Dynamic mixed-mode I/II delamination fracture and energy release rate of unidirectional graphite/epoxy composites

Transcription

1 Engineering Fracture Mechanics 7 (5) Dynamic mixed-mode I/II delamination fracture and energy release rate of unidirectional graphite/epoxy composites Sylvanus N. Wosu a, *, David Hui b, Piyush K. Dutta c a Department of Mechanical Engineering, University of Pittsburgh, Pittsburgh, PA 56, United States b Department of Mechanical Engineering, University of New Orleans, New Orleans, LA 748, United States c US Army Cold Region Research and Engineering Laboratory, Hanover, NH 3755, United States Received 3 December 3; received in revised form August 4; accepted 4 August 4 Abstract Mixed-mode open-notch flexure (MONF), anti-symmetric loaded end-notched flexure (MENF) and center-notched flexure (MCNF) specimens were used to investigate dynamic mixed I/II mode delamination fracture using a fracturing split Hopkinson pressure bar (F-SHPB). An expression for dynamic energy release rate G d is formulated and evaluated. The experimental results show that dynamic delamination increases linearly with mode mixing. At low input energy E i 6 4. J, the dynamic (G d ) and total (G T ) energy rates are independent of mixed-mode ratio. At higher impact energy of 4. 6 E i J, G d decreases slowly with mixed I/II mode ratio while G T is observed to increase more rapidly. In general, G d increases more rapidly with increasing delamination than with increasing energy absorbed. The results show that for the impact energy of 9.3 J before fragmentation of the plate, the effect of kinetic energy is not significant and should be neglected. For the same energy-absorption level, the delamination is greatest at low mixed-mode ratios corresponding to highest Mode II contribution. The results of energy release rates from MONF were compared with mixed-mode bending (MMB) formulation and show some agreement in Mode II but differences in prediction for Mode I. Hackle (Mode II) features on SEM photographs decrease as the impact energy is increased but increase as the Mode I/II ratio decreases. For the same loading conditions, more pure Mode II features are generated on the MCNF specimen fractured surfaces than the MENF and MONF specimens. Ó 4 Elsevier Ltd. All rights reserved. Keywords: Mixed mode; Delamination; Dynamic interlaminar fracture; Split Hopkinson pressure bar; Energy release rate * Corresponding author. Fax: address: nwosu@engr.pitt.edu (S.N. Wosu) /$ - see front matter Ó 4 Elsevier Ltd. All rights reserved. doi:.6/j.engfracmech.4.8.8

2 53 S.N. Wosu et al. / Engineering Fracture Mechanics 7 (5) Introduction Dynamic delamination as an interlaminar fracture has been identified as the dominant failure mode in laminated carbon/epoxy composites [ 3]. The increasing use of composite materials in aerospace and military applications necessitates a better understanding of the dynamic modes of failure. Some progress has been made in the determination of both the statical and quasi-statical interlaminar fracture toughness [4 7]. However, progress in developing dynamic fracture tests has been slow, and data very limited because of the difficulties in obtaining accurate loading-point displacement of the specimen, speed of the crack tip at high strain rate, development of accurate closed form model of analysis, and reliable specimen configurations that could be adopted to the existing dynamic testing systems such as the split Hopkinson pressure bar (SHPB) and Charpy impact test systems. The use of these methods also generates delaminated fracture damage modes that are usually convoluted with mixed and multiple fractures and delamination. The use of impact tests for fracture toughness measurements, for example, assumes that the impact test on a stationary crack will yield the same result as an unstable propagating crack. The analysis is further complicated by the difficulties of separating the individual modes from the mixed modes or evaluating their contributions to the composite failure. Todo et al. [7] reported the development of a displacement measuring apparatus to measure the dynamic fracture toughness. However, the experiment was done at such a low speed and strain rate that static fracture toughness formulation was again used to estimate the dynamic fracture as in the cases of other researchers [6 8,6,8 3]. Such approximations have been successfully applied by others [,3], assuming the strain rate is low and the sample attains a quasi-static state of the stress in the vicinity of the crack tip. However, it still leaves us with the challenge of determining fracture toughness at high strain rates based on a stress-field that drives the extension of the crack tip. Zhang et al. [4] measured the fracture toughness of marble and other samples over a wide range of strain rates and showed that while the fracture toughness increased with loading rate for the marble sample, at high stain rate, the fracture toughness was definitely different from that of the static value. Using modified ENF loaded with MTS, Tsai and Sun [5] observed that the dynamic fracture toughness was the same as the static fracture toughness up to the crack speed of m/s. Some investigators [9] reported an increase with crack speed while others [6] reported a decrease at certain speeds. These apparent inconsistencies are due to the lack of actual knowledge of the true stress field at the crack tip and show the complexity of dynamic measurements. However, in the absence of a closed form model for dynamic fracture studies, these approximations of dynamic behavior continue to be useful in providing fracture toughness data that are applicable to structures in general. Sohn and Hu [7] studied dynamic delamination at high strain rates using Charpy and Izod impact tests and specimen configurations for Modes I, II and mixed-mode tests. The major difficulties with the test methods are the specimen configuration which required two laminate specimens to accomplish the Mode II test and for which constrained movement only along the specimen length. Glue used to bond the specimen to steel bars could have compromised the integrity of the plate and the shearing movement in the Mode II test. The analysis also ignored the effects of kinetic energy in the dynamic process. The purpose of the studies presented in this paper was to formulate an approximate closed form extension of Sohn and Hu [7] and Reeder [8] MMB equations to characterize the dynamic mixed-mode delamination fracture of composite materials using a mixed-mode open-notched flexure (MONF) that allows mode mixing by simply varying the loading position. These difficulties are accounted for in the present study. In contrast with Sohn and Hu [7], the pre-crack is placed at the mid-plane of the specimen notched-edge and simply supported with limited movement along the specimen length. Flexing in the specimen thickness direction is allowed. This permits shearing at the notched-edge since bending is an important feature of Mode II fracture. The cases of pure mode II have been reported earlier using end-notched (ENF) and center-notched (CNF) flexure specimens simply supported at the ends and loaded at the center. The use of a fracturing split Hopkinson pressure bar (F-SHPB) rather than Chary or Izod has the added advantage

