epoxy laminates

Composites Science and Technology 63 (2003) 695–713 www.elsevier.com/locate/compscitech

Influence of fiber direction and mixed-mode ratio on delamination fracture toughness of carbon/epoxy laminates Ben W. Kim*, Arnold H. Mayer1 Structural Design and Development Branch, Air Vehicle Directorate, Air Force Research Laboratory, Wright-Patterson Air Force Base, OH 45433, USA Received 10 June 2002; received in revised form 18 September 2002; accepted 3 October 2002

Abstract Surface free energy has been often treated as a scalar constant without considering its dependence on propagation direction. It is desirable, however, to investigate how surface free energy or fracture toughness of delamination in a single interface varies with both the local mismatch angle of fiber directions and the direction of crack propagation in polymeric laminate composites. As a materials constant, fracture toughness is effectively used for various mechanical analyses of fiber-reinforced composites as well as conventional materials. This study investigates quantitatively and qualitatively the dependence of delamination fracture toughness on mismatch angle and crack propagation direction in laminated structures. AS4-Carbon/Epoxy prepregs were used for fabricating test specimens, and 50 different mismatch angles of fiber direction were applied on the delaminated interface of laminates. Fracture toughness was measured using the Mixed-mode Bending (MMB) test. This test method is composed of fracture mode-I and modeII, and the mixed-mode ratio (GII/G) can be controlled. The mixed-mode ratios used here were 20, 35, 50, 65, and 80%. The crack path and the delamination fracture toughness were observed and calculated for specimens, and the dependence of the toughness was shown to be related to the mismatch angle of ply fibers at the delaminated interface. The relationship between mismatch angle and delamination fracture toughness was newly revealed and discussed for various angles. These results can be usefully applied to various fracture mechanics analyses in fiber-reinforced laminated composites. Published by Elsevier Science Ltd. Keywords: Mixed-mode bending (MMB)

1. Introduction Fiber-reinforced polymeric composites have been studied as advanced materials for many applications such as aircraft, ships, cars, leisure, and sports equipment, etc., because of their potential for outstanding mechanical properties. Fiber-reinforced composites are made of high-modulus fibers and relatively ductile polymer matrices. Fibers and polymeric matrices produce excellent mechanical properties when they are combined together. As a result, composites have the high strength and modulus similar to the fibers and the * Corresponding author. Tel.: +1-937-255-3709; fax: +1-937-6566321. E-mail addresses: [email protected] (B. W. Kim), [email protected] (A. H. Mayer). 1 Tel.: +1-937-255-5232; fax: +1-937-656-4945 0266-3538/03/$ - see front matter Published by Elsevier Science Ltd. PII: S0266-3538(02)00258-0

light weight and the chemical resistance of the polymeric matrix. Similar to other materials, the fracture modes of polymeric composites should also be investigated so that the materials can be applied into the real products. There are several fracture modes in fiberreinforced composite materials: fiber breakage, matrix cracking, fiber/matrix debonding, fiber pull-out, plugging, and interlaminar delamination, etc. [1]. The main fracture mode that should be emphasized may be different for each damage circumstance, but generally delamination is one of the major fracture modes of many advanced laminated composite structures [2]. Therefore, better understanding of the interlaminar fracture resistance of laminates is very useful for the structural design and development of materials. Delamination is the separation between two laminae in a laminated structure. Delamination is usually evaluated or estimated as the type of fracture toughness that

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is the constant property of a material. Fracture toughness is determined as the value of critical stress intensity factor or critical strain energy release rate of a material. Fracture toughness of composite materials is believed to depend on the mismatch angles of fiber directions of two neighboring laminae as well as the directions of crack propagation [3,4]. Thus, in order to investigate the relationship between the fiber direction of interfacial plies and the delamination fracture toughness, it is required to measure and analyze the toughness quantitatively and qualitatively for various mismatch angles, or fiber directions, and crack growth directions. Also, the fracture toughness of delamination has to be considered for a wide range of mixed-mode ratios (GII/G), or the fraction of mode-II, because delamination could occur at any ratio of mode-I and mode-II. Many researchers have investigated delamination fracture toughness of composite laminates for mode-I [5–8], mode-II [9–13], and mixed-mode of mode-I and II [14,15]. Recently, the Mixed-mode Bending (MMB) test has been popular to measure the delamination fracture energies of laminates [16–26]. The MMB apparatus was first designed and developed at NASA Langley Research Center [27,28]. In this research, critical energy release rate of delamination, or delamination fracture toughness, was measured using the MMB apparatus. The MMB test was designed to observe the strain energy release rate of laminates for various mode ratios of modes I and II. Since delamination is not only the result of pure mode-I loading but also the result of pure mode-II loading, the fracture toughness of delamination should be considered for mode-I, mode-II, and their mixed-mode. The mixed-mode ratio of the MMB can be easily controlled for a wide range of mode ratios using the lever length of the apparatus, but pure mode-I and pure mode-II cannot be obtained. The purpose of this research is to investigate and analyze the interlaminar fracture toughness of continuous fiber reinforced composite materials for various mixed-mode loading ratios of mode I and II, fiber directions of plies at the delaminated interface, and crack propagation directions. The results reveal how mixed-mode ratio and mismatch angle are relevant to delamination fracture toughness, and the failure criteria of delamination are also developed for composite damage tolerance and durability. Also, information on the delamination fractography and crack growth is presented. The MMB specimen consists of unidirectional carbon fiber laminates with single-phase polymer matrices, AS4-3506 carbon/epoxy. The delamination fracture toughness of carbon/epoxy laminates was measured and characterized at various mixed-mode ratios and local mismatch angles at the interface. Even though many researchers have studied delamination fracture toughness of laminates, most of them performed the tests using unidirectional composite laminates which

have no mismatch angle at the delaminated interface. A few researchers tried to analyze fracture toughness using laminates of multidirectional fibers [5,21,22], but this data is too limited to clearly reveal the dependence of fracture toughness on various mismatch angles and mixed-mode ratios. In this research, the relationship between the delamination fracture toughness and the mismatch angle and the mode ratio were clearly presented. The quantitative and qualitative analysis of fracture toughness can be applied to the various fracture phenomena of composite laminates, such as the ballistic impact phenomena in which fracture toughness of delamination may play an important role [29,30].

