crack®ber interactions, tractions in the bridging ®bers, interface debonding and sliding, energy dissipation in the matrix material, etc., to the overall response of composite materials. The interaction of these mechanisms is the main obstacle to the characterization and modeling of fracture in a large class of modern composite materials. As an eort to further understand the role of the various elementary events leading to the fracture of composites and other heterogeneous materials, Botsis and co-workers have reported experimental and numerical research on composites with well-controlled ®ber spacing [1±4]. This approach oers certain advantages. By controlling the ®ber distribution, the properties of the constituent materials and the interface, various elementary events responsible for the non-linear response and fracture can be examined experimentally. Thus some of the pertinent parameters (®ber size, spacing, interface, etc.) aecting strength, crack growth and debond development can be identi®ed and thoroughly
0266-3538/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S0266 -3 538(99)00044 -5
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characterized. Such characterization is particularly important when a continuum approximation of the reinforcement is not realistic. These cases may arise when studying: (a) crack initiation, (b) driving forces of relatively short cracks, (c) the area around the tip of a long crack, (d) criteria for fracture, etc. Several studies have been reported in the literature that investigate the strength and fracture properties of model composites systems (e.g. polyester resin with glass ®bers [5]; epoxy resin with ductile metal wires [6]; PMMA with aluminum ®bers and other brittle systems [7]; metallic glass ribbon in epoxy resin [8]; polycarbonate or Nylon inclusions in epoxy resin [9], epoxy resin with aluminum [10]). Most of these studies, however, do not address the eects of ®ber spacing on deformation and fracture behaviors. In this paper, we report experimental and numerical results related to the mechanical and fracture properties of a composite with well-controlled ®ber spacing. In all studies the matrix material was an epoxy resin and the reinforcement consisted of mono®lament ®bers. Specimens reinforced with one layer of equally spaced ®bers (monolayer specimen) and also with several layers (multilayer specimen) were used in the experimental studies that were fractured under low, displacementcontrolled, tension ramp testing. The main dierences between the present studies and those reported previously [1±4] are in the relatively weak interface of the present composite and the geometries of the specimens. Eorts were concentrated on understanding the nonlinear deformation behavior and fracture response of two specimen geometries as a function of ®ber spacing with constant external specimen dimensions. To further elucidate the eects of ®ber spacing on the multilayer specimens on strength, a 2-dimensional boundary element method (BEM) analysis was used to simulate an eective crack-initiation stress. The paper is arranged so that in each section the results on the monolayer specimens are presented ®rst followed by those on multilayer specimens. Materials and methods are outlined in Section 2. Results and discussion of the mechanical and fracture behaviors of both specimen types are presented in Section 3, based on macro- and microscopic observations, and acoustic emission (AE). Analysis of the experimental data, including numerical simulations for the stress-intensity factor and crack-initiation stress on the CT specimens are presented in Section 4. The results are summarized in Section 5.
matrix. FiberGuide Industries Anhydroguide G optical ®bers, referred to as `®bers', were used as the reinforcing material. The ®bers consisted of a fused silica core, ¯uoride doped silica cladding and polyimide jacket with a total diameter of 465 mm [Fig. 1a]. Manufacturer data for the Young's modulus of the core and strain to failure were 66.67 GPa and 5±10%, respectively. There were two reasons that lead to the selection of the particular matrix and ®bers. The ®rst one was that the resulting interface (i.e. epoxy/polyimide) was relatively weaker as compared to the previous studies of glass/ epoxy and Kevlar2/epoxy systems [2,11]. The second was that the present ®bers have a precise circular crosssection as compared to the tows used previously. Thus, ®ber pull-out and post-failure analysis could be easily observed. The matrix material was a three-part epoxy system, supplied by the Dow Chemical Company. It consisted of a `hard' epoxy resin (D.E.R. 383), `soft' epoxy resin (D.E.R. 732) and an epoxy curing agent (D.E.H. 24). A unit mixture of the epoxy consisted of a hard resin, soft resin and curing agent mixture in a weight ratio of 3.5:1.5:1. Curing of all specimens was carried out at laboratory atmosphere conditions (temperature 22 C and relative humidity 50±74%) for a minimum of 24 h. The ascured specimens were transparent with negligible void formation and small shrinkage. Using standard ASTM methods, the fracture toughness and Young's modulus of
2. Materials and methods 2.1. Materials The composite system investigated consisted of continuous uniaxially aligned optical ®bers in an epoxy
Fig. 1. (a) Cross-section of the ®ber, (b) schematics of molds for the monlolayer and (c) multilayer specimens.