3 S.N. Wosu et al. / Engineering Fracture Mechanics 7 (5) of allowing the measurement of the stress field for the determination of energy absorption or fracture energy at actual dynamic conditions. The F-SHPB apparatus used in this study differs from impact tests in the sense that the specimen is loaded in sudden compression at a high strain rate. The only impact is between the striker bar and the input bar. Fracture damage on the specimen is mainly caused by the interaction of the resulting incident compressive wave and the specimen. Thus, the flexure vibration generated due to the sudden compressive loading of the specimen results in stress distribution in the vicinity of the crack tip that will account for the dynamic crack propagation. In the presence of the stress, the crack will propagate at a certain speed, making the determination of stress intensity factors or energy release rates based on static or quasi-static formulation less appropriate. For complete dynamic test analysis, we acknowledge that the exact crack speed and rate effects are necessary considerations for the total energy release rate. As a first approximation, however, we assume that () the specimen is in a state of uniform stress and strain for the duration of the measurement, () the crack propagates at a constant velocity along the delamination path, and (3) the average dynamic behavior of an initially stationary crack is the same as a crack propagating at a uniform rate for the same conditions of loading. Assumption 3 is a consequence of assumptions and. Assumptions and are valid, and the method of analysis valid, since the particle velocity and strain rate are both constant for the time duration of the measurements [8].. Mixed-mode model description and analysis.. Quasi-static mixed-mode bending specimen Generally, the interlaminar fracture toughness or energy release rate, G, is a measure of the specimen crack growth resistance. Reeder [8] in his studies on failure criteria for mixed-mode delamination, introduced a closed form modified beam analysis of the delaminated surface and included the effects of transverse shear deformation, rotation of the specimen at the delamination tip, and non-linear effects through the specimen. The energy release rate from that study can be expressed as: G m I ¼ G ki a þ a k þ k þ h E P c ðþ G 3 G m II ¼ G kii a þ h E P c ðþ 5G 3 where P c is critical load where load displacement curve deviates from a linear response [8]. The delamination length, a, is determined by measuring the length of lamination or mid-thickness crack length, D is the perpendicular distance from the geometric center to the the line of action of the loading force, L is the specimen span length, and the constants k, G ki and G kii are defined as: 3ð3D LÞ G ki ¼ 4L B h 3 L E 9ðD þ LÞ G kii ¼ 6L B h 3 L E k ¼ 6E 4 ð3þ h E Reeder et al. [9,] had earlier noted the importance of including mixed-mode toughness testing when characterizing failure modes of a composite material. This is because laminated composite materials are often subjected to mixed-mode loading that cannot be determined from a pure mode toughness test when

4 534 S.N. Wosu et al. / Engineering Fracture Mechanics 7 (5) following the standard procedure for interlaminar fracture toughness tests [,]. Thus, even in cases where pure Mode II is certain to be the dominating mechanism, we maintain that a small fraction of Mode I could be expected and should be evaluated. From Eqs. () and (3), the G I /G II ratio can be written as: c ¼ Gm I G m II ¼ 4 3 3D L F c ; D P L D þ L 3 where F c is a correction factor that accounts for deformation caused by shearing and the specimen rotation at the crack and given as: " # F c ¼ ða þð=kþþ þðh E =G 3 Þ ð5þ a þðh E =5G 3 Þ Setting F c =, Eq. (4) predicts a mixed-mode ratio of 4/3 when D = and D = L, a situation that is impossible... Dynamic mixed-mode opening notch flexure (MONF) test from modified quasi-static equations The configuration for the MONF specimen is shown in Fig. in which the span of the upper half plane is slightly (3 mm) longer than the lower plane and loaded by the reaction (P ) at the support. The specimen was fabricated with a.3 lm Teflon pre-crack placed in the mid-thickness from the opening edge. A line edge loading applied from the center simultaneously bends the specimen similar to the ENF, and causes a reaction in the opposite direction at the opening section. Combination of the loading and the reaction pushes the specimen upper section open in tension, creating an anti-symmetric Mode I similar to the ELS specimen while the flexure at the center and reaction (P ) contribute to shearing at the edges. Mixed-mode ratio effect is introduced by varying the loading position, D, from the center and away from the pre-cracked side. Using beam theory analysis, general expressions for a symmetric a plate can be expressed as [6]: G ELS I ¼ P ða þ vhþ ð6þ 4 BE I G ENF II ¼ 3 P ða þ vhþ ð7þ 6 BE I where I = Bh 3 / is the plate at the moment of inertia and v is a correction for some deflection, curvature effect, and rotation at the crack tip, and given as: "!# = C 3 þ C v ¼ E G3 C ¼ :8 p ffiffiffiffiffiffiffiffiffiffiffiffiffi E E G 3 where D is loading position with respect to the plate center as shown in Fig. with the anti-symmetric loading force on the specimen expressed as: ð4þ ð8þ P ¼ P mðl DÞ L P ¼ P mðl þ DÞ L ð9þ

