Adhesional pressure as a criterion for interfacial failure in fibrous microcomposites and its determination using a microbond test

COMPOSITES SCIENCE AND TECHNOLOGY Composites Science and Technology 66 (2006) 2610–2628 www.elsevier.com/locate/compscitech

Adhesional pressure as a criterion for interfacial failure in fibrous microcomposites and its determination using a microbond test Serge Zhandarov a

a,b

, Yulia Gorbatkina c, Edith Ma¨der

b,*

V.A. Belyi Metal-Polymer Research Institute of the National Academy of Sciences of Belarus, Kirov Str. 32a, 246050 Gomel, Republic of Belarus b Leibniz Institute of Polymer Research Dresden, Hohe Strasse 6, D-01069 Dresden, Germany c N.N. Semenov Institute of Chemical Physics, RAS, Kosygina Str. 4, 117977 Moscow, Russia Received 3 February 2006; received in revised form 24 March 2006; accepted 25 March 2006 Available online 6 May 2006

Abstract Typical interfacial strength parameters calculated from the microbond and pull-out tests are the ultimate (local) interfacial shear strength, sd, and the critical energy release rate, Gic. These two parameters are often considered as criteria for interfacial failure. Several years ago, we proposed a new interfacial parameter, adhesional pressure (rd), which is the normal (tensile) stress component at the fiber– matrix interface at the moment of the debonding onset near the crack tip. Adhesional pressure has the following advantages as an interfacial parameter and eventual failure criterion: (a) it corresponds to the actual mechanism of crack initiation in the microbond test (interfacial debonding starts in tensile mode), and (b) the rd value is directly proportional to the work of adhesion, WA, between the fiber and matrix surfaces. In this paper, we investigated, both theoretically and experimentally, the applicability of the adhesional pressure as an interfacial failure criterion. Several fiber–polymer pairs have been tested using a microbond technique, and local interfacial parameters (sd, Gic and rd) for these systems have been measured. Assuming rd = const, we analyzed the initiation of interfacial crack (the debond force, Fd, as a function of the embedded length, le) as well as crack propagation (variation of the crack length with the load applied to the fiber) and dependencies of the peak force, Fmax, and the apparent shear strength, sapp, on the embedded length. Residual thermal stresses and interfacial friction were included in our analysis. A comparison with two other interfacial failure criteria (sd and Gic) was made depending on specimen geometry (cylindrical specimens; spherical matrix droplets). It was found that all three parameters satisfactorily describe interfacial failure in a microbond test, yielding asymptotically (at large embedded lengths) very close predictions for Fmax and sapp as functions of the embedded length. They also predict equally well the interfacial crack growth under increasing external load for not very large crack lengths. For shorter embedded lengths, the three interfacial criteria yield substantially different results, e.g. finite sapp value as le ! 0 from the sd criterion, infinite sapp from Gic = const, and zero sapp from rd = const. In this range of le’s, the local interfacial strength, sd, appears to be the best failure criterion.  2006 Elsevier Ltd. All rights reserved. Keywords: B. Debonding; B. Fiber-matrix bond; B. Interfacial strength; C. Failure criterion; B. Adhesional pressure

1. Introduction Micromechanical techniques, such as the microbond and pull-out tests, are widely used to characterize the quality of interfacial bonding between matrices and reinforcing fibers in fiber-reinforced plastics [1–18]. These techniques

*

Corresponding author. Tel.: +49 351 4658 305; fax: +49 351 4658 362. E-mail address: [email protected] (E. Ma¨der).

0266-3538/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2006.03.023

have been shown to be relatively simple, reproducible and very sensitive to the state of the interface, e.g., to various fiber surface treatments [5,6,19–23]. They are based on the measurement of the force, Fmax, required to pull out a single fiber with one end embedded in a droplet of the matrix material. Traditionally, the interfacial bond strength has been characterized by the value of the apparent interfacial shear strength (apparent IFSS, sapp), which is the ratio of the measured peak force, Fmax, to the embedded area, S:

S. Zhandarov et al. / Composites Science and Technology 66 (2006) 2610–2628

sapp ¼

F max F max ¼ ; S 2prf le

ð1Þ

where rf is the fiber radius and le is the embedded strength. The sapp parameter is sufficient for qualitative estimation of the interfacial bond strength and for comparative study of different interfaces (obtained, for instance, using surface modification of the fiber and/or the matrix), when mechanical properties of both components and the test geometry remain unchanged. However, numerous experiments have shown that the apparent IFSS depends on the embedded length [4–6,14,18,20,23–27]. Moreover, the sapp values have been found to be highly affected by interfacial friction between the fiber and the matrix which develops during the debonding process [9,28–34]; thus, sapp appeared to represent a complex combined effect of interfacial bonding and friction rather than pure bond strength. Therefore, it is obvious that the apparent IFSS cannot be regarded as a true interfacial parameter or a criterion of interfacial failure. What do we mean when speaking about a ‘‘true interfacial parameter’’ or ‘‘failure criterion’’? In an ideal case, it is a parameter which characterizes the intensity of interfacial bonding only. It must depend on the energy of interfacial interactions per unit interfacial area (and, consequently, on the conditions of the interface formation, such as the temperature and time of thermal treatment in composite production), but not on the mechanical or geometrical features of test specimens. In particular, the true interfacial parameter should be independent of 1) elastic moduli and other mechanical properties of the fiber and the matrix; 2) embedded fiber length; 3) diameters of the fiber and the matrix droplet; 4) crack length during the debonding process; 5) testing technique itself. The last item requires further comments. It means that the ‘‘true interfacial parameter’’ should have the same value for a specific fiber–matrix pair, regardless of whether it was tested using a microbond, pull-out, push-out, fragmentation or any other possible technique. The ideal aim of any of these test methods is to deduce a value of this parameter from its own experimentally measured parameters; then, this value could be used to predict the results of other tests with the same fiber–matrix system. The term ‘‘failure criterion’’ may seem to have a more particular, test-specific meaning. It is well known that failure modes for the microbond, pull-out, push-out and fragmentation tests are different [8,12,14,19,35–37]; should the failure criteria for these tests differ as well? The ample literature available on this subject cannot give a definitive answer to the question. As a rule, investigators try to find a parameter (criterion) which is more or less constant (first of all, independent of specimen geometry and crack length) within the frames of a given particular test. However, some

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of the parameters proposed as failure criteria, such as the critical energy release rate (Gic), pretend to be universal, i.e., valid for different test configurations and failure modes. From a theoretical point of view, the most adequate interfacial parameter for bond strength characterization is the work of adhesion, WA. Indeed, it accounts for both non-local (long-range) molecular interactions at the interface, such as van der Waals forces, and local interfacial bonds (acid–base interactions) [38–40]. The molecular kinetics theory shows that the intensity of these interactions, in particular, the number of local bonds per unit interfacial area, depends on the conditions of the contact formation [41,42]. And, finally, since the amount of external work in the process of separation of two contacted solid surfaces under specimen loading spent to overcome all these local and non-local interactions is equal to the work of adhesion, the latter can also be regarded as a failure criterion. It only should be noted that for fiber–matrix pairs able to chemical interaction, such as epoxy–glass and some other systems [43,44], the energy of chemical bonding per unit area must be added to the work of adhesion. However, the measurement of the work of adhesion between two solid surfaces is a very complicated experimental task. Of course, several techniques for direct [45– 50] and indirect [51–53] WA measurement exist, but the former cannot be applied to thin fibers, and the latter cannot account for eventual chemical bonds. Therefore, experimentalists deduce various parameters of mechanical origin from destructive tests (such as pull-out and microbond tests) and then try these as criteria for interfacial failure. Among these parameters, the most popular are the local interfacial shear strength, sd [4–6,14,54] and the critical energy release rate, Gic [29,55,56]. Many papers have been published in support of each, referring both to different theoretical models of interfacial failure and experimental results obtained in micromechanical tests. In our papers [18,26,32,57–59], we consistently support the conclusion that these two parameters can equally well describe the behavior of real interfaces in micromechanical tests, but neither of them can be regarded as a ‘‘universal’’ or ‘‘true’’ interfacial parameter which is constant under any conditions. In Refs. [7,33,40], we proposed to consider the ‘‘adhesional pressure’’, or the ultimate normal (radial) component of interfacial stress in fiber–matrix systems, rd, as a new stress-based failure criterion and a tool for estimating the work of adhesion between a fiber and a matrix (including the contribution of chemical interactions). The introduction of this criterion can be justified by the fact that, as has been demonstrated by Scheer and Nairn [55], Piggott [60] and Marotzke [61], crack initiation in the microbond test occurs in Mode I (normal tension); therefore, it was quite natural to suppose that interfacial debonding starts when normal stress at some point at the interface reaches its ultimate value, namely, adhesional pressure (rd). Physical meaning of this parameter and the algorithm of its

S. Zhandarov et al. / Composites Science and Technology 66 (2006) 2610–2628

determination in a microbond test is presented in more detail in Section 2. Since the adhesional pressure is directly proportional to the work of adhesion, this concept could help to find common features in the theoretical and experimental approaches to the problem of failure criteria. The aim of this study was to assess the applicability of the adhesional pressure as a failure criterion from a viewpoint of micromechanical tests and compare it, in this aspect, with other interfacial parameters (sd and Gic). 2. Criteria of interfacial failure in a microbond test: theory 2.1. The difference between the microbond and pull-out tests A schematic view of the microbond and pull-out tests is shown in Fig. 1. The main difference between them is the test geometry: while in the pull-out test the matrix droplet is fixed at a solid substrate and kept from the bottom, in the microbond test it is placed directly on the fiber and is held by the knife edges from the top side of the fiber during the test. It has an important consequence concerning the stress state at the interface. In the pull-out test, interfacial stresses at the point where the fiber enters the matrix droplet is a combination of shear and tensile contributions; but in the case of the microbond test it is pure compression [55,61]. This enables one to calculate the normal stress at which interfacial debonding starts (equal to the adhesional pressure) from the value of the current force applied to the fiber top at this moment, and justifies the use of this value of the normal stress as a failure criterion. All experimental data reported in this study have been obtained using the microbond test. The models of interfacial failure can be classified according to the basic assumption concerning the condition for debonding; the parameter chosen as the failure criterion should be constant under wide variety of conditions. 2.2. Local interfacial shear strength (sd) There are many models which describe the distribution of interfacial shear stress along the embedded fiber length. Some of them are based on different versions of the shear-

