matrix interface using the single-fiber fragmentation test

Composites: Part A 40 (2009) 679–686

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Composites: Part A journal homepage: www.elsevier.com/locate/compositesa

Review

A method to measure fracture toughness of the fiber/matrix interface using the single-fiber fragmentation test F.A. Ramirez a, L.A. Carlsson a, B.A. Acha b,* a b

Department of Mechanical Engineering, Florida Atlantic University, 777 Glades Road, Boca Raton, FL 33431, USA Department of Ocean Engineering, Florida Atlantic University, 101 N. Beach Road, Dania Beach, FL 33004, USA

a r t i c l e

i n f o

Article history: Received 25 June 2008 Received in revised form 17 March 2009 Accepted 15 April 2009

Keywords: B. Adhesion B. Debonding B. Fracture toughness

a b s t r a c t A fracture mechanics model was developed for determining the fracture toughness of the fiber/matrix (F/ M) interface based on a modified test procedure for the single fiber fragmentation test (SFFT). After loading the specimen until the first fiber fracture and instantaneous debonding events occur, the specimen is unloaded and loaded again until the debond propagates. The critical load for debond propagation is measured and is used with a fracture mechanics analysis to determine the interface fracture toughness. The analysis considers also friction between the fiber and matrix in the debonded region. To obtain the necessary data for calculation of residual radial stress at the F/M interface due to matrix cure shrinkage, simultaneous measurements of dynamic modulus and cure shrinkage were conducted on the matrix (vinylester) during cure. Tests employing E-glass/vinylester SFFT specimens provided fracture toughness values of Gcd = 62 J/m2 (frictionless) and 48 J/m2 (friction). Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction The structural integrity and lifetime of polymer composites are critically dependent on the stability of the fiber/matrix (F/M) interface region [1,2]. Therefore, it is extremely important to characterize the F/M interface to understand the overall performance of polymer matrix composites. Among the test methods to evaluate the bonding between fiber and matrix, the single fiber fragmentation test (SFFT) method is the simplest in terms of the experimental setup, and therefore the most commonly used [2]. The SFFT consists of a single fiber embedded in a resin matrix molded into a dog-bone specimen. The specimen is loaded in tension and, if the specimen is properly designed, the fiber will break into more and more fragments until fiber break saturation occurs. The final fragmentation length is called critical length, ‘lc’ [3]. Several models such as the Kelly–Tyson shear lag analysis [4] use the critical length to determine the F/M interface shear strength [3,4]. The Kelly–Tyson model, however, assumes a constant shear stress along the F/M interface and does not account for other failure processes that are involved in the fragmentation test such as F/M interface shear failure (debonding). It must be pointed out that experimental observations reveal a finite amount of F/M debonding (or instantaneous debonding) [5,6] around each fiber break and therefore, a fracture toughness quantity, such as the energy release rate for propagation of an interface debond, may be better fitted to quantify the F/M interface adhesion than the average shear strength. * Corresponding author. Tel.: +1 954 924 7302; fax: +1 954 924 7270. E-mail addresses: [email protected], [email protected] (B.A. Acha). 1359-835X/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesa.2009.04.011

Currently there are several fracture models to evaluate SFFT test results [7–10]. These fracture models integrate the effects of energy released by a fiber fracture with debond propagation. However, determination of the interface fracture toughness from the fiber fracture and subsequent instantaneous F/M debonding events may not be accurate since the energy balance is not straightforward. The main objective of this work is to propose a modified SFFT procedure and develop an associated fracture mechanics model capable of separating the event of fiber break from the event of F/M debond growth to be able of determining the fracture toughness of the F/M interface without the need to consider the energy released upon fiber fracture. In order to focus the attention to the debond propagation event, the SFFT specimen is loaded under a microscope until the first fiber break and associated instantaneous F/M debonding occur. Following these events, the specimen is unloaded and then loaded again until the existing debond is observed to grow. Based on the critical load required to propagate the existing F/M debond, the energy release rate for debond propagation is determined using a fracture mechanics model developed herein. Friction over the debond zone opposing the sliding displacements between the fiber and the matrix is also considered. When friction is considered it becomes necessary to estimate the interfacial residual stresses due to matrix cure shrinkage. Dynamic-mechanical testing was conducted on the resin from the time of mixing until complete cure to allow determination of the gel point and the residual stress at the F/M interface. Experiments are performed on SFFT specimens with E-glass fiber in a vinylester matrix.