2. Mixed-mode bending (MMB) test 2.1. MMB apparatus Recently, the MMB test has become popular for delamination testing of materials, because with one specimen geometry, it is possible to characterize the delamination initiation and growth for a wide range of mixed-mode ratios. Since the fracture of real products occurs as a type of mixed-mode rather than only one mode, the mixed-mode test is necessary in the actual applications of fracture analyses. The Assymetric End Loaded Split (AELS) test was often used to investigate the fracture toughness for the various configurations of fiber-reinforced composites [31]. The apparatus was the mixing of both the Double Cantilever Beam (DCB) test for mode-I and the End Loaded Split (ELS) test for mode-II. Later, Reeder and Crews designed the Mixedmode Bending test apparatus, which is very similar to AELS, by combining fracture mode-I and mode-II [27] as shown in Fig. 1, and redesigned to reduce the nonlinear effects [28] as shown in Fig. 2. Although the MMB was created for thin, unidirectional, symmetric laminates, it has been used for thick, asymmetric, offaxis ply laminates, and thus delamination fracture toughness could be affected by other factors which are not explained properly [22]. It has been said that a delamination between off-axis plies could complicate the fracture mechanics analysis because of the possibility for oscillations of stresses and displacements near the crack tip, and that the delamination typically possesses bending/twist coupling and bending/bending coupling and thus coupling results in non-uniform toughness value distribution, local mixed-mode effects, and skewed and curved crack fronts in fracture testing [22]. Nevertheless, the off-axis ply and asymmetric specimens were designed here and tested using the MMB apparatus, because the apparatus currently appears to be the best alternative to characterize the fracture toughness of delamination. Since the MMB test is used to measure the fracture toughness of mode-I and mode-II simultaneously and

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Fig. 1. Schematic figures of (a) DCB, (2) ENF, and (c) MMB.

thus mixed mode, the mixed-mode ratio of mode-I and mode-II is arbitrarily chosen and set up easily without changing the specimen geometry. The new hinge was also introduced for better loading of this improved delamination beam testing [32]. The apparatus of the delamination test is composed of four main parts: MMB fixture, loading machine, load-displacement recorder, and traveling microscope. The MMB fixture is a main part of the apparatus, and the fixture used in this research was fabricated by Dr. Uday Vaidya’s group at the Dept. of Mechanical Engineering in North Dakota State University (currently, he works at the University of Alabama, at Birmingham) and installed in the Wright-Patterson Air Force Research Lab. Fig. 2 shows the MMB fixture, which was fixed in the MTS loading machine for applying load to the specimen.

Fig. 2. Mixed-mode bending fixture.

2.2. Strain energy release rates from the MMB test The energy release rate, G, is the rate of change in potential energy,
, with crack area, A, as expressed in Eq. (1). G¼

d
dA

ð1Þ

For linear elastic materials, G can also be written in terms of total strain energy: G¼

dU ; bda

ð2Þ

where a and b are delamination length and specimen width, respectively. That is, G is the loss of strain energy, dU, per unit width for an infinitesimal increase of delamination length, da. Fracture toughness of mixed-mode is defined as the critical value of strain

energy release rate for delamination growth in mixedmode. The strain energy release rate of mixed-mode is the sum of mode-I and mode-II strain energy release rates. G ¼ GI þ GII ;

ð3Þ

where GI and GII are strain energy release rates of mode-I and mode-II, respectively. The MMB loading is presented by the simple addition of pure mode-I and pure mode-II, such as DCB and ENF, respectively. Strain energy release rate equations can be formulated using the beam theory and shear deformation component. Reeder and Crews [27,28] developed the theoretical equations of strain energy release rate for the MMB test by superposing strain energy release rate expressions of the DCB and ENF:

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GI ¼

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4ðPð3c  LÞÞ2 ða þ hÞ2 64bL 2 E1f I

ð4Þ

3ðPðc þ LÞÞ2 ða þ 0:42hÞ2 ; 64bL 2 E1f I

ð5Þ

and GII ¼

where P, c, and a are the applied load, lever length, delamination length, respectively. L is the half-span length of the MMB apparatus, and b and h are the width and the half thickness of the specimen, respectively. E1f is the bending elastic modulus of the laminate in a fiber direction and can be obtained as
 8ða0 þhÞ3 ð3cLÞ2 þ 6ða0 þ0:43hÞ3 þ4L 3 ðcþLÞ2   ; E1f ¼ 1 2 3  Csys 16L bh m ð6Þ where a0, Csys, and m are the initial delamination length, the compliance of the loading system and the slope of load-displacement plot of the calibration specimen, respectively. I is the area moment of inertia of one delaminated half of a specimen. w is the crack length correction for crack tip rotation and is defined as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u  2 ! u E11 ¼t ; ð7Þ 32 1þ 11G13 where is the transverse modulus correction parameter, and E11 and G13 are the longitudinal tensile modulus of elasticity and the out-of-plane shear modulus, respectively. Also is determined as pffiffiffiffiffiffiffiffiffiffiffiffiffiffi E11 E22 ¼ 1:18 ; ð8Þ G13