a

b

F

F Fiber

Knife edge

Matrix droplet le

Substrate le

lag approach [4,9,14,32,54], others on variational mechanics [55] or finite element analysis [56,61,62]. The use of the local interfacial shear strength, sd, as a failure criterion, implies that interfacial debonding at a given point of the interface occurs when the local interfacial shear stress at this point reaches its critical value (local IFSS, sd). This results in two important features: for shear stresscontrolled debonding, the sd value for crack initiation for different specimens should not depend on the embedded length, le, and the sd value for crack propagation in a given specimen should not depend on the current crack length, a. The latter is illustrated in Fig. 2, where the interfacial shear stress distribution along the embedded fiber is shown. Curve 1 corresponds to the moment of crack initiation; the interfacial shear stress is sd at the crack tip (x = 0) and decreases along the embedded length. Curve 2 presents shear stress distribution for an advanced stage of debonding (crack length a = 50 lm). In the debonded region (x < a), the shear stress is determined by interfacial friction and is assumed to be constant. In the intact zone, shear stress decreases from the same sd value; in other words, the peak value is constant and does not depend on the crack length. In this paper, we shall use our own shearlag model described in detail in Refs. [18,62,63], which includes, in addition to external loading, also residual thermal stresses and friction in debonded regions. It relates the debond force, Fd, at which interfacial crack is initiated, to the value of the local IFSS, sd, as   2prf ble tanhðble Þ sd  sT tanh Fd ¼ ; ð2Þ b 2 where b is the shear-lag parameter as determined by Nayfeh [64], and sT is a stress term due to thermal shrinkage [18], which is typical to polymer composites formed at high temperatures. Analytical expressions for b and sT can be found in Appendix A1. Note that Fd is not the peak force measured in the test, but the force at which debonding starts; it corresponds to

70

Interfacial shear stress (MPa)

2612

τd

60 50

2 40 30

1

a

20

τf

10 0 0

25

50

75

100

125

150

175

200

Coordinate along the fiber (µm)

Fig. 1. Schematic view of the microbond (a) and pull-out (b) tests.

Fig. 2. Interfacial shear stress distribution along the embedded length: at the moment of crack initiation (1) and at an advanced stage of debonding when crack length is a = 50 lm (2).

S. Zhandarov et al. / Composites Science and Technology 66 (2006) 2610–2628

the ‘‘kink’’ in the force–displacement curve (Fig. 3). This kink is easily discernible (and the Fd value measurable) if the test installation is stiff enough and the free fiber length is short [12,65]. Our model also gives a direct expression for the current load, F, applied to the fiber end, versus the crack length value at this moment:  pd f F ¼fs ¼ sd tanh½bðle  aÞ  sT tanh½bðle  aÞ b    ½bðle  aÞ  tanh þ basf ; ð3Þ 2 where sf is the frictional stress in debonded regions, which is assumed to be independent of le and a.

2613

In our prior paper [57], we solved Eq. (4) for F, assuming G = Gic = const, and thus obtained the current load as a function of the crack length for Gic as a failure criterion (similarly to Eq. (3) for sd): 2 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 c ðaÞ c ðaÞ c ðaÞ  G 1 1 0 ic 5 þ ; ð5Þ  F ¼ fe ¼ pr2f 4 2c2 ðaÞ 2c2 ðaÞ c2 ðaÞ where ci(a) are functions of the crack length which also depend on specimen geometry, interfacial friction and residual thermal stresses. These functions in explicit form are also given in Ref. [34]. Note that substituting a = 0 into Eq. (5) we obtain the debond force, F(0) = Fd, for the critical energy release rate criterion.

2.3. Critical energy release rate (Gic)

2.4. Adhesional pressure (rd)

Energy-based models of interfacial failure in micromechanical tests are also numerous [12,34,55,66,67]. A consistent approach should assume that interfacial crack is initiated and propagated at a constant energy release rate G = Gic, which is referred to as a ‘‘critical energy release rate’’ and can be considered as a failure criterion. The most comprehensive theory, which also includes friction and thermally induced stresses, has been developed by Nairn et al. [12,34,55]. Their approach is based on generalized fracture mechanics of composites [68]. Liu and Nairn [34] expressed the energy release rate, G, as a function of the crack length, a:

Scheer and Nairn [55] investigated stress distribution in a microbond specimen using a three-dimensional variational mechanics analysis. Fig. 4 presents the variation of normal (radial), rrr and shear, srz, interfacial stresses with the axial coordinate, z. This distribution takes place for the intact interface right up to the moment of debonding onset. It can be seen that shear stress is zero at the point where the fiber enters the matrix (maximum srz is reached in the vicinity of this point but at z 5 0), while radial stress, rrr, is at its maximum here. Pisanova et al. [40] assumed that interfacial crack is initiated when rrr reaches its critical value, rd, at z = 0, and proposed rd as a new stress-based criterion for interfacial failure. At the moment of crack initiation, the radial tensile stress, rrr, balances the adhesional pressure (interfacial pressure produced by adhesion forces), and therefore rd is numerically equal to this adhesional pressure and directly proportional to the work of adhesion, WA. This proportionality can easily be understood when we consider a

G ¼ gða; le ; F ; DT ; other factorsÞ;

ð4Þ

where DT is the temperature difference between the test temperature and the stress-free temperature, and ‘‘other factors’’ include interfacial friction and specimen geometry. The explicit expression for G and its detailed derivation can be found in Ref. [34].

0.30

Fiber: E-glass Matrix: epoxy

Fmax 0.25

Force (N)

0.20

Fd 0.15 0.10 0.05

ukink 0.00 0

10

20

30

40

50

60

70

80

Displacement (µm) Fig. 3. A typical force–displacement curve recorded during a microbond test. Fd is the force at which interfacial debonding starts and the curve shows a ‘‘kink’’ (ukink is the displacement of the loaded fiber end at this moment), and Fmax is the peak force.

Fig. 4. Interfacial stress distribution along the embedded length in the variational mechanics model by Scheer and Nairn [55]: srz, shear stress; rrr, radial stress.

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S. Zhandarov et al. / Composites Science and Technology 66 (2006) 2610–2628

process of quasi-equilibrium separation of two contacting phases (in our case, the fiber and the matrix) at the molecular level. The work required for separation of two adjacent molecular layers of the fiber and the matrix is equal to the Rwork of adhesion, WA, and can be expressed as þ1 W A ¼ x0 rrr ðxÞdx, where x is the current separation between the centers of atoms in the contacting layers (x0 is the equilibrium interatomic distance in the absence of external load) and rrr is the current normal stress at the interface. This equation can be rewritten in the form WA = rd Æ k, where k is the effective normal displacement between the contacting surfaces required for their separation, and rd is the interfacial normal stress at which separation occurs, i.e., the maximum value of the function rrr (x) reached somewhere between x0 and +1). For any explicit law for interatomic interactions, e.g. van der Waals forces, direct proportionality between rd and WA can easily be derived. Having determined rd from a microbond test, it is possible to estimate the work of adhesion in a real fiber–matrix system [7,33,40] as well as the thickness of effectively loaded matrix layer, or effective thickness of the interphase [7]. Scheer and Nairn [55] derived an analytical expression for rrr. The rrr value at the moment of crack initiation and z = 0 (i.e., adhesional pressure rd) depends on specimen geometry, residual thermal stresses and the value of the force applied to the fiber at this moment (the debond force, Fd): rd ¼ sðle ; F d ; DT ; geometryÞ.

ð6Þ

From this equation, Fd can be expressed in similar way as in Eqs. (2) and (5): F d ¼ fr ðle ; rd ; DT ; geometryÞ.

ð7Þ

For a given fiber–matrix pair (the same geometry and constant DT, as well as constant value of the failure criterion rd), Eq. (7) gives the variation of the debond force with the embedded length, le. If we assumed, like in Subsections 2.2 and 2.3, that the friction in debonded regions can be expressed in terms of interfacial frictional stress, sf, which does not depend on the axial coordinate, we also can find the current value of the force, F, applied to the fiber during the debonding process, as a function of the crack length: F ðaÞ ¼ sf  2prf a þ F d ðle  aÞ;

ð8Þ

where Fd(le  a) is given by Eq. (7). However, the mode of interfacial failure in microbond specimens was found to change from Mode I (tensile) at short crack lengths to the mixed mode and predominantly Mode II at large crack lengths [55,61]; therefore, Eq. (8) is only valid for a  le. We should also note that the analysis carried out by Scheer and Nairn is valid, strictly speaking, for the microbond test only, where the matrix undergoes compression (cf. Subsection 2.1); all rd data obtained using similar but different tests (pull-out, push-out) can be only regarded as rough estimations [7,40].

2.5. Peak force and apparent IFSS as functions of the embedded length for different failure criteria Equations which express the current force, F, versus the crack length, a (Eqs. 3,5,8), can be used to obtain the maximum force measured in a microbond test, Fmax, and the apparent IFSS, sapp, as functions of the embedded length, le, for a chosen interfacial failure criterion (sd, Gic and rd, respectively). For sd, this problem has been solved analytically [18,63]; the corresponding equations can be found in Appendix A2. For Gic and rd, the Fmax(le) and sapp(le) dependencies can be found only numerically. The algorithm for this consists in generation of many F(a) curves for different embedded lengths (e.g. for the range 0 < le < 1 mm, with a 5–10 lm increment in le) and plotting their maximum values versus le; for more details, see Ref. [18]. This algorithm for Gic and rd criteria was realized using the Mathematica 4.2 software [69]. We would like to emphasize again that, for rd as a failure criterion, the Fmax(le) and sapp(le) curves have a physical meaning only where Fmax is reached at short crack lengths (a  le). In the case of high interfacial friction, it corresponds to a le’s range limited to several fiber diameters; however, for small sf values it can be rather wide. 2.6. Specimen geometries Most of parameters and functions discussed in this paper depend on ‘‘specimen geometry’’. Practically, it means that they require the value of fiber volume fraction, Vf, for their calculation. We used microbond specimens with three different geometries and, correspondingly, different equations for the fiber volume fraction: (a) cylindrical specimen, the fiber runs along the cylinder axis: Vf ¼

r 2 f ; R

ð9Þ

where R is the radius of the matrix cylinder; (b) spherical droplet with the fiber running through its center [18]: Vf ¼

1 2

1 þ 6rle2

;

ð10Þ

f

(c) cylindrical specimen (disc) in which the fiber runs through its center parallel to its bases (Fig. 5). For rf  l e, Vf ¼

4r2f ; le H

ð11Þ

where H is the height of the cylinder (disc thickness). The problems concerning a correct account for specimen geometry are discussed in more detail in Ref. [63].