5 S.N. Wosu et al. / Engineering Fracture Mechanics 7 (5) (a) (b) Following Zhang et al. [4], we define the critical loading force P m as the peak value of the load displacement curve at time t m as P m (t)=e Ae t (t m ). Thus, replacing P in Eqs. (6) and (7), the mode contributions can be expressed as: G m I ¼ 3 ðe Ae t ðtþþ ða þ vhþ ðl DÞ 4 E B h 3 L G m II ¼ 9 ðe Ae t ðtþþ ða þ vhþ ðl þ DÞ 6 E B h 3 L where the upper and lower sections of the plate are of equal thickness h with the Teflon pre-crack between them and B is the l thickness of the composite plate. The delamination length, a, is determined by measuring the length of lamination or mid-thickness crack length. It is clear from Eq. () that Mode I is at the maximum when the loading is at the center (D = ) and decreases as the loading position increases toward the edge due to increased movement in the upper section of the plate. Moderate incident energy is needed to initiate the crack in the MONF specimen. Note that G m II in Eq. () reduces to pure ENF Mode II at D = and zero when D = L and predicts a mixed ratio expressed as: c ¼ Gm I G m II ¼ 4 3 (c) Fig.. Anti-symmetric loading for mixed-mode I/II (a) end-notched flexure testing (MENF), (b) center-notched flexure (MCNF), and (c) open-notch flexure test (MONF specimens). L D ; D 6 L ðþ D þ L ðþ

6 536 S.N. Wosu et al. / Engineering Fracture Mechanics 7 (5) One major difference and advantage of the present mixed-mode formulation can easily be seen: The MMB specimen is valid for Mode I fracture when D is greater than L/3 compared to the proposed MONF specimen that is valid for any value of D < L, where D = L corresponds to no opening and crack propagation and G I =;D = corresponds to greater crack opening in which Mode I is the dominating fracture mode. Other specimen configurations such as mixed-mode end-loaded specimen (MELS), mixed-mode flexure (MMF), and cracked lab shear (CLS), and others [3 8] have been used to investigate mixed-mode fracture in composite materials. The present study postulates that total energy release rate is equal to the sum of the individual contributions, mainly from opening and sliding shear fractures in this case, and expressed as: G T ¼ G m I þ Gm II ðþ Delamination crack extension is initiated at a given mixed-mode ratio when G T reaches the materialõs fracture toughness (G Tc ). Experimental observations show that the energy release rate is not a linear function of fracture energy absorbed, impact energy or delamination [8]. A fracture behavior is modeled as a power law function of the fracture controlling variable X expressed as: G m I ¼ G Ic X a G m II ¼ G ð3þ IIcX a where the variable X is taken as the relative fracture energy absorbed by the specimen or delamination crack extension. The parameters G Ic and G IIc are mode-dependent critical values for each mode and can be determined from the curve fitting, and the exponent a is a measure of the dependence of the energy release rate on the variable X. Substituting Eq. (3) into (), the total critical energy release rate is given as: G T ¼ðG Ic þ G IIc ÞX a ¼ G Tc X a ð4þ where G Tc = G Ic + G IIc, and G Tc must exceed a critical value for the crack to grow..3. Dynamic energy release rate from energy balance In general, the total energy of an elastic body whose ends are free to move under dynamic crack growth can be partitioned into three energies: the potential (elastic) energy contained in the body which is proportional to square of the stress field and increases as the crack grows; the fracture energy, U F which is proportional to the crack length and drives the breaking of bonds and creation of new surfaces and heat; and the kinetic energy U k due to the crack motion [9]. These energies are accounted for by the total energy balance of the body, E, for dynamic crack growth given as: E ¼ðU F F ÞþU k þ W ð5þ where F is the work done by the external forces, U F is the total fracture energy including elastic strain energy of the plate, W is the energy required for the crack extension, and U k is the kinetic energy to drive the rapid extension of the crack to length da on the crack path. At high strain rate, the crack propagates at a high speed and fracture instability occurs when the energy release rate, G, is greater than the crack resistance (energy consumption rate), R. The force to drive this per unit thickness is obtained after differentiating Eq. (5) with respect to the crack length as: ¼ dðu F F Þ þ du k da da þ dw ð6þ da subject to conservation of total energy of the body and its surrounding, i.e.: de da ¼ ð7þ

7 S.N. Wosu et al. / Engineering Fracture Mechanics 7 (5) A sufficient condition given in Eq. (6) for crack extension is that the energy release rate must exceed the rate at which the energy is consumed (G d > R) such that: G d ¼ dðf U FÞ du k > dw da da da ¼ R ð8þ Defining U A =(F U F ) as the fracture energy absorbed for crack extension, the dynamic strain energy release rate for a plate of B thickness is written in differential form as: G d ¼ BDa ½DU A DU k Š max > dw da ¼ R ð9þ where B is the specimen thickness and Da is the crack extension from the critical crack length, and the terms are evaluated at the peak or average values. For a limiting case given by assumption (), it is assumed that specimen has reached a state of uniform stress at a constant energy consumption rate, and an approximate expression for the kinetic energy can be written as [3]: DU k ¼ pr a ða a c Þ ðþ E where r a is equal to average stress on the specimen, a is the total delamination crack length, and a c is the critical crack extension length above which the extension will be unstable. For the first estimation, a c is taken as the initial pre-crack length equal to 6 mm. The term DU A in Eq. (9) is determined from classical wave mechanics [8] and summarized as follows: After the impact of a striker bar with the incident (input) bar, a fraction of the compressive wave generated is reflected at the surface of the plate and others are transmitted through the plate. An elastic wave traveling through the specimen for time t pumps this energy into the crack tip in the direction of crack propagation. Delamination damage to the laminate occurs by the transfer of sufficient energy to the delamination surface. Thus, neglecting energy losses within the fixture, the total energy dissipation history for the damage process can be partitioned as follows: DU e ¼ DU i ¼ AC E DU ss ¼ DU r ¼ AC E DU is ¼ DU t ¼ AC E Z t Z t Z t r i ðtþdt r r ðtþdt r t ðtþdt ðþ where DU e is the incident energy due to the incident compressive wave, DU ss is the surface strain due to the reflected wave resulting from surface impedance mismatch, and DU is is the internal strain energy in the specimen. Thus, the energy dissipated in fracture can be expressed as: DU A ¼ DU e DU ss DU is Substituting Eq. () into (), gives the total energy dissipated in the fracture process as: DU A ¼ AC E DU min A DU max A Z t ¼ DU a ¼ DU a þ DU s ðr i r r r t Þdt ðþ ð3þ