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p the matrix material were measured to be 3.3 MPa m and 2.2 GPa, respectively. 2.2. Experimental methods In this work two specimen-loading con®gurations were examined. The ®rst was a specimen with one layer of equally spaced ®bers under a remote load. The second one was a compact tension (CT) specimen reinforced with several layers of equally spaced ®bers along the ligament and thickness directions. These testpieces are hereafter called monolayer and multilayer specimens, respectively. 2.2.1. Monolayer specimens Monolayer specimens were prepared using the mold shown in Fig. 1(b). Grooves 0.25 mm deep, 0.5 mm wide and 0.75 mm center-to-center were machined into each half of the mold ends. This allowed the 0.465 mm diameter ®bers to be well aligned and held ®rmly in place in center-to-center increments of 0.75 mm. Commercially available mold release for epoxy (Crown 3070) was sprayed on the clean mold surface before placing the ®bers. Afterwards, the ®bers, 50 mm longer than the length of the mold, were aligned in the grooves at the desired center-to-center spacing. The mold halves were then screwed together and all joints and grooves sealed with silicone gel (General Electric) to prevent leakage of the epoxy. To eliminate sagging of the ®bers across the length of the mold, their ends were gripped and put under tension. Rubber gasket strips, 2.8 mm thick, were glued to the inside of two 25 mm binding clips. The clips were placed on the ®ber ends in such a manner that the rubber gasket strips ®rmly gripped the ®bers. One clip was then anchored while a dead load of approximately 2 N was applied to the other clip. The dead load resulted in an average pre-stress of 2 MPa on the ®bers. This stress was neglected in subsequent analysis. The epoxy mixture was then prepared and slowly poured into the mold. Two specimen types were machined from the as-cured composite bars, single-edged notched and smooth. For the notched specimens three ®ber spacings were investigated, 0.75, 1.5 and 2.25 mm, corresponding to ®ber volume fractions of 7.3, 3.8 and 2.5%, respectively. The
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machined notch was 1 mm deep, angled at 60 and about 1 mm from the ®rst ®ber. For the smooth specimens two ®ber spacings were investigated, 1.5 and 2.25 mm, corresponding to ®ber-volume fractions of 4.1 and 2.5%, respectively. The overall specimen dimensions are shown in Fig. 2(a) and (b). Using experimental and manufacturer's data for the matrix and ®ber, the rule of mixtures (ROM) was used to calculate the Young's moduli of the composite specimens. Pertinent data for the monolayer specimens are shown in Table 1. Preliminary studies indicated that for the range of 1± 50 mm/min tension tests, the mechanical and fracture response of the monolayer specimens were practically the same (Table 1). Therefore, in the present investigations, displacement controlled experiments with 1 mm/ min for the smooth specimens and 30 mm/min for the notched specimens were performed. The lower rate for the smooth specimens was intentionally chosen to facilitate use of AE sensors in the study of the micromechanics of deformation prior to specimen fracture. Similar studies on the notched specimens were not carried out because all experiments showed that fracture initiated at the notch and subsequently arrested at the interface of the ®ber closest to the notch tip. 2.2.2. Multilayer specimens To prepare the multilayer testpieces, the mold shown in Fig. 1(c) was designed and used. Brass screens with 0.5 mm diameter holes and 0.75 mm center-to-center hole spacing were used to align the ®bers (screens with smaller or larger ®ber diameter can be used). The screens were attached to frames, one which was anchored to the mold and one which was part of a mobile screw mechanism. The frames were positioned apart to approximately the desired length of the composite bar. Prior to ®ber placement, a mold release for epoxy was sprayed on the bottom and side plates of the mold. Fibers approximately 50 mm longer than the desired length of the bar were aligned through the screens at the desired spacing. Silicone gel was used to seal completely the screens and encase the extra length of the ®bers, 25 mm at each end, in one large mass of gel. The gel was allowed to cure for a minimum of 6 h. Afterwards the mobile frame was moved outwards for a few millimeters. The silicone gel massed on the extra length of
Table 1 Testing conditions and results for uniaxial-loaded monolayer composite geometries. Specimen Test rate Fiber spacing type mm/min lx (mm)
Fiber content vol %
Young's modulus (GPa) (experiment)
Young's modulus (GPa) (ROM)
Stress at onset of nonlinearity (MPa)
Fracture strength (MPa)
Notched Notched Notched Smooth Smooth
7.3 3.8 2.5 4.1 2.5
5.8 4.2 3.7 4.6 3.7
6.8 4.6 3.7 4.8 3.7
100 75 53 67 52
2485 1638 1215 16110 1165
30 30 30 1 1
0.75 1.50 2.25 1.50 2.25
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the ®bers and also prevented them from slipping through the screen, thus, putting them under slight tension. Next, the epoxy was slowly poured into the mold producing a composite bar which was machined into
Fig. 2. (a) Geometries of notched monlolayer, (b) smooth monolayer and (c) multilayer CT specimens.