load to the specimen, two tabs were attached into the top and bottom of one end of a specimen. The endblock was used because it was more convenient and less erroneous than the piano hinge. End-blocks (or piano hinges) should have an elastic modulus greater than 60,000 MPa and be attached strongly enough to endure the maximum loading of MMB apparatus. Mild steel was used for the end-block and its dimensions were 25 mm  25 mm  9.5 mm. The specimen is made up of 16 plies of carbon/epoxy laminas. The top seven plies and bottom seven plies are unidirectional parallel to the beam direction, and the two innermost plies are aligned to contact each other and to form mismatched angles at the interface. Specimens were designed with respect to various incline angles (inc) and mismatch angles (mis). The mismatch angle is defined as the angle of the interface between the 8th and 9th plies. The incline angle is defined as the angle from the beam axis (0 ) to the bisection of the mismatch angle as defined in Fig. 4. Two plies (8th and 9th plies from top ply of a specimen) at the interface form the mismatch angle of the fiber direction with respect to each other. The incline angle, therefore, plays the role of the bisector of the mismatch angle. Ten incline angles (0, 10, 20, 30, 40, 50, 60, 70, 80, and 90 ) and five mismatch angles (0, 30, 45, 60, and 90 ) were selected for the lay-up design of specimens. The combination of incline angle and mismatch angle created fifty different types of mismatched angles at the delaminated interface. [(10 incline angles)  (5 mismatch angles)=(50 angles)]. The standard lay-up formulation is as follows:

where E22 is the transverse modulus of elasticity. 2.3. Materials and specimen The MMB specimen is a rectangular, uniform thickness, and symmetrical or asymmetrical laminate of AS43506 carbon/epoxy which has a longitudinal modulus of 129 GPa and a transverse modulus of 11.0 GPa. A nonadhesive insert is placed at the mid-plane which serves as a delamination initiator as shown in Fig. 3. A polymer film is recommended for the insert, and the folding or crimping problems should be avoided at the cut end of the insert. Teflon was used as an insert material and its thickness was no greater than 13 mm. The overall length and width (b) of the specimen are 135 and 25 mm, respectively. The thickness of the specimen (2h) is about 3 mm for 16 plies of prepregs. In order to apply

Fig. 3. MMB specimen.

Fig. 4. Mismatch angle (mis) and incline angle (inc).

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 3 2 mis 0 =ð þ mis =2Þ= inc  6 7 inc 2 7 6 7 4 5 07 The laid-up laminates were placed in the autoclave and cured at 250  F for 1 h and 350  F for 1 h under a pressure of 30 psi. Next, end-blocks were attached into the cured laminates using Hysol adhesives from Loctite Corporation. Even though the MMB specimen was designed for thin, unidirectional, symmetrical specimens, the fracture criterion from the MMB test was validated for various thick [19] and non-symmetric off-axis ply specimens [22,23]. Thus, in this research, the 16 plies were used to fabricate a specimen, and the specimens were laid up symmetrically or asymmetrically according to the lay-up design of the specimen. 2.4. Test procedure The test begins with measuring the physical dimensions of a specimen, including width, thickness, and height. Then, the edges of the specimen are coated with a thin layer of water-soluble typewriter correction fluid to enable visual observation of the delamination propagation during the test. The ends of an insert on both edges are marked with a thin vertical line every millimeter for the first 5 mm past the end of the insert and every 5 mm thereafter up to 25 mm. Mode-II fraction (GII/G) is chosen, and lever length, c, is set for mode-II fraction. This research dealt with five different mode-II fractions: 20, 35, 50, 65, and 80%, and the corresponding lever lengths were 107.7, 59.9, 43.7, 34.6, and 27.9 mm, respectively. The compliance of the loading system was measured using a calibration specimen for each mode-II fraction. A pure aluminum beam was used for the calibration specimen in this research. A specimen is placed in the text fixture when all parameters for the apparatus are set up. A traveling optical microscope is installed by the apparatus to effectively observe the crack growth. The microscope pinpoints the crack front with an accuracy of at least 0.5 mm. Loading is applied to the specimen continuously in displacement control, and the load-displacement data are recorded. The crack front of one edge of the specimen is also visually observed from the end of the insert up to 25 mm. When the crack begins at the end of the insert, the loading value is taken and recorded from the load–displacement plot. Additional indications are recorded as crack grows past each point marked on the specimen edge. When crack has extended past the last mark, the loading is released and the specimen is removed from the apparatus. To measure delamination area, white paint is sprayed on the delaminated surface and then the specimen is placed under

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the vacuum until the paint penetrates evenly into the cracked area. When the paint becomes dry, the specimen is split completely by hand so that the delamination shape can be observed precisely. One half of the specimen is taken and initial delamination length (pre-crack length) is measured more exactly from the center of the loading pin hole of the tab to the end of the insert. Even though the 50 mm long insert is placed as a delamination initiator during specimen fabrication, insert length and position can vary. The specimen with a mismatch angle at the interface would show various shapes of delamination area. Then, using the parameters taken from specimen properties and delamination results, the energy release rates for mode-I (GI), mode-II (GII), and mixed mode (G) are calculated. Delamination fracture toughness (GIc,GIIc, andGc), or critical energy release rates, for each mode are determined from the load point at which the initiation of delamination is microscopically observed on the specimen edge. All experiments were done at room temperature (25  C). The test apparatus is operated in a displacement control mode with a constant displacement rate of 0.5 mm/min. This slow rate enables the observation of the crack propagation more effectively. Since the loading rate hardly influences the interlaminar fracture properties [4], the same load rate was applied to all specimens in this research.