8 538 S.N. Wosu et al. / Engineering Fracture Mechanics 7 (5) where Eq. (3) shows the partitioning of the energy absorption into the residual energy absorbed by the system, DU a, and the total stored elastic strain energy, DU s of the composite plate. The numerical integration of Eq. (3) is carried out with all time shifted to zero and all three waves beginning at the same time and for the same time duration, t. Assuming that the energy released by the Hopkinson input bar is absorbed in fracture (DU R ), the total energy dissipated in the fracture process will be the same as energy released by the input bar (neglecting other losses) and can be expressed from elementary classical wave formulation as [8]: Z DU R ¼ P in ðtþdu n ðtþ ¼DU A ð4þ where P in (t)=a(r i (t) +r r (t)) is the external loading force at the input side of the plate and du n = (C /E)(r i (t)+r r (t))dt is the net specimen displacement in the direction of the net force. Thus, the total fracture energy absorbed is obtained from the energy-absorption curve (Eq. (3)) as the difference between the peak energy dissipated ðdu max A Þ and residual energy absorbed by the system, DU a DU min A. Substituting Eqs. () and (3) into (9) gives the dynamic (high strain rate) energy release rate as: G d ¼ Bða a c Þ DU max A pr t E ða a c Þ As stated earlier by Eq. (), Eq. (5) represents a total energy release rate equal to the sum of the individual mode contributions, mainly from opening (Mode I-d) and sliding shear (Mode II-d) fractures in this case, and expressed as: G T ¼ G d ¼ G I-d þ G II-d ð6þ The dynamic energy release rate, G d is dominated by Mode I-d energy release rate when D = and by Mode II-d energy release rate when D = L which is in agreement with earlier observation and that of Sohn et al. [7]. ð5þ 3. Experimental configuration The experimental set-up shown in Fig. consists of () a stress-generating system which is comprised of a split Hopkinson pressure bar and the striker, () a special specimen fixture consisting of a specimen holder and line edged impactor, (3) a stress measuring system made up of sensors (typically resistance strain gages), and (4) a data acquisition and analysis system. Each component of the system is described by Nwosu [8]. Dynamic loading of the composite plates is provided by a split Hopkinson pressure bar modified for fracture tests using the appropriate specimen fixture. The compressive wave is generated on the Hopkinson input bar by the longitudinal impact between the input bar and the striker bar at a given impact energy determined by the compressor air pressure. Upon the arrival of the incident wave at the incident bar/specimen interface, the wave is partially reflected (because of the impedance mismatch) and partially transmitted through the specimen. The loading is accomplished by a line edge loading fixture (attached to the input bar) that suddenly compresses the specimen by the forward motion of the input bar due to the energy of the striker bar. Thus, the initial energy of the striker bar transferred to the input bar as impact energy determines the incident energy to the bar specimen interface. Since the fracture energy released by the input bar is a direct function of the striker impact energy, it is conceivable that the effect of increasing the striker impact energy will be the same as increasing fracture energy. The specimens are fabricated from AS4/35-6 toughened epoxy unidirectional [] n composites also used by Reeder [8]. The experimental parameters are 3 GPa, 9.7 GPa, and 5.9 GPa for the longitudinal modulus (E ), transverse modulus (E ), and shear modulus (G 3 ), respectively. The dimensions of the graphite/ epoxy specimens used in this present study are 5 mm in total span (L), 5.4 mm in width (b), and.7 mm/ply in thickness (h). Dimensions were chosen to be of the same (L/b) scale with Reeder [8].

9 S.N. Wosu et al. / Engineering Fracture Mechanics 7 (5) Fig.. Experimental set up for MONF mixed mode test showing (a) sample holder fixture and (b) fracturing split Hopkinson pressure bar and associated instrumentation. The stress wave loading force is determined as the average peak loading force in the force displacement curve between the input and output bar interfaces, and is related to the stress field transmitted to the crack tip. This force is stress wave dependent, and is the driving force for the crack propagation. The delamination length, a, is determined by measuring the length of the mid-thickness crack along the specimenõs edge. A microscope is used for clearer viewing of the extent of the delamination. When the specimen was pulled apart through the middle, it was observed that the length of the delamination zone measured from the edge to the crack tip was a little longer than the edge crack length due to the effects of deflection and curvature. In cases where the specimen is completely split through in mid-thickness by the loading force, the delamination length is taken as the total specimen span (L). The strain measurements (in Volts) are converted to stress using appropriate system calibrations and known value of the YoungÕs modulus of the maraging steel of which the bar is made. With the dynamic process confined within the mid-plane containing the initial Teflon pre-crack in the test specimen, the experiment was considered successful. 4. Experimental results and discussion 4.. Dynamic responses 4... Effect of mode mixing on the stress wave form Fig. 3(a) and (b) shows the stress field for varying mode mixing for. J and 4. J threshold impact energies for the crack opening in 6-ply (4.3 mm) and 4-ply (6.48 mm) composite specimen, respectively. No significant difference in the shape of the wave form is observed at these low impact energies. However, as the crack propagates at a higher impact energy (9.3 J) for the 6-ply specimen, the amplitude of the reflected wave increases with decrease in mode mixing as shown in Fig. 3(c). This is a mode dependent effect since the amplitude of the incident wave remains independent of the mode mixing as expected. When the I/II mode

