Engineered Interfaces in Fiber Reinforced Composites Part 6 ppt

iii In the extreme case of z,,, value approaching zero, as in some ceramic matrix composites, the debond process is always stable until complete debonding independent of embedded fiber

134

(mm)

Fig 4.24 Plot of partial debond stress, uz, as a function of debond length, e, for untreated Sic fiber-glass

matrix composite After Kim et al (1991)

In light of the foregoing discussion concerning the functional partitioning of the partial debond stress, the characteristic debond stresses can be evaluated The initial debond stress, ao, is obtained for an infinitesimal debond length where the frictional

stress component is zero, i.e.,

(4.103)

In Eq (4.103), it is assumed that the influence of the instantaneous fiber

displacement relative to the matrix due to the sudden load drop after instability is negligible

Chapter 4 Micromechanics of sfi-ess transfer 135

4.3.4 Instability of debond process

The instability condition requires that the derivative of the partial debond stress with respect to the remaining bond length (z = L - e) is equal to or less than zero, i.e., do$'dzdO (Kim et al., 1991) Therefore, the fiber debond process becomes

unstable if ( L - C) is smaller than a critical bond length, z,,,, where the slopes of the curves become zero in Figs 4.23 and 4.24 At these bond lengths, the partial debond

stress, a : corresponds to the maximum debond stress, CT; The zmax value is

determined from Eq (4.102) as

the fiber surface treatments and when compared to other epoxy matrix based composites

To show clearly how and to what extent the parameter, zmax, varies with the properties of the interface and the composite constituents, a simple fiber pull-out model by Karbhari and Wilkins (1990) is chosen here This model is developed based on the assumption of a constant friction shear stress, zfr, in the context of the shear strength criterion for interface debonding In this model, the partial debond stress may be written as

where the frictionless debond stress, (TO, is given by

(4.105)

(4.106)

Eq (4.106) is essentially similar to the solution of the debond stress derived earlier

by Takaku and Arridge (1973) The above instability condition for the partial debond stress of Eq (4.105) gives a rather simple equation for zmax as

where p4 is a complex function of o! and y, and is given by

(4.107)

(4.108)

136 Engineered interfaces in jiber reinforced composites

whose approximate solution for b >> a is identical to /3, given in Eq (4.3) Eq (4.107) suggests that the ratio of the bond strength at the bonded region to that at the debonded region, q,/Zfr, and the Young’s modulus ratio, CL = E m / & , are key material properties that determine zmax and thus control the stability of the debond

process It should be noted here that in the early work of Lawrence (1972), Laws

et al (1973) and later Gopalaratnam and Shah (1987) the maximum debond stress is found to be dependent on these properties Eq (4.107) has a limiting value zmax = 0 when q, N zfr and y N 0 in which the debond process becomes totally stable as in some ceramic matrix composites (e.g S i c fiber-glass matrix composites (Butler

et al., 1990))

From the discussion presented above, it is clear that the stability of the debond process can be evaluated by a single parameter, zmax, which is the shortest (remaining) bond length needed to maintain the debond process stable, and is a constant for a given composite system Therefore, three different interface debond processes are identified in the following: totally unstable, partially stable and totally stable debond processes The schematic plots of the applied stress versus displacement curves are illustrated in Fig 4.25 for these debond processes

(i) If L <zmax, the debond process is totally unstable and the initial debond leads immediately to complete debonding (i.e GO = ni) Therefore, the corresponding stress-displacement curve shows a monotonic increase in stress until debonding is initiated, followed by an instantaneous load drop (Fig 4.25(a)) Totally unstable debonding may also occur when the frictional resistance in the debonded region is negligible (i.e either zero residual clamping stress, 40, or negligible coefficient of

friction p) such that zmaX approaches an infinite value as can be envisaged from Eq (4.107) However, this situation seems most unlikely to occur in practical composites

(ii) If L > z,,,, which is the most common case where practical fiber pull-out tests are performed, the stress increases linearly until debond initiates Then, the debond crack propagates in a macroscopically stable manner, leading to a non-linear increase in the debond stress, though ‘stick-slips’ are normally observed in the rising stress-displacement curve (Fig 4.25(b)) Stable debonding proceeds until the

Fig 4.25 Schematic presentations of applied stress versus displacement (0-6) relationship in fiber pull- out test: (a) totally unstable, (b) partially stable and (c) totally stable debond processes After Kim et al

(1992)

Chapter 4 Micromechanics trunsfer 137 debond length reaches e = L - z,,,, followed by unstable debonding leading to complete debonding Therefore, this debond process is partially stable