3. Experimental results and discussion 3.1. Interlaminar fracture 3.1.1. Crack propagation When load is applied to a specimen in the MMB apparatus, the crack propagates along the pre-cracked plane which is the exact mid-plane of the specimen thickness. The mid-plane is the interface between the 8th and 9th plies when plies are numbered from the top layer to the bottom layer in a specimen. The pre-delaminated area is formed by placing a Teflon film insert in the midplane during specimen fabrication. The insert film is approximately 50 mm long, and delamination initiates at the end point of the insert. Fig. 5(a) schematically shows the typical route and shape of crack propagation in the thickness of a specimen. After passing by the pre-delaminated area, the crack propagated by jumping up inside an 8th ply rather than going along the mid-plane. The crack went through the 8th ply and approaches the interface of the 7th and 8th plies. Therefore, the fiber direction of the 8th ply appears to be the main factor for the variation of delamination fracture energy, because the delamination occurred inside the 8th ply and at the interface between the unidirectional 7th ply and the mismatched 8th ply. This phenomenon of crack jumping was observed for all

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Fig. 5. Delamination path and shape: (a) side-view (thickness view) and (b) top-view (delaminated area view) of a specimen.

specimens tested in this research. This kind of crack jumping at interlaminar experiments is considered a result of elastic coupling effects [22], and it also appears to influence the R-curve of interlaminar fracture especially for fracture mode-I [21]. The crack jump may also be due to the specimen geometry. In the MMB apparatus, the upper eight plies are in much more bending than lower 8 plies for all specimens. More bending causes to decrease the toughness of the 8th ply around the crack tip area and then the crack propagates along the weaker points. As a result, the crack moves up to the interface between the 7th and 8th plies. Fig. 5(b) shows a schematic of two completely split pieces of a specimen. The delaminated area includes three different areas: a pre-crack area, a triangularshaped crack area, and an interlayer crack area. The pre-crack area is the specimen midplane which is the interface between the 8th and the 9th plies. When crack initiated at the end of the pre-crack area, the crack path deviated from the specimen midplane and jumped into the 8th ply to form a triangular-shaped area. This triangle area did not form at the midplane of the 8th and 9th plies, but rather inside 8th ply. That is, the 8th ply is split into two pieces in thickness, which are attached at the top half and the bottom half of a specimen, respectively. Thus, two of the triangles in the top and bottom halves are all 8th ply itself. Triangular shapes varied with respect to the fiber direction of 8th ply, and the triangle area became smaller as the fiber angle of the 8th

ply increased. The triangular area, however, did not form for fiber directions below 25 . The size of triangular area was fully dependent on the fiber angle of the 8th ply, and it became smaller as the angle increased. After passing by the triangle area, the crack approached the interface between the 7th and the 8th plies. It is important that the actual delamination does not propagate at the midplane, or the interface between the 8th and the 9th plies, but it rather propagates near the interface between the 7th and the 8th plies. The fiber direction of the 7th ply is 0, or parallel to the beam axis, and the fiber of the 8th ply is off-axis, or mismatched. Although specimens of this research were fabricated by giving various mismatch angles into the interface of the 8th and 9th plies, the fiber direction of the 9th ply appeared not to directly influence the delamination procedure. This is because the delamination crack passes through near the interface of 7th and 8th plies rather than along the midplane, or the interface of 8th and 9th plies. 3.1.2. Delamination shape The delamination shape varies according to the fiber direction resulting from the mismatch and incline angles. Since the angle of the fibers at the interface is different for each configuration of the specimen, the delamination shape is also different. Generally, three cases of delamination shape were considered. Fig. 6 shows schematics of typical delamination shapes. If there is no mismatch angle but symmetric lay-up, the

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Fig. 6. Schematic shapes of delamination area.

delamination shape for the specimen with the principal angles (0 or 90 ) of fiber is relatively simple as shown in Fig. 6(a). The delamination crack grows regularly and keeps its rectangular shape, and the crack growth direction is either parallel or perpendicular to the fiber direction for 0 or 90 lay-ups, respectively. When the fiber direction is oblique, however, the crack shape changes according to the fiber angle from the beam direction [Fig. 6(b) and (c)]. As mentioned in the previous subsection, the triangular shape is created for the specimens with the fiber angle of over 25 , whereas, for the ones with the fiber angle of below 25 , no triangular shape forms. Similar to the case of principal fiber direction, the forward edge line of delamination area is straight. For the specimen with the interfacial mismatch angle, however, the edge line becomes concave as shown in Fig. 6(d). Most specimens with a mismatch angle at the interface showed this type of delamination growth, while still keeping the triangular area for the fiber angle of over 25 . The reason of this round edge of the delamination area is believed to be due to the asymmetric lay-up of the specimen. When the interfacial mismatch angle is engaged in the lay-up procedure of prepregs, the specimen becomes asymmetric and thus there will be a non-even stress distribution during the high temperature

curing. It can cause the unbalance of local fracture toughness of specimen width. As a result, the crack, or delamination, does not grow in a straight shape, but rather in a round shape. The concave shape of the delamination area indicates that the center of specimen width shows a higher local toughness than around the edges. 3.1.3. Load–displacement relationship It is not easy to determine the load point at which a crack initiates. Several methods have been used for determining the initiation load value, such as nonlinearity (NL), acoustic emission (AE), 5% off slope, 5% off maximum load, and visual observation (VO). The nonlinearity method determines the load of crack initiation as a point where the slope deviates from the linearity, and the acoustic emission takes the load point where the first obvious signal occurs using an acoustic emission recorder. The nonlinear method is difficult to identify the load point exactly, but it is commonly used. Acoustic emission correlates more closely with non-linearity on the load–displacement curve, and both methods show very similar results [19,23]. The 5% Off Slope determines crack initiation as the load point where 5% of the initial slope is offset. This method has been pre-

