Methods and models for fibre–matrix interface characterisation in fibre-reinforced polymers: a review

Matrix cracking

The (co-)existence of interfacial debonding and transverse and/or bi-conical matrix cracks in SFFT affect the stress transfer to the fibre. The debonding occurrence can minimise the matrix crack initiation probability or the extent of its propagation [33]. The small matrix cracks are predominantly associated with the instant energy release due to a fibre break and do not significantly propagate during SFFT. The three common damage modes at the fibre fracture site, based on experimental observations [2,34,35] and interfacial adhesion level, are:

Mode

${\rm \alpha }$ α

Mode

${\rm \beta }$ β

: For a weak interface, the fibre breaks hardly damage the matrix and reveal an immediate widening of the breaking gap due to extensive debondings. Some of these breaking gaps can even become larger than 5

$d_f$ d f

with additional loading [38] (Figure 3b). This failure mode has been detected in GF and unsized CF model composites [35,39]. A characteristic feature of this mode is that a portion of the fibre load-bearing capability is sustained via the frictional stress transfer across the debonded interface.

Mode

${\rm \gamma }$ γ

: A conical matrix crack initiates at the fibre end and propagates into the matrix at an angle with respect to the fibre axis (see Figure 3(c), with 45° or 21° as in [2]). This failure mode is probable in high-modulus CF-epoxy systems [2,40] and in a matrix material that exhibits a lower shear strength relative to its tensile strength.

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Figure 3.

Three types of fracture in the matrix in SFFTs: (a) a disk-shaped matrix crack indicating a strong interfacial bond, (b) fibre–matrix debonding which implies a rather weak interface, (c) double-cone matrix crack referring to a matrix with lower shear strength than the tensile strength. Note that crack (c) becomes nearly normal to the fibre axis as it propagates (redrawn from [37]).

Fibre surface treatment and sizing

There are two primary methods for achieving an optimal interfacial bond: treating the surface of the fibre/reinforcement or modifying the composition of the matrix (see Figure 2). However, it is important to note that the objective is not always to maximise the bond strength. In brittle matrix composites, an excessively strong bond could lead to embrittlement.

The interfacial bond strength significantly relies on the surface treatments applied to the fibre. Size (also known as fibre finish or coating) is a protective coating applied to the fibre surface that improves the fibre handling during processing and enhances the interfacial adhesion. The process of coating fibres with size is termed sizing. Nonetheless, many authors use the sizing terminology instead of size, which can often save confusion in ‘fibre size’ not relating to the ‘fibre dimension’ [41]. Fibre sizing is arguably the most crucial element involved in GFs and GFRPs manufacturing and their optimal functioning. Yet due to the high level of secrecy surrounding size formulations, only a limited number of individuals within the extensive supply chain of FRPs suppliers, processors and end-users have a profound grasp of GF sizings. An inquisitive reader may refer to a recent review article by Thomason [41] that provides an in-depth analysis of GF sizing. This article explores various aspects, such as sizing formulations, their impact on the performance of both the fibre and the composite, as well as the challenges and advancements in GF sizing technology.

SEM and X-ray photoelectron spectroscopy can be utilised to characterise the surface morphology (or roughness) and to examine the chemical structure of the fibre surface, respectively [42]. The IFSS enhancement can be directly related to the number of chemical functional groups on the fibre surface [43] or micromechanical interlocking (Figure 1f) due to surface roughness [42]. Note that the CF structure has distinct ‘skin-core’ characteristics. The graphite crystallites on the CF skin layer are in a compact arrangement and lack active carbon atoms. This results in low surface energy and inadequate functional groups for chemical reactions [44]. Based on experimental evaluations on a CF-epoxy system [45,46], the IFSS is influenced chiefly by the local morphology of the outermost graphite surface layers and the number of active sites. Apart from sizing [47,48], various methods have been developed to modify the fibre surface activity, roughness and wettability and hence improve the interfacial adhesion. For CFs, these treatments include electrochemical [49] or wet chemical oxidation (with aqueous ammonia) [50], gas-phase oxidation in ozone or oxygen mixtures [51],

${\rm \gamma }$ γ

Oxidative surface treatment chemically activates the surface layer of CFs (see Figure 1d), increasing the interaction with the matrix [58]. Plasma treatment increases both the surface energy and roughness of the fibre to improve the wettability of the fibre by the matrix and the micromechanical interlocking (see Figure 1f) [59]. The fibre sizing, succeeding surface treatment, creates a brittle interface/interphase layer which increases the IFSS but could potentially alter the failure mode from interfacial debonding to matrix cracking (see Figure 2) [60]. The grafting of CNTs onto the CFs enhanced the IFSS between the CNT-CF hybrids and the epoxy resin in [56,61]. Contrarily, a lower IFSS with CNT-grafting was achieved in [62], which was linked to the smaller size and lower density of the grown CNTs, lowering the contribution of fibre surface roughness. Moreover, CNTs were more hydrophobic than the unmodified CFs, resulting in poorer wettability (larger contact angle) with the epoxy resin [63]. The advantage of CNT-grafting chiefly relies on the tensile strength of CNTs and their interfacial cohesion with the matrix. The utmost enhancement is acquired when the matrix is allowed to absorb energy through microscale damage, and the CNT debonding at the nanoscale has only a partial contribution to it [64]. According to Godara et al. [65], the most efficient strategy for enhancing the IFSS is achieved by exclusively incorporating CNTs in fibre sizing [65,66].

Graphene oxide and other 2D materials have gained attention due to their desirable properties, including high specific area, flexibility and good mechanical, thermal, and electrical conductivity [67,68]. Graphene oxide exhibits higher chemical reactivity with both the fibre and epoxy resin compared to graphene. This is attributed to the presence of epoxide and hydroxyl groups within the graphene oxide sheets, as well as carbonyl and carboxyl groups at their edges [69]. This property enables improved mechanical interlocking at the fibre–matrix interface (see Figure 1f), making fibre surface modification with graphene oxide a viable strategy [9,70]. A similar approach, treating the CF surface with polydopamine and graphene oxide, yielded enhanced IFSSs in [9].

Both short- and long-term FRP performances are critically influenced by the optimisation of fibre sizing [41]. Significant reductions in glass-fibre-reinforced polymers (GFRPs) performance were directly correlated with a loss of fibre–matrix adhesion caused by thermal degradation of some of the primary sizing components in [71]. Reportedly [39], the interfacial failure of their sized system consisted mainly of mixed-mode cracking, whereas clear fibre–matrix debonding was observed in the unsized system. Reportedly and based on LRS and SEM observations [39], sized CF-epoxy systems can withstand higher interfacial shear stresses than unsized ones. However, this might lead to a mixed-mode interfacial failure which is undesirable since only a portion of the crack faces are in full contact, and, therefore, the effective length of the fibre reduces heavily. On the contrary, a clear mode II debonding was observed for the unsized systems, in which the radial compression, which was developed throughout the curing, effectively kept the interface closed [39].

As a plausible scenario in UD FRPs, an improved surface treatment could give rise to shorter debonded and ineffective lengths, potentially leading to the development of transverse matrix cracks. The transverse matrix crack results in an increased stress/strain concentration factor (SCF) on the neighbouring fibres, causing brittle failure of the FRP. Consequently, it was shown that the tensile strength of a UD CF-epoxy FRP was maximised when 50% of the standard commercial fibre surface treatment was used [72]. Hoecker and Karger-Kocsis [73] studied the effect of the interface on the transverse and impact properties of the CF-epoxy FRPs. The effect of the adhesion quality was observed in a transverse tensile test, even though the loading direction in this test is very different from the SFFT. The improvement in properties was attributed to differing mechanisms of failure, which changed from a clean fibre surface fracture to a cohesive matrix failure (similar to Figure 2c) as the interfacial adhesion improved. The relevance of the interface to the transverse flexural properties was not observed. Furthermore, short beam shear tests did not show the effect of the adhesion, while the transverse Iosipescu test shows improved shear properties for the composite with the improved adhesion, as suggested by the micromechanical tests. The Charpy impact resistance of the UD laminates were a complex function of the interfacial adhesion. An improvement in impact properties of an FRP with a ‘good’ interface (as predicted by the micromechanical tests) was observed, which is contrary to the popular belief that improved adhesion will cause brittle failure of the composite [73]. Moreover, the effect of interfacial adhesion can be substantial in the case of compression-moulded or injection-moulded short-fibre composites since a major portion of the fibres will not be aligned with the loading direction [74].

Based on the above discussion, it can be concluded that the macromechanical properties are a complex function of the interface properties as measured from the micromechanical tests, in particular, the SFFT. A clear understanding of the translation of the micromechanical results to the macromechanical properties is still being developed. A proper consideration of the loading scenario of the ultimate composite application is therefore indispensable for the design of the optimal interfaces for the improved mechanical properties of the FRPs [74].

Curing cycle

The type of curing cycle utilised can impact interfacial properties. Cross-linking occurs during the curing of thermosets, and there may be various alterations, such as changes in elastic modulus, thermal expansion coefficients, expansion paired with contraction, and increases/decreases in toughness and plasticity. With multiple variables throughout the curing process, it is challenging to fully understand how curing schedules alone affect adhesion without considering factors such as the resin's degree of cure or crosslinking [75].

Ideally, the curing cycle for microcomposites should align with that of conventional FRPs. Not only do cure pathways influence the interfacial properties, but the cure paths required to fabricate the specimens for each of the testing methods is fundamentally different to those recommended by manufacturers for making FRP laminates due to the different geometries and small amounts of materials used. Furthermore, even if the exact same cure cycle were used for SFFT (or other) specimens, the resulting microstructure would likely be different due to thermal effects, heat transfer limitations and material volume-to-surface area ratio. These factors, in turn, impact the cure kinetics and environmental factors at play. The higher the temperature of post-curing, the higher the

$T_g$ T g

ElKhoury and Berg [75], for CF-epoxy SFFT specimens, investigated the effects of a number of curing cycles, curing temperature and schedule, degree of cure, use of accelerants, annealing and the use of fibre handling agents. To achieve the highest apparent adhesion, curing at the highest temperature in a single stage is the most effective method. The addition of accelerants can speed up the curing process without affecting the level of adhesion, but it may reduce the plastic yield strength in some cases. Annealing of thermoset composites can reduce both the induced residual stress and the apparent adhesion but cannot lower it below the level attained at lower curing temperatures. For thermoplastic composites, Wang et al. [80] indicated that post-process annealing could significantly affect apparent adhesion due to the development of internal stress. This suggests that the thermal history of polymers must be considered during the design process.

Data reduction schemes

Cottrell–Kelly–Tyson model

Cottrell (1964) [81], Kelly and Tyson (1965) [82] (CKT), concurrently but independently, used a simple force balance to calculate the average IFSS. They assumed that the matrix is perfectly plastic, and the interfacial shear stress is constant and equal to the shear yield strength of the matrix,

$\tau _y$ τ y

, in the recovery region. This results in a linear axial stress build-up from 0 at the fibre fragment end to the applied stress level farther from the fibre end, where it reaches a plateau. At saturation, the fragments are too short to reach the applied stress, and thus the plateaus disappear (see Figure 4). In this model (also known as the simple shear-sliding model), the interface bears any shear stress up to the IFSS, after which the deformation occurs through interfacial sliding. This model assumes that all the stress recovery in the broken fibre occurs through shear sliding. The primary shortcoming or uncertain presumption is that the interfacial shear stress is constant over the fibre length. Additionally, real-world materials are seldom perfectly plastic.

Due to the assumption of constant interfacial shear stress in the CKT approach, this scheme is not suitable for fibre-polymer systems, in which friction is not the primary mechanism for axial stress transfer. However, for ceramic-/metal–matrix composites, with absent interfacial chemical bonding, the CKT reduction scheme may provide a reasonable estimate of the interfacial yield stress. This, in principle, is due to the underdeveloped constitutive models. The response of metallic materials to applied loads can be described with greater precision compared to that of thermoset matrices. This is due to the fact that the mechanical behaviour of amorphous thermoset matrices possesses a degree of atomic-level long-range order [6].

The critical length of the fibre,

$l_c$ l c

, the length below which the fibre cannot fragment any further due to insufficient load transferred to it via the interface, is calculated as

$$l_c = \displaystyle{{d_f\sigma _f\lpar {l_c} \rpar } \over {2\tau _y}}$$ l c = d f σ f ( l c ) 2 τ y

(1)

with

$d_f$ d f

denoting the fibre diameter and

$\sigma _f$ σ f

being the fibre failure strength at

$l_c$ l c

. Based on this model, the

$l_c$ l c

is often considered to be a material constant and independent of the applied strain [87,88]. Figure 5 depicts the effect of fibre length on the axial and shear stress profile along the fibre. The fragment lengths are measured conventionally either directly by transmission optical microscopy [58] or indirectly via acoustic emission (AE) [89,90]. The modified versions of this relation (such as in [91]) vary by the definition of the critical length or the incorporation of the length-dependence of

$\sigma _f$ σ f

. The

$l_c\;$ l c

can be:

the same as the arithmetic mean of the fragment lengths at saturation,

$l_c = l_{{\rm avg}}$ l c = l a v g

[92,93],

the widely used definition of

$l_c = {\rm 4/3}\; l_{{\rm avg}}$ l c = 4 / 3 l a v g

(also called the corrected length) [94],

the maximum fragment length,

$l_c = l_{{\rm max}}$ l c = l m a x

[92,93] or

equal to the fibre fragment length at which the maximum fibre axial stress coincides with its nominal strength from the single-fibre tensile tests [91].