S. Zhandarov et al. / Composites Science and Technology 66 (2006) 2610–2628

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ure’ and ‘true interfacial parameters’ will be discussed from the viewpoint of requirements to such universal parameters described in Section 1.

H

3. Modeling results and theoretical comparison of interfacial parameters

le

Fig. 5. A cylindrical microbond specimen in which the fiber runs parallel to its bases.

2.7. Final remarks on different failure criteria Our next step (Section 3) will include theoretical prediction of microbond test results (current force as a function of the crack length; debond and peak forces as functions of the embedded length) provided that the failure criterion is rd: rd(le) = const. Than we shall compare these results with theoretical curves for two other failure criteria (sd and Gic) and study the interrelations between all three parameters. Section 5 will provide experimental data from real microbond tests with different specimen geometries and their fitting by theoretical curves with rd, sd and Gic as the failure criteria. The applicability of adhesional pressure and two other parameters as ‘criteria of interfacial fail-

Applying the approach described in Subsection 2.4 to a model system (glass fiber–polystyrene), we obtained theoretical results of a virtual microbond test for this fiber– matrix pair with rd as a failure criterion. The mechanical and thermal properties of the components required for our calculations are listed in Table 1. In our prior paper [18] we set the interfacial parameters in this model systems equal to typical experimental values, namely, local IFSS sd = 60 MPa and interfacial frictional stress sf = 2 MPa. If we consider a cylindrical microbond specimen with R/ r = 10 in which the local IFSS is 60 MPa, the analysis according to Scheer and Nairn’s approach [55] shows that rd value quickly tends to approximately 92.5 MPa when le ! 1 (the difference is less than 1% when le > 115 lm, i.e. ble > 2.4). Therefore, in this study we used a close constant rd value of 90 MPa. We performed our ‘virtual microbond test’ for different ‘experimental’ conditions, varying the following factors: specimen geometry (cylindrical matrix with R/r = 10 and spherical matrix droplets); residual thermal stresses (real stresses, calculated for DT = 75 K, and zero residual stress); interfacial friction (0 and 20 MPa). The value sf = 20 MPa is much greater than interfacial frictional stress in real specimens; we have chosen it just for illustration of the effect of friction, since the microbond test modeling with an experimental value (2 MPa) showed little difference from the frictionless case. The calculated debond force value, Fd, as a function of the embedded length for cylindrical specimens is shown in Fig. 6a. Fig. 6b and c present the variations of interfacial parameters sd and Gic, respectively. As can be seen, all three functions grow nearly linearly at short embedded lengths, then level out and, at le ! 1, tend to limits whose values depend on residual thermal stresses (Gic limit values

Table 1 Properties of fibers and matrices and specimen dimensions required for the microbond data treatment Property

E-glass fiber [32,34]

Steel wire [5,6,70]

Kevlar49 fiber [34]

Polystyrene [26]

EDT-10 epoxy [5,6,70]

Epon 828 epoxy [55]

Polycarbonate [33]

Polypropylene [26]

Fiber diameter, 2rf (lm) Diameter of the matrix cylinder or droplet (lm) Axial tensile modulus, EA or Em (GPa) Transverse tensile modulus, ET (GPa) Axial Poisson’s ratio, mA or mm Transverse Poisson’s ratio, mT Axial CTE, aA or am (ppm/C) Transverse CTE, aT (ppm/C) Stress-free temperature (C) Embedded fiber length, le (lm)

14 and 21 –

150 –

11.7 –

– 20. . .600

– 8000

– 60. . .450

– 20. . .600

– 20. . .600

75

207

130

3.2

2.4

3.3

2.3

1.3

75

207

10

3.2

2.4

3.3

2.3

1.3

0.17 0.17 5 5 – 80. . .450

0.33 0.33 11.7 11.7 – 150. . .2500

0.2 0.35 2 60 – 60. . .450

0.32 0.32 70 70 100 –

0.34 0.34 75 75 160 –

0.35 0.35 48 48 125 –

0.38 0.38 65 65 149 –

0.34 0.34 150 150 25 –

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S. Zhandarov et al. / Composites Science and Technology 66 (2006) 2610–2628 0.16

a

0.14

3, 4

Debond force (N)

0.12

1, 2 0.10 0.08 0.06 0.04

Glass fiber + polystyrene σd = 90 MPa Cylindrical specimens

0.02 0.00 0.0

0.1

0.2

0.3

0.4

0.5

0.6

Embedded length (mm) 100

b

3, 4

Local IFSS (MPa)

90 80 70 60

1, 2

50 40

Glass fiber + polystyrene σd = 90 MPa Cylindrical specimens

30 20 0.0

0.1

0.2

0.3

0.4

0.5

0.6

2

Critical energy release rate (J/m )

Embedded length (mm) 50

3

c

4

40

30

2

20

1 Glass fiber + polystyrene σd = 90 MPa Cylindrical specimens

10

0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

Embedded length (mm) Fig. 6. The debond force (a), local interfacial shear strength (b) and critical energy release rate (c) versus the embedded length for a microbond test with cylindrical specimen geometry. The adhesional pressure is assumed to be a failure criterion (rd = 90 MPa). Curves 1 ignore both residual thermal stresses and interfacial friction; curves 2 include friction (sf = 20 MPa) but ignore residual stresses; curves 3 include residual stresses (DT = 75 K) but ignore friction, and curves 4 include both residual stresses and friction.

are also slightly sensitive to interfacial friction, as follows from the model used [55]). For le > 0.15 mm (ble > 3), Fd, sd and Gic are practically constant. This behavior is quite expectable if we note that all these functions vary with

the embedded length as tan h(ble) and tan h(ble/2), which are linear in ble at small argument values but are very close to 1 and nearly constant for ble > 3. In this sense, all three interfacial parameters (rd, sd and Gic) seem to be practically ‘equivalent’ as failure criteria for ‘good’ specimens (with large embedded lengths). However, the question remains open for shorter embedded lengths, which also occur in micromechanical tests. The curves in Fig. 7 are plotted for the same combinations of thermal and frictional parameters as in Fig. 6 but for spherical microbond specimens. As has been shown in Ref. [63], the shear-lag parameter, b, for this type of specimen geometry decreases monotonically with le, and ble has a minimum at some le value. In our example, (ble)min  2.68 at le  65 lm. For spherical specimens, the debond force steadily increases over the whole range of embedded fiber lengths. In Ref. [18], a similar behavior was reported for sd = const; but in our case (constant rd), sd also appear to grow over the most part of the le axis, so that the increase in Fd becomes even more pronounced. Note, however, that the local IFSS has a minimum at some le value whose position corresponds to medium ble’s (neither ble  1 nor ble  1) and depends on residual thermal stresses. This relationship between rd and sd could not be predicted a priori, from general considerations; it is a fine consequence resulting from the variational mechanics model [55] and a specific geometry. Curves in Figs. 6b and 7b belong to key plots in our study; these illustrate a very important relationship between two interfacial stress parameters, namely, the shear (sd) and normal (rd) stresses at which interfacial crack initiates and debonding starts. In cylindrical geometry, these parameters remain nearly constant at large embedded lengths; however, for spherical specimens adhesional pressure as a failure criterion (rd = const) means that the calculated local IFSS (sd) increases with le, and, vice versa, if sd is the failure criterion (sd = const), the calculated rd values should decrease with the embedded length. Of course, a more substantiated comparison of these two criteria can only be made on the basis of real experimental data. However, the most important and interesting thing in modeling a microbond test with rd as a failure criterion was to obtain ‘experimental’ dependencies of the peak force measured in the test (Fmax) and the apparent interfacial shear strength (sapp, Eq. (1)) on the embedded length. As follows from the algorithm described in Subsection 2.5, the calculation of Fmax and sapp is based on the variation of the current applied force with the crack length, i.e. function F(a) (Eq. (8)). The plots of this function for rd = const and different combinations of thermal and frictional parameters are presented in Fig. 8a (cylindrical geometry) and b (spherical droplets). The embedded length in both cases was 0.3 mm. When the fiber–matrix interface is assumed to be frictionless (curves 1 and 3), F(a) decreases monotonically with a. This means that the peak force, Fmax, is reached already at crack initiation, and crack

S. Zhandarov et al. / Composites Science and Technology 66 (2006) 2610–2628

propagation is unstable from the very beginning. If interfacial friction in debonded regions is present, F(a) increases with a up to a maximum which corresponds to a finite crack length value. However, though this crack length

a

Debond force (N)

0.25

3, 4

a

4

0.30

0.20

0.25

2

0.20

3

0.15

4'

0.15 0.10

Glass fiber + polystyrene σd = 90 MPa, le = 0.3 mm Cylindrical specimens

0.05

1, 2

0.00 0.00

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0.10

1

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Crack length (mm)

Glass fiber + polystyrene σd = 90 MPa Spherical specimens

0.05

0.35

b

4

0.30

0.00 0.0

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0.2

0.3

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Applied force (N)

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b 120

Local IFSS (MPa)

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Applied force (N)

0.30

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3, 4

100

1, 2

0.10

20 0.0

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Glass fiber + polystyrene σd = 90 MPa, le = 0.3 mm Spherical specimens 0.05

0.10

0.15

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Crack length (mm)

Glass fiber + polystyrene σd = 90 MPa Spherical specimens

40

4'

1

0.15

0.00 0.00

60

3

0.20

0.05

80

2

0.25

0.6

Fig. 8. Theoretical plots of the current force applied to the fiber in a microbond test versus the crack length: (a), cylindrical specimens; (b), spherical specimens. Curves 1–4 have been plotted for the same conditions as in Figs. 6 and 7. Curve 4 0 corresponds to sDT = 75 K and sf = 2 MPa.