three to four identical specimens. This mold can be used to fabricate composite bars, not only with a regular ®ber distribution, but also with a 3-dimensional random distribution, particularly with mono®lament ®bers. The as-cured composite bar was machined into CT specimens with characteristic external dimensions as speci®ed by the ASTM standards for fracture testing of isotropic materials. A two step notching procedure was adopted for the CT con®guration: ®rst, a straight notch was cut using a straight head disk cutter 1.5 mm in thickness and, second, a razor blade was used to give a symmetrical and relatively sharp tip. The specimen surfaces were hand-polished to allow for in situ observations of crack growth. In the present study, three specimen types were tested: (a) unreinforced matrix specimens with W=36 mm for determining the fracture toughness of the matrix, (b) specimens with W=32 mm for the deformation characteristics and strength of the composite under single ramp testing, and (c) specimens with W=24 mm for AE studies of the deformation and fracture mechanisms, under quasi-static loading/unloading conditions. In all multilayer specimens the reinforcement in the thickness direction had the same center-to-center distance, ly =1.5 mm, while the spacing, lx , was varied along the ligament direction [Fig. 2(c)]. For the deformation and strength characteristics (single ramp testing), three ®ber spacings were investigated, lx 0:75, 2.25 and 3.0 mm. For the AE observations, specimens with two ®ber spacings, lx 1:5 and 2.25 mm were tested. Parameters pertaining to the multilayer specimens are summarized in Table 2. All experiments on the multilayer specimens were displacement controlled with a rate of 1 mm/min. During single ramp testing the actuator was moved at the aforementioned rate until specimen failure. For the quasi-static loading/unloading experiments, the specimens were stretched up to a given level of grip displacement. The level of the grip displacement at the end of the unloading phase was not brought exactly to zero. This was intentionally chosen to avoid compression of the specimens. Corresponding readings of load levels were about 50 N for all specimens. The initial loading phase was programmed for a displacement of 1 mm with subsequent increments of 0.5 mm until the specimen fractured.
Table 2 Average values of parameters for the multilayer CT specimens (test rate 1 mm/min) Fiber spacing lx mm
Specimen length, W (mm)
Crack length, a (mm)
Crack tip to ®ber, (mm)
Stress at onset of nonlinearity
Eective Fracture p toughness, (MPa m)
Fracture strength (MPa)
Matrix 0.75 2.25 3.00 1.50 2.25
36 32 32 32 24 24
16.5 18.5 17.1 16.2 ±a 15.5
n/a 0.50 0.65 0.55 ±a 0.50
33.52 68.07 48.55 57.06 n/a n/a
3.30 6.90 4.55 5.80 n/a n/a
33.52 1368 1095 1055 n/a n/a
a
Fracture morphology prevented post-mortem measurements.
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All experiments were performed on an Instron 8500 with Instron Series IX software to collect load/displacement data. A Physical Acoustics Corporation (PAC) Spartan 2000 and PAC SA-DAQ software were used to record AE signals during dual ramp testing. Two 6 mm diameter PAC S9220 sensors were attached to the specimens with silicone gel as the couplant and connected to a PAC 1220A preampli®er with 40 dB gain. Acoustic emission recording parameters were set at 40 dB threshold and peak de®nition, hit de®nition and hit lock-out times of 50, 200 and 300 ms, respectively [12]. Fracture events at the crack tip and the extent of debonding along the ®bers were monitored using a traveling optical microscope with magni®cations up to 200. A video camera was attached to it for real time recording. The extent of debonding along the ®ber was observed with the aid of a light beam directed at an angle of about 30 to the specimen's plane. Optical and SEM microscopy techniques were combined with mechanical and AE data to explain the deformation and failure mechanisms. The reproducibility of the results was examined by performing three to four experiments for each ®ber spacing. A variation of the experimental data less than 5% was seen in the monolayer specimens and less than 10% on the data of the multilayer specimens tested under the same conditions. 3. Results and discussion 3.1. Mechanical and fracture behavior 3.1.1. Monolayer specimens Experimental stress/strain curves for the notched monolayer specimen including the matrix material are shown in Fig. 3. At approximately 1% strain the behavior of the matrix became non-linear with fracture occurring at about 3.6% strain. The data in Fig. 3 show that the addition of reinforcing ®bers not only resulted in stier specimens but also increased the range of linear elastic behavior. The addition of the ®bers increased the strain to failure of the composite specimens by about 20±25%, as compared to that of the pure epoxy, but, without a de®nite trend as a function of ®ber spacing. As shown in Table 1, the elastic moduli as well as the strength characteristics of notched and smooth specimens are practically the same. In all notched monolayer specimens, crack growth started from the notch at a very early stage of the loading sequence. As the crack approached the ®rst ®ber, interface separation was observed along that ®ber. Growth of the interface separation and/or sliding for a few ®ber diameters took place before the crack front reached the ®ber. Upon continuing loading, the crack and the interface separation did not grow any longer and the specimen fractured at some other place of the
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Fig. 3. Typical stress/strain curves for the monolayer notched specimens.