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2. Fracture mechanics model The fundamental fracture events occurring in the fragmentation test are illustrated in Fig. 1. In the initial state (0), there is no damage, and the specimen compliance is C0. The energy released by the first fiber fracture, G, is determined from the difference in specimen compliance after (1) and before the first fiber break (i.e. C1C0). The second fracture state (2) considers debond growth following the first fiber fracture. An expression for G for debonding following a fiber break can be derived in terms of compliance difference between C2 and C1. Finally, in the third state (3), the specimen is supposed to be unloaded and then loaded again until the existing debond starts to grow. The energy release rate, Gd, corresponding to growth of the debond, after unloading and loading again the specimen, is determined from the compliance change (i.e. C3C2). Notice that a condition for this analysis is that the debond grows before further fiber breaks occur in the SFFT specimen. We will first consider the event of the first fiber failure (i.e. 0–1 event in Fig. 1). The break is assumed to occur in the center of the specimen enabling consideration of only the half symmetry section shown in Fig. 2. This assumption, however, does not limit the viability of the analysis which should be valid for an arbitrary fracture site away from the gripping regions. The axial fiber stress distribution is illustrated in Fig. 2. After the fiber breaks, it carries no load at the break point. The regions of the fiber near the break are subjected to interfacial shear stress which will reload the fiber fragments. The zone over which the fiber stress builds up to its farfield value has been termed the ineffective zone. The length of this zone, Li, is defined as the distance required to recover the stress in

the fiber to 95% of the far-field stress [11] (the total ineffective length is 2Li). The zone where the fiber is fully loaded away from the break point is considered as the effective zone. The fiber tensile stress distribution in the ineffective length region is given by an axisymetric model developed by Whitney and Drzal [11],

#    4:75Lx x i A1 e0 rf ðxÞ ¼ 1  4:75 þ 1 e Li "

ð1Þ

where x is the distance from the fiber break point along the fiber axis (0 6 x 6 Li), see Fig. 2, and A1 is a material property constant defined as:

A1 ¼ E1f þ

4K f Gm v1f ðm1f  vm Þ ðK f þ Gm Þ

ð2Þ

where Kf is the plane strain bulk modulus of the fiber,

Kf ¼

E2f 2ð2  E2f =2G2f  2v2f E2f =E1f Þ

ð3Þ

E1f, E2f, m1f, and m2f are the axial and transverse moduli and Poisson’s ratios of the cylindrically orthotropic fiber, G2f is the transverse shear modulus of the fiber, and Gm is the shear modulus of the matrix. The far-field strain, eo, of the specimen is defined as:

eo ¼

P Ac Ec

ð4Þ

where P is the axial load, and Ac and Ec are the cross section area and effective Young’s modulus of the SFFT specimen,

Ac ¼ b

2

ð5aÞ

Ec ¼ V f E1f þ V m Em

ð5bÞ

where b is the side length of the square cross section of the SFFT specimen, Em is the Young’s modulus of the matrix, and Vf and Vm are the volume fraction of the fiber and matrix calculated as:

Vf ¼

Af Ac

and V m ¼

Am Ac

ð6a; bÞ

where Af and Am are the cross section areas of fiber and matrix in the SFFT specimen,

Af ¼

p 4

2

df

and Am ¼ Ac  Af

ð7a; bÞ

The stress analysis developed by Whitney and Drzal provides the following expression for the ineffective length [11]:

Fig. 1. Schematic of fracture events and associated compliances of SFFT specimen.

df Li ¼ 2:375 2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E1f  4v1f Gm

ð8Þ

A fracture mechanics analysis of the first fiber fracture event in terms of the specimen compliance (displacement/load) requires the displacements of the SFFT specimen before and after the first fiber break. Before a fiber break, the total specimen displacement, do, is given by Hooke’s law,

do ¼

2PL Ac Ec

ð9Þ

After the first fiber break (see Fig. 2), the displacement, d1, is given by the sum of the displacements of the effective (e) and ineffective (i) regions,

d1 ¼ d1;e þ d1;i

ð10Þ

The displacement of the effective region is given by: Fig. 2. Schematic of the fiber stress distribution in a SFFT specimen after a fiber break for the case of no debonding.

d1;e ¼ 2

PðL  Li Þ Ac E c

ð11Þ

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The displacement of the ineffective region is given by:

Z

d1;i ¼ 2

Li

ei ðxÞdx

ð12Þ

0

The strain variation is here based on the continuous phase (matrix) (the fiber carries zero load at the break (x = 0)).

ei ðxÞ ¼

rm;i ðxÞ

ð13Þ

Em

where the matrix stress, rm,i, is obtained from equilibrium of force,

ei ðxÞ ¼

P  rf ðxÞAf Am E m

ð14Þ

Substitution of Eqs. (14) and (7a) into (12) yields: 2

d1;i

0:1478pA1 df PLi ¼2 1 Am E m Ac Ec

! ð15Þ

The total displacement of the specimen after the fiber breaks becomes: 2

0:1478pA1 df PðL  Li Þ PLi d1 ¼ 2 þ2 1 Ac E c Am E m Ac Ec

! ð16Þ

The energy release rate, G, available for fracturing a fiber in the SFFT specimen is given by:

G¼

4

2 df

ð18Þ

and DC is the difference in the SFFT specimen compliance after and before the first fiber break, i.e. DC = C1C0, obtained from Eqs. (9) and (16) as,

C1  Co ¼

2

ð17Þ

where DA is the new surface area considered as the cross section area of the fiber,

DA ¼

Lid and Lib are the debonded and bonded lengths of the ineffective zone (the total debonded and bonded lengths are 2Lid and 2Lib, respectively). In the absence of friction shear stress transfer in the ineffective region occurs only along the bonded length and thus the length Lib shown in Fig. 3 equals the length Li shown in Fig. 2. The displacement of the bonded portion of the ineffective region is obtained from Eq. (15)

d2;ib ¼ d1;i

P 2 DC 2 DA

p

Fig. 3. Schematic of the fiber stress distribution in a SFFT specimen after a fiber break with F/M debonding (frictionless analysis).

2

0:1478pA1 df ðL  Li Þ Li ¼2 þ2 1 Ac Ec Am E m Ac Ec

! 

2L Ac E c

d2;id ¼ 2

4P2 Li 2 df

p

2

0:1478pA1 df 1 1   Am Em Ac Ec Ac Am Ec Em

ð25Þ

Substitution of Eqs. (25) and (6b) into (24) yields,

PLid Am Em

ð26Þ

2

C2 ¼

ð20Þ

In this event, a modified SFFT procedure, consisting of unloading the specimen after the first fiber break and instantaneous debonding event, is proposed. Upon loading the specimen again, the existing debond length is assumed to extend (from state 2 to state 3 in Fig. 1) at a certain critical load. The displacement, d2, of the specimen with a fiber break and debond (state 2 in Fig. 1), is obtained by adding the displacements of the effective region, d2,e, bonded portion of the ineffective region, d2,ib, and the debonded portion of the ineffective region, d2,id (debonded region), see Fig. 3.

ð21Þ

where the displacement of the effective region is given by:

PðL  Lib  Lid Þ Ac Ec

ð24Þ

The compliance of the SFFT specimen at state 2 (Fig. 1) becomes:

2.1. Debonding: frictionless analysis

d2;e ¼ 2

PLid Ac Ed

Ed ¼ V m Em

ð19Þ

!

d2 ¼ d2;e þ d2;ib þ d2;id

ð23Þ

where the effective modulus of the debond region is:

Hence, the energy release rate when the fiber breaks in the SFFT specimen becomes:

G¼

!

and the displacement of the debonded region is:

d2;id ¼ 2

d1 do  P P

0:1478pA1 df PLi ¼2 1 Am E m Ac E c

0:1478pA1 df d2 ðL  Li  Lid Þ Li þ2 1 ¼2 Ac Ec P Am Em Ac E c

! þ2

Lid Am Em ð27Þ

In state 3 (Fig. 1), the SFFT specimen is loaded (after unloading at state 2) until the debond grows. If the F/M debond length increases by an amount a, the length of the effective region will decrease by the same amount a (at each side of the fiber break point). Therefore, the SFFT specimen compliance after debond propagation, C3, is given by: 2

0:1478pA1 df ðL  Li  Lid  aÞ Li C3 ¼ 2 þ2 1 Ac E c Am E m Ac Ec

! þ2

Lid þ a Am Em ð28Þ

The compliance change is:

  1 1 DC ¼ C 3  C 2 ¼ 2a  A m E m Ac E c

ð29Þ

and the increment in debond area is given by:

DA ¼ pdf ð2aÞ

ð30Þ

The energy release rate available for debond propagation is then,

ð22Þ

Gd ¼

P2 2pdf



1 1  Am Em Ac Ec

 ð31Þ

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From the facture criterion, the existing debond length will start propagating at a load P = Pc when Gd reaches the interface fracture toughness, Gcd, given by:

Gcd ¼

P2c 2pdf



1 1  Am Em Ac Ec

 ð32Þ

Note that Gcd does not depend on the debond length which would contribute to stable debond growth in the actual test. 2.2. Debonding: friction analysis The fracture toughness, Gcd, determined using the frictionless analysis (Eq. (32)) is an apparent toughness because the actual test may incorporate a contribution from interfacial friction. When a debond propagates, it generates a relative displacement (sliding) between the fiber and the matrix as they tend to retract from their previous stressed states. Friction between the surfaces of the fiber and matrix opposes this sliding and is therefore an energy absorbing mechanism in addition to the energy dissipation due to the formation of a new surface area (debonding). Therefore, the purpose of the friction analysis is to quantify the friction component. The crack growth criterion may be modified by considering that the energy available for debond propagation is reduced due to the presence of friction [12],

GðlÞ ¼ Gðl ¼ 0Þ 

dW f dA

ð33Þ

Fig. 4. Schematic of fiber/matrix sliding caused by debond propagation.

Z

uf ¼ 

a

ef ðxÞdx

ð39Þ

0

where integral extends over the region of the debond growth (0 6 x 6 a) and ef is the strain in the fiber (rf (x)/E1f), where the fiber stress is given by Eq. (1). Integration yields,

where l is the coefficient of friction, G(l) is the energy release rate available for debond propagation, G(l = 0) is the apparent energy release rate given by Eq. (32), and Wf is the frictional work during debond growth. The increments in area, dA, and frictional work are given by:

    " ! !# 4:75a 4:75a PA1 Li Li 1 þa e þ1 uf ¼  0:421Li e Ac Ec Af

dA ¼ pdf da

ð34aÞ

um ¼

W f ¼ F r Du

ð34bÞ

ð40Þ

The corresponding matrix displacement is given by:

Z

a

em ðxÞdx

ð41Þ

0

where Du is the relative sliding displacement between fiber and matrix over the new debond area (dA) and Fr is the frictional force opposing the sliding,

where the strain in the matrix is, em = rm/Em with the matrix stress, rm, given by Eq. (14). Integration yields, h i  4:75a   4:75a  P   um ¼ 0:421A1 Li Af e Li  1  aA1 Af e Li þ 1 þ aAc Ec Ac Am Ec Em ð42Þ

F r ¼ lrR pdf Lid

The total sliding is then,

ð35Þ

rR is the compressive radial pressure at the F/M interface caused by residual stress introduced during cure of the resin and differential Poisson’s contractions of the fiber and matrix upon axial loading. Assuming that the fiber is embedded in an infinite matrix, the radial pressure due to Poisson contraction of fiber and matrix, rR,P is given by: [13],

rR;P ¼ 

eo ðmm  m1f Þ ð1m2f Þ E2f

þ ð1þEmmm Þ 

2m21f

ð36Þ

e ð1 þ v1f Þ ð1m2f Þ E2f

þ ð1þEmmm Þ 

2m21f

ð37Þ

E1f

where e* is the magnitude of the matrix cure shrinkage after the gel point expressed as a linear strain. The magnitude of the total compressive radial pressure, jrR;P þ rR;S j, at the interface is then,

rR ¼

eo ðmm  m1f Þ þ e ð1 þ v1f Þ ð1m2f Þ E2f

þ ð1þEmmm Þ 

2m2

ð38Þ

1f

E1f

The F/M sliding, Du, is given by the fiber and the matrix displacements caused by the debond propagation. Notice that the fiber and the matrix retract in opposite direction when debond propagates as indicated in Fig. 4. The fiber displacement is given by:

ð43Þ

Substitution of Eqs. (43) and (35) into (34b) yields the frictional work, Wf,



   4:75a  Af h A1 P 1   0:421Li e Li  1 Ac Ec E1f Am Em

 4:75a i Pa  L þa e 1 þ 1 þ Am Em

W f ¼ lrR pdf Lid

E1f

The radial pressure, rR,S, caused by matrix cure shrinkage is [13]:

rR;S ¼ 

   4:75a  Af h A1 P 1  0:421Li e Li  1  Ac Ec E1f Am Em  4:75a i Pa  L þa e 1 þ 1 þ Am E m

Du ¼ um  uf ¼

ð44Þ

Differentiation with respect to dA = pdfda yields the rate of energy dissipated by the friction.