2

Critical energy release rate (J/m )

Embedded length (mm) 120

c

3

100

4

80 60 40

2

20

1

0 0.0

0.1

0.2

Glass fiber + polystyrene σd = 90 MPa Spherical specimens 0.3

0.4

0.5

0.6

Embedded length (mm) Fig. 7. The debond force (a), local interfacial shear strength (b) and critical energy release rate (c) versus the embedded length for a microbond test with spherical matrix droplets. The adhesional pressure is assumed to be a failure criterion (rd = 90 MPa). Curves 1 ignore both residual thermal stresses and interfacial friction; curves 2 include friction (sf = 20 MPa) but ignore residual stresses; curves 3 include residual stresses (DT = 75 K) but ignore friction, and curves 4 include both residual stresses and friction.

can be rather large (60% of the embedded length and more), the increase in the peak force due to interfacial friction is surprisingly small. At a real level of friction at the glass fiber – polystyrene interface (sf = 2 MPa, curves 4 0 ), Fmax increased (with respect to Fd = F(0)) only by 5.65% for cylindrical specimens (at a = 0.135 mm) and by 2.24% for spherical droplets (at a = 0.094 mm). Even a very large interfacial frictional stress sf = 20 MPa could only double the Fmax value (curves 4 in Fig. 8a and b). Since the effect of interfacial friction in most real fiber–matrix systems is small or moderate, the Fmax(le) curves plotted used the modeled F(a) dependencies do have physical meaning and should be good approximations of experimental datasets. Fig. 9 presents the peak force (a) and the apparent IFSS (b) calculated for our model system in a microbond test with cylindrical specimen geometry as functions of the embedded fiber length. In the case of frictionless interface, Fmax(le) plot levels out at ble  1 and tends to a limit whose value depends on residual thermal stresses (curves 1 and 3). The presence of friction imparts the plot a nearly constant slope which is proportional to the sf value, similarly as for

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S. Zhandarov et al. / Composites Science and Technology 66 (2006) 2610–2628 0.7

0.6

Glass fiber + polystyrene σd = 90 MPa Cylindrical specimens

Peak force, Fmax (N)

0.5

0.6

2

0.4 0.3

4'

3 0.2 0.1

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0.2

0.3

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0.5

0.4

3

1

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Apparent IFSS (MPa)

Apparent IFSS (MPa)

26

Glass fiber + polystyrene σd = 90 MPa Cylindrical specimens

4 25

2

20 15

3

10

1

5 0 0.0

4'

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4

b

24

2

Glass fiber + polystyrene σd = 90 MPa Spherical specimens

22 20 18

1

16

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4'

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40 35

4'

0.3

Embedded length (mm) b

2

0.5

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1

0.0 0.0

4

Glass fiber + polystyrene σd = 90 MPa Spherical specimens

a

4 Peak force, Fmax (N)

a

0.4

0.5

0.6

Embedded length (mm) Fig. 9. Theoretical plots of the peak force (a) and apparent interfacial shear strength (b) as functions of the embedded length for the glass fiber – polystyrene system (cylindrical specimens). Curves 1–4 and 4 0 have been plotted for the same conditions as in Figs. 6–8.

sd as a failure criterion [18]. However, in contrast to the case sd = const, apparent IFSS does not decrease monotonically with le but has a maximum at some finite le value. Fig. 10 shows Fmax(le) and sapp(le) curves for spherical droplets. In spherical geometry, the increase of the embedded length is accompanied by a reduction of the fiber volume content (Vf) in the specimen; therefore, the behavior of theoretical curves is more intricate. For instance, Fmax substantially increases with le even for specimens with no friction between the fiber and the matrix, and sapp(le) plots, in addition to a maximum at a medium le value, have a minimum at a very short embedded length. Fig. 11 compares the behavior of Fmax(le) and sapp(le) functions calculated for different failure criteria: sd [18,63], Gic [57] and rd. As can be seen, in the case of cylindrical geometry (Fig. 11a and b), all three criteria give asymptotically the same result (constant Fmax and sapp decreasing as l1 e already at ble P 2). However, these predict three substantially different behaviors of the virtual microbond test results in the range of short embedded length, and especially for ble  1, which is best of all illus-

0.1

0.2

0.3

0.4

0.5

0.6

Embedded length (mm) Fig. 10. Theoretical plots of the peak force (a) and apparent interfacial shear strength (b) as functions of the embedded length for the glass fiber – polystyrene system (spherical droplets). Curves 1–4 and 4 0 have been plotted for the same conditions as in Figs. 6–8.

trated by Fig. 11b. The local IFSS (sd) criterion yields limle !0 sapp ¼ sd , while the limit of the apparent IFSS derived under the condition Gic = const is infinity, and for the rd = const condition, zero. Which criterion better describes interfacial failure in the microbond test, can only be verified experimentally. The first two criteria (sd and Gic) have been repeatedly compared in the literature [32,57,59], and it has been found that the local IFSS failure criterion gives better results for initiation of debonding than the critical energy release rate criterion but the situation may be different for analysis of propagation of debonding. In this context, ‘better results’ means the independence of the specimen geometry as well. However, if we try to transfer the values of interfacial parameters found for cylindrical geometry to spherical droplets, poor correlation between the three criteria will be obtained (Fig. 11c and d). Nevertheless, the sd, Gic and rd values can be chosen so that the corresponding Fmax(le) and sapp(le) curves were in good mutual agreement over a considerable part of the range of the embedded lengths, for both cylindrical and spherical geometries, even in the presence of residual thermal stresses

S. Zhandarov et al. / Composites Science and Technology 66 (2006) 2610–2628 80

0.16

a

0.14

Apparent IFSS (MPa)

Peak force, Fmax (N)

b

70

2 0.12 0.10

1 0.08 0.06

(1) τd = 60 MPa

3

0.04

2

(2) Gic = 21.3 J/m (3) σd = 92.5 MPa

0.02 0.00

0.05

0.10

0.15

0.20

(1) τd = 60 MPa

2

2

(2) Gic = 21.3 J/m (3) σd = 92.5 MPa

60 50

1

40 30

3

20 10

0.25

0 0.00

0.30

0.05

0.10

Embedded length (mm)

0.20

0.25

0.30

40

(1) τd = 60 MPa

3

35

0.20

1 0.15

2 0.10 0.05 0.00 0.0

0.1

(1) τd = 60 MPa

d 2

(2) Gic = 21.3 J/m (3) σd = 92.5 MPa

0.25

Apparent IFSS (MPa)

c Peak force, Fmax (N)

0.15

Embedded length (mm)

0.30

0.2

0.3

0.4

2

30 25

1 20 15

3 10 5 0.0

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(2) Gic = 21.3 J/m (3) σd = 92.5 MPa

2

0.1

Embedded length (mm)

0.2

0.3

0.4

0.5

Embedded length (mm) 80

0.15

e 2

0.10

1 0.05

(1) τd = 60 MPa 2

(2) Gic = 20.5 J/m (3) σd = 67.7 MPa

3 0.00 0.0

f

70

Apparent IFSS (MPa)

Peak force, Fmax (N)

2619

0.1

0.2

0.3

0.4

0.5

(1) τd = 60 MPa

2

2

(2) Gic = 20.5 J/m (3) σd = 67.7 MPa

60 50

1 40 30

3 20 10 0 0.00

0.6

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3

g

Apparent IFSS (MPa)

Peak force, Fmax (N)

1

2 0.10

(1) τd = 60 MPa

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(2) Gic = 23.5 J/m (3) σd = 56.2 MPa 0.00 0.0

(1) τd = 60 MPa

h 30

0.15

0.1

0.2

0.3

Embedded length (mm)

0.4

2

2

(2) Gic = 23.5 J/m (3) σd = 56.2 MPa

25 20 15

3 10 5 0 0.0

1 0.1

0.2

0.3

0.4

Embedded length (mm)

Fig. 11. Comparison of theoretical plots of the peak force and apparent interfacial shear strength as functions of the embedded length plotted for different failure criteria. Specimen geometry: cylindrical (a), (b), (e), (f) and spherical (c), (d), (g), (h). Interfacial friction and residual thermal stresses: both ignored (a), (b), (c), (d) and both included (e), (f), (g), (h).

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S. Zhandarov et al. / Composites Science and Technology 66 (2006) 2610–2628

(Fig. 11e–h). The applicability of these parameters as failure criteria should be judged from how do they fit experimental microbond results; this will be discussed below in Section 4.

lengths, interfacial friction does not affect the Fmax values. This value has been obtained assuming that sd = const; however, below we shall see that for rd = const the specimen behavior is very similar. Fig. 12a presents the experimental Fmax dependency on the embedded length and its

4. Real microbond test results and their fitting using different failure criteria

4.1. Measurement of the debond force and comparison of its experimental values with theoretical predictions based on different failure criteria We shall begin with cylindrical geometry, which is the simplest for theoretical analysis and calculation and yields the most illustrative experimental results. Gorbatkina and Ivanova-Mumjieva [70] studied the system composed of epoxy matrix EDT-10 and steel wire. A microbond specimen was fabricated in an aluminum mold in the middle of which the steel wire was placed. The mold diameter was 8 mm, the wire diameter 0.15 mm, and the embedded length varied within the range of 0.08–2.5 mm. After thermal treatment (8 h at 160 C), microbond specimens were tested at room temperature using a microtensile test machine designed at the Institute of Chemical Physics [5,6,71]. The loading rate was constant and equal to 1 N/s. In the tests, the peak force, Fmax, was determined for all specimens and then plotted as a function of the embedded length. However, in our prior paper [63] we have demonstrated that for cylindrical geometry a ‘transitional’ embedded length, lf, exists, such that for specimens with le < lf the fiber–matrix interface fails in a catastrophic way (instable debonding) as soon as the force applied to the fiber reaches the Fd value (and, consequently, Fmax = Fd for these specimens). For the steel wire – EDT-10 system considered, lf = 1.44 mm [63]; for shorter embedded

Peak force, Fmax (N)

30

20

2

Steel wire + EDT-10 (1) τd = 85.4 MPa

10

2

(2) Gic = 180 J/m (3) σd = 158 MPa

3

1 0 0.0

0.4

0.8

lf = 1.44 mm

1.2

1.6

2.0

Embedded length (mm) 120

b

2

100

Apparent IFSS (MPa)

• Crack initiation: prediction of the debond force, Fd, as a function of the embedded length. • Crack propagation: crack length, a, as a function of the current applied load, F. • Dependencies of the peak force, Fmax, measured in the microbond test, and the apparent interfacial shear strength, sapp, on the embedded length.

a

40

80

1

60 40

Steel wire + EDT-10 (1) τd = 85.4 MPa

lf = 1.44 mm

2

20

3 0 0.0

(2) Gic = 180 J/m (3) σd = 158 MPa 0.4

0.8

1.2

1.6

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Embedded length (mm) 0.5

Crack length at F = Fmax (mm)

In this paper we consider the results of microbond tests published in our earlier papers [5,6,18,57,70], as well as experimental data presented by other authors [55]. We shall not describe detailed experimental procedures unless it is absolutely necessary (e.g. for understanding the specimen geometry), but refer to the original papers. The mechanical and thermal properties of matrices and fibers (wires, rods), as well as specimen dimensions required for microbond data treatment, are listed in Table 1. We compared the applicability of the three interfacial parameters (sd, Gic and rd) as eventual failure criteria (special attention being paid to the adhesional pressure, rd, as a newly proposed criterion) analyzing the following situations (tasks).

c

Steel wire + EDT-10 σd = 158 MPa

0.4

0.3

0.2

0.1

lf = 1.44 mm 0.0 0.0

0.4

0.8

1.2

1.6

2.0

Embedded length (mm) Fig. 12. Experimental plots of the peak force (a) and the apparent IFSS (b) versus the embedded length for the steel wire – EDT-10 epoxy system (according to original data from Ref. [70]) and their theoretical fits for different failure criteria; (c) crack length at which instable debonding starts as a function of the embedded length (see text for details).