specimen with an audible sound in two pieces and multiple matrix cracking (Fig. 4). Limited fragmentation and secondary matrix cracking followed an inverse trend to the ®ber spacing, i.e. lower ®ber spacing yielded more secondary matrix cracking. The mechanisms of deformation and fracture are discussed in Section 3.2. 3.1.2. Multilayer specimens Load/displacement curves for single ramp loading of the matrix and the multilayer CT composite specimens are shown in Fig. 5. Linear behavior up to fracture prevailed in the specimens of the matrix material since the specimen dimensions were chosen according to ASTM standards for fracture toughness. As for the composite specimens, crack initiation from the notch was observed in a range of 400±500 N. Subsequently, non-linear load/displacement behavior was recorded. Before the crack tip reached the ®rst row of ®ber, the clear epoxy matrix started to become opaque ahead of the crack tip at the location of the ®rst ®ber row. With continued loading, the opaqueness propagated vertically along the row of ®bers. As the crack approached the ®rst ®ber row, similar opaqueness started to grow along the second row of ®bers and propagated parallel to the ®bers. The observed opaqueness indicated a form of interface separation and/or plastic deformation of the matrix material growing along the ®bers. The process of fracture described above, schematically shown in Fig. 6(a) suggests the presence of a crack arrest mechanism, similar to one described earlier for the monolayer specimens. Some details on the underlying mechanisms are given in the next section. Crack growth in the ligament direction was arrested at, or within fractions of a millimeter after the ®rst ®ber
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Fig. 6. Schematics of (a) crack growth, (b) fracture path and (c) a photograph indicating the ®ber surface after fracture on a typical multilayer specimen.
Fig. 4. Schematic of fracture sequence on a monolayer specimen. Phases (2) to (4) were not directly observed and ®ber fractures were seen during post-failure observations.
Fig. 5. Typical load/displacement curves for the multilayer CT specimens.
row. As the grip displacement was increased past the point of crack arrest, fracture began to propagate along the ®ber direction. The load decreased slightly before specimen fracture in the ®ber direction. Fast fracture followed a curved path and the specimen separated into two or three pieces [Fig. 6(b)]. The experimental data in Fig. 5 also show that the addition of the ®bers increased the overall ductility of the specimens. Although a small dierence in the crack lengths between the specimens may have contributed to the observed behavior, note that a weak ®ber/matrix interface existed in these composite specimens. Moreover, as the ®ber spacing was decreased, the amount of ®ber-matrix interface was increased. Therefore, the interface area also has contributed to an increase in ductility and toughness of the composite CT specimens. Other sources contributing to the stiness reduction may be irreversible deformation of the matrix material between the ®bers as well as crack growth. Recall that the external specimen dimensions are the same in all cases with small dierences in the initial crack length (Table 2). Thus, a decrease in the ®ber spacing lx [Fig. 2(c)] resulted in a decrease of the thickness of the matrix material between the vertical layers of ®bers. As a result the extent of non-elastic deformation of the epoxy matrix was more pronounced on the specimens
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with smaller lx . The extend to which the amount of interface, matrix deformation and crack growth contributed to the load/displacement behavior shown in Fig. 5, was not determined in the present studies. It is also interesting to note that the load to fracture was practically independent of the ®ber spacing (Fig. 5). This behavior was attributed to the weak interface and the mechanisms preceding global fracture. It is well known that the sensitivity of crack initiation and growth to local notch geometry decreases with a decrease in interface strength and damage around the vicinity of the notch tip [13]. For the present material, even though the number of ®bers was increased, it did not aect the load to fracture since their eects were diminished or even eliminated by the weak interface and possibly plastic deformation of the epoxy around the crack tip towards the end of the loading course. 3.2. Microscopic observations 3.2.1. Monolayer specimens One of the most interesting features on both specimen types was the onset of non-linear deformation behavior and its dependence on ®ber spacing. To shed light onto the mechanisms controlling the observed behavior, AE methods were coupled with mechanical testing. The number of hits and the associated energies superimposed on the load/displacement curve for a
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typical monolayer smooth specimen are shown in Fig. 7. Note that acoustic activity began as the stress/strain behavior started to deviate from linearity. Both the number of hits and energy were observed to increase proportionally, up to a displacement of about 5 mm, suggesting that only one damage process was dominant. A sharper increase in energy was seen at about 6 mm which continued until specimen fracture. The exact damage processes which result in acoustic activity are usually unknown, especially in new materials. Correspondence of an acoustic signal with a particular event may be achieved by specifying the event with the pertinent acoustic signal through detailed experimentation. While attempts have been made in the literature to correlate acoustic activity with speci®c damage processes no absolute relationships have been found. On the other hand, it is commonly reported that matrix damage, such as plastic deformation, is a source of low acoustic energy while ®ber/matrix separation, debonding, delamination and friction are sources of medium acoustic energy, and ®ber and matrix fracture are sources of high acoustic energy [12]. Yet, without identifying the AE behavior of each mechanism separately as well as in composite form, one cannot make speci®c statements as to the source of acoustic activity. Although an identi®cation was not performed in the present studies, based on the optical observations it is possible that acoustic activity that started at the onset of
Fig. 7. Acoustic emission signals [(a) Hits and (b) Energy versus displacement] of smooth monolayer specimen superimposed on load displacement curve. `n' indicates the onset of nonlinearity (®ber spacing lx 2:5 mm).