10 Incident and Reflected Stress (MPa) Incident and Reflected Stress (MPa) Incident and Reflected Stress (MPa) 54 S.N. Wosu et al. / Engineering Fracture Mechanics 7 (5) Ply at E i =. J Incident Wave I/II=4/3 I/II=/5 I/II=4/5 I/II=/ Ply at E i = 4. J I/II=4/3 I/II=/5 I/II=4/ Reflected Wave (a) Time (microsec.) (b) Time (microsec.) Ply at E i = 9.3 J Reflected Wave Incident Wave I/II=/5 I/II=4/3 I/II=/5 I/II=4/ (c) Time (microsec.) Incident and Reflected Stress (MPa) 6 I/II=/5, 6-Ply J 4. J. J (d) Time (microsec.) Fig. 3. Stress wave forms for (a) 6-ply MONF specimen at. J, (b) 4-ply MONF at 4. J, (c) 6-ply MONF at 9.3 J for varying mixed mode ratios and (d) 4/5 mixed mode ratio at varying impact energies. ratio is fixed at 4/5 (D = 6 mm) while varying the incident impact energy, a significant dependence of the amplitude of the waveform on the impact energy is observed in Fig. 3(d) Effect of mode mixing on the force time history Fig. 4 displays the force time curves at varying mixed-mode ratios for. J and 9.3 J incident impact energies and shows that the peak force is slightly dependent on mode mixing (due to the observed dependency of reflected wave on mode mixing) but strongly dependent on the incident impact energy. Similarly, the amplitude of the loading force in the force displacement curves in Fig. 5 depends slightly on mode mixing but depends strongly on the incident energy. It is also noted that the maximum specimen displacement depends on the incident energy with.5 mm and.5 mm at. J and 9.3 J, respectively Effect of mode mixing on energy-absorption time history Fig. 6 shows fracture energy-absorption time histories for varying mode mixing ratios at three impact energies for the 6-ply specimen. The results for the same impact energies. J, 4. J or 9.3 J are summarized in Fig. 6(c) and show that mode mixing above I/II = has no significant effect on the peak fracture

11 Loading Force (kn) Loading Force (kn) Loading Force (kn) Loading Force (kn) S.N. Wosu et al. / Engineering Fracture Mechanics 7 (5) I/II=4/3 I/II=/5 I/II =4/3 I/II=/5 I/II=4/5 I/II=/5 E i =. J 5 5 I/II=4/5 I/II=/5 I/II =4/3 I/II=/5 I/II=4/5 I/II=/5 E i =9.3 J (a) Time (microsec.) (b) Time (microsec.) Fig. 4. Force time histories at (a). J and (b) 9.3 J impact energies for varying mode ratios for 6-ply MONF specimen. (a) 6 4 I/II=4/3 I/II=/ Displacement (mm) I/II=4/3 I/II=/5 I/II=4/5 I/II=/5 6-Ply at E i =. J (b) 5 5 I/II=/5 I/II=4/3 I/II=4/3 I/II=/5 I/II=/5 I/II=/5 6-Ply at E i =9.3 J...3 Displacement (mm) Fig. 5. Force displacement curves for 6-ply MONF specimen at (a). J and (b) 9.3 J impact energies for varying mixed mode ratios. energy absorption. A slight decrease in fracture energy absorption observed with increasing mode mixing for I/II <.5 where Mode I is more dominant mode at the higher impact energy of 9.3 J. This implies that less energy is absorbed in shearing fracture (Mode II) than in opening mode due to greater strain energy and breaking of bonds in Mode I than in Mode II. The result also shows that the residual energy retained by the specimen is also more mode dependent at I/II <.5. At a higher impact energy above 4. J, approaching unstable crack propagation state, this region (between 5 and 3 ls) decreases with time. This is because at this energy, the crack length has more than exceeded the critical crack length at which point the potential energy exceeds the fracturing energy. Thus, the fracture energy absorbed decreases because more energy is released than consumed by the crack growth which is shown to be rapid at these conditions, and crack propagation dissipates less energy during the period of rapid propagation and instability than during initiation. As shown in Fig. 6(c), the energy loss to the specimen decreases with increasing mode mixing with greater residual energy at higher impact energy above the 4. J threshold energy. Comparing Fig. 6(a) and (b) at lower energy (. and 4. J) with the curve at 9.3 J (Fig. 6(c)), it is clear that the peak energy absorbed at lower impact energies E i 6 4. J exhibits a plateau indicating the region of constant velocity. This is also observed for each mode ratio tested, implying that the observed behavior is

12 Peak Energy Absorbed, Ea (J) Energy Loss to Specimen, EL(J) Energy Absorbed (J) 54 S.N. Wosu et al. / Engineering Fracture Mechanics 7 (5) E i =. J I/II=/5 I/II=4/5.8 E i =4. J I/II=/5.3.. I/II=/5 I/II=4/3 Energy Absorbed (J).6.4. I/II=4/5 I/II=/5 I/II=4/ (a) Time (microsec.) (b) Time (microsec.) Energy Absorbed (J) (c).5.5 E i =9.3 J I/II=/5 I/II=/5 I/II=4/5 4. J I/II=4/3 J J Time (microsec.) (d) Mixed Mode Ratio (G I /G II ) J 9.3 J 4. J.5.5 Fig. 6. Fracture energy absorption time histories for (a). J, (b) 4. J, (c) 9.3 J impact energies for varying mixed mode ratios, and (d) peak energy and residual energy loss to specimen as functions of mode mixing. an energy-dependent factor rather than a mode-dependent factor. At a higher impact energy or as more energy is transferred to the crack tip, the energy absorbed in the initial plateau region decreases as in Fig. 6(c). This shows a decease in energy required to sustain the crack extension once the crack is initiated. Fig. 7 shows the energy-absorption time histories for varying impact energies. Points A and B on the curve represent the peak fracture energy absorbed and residual energy absorbed by the specimen, respectively. The difference between these two points is the elastic strain energy released. For a fixed mode ratio, the energy absorption depends strongly on the impact energy, with more energy absorbed from 4. J to 9.3 J than from. J to 4. J. The summary plot of peak values in Fig. 7(d) clearly shows energy absorbed is slightly dependent on mode mixing with more energy absorbed as Mode II dominates. In contrast, the residual energy absorbed is strongly dependent on mode mixing. As in Fig. 6, the summary plot also shows that the residual energy retained by the specimen is a more mode dependent factor than peak energy absorbed. The residual energy absorbed by the system depends on the properties of the specimen, laminate configuration, damage energy threshold, and the mode of damage generated. Since the mode mixing effect is a