S. Zhandarov et al. / Composites Science and Technology 66 (2006) 2610–2628 1.0

Glass fiber + Epon 828 (1) τd = 177 MPa 0.8

Peak force, Fmax (N)

fits using three different failure criteria, sd, Gic, and rd. All three fits yielded satisfactory approximations of experimental data for large embedded lengths (le > 0.4 mm), which is in agreement with theoretical predictions (cf. Fig. 11e). Only for le < 0.4 mm, which corresponds to a rather low ble value of 0.35 (b = 686 m1 for these specimens), the difference becomes significant. This is due to a fundamental difference between the three failure criteria, which can be illustrated by Fig. 12b. This figure shows the apparent interfacial shear strength, sapp, as a function of the embedded length. While for the sd failure criterion limle !0 sapp ¼ sd (this is true for cylindrical specimens but may be not for other specimen geometries if the fiber volume fraction, Vf, varies substantially with le [63]), the sapp limit as le ! 0 for the Gic criterion is infinity [57], and for the rd criterion, zero. We should note again that this difference is observed for low le values (ble  1) (see theoretical plots in Fig. 11a, b, e and f); at larger ble’s, theoretical curves asymptotically coincide, and fits of experimental data are very close to each other. An important intermediate result from the procedure of theoretical Fmax(le) calculation is the critical crack length, acr, at which the applied force is at its maximum, Fmax, in a given specimen. A plot of this parameter against the embedded length can be very illustrative. For the shear strength failure criterion (sd = const), as was shown in Ref. [63], acr is exactly zero at le < lf and increases fast when le > lf. Our calculation made assuming rd = const gave a very similar result (Fig. 12c). For le < lf, acr grows very slowly (at le = 1.44 mm acr  0.02 mm, which is much smaller than the wire diameter) and nearly linearly; at le > lf, the plot is also practically linear but its slope increases drastically. Therefore, we can recognize that interfacial friction has little effect on Fmax values at le < lf, and Fmax  Fd within this range of embedded lengths for the adhesional pressure (rd) criterion as well. Moreover, the lf values from the sd and rd criteria are very close to each other, though calculated using different models. For le > lf, the Fmax value is considerably affected by interfacial friction, which is rather large for the steel wire – EDT-10 system (the interfacial frictional stress sf  15 MPa). The measured Fmax value can also be regarded as a debond force in a wide range of embedded lengths if the interfacial frictional stress in debonded regions is negligible. In Fig. 13, Fmax is plotted as a function of le for the glass fiber – Epon 828 system. Experimental data were taken from Ref. [55]. The authors used a microbond test with ellipsoidal matrix droplets; the fiber diameter was 21 lm, and the embedded length varied from 90 to 420 lm. In this paper, as well as in our two previous papers [18,57], we approximated these droplets by spherical ones. For this fiber–polymer pair, our estimations using different models yielded sf < 2.1 MPa [18,57]. Therefore, the debond force, Fd, can be successfully approximated by the peak force values, Fmax, measured in the microbond test. Since the fiber content in the specimen varied with le, the Nayfeh’s shear-lag parameter b also varied. For spherical

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(2) Gic = 220 J/m (3) σd = 182 MPa

2

0.6

0.4

0.2

0.0 0.0

3

1 0.1

0.2

0.3

0.4

0.5

Embedded length (mm) Fig. 13. Experimental plot of the peak force versus the embedded length for the glass fiber – Epon 828 epoxy system (from Ref. [55]) and their theoretical fits for different failure criteria.

microbond specimens, the plot of ble versus the embedded length is a curve having a minimum at some le value [63]. For the system studied, ble was greater than 2.5 over the whole region of embedded lengths. As can be seen in Fig. 13, all three fitting curves approximate the experimental data satisfactorily (curve 2 for Gic = const seems to be the best one) and are in agreement with the theoretical plot in Fig. 11g. Recognizing that the rd-based fits in Figs. 12 and 13 also are good approximations to experimental dependencies, we can conclude that the adhesional pressure, rd, can be used as a criterion for the initiation of debonding. 4.2. Relations between the crack length, a, and the current applied load, F: Resistance curves for different failure criteria To study the adequacy of sd, Gic and rd as interfacial parameters describing crack propagation within a microbond specimen, we calculated these parameters as functions of the crack length for individual microbond trials. Liu and Nairn [34] carried out such calculations for G(a) in model microbond specimens (steel rod in epoxy matrix) using Eq. (A3). They considered the interfacial shear stress, sf, as an additional fitting parameter and plotted a set of resistance curves (R-curves) G ¼ f ðaÞjsf ¼const for individual specimens. The curve giving the ‘flattest’ (i.e. the closest to constant) plot was considered to be the best fit, and the corresponding G value was regarded as the critical energy release rate (Gic) for crack propagation in this specimen. Zhandarov et al. [58] extended this approach to local interfacial shear strength (sd): a family of s(a) curves corresponding to different sf values was plotted for a microbond specimen using the equation     F bðle  aÞ sðaÞ ¼ b coth½bðle  aÞ  sf a þ sT tanh 2prf 2 ð12Þ

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S. Zhandarov et al. / Composites Science and Technology 66 (2006) 2610–2628

which immediately follows from Eq. (3), and the ‘optimally flat’ plot was used to determine the best-fit values of the local IFSS (sd) and the interfacial frictional stress (sf). In this paper, we used a similar approach to estimate the best-fit adhesional pressure value (rd). We plotted a set of r(a) curves for different sf’s using Eqs. 6,7,8 and (A3), and took the ‘optimal’ r value from the flattest curve as the adhesional pressure (r = rd). We also plotted G(a) and s(a) curves for the same microbond specimens in order to compare all three parameters as criteria for crack propagation. In our experiments, we used cylindrical specimens (discs) with the fibers running parallel to their bases (Fig. 5). The specimens were prepared as follows. A small thermoplastic polymer bead was placed on a thin cover glass placed on a heated stage adjustable for temperature. The temperature of the glass surface was 200 C for polypropylene, 230 C for polystyrene, and 280 C for polycarbonate. After the bead had melted, a glass fiber was placed on the top of the droplet formed, and covered with a second glass plate. The specimen was kept in this state for about 2 min, which was sufficient for the droplet to flatten and for the fiber to immerse in the polymer. Then, the specimen was cooled down, and both glass plates were carefully separated from the polymer droplet (in contrast to pull-out and fiberstretching tests described in Ref. [33]). The finished specimen was a polymer disc (flattened droplet) having the diameter of 0.3–1.0 mm and thickness about 0.15 mm, in which a glass fiber was embedded parallel to the disc face (see Fig. 5). All specimens were examined in a microscope, and those containing non-straight fibers or interfacial defects (air bubbles, partially immersed fiber, etc.) were discarded. A mini-tensile machine (Kammrath and Weiss GmbH, Germany) was used to apply tensile load to microbond specimens. The tested specimen was placed on a horizontal support plate clamped in the frame of the tensile machine, as shown in Fig. 14. One fiber end was fixed to a movable plate connected to the force sensor by a cyanoacrylate glue

over a length of more than 1 mm in order to prevent a fiber pull-out from the glue. The free fiber length was 0.2–2 mm. During the loading, the droplet was held in place by knife edges on the support plate. The test speed was set to the minimum available with this tensile machine, 0.25 lm/s, to ensure slow crack propagation. The force applied to the fiber was recorded as a function of time. The test device was placed on the stage of an optical microscope (Leica MZ12) equipped with a video camera connected to a computer, which made possible to record the picture of crack propagation in step with the applied load. Thus, the crack length, a, was obtained as a function of the applied external force, F. For the above-described specimens, the geometry was strongly different from the system of two concentric cylinders, so that the theoretical approach described in this paper can be applied to these specimens only approximately. Moreover, accounting for the fiber volume fraction in these specimens becomes a separate important problem. The usual procedure is to determine the ‘effective’ fiber content [32,58]. By varying the Vf value, the theoretical displacement of the loaded fiber end at the moment of crack initiation is calculated using the described models. For some Vf, it is equal to the experimentally measured displacement from the recorded force–displacement curve. This ‘effective’ Vf value is then used to calculate b and to plot interfacial parameters against the crack length. However, in this paper we calculated the nominal fiber volume fraction in the specimens using Eq. (11), and these nominal values appeared to be surprisingly close to their effective values reported in Ref. [58]. This evidences again that the theoretical shear-lag approach with Nayfeh’s b parameter can be successfully used for microbond specimens even when their shape is not cylindrical. Moreover, an important advantage of the above-described version of the microbond test is that it allows continuous monitoring, including synchronized recording of the applied force, displacement and crack length as functions of time. As can be seen in

Knife edges

Fiber

To the force sensor

Matrix Glue

droplet

Fig. 14. Schematic view of the microbond test configuration with a horizontal matrix droplet (considered as a cylindrical disc) and a fiber running parallel

S. Zhandarov et al. / Composites Science and Technology 66 (2006) 2610–2628

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Fig. 15. Photograph of a typical microbond specimen with a growing crack.