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ferred in standards for homogeneous materials and pure mode loadings, but it is considered unsuitable for the MMB test [19]. The 5% Off Max takes the crack initiation as the point of 5% offset from maximum load. The VO observes the crack initiation visually and takes the load point of crack initiation. NL and AE result in the lower values of fracture toughness, while 5% Off Slope and 5% Off Max show higher toughness values. The VO technique usually shows the median value of NL and 5% Off Max loads. In this research, the load of crack initiation for delamination fracture toughness was determined by 5% Off Max in order to give the same condition to all specimens. Fig. 7 shows an example of load–displacement plots for five mixed mode ratios. A load was applied to zero incline angle and zero mismatch angle specimens, i.e., symmetric and unidirectional specimens, and the load was measured according to mode-II ratios (GII/G): 20, 35, 50, 65, and 80%. As shown in the figure, the load of crack initiation looks relatively similar to 5% Off Max and NL methods, except for 20% of mode-II ratio. For the higher mode-II ratios (80, 65, and 50%), the load drops suddenly. This phenomenon indicates that the crack propagates quickly after the crack initiates for higher mode-II specimens. For the lower mode-II ratios (35 and 20%), the load increases and decreases slowly after the peak. 3.2. Dependence of fracture toughness 3.2.1. Dependence on mixed-mode ratio (GII/G) Fig. 8 indicates the relationship of mode-I and modeII fracture toughness at a 50 incline angle. The plot

includes three mismatch angles of 0, 30, and 45 , and five different mixed-mode ratios. Strain energy release rates for delamination were calculated for mode-I and mode-II using Eqs. (4) and (5), respectively, and total energy release rates were obtained from Eq. (3). Fig. 9 presents delamination fracture toughness as a function of mode-II ratios (GII/G) for zero incline and mismatch angles, and it shows the effect of mode-II fraction on total mixed-mode fracture toughness. The fracture energy of mode-I does hardly change for all ranges of mixed-mode ratios, but the mode-II fracture energy increases substantially as the mode-II ratio increases. Below 50% of mode-II ratio, the total fracture energy mainly comes from mode-I loading and the highest stresses are near the crack tip. Above 50% of mode-II ratio, the total fracture energy is strongly dependent on mode-II loading rather than mode-I loading. The highest stresses of the mode-II loading are away from the crack tip and it is loaded more in shear than in peel. The reason for higher mode-II toughness may be due to the longer plastic zone than mode-I, and thus the longer plastic zone dissipates more energy [24]. As a result, the mode-II loading causes higher energy and thus higher fracture toughness than mode-I. Thus, the increase of total delamination fracture toughness with respect to mode-II ratio is mainly due to mode-II fracture energy. Fig. 10 presents total fracture toughness of delamination as a function of mode-II fraction for three mismatch angles (mis=0, 60, and 90 ) at 20 of incline angle. The toughness clearly increases as mode-II fraction increases. The experimental data were well fitted using power equations for all three data curves. The power fitting equations are follows:

Fig. 7. Load versus displacement for mode-II ratios at zero incline and mismatch angles: GII/G=20, 35, 50, 65, and 80%.

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Fig. 8. Mode-I fracture toughness (GIc) versus mode-II fracture toughness (GIIc) at 50 incline angle (inc=50 ).

Fig. 9. Fracture toughness of GIc, GIIc, and Gc for zero incline and mismatch angles (inc=0 , mis=0 ).

Gc ¼ 0:069 þ 0:491ðGII =GÞ1:810 for mis ¼ 0 at inc ¼ 20 ;

Gc ¼ 0:072 þ 0:366ðGII =GÞ1:921 for mis ¼ 60 at inc ¼ 20 ; and

Gc ¼ 0:063 þ 0:260ðGII =GÞ1:856 for mis ¼ 90 at inc ¼ 20 where Gc is the delamination fracture toughness. On this basis, the general relationship between delamination fracture toughness and mode ratio can be formulated as  m GII Gc ¼ A þ B

; ð9Þ G

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Fig. 10. Fracture toughness (Gc) as a function of mode-II fraction (GII/G) for three mismatch angles (mis=0, 60, and 90 ) at 20 incline angle (inc ¼ 20 ).

where A, B, and m are parameters and Table 1 presents the parameters for each specimen configuration. As described in Section 2.3, there are 50 different kinds of mismatched angles at the delaminated interface. Each specimen configuration has its own parameters for the relationship between the toughness (Gc) and the modeII fraction (GII/G). For each configuration, 15 specimens were tested in the range of GII/G from 20 to 80%. The ranges of A, B, and m values are0.997–0.106, 0.182–1.232, and 0.093–2.970, respectively. Using these values and the relationship of Eq. (9), the delamination fracture toughness for untested mode ratios and mismatched interfacial angles can be easily predicted for each configuration. Previous studies present the relationship between interlaminar fracture toughness and mode-II fraction (GII/G) for glass fiber laminates [18,19,22–24] and carbon fiber laminates [17] as a failure initiation criterion. The relationship was written as [18]  m GII Gc ¼ GIc þ ðGIIc  GIc Þ

; ð10Þ G where GIc and GIIc are the delamination fracture toughness for mode-I and mode-II, respectively. From this equation, m was deduced as  2.60 for E-glass/ epoxy [18] and  1.56 for carbon/epoxy composites [17]. In this research, however, the more precise mathematical relationship was obtained from the experimental data as  k GII Gc ¼ 2GIc þ ðGIIc  GIc Þ

; ð11Þ G where k is the exponential parameter of the model. Comparing Eq. (11) with Eq. (10), the first term of right

hand side of the equation is doubled. This model presents a much closer relationship to experimental data points and can be a good fracture criterion for mixed mode delamination testing. The parameter k values for each configuration are also shown in Table 1. 3.2.2. Dependence on incline angle (inc) As described earlier, the incline angle (inc) is the angle between the beam direction (0 ) and the bisection of the mismatch angle (mis). The dependence of fracture toughness on incline angle varies with respect to mismatch angle. For the specimens with zero mismatch angle (mis =0), the toughness shows a linear dependence on incline angle for all mixed-mode ratios (GII/G), as shown in Fig. 11. The toughness is inversely proportional to incline angle and varies the most as mode-II fraction is 80%. The fitting slope becomes smaller as the mode-II fraction decreases. When GII/G=50%, the toughness varies little, and, for GII/G=35% and GII/ G=20%, the toughness is essentially constant. This is attributed to the trend that the variation of toughness is small for lower fractions of mode-II. Similar to the specimens without a mismatch angle, the fracture toughness of specimens with mismatch angles (mis6¼0) decreases as the incline angle increases. The toughness, however, shows bilinear variation against incline angle rather than linearity. Fig. 12 shows the variation of fracture toughness as a function of incline angle for each mixed-mode ratio at the mismatch angle of 60 (mis=60 ). As the incline angle increases, the toughness of all mixed-mode ratios except GII/ G=20% decreases until inc=60 , shows the lowest value around inc=60 , and increases in later regions of