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non-linear stress/strain behavior was due to ®ber/matrix debonding and/or sliding. Shortly before specimen fracture, a relatively high energy signal was recorded (displacement 6.85 mm, Fig. 7). This signal may be due to crack initiation around a ®ber as shown in Fig. 8. The specimen did not fracture at this point because the ®bers around the crack reduced the stresses at its tip. Upon continued loading, the recorded signals may be due to either ®ber fracture and sliding or crack growth. The largest peak was seen at specimen failure shortly after the presumed crack initiation and with a very small additional load and audible sound. The scenario of deformation and fracture outlined above suggests that matrix-crack initiation occurred well after the proportional limit. This observation is dierent from what is seen in brittle materials. For example, it has been
Fig. 8. Photographs of fracture surfaces of monolayer specimens: (a) and (b) indicate morphologies of fracture initiation and (c) indicates a typical morphology of the surface at fast fracture (arrows point at ®bers).
reported by Kim and Pagano [14] that matrix cracking in a brittle matrix composite may occur, not at the proportional limit as commonly assumed, but before this limit is reached. In every monolayer specimen, catastrophic failure occurred at the peak load. Composite failure initiated with matrix cracking which propagated along the entire width of the specimen. The ®bers were unable to support the excess load thrust upon them in the moments after matrix cracking (Fig. 8), therefore composite failure occurred in a very short time after matrix cracking. Interestingly, in all notched specimens fracture of the specimens into two pieces occurred away from the notch and shortly after matrix cracking. Failure of the specimens at a location dierent from the notch can be explained by the weak interface present in the specimens. The observed debond growth along the ®rst ®ber caused crack blunting and, thus, a reduction of the stresses at that point. Other defects in the specimen became more severe leading to crack initiation and specimen fracture. The phenomenon of interface failure in the path of a growing crack is typical in composites with `weak' interfaces and has been described by Cook and Gordon [15]. Inspection of the fracture surfaces revealed that the fracture mechanism was identical in all specimens. Fig. 8(a), (b) and (c) show the fracture surfaces of two typical specimens around the location of fracture initiation. The photographs in Fig. 8(a) and (b) indicate that a radial crack initiated at the ®ber/matrix interface, i.e. interface between the ®ber polyimide coating and the epoxy. Note here the smooth fracture surface associated with slow growth of the radial crack and the relatively rough surface indicative of fast fracture [Fig. 8(c)]. Combining the in-situ observations, AE signals and post failure analysis, the fracture process sketched in Fig. 4 was presumed for the monolayer specimens. 3.2.2. Multilayer specimens Fig. 9 shows a sequence of load/time curves superimposed on the AE hits for a specimen with ®ber spacing 1.5 mm obtained during the loading/unloading experiments. A similar behavior was seen on a specimen with ®ber spacing 2.25 mm. Thus, only the results from the specimen with a 1.5 mm ®ber spacing are discussed herein. In the ®rst phase `a', the specimen was loaded up to 800 N. During this phase, low activity was observed at approximately 100, 450 and 800 N. The activity at 100 N was attributed to mechanical noise associated with the alignment of the specimen and grips. Based on the in±situ observations described in Section 3.1, the activity at 450 and 800 N was presumed to be due to crack initiation and growth from the notch. In the second phase `b', no activity was seen until 800 N, the maximum load of phase `a'. The absence of acoustic activity until some previous threshold load has
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Fig. 9. Acoustic emission signals of CT multilayer specimen superimposed on load displacement curve (®ber spacing lx 1:5 mm).
been reached is known as the `Kaiser' eect and is an important tool in identi®cation [12]. After this point, the rate of activity was fairly constant until shortly after unloading and was attributed to crack growth and interface sliding. In phase `c', the activity began at 800 N, 200 N less than the maximum load of phase `b'. This observation coupled with the events in phase `b', i.e. activity after unloading, suggested that friction and separation of the interface were taking place after 800 N. Yet the activity rate did not greatly increase until about 1000 N, the maximum load of the previous loop. The rapid increase in activity afterwards suggested that crack growth and interface fracture were taking place (recall the opaqueness growing parallel to the ®ber direction). Similar to phase `b', activity continued until just after unloading began. A similar scenario was repeated in loop `d'. Activity attributed to interface friction and separation began at approximately 150±200 N and continued at a low rate until 1200 N. At this point crack growth began. Interestingly the activity ceased just prior to the maximum
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load. This may have been a sign of crack arrest at the ®rst ®ber row. Note that at all loading steps, crack growth did not resume until the maximum load of the previous loading loop was reached. The acoustic behavior of loop `e', in which specimen fracture occurred, was quite unique. At a load of about 200 N the greatest amount of activity was recorded. Based on the behavior of previous loading loops it was concluded that the intensity of this burst of activity occurred at a load too low to be due to crack growth. Combined with optical observations it was assumed that this activity was due to the interface sliding, most of it already in place from the loading and subsequent unloading of the previous phases. The activity spike at about 1300 N was due to specimen fracture along planes parallel to the ®ber row (Fig. 6). Interestingly, a decrease in the load was recorded prior to reaching the programmed maximum displacement. This correlated well to the observation that the peeling behavior just prior to fracture was accompanied by a slight decrease in load (Fig. 5). The experimental evidence so far in this composite system points to a weak interface. Further evidence of a weak interface was found during SEM observations of the multilayer specimens. The photograph in Fig. 6(c) shows evidence of the adhesive bond between the ®ber and matrix. Therefore, it was assumed that the change in optical properties along the ®bers during loading was due to ®ber/matrix separation (Section 3.1). According to these observations the following scenario of events was assumed. As the crack tip approached the ®rst row of ®bers, the crack induced higher normal and shear stresses at the ®ber/matrix interface causing debonding and interface separation [Fig. 6(a)]. This process is similar to the one well explained in by Cook and Gordon [15]. As the loading continued, interface separation and crack growth occurred prior to fracture along the ®ber direction leading to specimen fracture [Fig. 6(b)]. 