   

dW f Af A1 P 1 4:75a P 4:75a 1  e L1 1 þ þ  ¼ lrR Lid Ac Ec E1f Am Em Li Am E m dA

ð45Þ At the onset of debond propagation, the debond increment approaches zero (a ? 0), therefore:

    dW f Af A1 Pc 1 4:75a 4:75a  ¼ lim lrR Lid 1  e L1 1 þ a!0 Li dA Ac Ec E1f Am Em

Pc lrR Pc Lid ¼ ð46Þ þ Am E m Am Em

F.A. Ramirez et al. / Composites: Part A 40 (2009) 679–686

683

Substitution of Eqs. (46) and (32) into (33) and P = Pc yields to the following expression for the interface fracture toughness in the presence of friction,

Gcd ðlÞ ¼

P2c 2pdf



1 1  Am Em Ac Ec

 

lrR Pc Lid Am E m

ð47Þ

3. Experimental 3.1. Materials A rubber modified vinylester system, Ashland Derakane vinylester 8084 (VE D8084), was used as the matrix. It consists of the following components: 1.5 parts per hundred resin weight (phr) of methylethylketone peroxide (MEKP), 0.18 phr of cobalt naphthenate-10% (CoNap 10%), and 0.05 phr of dimethylaniline (DMA). The cure schedule consisted of 24 h cure at room temperature followed by a two hour post cure at 99 °C. The fiber was an E-glass fiber from 3TEX with silane sizing. The properties of fiber and matrix are listed in Table 1 [14–16]. 3.2. Test procedures In order to determine the contribution to the fracture work due to friction, it is necessary to determine the residual stress at the F/ M interface after cure. This was approached by dynamic-mechanical analysis on a resin during cure and simultaneous direct measurement of the cure shrinkage. The second part of the test program was the actual single fiber fragmentation test procedure, conducted according to the modified procedure outlined above. 3.2.1. Characterization of the development of residual stresses by matrix shrinkage After mixing the thermoset components and the resin, the polymer mix can flow. It is therefore assumed that before gelation the material will shrink without generating stresses. Hence, only after the resin gels, the shrinkage is considered important for determination of residual stresses at the F/M interface [17]. To detect gelation of the neat vinylester resin, the cure process was monitored using dynamic-mechanical analysis (DMA). A TA-Q800 dynamicmechanical analyzer configured as illustrated in Fig. 5 was used. A small cylindrical container of 14 mm internal diameter was attached to the vibrating part of the DMA fixture. A vertical steel rod of 4.7 mm diameter was clamped to the fixed part of the fixture. In this configuration, the container is vibrating while the steel rod, partially dipped in the resin, remains stationary. A small amount of the vinylester resin was poured into the container immediately after mixing. This point defines zero time. The test was performed at room temperature using small constant load amplitude. This configuration permits monitoring of the viscoelastic properties, i.e. the in-phase and out-of-phase stiffnesses (S0 and S00 ), and loss tangent (tan d), from the time of mixing (phase angle

Fig. 5. Schematic of dynamic-mechanical analysis test configuration for cure monitoring of the resin.

d  90°) to the final cured solid state (d  0°). The gel point of the resin was here defined as the cross-over point; S0 = S00 (i.e. d = 45°). To determine the amount of shrinkage in situ, resin was poured into a test tube (with the same internal diameter as the container used for DMA testing) immediately after mixing. A Bausch–Lomb long focal distance horizontal microscope with a high precision measuring scale was used to measure the amount of shrinkage during cure. By the use of transmitted light, the bottom of the meniscus could easily be seen and used as a reference. 3.2.2. Single fiber fragmentation test Experiments were performed on the single fiber fragmentation test specimen shown in Fig. 6 consisting of an E-glass fiber embedded in a vinylester Derakane 8084 (VE D8084) matrix. 10 replicate SFFT specimens were used. The specimen was loaded in tension using a small tensile stage (from Ernest F. Fullam, Inc.) equipped with a 1 kN capacity load cell. The initiation of fiber fracture and F/M debonding, and propagation of the debond were examined using the photoelastic patterns observed in optical transmission microscopy (Olympus BX41 with a QICAM-FAST 1394 camera). The region around a fiber break observed between crossed polarizers exhibits a colored pattern called birefringence which is a phenomenon is caused by shear stresses in the matrix [19]. Photoelastic patterns permit visualization of the stress field around a fiber break. Fig. 7 is a schematic illustration of birefringence patterns around a fiber break for the cases of no F/M debonding (top) and debonding (bottom) [5]. It can be seen that when debonding accompanies a fiber break, the intensity of the birefringence over the debond zone near the fiber break is weak. The end of the debonded region can be seen, with careful observation, as a photoelastic effect close to the fiber/matrix interface. The debond tip is located between the maximum width of the photoelastic pattern and the fiber break. The photoelastic effect indicating the debond tip does not always show up well in the pho-