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B.W. Kim, A.H. Mayer / Composites Science and Technology 63 (2003) 695–713 Table 1 Parameter values for the relationship of mixed-mode delamination fracture toughness (Gc) and mode mixture (GII/G) Gc ¼ A þ B ðGII =GÞm [Eq. (9)]

Configuration ( )

Gc ¼ 2GIc þ ðGIIc  GIc Þ ðGII =GÞk [Eq. (11)]

inc

mis

A

B

m

k

0

0 30 45 60 90 0 30 45 60 90 0 30 45 60 90 0 30 45 60 90 0 30 45 60 90 0 30 45 60 90 0 30 45 60 90 0 30 45 60 90 0 30 45 60 90 0 30 45 60 90

0.061 0.106 0.028 0.071 0.103 0.050 0.040 0.071 0.042 0.041 0.069 0.018 0.121 0.072 0.063 0.065 0.082 0.027 0.061 0.064 0.003 0.004 0.050 0.034 0.010 0.063 0.036 0.060 0.067 0.006 0.036 0.012 0.064 0.032 0.071 0.012 0.034 0.015 0.005 0.036 0.828 0.346 0.067 0.002 0.017 0.997 0.054 0.074 0.140 0.038

0.656 0.603 0.341 0.368 0.328 0.601 0.577 0.523 0.444 0.303 0.491 0.455 0.458 0.366 0.260 0.426 0.322 0.375 0.296 0.256 0.386 0.287 0.318 0.401 0.222 0.337 0.300 0.247 0.290 0.251 0.302 0.266 0.304 0.226 0.314 0.291 0.227 0.194 0.241 0.222 1.072 0.569 0.182 0.277 0.290 1.232 0.242 0.274 0.439 0.342

2.287 2.970 0.960 1.532 2.561 1.722 1.756 1.954 1.444 1.168 1.810 0.909 2.942 1.921 1.856 1.662 1.498 1.074 1.408 1.770 0.826 0.712 1.473 2.062 0.717 1.406 1.223 1.705 1.930 0.829 0.969 0.998 1.763 1.251 2.412 0.872 1.363 0.787 0.724 1.219 0.107 0.191 2.021 0.962 1.076 0.093 1.181 1.718 0.463 1.556

4.85 3.69 9.36 6.91 5.14 2.54 4.93 1.85 2.15 6.21 6.24 8.37 2.03 1.34 3.40 9.30 4.62 3.36 3.11 1.82 3.28 2.65 3.23 2.10 3.03 4.90 1.19 5.11 6.44 4.47 3.58 3.91 1.84 5.17 4.23 3.30 2.28 8.22 1.03 3.41 8.87 1.31 2.43 4.85 4.50 1.13 1.09 8.10 1.22 1.99

10

20

30

40

50

60

70

80

90

incline angle. As for the specimens tested under GII/ G=20%, there was no remarkable change of toughness, because toughness dependence on incline angle becomes very small as mode-II fraction reduces. For mis=60 , the fiber direction of the 8th ply lie at the angles of 30, 40, 50, 60, 70, 80, 90, 100, 110, and 120 each corre-

                                                 

103 103 103 103 103 103 103 102 1014 105 103 104 102 102 103 103 104 103 103 104 103 103 104 102 103 103 102 102 103 103 103 103 102 103 103 103 1014 103 104 103 103 102 1014 103 103 102 102 103 103 1014

sponding to inc=0, 10, 20, 30, 40, 50, 60, 70, 80, and 90 , respectively. The fiber direction of the 8th ply at 90 corresponds to inc=60 , and, at this point, the toughness value shows the lowest point for most mixed mode ratios except GII/G=20%. After inc=60 , the fracture toughness increases slowly. As a result, as the fiber angle

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Fig. 11. Fracture toughness (Gc) as a function of incline angle (inc) for several mixed-mode ratios at zero mismatch angle (mis=0 ).

Fig. 12. Fracture toughness (Gc) as a function of incline angle (inc) for several mixed-mode ratios at 60 mismatch angle (mis=60 ).

of the 8th ply is around 90 , the delamination fracture toughness was lowest for each configuration. 3.2.3. Dependence on mismatch angle (mis) The dependence of fracture toughness on mismatch angle can be classified as three cases according to the

fiber direction of the 8th ply: the range smaller than 90 , the range including 90 , and the range greater than 90 . Fig. 13 shows the variation of delamination fracture toughness as a function of mismatch angle in the range smaller than 90 of the fiber angle of the 8th ply at zero incline angle. As a result, the fiber angle of the 8th ply

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707

Fig. 13. Fracture toughness (Gc) as a function of mismatch angle (mis) for several mixed-mode ratios at zero incline angle (inc=0 ).