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Figure 5.

Effect of fibre length on axial and shear stress profile along the fibre according to the CKT model.

$E_{\,f1}\sigma _{c1}/E_1$ E f 1 σ c 1 / E 1

is the fibre stress in a continuous fibre composite under longitudinal composite stress,

$\sigma _{c1}$ σ c 1

, with the longitudinal modulus of

$E_1$ E 1

.

The choice of corrected length is based on the premise that the lengths of the fibre fragments from SFFT data fall within the range of

$l_c/2$ l c / 2

to

$l_c$ l c

. However, it has been documented that the distribution of fragment lengths does not conform to this range, as evidenced by studies such as [95–97]. Drzal et al. [97] utilised a two-parameter Weibull distribution, discussed in [95], to model the fragment length distribution during the saturation stage. Therefore, the IFSS values obtained using any traditional approaches for the critical length (options (1)–(3)) are of relatively low importance, as emphasised in [95]. The ‘nonlinear stress transfer model’ [91] warns against using

$l_c$ l c

alone to determine IFSS. Instead, it suggests that SFFT data analysis should be used with caution, taking into account both the fragment lengths and test specimen loading to avoid overlooking possible changes in fibre strength. This approach will help prevent potential changes in fibre strength and ensure accurate results [91].

It should be pointed out that the CKT model solely considers the load transfer that transpires within the plastic zone at the fibre ends, disregarding the effects of elastic load transfer in the central portion of the fibre (

$\tau _i = 0$ τ i = 0

). This approach remains valid under the assumption that the plasticity threshold of the matrix (or the interfacial sliding threshold) is minimal. Accordingly, plastic deformation (or slippage) occurs at the onset of loading, without an elastic load phase.

Meeting the saturation condition is primarily regulated by the yield strain of the matrix rather than the fibre failure strain [98]. This indicates the necessity of a matrix system with a higher failure strain than the embedded fibre. A typical recommendation is for the matrix failure strain to exceed the fibre failure strain by at least three times [25]. Alternatively, the coaxial geometry fragmentation test (bimatrix SFFT) can be used to extend the applicability to brittle resin systems that cannot attain saturation. This intricate method applies a thin coating of the brittle resin of interest to the fibre surface before being enclosed in a tough resin [99,100].

Regarding the standardisation and round-robin efforts, an inter-laboratory round-robin on the SFFT was accomplished with the participation of seven laboratories [101] (similar to [25]). To conduct the test with minimum operator dependence, a set of protocols is essential. The principal steps proposed by the round-robin were:

Measure

$d_f$ d f

at five different locations in the gage section with an accuracy of at least ±1

${\rm \mu m}$ μ m

Place fiduciary marks on the specimen to facilitate strain measurements

Increment coupon strain by 0.2%, hold for 8

${\rm min}$ m i n

before counting fibre breaks

After a 10-

${\rm min}$ m i n

interval from the completion of the previous strain increment, apply the subsequent strain step and repeat till saturation

Document the images of the fibre under polarised light at saturation

Measure the fragments lengths pre-unloading.

The CKT approach of measuring an average IFSS from the SFFT is an oversimplification that can only be justified in full debonding or full matrix-yielding cases. The search for a better data reduction scheme resulted in partially-elastic models that divided the interface into inelastic and elastic regions and disconfirmed the CKT critical length as a strain-independent material constant [45,88]. Moreover, the constant interfacial shear stress assumption was discredited by experimental observations [102]. Further experiments invalidated the intrinsic assumptions of the CKT model, such as meeting the perfect saturation state [103], its inability to differentiate bonded/debonded surfaces and disregarding the effect of matrix plasticity and interfacial friction [104]. Recently, modified versions of CKT and Bader–Bowyer (derived from the original CKT model) [105,106] models were proposed [107]. These versions calculate the IFSS for short (aspect ratio

$\lt 20$ < 20

) and ultra-short (aspect ratio

$\lt 10$ < 10

) fibre composites and consider the end effects that were excluded in the original model.

Cox (shear-lag) model

The shear-lag approach, developed by Cox (1952) [108], assumes that the matrix in a polymer composite is exceedingly compliant compared to the fibre. The matrix, therefore, does not contribute significantly to the tensile stiffness of the composite. Accordingly, the acting longitudinal loads in the matrix can be neglected, simplifying the calculation of fibre stresses. The matrix is exclusively responsible for transferring shear loads occurring at the fibre ends of a discontinuous fibre or a fibre break site. These shear loads occur due to the mismatch in longitudinal strain between the fibre ends and the surrounding material. To generate an analytical formulation, a matrix volume around the fibre, whereupon the shear stresses act, is required. This volume commonly takes the form of an arbitrarily drawn cylinder with the radius

$R_m$ R m

around the fibre with a radius

$r_f$ r f

. This model still forms the basis for many recent models. The main assumptions of these shear-lag-based stress transfer models (1D analytical models) are linear-elastic isotropic and well-bonded constituents, non-interacting fibre fragments and the absence of residual stresses and plasticity.

Lacroix et al. [109] used a unimodal Weibull distribution for the fibre strength to predict the critical length in SFFTs. They compared the results for fully-elastic (perfect adhesion), partially-elastic and total-debonding 1D shear-lag-based models. The fully-elastic model predicts that the critical length decreases as the applied strain increases, while the total-debonding model considers it a material constant (assuming frictional sliding stress to be strain-independent). The more realistic partial-debonding model predicts lower values of critical length compared to the overestimated lengths by the CKT model, underlining the significance of considering both IFSS and frictional sliding in processing the SFFT data [109].

The Cox-based models, limited by the introduction of the effective matrix radius,

$R_m$ R m

(whose physical existence has not been defined clearly in the case of single-fibre composites), and disregarding hoop and radial stresses, were more qualitative than quantitative [110]. Nairn [111] comprehensively examined the potential of the shear-lag methods (SLMs). The shear-lag inaccuracy grows as

$E_f/E_m$ E f / E m

decreases or at low fibre volume fractions,

$V_f$ V f

(approaching the case of a fibre embedded in an infinite matrix). The shear-lag parameter,

$\beta$ β

, employed in various shear-lag analyses of composites (see Appendix Table A1), depends on the material properties and the geometry of the problem. To have a reasonable agreement with FEM predictions, the erroneous

$\beta$ β

suggested by Cox,

$\beta _{{\rm Cox}}$ β C o x

, must be replaced by the

$\beta$ β

's proposed notably by Nairn [111], Nayfeh [112] or McCartney [113] for two concentric cylinders case. Nairn [114] extended his optimal shear-lag method (for perfect interfaces) [111] to a generalised analysis to include imperfect interfaces. Nairn's analysis also handles any fibre volume fractions and anisotropy levels of the fibres. Figure 6 compares the CKT and Cox model predictions for the axial stress recovery of a broken fibre.OPEN IN VIEWER

Figure 6.

Predicted stress recovery profiles by CKT and Cox models (adapted from [115]).

Following the initial shear-lag analysis, the extended SLMs include models with a matrix carrying tensile stress [116], elastoplastic matrix [117,118], debonded fibre–matrix interfaces [119,120] and matrix with linear damage evolution along with interfacial slipping [121]. The shear-lag analyses have also been supplemented with fibre-strength statistical models [120,122,123] to predict composite failure. The polymeric matrix properties can also affect the measured interfacial properties. Based on Cox's [108] prediction (see Table A1), the IFSS increases with increasing matrix shear modulus,

$G_m$ G m

, and presumably elucidates why amorphous thermoplastics, typically possessing a lower

$G_m$ G m

than thermosetting matrices, have a weaker adhesion to CF [124]. In an elastoplastic SLM [125], the addition of interfacial debonding to matrix cracking slowed down the fibre axial stress build-up [126]. Having a perfectly plastic matrix rather than plastic deformation with hardening had a similar effect [126].

Other stress-based models

The 2D analytical models (based on axisymmetric micromechanics analyses) for SFFT were premiered by Whitney and Drzal [127] and were later refined by the variational mechanics’ approach by assuming the axial stress to be independent of the radial direction either in both constituents [128] or merely in the fibre [129]. The axisymmetric model of Whitney and Drzal [127] (relevant equations given in Table A1) was competent at estimating all the stress components (also for transversely isotropic fibres) and reforming the shear stress distribution of the SLMs. The duo employed a stress function method that ensured a valid solution, but it had two drawbacks. First, their stress function was not based on one that minimises the complementary energy. However, this solution showed good agreement with experimental data for high aspect ratios of fibre length over

$r_f$ r f

[110]. Second, their evaluation only considered an isolated fibre break, whereas in actual SFFTs, the fibre breaks can interact [128]. Consequently, to improve the limitations of SLM and the Whitney and Drzal approach, Nairn [128] presented a closed-form, 3D, variational mechanics-based axisymmetric solution for the stresses around breaks in an embedded single fibre (the pertinent equations are not presented here, but can be found in [128]). This model implemented thermal residual stresses and interacting fibre breaks but was only relevant for perfectly bonded fibres (no debond nor frictional effects). Thereafter, the matrix plasticity exclusion was resolved by the plasticity effect model of Tripathi et al. [130]. The variational model of Wu et al. [18] considered both a perfectly bonded region and a discontinuous interface of a two-phase composite. Johnson et al. [19] in their plasticity model, by addressing the shortcomings of the plasticity effect model (for instance, no radial stress variation from bonded to debonded regions), integrated matrix non-linearity into the model of Wu et al. [18]. Nevertheless, it could not predict the debond initiation/propagation and required a predefined debond length.

The location of the debond tip reveals itself as an inflection point in the fibre axial stress profile, with the debonded region having a linear stress build-up, while the rest displaying a Cox-type stress profile [35]. A key discrepancy between models is the shape of this transition region between the debonded and intact regions. Piggott (or similarly in [110]) claims that this shift is an unsmooth point of mathematical discontinuity, while Nath et al. [35] argue that the transition is smooth. This discrepancy emerges due to the linear elasticity assumption in Piggott's analysis, leading to stress singularities at the crack tip and a discontinuity in the stress recovery profile. Contrarily, the plastic zone at the crack tip in [35] creates a smooth transition in the fibre stress recovery profile. Figures 7 and 8, respectively, compare the axial stress recovery of the broken fibre and the interfacial shear stress profiles for the full-bonding and partial-debonding models.OPEN IN VIEWER

Figure 7.

Schematics of fibre stress distributions predicted by the full-bonding (Cox model [108]) and partial-debonding [20,131] models (inspired by [110] and redrawn from [132], with permission from Elsevier).

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Figure 8.

Interfacial shear stress as determined by Cox [108], Nairn [128] and partial-debonding models (redrawn from various figures in [110]).

To conclude, Cox [108] and CKT [81,82] models offer analytical solutions to the interfacial stress transfer problem. Cox's SLM, presuming a linear elastic behaviour for both constituents, enables the determination of the fibre axial stress and shear stress between the fibre and matrix. This model is only viable at low applied strains before the inherent non-linear behaviour of epoxy resins at higher strains. The CKT model uses a perfectly-plastic matrix and determines the critical length by assuming a constant interfacial shear stress. Table A1 in the Appendix elaborates on the major stress-based analytical models.

Assuming the perfect interfacial bonding and disregarding the existence of matrix axial stress can lead to inaccuracies in predicting the load transfer mechanisms and overall mechanical behaviour of FRPs using SLMs. Various solutions and alternatives to traditional SLMs have been proposed to address this issue, such as FE models and improved analytical models. For instance, Nardone and Prewo [133] considered the load transferred from the matrix to the end faces of the fibre, which was overlooked in Cox's original SLM that assumed a continuous or significantly long fibre compared to its

$d_f$ d f

. A modified SLM [134] accounted for the influence of fibre shape on the fibre tensile stress distribution. Other studies, such as [135,136], have developed modified SLMs with a cohesive interface that can incorporate the effect of various factors such as cohesive interfacial shear stiffness, elastic properties of the fibre and matrix, fibre aspect ratio and average fibre axial stress at the embedded end face on the stress-transfer characteristics and axial tensile properties of FRPs.