13 Peak Energy Absorbed, Ea(J) Energy Loss to Specimen, El(J) S.N. Wosu et al. / Engineering Fracture Mechanics 7 (5) Energy Absorbed (J) I/II=4/3 (D= mm) 9.3 J 4. J J Energy Absorbed (J) I/II=/5 (D=8 mm) 9.3 J 4. J (a) J Time (microsec.) (b) J Time (microsec.) Energy Absorbed (J).5.5 I/II=/5 (D= mm) 9.3 J 4. J.5 J (c) Time (microsec.) (d) I/II=4/3 I/II=/5 I/II=4/5 I/II=/5 I/II=4/3 I/II=/5 I/II=4/5 I/II=/ Impact Energy, E i (J) Fig. 7. Fracture energy absorption time histories for (a) I/II = 4/3, (b) I/II = 4/5, (c) I/II = /5 for varying impact energies, and (d) peak energy absorbed and residual energy loss to specimen as functions of impact energy. material property, it is conceivable that the residual energy is a more mode-dependent factor than the energy absorbed. 4.. Variation of dynamic delamination with impact energy The expected specimen response to energy absorption is an increase in delamination crack length. Fig. 8 shows variations of dynamic delamination with increasing impact energy and loading positions for MENF and MONF configurations. It is clearly evident in the figure that delamination, once initiated, increases as impact energy is increased until the delamination length approaches the specimen span. For the MONF specimen,. J and 4. J of impact energies were required to initiate an interlaminar crack opening in 6-ply and 4-ply specimens, respectively, and J for the crack to propagate the entire span (L) of the 6-ply specimen. The results show that delamination crack growth is slow at low energy and increases very sharply as more energy is pumped into the crack tip. Note that the delamination becomes constant and independent of impact energy after 9.3 J for I/II = 4/5 and /5, and 3 J for both G I /G II = 4/3 and I/II = /5, and approaches a maximum value at incident energy of 3 J. Similarly for the MENF specimen, delamination is higher for mixed-mode loading (D = mm) than for pure mode loading (D = mm) and becomes constant at 8 J incident impact energy. It was observed that energy above this value (9.3 3 J

14 Delamination Crack Length, a (mm) Delamination Crack Length, a (mm) 544 S.N. Wosu et al. / Engineering Fracture Mechanics 7 (5) MONF, 6-Ply 4 (a) 3 D= mm (I/II=4/3) D = 8 mm (I/II=/5) D = 6 mm (I/II=4/5) D = mm (I/II=/5) 5 5 Impact Energy, E i (J) ENF, 6-ply, (D= mm) MENF, 6-ply, (D= mm) (b) Impact Energy, E i (J) Fig. 8. Effect of impact energy on delamination crack growth for (a) MONF and (b) MENF/ENF specimens. for MONF and 8 J for MENF) resulted in fragmentation and a reduced delamination length of the specimen as the specimen merely breaks into fragments [8] Variation of dynamic delamination with energy absorbed The delamination crack propagation increases non-linearly as the fracture energy absorption is increased as shown in Fig. 9(a). A non-linear curve fit to the results shows that the dynamic delamination is a power function of energy absorbed and is dependent on mode mixing for all the mixed-mode ratios tested. The higher the mixed-mode ratio, the lower the delamination for the same energy absorbed. Thus, for the same energy-absorption level, the delamination is greatest at a lower mixed-mode ratios corresponding to highest Mode II contribution. That a shearing mode contributed to more delamination than an opening mode for the same energy absorbed is contrary to our predictions. It implies that shear forces play a more dominant

15 Peak Energy Absorbed, Ea(J) Energy Loss to Specimen, El(J) Delamination Crack Length, a (mm) Loading Position from the Center, D (mm) S.N. Wosu et al. / Engineering Fracture Mechanics 7 (5) J J 5 J Loading Position 5 (a).5.5 Mixed Mode Ratio (G I /G II ) J 4. J 9.3 J J 4. J 9.3 J (b).5.5 Mixed Mode Ratio (G I /G II ) Fig. 9. Effect of mixed mode mixing on (a) delamination crack growth and (b) peak energy absorbed and residual energy absorbed by specimen for varying impact energies. role in the delamination of uni-directional composite material than tensile forces. An areal plot of Fig. 9(b) shows the partitioning of energy absorption into critical regions of delamination crack growth for the same (I/II = 4/3) mode ratio. Note the sudden increase in energy absorption after 4. J for the same mode ratio. The areal plot also indicates the main energy-absorption mode for the fracture process. The first region (labeled I in the figure) represents the internal energy of the composite system due to fracturing and breaking of the bonds while the second larger region (II) represents the elastic energy for crack propagation which appears to increase in proportion to crack propagation and decrease as the crack propagation decreases. The sudden jump in region III indicates high kinetic energy as the crack propagates uncontrollably, indicating the region of unstable crack growth. These observations indicate a smooth increase in crack propagation for. 6 E i 6 4. J, with beginning of unstable delamination at an impact energy of E i P 4. J for this 6-ply specimen.