(iii) In the extreme case of z,,, value approaching zero, as in some ceramic matrix composites, the debond process is always stable until complete debonding independent of embedded fiber length, L The rising portion of the debond stress versus displacement curve (Fig 4.25(c)) is typically linear without apparent ‘stick- slips’ and there is no appreciable load drop after complete debonding (Bright et al., 1991) This is because the interface is in principle frictionally bonded and there is little chemical bonding That is Gi,, or Tb is very small Therefore, the linear increase

in stress represents primarily the frictional shear stress transfer across the interface without virtual debonding until the frictional resistance over the entire embedded fiber length is overcome The maximum debond stress, cri, is then approximately equal to the initial frictional pull-out stress, qr, because the frictionless debond

stress, op, is negligible (due to small Gi, or Q,)

The concept of z,,, with regard to the issue of the stability of the debond process has practical implications for real composites reinforced with short fibers There is a minimum fiber length required to maintain stable debonding and thus to achieve maximum benefits of crack-tip bridging between fracture surfaces without the danger of catastrophic failure It should also be mentioned that in practical fiber pull-out experiments the stability for interface debonding deviates significantly from what has been discussed above, and is most often impaired by adverse testing conditions (e.g soft testing machine, long free fiber length, etc.) Therefore

debonding could become unstable even for L > z,,, and in composites with zmay = 0 Moreover, when L is very short, as is the normal case in the microdebond test, the precipitous load drop after complete debonding may be aggravated by the release of the strain energy stored in the stretched fiber The load drops to zero if the fiber is completely pulled out from the matrix Alternatively, if the fiber is regripped

by the clamping pressure exerted by the surrounding matrix material frictional pull- out of the fiber is possible to resume

Another important parameter related to the fiber length in the fiber pull-out test is the maximum embedded fiber length, L,,,, above which the fiber breaks instead of being completely debonded or pulled out L,,, value for a given composite system can be evaluated by equating 02 of Eq (4.102) to the fiber tensile strength, CJTS,

(which is measured on a gauge length identical to the embedded fiber length used in fiber pull-out test), Le.,

(4.109)

where (J[ is the crack tip debond stress determined for bond length z,,, = ( L ~ t )

L,,, values calculated for a constant fiber tensile strength CJTS = 4.8, 1.97 and 2.3 GPa for carbon fiber, steel fiber and S i c fiber, respectively, are included in Table 4.3 These predictions are approximately the same as the experimental L,,, values,

e.g., the predictions for L,,, = 49.3 and 23.4 mm compare with experimental values

L,,, = 5 1 .0 and 21.7 mm, respectively, for the untreated and acid treated S i c fibers

138 Engineered inlerfaces in Jiber reinforced composites

(Fig 4.28) It is worth noting that the L,,, value decreases significantly when the fiber surface is treated to improve the interfacial bonding (and thus the interface fracture toughness, Gic), e.g acid treated S i c fibers versus untreated fibers This observation is analogous to what is expected from the fiber fragmentation test of single fiber composites: the stronger the interface bond the shorter is the fiber fragment length at the critical stage (see Section 4.2)

4.3.5 Characterization of interface properties

Microcomposite tests including fiber pull-out tests are aimed at generating useful information regarding the interface quality in absolute terms, or at least in comparative terms between different composite systems In this regard, theoretical models should provide a systematic means for data reduction to determine the relevant properties with reasonable accuracy from the experimental results The data reduction scheme must not rely on the trial and error method Although there are several methods of micromechanical analysis available, little attempt in the past has been put into providing such a means in a unified format A systematic procedure is presented here to generate the fiber pull-out parameters and ultimately the relevant fiber-matrix interface properties

In single fiber pull-out experiments, the most useful data that are readily obtained from the load-deflection records are the maximum debond stress, 02, and the initial frictional pull-out stress, ofr, as a function of L If the debond process is carefully

monitored for a large embedded fiber length, L , the initial debond stress, 00, can also

be determined directly in the average sense, depending on the composite system Most important properties to be calculated are the fracture toughness, Gi,, at the

bonded region, and the coefficient of friction, p, and the residual clamping stress, 40,

at the debonded region, by evaluating the pull-out parameters o f , i and r ~ There are several steps to be followed for this purpose

(i) Firstly, ofr versus L data allow the initial slope at L = 0 to be determined based

where the difference between the stresses obtained immediately

the load instability is given by

Ao = o - ofr = {of + Tj[exp(-;lz,,,) - 11) exp[-A(L - zmax)]

(4.111) before and after

(4.112)

Chapter 4 Micromechanics of stress transfer 139

(iii) Thirdly, combining Eqs (4.1 10) and (4.11 1) allows 2 and 8 (and thus p and qo

from Eqs (4.23) and (4.24)) to be determined Alternatively, the asymptotic debond stress, 5, can be directly estimated at a long embedded length through linear regression analysis of the maximum debond stress, 0; Once ;2 and are known, Eq (4.102) may be used to evaluate the optimum value of Gi, (and also

for zmax) that would give the best fit to the 0; versus L experimental results In this

procedure theoretical values for the maximum debond stress, o : have to be obtained at instability Alternatively, data for the initial debond stress, G O , versus