Energy-based approaches

The CKT model is not always accurate in calculating the IFSS when saturation is not attained. This drove the development of alternative data reduction methods, such as energy-based approaches [98]. Interfacial debonding is a major energy absorber during composite damage and failure [137]. An energy-based approach (fracture mechanics) is an eligible alternative to the traditional stress-based methods (strength of materials) for interpreting the SFFTs. The energy-based method was instigated using approximate Bessel–Fourier [138] or shear-lag-based (Cox-based [139] or Nayfeh-based [140]) analyses. With IFFT (

$G_{ic}$ G i c

) being a reliable parameter, the energy-based method allows the interfacial bond to be quantified even in SFFTs without saturation. Moreover, the accurate evaluation of the

$l_c$ l c

is crucial for the design of FRP since it affects the FRP's fracture toughness, impact resistance, notched strength and notch sensitivity. This property is relevant to loading along the fibre direction and can be used to measure the IFSS, which plays a role in transverse cracking in multidirectional laminates. However, some micromechanical models for transverse cracking have revealed that IFFT may be even more essential than IFSS [22,23].

In FRPs with relatively poor adhesion, the released energy during fibre breakage typically initiates a short interfacial debond at the vicinity of the break [141]. However, the debond remains arrested at a certain distance since the strain energy release rate (SERR) decreases as the debond grows. Owing to the larger Poisson's ratio and coefficient of thermal expansion (CTE) of the matrix compared to the fibre, the debond crack surfaces are continuously in contact. Thus the debond propagation is mode II-dominant [21,85,142] under tensile loading and below the curing temperature. Additional debond growth would require an increase in the applied strain [21]. Note that due to the brittle nature of the epoxy matrix, it is inclined to crack under tensile stress, which is the predominant mode of stress in mode I loading. The mentioned IFFT in the majority of the interface characterisation tests is more representative of a (near-) pure mode II cracking since the applied stress in mode II loading is parallel to the interface, which puts more stress on the interface itself rather than the matrix. However, in transverse cracking, both mode I and II components play a role.

Wagner et al. [143] use a 1D approach to determine the interfacial fracture energy caused by the initial debond. Their approach compares the energies before and after fibre and interface failure and considers the matrix contribution to the energy balance. The debonded region was assumed to be frictionless, and the residual stresses were neglected. However, the Boundary Element Method (BEM) [85] included the effect of these two key debond propagation suppressors. Using the solution of the BEM model (investigated for GF-epoxy), Graciani et al. [144] speculated that SERR evolves linearly with applied strain,

$\varepsilon$ ϵ

. In this manner, the interfacial friction coefficient,

$\mu _i$ μ i

, and

$G_{ic}^{{\rm II}}$ G i c I I

(the debond crack growth takes place when the SERR = 

$G_{ic}^{{\rm II}}$ G i c I I

) can be jointly determined by parametrical variation of

$\mu _i$ μ i

on SERR-

$\varepsilon$ ϵ

numerical solutions. Using the SFFT (GF-epoxy) data of Kim and Nairn [145], the BEM model led to

$\mu _i = 1 .0$ μ i = 1 .0

and

$G_{ic}^{{\rm II}} = 12 .12\, {\rm J}/{\rm m}^2$ G i c I I = 12 .12 J / m 2

[144], while Kim and Nairn model [145] (based on [146]) determined that

$\mu _i = 0 .01$ μ i = 0 .01

and

$G_{ic}^{{\rm II}} = 120\; {\rm J}/{\rm m}^2$ G i c I I = 120 J / m 2

.

It is peculiar that the two models identify widely different parameters despite using the same raw data, whereas both approaches incorporate Poisson's effects as well as frictional and residual stresses. In pursuit of a practical approach for more explicit identification of the interfacial parameters, Sørensen [147] developed a simple 1D SLM that determines the interfacial fracture energy and frictional sliding shear stress using the documented applied strain, debond length, and opening displacements of the fragment ends in the SFFTs. Using the potential energy approach of Budiansky et al. [148], Sørensen [147,149] considered friction and residual thermal stresses but dismissed the effect of matrix plasticity. Consequently, a more inclusive model that incorporates all relevant interfacial and material features is required to investigate the detailed stress transfer attributes of the constituents.

Numerical methods

Overly simplified perfectly-bonded or fully-debonded models disregard realistic debonding and pull-out mechanisms. The majority of the existing analytical interface models, such as Cox's SLM, assume a well-bonded condition and no interfacial friction (

$\mu _i = 0$ μ i = 0

). These assumptions lead to overestimated IFSS and IFFT values. A more accurate interfacial analysis asks for the addition of a third state between full-bonding and debonding states. The cohesive zone model (CZM) introduces partial-debonding as an intermediate state when stiffness degrades [150]. For the CZM, compared to the fracture toughness, it is less straightforward to determine damage onset and penalty stiffness (

$K_i$ K i

) parameters for FE modelling. The cohesive strength is difficult to determine from experiments and is usually assumed based on mesh size considerations [151–153] as a compromise between computational efficiency and numerical accuracy [154].

$K_i$ K i

is a numerical parameter employed to preserve the inter-ply connection before delamination initiation [155–157] and is vital in FE modelling [158]. Selecting an appropriate value for

$K_i$ K i

is essential to ensure that it does not significantly alter the overall compliance of the structure or lead to non-convergence issues [151]. However, a value within the acceptable range, which balances numerical accuracy and feasibility, has been found to affect numerical predictions [158,159], particularly under mixed-mode loading [160,161]. Figure 9 illustrates the traction-separation law, with several common softening laws, required to establish a CZM. Based on the stress concentration around a fibre break in an SFFT, Budiman et al. [150,163] linked the photoelastically observed matrix stress contours to the IFSS. Using an axisymmetric elastic FEM and iterating the simulation with various interfacial properties (

$\mu _i$ μ i

, IFSS, IFFT and

$K_i$ K i

) for the studied CF-epoxy system, the resulting stress contour can match the experimental stress contour [150]. The discrepancy between their FEM-predicted IFSS and IFFT values and the predictions of CKT and an energy-based model [164] was justified by disregarding debonding and frictional energies in the analytical models, respectively. The key benefit of their method is that only a single fibre break is required to evaluate IFSS. This reduces the matrix plasticity effect, whereas the CKT approach requires saturation [150,163].OPEN IN VIEWER

Figure 9.

Cohesive zone model for mode II fracture with various softening laws (inspired by [162]).

Based on an axisymmetric single-fibre FEM [150], the limiting radius into the matrix beyond which no stress transfers is reported to be at least 20 times the fibre radius (

$R_m/r_f \ge 20$ R m / r f ≥ 20

) to avoid the radius ratio effect. Different features and phenomena were considered in various numerical models. Varna et al. [165] considered the residual stresses in their axisymmetric FE model but adopted an elastic matrix. For a material system consisting of CF-epoxy, Nishikawa et al. [103] included matrix plasticity, debonding and matrix cracks but neglected the effect of thermal residual stresses and interfacial friction. Graciani et al. [85] used the BEM and included the interfacial friction and thermal residual stresses but neglected matrix plasticity. Recently, Van der Meer et al. [104] assessed the authenticity of the simplified shear-lag assumptions, including constant friction, non-interactive debond zones (no interaction between different fibre breaks) and neglected plasticity in a simulated GF-epoxy SFFT. Their numerical results revealed the absence of debonding interactions and the insignificance of the matrix plasticity in the debonding process. In addition, for an excessively non-linear matrix material, the fibre strain was found to lose its proportionality to the applied stress. Therefore, it is highly recommended that the

$l_d$ l d

is monitored as a function of the applied strain rather than the applied stress [104]. Reportedly, the constant frictional stress assumption in debonded regions upholds only in the presence of matrix plasticity in the numerical models [104].

The SLM is widely employed for calculating the ineffective length (twice the length at which 90% of the broken fibre stress is recovered) and/or SCFs in FRPs. SCF is the relative increase of the cross-sectional average stress,

$\bar{\sigma }_z$ σ ¯ z

, at

$z = z^\ast$ z = z ∗

with respect to the average far-field stress (at

$z = L$ z = L

)

$\left({{\rm SC}{\rm F}_{z = z^\ast } = \displaystyle{{{\bar{\sigma }}_z\lpar {z = z^\ast } \rpar -{\bar{\sigma }}_z\lpar {z = L} \rpar } \over {{\bar{\sigma }}_z\lpar {z = L} \rpar }} \times {\rm 100\percnt }} \right)$ ( S C F z = z ∗ = σ ¯ z ( z = z ∗ ) − σ ¯ z ( z = L ) σ ¯ z ( z = L ) × 100 % )

[166]. Green's function [167] and FEM [168] methods have been compared with SLM for their competence in stress transfer analysis. SLM is computationally less expensive than FEM, owing to its simplified portrayal of the constituents. However, its accuracy is limited to the systems with a high

$E_f/E_m$ E f / E m

ratio and high

$V_f$ V f

. Compared to FEM, the ineffective length is well estimated by the SLM. However, the predicted SCFs from SLM and FEM are only identical for a high

$E_f/E_m$ E f / E m

ratio and high

$V_f$ V f

since SLM ignores the matrix load bearing capacity, its plasticity and the shear deformation of the fibre [168].

A new interface model based on continuum damage mechanics was established to investigate interfacial debonding in CF- and GF-epoxy composites in 2D [169]. The discrete element method (DEM), developed initially to model movements within granular media [170], has been extended to laminated FRPs and interfacial debonding issues [171–173]. To better describe the post-failure frictional behaviour [173], 3D DEMs were conceived. Recently, a 3D DEM with an interfacial cohesive contact model was developed based on experimental SFFTs [174]. The novel 3D FEM of AhmadvashAghbash et al. [175,176] considers the significant interrelated phenomena during CF-epoxy debonding, including thermal residual stresses, frictional sliding, matrix crack and matrix plasticity, and does not pre-impose the debond length (see Figure 10). Their predicted stress recovery profiles were verified versus those obtained with Raman spectroscopy. Using this 3D FEM, a reliable simulation of a debonding single fibre can be tailored to multi-fibre models, whereas the improved accuracy of the stress concentration factor predictions enables generating authentic numerical fibre break models and optimising composites at fibre level for a preferential behaviour.OPEN IN VIEWER

Figure 10.

3D FE model of single-fibre debonding model: (a) reduced wedge model (due to symmetry) and assigned boundary conditions, (b) mesh refinement biased towards the interface and the debonding zone and (c) predicted stress recovery profile for the broken fibre throughout the applied strain of 2%. Note that the indicated half-debond length corresponds to the extent of the initial linear segment of the stress profile (reprinted from [175], with permission from Elsevier).

Experimental characterisation techniques

Laser Raman spectroscopy

A Raman spectrum can be acquired from materials that scatter light inelastically. Laser Raman spectroscopy (LRS or micro-Raman (MRS)) enables the evaluation of the stresses in CF-reinforced polymers (CFRPs) using the stress/strain dependence of certain vibrational modes of the fibres [39,177,178]. Although simplified, the following explanation helps to grasp the fundamental concepts of using LRS for conducting microstrain measurements (for further details regarding the basics of LRS and its utilisation for strain measurements, as well as the necessary instrumentation, the interested readers may refer to [179]):

Vibrational energy states exist between the electronic states of a molecule or crystal, and when a group of atoms vibrates, it causes instantaneous changes in the electron cloud and polarisability. This periodic change in the charge distribution interacts with an external electromagnetic field, leading to various light scattering phenomena, including infrared and Raman effects. If the incoming light frequency matches the frequency of a normal vibrational state in the material and results in a dipole moment alteration, the light is absorbed, and this is how infrared spectroscopy originates. When the incoming frequency does not match a normal vibrational frequency, the material can be excited to a virtual excited state and scatter light either elastically (Rayleigh scattering) or inelastically. Raman scattering is a type of inelastic scattering. The Raman effect occurs when the molecule or crystal drops from the excited state into a normal vibration mode with either higher or lower energy than its initial state and if there is a change in the molecule's polarisability. The scattered photon's energy equals the initial energy plus or minus the molecule's vibrational energy. Various types of reinforcing fibres, including graphite, aramid, rigid rod polymer fibres, silicon nitride and highly aligned organic fibres like polyethene, display a shift in their Raman peak when subjected to compressive or tensile strain [179]. These shifts in amorphous materials, such as CF, are mainly due to strain-induced lattice deformation, changes in the degree of disorder and interfacial debonding.