16 546 S.N. Wosu et al. / Engineering Fracture Mechanics 7 (5) Variation of dynamic delamination with Mode I/II mixing Fig. shows delamination crack length as a function of mixed-mode I/II mixing for 6-ply MONF specimen. In general, delamination in the MONF specimen configuration decreases non-linearly as the mixed-mode ratio increases (or loading position decreases from center toward the specimen edge). At low incident energy, an increase in impact energy from. to 4. J results in only a % increase in delamination length for all the mixed-mode ratios tested. A higher impact energy above 4. J (from 4. to 9.3 J) results in a 73% increase in delamination. This observed low delamination growth at lower energy is due mainly to the greater fraction of the initial energy expended in the early stage to overcome friction and other effects of inertia, breaking of bonds, opening and propagation of the crack. As more energy is Fig.. (a) Delamination crack growth as a function of peak energy absorbed at 9.3 J impact energy for varying mode ratios and (b) regions of crack growth and area plot of energy absorbed delamination curve for varying impact energy showing.

17 Total Energy Release Rate, G T (kj/m ) Dynamic Energy Release Rate, G d (kj/m ) Energy Release Rate, G m, G m (kj/m ) S.N. Wosu et al. / Engineering Fracture Mechanics 7 (5) pumped into the crack tip, delamination increases faster as the crack propagates. Fig. (a) also shows that the loading position is a power function of mode mixing. It is observed (Fig. (b)) that the energy absorption as a function of mode mixing follows the same behavior as for delamination. This supports our initial assertion that dynamic delamination is driven mainly by the amount of energy available at the crack tip Variation of energy release rate with mode mixing The variations of energy release rates with loading position or mixed-mode ratios determined from Eqs. (5) and () are shown in Fig. for the three impact energies,. J, 4. J and 9.3 J. As predicted, Fig. (a) shows that G m II decreases as mixed-mode ratio increases and loading position decreases. In contrast, Gm I 5 II-MONF-9.3 J I-MONF-. J I-MONF-4. J I-MONF-9.3 J II-MONF-. J II-MONF-4. J II-MONF-9.3 J I II 5 I-MONF-9.3 J II-MONF-4. J I-MONF-4. J II-MONF-. J I-MONF-. J (a).5.5 Mixed Mode Ratio (G I/ G II ) J 9.3 J J J 4. J J.5 (b).5.5 Mixed Mode Ratio (G I /G II ) Fig.. Comparison of variations of energy release rates (a) G m I positions) for MONF 6-ply graphite/epoxy specimen. and G m II and (b) G T and G d with mixed I/II mode ratio (loading

18 548 S.N. Wosu et al. / Engineering Fracture Mechanics 7 (5) increases as mixed-mode ratio increases, and equal to G m II at I/II = / as predicted by the model. Thus, Mode I contribution decreases (and Mode II contribution increases) in the direction away from the center (D = ), that is, as the crack front moves toward the edge of the specimen (D L, or I/II = /5). The result shows that G m I and G m II are linearly dependent and vary with mixed-mode ratio. The study postulates that total energy release rate given by the failure criteria, G T, and dynamic energy release rate, G d are materials properties and are therefore expected to be independent of mixed-mode ratio. This is partly supported by Fig. (c) which shows that at low energy below the region of instability, the dynamic (G d ) and total energy (G T ) rates are approximately constant. At higher impact energy, the dynamic energy release rate decreases slowly with mixed-mode ratio due to an increase in crack velocity and associated kinetic energy which reduces the net energy available for crack propagation. This behavior implies that the resistance for crack growth decreases with particle velocity. However, the G T is observed to decrease more rapidly since the kinetic energy of the crack extension is neglected. A formulation similar to the above was first proposed by Sohn et al. [7] with kinetic energy neglected Variation of energy release rate with delamination and energy absorbed Fig. displays the variation of energy release rate with normalized delamination length, displaying a typical R-curve under plane stress. The high strain rate energy release rate, G d increases with normalized delamination crack extension with a change in slope at above four times the critical crack length for opening of the crack. The curve shows that the energy release rate as function of normalized delamination crack length is mode dependent, increasing as the loading position decreases (or as mixed-mode ratio increases). The fact that this curve is shifted from zero means that a critical crack initiation point exists above which the delamination growth will be unstable. The values can be estimated by fitting an appropriate non-linear function to the G d -crack extension curve. Such a curve when extrapolated to zero G d -value gives the critical values of 4. < (a/a c ) < 4.4 depending on loading position. In Fig. (b), it can be observed that G d increases with normalized energy absorbed more slowly than in the case of delamination, with the change in slope occurring at about two times the critical energy for the crack opening. Below this value, G d appears to be independent of energy in agreement with earlier results that show a small variation at energy below 4. J. However, in contrast to the delamination case, the result shows that the initial energy for crack growth initiation is independent of mixed-mode ratio. In both delamination and energy absorbed curves, it is clear that the higher the loading position (lower mixed-mode I/II ratio), the lower the dynamic energy release rate. Since the energy release rate is the crack driving force, it means that the force required to sustain a unit length of shearing (represented lower mode I/II mixing) in this uni-directional composite material is more dependent on crack extension than on the energy absorbed Comparison of fracture laws for MONF and MMB for varying mixed-mode ratios and impact energies Variation of G I with G II for varying mode ratios is shown in Fig. 3(a) comparing the MONF and MMB formulations for various mixed-mode ratios. An attempt was made to eliminate the scaling effect by fabricating the specimen with same span/width (L/B) ratio. Although the present study compares reasonably with the Reeder [8] model for G m II at both 4. J and 9.3 J incident impact energies, there is a significant difference between the two formulations in the case of G m I for both energies. The variation of the mixed-mode ratio with incident impact energy in Fig. 3(b) shows that the ratio decreases slightly with energy in MMB specimen but independent of energy in MONF specimen. This is because Eq. (4) for MMB included a specimen deformation and rotation correction term that depends on energy. As shown in Fig. 3(c), the difference between mode ratio for MONF model and MMB model is maximum when D is close to L or when the I/II ratio is the smallest (maximum Mode II contribution).