0.28

Glass fiber + polycarbonate Experimental points 1 τd = 92 MPa

0.24

Applied load (N)

Fig. 15, interfacial crack is easily discernible in recorded video frames. The nominal and effective Vf values, as well as all three interfacial parameters (sd, Gic and rd) and the frictional interfacial stress for the investigated fiber–polymer systems are presented in Table 2. Fig. 16 shows a typical plot of the applied load, F, required to cause debond growth (crack propagation), versus the current crack length, a, for a glass fiber – polycarbonate microbond specimen. The position of the last experimental point indicates that crack propagation was stable up to a rather large crack length (about 80% of the embedded length). Further crack propagation was nearly instantaneous (catastrophic). This is in good agreement with a model based on the local IFSS as a failure criterion (curve 1). Curve 3, based on adhesional pressure, adequately predicts the debond growth up to about a half of the embedded length, but fail at larger crack lengths. In all probability, this is due to the fact that the failure mode in a microbond specimen changes from pure

2 Gic = 37.4 J/m 3 σd = 64 MPa

0.20

2

2

1 3

0.16

0.12

le = 344 µm 0.08 0

50

100

150

200

250

300

350

Crack length (µm) Fig. 16. Experimental plot of the current force applied to the glass fiber embedded in a polycarbonate matrix versus the crack length and its best fits using three different failure criteria.

Table 2 Interfacial parameters determined for fiber–polymer systems using models based on different failure criteria Fiber

Steel wire Steel wire E-glass E-glass E-glass E-glass Kevlar 49 a b

Matrix

EDT-10 EDT-10 Polycarbonate Polystyrene Polypropylene EPON 828 EPON 828

Reference

[70] [5,6] This paper This paper This paper [55] [55]

Fiber volume fraction

Nominal

Effective

0.00035 0.00035 0.0037 0.0033 0.0041 0.003–0.094 0.003–0.054

– – 0.0034 0.0032 0.0025 – –

Method of measurementa

1 3 2 2 2 1 3

Interfacial parameters determined using failure criteria Local IFSS (sd)

Critical energy release rate (Gic)

Adhesional pressure (rd)

sd (MPa)

sf (MPa)

Gic (J/m2)

sf (MPa)

rd (MPa)

sf (MPa)

85.4 74.2 91.8 81.4 17.8 177 31.8

14.7b [63] 14.3 8.3 3.7 4.7 2.05b [18] 33.1

180 279 37.4 21.6 2.96 220 35.5

– 19.7 8.3 3.3 4.2 0.14b [57] 20.4

158 128 64 96 61 182 111

– 22 13 6.4 5.8 – 19

1, from the debond force value; 2, from resistance curves; 3, from the best fit of Fmax(le) plots. Estimated in our prior papers using best fits of Fmax(le) plots.

S. Zhandarov et al. / Composites Science and Technology 66 (2006) 2610–2628

τf = 0 τf = 4 MPa τf = 8.3 MPa τf = 12 MPa

Local IFSS (MPa)

160

120

2

Gic = 37.4 J/m

80

40

σd = 64.0 MPa

a 0 0

50

100

150

200

250

300

350

Crack length (μm) 100

τf = 0 τf = 4 MPa τf = 8.3 MPa τf = 12 MPa

2

Critical energy release rate (J/m )

Mode I (tensile) to the mixed mode (tensile + shear) and then to almost pure shear Mode II with the increasing crack length [55,61]. The energy-based model (curve 2) yields the force increase over the whole embedded length, as was considered in detail in Refs. [57,58]; nevertheless, in the experimentally stable region of crack propagation it predicts the crack length as a function of the applied load with good accuracy. Fig. 17 presents the dependencies of the local IFSS (a), energy release rate (b) and adhesional pressure (c) on the crack length calculated for different sf values. Separate symbols correspond to experimental points; solid lines show the behavior of the parameter on the vertical axis under the assumption that one of the two remaining parameters is constant. Sets of experimental points in Fig. 17a–c differ by values of frictional stress in debonded zones. For sf = 0, all three calculated interfacial parameters increase fast with the crack length. However, with the increase in the frictional stress, the curves become more and more flat, and for some sf value can be successfully approximated by a constant. If the parameter on the vertical axis is an adequate criterion for crack propagation, the corresponding curve should be flat or at least include a flat plateau over most of the crack length; this plateau value can be considered as interfacial strength parameter, or criterion for crack propagation. For the specimen under consideration (glass fiber – polycarbonate), interfacial strength parameters determined from the flattest curves are: sd = 91.8 MPa for sf = 8.3 MPa; Gic = 37.4 J/m2 for sf = 8.3 MPa; and rd = 64.0 MPa for sf = 13 MPa. It should be noted that the curves for sd and Gic (Fig. 17a and b) are nearly flat over the whole range of crack lengths where debond growth is stable (amax  0.27 mm in our example), while the optimal experimental curve for rd (Fig. 17c) falls off from horizontal straight line already at a  0.22 mm. This is in good agreement with Fig. 16 and is due to the change in the interfacial failure mode from tensile to shear as the crack grows. Thus, the analysis of Fig. 17 and similar plots for other fiber–polymer systems (not presented here) shows that both sd and Gic are equally good criteria for crack propagation. The values of interfacial frictional stress determined from these two models are also close (8.3 MPa). The ‘equivalence’ of these interfacial parameters as failure criteria was discussed in detail in Refs. [12,26,32,57–59] (crack initiation) and Ref. [12,59] (crack propagation). It was found that for crack propagation, especially at advanced stages of debonding (large crack lengths), the energy-based criterion (Gic) gives the best fit to experimental data from micromechanical tests [12,57,59]; in this paper, this can be clearly seen, e.g., in Fig. 13. Constant adhesional pressure (rd) as a failure criterion describes crack initiation and the initial part of crack propagation, but is not valid for larger crack lengths when failure mode changes to shear. The sf value obtained under assumption rd = const is somewhat overestimated (13 MPa), while the value of the adhesional pressure itself looks too low as compared with our previous estimations [7,40].

80

60

τd = 91.8 MPa 40

20

σd = 64.0 MPa

b 0 0

50

100

150

200

250

300

350

Crack length (μm) 300

Adhesional pressure (MPa)

2624

τf = 0 τf = 4 MPa τf = 8.3 MPa τf = 13 MPa τf = 17 MPa

250 200

G ic = 37.4 J/m

2

150 100

τd = 91.8 MPa

50

c 0 0

50

100

150

200

250

300

350

Crack length (µm) Fig. 17. Local interfacial shear strength (a), critical energy release rate (b) and adhesional pressure (c) as functions of crack length for glass fiber – polycarbonate specimen. Solid lines are theoretical curves plotted on the assumption that the corresponding interfacial parameter is independent on the crack length. Separate symbols were calculated from experimental results for different values of interfacial frictional stress sf in the debonded regions.

It should also be noted that for any of the three interfacial parameters investigated, extrapolation of resistance curves (Fig. 17) to zero crack length can be a reliable technique for the measurement of the debond force, Fd. The

S. Zhandarov et al. / Composites Science and Technology 66 (2006) 2610–2628 0.35 0.30

1

a

2 Peak force, Fmax (N)

common practice is to determine Fd from the ‘kink’ in the force–displacement curve (see Fig. 3). However, for some fiber–matrix systems this kink may be hardly discernible. In this case, if crack propagation was continuously monitored and the current applied force, F, was plotted against the crack length, the Fd value can be obtained by extrapolating the plot F = F(a) to zero crack length [33,58] or from the extrapolation of resistance curves to a = 0. Since the contribution of friction is proportional to the crack length, the extrapolated s, G and r values at a = 0 are not affected by friction. In other words, all resistance curves for different sf should converge to the same sd, Gic and rd values.

2625

0.25 0.20 0.15

Kevlar 49 + Epon 828 (1) τd = 31.8 MPa

0.10 0.05 0.00 0.00

4.3. Peak force and apparent IFSS as functions of the embedded length: experimental data and three theoretical fits

(2) Gic = 35.5 J/m (3) σd = 111 MPa

3 0.05

0.10

0.15

0.20

2

0.25

0.30

Embedded length (mm) 45

b

Apparent IFSS (MPa)

40 35

3 1

30

2 25

Kevlar 49 + Epon 828 (1) τd = 31.8 MPa

20

2

(2) Gic = 35.5 J/m (3) σd = 111 MPa

15 10 0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.25

0.30

Embedded length (mm) 0.0030

c

Crack length at F = Fmax (mm)

A ‘good’ model based on an adequate failure criterion should also predict dependencies of the peak force, Fmax, and the apparent IFSS, sapp, on the embedded length. In Refs. [18,63] we derived analytical expressions for Fmax(le) and sapp(le) assuming sd = const, and compared these dependencies with experimental plots from real microbond and pull-out tests with different specimen geometries. In Ref. [57], Fmax and sapp were calculated for Gic = const; in this case, resulting equations cannot be obtain in explicit analytical form, so that theoretical curves were obtained using numerical simulation. We found that both models (based on sd and Gic as failure criteria) described well the Fmax(le) and sapp(le) behavior and could predict even such rarely observed details as decreasing sapp in the case of spherical microbond specimens as le ! 0. In this paper, we calculated theoretical dependencies of Fmax and sapp on the embedded length for the adhesional pressure criterion (rd = const), using the algorithm described in Subsection 2.5. To fit the experimental data, we used a standard least-squares method with two fitting parameters, rd and sf, similarly to approaches presented in Refs. [18,57]. Fig. 18a and b presents experimental plots of the peak force, Fmax, and the apparent IFSS, sapp, versus the embedded length and their best fits obtained using three different interfacial failure criteria (sd, Gic and rd) for Kevlar 49 – Epon 828 system (ellipsoidal droplets close to spherical). Experimental data were taken from Ref. [55]. The fiber diameter was 11.7 lm, the embedded length varied from 0.06 to 0.26 mm. As can be seen in Fig. 18a, all three theoretical plots fit experimental Fmax points with good accuracy. The difference between the three criteria manifests itself in the sapp(le) plot (Fig. 18b). The model based on the adhesional pressure predicts that sapp should have a local minimum at some embedded length (le = 24 lm for this example), while for sd = const and rd = const sapp decreases monotonically as le tends to zero. However, the presence of this minimum can hardly be verified experimentally, since the embedded length of 24 lm requires such small matrix droplets whose testing is very difficult. Moreover, the model itself may be not valid at such short embedded lengths (le < 2df). For larger le’s, all three models,