L , if available from experiments, can be directly evaluated to determine Gi, based

on the debond criterion of Eq (4.99) for infinitesimal debond length Application

of this procedure to obtain Gic, 11 and 40 have been demonstrated in fiber pull-out for several fiber composites materials (Kim et al., 1992, Zhou et al., 1993) Having determined the relevant interface properties (Table 4.3), the maximal debond stress, a: and the initial frictional pull-out stress, ofr, are compared with experimental data in Figs 4.26-4.28 for three different composite systems of carbon fiber-epoxy matrix, steel fiber-epoxy matrix and S i c fiber-glass matrix In general, there is very good agreement between theories and experiments over the whole range

of the embedded fiber length, L, for all the composite systems considered A new

methodology has also been proposed recently by Zhou et al (1994) to determine

systematically the longest embedded fiber length for instability, zmax, without

iteration and curve fitting of Eq (4.102)

4.3.6 Multiple~fiber composite model

From the review of the theoretical studies of the fiber pull-out test as discussed in Section 4.3.1, it is identified that most micromechanics theories are developed based

on a shear-lag model of single fiber composites where the cylindrical surface of the

matrix is invariably assumed to be stress free Although this assumption is required

to obtain the final solutions in closed form for the stress distributions it often leads

to an unacceptably high applied stress required to initiate/propagate interface debonding when the radial dimension of the matrix is similar to that of the fiber (Le for a high fiber volume fraction, F), This in turn implies that the application of the conventional models to practical composites is limited to those with a very small Vi

where any effects of interactions between neighboring fibers are completely

neglected Therefore, a three-cylinder composite model is developed (Kim et al., 1994b) to simulate the response of practical composites of large vf and thus to accommodate the limitation of the shear-lag model of single fiber microcomposite test properly Both the micromechanics analysis and the FE method are employed

in parallel for fully bonded interface to validate the results obtained from each model

To analyze the stress transfer in the fiber pull-out test of a multiple fiber composite, the specimen is treated as a three-cylinder composite (Zhou and Mai,

1992) where a fiber is located at the center of a coaxial shell of the matrix, which, in turn, is surrounded by a trans-isotropic composite medium with an outer radius B ,

Fig 4.26 Comparisons between experiments and theory of (a) maximum debond stress, c$, and (b) initial

frictional pull-out stress for carbon fiber-epoxy matrix composites After Kim et al (1992)

as schematically illustrated in Fig 4.29 The radii of the fiber and matrix, a and b,

are related to the fiber volume fraction vf = a2/b2, which is the same as that of the composite medium When the fiber is subjected to an external stress, 0 , at the loaded end ( z = 0) while the matrix and composite medium are fixed at the embedded end

( z = L ) , stress transfers from the fiber to the matrix and in turn from the matrix to

the composite medium via the IFSSs, zi(a,z) and zi(b,z), respectively For the

cylindrical coordinates of the three-cylinder composite, the basic governing equations are essentially the same as those for the single fiber composite However, the equilibrium equations between the external and the internal stresses have to be modified to take into account the presence of the composite medium Eq (4.87) is

now replaced by:

where y , = b 2 / ( B 2 - b2), and B is the outer radius of the composite medium The

subscript c refers to the composite medium In addition to Eq (4.12) for the relationship between FAS and IFSS, equilibrium between IFSSs and MAS requires

142 Engineered interfaces in jiber reinforced composites

where y2 = a 2 / ( B 2 - a 2 ) The additional radial stress, q 2 ( a r z ) , acting at the fiber-

matrix interface, which is caused by Poisson contraction of the fiber when subjected

UYl

Chapter 4 Micromechanics of stress transfer I43

Fig 4.29 Schematic illustration of fiber pull-out test on a three cylinder composite After Kim et al

(l994b)

to an axial tension, is obtained from the continuity of tangential strain at the interface

(4.117)

where c q = E,/Ec and kl = 1 + 2y - v, + a1 (1 + 2 y l + vc) Eq (4.1 17) replaces

ql ( a , z ) given by Eq (4.18) applied for the single fiber composite model Combining

Eqs (4.12) and (4.1 13) to (4.117) yields a differential equation for the FAS

Therefore, the solutions for the FAS, MAS, MSS and IFSSs normalized with the

applied stress 0, are obtained:

@+ 1) s i n h [ f i ( L -z)] +%sinh(&z)

144 Engineered interfaces in fiber reinforced composites

to the partially embedded fiber at the surface (z = 0) The boundary conditions are imposed such that the bottom surfaces of the matrix and composite medium are fixed at z = 2L, and the axis of symmetry ( r = 0) is fixed where there is no displacement taking place