4. Analysis 4.1. Monolayer specimens In previous works [1,3] the fracture strength, c , of similar composite geometries was correlated with ®ber spacing, lx , on the basis of dimensional analysis arguments. The behavior of those specimens was linear elastic up to fracture and had the same mechanisms of deformation throughout the entire loading phase. A dierence between the present results and those reported earlier [1,3] is that the composite specimens used herein displayed a nonlinear response after a certain load level. These eects were attributed to a relatively weak interface and the nonlinear behavior of the epoxy matrix. Using dimensional analysis for the fracture
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strength of the data in Fig. 3 may not be appropriate since the mechanisms of deformation and their extend, especially in the non linear part of the curve, depend on lx . Note that for lx 0:75 mm, the entire range of the stress/strain curve is linear. Therefore, in this work the same expression, as in earlier studies [1,3], was used to correlate the stress at the onset of the non linear behavior, ci , with ®ber spacing, lx p ci l = lx 1 p where 1 (Pa m) is a parameter. The data in Fig. 10(a) show that the experimental values for i c follow expression (1).
As a matter of a qualitative comparison, the fracture p stress was also plotted against 1= lx [Fig. 10(a)]. Interestingly, when a linear relationship was forced through the origin, the deviation from the linearity diminished with increased ®ber spacing. Based on the later observation, the linear region of each stress/strain curve was extrapolated until the experimental failure strain of the composite specimens [approximately 4.3% in every case, pFig. 3(a)]. This stress was then plotted against 1= lx . As can be seen in Fig. 10(a), the extrapolated fracture stress displayed the same trend as ci , thus suggesting that non-linear eects may be responsible for the deviation of the fracture stress from relation (1). Fiber spacing was also expected to have an in¯uence on the stress/strain behaviorpof p composite speci the mens. The scaling parameter lx = B was chosen over the entire stress range. It is worth mentioning that a similar ratio is often used in fracture mechanics, i.e. crack length versus specimen width. Scaling or `collapse' of the stress/strain curves in the linear region was observed for the range of ®ber spacing used here [Fig. 10(b)]. The deviation of the scaled curves after the proportional limit is attributed to the dierent extent of non linear deformation in each specimen. 4.2. Multilayer specimens The stress intensity factor (SIF) for a CT specimen can be expressed as [16] p 2 KA A W ÿ aF a=W where A is a nominal stress that accounts for both tension and bending of the ligament of the specimen and, for a specimen of unit thickness, is given by A Atension Abending
PA 6PA a 0:5 W ÿ a W ÿ a W ÿ a2 3
p Fig. 10. (a) Various characteristic stresses plotted against 1/ lx for the monolayer specimen (see text for details.) (b) Scaled stress/strain curves of the notched monolayer specimens.
where PA is the applied pin load; a and W are the crack length and specimen width, respectively [Fig. 2(c)]; F a=W is a dimensionless parameter that accounts for the specimen geometry. To analyze the mechanical and fracture responses of the CT specimens as a function of ®ber spacing, the stress given by Eq. (3) was used. Recall that the only variable ®ber spacing in the multilayer CT specimens was lx [Fig. 2(c)]. From the load/displacement curves (Fig. 5) the load at the onset of non-linearity was measured and subsequently used to determine the corresponding stresses i A from Eq. (3). The results of these calculations as a function of lx are shown in Fig. 11. Each datum represents the average value of stress from specimens with a characteristic length of 32 mm with a particular ®ber spacing while the bars represent a 10%
K.M. Ganey, J. Botsis / Composites Science and Technology 59 (1999) 1847±1859
error. The data in Fig. 11 can be approximated by the following equation p Ai Ao 2 = lx 4 p m) and Ao (Pa) are empirical parawhere 2 (Pa meters. Interestingly, a non zero intercept [as compared to relation (1)] was found for the multilayer specimens and the value of the slope 2 was about 3 times smaller than 1 . Note also that the fracture stress of the CT specimens, calculated from Eq. (3), can be described by a similar equation (Fig. 11). In this work, we do not attempt to give a physical interpretation of the intercept. However, for the data shown in Fig. 11, relation (4) gives Ao =37 MPa. This value is about 4 MPa higher than the specimen fracture stress found experimentally using Eq. (3) for the unreinforced matrix CT specimens. Moreover, it is interesting to point out that in previous studies on multilayer CT specimens of epoxy matrix/E-glass ®bers, the crack initiation stress as a function of ®ber spacing was described by relation (4), or for that matter relation (1), with Ao 0. A key dierence between the present studies and those reported in previous works was that the external geometry varied with the ®ber spacing [3] while here the external geometry was constant. Investigations of strength as a function of a characteristic length in polycrystalline materials (ceramic and metals) have also yielded relationships similar to Eqs. (1) or (4). Two types of mechanisms have been discussed in attempts to explain the strength of polycrystalline solids. When Ao 0, it is assumed that
Fig. 11. Stress at the onset of nonlinearity and fracture stress of the CT multilayer specimens.