Table 1 Mechanical properties of E-glass fiber and vinylester matrix Fiber Axial modulus, E1f (GPa) Transverse modulus, E2f (GPa) Axial Poisson’s ratio, v1f Fiber fracture toughness, Uf (J/m2) Diameter, df (lm)

73 73 0.2 10 14

Matrix Young’s modulus, Em (GPa) Poisson’s ratio, mm Strain to failure, e (%)

2.7 0.34 8–10

Fig. 6. Single fiber fragmentation test (SFFT) specimen [18].

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F.A. Ramirez et al. / Composites: Part A 40 (2009) 679–686

Fig.

9. Non-dimensional

(S0 =jS j and S00 =jS j where jS j ¼

components of the dynamic stiffness qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðS0 Þ2 þ ðS00 Þ2 ), and shrinkage evolution curves

during cure of vinylester resin.

Fig. 7. Schematic photoelastic patterns around a fiber break. Top: without debonding and bottom: with debonding (after Kim et al. [5]).

tographs but is detectable by visual inspection during testing. Fig. 8 shows an example of an actual photoelastic pattern indicating the debond tip. The SFFT specimens were loaded until the first fiber break was observed. The load required to break the fiber was recorded. The instantaneous debond length created by the energy released in this event was measured, using a computer software (Q-capture Pro)). The gap between the two fiber fragments was subtracted from the total length measured. The instantaneous debond length was used as a reference for the later study of debond propagation. Then, the SFFT specimen was unloaded, and loaded again until debond growth was observed. The load required to extend the debond, Pc, was recorded. Additionally, in order to verify the experimental results, the test was repeated using a small hand-operated tensile stage incorporating a dial gage providing the applied strain. In this case, the SFFT specimen was gradually loaded in incremental load (strain) steps. 4. Results and discussion 4.1. Cure monitoring The evolution of dynamic stiffness (S* = S0 + S00 ) and linear shrinkage (eS) of the vinylester resin as it cures at room temperature is illustrated in Fig. 9. The gel point, estimated by the cross over point of S0 and S00 , was reached approximately 48 min after mixing the vinylester components. Fig. 9 shows that a dominating

Fig. 8. Photoelastic stress pattern around a fiber break showing F/M debonding.

fraction of the resin shrinkage occurred before the gel point. Only about 1.1% (out of the total 2.7%) of the shrinkage occurred after gelation. Therefore, it is assumed that the linear shrinkage contributing to the formation of residual stresses at the F/M interface is e* = 1.1%. Calculation of the residual stress, rR,S, at the interface due to the resin shrinkage (Eq. (37)) yields a magnitude of rR,S = 26 MPa which is quite high considering that the tensile and compressive strengths of the resin are 58 [14] and 95 MPa [20], respectively. Rosso et al. [21] calculated the residual stress in a carbon/vinylester composite with a fiber volume fraction of 50% using finite element analysis. For a liner shrinkage, e* = 0.42%, they obtained rR,S  15 MPa which is in close proportion to the stress obtained here (based on e* = 1.1%). 4.2. Single fiber fragmentation test Ten replicate E-glass/vinylester SFFT specimens were tested. The specimens were loaded in tension until the first fiber break occurred. Fig. 10 shows a typical birefringence pattern at the first fiber break which occurred at 2.1% strain (as calculated from the applied load, P, using Eq. (4)). After determining the fiber breaking load and the length of the debond region, 2Lid, the SFFT specimen was unloaded. Fig. 11 illustrates the region near the fiber break point after the external load was removed. Even though the birefringence patterns partially disappeared after unloading the SFFT specimen, there are indications that the fiber/matrix interface was damaged by the energy released at fiber fracture. From Figs. 10 and 11 it can be seen that the gap between the two fiber fragments closes up as the specimen is unloaded indicating minor extent of matrix yielding.