becomes half of the mismatch angle. In this range, delamination fracture toughness is obviously dependent on the mismatch angle between fiber directions of two adjacent plies (8th and 9th). The fracture toughness is inversely proportional to mismatch angle, i.e., as the mismatch angle increases, the toughness linearly decreases for higher mode-II ratios (GII/G=80 and 65%), but, for lower mode-II ratios (GI/GII=50, 35, and 20%), the toughness barely varies with mismatch angles. This means that the effect of mismatch angle is greater in shear mode-II than opening mode-I, and the dependency of fracture toughness on mismatch angle becomes greater as the mixed-mode approaches more closely to pure shear mode-II. Second, in the range of 90 fiber angle of the 8th ply, the delamination fracture toughness behaves bi-linearly with respect to mismatch angle. Fig. 14 shows the variation of fracture toughness as a function of mismatch angle at the incline angle of 70 . The corresponding fiber angles of the 8th ply were indicated as the second x-axis. As the mismatch angle increases, the toughness decreases with respect to the mismatch angle and shows the lowest value around 45 of the mismatch angle, or around 90 of the fiber angle of the 8th ply. Then, the toughness increases as the mismatch angle goes up. Similar to the previous case, for GII/G=20%, there is no remarkable dependence of fracture toughness on mismatch angle. For the range greater than 90 of fiber angle, the fracture toughness linearly increases as the mismatch angle goes up as shown in Fig. 15. The dependence of toughness on mismatch angle is the

greatest for the mode ratio of GII/G=80%, but, for lower mode-II ratios, the toughness dependency becomes smaller. As a result, the fracture energy of opening mode-I hardly affects the total delamination fracture toughness. The main reason for the influence of mismatch angle can be found from the stiffness of the fibers embedded in the laminates. The fibers generally have higher stiffness than the polymeric matrix. In this research, the carbon fiber shows a 10 times larger modulus than that of the epoxy matrix. Most of the high-strength property of polymeric composites comes from the fibers embedded as a reinforcing agent. Therefore, the alignment of fibers plays an important role for composites’ mechanical properties. According to the tensile test results of a graphite/epoxy off-axis laminate [12], the modulus of composites initially decreases quite rapidly as the offaxis (or, mismatch) angle increases from 0 , and it stays nearly constant from 45 to 90 . This indicates that the fiber alignment influences the mechanical properties of composite laminates. The dependence of delamination fracture toughness may be relevant to the flexural modulus of the laminae, since fiber alignment influences the flexural modulus of laminates especially for higher mode-II fractions. For higher mode-II fraction, fracture energy is much more dependent on flexural, or bending, modulus of laminates than higher mode-I fraction. Current research also shows that fracture toughness of delamination clearly varies with the fiber orientation. That is, fracture toughness was inversely proportional to the mismatch angle of the embedded fiber.

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Fig. 14. Fracture toughness (Gc) as a function of mismatch angle (mis) for several mixed-mode ratios at 70 incline angle (inc=70 ).

Fig. 15. Fracture toughness (Gc) as a function of mismatch angle (mis) for several mixed-mode ratios at 90 incline angle (inc=90 ).

In Ref. [22], it was concluded that fracture toughness increased with ply angle. The result appears to contradict only to the results of this research, but, in Ref. [22], the number of plies was different for each specimen. The lay-up used in Ref. [22] were [0]6, [ 30]5, and [ 45]5. Thus, six plies, ten plies, and ten plies were used for [0]6,

[ 30]5, and [ 45]5, respectively. These three kinds of specimens were compared directly for accounting for the dependency of toughness on ply angle. Ply number and specimen thickness should be the same to directly compare the toughness values to one another, because fracture energy is related to the flexural (bending) mod-

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ulus of a specimen. Also, the loading forces at the crack initiation were about 180 N and 120 N for [ 30]5 and [ 45]5, respectively. Since the load is one of the biggest factors to fracture toughness derived from beam theory, the toughness value of the [ 30]5 laminate would be greater than that of the [ 45]5 laminate. Therefore, it is inappropriate that fracture toughness was proportional to the ply angle. The number of plies, or specimen thickness, is normally a stronger factor than the ply angle for measuring delamination fracture energies. Rather, as stressed in this research, fracture toughness decreases as the fiber angle increases. The result of this research can be justified because the physical conditions of specimen fabrication were uniform for every specimen and can be compared with each other directly. 3.2.4. Dependence on fiber angle of 8th ply As shown in previous Figs. 11–15, the fiber direction of the 8th ply is very relevant to the value of fracture toughness. Fig. 16 indicates the toughness values of all specimens according to the mixed-mode ratio, and the toughness is obviously dependent on the fiber direction of the 8th ply for lower mixed-mode ratios. As described above, however, the toughness is less dependent on the 8th ply for higher ratios. Generally, delamination fracture toughness for all mode ratios decreases until the fiber angle of the 8th ply approaches to 90–100 . After 90–100 of the 8th ply, the toughness for the two higher mode-II ratios (GII/G=80 and 50%) increases, but the toughness for the three lower mode-II ratios (GII/G=50, 35, and 20%) does not increases but decreases continuously. It is because the effect of fiber direction becomes lower as the portion of shear mode-II in the mixed-mode ratio decreases. In the case of GII/G=35 and 20%, the fiber direction hardly affects the delamination fracture toughness. The mismatched 9th ply might influence the global bending energy which is related to the load of crack initiation and thus affects failure energy of delamination. The effect of the 9th ply on fracture energy was, however, revealed to be infinitesimal compared to the effect of mismatch angle of the delaminated interface between the 7th and 8th plies. As described in Section 3.1.1, even though the delamination is initiated at the interface between the 8th and 9th plies, the crack jumps up to near the interface between the 7th and 8th plies. This phenomenon decreases the effect of the fiber orientation of the 9th ply on the total fracture toughness. 3.2.5. Dependence on triangular area During the delamination process, triangular shapes form at the delaminated interface. These shapes were observed very clearly with the naked eyes. Fig. 17 shows the variation of triangular area corresponding to the fiber angle of the 8th ply. There is no data below 25