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strength is controlled by a Grith-type ¯aw with size proportional to the grain size. When Ao > 0 it is implied that plastic deformation takes place prior to fracture. Expression (4) is known as the Hall±Petch relationship for metals [17] or the Orowan±Petch relationship for ceramic materials [18]. To investigate further the strength behavior of the CT specimen and its dependence on ®ber spacing, numerical analysis of crack initiation in a 2-D model of the actual CT composite specimen, as shown in Fig. 12(a) developed previously [4] was used herein. The remainder of the paper outlines the essential elements of the numerical scheme and presents results pertaining to the eects of the ®ber spacing on the fracture initiation stress on a CT specimen with constant external dimensions. The BEM was used to obtain the strain energies U1 and U2 of a specimen at two crack lengths, a1 and a2 a1 0:1 mm. From these values the energy release rate for mode I loading was calculated as U2 ÿ U1 5 a2 ÿ a1 Assuming linear elasticity, the SIF for mode I loading was found from G1
Fig. 12. Schematics of approximations for the reinforcement. (a) discrete model; (b) eective model; (c) results of numerical analysis using the two 2-D approximations.
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p K1 G1 E0
K.M. Ganey, J. Botsis / Composites Science and Technology 59 (1999) 1847±1859
6 0
Plane-strain conditions were considered, i.e. E Em / ÿ 1 ÿ 2 . In the present simulations it was assumed that (a) the crack initiated from the pre-crack at the matrix material, (b) the crack front was suciently far from the interface of the ®rst layer of ®bers so that an inverse square root singularity for the stresses can be assumed, (c) the interface behaves as a perfect one, (d) the Young's modulus, Em , was taken as that of the matrix and the Poisson's ratio, was equal to 0.3 for both matrix and ®bers. The stress at crack initiation was determined as follows. Relation (2) can be written as KA P KO P1
7
where KO P1 is the SIF due to the unit force. Usually the applied force, PA , is measured experimentally. However, it can be calculated numerically by assuming that the crack initiates at the matrix and imposing a Grith fracture criterion. Namely, crack initiation begins when the total SIF, Kr, of the composite specimen equals the fracture toughness of the matrix material, Kmc. If the SIF in the composite is proportional to the applied force at crack initiation one ®nds PA Kmc = Kr P1
8
Here Kr P1 is the stress-intensity factor due to a unit applied force in the composite specimen which is determined numerically from Eqs. (5) and (6). It has been reported that when using an experimental value for the Kmc the numerical scheme correlates very well with experimental data of crack initiation stress of a CT epoxy/glass specimen [4]. In view of the weak interface present in the composite specimen examined here, and the perfect interface assumed in the numerical model, it would be unrealistic to correlate the experimental data shown in Fig. 11 with results from numerical analysis. However, using the numerical studies we can examine if expression Eq. (4) is the result of the spacing variation or the result of the specimen geometry. Approximating the actual 3-D ®ber architecture with the 2-D geometries shown in Fig. 12(a) and (b), crack initiation stress [Eq. (3)] as a function of reinforcement for each approximation can be easily simulated numerically as a function of the various geometrical parameters of the CT specimen. In the present studies three discrete approximations for the reinforcement, with spacing 1.5, 2.25 and 3.0 mm, were examined. A layer thickness of 0.5 mm, approximately equal to the ®ber diameter, D, was used as the reinforcement. The modulus of a layer was calculated
using the rule of mixtures, with the elastic properties of the constituent materials and the ®ber spacing taken from the actual specimen geometry. For the second geometry, the discrete reinforcing layers were assumed to behave as a single eective material equal in dimension to the ligament of the specimen minus the distance of the crack tip to the ®rst layer of ®bers. The rule of mixtures also was used here to calculate this `eective' modulus of the reinforcement from the actual discrete ®ber distribution. Another important length parameter in the present simulations was the distance of the crack tip to the ®rst layer of ®bers, . Although care was taken during specimen preparation to assure that this distance was the same in all specimens, small variations from specimen to specimen were not easy to avoid. This distance was between 0.5 and 0.65 mm (Table 2). Since the present simulations were aimed at examining the trend of the initiation stress on a CT composite specimen as a function of ®ber spacing, and not modeling the experimental data, was taken equal to 0.8 mm. The results of the simulations [Fig. 12(c)] show that the dierences between the two approximations are small. As expected, for large values of ®ber spacing, there are some dierences between the two solutions, with the eective model giving lower critical stress. The trend is reversed for small ®ber spacing. Some of these dierences may be attributed to the numerical approximation in the