Galiotis et al. [180] were the first to adopt LRS to evaluate the strain dependence of Raman peaks in polydiacetylene fibres rather than crystals along with the Raman response of a crystalline urethane resin system [181]. Robinson et al. [178] and Galiotis et al. [177] subsequently analysed and evaluated the Raman spectrum response of various types of CFs under uniaxial stress. This was followed by a series of studies by Galiotis et al. [36,179,182–184] regarding fibres embedded in polymer resins and identifying interfacial stress transfer efficiency. A review [185] of the relevant work on FRPs was presented in the 1990s. Figure 11 displays a diagram depicting the standard LRS equipment along with strain profiles that have been fitted for both the broken and an adjacent fibre. LRS is capable of measuring the fibre stress with a spatial resolution in the order of 1

${\rm \mu m}$ μ m

. The fibre is required to be positioned in the close vicinity of the model composite surface. Once the Raman spectra are taken along the fibre fragment and the maximum of a Raman band is plotted against the distance from the fragment ends, the stress or strain point-to-point variation along the embedded fibre in a matrix can be determined by evaluating the maximum of that Raman band using an appropriate calibration curve [188]. The Raman band frequency and intensity depend on the manufacturing method and the subsequent surface treatment of the fibres [177].OPEN IN VIEWER

Figure 11.

(a) Schematic of standard LRS equipment (adapted from [186]) and fitted strain profile for (b) the broken fibre, and (c) a neighbouring fibre positioned at a relatively small inter-fibre spacing of 0.8

$d_f$ d f

in a 2D CF-epoxy microcomposite [187].

LRS can yield a detailed understanding of the stress transfer micromechanics in SFFT. The obtained stress variation along the fibre assists in discerning between bonded and debonded interface regions and can be employed to validate fibre axial stress predictions by straightforward stress transfer models (CKT, SLM) or more elaborate ones which consider plasticity, friction and so forth. Hence, after acquiring raw experimental data via SFFT augmented with LRS or optically assisted SFFT, estimating the interfacial properties requires using a reliable model to predict the interfacial stress transfer.

Huang and Young [42] utilised both strain-induced Raman band shifts and stress birefringence patterns during SFFTs. Their experiments confirmed that for low strain levels (pre-debonding), the fibre strain distribution follows a Cox-type SLM. Recently, Zhu et al. [189] developed an analytical model for inverse identification of all three interfacial properties: IFSS, IFFT and

$\tau _f$ τ f

(compared to only IFSS and

$\tau _f$ τ f

by [131]) for both micro-Raman-SFFT and the conventional SFFT data. Interfacial cohesive damage and frictional sliding were integrated into a trilinear constitutive law, and the fibre/interface analytical stress profile predictions were validated by axisymmetric linear-elastic FEM and experiments. Figure 12 shows the fibre stress recovery profile over one-half of an untreated T50-U CF embedded in epoxy. The partial-debonding and full-debonding models were fitted to the experimental data by [108,131]. In addition to a recent 3D FEM [175], the partial-debonding, in particular, matches well with the experiments.OPEN IN VIEWER

Figure 12.

Axial stress recovery profile along a T50-U fibre in an epoxy matrix at

$\varepsilon _m = 0{\rm .37\percnt }$ ϵ m = 0 .37 %

, obtained from Raman spectroscopy [132], and predicted by analytical models [108,131] and an FEM [175]. ↔ indicates half of the total debond length (recreated by the digitised data from [132] and existent FEM results from [175], with permissions from Elsevier).

In addition to its quantitative precision in yielding fibre axial stress, another experimental advantage of the LRS approach is its application versatility. In tensile procedures for determining IFSS, as mentioned earlier, the strain-to-failure of the matrix should preferably be up to three times greater than that of the fibre to ease measurement [153]. However, for Raman-active fibres, the LRS can be employed for matrix/fibre strain-to-failure ratios much smaller than this [154], which is more indicative of engineering applications.

Photoelasticity

Polarised light can be used to monitor the in-situ fragmentation process. Since the epoxy resin is stress-birefringent, cross-polarisers enable the observation of the stress state near fibre breaks and the interface [38]. A schematic illustration of a typical fragmentation test setup is shown in Figure 13. Once the fibre breaks, the specimen exhibits birefringence colours (or photoelastic patterns) surrounding the broken fibre ends due to the developed shear stresses at the interface as a result of stress transfer from the matrix to the fibre. The tensile stress drops to zero in the break gap between the fragments, creating a zone with no birefringence. This phenomenon offers a valuable tool to observe shear strain patterns and the fracture of opaque CFs. The birefringence usually exhibits symmetric features around the given fibre break and becomes more prolonged and flatter for higher load levels (see Figure 14b). Upon saturation, the birefringence pattern ends almost touch each other, indicating that shear stress transfer takes place over the whole fragment length and further fibre breakage is then unlikely [38].OPEN IN VIEWER

Figure 13.

Schematic feature of the fragmentation test apparatus (adapted from [86]). The recorder displays the applied load level. Using polarisers and a camera, events such as fibre breaks, debonding, matrix cracking and the resulting photoelasticity (or birefringence) are observable.

OPEN IN VIEWER

Figure 14.

The schematics of photoelastic birefringence patterns indicating the shear stress originated from an E-glass fibre break inside an epoxy matrix: (a) without interfacial debonding, (b) with debonding and (c) unloaded state with vanished larger birefringence (redrawn from [86]).

Reportedly, as-received (sized) CFs display a highly concentrated birefringent pattern within the adjacent matrix [62], indicating strong adhesion. However, unsized CFs exhibit noticeable break gaps and long debonded interfaces, with some debond zones even connecting to form ‘chain-like’ breaks. Epoxy systems exhibit extensive birefringence, which persists even after unloading the specimen (see Figure 14c). However, in polyester matrix systems, the birefringence is less prominent and promptly disappears once the specimen is unloaded [38].

The birefringence patterns were employed to determine the debond length at each fibre break in epoxy resins in [86]. Figure 14 compares the two expected birefringence patterns for fibre breaks with and without debonding. Based on Kim and Nairn's [86] observations, the birefringence around the fibre break includes two distinct colours, representing the inner and outer zones (see Figure 14b). Upon unloading, the outer birefringence disappears, while the inner band remains visible (albeit contracted). The authors assumed that, throughout loading, the debond zone length was therefore equal to the length of the inner zone at the interface, as indicated in Figure 14(b). Commonly, half of the distance between the two brightest light spots observed in the photoelastic birefringence is regarded as the ‘experimentally measured

$l_d$ l d

’. Debond growth in an AS4 CF-epoxy SFFT is shown in Figure 15, wherein the debond zones lengthen with an increase in the externally applied load.OPEN IN VIEWER

Figure 15.

Debond propagation for AS4 CF-epoxy as a function of applied strain (adapted from [86]).

Feih et al. [38] suggested an alternative definition for the debond zone extent by comparing the birefringence and white light images. The

$l_d$ l d

in their definition corresponds to the maximum of the two colours in the birefringence pattern. Zhu et al. [189] recently showed that the shear stress field from FEM has a very similar morphology to the photoelastic birefringence. Their justification is that the photoelastic birefringence reflects the disparity between the minimum and maximum principal stresses in the material (i.e. twice the maximum shear stress). The photoelastic birefringence can therefore be analogous to the shear stress field from a fragmentation FE model.

Multi-fibre fragmentation test (MFFT)

Conducting experimental investigations on stress concentrations in neighbouring fibres resulting from fibre fracture (and accompanying matrix failure) necessitates using model composites consisting of multiple fibres arranged in a precisely defined geometry and embedded within a matrix. Of specific significance is the requirement that the distance between adjacent fibres is limited to a few fibre diameters or less. This strict proximity criterion ensures the accurate examination and analysis of stress distribution and the intricate interplay between fibres within the composite structure. Except for the layout of testing frames and equipment, which is adjusted to ensure constant inter-fibre spacings, the multi-fibre fragmentation test (MFFT) closely mimics SFFT.

The interfibre spacing directly influences the effective stiffness experienced locally by fibres and thus impacts fragmentation length. An interfibre distance of 4

$d_f$ d f

$d_f$ d f

$d_f$ d f

$R_m$ R m

In addition to enabling the in-situ strain measurements at the fibre level in single-fibre model composites, LRS is a valuable method for determining SCFs and also investigating the effect of the SCF on the subsequent failure process in multi-fibre model composites. The SCF and ineffective length have been measured experimentally using LRS [192–194] and are often calculated using SLMs. For instance, for 2D Kevlar multi-fibre micro-composites, the strain along the fibres was mapped, via LRS, at different load levels, and specimens with various inter-fibre spacings were utilised to investigate the effect of fibre content. The obtained experimental SCFs were compared with values predicted by different theoretical models, and in general, they aligned with the literature models that considered inter-fibre distance and matrix effects [192]. Similarly, CF-epoxy micro-composites with an inter-fibre spacing ranging from 0.8 to 19

$d_f$ d f

$d_f$ d f

$d_f$ d f

To obtain the maximum SCF from the experimental data, the strain profiles of the SCF in the fibres adjacent to a broken fibre, for instance, might be fitted to a Gaussian distribution function and the IFSS (using equation (34)). A Piggott-type stress transfer model [195], consisting of two linear stress build-up regions (CKT), was found to fit the experimental strain profiles better [187]. The suggestion made by the authors [187] is to enhance the existing theories by considering the local yielding of the interface in the area surrounding the broken fibre, as indicated by the strain profiles. A preliminary FE study that included this local yielding showed encouraging outcomes [187]. Concerning the effect of the inter-fibre spacing on the interfacial shear stress at the fibre adjacent to a break, an increase in the inter-fibre spacing led to a reduction in the interfacial shear stress [187].

Grubb et al. [196] suggested that SLMs, designed to model the behaviour of single-fibre fragmentation along the fibre axis, were inherently incompetent in modelling stress transfer between fibres. Nairn's SLM [111] offers the advantage of relying heavily on parameters such as

$V_f$ V f

$V_m$ V m

$E_f$ E f

$E_m$ E m

Typically, the difficulty associated with MFFTs has been embedding small, brittle fibres at uniform inter-fibre intervals within a matrix [6]. Wagner and Steenbakkers [197] used a fibre spacing pin-array system, and fibres were placed between pin arrays, which were then rotated to deliver precise and reproducible interfibre distances in the eventual specimen. This approach performed well for Kevlar and other polymer fibres, but CFs and GFs broke too easily under rotation. Jones and DiBenedetto [198] employed rotating brass combs with 101 µm spacings. The interfibre spacing was controlled by adjusting the rotation angle on the device before depositing the aligned dry fibres in a silicone mould to cure. Li et al. [190] introduced another fibre-spacing tool that uses spacers to maintain fixed and settled fibre spacings. The MFFT specimens in [199] were prepared using the rotation device concept developed by Wagner and Steenbakkers [197] with modifications [200] made to achieve uniform interfibre spacings of about 1 µm. While MFFT test equipment has achieved increasingly more precise control over fibre uniaxial testing, fibre tension handling during fibre embedding is arduous due to the absence of an effective in-situ fibre local tension measurement. Additionally, the embedment process is typically manual, which introduces variability in initial fibre tension and potential undetected crazes on fibre surfaces that might trigger premature failure. Moreover, the literature lacks comprehensive consideration of the stress relaxation state of individual fibres within an array, highlighting the necessity for in-situ tension measurement during the embedment process. However, advancements in micro-actuators, sensors, and programming offer promising opportunities to overcome these challenges [6].

Conclusion

An SFFT moderately replicates the stress transfer characteristics in real FRPs and allows a large statistical interface sampling. Additionally, the critical length is sensitive to and reflects the changes in the fibre–matrix adhesion level, which is vital for a decent interface characterisation method. Fracture mechanics-based data reduction methods have been established which do not require fragmentation saturation. However, a highly complex/non-uniform stress state at the interface due to interfacial shear stress concentration near the fragment ends complicates the data analysis. Additionally, nearby fractures and fibre pre-tension integrate more difficulty into the analysis. The majority of the existing data reduction models ignore at least one of the key features of the SFFT. None of the developed approaches for SFFT has yet achieved adequate credibility to form the basis of a universal standard for the measurement of IFSS that could later be used to estimate the ILSS of a laminate [6]. An optimal data reduction method considers all the major interrelated phenomena during fibre–matrix debonding, including thermal residual stresses, frictional sliding and matrix plasticity, and avoids a pre-imposed debond length. Although it is possible to design a fragmentation test for a high

$V_f$ V f

The SFFT is severely limited by the elastic mismatch requirements between the fibre and the matrix. The test evaluations are consistent and reliable only if the fibre failure strain is several times lower than the matrix failure strain. The test is cumbersome to perform, and the specimen quality and success of SFFTs largely depend on the skill and experience of the operator. The process of isolating single fibres requires a high level of precision, and any errors can compromise the accuracy of the results. However, with careful preparation and attention to detail, SFFT can provide valuable insights into the mechanical properties of the interface. The key factors for the prevalence of the SFFT are the less intricate test setup and its applicability to brittle fibre systems and more frequent (relative to the other three main methods) attempts to ameliorate the data reduction methods. The SFFT is more suitable for CFRPs due to the small diameter and brittleness of CFs [42]. The SFFT, the only test involving fibre breakage while assessing the interfacial properties, is restricted to transparent or highly-translucent matrices with high failure strain. However, acoustic emission can be used to determine the length distribution of the fibre fragments, even for specimens with opaque polymer matrices and fibrillar fibre breaks [90]. The majority of the in-situ measurements of the fibre strain and debond length in SFFTs is acquired either by optical microscopy, photoelastic birefringence patterns (using polarisation filters) or LRS. The key advantage of LRS-derived axial fibre strain data is that

$\sigma _f\lpar x \rpar$ σ f ( x )

Note that the direction (tension vs compression) or type of applied loading (static vs fatigue) may affect IFSS. A comparative assessment of stress transfer efficiency in tension and compression for a CF embedded in epoxy revealed that the rate of stress transfer from a fibre break was extremely high in compression, resulting in a short ineffective transfer length compared to tension. By further loading of the system, stress was transferred in the fibre not only by interfacial shear but also by fibre–fibre contact at the compressive failure location [201]. To observe the debond growth with increasing cycles, a single-fibre specimen containing a fragmented fibre can be subjected to axial fatigue loading. Previous fatigue tests on model composites with only 2–5 fibres have been conducted, such as in reference [202]. When a simulation tool is available for debonding process, combining experimental and simulation approaches can help identify the fatigue law and its parameters for the interface being studied. This information can then be used to analyse the fatigue of UD FRPs. The approach in [203] assumes that the propagation of an individual interfacial debond follows the Paris law. This law describes the growth rate of the debond as a power-law function of the difference between strain energy release rates at the highest and lowest load levels during cyclic loading. Moreover, approaches such as analytical, FEM and virtual crack closure techniques have been employed to analyse the debond growth in tension-tension fatigue using Paris law [21,142].