709

because no triangle forms at those lower angles. The area becomes zero at 90 and then increases as the fiber angle of the 8th ply goes up. The triangular area is related to the delamination fracture toughness and could be used as an indicator of the trend of fracture toughness. The shape of the triangle varies with respect to the fiber direction of the 8th ply, as shown in Fig. 17. The triangular area decreases as the fiber angle increases, and the area becomes zero at 90 of 8th ply fiber. After 90 , the triangular area is proportional to the fiber direction of the 8th ply. Fig. 18 shows the trend of dependency of delamination fracture toughness on the triangular area. For all mixed-mode ratios, or GII/G, the toughness linearly increases with respect to the triangular area. Similar to the angular dependence of toughness, as the mode-II ratio increases, the dependence of toughness on the triangular area becomes higher. When the initial delamination crack is created at the interface between the 8th and 9th plies, the crack propagates into the 8th ply rather than the interface between the 8th and 9th plies, and then the 8th ply is split into two parts in its thickness making a triangular shape in each piece according to the fiber direction of the 8th ply. Then, the crack grows near the interface between the 7th and 8th plies. Since critical strain energy release rate is affected by the initial crack, or triangular delamination, the formation of triangles is considered to be relevant to delamination fracture toughness. 3.2.6. Three-dimensional plots Figs. 19–21 are three-dimensional surface plots showing the dependence of delamination fracture toughness on mode-II ratio, incline angle, and mismatch angle. Fig. 19 indicates the toughness dependence on mode-II ratio (GII/G) and incline angle (inc) for zero mismatch angle (mis) of fibers at the interface between the 8th and 9th plies. The fracture toughness clearly decreases, as the mode-II ratio decreases. The toughness shows the highest value at the highest mode-II ratio (GII/G=80%) for zero incline angle and decreases as the incline angle increases. Consequently, as the mode-II ratio is highest and the incline angle is smallest, the fracture toughness becomes higher. Conversely, the lowest fracture toughness value is obtained for the lowest mode-II ratio and the highest incline angle. Fig. 20 shows the fracture toughness as functions of mode-II ratio and mismatch angle at zero incline angle. Similar to the relationship of mode ratio–incline angle– fracture toughness, the fracture toughness decreases as the mismatch angle increases. Even though there is some unstable data for GII/G=80% and mis=45 , the fracture dependence appears to decrease for higher mismatch angles. Like incline angle, however, the variation of mismatch angle does not influence the delamination fracture toughness for lower mode-II ratios. The

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Fig. 16. Fracture toughness (Gc) as a function of the fiber angle of 8th ply for several mixed-mode ratios.

Fig. 17. Triangle area versus the fiber angle of 8th ply.

dependence of delamination fracture toughness on both incline angle and mismatch angle at GII/G=80% is shown in Fig. 21. Generally, fracture toughness decreases as incline and mismatch angles increase.

4. Conclusion The delamination fracture toughness of AS4-3506 carbon/epoxy laminates was measured using the MixedMode Bending test, an improved test method. The delamination crack was expected to grow along the

midplane of a beam specimen but propagated to the interface one layer above the midplane making triangular-shaped area. The triangle size was inversely proportional to the mismatch angle but was proportional to the fracture toughness of delamination. Delamination fracture toughness varied clearly with mismatch angles of fiber direction in two adjacent plies, and also depended on mixed-mode ratios. The mismatch angle had much influence on fracture toughness for the relatively higher mode-II ratios such as GII/G =65 and 80%: i.e., fracture toughness decreased as the mismatch angle increased. For lower mode-II ratios, however, such as

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711

Fig. 18. Delamination fracture toughness (Gc) as a function of the triangle area formed during the delamination procedure.

Fig. 19. Fracture toughness (Gc) as functions of mixed-mode ratio (GII/G) and incline angle (inc) at zero mismatch angle (mis=0 ).

GII/G =20, 35, and 50%, the effect of mismatch angle reduced. Generally, delamination fracture toughness decreased as mismatch angle increased for all mixedmode ratios. The reason of fracture energy dependence on mismatch angle is considered to be stiffness or modulus of fiber and thus laminate. More failure energy dissipates when delamination occurs at the unidirectional laminates because of higher flexural, or bending, modulus of laminates. Bending modulus plays more important role in mode-II rather than mode-I. As the mismatch angle increases, therefore, the bending modulus of laminates decreases and thus fracture energy decreases. Fiber

bridging and interlaminar friction effect may also influence fracture toughness dependence on mismatch angle. The results of this work can be effectively used for analyzing the delamination fracture phenomena like ballistic impact damage and for designing the alignment of fiber in materials development. Analytical equations [Eqs. (4)–(8)] for calculating the strain energy release rate are only applicable to unidirectional laminates, and mismatch layers were not considered in the equations. Therefore, there must be limitation in applying those equations to interpret the fracture energy of the specimen with mismatch layers. However, the trends of fracture energy were revealed

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Fig. 20. Fracture toughness (Gc) as functions of mixed-mode ratio (GII/G) and mismatch angle (mis) at zero incline angle (inc=0 ).

Fig. 21. Fracture toughness (Gc) as functions of incline angle and mismatch angle at mixed-mode ratio of GII/G=80%.

indirectly and the results are valuable in understanding the effects of mismatch layer on interlaminar fracture energy. For the more accurate characterization of fracture energy, it is strongly suggested to analyze the more exact fracture mechanics of the specimen geometry with mismatch layers.

grant (2302DW01) from the Air Force Office of Scientific Research (AFOSR), Dr. Dan Segalman, Technical Manager, for financial support of this work. The authors also thank Dr. Uday Vaidya from the Department of Materials and Mechanical Engineering at the University of Alabama, Birmingham, and his graduate student, Mr. Shane Bartus, who fabricated and installed the MMB apparatus.

Acknowledgements This work was performed while the author held a National Research Council Research Associateship Award at the Air Force Research Lab, Wright-Patterson Air Force Base, OH, USA. The authors gratefully acknowledge the National Research Council and the

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