BEM scheme. Note also in Fig. 12(c) the presence of the intercept and the relatively weak dependence of the strength on the spacing of the reinforcement, as in fact was shown from the experimental results (Fig. 11). For a composite specimen with W=32 mm and matrix fracture toughness equal to 3.3 p MPa m, the numerical results shown in Fig. 12(c), extrapolate to a stress of 35 MPa, a value very close to the experimentally measured strength of an unreinforced specimen. The experimental results presented in this paper and in Refs. [3,4], as well as the numerical analysis, demonstrate that not only the spacing of the reinforcement but also the external geometry is important to the strength behavior of the CT composite specimens. Further work is needed to establish the limits of relation (4) and the physical meaning of the intercept. 5. Summary The deformation behavior and strength characteristics of a composite with long aligned ®bers as a function of ®ber spacing were investigated. Two types of specimens were used in the experimental studies: monolayer specimens under remote load and multilayer specimens in a compact tension loading con®guration. For the range of parameters used, the following conclusions can be drawn:
K.M. Ganey, J. Botsis / Composites Science and Technology 59 (1999) 1847±1859
1. In the monolayer specimens, the proportional were limit, ci , and the ®ber pspacing, lx , p prelated according to ci 1 = lx . The ratio lx = B was found to scale the linear stress/stain behavior of the composite. 2. In the multilayer specimens the proportional limit (an eective stress accounting for tension and bending) and the ®ber spacing along the ligament p direction, lx , were related by Ai Ao 2 = lx , where Ao is a stress parameter close to the stress for crack initiation of a CT-matrix specimen. It may be argued that the experimental data, presented in Fig. 11, are limited and relation (4) must be further examined with more experiments. Although this should be the objective of a thorough experimental program, relation (4) is also obtained using a 2-D numerical scheme for the crack initiation stress on a CT specimen with a constant external geometry and variable lx . 3. Real-time observations, combined with acoustic emission data and fracture-surface observations, suggested that weak ®ber/matrix interface damage may have contributed to the nonlinear deformation behavior. 4. The results of the present studies and those reported earlier [3,4] suggest that the intercept Ao depends primarily on the external specimen geometry. 5. The results of the present studies may be very useful in characterizing the in¯uence of the various length parameters on deformation and strength of various heterogeneous materials. They are also directly applicable to deformation and fracture of layered composite materials. Acknowledgements Part of this work was carried out at the Department of Civil and Materials Engineering, University of Illinois at Chicago, IL. The authors wish to acknowledge the ®nancial support from the AFOSR under grant
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93-1-0319 at the University of Illinois. Dr. W. F. Jones was the program monitor. Thanks are also due to DOW Chemicals for providing the epoxy. References [1] Botsis J, Beldica C, Zhao D. On strength scaling of composites with long aligned ®bers. Int J Fract 1995;68:375±84. [2] Botsis J, Beldica C. Strength characteristics and fatigue crack growth in a composite with long aligned ®bers. Int J Fract 1994;69:27±50. [3] Zhao D, Botsis J. Experimental and numerical studies in model composites, part IÐexperimental results. Int J Fract 1996; 82:153±74. [4] Beldica C, Botsis J. Experimental and numerical studies in model composites, part IÐnumerical results. Int J Fract 1996;82:175±92. [5] Harris B, Ankara AO. Cracking in composites of glass ®bres and resin. Proc Royal Soc London 1978;A.359:229. [6] Bowling J, Groves GW. J Mater Sci 1979;14:443. [7] Zok F, Hom CL. Large scale bridging in brittle matrix composites. Acta Metall Mater 1985;38. [8] Friedrich K, Fels A, Hornbogen E. Fatigue and fracture of metallic glass ribbon/epoxy matrix composites. Compos Sci Technol 1985;23:79. [9] Mower TM, Argon AS. Experimental investigations of crack trapping in brittle heterogeneous solids. Mechanics of Materials 1995;19:343. [10] Walter ME, Ravichandran G. Experimental simulation of matrix cracking and debonding in a model brittle matrix composite. Experimental Mechanics 1997;37:126. [11] Beldica C, Ganey K, Botsis J. unpublished data. [12] MISTRAS 2000, users manual. Princeton (NJ): Physical Acoustics Corporation, 1995. [13] Hull D. An introduction to composite materials. Cambridge Solid State Science Series. New York: Cambridge University Press, 1981. [14] Kim RY, Pagano NJ. Crack initiation in unidirectional brittlematrix composites. J Am Ceram Soc 1991;74:1082±90. [15] Cook J, Gordon E. A mechanism for the control of crack propagation in all brittle systems. Proc Royal Soc London 1964;A-282:508±20. [16] Tada H. The stress analysis of cracks handbook. Hellertown, (PA): Del Research Corp., 1973. [17] Hirth JP, Lothe J. Theory of dislocations. New York: McGraw± Hill, 1968. [18] Lawn B. Fracture of brittle solids. 2nd ed. Cambridge Solid State Series. New York: Cambridge University Press, 1993.