Additionally, the dependence on the load direction and specimen size is a distinct feature of thermosets matrices compared to thermoplastics. The stress–strain behaviour of macro-scale bulk epoxy resin specimens exhibits an over-reliance on the loading direction [204,205]. Performing tensile tests typically leads to a brittle behaviour [206] and, contrarily, specimens under uniaxial compression exhibit notable non-linearity and plasticity on both the micro- and macro-scales. The micro-scale shear behaviour, as the most relevant description of the matrix in SLMs, displays a similar stress–strain diagram as obtained by macroscopic compressive tests. However, there is currently no agreement on the best practice to measure the relevant epoxy constitutive properties for use in micro-scale composite models and, therefore, also on the correct constitutive model. Conceivably, a first-order approximation can be achieved by simulating the constitutive behaviour based on compressive stress–strain data through the use of a yield criterion [207,208].

Table A2 highlights the scatter in the reported interfacial properties measured with SFFT for CF or GF (separated by a horizontal dashed line) embedded in thermosetting or thermoplastic (TP) matrices. The immense existent ranges for the reported interfacial properties have received various rationalisations. High radial compression stresses can give rise to overestimated IFSSs (especially in TP matrices, related to the higher processing temperatures). Reportedly [209], the IFSS never exceeds the shear yield strength of the matrix. However, a widely used explanation for inflated IFSSs is that the polymer layer formed with a finite thickness around the fibre surface exhibits substantially different properties from those of the bulk matrix, which was criticised in [210]. The large

$\mu _i$ μ i

$\mu _i$ μ i

$\mu _i$ μ i

$\mu _i$ μ i

$\tau _f$ τ f

$\mu _i$ μ i

Reportedly, in the case of intermediate-modulus (IMD) FRPs, the matrix tends to yield first rather than debonding along the interface, whereas in high-modulus (HM) FRPs, debonding may occur first. However, when the CKT model is adopted for IFSS calculation, its value is approximately half of the Tresca matrix yield stress. If the LRS-derived IFSS is considered more accurate, a clear disparity arises between credible experimental data for IFSS and the calculations of a classical theoretical model, with the LRS-measured IFFS tending to be twice that of the CKT IFSS [213]. The decrease in IFSS with increasing

$E_f\;$ E f

$E_f$ E f

$E_f$ E f

$E_f$ E f

Load–displacement response

The pull-out load–displacement curve generally contains the typical linear elastic, crack propagation and fibre pull-out zones. According to Piggott [224], the evident regions can be distinguished in each plot as:

a linear increase in force with displacement;

For the reported experiments in [224], all systems have a region (i) to a greater or lesser extent, and Kevlar-epoxy and GF and CF in thermoplastics have a distinct curved region (region ii). GF-polyester and GF-epoxy [212] have no region ii, whereas CF-epoxy does. Reportedly, all systems display a sudden decrease in force and only the systems with shorter

$l_{{\rm emb}}$ l e m b

OPEN IN VIEWER

Figure 17.

Schematic illustrations of three typical force–displacement curves for the single-fibre pull-out test of brittle fibre-resin systems: (a) for strong interfaces or weak interfaces with small

$l_{{\rm emb}}$ l e m b

, (b) commonly observed for systems with weak interfaces, and (c) observed for Kevlar-epoxy where the peaks in the ascending region were related to friction or a stepwise debonding phenomenon (redrawn from [225], with permission from Elsevier).

It is customary, although not necessarily correct, to associate

$F_{{\rm max}}$ F m a x

$F_{{\rm max}}$ F m a x

$F_{cat}$ F c a t

$l_{{\rm emb}}$ l e m b

$l_{{\rm max\comma \; cat}}$ l m a x , c a t

$l_{{\rm emb}}$ l e m b

$l_{{\rm max\comma \; cat}}$ l m a x , c a t

$l_{{\rm max\comma \; friction}}$ l m a x , f r i c t i o n

$l_{{\rm emb}}$ l e m b

$l_{{\rm emb}}$ l e m b

$l_{{\rm max\comma \; friction}}$ l m a x , f r i c t i o n

$l_{{\rm emb}}$ l e m b

OPEN IN VIEWER

Figure 18.

Theoretical relationship between the embedded length and the load required to debond (redrawn from [220], with permission from Elsevier).

The most basic model for the fibre pull-out process is the constant interfacial shear stress (linear

$F_{{\rm max}}-l_{{\rm emb}}$ F m a x − l e m b

$l_{{\rm emb}}$ l e m b

The apparent maximum debonding force increases with

$l_{{\rm emb}}$ l e m b

$E_f$ E f

$E_m$ E m

$E_f/E_m$ E f / E m

$\lpar {R_m/r_f} \rpar ^2-1$ ( R m / r f ) 2 − 1

$R_m$ R m

$l_{emb}$ l e m b

Specimen design and test setup

A carefully designed experiment, in addition to IFFT, can yield data on the frictional region of the pull-out curve. Regarding the test configuration, different boundary conditions would provoke different stress states in the constituents [229]. Figure 19 demonstrates the schematics of various fibre pull-out configurations in analytical models. Commercial devices have been developed to (partially) automate the pull-out procedure [230]. For the pull-out process, there are advantages to using a stiff loading system (including fibre grips, drivetrain and so on) to achieve controlled and stabilised debond propagation [218]. To ensure a highly stiff test setup, the Federal Institute for Materials Research and Testing (BAM), in addition to piezo force sensors, has utilised piezo translators for precise displacement generation [217]. The components were mounted on a stiff steel frame and reported to yield stable crack propagation during the pull-out test. Often the most compliant part of the test setup is the fibre itself since a significant length of fibre (the free length, see Figure 16) connects the embedded part of the fibre and the part in the grips. This free length mainly reduces the reliability of the post-debonding frictional data [218]. Even moderately short free lengths (a few

${\rm mm}$ m m

${\rm \mu m\ }{\rm s}^{-1}$ μ m s − 1

${\rm s}^{-1}$ s − 1

OPEN IN VIEWER

Figure 19.

Schematics of various fibre pull-out configurations in developed analytical models: (a) fixed matrix bottom, (b) fixed fibre and matrix bottom, (c) restrained matrix top [229] and (d) bounded bottom and sides, after Piggott [211] (subfigures (a–c) are redrawn from [229], with permission from Elsevier).

In a pull-out test, it is crucial that interfacial debonding occurs ahead of the fibre break. Since the interfacial debond force is directly related to the

$l_{{\rm emb}}$ l e m b

$l_{{\rm emb}}$ l e m b

$l_{{\rm emb}}$ l e m b

$l_{{\rm emb}}$ l e m b

$200\, {\rm \mu m}$ 200 μ m

$l_{{\rm emb}}$ l e m b

${\rm \sim \; 1}\, {\rm mm}$ ∼ 1 m m

$l_{{\rm emb}}$ l e m b

${\rm \pm 10}\, {\rm \mu m}$ ± 10 μ m

$l_{{\rm emb}}$ l e m b

The in-situ observations could include LRS [235,236] and photoelasticity (isochromatic fringe patterns) [232]. The single-fibre pull-out test was analysed for Kevlar-cold-cured epoxy using both conventional pull-out and LRS-coupled tests in [225,237]. At low strains, the behaviour follows the elastic shear-lag analysis, but as the fibre strain increases, interfacial debonding occurs. Reportedly, the conventional pull-out test only produces an apparent IFSS value, and employing a partial-debonding model allows using the interfacial parameters obtained from Raman spectroscopy to predict the data from the conventional pull-out test [225].

Data reduction schemes

Stress-based methods

The stress-based models, being plainer than the energy-based approaches, target estimating the interfacial stress profile in accordance with the applied load. It is generally agreed upon that the strength approach, neglecting friction, can be applied to a totally unstable debonding process with short

$l_{{\rm emb}}$ l e m b

. The energy method, however, can describe stable debonding, usually with long

$l_{{\rm emb}}$ l e m b

[238,239]. Uniform interfacial shear stress (or mean stress model) [82] is often presumed as the first approximation. In fact, this is only valid when the matrix is fully plastic. Consequently, based on force equilibrium on the fibre, the IFSS,

$\tau _{ic}$ τ i c

, is:

$$\tau _{ic} = \displaystyle{{F_d} \over {2\pi r_fl_{{\rm emb}}}}$$ τ i c = F d 2 π r f l e m b

(2)

where

$F_d$ F d

is the debond load. Greszczuk [240,241], by modifying Cox's SLM [108], derived the shear stress distribution along the fibre for an elastic matrix. The test geometry studied by Greszczuk is similar to the one considered by Piggott (Figure 19d). The interface was assumed to be perfectly bonded (no interfacial sliding was allowed), the embedded fibre-end carried no load, and radial effects were neglected [239,240]. The non-constant shear stress along the fibre incorporates interfacial shear modulus and the effective interphase thickness, which are assumed to be

$G_m$ G m

and

$r_f\ln \lpar {R_m/r_f} \rpar$ r f ln ⁡ ( R m / r f )

, respectively, for convenience. The IFSS, at the point of emergence of the fibre, equals to:

$$\tau _{ic} = \displaystyle{{F_d} \over {2\pi r_fl_{{\rm emb}}}}\displaystyle{{l_{{\rm emb}}\sqrt {\displaystyle{{2G_m} \over {r_f^2 E_f\ln \lpar {R_m/r_f} \rpar }}} } \over {\tanh \left({l_{{\rm emb}}\sqrt {\displaystyle{{2G_m} \over {r_f^2 E_f\ln \lpar {R_m/r_f} \rpar }}} } \right)}}$$ τ i c = F d 2 π r f l e m b l e m b 2 G m r f 2 E f ln ⁡ ( R m / r f ) tanh ⁡ ( l e m b 2 G m r f 2 E f ln ⁡ ( R m / r f ) )

(3)

Note that

$R_m$ R m

is the radius of the matrix involved in the shear strain. Greszczuk's analysis, possibly being the oldest, has inspired many authors [20,242] to extend this SLM for the interpretation of pull-out tests.

Lawrence et al. [243] introduced the partial interfacial debonding concept with a debond initiating at the loaded fibre-end and propagating towards the embedded fibre end, while the effect of frictional shear stress was included throughout the debonding process. The authors recalculated the shear stress distribution to an expression analogous to that given by Greszczuk [240] but with a different elastic constant [239]. The Lawrence model predicts the debond stress satisfactorily for short

$l_{{\rm emb}}$ l e m b

, but tends to underpredict it as the

$l_{{\rm emb}}$ l e m b

increases [239].

The analytical model of Zhang et al. [244] (developed for continuous fibre-reinforced cementitious composites) divides the pull-out curve into three segments: perfect bonding, debonding and pure friction. The matrix was considered as a rigid non-deformable body, and the constituents were isotropic and linearly elastic. Their proposed shear stress profile is reminiscent of a CZM with a bilinear traction-separation law coupled with a frictional sliding after debond completion. Sørensen and Lilholt [24] developed a relatively simple 1D analytical SLM that incorporates the residual stresses but neglects the Poisson effects. Hsueh [245,246] included the effects of interfacial debonding, residual radial and axial stresses and fibre sizing and determined the axial strains of the constituents only by the axial stresses and disregarded the effect of Poisson contraction [247]. Regarding the micromechanics of elastic stress transfer, a two-cylinder model for the single-fibre pull-out test and a three-cylinder model for the multi-fibre pull-out test were compared in [248]. This comparison revealed the importance of neighbouring fibres (local fibre volume fraction). Concisely, none of the current and prevalent stress-based models capture all relevant features within interfacial failure in FRPs.