Fig. 10. Photoelastic pattern of the first fiber break in a SFFT specimen.

F.A. Ramirez et al. / Composites: Part A 40 (2009) 679–686

685

Fig. 11. Photoelastic pattern near a fiber break point after unloading.

The SFFT specimen was loaded again and the F/M debond length was measured at different applied strains. Fig. 12 shows examples of photoelastic patterns near the fiber break at increasing levels of the externally applied strain. It is evident that the existing debond starts to propagate at certain critical strain (load). More fiber breaks started to occur, but at loads above the critical load for debond propagation making the fracture analysis suitable for this glass/vinylester system. 4.2.1. Determination of debond fracture toughness The load, P, at first break of the E-glass fiber was determined to be 237 ± 9 N (Ac = 4.0 ± 0.4 mm2). The energy release rate, G, corresponding to the fiber failure event (Eq. (20)) is 2.04 ± 0.44 kJ/m2. Nairn [7] reported that the corresponding energy release rate during fragmentation testing of T50 carbon fibers in an epoxy matrix is around 4.6 kJ/m2. A fraction of this released energy is consumed by actually fracturing the fiber (the fracture toughness of a glass fiber is only around 10 J/m2 [16]). Most of the energy released is consumed by initiation of fiber fracture, F/M debonding, and other dissipative processes [7]. Fig. 13 depicts the total debond length, 2Lid, vs. applied strain for specimens that were previously unloaded after the first fiber fracture. From Figs. 12 and 13, it can be deduced that, for this composite system, debond growth starts to propagate at an external

Fig. 13. Debond length as a function of applied strain.

load, Pc, approximately equal to the load required to break the fiber. The energy release rate required to grow an existing debond (or interface fracture toughness, Gcd) determined from Eq. (32) (assuming a frictionless debond zone) is 62 ± 13 J/m2. For the case where friction is considered, the magnitude of the total radial pressure determined from Eq. (38) is 32 MPa (rR,P = 6 MPa and rR,S = 26 MPa). Fracture toughness values, calculated from Eq. (47), for a range of coefficients of friction (l = 0–1) are summarized in Table 2. When friction is considered, the interface fracture toughness is less than that for a frictionless analysis since part of the energy released is consumed in frictional work. Several researchers [22,23] have estimated the coefficient of friction at the interface. Chua and Piggott [22] reported l  0.5– 0.8 for glass/epoxy. Using l = 0.8 in Eq. (47) gives an interface fracture toughness, Gcd = 47.9 ± 20 J/m2. This value is comparable to data for similar composite systems obtained by other authors using different tests. For instance, based on pull-out experiments, Piggott et al. [22] reported Gcd, of 50 J/m2 for glass/epoxy; Farooq et al. [24], using the Outwater-Murphy single-fiber test, reported Gcd = 39 J/m2 for the glass/vinylester used herein.

Fig. 12. Photoelastic pattern near a fiber break at increasing levels of applied strain.

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Table 2 Effect of friction on the interface fracture toughness: Pc = 237 N, 2Lid = 48 lm, rR = 32 MPa, and Ac = 4 mm2 (averages).

l

Gcd (J/m2)

0 0.3 0.5 0.8 1.0

61.5 56.4 53.0 47.9 44.5

5. Conclusions The main objective of this research was to develop a test methodology and a simple fracture mechanics model for determining the toughness of the fiber/matrix interface from single fiber fragmentation test results. A test procedure that focuses solely on the event of debond growth is proposed. A fracture mechanics model that includes the effects of friction over the debond zone was developed to quantify the debond fracture toughness. In order to determine the interface fracture toughness in the presence of friction, the residual radial stress at the fiber/matrix interface is needed. An experimental approach based on simultaneous dynamic-mechanical analysis and precision shrinkage measurements was developed for this purpose. The fracture toughness value, Gcd = 62 J/m2 (frictionless) and 48 J/m2 (friction), was in good agreement with results obtained for similar systems by other investigators and test methods. Acknowledgements The authors acknowledge support for this work at Florida Atlantic University from ONR Grant No. N00014-05-1-0341 managed by Dr. Yapa Rajapakse. Thanks are due to Shawn Pennell for the art work. References [1] Ray BC. Temperature effect during humid ageing carbon fibers reinforced epoxy composites. 2006;298:111–7. [2] Rao V, Herra-Franco PJ, Ozzelo AD, Drzal LT. A fragmentation test and the microcond pull-out interfacial shear strength. J Adhes 1991;34:65–77.