Kevlar 49 + Epon 828 σd = 111 MPa

0.0025 0.0020 0.0015 0.0010 0.0005 0.0000 0.00

0.05

0.10

0.15

0.20

Embedded length (mm) Fig. 18. Experimental plots of the peak force (a) and the apparent interfacial shear strength (b) versus the embedded length for the Kevlar 49 fiber – Epon 828 epoxy system (from Ref. [55]) and their theoretical fits using different failure criteria; (c) crack length at which instable debonding starts as a function of the embedded length (see text for details).

including rd = const, yield very similar sapp(le) curves; this evidences that the unstable stage of debonding starts for this system at short crack lengths, and thus the Fmax value is affected by interfacial friction only marginally. Fig. 18c presents the plot of the crack length, at which F = Fmax, as a function of the embedded length (calculated for rd = const). Indeed, the maximum value is reached at very

2626

S. Zhandarov et al. / Composites Science and Technology 66 (2006) 2610–2628 35

Peak force, Fmax (N)

30 25 20

2

15

1

Steel wire + EDT-10 (1) τd = 74.2 MPa

10

2

(2) Gic = 279 J/m (3) σd = 128 MPa

5

3 0 0.0

0.4

0.8

1.2

1.6

2.0

Embedded length (mm) Fig. 19. Experimental plot of the peak force versus the embedded length for the steel wire – EDT-10 epoxy system (according to the data taken from Ref. [6]) and its theoretical fits using different failure criteria.

short crack lengths (0.5–3 lm), so that the effect of interfacial friction in debonded regions on the peak force is negligible. And, finally, Fig. 19 shows Fmax(le) for a steel wire - EDT-10 epoxy pair (cylindrical specimens; the wire diameter was 150 lm [6]). It can again be concluded that all three models give satisfactory fits to experimental microbond test data, and all three parameters can be considered, with the restriction discussed above, as interfacial failure criteria in the microbond test. Table 2 summarizes numerical values of interfacial parameters measured in our study using all three methods described in Section 4. 5. Conclusion The adhesional pressure (rd), an important interfacial parameter, which characterizes strength in normal (radial) direction at fiber–matrix interfaces and relates interfacial strength properties to the work of adhesion, can also be considered as a criterion for interfacial failure. A model based on rd as a failure criterion predicts crack initiation and the initial stage of crack propagation in microbond specimens equally well as models with two other popular interfacial parameters, local interfacial shear strength (sd) and critical energy release rate (Gic). All three approaches describe with good accuracy the dependencies of the peak force, Fmax, and the apparent IFSS, sapp, on the embedded length; by fitting experimental plots with theoretical Fmax(le) and sapp(le) curves, the best-fit values of all parameters (sd, Gic and rd) and the interfacial shear stress, sf, can be determined. However, adhesional pressure is not valid as a criterion for large crack lengths, where the failure mode changes from Mode I (tensile) at crack initiation to predominantly Mode II (shear) at advanced stages of debonding; in this case, Gic yields the best fit. For microbond specimens with very small embedded length (ble  1), the three models considered predict substantially different

behavior. The apparent IFSS (sd) seems to be the best approximation to experimental data for these specimens. Proceeding from our comparison of different interfacial parameters as eventual failure criteria, we should recommend that, in order to reliably measure the adhesional pressure at a fiber–matrix interface (and then estimate the work of adhesion between the matrix and the fiber surface), the debond force in a microbond specimen with sufficiently large embedded fiber length (ble  1) must be measured. Our study also demonstrated that interfacial parameters, sd, Gic and rd, though yielded good enough fits to experimental data, appeared to be sensitive, to a certain extent, to specimen geometry, loading conditions and failure mode, so that none of these parameters could pretend to be a universal (absolute) failure criterion which would valid under arbitrary conditions. This requires a further detailed investigation which should be extended to a much wider class of physical and mechanical phenomena than micromechanical tests. It seems to be obvious that the work required to separate two dissimilar surfaces, or generalized work of adhesion (including eventual irreversible components such as chemical bonds) can be considered as the above-mentioned ‘absolute’ interfacial parameter which, of course, manifests itself differently under different loading (surface separating) conditions. The problem is to create appropriate models which correctly account for these conditions. This also concerns the adhesional pressure, which is, according to its physical meaning, proportional to the generalized work of adhesion and thus could be a universal mechanical parameter to be determined from mechanical tests. The variational mechanics model derived by Scheer and Nairn [55] works well in a wide range of conditions but also has its own restrictions. There is a need for a more comprehensive model which would be adequate for a wider set of stress states occurring in composite materials under load. Appendix A. A.1. Nayfeh’s shear-lag parameter and the residual thermal stress In this paper, b denotes the shear-lag parameter obtained by Nayfeh [64] and rediscovered by Nairn [72], defined as 0 1 2 E V þ E V A f m m @

A; b2 ¼ 2 ðA1Þ 1 1 r f EA E m V m þ 1 ln  1  V f 4GA

2Gm

Vm

Vf

2

where EA and GA are the axial tensile and shear moduli of the fiber and Em and Gm, the tensile and shear moduli of the matrix, and Vf and Vm are the volume fractions of fiber and matrix within the specimen. The stress term sT is due to unequal thermal shrinkage of the fiber and the matrix during cooling the specimen from the temperature of thermal treatment down to room temperature. It is given by the equation [18]

S. Zhandarov et al. / Composites Science and Technology 66 (2006) 2610–2628

sT ¼ EA brf ðaA  am ÞDT =2

ðA2Þ

where aA is the axial coefficient of thermal expansion for the fiber; am, the coefficient of thermal expansion for the matrix; and DT is the difference between the testing temperature and the stress-free temperature. A.2. Analytical expressions for Fmax(le) and sapp(le) as functions of the embedded length for the local IFSS as a failure criterion (sd = const) The peak force recorded in a microbond test, Fmax, and the apparent interfacial shear strength, sapp, can be expressed analytically as functions of the embedded length, le, if we assume that the interface fails in shear when the local shear stress at a given point reaches its critical value, sd (the local interfacial shear strength as a failure criterion) [63]: 8 2pr   f sd tanhðble Þ  sT tanhðble Þ tanh bl2e ; > b > > > pffiffiffiffiffiffiffiffiffiffiffiffiffi > > ble < lnðu þ u2 þ 1Þ; > > >    < u 1 f p ffiffiffiffiffiffiffi p ffiffiffiffiffiffiffi ðA3Þ F max ðle Þ ¼ 2pr  s s 1  d T b > u2 þ1 u2 þ1 > > pffiffiffiffiffiffiffiffiffiffiffiffiffi 

> > > þsf ble  lnðu þ u2 þ 1Þ ; > > > pffiffiffiffiffiffiffiffiffiffiffiffiffi : ble P lnðu þ u2 þ 1Þ;  8 tanhðble Þ eÞ tanh bl2e ; sd ble  sT tanhðbl > ble > > pffiffiffiffiffiffiffiffiffiffiffiffiffi > > > ble < lnðu þ u2 þ 1Þ; > > >   < sd sT u 1 p ffiffiffiffiffiffiffi p ffiffiffiffiffiffiffi ðA4Þ sapp ðle Þ ¼ ble  ble 1  2 > u2 þ1 u þ1 > > p ffiffiffiffiffiffiffiffiffiffiffiffi ffi > > > þ blsfe ½ble  lnðu þ u2 þ 1Þ; > > > pffiffiffiffiffiffiffiffiffiffiffiffiffi : ble P lnðu þ u2 þ 1Þ; where u is the interfacial parameter given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2T þ 4sf ðsd  sf Þ  sT u¼ . 2sf

ðA5Þ

References [1] Shiriajeva GV, Andreevskaya GD. Method of determination of the adhesion of resins to the surface of glass fibers. Plast Massy (Polymer Compounds USSR) 1962(4):43–6 [in Russian]. [2] Penn LS, Bowler ER. A new approach to surface energy characterization for adhesive performance prediction. Surf Interf Anal 1981;3:161–4. [3] Miller B, Muri P, Rebenfeld L. A microbond method for determination of the shear strength of a fiber–resin interface. Compos Sci Technol 1987;28:17–32. [4] De´sarmot G, Favre JP. Advances in pull-out testing and data analysis. Compos Sci Technol 1991;42:151–87. [5] Gorbatkina YA. Adhesive strength of fiber–polymer systems. New York: Ellis Horwood; 1992. [6] Gorbatkina YA. Adhesive strength of fiber–polymer systems. Moscow: Khimia; 1987 [in Russian]. [7] Zhandarov S, Ma¨der E. Characterization of fiber/matrix interface strength: applicability of different tests, approaches and parameters. Compos Sci Technol 2005;65:149–60.