Energy-based methods

The strong heterogeneous interfacial stress distribution, exhibiting stress singularities, suggests that a stress-based evaluation of the test data is not entirely appropriate [249]. Based on Griffith's fracture criterion [250], an interfacial crack propagates once SERR (

$G$ G

) is just equal to the critical fracture energy release rate. The energy method includes the early work of Outwater and Murphy [251] in 1969. The basis of their work, and many to follow, was the general compliance equation expressed as:

$$G = \left({\displaystyle{{P^2} \over {2d}}} \right)\left({\displaystyle{{{\rm d}C} \over {{\rm d}a}}} \right)$$ G = ( P 2 2 d ) ( d C d a )

(4)

where

$P$ P

is the load,

${\rm d}C/{\rm d}a$ d C / d a

the change in compliance with crack length and

$d$ d

the crack width. The appeal of this type of approach was highlighted, notably by Piggott [252,253]. Initially employing a stress-based criterion in earlier studies, Piggott advocated an energy-based approach to interpret tests where brittle fracture develops. This is particularly relevant for most GF or CF-epoxy systems. The geometry used in this model is that of pull out in Figure 19(d), in which fibre of radius

$r_f$ r f

has an embedded length of

$l_{emb}$ l e m b

within a resin cylinder with a radius of

$R_m$ R m

. When the fibre is subjected to a tensile force, the matrix is sheared elastically. The fibre is predicted to debond at a force

$F_d$ F d

:

$$F_d = \pi d_f\sqrt {r_fE_fG_{ic}} \sqrt {\displaystyle{{\displaystyle{{2G_ml_{emb}} \over {E_f\ln \lpar {R_m/r_f} \rpar }}\tanh \left({\displaystyle{{\displaystyle{{2G_ml_{emb}} \over {E_f\ln \lpar {R_m/r_f} \rpar }}} \over {r_f}}} \right)} \over {r_f}}} $$ F d = π d f r f E f G i c 2 G m l e m b E f ln ⁡ ( R m / r f ) tanh ⁡ ( 2 G m l e m b E f ln ⁡ ( R m / r f ) r f ) r f

(5)

The geometry considered by Penn and Chou [254] is more comparable to that of the microbond test (in the absence of the blades, see Figure 20). By assuming an initial interfacial crack size of length

$a$ a

, the energy equilibrium indicates that the portion of the stored elastic energy releases to allow interfacial failure. This leads to the derivation of the debonding force,

$F_d$ F d

, as

$$F_d = \displaystyle{{\pi d_f\sqrt {r_fE_fG_{ic}} } \over {\sqrt {\left({1 + \displaystyle{1 \over {\cosh \left({\cosh \left({\displaystyle{{2G_m} \over {E_f\ln \lpar {R_m/r_f} \rpar }}\left({\displaystyle{{l_{emb}-a} \over {r_f}}} \right)} \right)} \right)}}} \right)} }}$$ F d = π d f r f E f G i c ( 1 + 1 cosh ⁡ ( cosh ⁡ ( 2 G m E f ln ⁡ ( R m / r f ) ( l e m b − a r f ) ) ) )

(6)

In the case where

$a$ a

is negligible compared to

$l_{emb}$ l e m b

, this expression can be simplified as

$$F_d = \displaystyle{{\pi d_f\sqrt {r_fE_fG_{ic}} } \over {\sqrt {\left({1 + \displaystyle{1 \over {\cosh \left({\cosh \left({\displaystyle{{2G_ml_{{\rm emb}}} \over {E_f\ln \lpar {R_m/r_f} \rpar }}} \right)} \right)}}} \right)} }}$$ F d = π d f r f E f G i c ( 1 + 1 cosh ⁡ ( cosh ⁡ ( 2 G m l e m b E f ln ⁡ ( R m / r f ) ) ) )

(7)

OPEN IN VIEWER

Figure 20.

Schematic of the microbond test set-up (redrawn from [255]).

Piggott's expression for the debond force as a function of the interfacial toughness and the

$l_{{\rm emb}}$ l e m b

(given in [219]) was not satisfactory since the strain energy in the fibre free length was neglected [239]. Penn and Lee [256] reformed his expression by considering the strain energy stored in the fibre free length within the energy balance equation [239]. Jiang and Penn [221] further improved the analysis by including the matrix compression contribution and the effect of the work of friction at the debonded interface in the energy balance.

The majority of energy methods assume a stable interfacial crack propagation with a constant value of the IFFT. Therefore, these approaches cannot be applied in the case of short

$l_{{\rm emb}}$ l e m b

, which displays an abrupt and complete debonding [239]. The important assumption of another group of models, namely the Lamé solution-based models [83,84,229,257–259], is that the axial strain in the matrix is independent of the radial distance and is approximated to an average strain in the matrix [247]. In addition, in the region near the loaded end, the stresses in both fibre and matrix are incapable of satisfying the equilibrium equation in the radial direction [247]. By extending the Lamé solution for a 2D-axisymmetric problem to the pull-out problem, Gao et al. [83] established a fracture-mechanics-based analysis, also known as the Gao–Mai–Cotterell model. This model incorporates the effect of interfacial friction in the debonding region, Poisson contraction and initial thermal residual stresses on the stress distributions. The SERR for crack propagation is given as

$$G_{\,prop.} = \displaystyle{{\lpar {1-2k\nu_f} \rpar {\lsqb {P-\lpar {1 + \beta } \rpar Q} \rsqb }^2} \over {4\pi ^2r^3E_f\lpar {1 + \beta } \rpar }}$$ G p r o p . = ( 1 − 2 k ν f ) [ P − ( 1 + β ) Q ] 2 4 π 2 r 3 E f ( 1 + β )

(8)

The unaccustomed parameters are defined in Equations 1A–8A of the Appendix.

Owing to its simplified equations and methodology in assessing the interface parameters based on experimental data, the Gao–Mai–Cotterell model is extensively exploited by other researchers. This model has been adapted to incorporate the effects of loading mode [229], fibre anisotropy [257], interfacial roughness [258] and fibre volume fraction [259]. Hutchinson et al. solution [84], based on the Lamé solution and considering the debond as a mode II fracture, was limited to systems featuring residual compressive stresses.

As mentioned earlier, debonding can be viewed as either complete or partial. In the case of partial debonding, the friction parameters must be considered in the pull-out analysis [239]. Accordingly, the partial debond stress was represented as a function of the debond length in [260], based on fracture mechanics. However, the expression for their debond stress was intricate, and the same interfacial parameters were obtained using a simpler model in [83], overshadowing its practicality [239].

The debonding stability is regulated by the elastic constants, the relative volume of fibre and matrix, the nature of the interfacial bonding and the

$l_{{\rm emb}}$ l e m b

[261]. The schematic representation of the stability state in load–displacement curves (i.e. totally unstable, partially stable and totally stable) for the interfacial debond process is given in [261].

The shear strength-based approach of Hsueh [262] and the energy-based approach of Gao et al. [83] were compared with experimental results in [261]. In Gao et al. [83], a pre-debonded interface is modelled as a stable interfacial crack propagating with a constant interfacial fracture toughness under plane strain conditions. Both models, considering an SLM, contain the effects of friction at the debonded region and Poisson contraction by assuming that the debond propagates from the loaded fibre end (no two-way debonding) [261]. For epoxy matrix composites, Gao et al. model [83] predicted the trend of maximum debond stress quite well for long

$l_{{\rm emb}}$ l e m b

, but it always overestimated it for systems with very short

$l_{{\rm emb}}$ l e m b

[239]. The expressions for partial debonding of Gao et al. [83] are summarised in the appendix of [239]. Contrarily, Hsueh's model [262] can predict the maximum debond stress for a short

$l_{{\rm emb}}$ l e m b

, but often a considerable adjustment was required for a better fit to the experimental results for cases with a long

$l_{{\rm emb}}$ l e m b

.

The model of Gao et al. [83] was later applied by Kim et al. [263] to determine the IFFT, friction coefficient and residual fibre clamping stress. A stochastic approach (Ising model combined with Monte Carlo simulation) was applicable to partial debonding, fibre breakage and matrix failure [238] in the course of a single-fibre pull-out. The energy-based models of the pull-out process can also be solved with the aid of FEM [249]. For instance, Atkinson et al. [264] calculated the energy release rate in a pull-out test using FEM.

Numerical methods

One of the earliest FEMs for the fibre pull-out test, based on the energy method, was developed by Beckert et al. [265]. They incorporated the thermal residual stresses, specimen geometrical features and a basic model for interfacial friction. In a perfectly bonded system, a parametric study with different fibre/matrix stiffness ratios and irregular fibre cross-sections was carried out in [266]. The effect of

$V_f$ V f

on the stress transfer was assessed both analytically and numerically by a three-cylinder composite model (coaxial fibre, matrix and composite) in [259].

The CZM bridges the gap between the stress- and energy-based approaches. Tsai et al. [267] used a CZM-friction interface model to simulate copper fibre-epoxy pull-out. In their axisymmetric FEM [268], CZM was used to explore the effect of various interfacial parameters in a single CF pull-out test. The debonding force was found to have a linear relationship with the IFSS and fibre geometric parameters (

$r_f$ r f

and

$l_{{\rm emb}}$ l e m b

), but slightly decreases with increasing initiation separation displacement (the

$\delta$ δ

that corresponds to

$\tau _{ic}$ τ i c

on Figure 9). Furthermore, the influences of

$E_f$ E f

, complete separation displacement and

$\mu _i$ μ i

are reported to be insignificant [268]. Additionally, the thermal residual stresses were shown to have a significant influence on the fibre pull-out during the frictional sliding stage, leading to higher specific pull-out energies (fibre pull-out energy per unit interfacial area) [269]. This highlights the importance of using an anisotropic material property for CFs in assessing the thermal residual stresses instead of simplified isotropic properties. Also, the specific pull-out energy increases with

$l_{{\rm emb}}$ l e m b

(but not with

$r_f$ r f

) due to the effect of these thermal residual stresses and interfacial friction. This observation implies that merely reducing

$r_f$ r f

would not improve the fracture resistance [269]. Moreover, this model certifies that the apparent IFSS is not constant and decreases with increasing

$l_{{\rm emb}}$ l e m b

[269]. According to a recent FE model [270], the most significant factors that affected the simulated pull-out test results and the corresponding force–displacement curves were the traction-separation law parameters of mode II and the free fibre length. The simulation of the crack indicated that although mode I had an influence on crack initiation, mode II dominated the further crack propagation.

Machine learning techniques have the potential to significantly enhance the accuracy, efficiency and automation of numerical models for interface characterisation methods. It is advisable to explore novel machine learning algorithms and techniques, develop hybrid models that integrate machine learning with conventional numerical methods, and examine the potential of machine learning in multi-scale modelling. For instance, the FEM with cohesive damage model and frictional contact for the interface was combined with Artificial Neural Network (ANN) to study the load-displacement behaviour during fibre pull-out in fibre-reinforced ceramic composites in [271]. The ANN, trained and tested using an analytical model and an FEM, can accurately predict load-displacement behaviour. By utilising a larger dataset of experimental results, the ANN can be thoroughly trained to capture the elaborate details derived from experimental observations, which otherwise are challenging to investigate through analytical models [271].