19 (kj/m ) (kj/m ) S.N. Wosu et al. / Engineering Fracture Mechanics 7 (5) Dynamic Energy Release Rate, G d (a).5.5 G d, D= mm G d, D= 8 mm G d, D=6 mm G d, D= mm Normalized Energy Absorbed, U f /E c Dynamic Energy Release Rate, G d G d, D= mm G d, D= 8 mm G d, D=6 mm G d, D= mm (b) Normalized Delamination Crack Length, a/a c Fig.. Variations of dynamic energy release rate with (a) normalized fracture energy absorbed and (b) normalized delamination crack extension for various loading positions (mixed mode ratio) for 6-ply MONF specimen. The mixed-mode energy release rate as function of loading position and mixed-mode ratios are plotted and compared for both MONF and MMB formulations in Fig. 4(a) and (b), respectively. For all the incident impact energies for the MONF, mixed-mode II contribution increases (and Mode I decreases) with loading position. It is clear from the figure that this observation is only true for MMB when D 6 8 mm. Although the variations of G m I and G m II energy release rates with mode mixing using MONF formulations is consistent with behavior with loading positions, Fig. 4(b) shows that both G m I and Gm II energy release rates using MMB formulations increase with mixed-mode ratio, reaching a maximum at I/II = 5/5 = / before decreasing. It is also observed that the mode mixing for the case of MMB does not have a consistent functionality with loading position, while in contrast in MONF, mode mixing increases with decreasing loading position. To understand why the models agree for G m II but disagree for Gm I, it is recalled that the present model is valid for all values of D 6 L compared to the Reeder [8] MMB model that is valid for D P L/3, and is particularly true for G m I according to Eqs. (4) and ().AtD = 8 mm (D < L/3), MMB predicts a mode ratio of zero (G I = ), implying that Mode I loading was not sufficient to open the crack. This is not the case in Mode II as been noted by previous investigators [6]. The similarity of fracture laws using MONF and MMB formulations for mixed-mode II with previous results [8 ] does show that the MONF configuration

20 55 S.N. Wosu et al. / Engineering Fracture Mechanics 7 (5) Fig. 3. SEM photographs of (a c) 4-ply MENF and (e f) 4-ply MCNF delamination fracture surfaces for varying loading positions at 75 J impact energy. proposed here can be used for mixed-mode testing using a fracturing Hopkinson bar. However, MONF formulation presented here adds the advantage of consistent fracture behavior for both Mode I and Mode II energy release rates, and the relationships between the loading position and the mode mixing Fractographic analysis Scanning electron microscope (SEM) techniques have been successful for fractographic studies of graphite/epoxy specimens to relate the detected fracture surface to the type of fracture mode involved [3 39]. Hackles features in SEM photographs are related to fracture induced delamination due to shearing fracture resulting from interlaminar stresses. In the present investigation, the fracture surfaces were generated by a fracturing split Hopkinson pressure bar under a variety of loading conditions and photographed using a scanning electron microscope (SEM) at regions indicated in Fig.. In mixed-mode testing using the MONF

21 S.N. Wosu et al. / Engineering Fracture Mechanics 7 (5) Fig. 4. SEM photographs of 6-ply MONF fractured surfaces between end of insert and near the edge for varying mixed-mode ratios at (a c) 4. J and (d f) 9.3 J impact energies. specimen, only the loaded upper section was photographed at the regions indicated by numbers in Fig. (where represents near insert end, 3 near the edge, and close to center). For MENF and MCNF loading configurations in Fig. 5, as the loading shifts from the center (D = mm) to the edge (D = mm), coarse hackles with some desegregation of very coarse hackles (Fig. 4(b) and (d)) are seen with some resins detached. An increased number of less coarse hackles are formed when the loading is near the edge (corresponding to low mixed I/II mode ratio) than at the center (D = mm). The central fracture surfaces at higher energy show fiber pull-out and evidence of fiber breakage. The hackle marks appear oriented perpendicular to the fiber with their tips bent over along the fiber in the direction of the relative motion of adjacent plies and with their width approximately equal to the distance between the fibers. Similarly, Fig. 6 for the MONF specimen shows smooth surface at higher mode ratios (4/3) indicating the absence of interlaminar shear force. Regularly spaced incipient to medium to fine hackles are observed at low impact energy (. J). As the energy is increased, the surface is mostly smooth with some fine to incipient hackles. At 9.3 J at which the crack propagates through the entire specimen

22 Hackle Marks Density (counts/µm) 55 S.N. Wosu et al. / Engineering Fracture Mechanics 7 (5) Hackle Marks Density (counts/µm) (a) Insert-end Center Edge Total Loading Position 9.3 J 4/3 /5 /5 Mixed Mode Ratio Loading Position from Center, D(mm) Hackle Marks Density (counts/µm).5 Insert-end Edge Center Total.5 MONF-6-ply, / (b) Impact Energy (J) Fig. 5. Quantitative summary of effect of (a) mixed-mode ratio and (b) impact energy on SEM hackle marks density for 6-ply MONF specimen. 4 3 MCNF-4 at 75 J Coarse Medium Fine Total (a) Loading Position from Center, D (mm) (b) Loading Position from Center, D (mm) Hackle Marks Density (counts/µm) (c).5.5 Coarse Medium Fine Total Hackle Marks Density (counts/µm) Loading Position from Center, D (mm) 4 3 MONF-6 Ply at 9.3 J Coarse Medium Fine Total MENF-4 at 75 J Fig. 6. Quantitative summary of effect of anti-symmetric loading on SEM hackle marks density for 4-ply (a) MCNF, (b) MENF specimens loaded at 75 J, and (c) 6-ply MONF specimen at 9.3 J.