Specific results are calculated for S i c fiber-glass matrix composites with the

elastic constants given in Table 4.1 A constant embedded fiber length L = 2.0 mm, and constant radii a = 0.2mm and B = 2.0mm are considered with varying matrix

radius b The stress distributions along the axial direction shown in Fig 4.31 are

predicted based on micromechanics analysis, which are essentially similar to those obtained by FE analysis for the two extremes of fiber volume fraction, fi, shown in Fig 4.32 The corresponding FAS distribution calculated based on Eqs (4.90) and (4.120), and IFSS at the fiber-matrix interface of Eqs (4.93) and (4.132) are plotted

along the axial direction in Fig 4.32

The three-cylinder composite model predicts that both the FAS and IFSS decrease from a maximum near the loaded fiber end towards zero at the embedded fiber end Increase in f i (and the equivalent improvement of stiffness in the composite medium) increases slightly both the maximum IFSS and the stress gradient, without changing the general trend of the stress fields For small fi, stress distributions in the single fiber composite model are equivalent to those of the three- cylinder model In sharp contrast, the stress fields change drastically in the single fiber composite model when vf is large The FAS values in the central portion of the fiber are approximately constant and do not diminish to zero at the embedded fiber end More importantly, the IFSS displays two peaks at the ends of the fiber, the one

at the embedded end being increasingly greater than the other at the loaded end with

Chapter 4 Micromechanics of stress transfer 145

- Fiber

Fiber Matrix Composite medium

Fig 4.30 Schematic illustrations of the finite element models of (a) single fiber pull-out specimen and

(b) a three cylinder composite After Kim et al (1994b)

increasing 6 It is also interesting to note that the single fiber composite model predicts that the IFSS obtained at the loaded end remains almost constant regardless of 6

The pronounced effect of fiber 6 is further manifested in Figs 4.33 and 4.34,

where the characteristic IFSS values obtained at the ends of the fiber are plotted as a function of 6 for the micromechanics and FE analyses, respectively It is clearly demonstrated for the three- cylinder model that these stresses vary only marginally with 6, and the magnitude of IFSS at the loaded end is always greater than that at the embedded fiber end This ensures that when the fiber is loaded continuously, debonding is always expected to initiate at the loaded fiber end for all 6, if the shear strength criterion is employed for the interface debonding However, for the single fiber composite model, IFSS at the embedded fiber end increases rapidly whereas

that obtained at the loaded fiber end either remains almost constant (Fig 4.33) or

decreases with increasing 6 (Fig 4.34) Therefore, there is a critical fiber volume

fraction above which the maximum IFSS at the embedded end exceeds that of the loaded end, allowing debond initiation from the embedded fiber end in preference to

146 Engineered interfaces in fiber reinforced composites

single fiber composite; (-) three cylinder composite model After Kim et al (1994b)

the loaded fiber end The critical fiber volume fractions vf M 0.15 and 0.26 are

estimated from the superimposed curves of the data points in Figs 4.33 and 4.34, respectively

One of the major differences between the results obtained from the micro- mechanics and FE analyses is the relative magnitude of the stress concentrations In particular, the maximum IFSS values at the loaded and embedded fiber ends tend to

be higher for the micromechanics analysis than for the FEA for a large vf This gives

a slightly lower critical vf required for the transition of debond initiation in the micromechanics model than in the FE model of single fiber composites All these

Fig 4.32 Distributions of (a) fiber axial stress and (b) interface shear stress along the axial direction

obtained from FEM calculations for two fiber volume fraction, V , = 0.03 and 0.6 Symbols a s in Fig 4.3 1

After Kim et al (1994b)

observations appear to be associated with the slightly different boundary conditions

used in these models

4.3.7 Two-M1ay debonding phenomenon

In the light of the discussion presented in Section 4.3.6, it is seen that the

surrounding composite medium in the three-cylinder composite model acts as a stiff

annulus to suppress the development of IFSS at the embedded fiber end by

constraining the radial boundary of the matrix cylinder This ensures that regardless

Fiber volume fraction, Vf

Fig 4.33 Interface shear stresses as a function of fiber volume fraction, 5, obtained from

micromechanics analysis Symbols as in Fig 4.31 After Kim et al (1994b)

of V, the maximal IFSS always occurs at the loaded fiber end where the interface

debond initiates and grows inward The maximum IFSS tends to increase slightly

with increasing 6, allowing debond initiation at a low external stress

In contrast, the single fiber composite model predicts that the IFSS concentration

becomes higher at the embedded end than at the loaded end if fiber vf is greater than

a critical value, suggesting the possibility of debond initiation at the embedded fiber