Frictional behaviour

Friction is described by two independent parameters: the friction coefficient (

$\mu _i$ μ i

$\mu _i$ μ i

$\mu _i$ μ i

$\sigma _n$ σ n

$\mu _i\sigma _n$ μ i σ n

Contrarily to the conventional Coulomb friction (

$\mu _i = {\rm constant}$ μ i = c o n s t a n t

$\mu _i$ μ i

Conclusion

The pull-out test can directly estimate the IFSS and provide information on friction coefficients and shrinkage pressures. Despite being an excellent choice for novel fibre–matrix configurations (e.g. fibres with interfacial topographical anchors), this method is challenging for brittle fibre systems. Apart from demanding specimen preparation and handling, this method is governed by critical embedded fibre length and is limited to maximum embedded lengths of tens of micrometres for certain systems (lower bound due to meniscus formation and upper bound constrained by fibre strength [100]). The existence of high shear stress concentration near the loaded fibre-end and lower shear stress concentration at the embedded fibre-end complicates the interfacial stress state. Another challenge associated with single fibre pull-out testing in FRPs, apart from a prerequisite thorough interpretation of the load-displacement trace, is that the interpretation varies between laboratories [220]. These interpretations have primarily centred around the maximum point in the load-extension diagram. If this maximum results from a catastrophic interfacial failure, the

$l_{{\rm emb}}$ l e m b

$l_{{\rm emb}}$ l e m b

The stress distribution of an appropriate test configuration must be similar to that in a bulk composite. For composites with brittle fibres and ductile matrices, the favoured candidate is the SFFT. Contrarily, for composites with brittle matrices, inducing failure via multiple transverse cracks, the pull-out test seems more appropriate [279]. Systems with strong interfaces entail very small critical

$l_{emb}$ l e m b

$\lt 100\, {\rm \mu m}$ < 100 μ m

$l_{{\rm emb}}$ l e m b

$l_{{\rm emb}}$ l e m b

${\rm \mu m}$ μ m

Table A3 summarises the interfacial properties from pull-out tests reported in the literature. The IFSSs estimated from shorter

$l_{{\rm emb}}$ l e m b

Contact angle

The surface energy of solids (and the surface tension of liquids), serving as a direct indication of intermolecular forces, encompasses Lifshitz–van der Waals (dispersive), basic and acidic components. By considering these components, one can assess the physical adhesion between two dissimilar materials and indirectly infer their compatibility [283,284]. The fibre surface energy, as an essential physical parameter to assess surface activity, and one of the key parameters in determining the final interfacial properties, is difficult to be directly measured due to the cylindrical shape and the small diameter of the fibre. Typically, the fibre surface energy, or wettability of the matrix on the fibre surface, is determined by the contact angle (

$\theta$ θ

Carroll [285] and Yamaki and Katayama [286], concurrently and independently, developed analytical models to determine

$\theta$ θ

$L$ L

$H$ H

$L\; {\rm \sim \; 300}\, {\rm \mu m}$ L ∼ 300 μ m

$H\; {\rm \sim \; 100}\, {\rm \mu m}$ H ∼ 100 μ m

$\theta$ θ

$L$ L

$H$ H

$\theta$ θ

$L$ L

$H$ H

$L$ L

$H$ H

$\theta$ θ

To provide a visual representation, a comparison of an elliptical fit, circular arc fit and Carroll's fit to an SEM image of a thermoplastic droplet on a sized CF is presented in Figure 21. Carroll's equation assumes the preservation of the unduloid shape of the droplet within both its initial liquid form and after its consolidation, considering negligible gravitational effect. The droplet profile (see Figure 22) on a cylindrical fibre is determined by the following two equations [285,291]:

$$y^2 = h^2\lpar {1-k^2{\sin }^2\varphi } \rpar $$ y 2 = h 2 ( 1 − k 2 sin 2 φ )

(9)

$$x = \pm \lpar {ar_fF\lpar {\varphi \comma \; k} \rpar + hE\lpar {\varphi \comma \; k} \rpar } \rpar $$ x = ± ( a r f F ( φ , k ) + h E ( φ , k ) )

(10)

OPEN IN VIEWER

Figure 21.

Comparison of elliptical fit, circular arc fit, and Carroll's fit to SEM image of a polypropylene homopolymer droplet on an epoxy-sized CF (TR50S) (adapted from [281,290]).

OPEN IN VIEWER

Figure 22.

Schematics of the droplet profile in a microbond test and the parameters for determination of the contact angle (redrawn from [285], with permission from Elsevier).

The optical measurement of

$\theta$ θ

$r_f$ r f

${\rm \mu m}$ μ m

$\theta$ θ

The ‘debonding cone’ phenomenon commonly occurs for a brittle matrix system (such as epoxy) when

$\theta \lt 20^\circ$ θ < 20 ∘

Specimen design and test setup

The absence of a standardised procedure for the microbond test has generated discrepancies in experimental setups among research groups, with universal testing machines equipped with low-capacity load cells being the mainstream approach. However, reliable measurement of the interfacial properties requires appropriately scaled exclusive equipment, as in [294]. In addition to having a highly stiff testing device, enabling a precise displacement measurement due to stabilised crack propagation, a shorter fibre free length,

$l_{{\rm free}}$ l f r e e

$$K\propto K_f = \displaystyle{{E_f\pi d_f^2 } \over {4l_{{\rm free}}}}$$ K ∝ K f = E f π d f 2 4 l f r e e

(11)

$K$ K

$K_f$ K f

Regarding the effect of the loading device types, [299] reports higher average IFSS values (and lower scatter) utilising a pin-holed steel film over microvise tips in quasi-disk type microbond specimens. Mendels et al. [292] reported that the numerically calculated IFSS is independent of the loading point location if it is positioned in between the inflection point of the droplet profile and the endpoint of the droplet. The inevitable discrepancies concerning the droplet configurations, as a consequence of

$\theta$ θ

$d_f$ d f

Concerning the deposition of multiple droplets on an individual fibre, pragmatically, it is recommended that placing more than two droplets on the fibre complicates the droplet identification/tracking, and in the case of fibre break, all the efforts will be in vain [280]. The measured

${\rm IFS}{\rm S}_{{\rm app}}$ I F S S a p p

The microbond technique is widely used for thermoset matrices but much less for thermoplastic composites. The preparation of thermoplastic droplets is, therefore, less established. An optimised method was proposed [301] using a cardboard frame with a single fibre glued to it and a rectangularly cut polymer film mounted on the fibre and melted in a convection oven. This is an alternative to those of V-shaped polymer strips [302,303], melting polymer pellets to form films and sandwiching GFs between two films [304] or knotting polymer fibre around CF [305].

Herrera-Franco and Drzal [293] provide comprehensive descriptions of the stepwise procedures for forming thermosetting or thermoplastic microdroplets. To pull the fibre out from the droplet without breaking the fibre (precondition:

$\sigma _f \gt \tau _{ic}$ σ f > τ i c

$l_{emb}$ l e m b

$l_{{\rm emb}}$ l e m b

$l_{{\rm emb}\comma \; c}$ l e m b , c

$l_{{\rm emb}}$ l e m b

$$l_{{\rm emb}} \lt \displaystyle{{\sigma _fd_f} \over {4\tau _{{\rm app}}}} = l_{{\rm emb}\comma \; c}$$ l e m b < σ f d f 4 τ a p p = l e m b , c

(12)

$\sigma _f$ σ f

$l_{emb}$ l e m b

${\rm \mu m}$ μ m

$\sigma _f$ σ f

$F_{{\rm max}}$ F m a x

$l_{{\rm emb}}$ l e m b

$l_{{\rm emb}} \gt l_{{\rm emb}\comma c}$ l e m b > l e m b , c

${\rm \mu m}$ μ m

$\sim 300\, {\rm \mu m}$ ∼ 300 μ m

$l_{{\rm emb}}$ l e m b

${\rm \mu m}$ μ m

${\rm \mu m}$ μ m

${\rm \mu m}$ μ m

$l_{{\rm emb}}$ l e m b

$l_{{\rm emb}}$ l e m b

Laurikainen et al. [294] reported that for small droplets (∼50–200

${\rm \mu m}$ μ m

$\beta _{{\rm Nayfeh}}^2$ β N a y f e h 2

$V_f$ V f

$\beta _{{\rm Cox}}$ β C o x

$l_{{\rm emb}}$ l e m b

${\rm \mu m}$ μ m

$d_f$ d f

The IFSS_app of a microbond test can vary by more than a factor of three depending on the atmospheric conditions during the preparation [310]. Schober [310] prepared the microbond specimens in a reduced oxygen atmosphere (oven flushed with inert argon gas) to reduce the polymer degradation (manifested by the discoloration) of the PP droplets. The resulting well-formed droplets with smooth menisci and limited asymmetries yielded much higher IFSS_app values. This indicates the need for expeditious standardisation.

The ratio of the droplet length and diameter and test speed repercussion on the measured IFSS were studied in a cylinder microbond test (droplet geometry is modified to a cylinder) complemented with in-situ acoustic emission monitoring [311]. Within the range of 10–100 mm min^–1, the IFSS was no longer altered by the test speed with embedded lengths of 120–300

${\rm \mu m}$ μ m

$l_{emb}$ l e m b

$l_{{\rm emb}}$ l e m b

Apart from the cylindrical droplet [311,314] (see Figure 23), other attempts in modifying the conventional ellipsoidal droplet geometry to diminish the scatter in IFSS values include forming quasi-disk droplets (averting high interfacial stress concentrations pliant to the geometrical variations) [299,316] and forming controlled symmetrical periodic microdroplets along the fibre through Rayleigh instability [317].OPEN IN VIEWER

Despite being referred to as ‘microbond’ test, the quantity of material contained within these microdroplets actually falls within the nanogram range, particularly when dealing with fine fibres like CF. This opens up the potential for scaling problems that can lead to unexpected and substantial disparities in the properties and performance of these droplets when compared to larger-scale specimens even with similar thermal and environmental exposure [318]. In an infrequently referenced study, Haaksma and Cehelnik [319] noted several problems with microdroplet specimen preparation. These issues suggest potential challenges in scaling up the polymerisation process for thermosetting resins at this size. Given that the droplets were not fully cured, the researchers concluded that achieving an IFSS value was ‘impossible’ using the cure cycle conditions employed for SFFT specimens. In the context of epoxy systems, it has been demonstrated that the performance of thin specimens undergoes notable variations in comparison to bulk specimens, primarily attributed to internal stress experienced during the curing process [320]. This observation holds relevance for both microbond and SFFT specimens. Thomason [318] elaborated on the scaling issues in the specimen preparation and data interpretation of the microbond test, for both thermosetting and thermoplastic droplets. An extensive evidence has demonstrated the strong reliance of microbond test results on the properties of the microdroplet, including its modulus and

$T_g$ T g

Droplets exhibit lowered

$T_g$ T g

$T_g$ T g

$T_g$ T g

$T_g$ T g

$T_g$ T g

$T_g$ T g

The interfacial properties are affected by cure kinetics. The volumetric shrinkage of the resin during the cool-down results in residual compressive stress on the fibre. A larger temperature difference results in higher compressive stress and hence a larger IFSS. Thomason and Yang [323] evaluated the temperature dependence of the IFSS_app in a GF-PP system by performing the microbond test under controlled temperature in a thermo-mechanical analyser. They reported an inverse dependence of the IFSS_app with the testing temperature, explained by the higher radial residual compressive stress at lower testing temperatures.

The effects of fibre sizing removal and the addition of coupling agents on the IFSS_app enhancement were investigated in [255] for a range of recycled CF-PP systems. Lastly, to envision the workload, the number of tested specimens for each matrix material ranged from 50 to 151 [280,306,324], and the reported success rates were 51% (excluding the specimens that failed through fibre failure) [255], 95%, 72%, 38% for the

$l_{{\rm emb}}$ l e m b

${\rm \mu m}$ μ m

${\rm \mu m}$ μ m

${\rm \mu m}$ μ m

$l_{{\rm emb}}$ l e m b

Data reduction schemes

Basic approach

A simplistic apparent (also known as overall, mean, or average) IFSS is defined as

$${\rm IFS}{\rm S}_{{\rm app}} = \tau _{{\rm app}} = \displaystyle{{F_{{\rm max}}} \over {2\pi r_fl_{{\rm emb}}}}$$ I F S S a p p = τ a p p = F m a x 2 π r f l e m b

(13)

where

$F_{{\rm max}}$ F m a x

is the maximum load. Alternatively, the debond force,

$F_d$ F d

, could be used:

$\tau _{{\rm app}} = F_d/2\pi r_fl_{{\rm emb}}$ τ a p p = F d / 2 π r f l e m b

[324]. The IFSS_app, as a comparative identification of a weak or strong interface, assumes a constant interfacial shear stress and indicates an abrupt failure by ignoring debond propagation. The IFSS_app decreases with increasing

$A_{{\rm emb}}$ A e m b

, however, the

$A_{{\rm emb}}$ A e m b

requires a severalfold increase to yield a discernible reduction in the IFSS_app values for CF- and GF-epoxy systems [300]. This was attributed to the non-uniform shearing force over the

$A_{{\rm emb}}$ A e m b

and its concentration within the upper part of the droplet. The dependence of this conventional IFSS_app on

$l_{{\rm emb}}$ l e m b

, triggered the development of energy-based or ‘local’ IFSS approaches as interfacial failure criteria [324]. Figure 24 illustrates the consecutive stages of the load–displacement diagram of a microbond test. The initiation stage detection, inferred by the slope change (stage 2 in Figure 24), is demanding.OPEN IN VIEWER

Figure 24.

A typical microbond force-displacement curve indicating the four stages of the test [325]. The peak force at stage 3 is used to calculate IFSS_app (adapted from [255]).

In addition to the probable fibre break (leading to a sharp peak succeeded by a force drop), droplet slippage is another likely undesired incident within the experiments, displaying frictional resistance with no sharp peak on the

$F-\delta$ F − δ

curve [280]. Note that the initial tension, i.e. friction-induced load as the shearing plates move down the fibre before contact with the droplet, does not affect the calculated IFSS_app [300].