2627

[8] Tandon GP, Pagano NJ. Micromechanical analysis of the fiber pushout and re-push test. Compos Sci Technol 1998;58:1709–25. [9] Kerans RJ, Parthasarathy TA. Theoretical analysis of the fiber pullout and pushout tests. J Am Ceram Soc 1991;74:1585–96. [10] Piggott MR. The interface in carbon fibre composites. Carbon 1989;27:657–62. [11] Piggott MR. Why interface testing by single-fibre methods can be misleading. Compos Sci Technol 1997;57:965–74. [12] Nairn JA. Analytical fracture mechanics analysis of the pull-out test including the effects of friction and thermal stresses. Adv Compos Lett 2000;9:373–83. [13] Ja¨rvela¨ P, Laitinen KW, Purola J, To¨rma¨la¨ P. The three-fibre method for measuring glass fibre to resin bond strength. Int J Adhes Adhes 1983;3:141–7. [14] Zhandarov SF, Pisanova EV. The local bond strength and its determination by the fragmentation and pull-out tests. Compos Sci Technol 1997;57:957–64. [15] Wagner HD, Gallis HE, Wiesel E. Study of the interface in Kevlar 49/ epoxy composites by means of the microbond and the fragmentation tests: effects of materials and testing variables. J Mater Sci 1993;28:2238–44. [16] Pitkethly MJ, Favre JP, Gaur U, Jakubowski J, Mudrich SF, Caldwell DL, et al. A robin-round programme on interfacial test methods. Compos Sci Technol 1993;48:205–14. [17] Tamuzˇs V, Apinis R, Vilks U, Tepfers R. Bond between glass fiber composite reinforcement and concrete in pull-out tests. Goteborg: Chalmers Univ. of Technology; Work No. 7, 1995. [18] Zhandarov SF, Ma¨der E, Yurkevich OR. Indirect estimation of fiber/ polymer bond strength and interfacial friction from maximum load values recorded in the microbond and pull-out tests. Part I: Local bond strength. J Adhes Sci Technol 2002;16:1171–200. [19] Hoecker F, Friedrich K, Blumberg H, Karger-Kocsis J. Effects of fiber/matrix adhesion on off-axis mechanical response in carbon-fiber/ epoxy-resin composites. Compos Sci Technol 1995;54:317–27. [20] Piggott MR, Reboredo MM. The durability of Kevlar-epoxy interfaces. Compos Interf 1994;2:457–72. [21] Jacobasch HJ, Grundke K, Uhlmann P, Simon F, Ma¨der E. Comparison of surface-chemical methods for characterizing carbon fiber-epoxy resin composites. Compos Interf 1996;3:293–320. [22] Sanchez M, De´sarmot G. In: Bathias C, Menke`s D, editors. 5e´mes Journe´es Nat. Comp. JNC-5. Paris: Pluralis; 1986. p. 290–3. [23] Auvray MH, Che´neau-Henry P, Leroy FH, Favre JP. Pull-out testing of carbon/bismaleimide systems in the temperature range 20–250 C. Composites 1994;25:776–80. [24] Scheer RJ, Nairn JA. Variational mechanics analysis of the stresses and failure in microdrop debond specimens. Compos Eng 1992;2:641–54. [25] Pisanova E, Dutschk V, Lauke B. Work of adhesion and local bond strength in glass fibre-thermoplastic polymer systems. J Adhes Sci Technol 1998;12:305–22. [26] Zhandarov S, Pisanova E, Lauke B. Is there any contradiction between the stress and energy failure criteria in micromechanical tests? Part I Crack initiation: stress-controlled or energy-controlled? Compos Interf 1998;5:387–404. [27] Takeda N, Song DY, Nakata K, Shioya T. The effect of fiber surface treatment on the micro-fracture progress in glass fiber–Nylon 6 composites. Compos Interf 1994;2:143–55. [28] Piggott MR, Xiong Y. Fiber, matrix, and interface properties. In: Spragg CJ, Drzal LT, editors. ASTM STP, 1290. Philadelphia: American Society for Testing and Materials; 1996. p. 84–91. [29] Hampe A, Marotzke C. The energy release rate of the fiber/polymer matrix interface: measurement and theoretical analysis. J Reinf Plast Compos 1997;16:341–52. [30] Chua PS, Piggott MR. The glass fibre-polymer interface: III ? Pressure and coefficient of friction. Compos Sci Technol 1985;22:185–96. [31] Fu SY, Zhou BL, Chen X, Xu CF, He GH, Lung CW. Some further considerations of the theory of fibre debonding and pull-out from an

2628

[32]

[33]

[34]

[35]

[36]

[37]

[38] [39]

[40]

[41]

[42]

[43]

[44]

[45] [46]

[47]

[48] [49]

[50]

[51]

S. Zhandarov et al. / Composites Science and Technology 66 (2006) 2610–2628

elastic matrix. Part 1: Constant interfacial frictional shear stress. Composites 1993;24:5–11. Zhandarov S, Pisanova E, Ma¨der E. Is there any contradiction between the stress and energy failure criteria in micromechanical tests? Part II Crack propagation: effect of friction on force–displacement curves. Compos Interf 2000;7:149–75. Zhandarov S, Pisanova E, Schneider K. Fiber-stretching test: a new technique for characterizing the fiber-matrix interface using direct observation of crack initiation and propagation. J Adhes Sci Technol 2000;14:381–98. Liu CH, Nairn JA. Analytical fracture mechanics of the microbond test including the effects of friction and thermal stresses. Int J Adhes Adhes 1999;19:59–70. Pisanova EV, Zhandarov SF, Dovgyalo VA. Adhesive strength in thermoplastic polymer/thin fibre systems. Effect of the testing techniques. Mech Compos Mater 1993;29:232–9. Ma¨der E, Zhou XF, Pisanova E, Zhandarov S, Nutt SR. Characterisation of the interfacial bond strength between glass fibre and epoxy resin using the pull-out and push-out techniques. Adv Compos Lett 2000;9:203–9. Zhou XF, Wagner HD, Nutt SR. Interfacial behavior of polymer composites measured by push-out and fragmentation tests. Composites Part A 2001;32:1543–51. Fowkes FM. Adhesion and adsorption of polymers. In: Lee LH, editor. New York: Plenum Press; 1980. p. A43. Nardin M, Schultz J. Relationship between work of adhesion and equilibrium interatomic distance at the interface. Langmuir 1996;12: 4238–42. Pisanova E, Zhandarov S, Ma¨der E. How can adhesion be determined from micromechanical tests? Composites Part A 2001;32:425–34. Pisanova E, Zhandarov S. Reversible and irreversible adhesion of polymers: possibilities for measurement and calculation. J Adhes 2001;75:89–127. Zhandarov S, Ma¨der E. Increase in the molecular component of the adhesive pressure in polymer composites induced by electric charge. Compos Interf, in press. Drzal LT, Sugiura N, Hook D. The role of chemical bonding and surface topography in adhesion between carbon fibers and epoxy matrices. Compos Interf 1997;4:337–54. Pisanova E, Ma¨der E. Acid–base interactions and covalent bonding at a fiber-matrix interface: contribution to the work of adhesion and measured adhesion strength. J Adhes Sci Technol 2000;14:415–36. Tabor D, Winterton RHS. The direct measurement of normal and retarded van der Waals forces. Proc R Soc Lond A 1969;312:435–50. Israelachvili JN, Tabor D. The treatment of van der Waals dispersion forces in the range of 1.5 to 130 nm. Proc R Soc Lond A 1972;331: 19–38. Israelachvili JN, Adams GE. Measurement of forces between two mica surfaces in aqueous electrolyte solutions in the range 0–100 nm. J Chem Soc Faraday Trans I 1978;74:975–1001. Parker JL. Surface force measurements in surfactant systems. Prog Surf Sci 1994;47:205–71. Schmitt FJ, Ederth T, Weidenhammer P, Claesson P, Jacobasch HJ. Direct force measurements on bulk polystyrene using the bimorph surface forces apparatus. J Adhes Sci Technol 1999;13:79–96. Mangipudi V, Tirrell M, Pocius AV. Direct measurement of molecular level adhesion between poly(ethylene terephtalate) and polyethylene films: determination of surface and interfacial energies. J Adhes Sci Technol 1994;8:1251–70. Saint Flour C, Papirer E. Gas-solid chromatography: a method of measuring surface free energy characteristics of short glass fibers. 2.

[52]

[53] [54]

[55]

[56]

[57]

[58]

[59]

[60] [61]

[62] [63]

[64] [65]

[66] [67] [68] [69] [70]

[71]

[72]

Through retention volumes measured near zero surface coverage. Ind Eng Chem Prod Res Dev 1982;21:666–9. van Oss CJ, Good RJ, Chaudhury MK. The role of van der Waals forces and hydrogen bonds in ‘‘hydrophobic interactions’’ between biopolymers and low energy surfaces. J Colloid Interf Sci 1986;111:378–90. Della Volpe C, Siboni S. Some reflections on acid–base solid surface free energy theories. J Colloid Interf Sci 1997;195:121–36. Greszczuk LB. Theoretical studies of the mechanisms of the fibrematrix interface. In: Interfaces in composites. ASTM STP, 452. Philadelphia: American Society for Testing and Materials; 1969. p. 42–8. Scheer RJ, Nairn JA. A comparison of several fracture mechanics methods for measuring interfacial toughness with microbond tests. J Adhes 1995;53:45–68. Beckert W, Lauke B. Fracture mechanics finite element analysis of debonding crack extension for a single-fibre pull-out specimen. J Mater Sci Lett 1995;14:333–6. Zhandarov S, Ma¨der E. Indirect estimation of fiber/polymer bond strength and interfacial friction from maximum load values recorded in the microbond and pull-out tests. Part II: Critical energy release rate. J Adhes Sci Technol 2003;17:967–80. Zhandarov S, Pisanova E, Ma¨der E. Is there any contradiction between the stress and energy failure criteria in micromechanical tests? Part III. Experimental observation of crack propagation in the microbond test. J Adhes Sci Technol 2005;19:679–704. Zhandarov S, Pisanova E, Ma¨der E, Nairn JA. Investigation of load transfer between the fiber and the matrix in pull-out tests with fibers having different diameters. J Adhes Sci Technol 2001;15: 205–22. Piggott MR. A new model for interface failure in fibre-reinforced polymers. Compos Sci Technol 1995;55:269–76. Marotzke C. Influence of the fiber length on the stress transfer from glass and carbon fibers into a thermoplastic matrix in the pull-out test. Compos Interf 1993;1:153–66. Beckert W, Lauke B. Finite element calculation of energy release rate for single-fibre pull-out test. Computat Mater Sci 1996;5:1–11. Zhandarov S, Ma¨der E. Peak force as function of the embedded length in pull-out and microbond tests: effect of specimen geometry. J Adhes Sci Technol 2005;19:817–55. Nayfeh AH. Thermomechanically induced interfacial stresses in fibrous composites. Fibre Sci Technol 1977;10:195–209. Pisanova E, Zhandarov S, Ma¨der E, Ahmad I, Young RJ. Three techniques of interfacial bond strength estimation from direct observation of crack initiation and propagation in polymer-fibre systems. Composites Part A 2001;32:435–43. Leung CKY. Fracture-based two-way debonding model for discontinuous fibers in an elastic matrix. J Eng Mech 1992;118:2298–318. Stang H, Shah SP. Failure of fiber reinforced composites by pull-out fracture. J Mater Sci 1986;21:953–8. Nairn JA. On the use of shear-lag methods for analysis of stress transfer in unidirectional composites. J Appl Mech 1997;64:804–10. Wolfram S. The Mathematica book. 4th ed. Cambridge: Cambridge University Press; 1999. Gorbatkina YA, Ivanova-Mumjieva VG. Variance of adhesive strength and fracture mechanism of fibre/matrix joints. Int J Adhes Adhes 2001;21:41–8. Gorbatkina YA, Ivanova-Mumjieva VG. Adhesive strength of fibre/ polymer joints upon loading in liquids: effect of liquid surface tension. Int J Adhes Adhes 1997;17:329–32. Nairn JA. On the use of shear-lag methods for analysis of stress transfer in unidirectional composites. Mech Mater 1997;26:63–80.