on interfaces of glass and J Colloid Interface Sci direct comparison of the test for determining the

[3] Drzal LT, Rich MJ, Camping JD, Park WJ. A single filament technique for determining interfacial shear strength and failure modes in composite materials. In: 35th Annual technical conference. New Orleans: Reinforced Plastics/Composites Institute, SPI; February 1980. Paper 20-C, 1. [4] Kelly A, Tyson WR. Tensile properties of fiber-reinforced metals: copper/tungsten and copper/molybdenum. J Mech Phys Solids 1965;13:329–50. [5] Kim BW, Nairn JA. Observation of fiber fracture and interfacial debonding phenomena using the fragmentation test in single fiber composites. J Compos Mater 2002;36:1825–58. [6] Ramirez FA, Carlsson LA, Acha BA. Evaluation of water degradation of vinylester and epoxy matrix composites by single fiber and composite tests. J Mater Sci 2008. doi:10.1007/s10853-008-2766-z. [7] Nairn JA, Liu YC. On the use of energy methods for interpretation of results of single-fiber fragmentation experiments. Compos Interface 1996;4:241–67. [8] Nairn JA. Exact and variational theorems for fracture mechanics of composites with residual stresses, traction-loaded cracks, and imperfect interfaces. Int J Fract 2000;105:243–71. [9] Liu HY, Mai YW, Ye L, Zhou LM. Stress transfer in the fiber fragmentation tests. Part III effect of matrix cracking and interface debonding. J Mater Sci 1997;32:633–41. [10] Wagner HD, Nairn JA, Detassis M. Toughness of interfaces from initial fiber– matrix debonding in a single fiber composite fragmentation. Appl Compos Mater 1995;2:107–17. [11] Whitney JM, Drzal LT. Axisymmetric stress distribution around an isolated fiber fragment. ASTM Special Technical Publication; 1987. p. 179–96. [12] Carlsson LA, Gillespie JW, Pipes RB. On the analysis and design of the end notched flexure (ENF) specimen for mode II testing. J Compos Mater 1986;20:594–604. [13] Nairn JA, Liu YC. Stress transfer into a fragmented, anisotropic fiber through an imperfect interface. Int J Solids Struct 1997;34:1255–81. [14] Ramirez FA. Master’s thesis, Dep Mech Eng, Florida Atlantic University, Florida; 2008. [15] Daniel IM, Ishai O. Engineering mechanics of composite materials. New York: Oxford University Press; 2006. [16] Harris B, Morley J, Phillips DC. Fracture mechanism in glass-reinforced plastics. J Mater Sci 1975;10:2050. [17] Schoch Jr KF, Panackal PA, Frank P. Real-time measurement of resin shrinkage during cure. Thermochim Acta 2004;417:115–8. [18] Feih S, Wonsyld K, Minzari D, Westmann P, Liholt H. Testing procedure for the single fiber fragmentation test, Riso National Laboratory, Riso-R-1483(EN), Roskilde, Denmark; 2004. [19] Timoshenko S. Theory of elasticity. New York: McGraw-Hill; 1934. [20] Farooq UM, Carlsson LA, Acha BA. Determination of fiber/matrix adhesion using the Outwater-Murphy single fiber specimen. In: Proceeding of 5th ESIS conference. Les diablerets, Switzerland; September 2008. [21] Rosso P, Fiedler B, Friedrich K, Schulte K. The influence of residual stresses via cure volume shrinkage on CF/VEUH-composites. J Mater Sci 2006;41:381–8. [22] Chua PS, Piggott MR. The glass fibre–polymer interface: III – Pressure and coefficient of friction. Compos Sci Technol 1985;22:185–96. [23] Baillie C. Ph.D. thesis, Dep Mater Sci and Eng, University of Surrey, Surrey; 1991. [24] Farooq MU, Carlsson LA, Acha BA. Design analysis of the Outwater-Murphy single fiber specimen. J Compos Mater, in press.