The

$F-\delta$ F − δ

curves obtained by pull-out tests and microbond tests are similar, except that the length over which the frictional sliding occurs reduces over further loading in the pull-out case, while it remains constant for microbond. The initial kink between the slopes of the

$F-\delta$ F − δ

curve (see stage 2 in Figure 24) indicates the debond initiation. Following the kink and preceding

$F_{{\rm max}}$ F m a x

, the debond propagation is stabilised by friction in the debonded regions [326]. As the crack length extends, the frictional force grows approximately proportional to the debonded area, and the cohesive contribution gradually decreases [327]. High frictional stress stabilises crack propagation [295].

$F_{{\rm max}}$ F m a x

solely represents the start of the unstable crack propagation and is affected by the friction in debonded sites [296]. The strong influence of the interfacial friction on the debonding process and

$F_{{\rm max}}$ F m a x

is confirmed in [295].

Other stress- and energy-based approaches

Reportedly [324], provided that

$d_f$ d f

is the same for all microbond specimens, the stress-based and energy-based failure criteria are equivalent; the discrepancy emerges only when specimens with different

$d_f$ d f

s are compared. Zhandarov et al. [312,324,325] used a 1D SLM to investigate the applicability of the energy- or stress-based approaches for debond initiation and propagation. Scheer and Nairn [328] adopted variational mechanics analysis and a modified SLM by incorporating the thermal residual stresses. However, they assumed a frictionless interface and an elastic matrix. Accordingly, the energy-based equation for the debonding initiation stress,

$\sigma _d$ σ d

, was expressed by

$$\sigma _d = -\displaystyle{{D_{3s}\Delta T} \over {C_{33s}}} + \sqrt {\displaystyle{{2G_{ic}} \over {r_fC_{33s}}} + \displaystyle{{{\rm \Delta }T^2} \over {C_{33s}}}\left({\displaystyle{{D_{3s}^2 } \over {C_{33s}}}-\displaystyle{{D_3^2 } \over {C_{33}}}} \right)} $$ σ d = − D 3 s Δ T C 33 s + 2 G i c r f C 33 s + Δ T 2 C 33 s ( D 3 s 2 C 33 s − D 3 2 C 33 )

(14)

Rearranging leads to IFFT (the corrected version, as given in [329], includes both friction and thermal residual stresses) for debond initiation:

$$\eqalign{ G_{ic} &#38; = \displaystyle{{r_f} \over 2}\Bigg[{C_{33s}\sigma_d^2 + 2D_{3s}\sigma_d\Delta T}\cr &#38; \quad{+ \left({\displaystyle{{D_3^2 } \over {C_{33}}} + \displaystyle{{V_m{\lpar {\alpha_{\,fT}-\alpha_m} \rpar }^2} \over {V_fA_0}}} \right){\rm \Delta }T^2} \Bigg]}$$ G i c = r f 2 [ C 33 s σ d 2 + 2 D 3 s σ d Δ T + ( D 3 2 C 33 + V m ( α f T − α m ) 2 V f A 0 ) Δ T 2 ]

(15)

and the shear-lag analysis leads to [328]

$$\sigma _d = -\displaystyle{{D_{3s}\Delta T} \over {C_{33s}}} + \sqrt {\displaystyle{{2G_{ic}} \over {r_fC_{33s}}}} $$ σ d = − D 3 s Δ T C 33 s + 2 G i c r f C 33 s

(16)

The interfacial friction was later included in

$G_{ic}$ G i c

in [329]. Using Equation (13) and the equation below:

$$\sigma _d = \displaystyle{{F_d} \over {\pi r_f^2 }}$$ σ d = F d π r f 2

(17)

IFSS_app becomes

$$\tau _{{\rm app}} = -\displaystyle{{D_{3s}\Delta Tr_f} \over {2C_{33s}l_{{\rm emb}}}} + \displaystyle{1 \over {l_{{\rm emb}}}}\sqrt {\displaystyle{{G_{ic}r_f} \over {C_{33s}}}} $$ τ a p p = − D 3 s Δ T r f 2 C 33 s l e m b + 1 l e m b G i c r f C 33 s

(18)

All the debuted parameters with their definitions can be found in the Appendix (Equations 13.A–18.A).

The deficiency of

$\tau _{{\rm app}}$ τ a p p

as an interfacial failure criterion is due to the dependence on

$l_{{\rm emb}}$ l e m b

and susceptibility to

$\mu _i$ μ i

[315]. Thereby, the concept of an ultimate IFSS,

$\tau _{{\rm ult}} = \mathop {\lim }\limits_{l_{{\rm emb}}\to 0} \tau _{{\rm app}}\lpar {l_{{\rm emb}}} \rpar$ τ u l t = lim l e m b → 0 ⁡ τ a p p ( l e m b )

, was conceived. As a stress-based criterion,

$\tau _{{\rm ult}}$ τ u l t

factors in the stress within the matrix induced by both thermal stresses and external load [330]. The concept of local and/or ultimate IFSS was proposed by Gorbatkina and Khazanovich [331]. The local IFSS,

$\tau _d$ τ d

, defined as the stress at which the local interfacial debonding occurs, is not necessarily equal to the ultimate IFSS [332,333], while some authors [330] have reported

$\tau _{ult} = \tau _d$ τ u l t = τ d

. However, near the debond tip, at the moment of debond initiation, the local IFSS equals the ultimate IFSS [279]. The main assumption is that the interfacial debonding initiates when the local interfacial shear stress at a point reaches the ultimate value (

$\tau _{ult}$ τ u l t

). Following the ultimate IFSS criterion, shear-lag analysis with a stress-based debonding [328], the derived equations with their definitions are given in Appendix (Equations 19.A–24.A).

Concerning the effect of friction throughout debond propagation, cohesive and frictional components of the microbond test were extracted by both stress- and energy-based criteria in [325]. Both frictional and residual thermal stresses regulate stable crack propagation. An energy-based debonding criterion (

$G_{ic}$ G i c

 = constant), unlike the stress-based approach, is incompatible with an unstable (catastrophic) debond propagation prediction at high debond lengths (

$G_{ic}$ G i c

≠ constant throughout the test). However, post-debonding interfacial friction is best described by the energy-based approach [325].

The most eminent inconsistency between different analyses of the microbond/pull-out tests is concerned with the point where the fibre enters the matrix (see Figure 25). A linear-elastic FEM yields a singularity at this point [334]; variational mechanics gives zero shear stress [328,335]; SLM estimations fall in between. However, a bit farther from this point, at a distance of

$l_m$ l m

, all the three approaches yield analogous interfacial shear stress values.OPEN IN VIEWER

Figure 25.

Interfacial shear stress profiles along the fibre embedded length in a microbond test, predicted by three different approaches – the

$l_m$ l m

(located at a distance of a few fibre diameters from the fibre end) is the point where the results of the three models converge (

$l_m\ll l_{{\rm emb}}$ l m ≪ l e m b

) (redrawn from [237], with permission from Elsevier).

An alternative to the

$\tau _{{\rm app}}$ τ a p p

approach is to exploit some of the numerous analytical models established for the classical pull-out test, even though the experimental configuration is different. For a ductile interfacial fracture,

$\tau _{{\rm app}}$ τ a p p

is reported to be uniform along the interface, thus affirming the applicability of Equation (13) (e.g. for GF-PPS system [302]). However, according to LRS [336], photoelastic analysis [293] and FEM [297,337–339], the interfacial shear stress is not constant but varies both circumferentially and axially, and manifests a peak at a 0.1–1

$d_f$ d f

away from the droplet top [340]. For a brittle interfacial failure, the shear stress diverges from the

$\tau _{{\rm app}}$ τ a p p

approach, and using the shear-lag analysis, leads to a stress-based criterion as in the work of Greszczuk [241]:

$$\tau _{{\rm ult}} = \displaystyle{{F_{{\rm max}}} \over {2\pi r_f}}\sqrt {\displaystyle{{2G_{{\rm int} .}} \over {b_ir_fE_f}}} \coth \left({\sqrt {\displaystyle{{2G_{{\rm int} .}} \over {b_ir_fE_f}}} l_{{\rm emb}}} \right)= \tau _{{\rm app}}\sqrt {\displaystyle{{2G_{{\rm int} .}} \over {b_ir_fE_f}}} \coth \left({\sqrt {\displaystyle{{2G_{{\rm int} .}} \over {b_ir_fE_f}}} l_{{\rm emb}}} \right)$$ τ u l t = F m a x 2 π r f 2 G i n t . b i r f E f coth ⁡ ( 2 G i n t . b i r f E f l e m b ) = τ a p p 2 G i n t . b i r f E f coth ⁡ ( 2 G i n t . b i r f E f l e m b )

(19)

This expression pertains to particular interface parameters, which restricts its applicability (in its current form) as these variables are typically unknown. To circumvent this issue,

$b_i$ b i

(width of the matrix under shear stress or the interface effective thickness) and

$G_{int.}$ G i n t .

(the interface shear modulus) are typically presumed to be

$r_f\ln \displaystyle{{R_m} \over {r_f}}$ r f ln R m r f

and

$G_m$ G m

, respectively [252]. Greszczuk's model and

$\tau _{app}\;$ τ a p p

provide analogous trends. The microbond test is reported to provide mixed-mode properties and not exclusively a mode II interfacial toughness or IFSS. Consequently, energy-based approaches were preferred over stress-based models [219,340] due to severe interfacial stress field heterogeneities. Following an energy-based approach (total energy criterion, i.e. the energy per unit length equals the

$G_{ic}$ G i c

), Piggott [219] concluded:

$$F_{{\rm max}} = 2\pi r_f\sqrt {E_fG_{ic}l_{{\rm emb}}n\tanh \left({\displaystyle{{nl_{{\rm emb}}} \over {r_f}}} \right)\; } $$ F m a x = 2 π r f E f G i c l e m b n tanh ⁡ ( n l e m b r f )

(20)

Penn and Lee [256] used a more precise energy-based criterion by considering a pre-crack (negligible compared to

$l_{emb}$ l e m b

):

$$F_{{\rm max}} = \displaystyle{{2\pi r_f\sqrt {r_fG_{ic}E_f} } \over {\sqrt {1 + {\rm csc}{\rm h}^2\left({\displaystyle{{nl_{{\rm emb}}} \over {r_f}}} \right)\; } }}$$ F m a x = 2 π r f r f G i c E f 1 + c s c h 2 ( n l e m b r f )

(21)

with

$$n = \sqrt {\displaystyle{{E_m} \over {E_f\lpar {1 + \nu_m} \rpar \ln \displaystyle{R \over {r_f}}}}} $$ n = E m E f ( 1 + ν m ) ln R r f

(22)

where

$R$ R

is an axial distance at which

$\tau _m = 0$ τ m = 0

. However, both approaches exhibit an incomplete energy balance (residual stresses being excluded). Additionally, all the above-mentioned models inaccurately adopt a linear elastic behaviour for the droplet, leading to an overestimated

$G_{ic}$ G i c

. The theoretical analysis must be reviewed to incorporate the energy dissipation via plastic flow (e.g. via the

$J$ J

-integral criterion) [340]. The inclusion of residual stresses, affecting the mechanical part of the interfacial bond, must not be overlooked. The eligibility of six theoretical models (

$\tau _{{\rm app}}$ τ a p p

, Greszczuk, Piggott, Penn and Lee, Scheer and Nairn shear-lag, and variational mechanics) was investigated in [340]. The Scheer and Nairn model [328] was appointed as the most authoritative model. The quality of fitting the experimental data by energy-based approaches can be graphically appreciated in Figure 26.OPEN IN VIEWER

Figure 26.

Comparison of the energy-based models predictions of the force-embedded length with the experimental data for the microbond test of GF-epoxy (redrawn from [340], with permission from Elsevier).

The three different methods of determining

$G_{ic}$ G i c

in a microbond test are [327]:

the ‘traditional’ method based on

$F_{{\rm max}}$ F m a x

and

$F_d$ F d

(debonding force, see Figure 27),

the ‘alternative’ method based on

$F_{{\rm max}}$ F m a x

and

$F_b$ F b

(initial post-debonding force, Figure 27),

the ‘indirect’ method based on

$F_{{\rm max}}$ F m a x

as a function of

$l_{{\rm emb}}$ l e m b

.

OPEN IN VIEWER

Figure 27.

An idealised force-displacement curve of a pull-out test. Segment

$OA$ O A

corresponds to an intact interface, and the interfacial debonding initiates at

$A$ A

and terminates at

$C$ C

, while meeting the peak force at

$B$ B

. The ‘tail’ force (segment

$CD$ C D

) is due to the interfacial friction. In the microbond test at this stage, the contact length between the fibre and the matrix is constant, and the measured force also remains constant, i.e. segment

$CD$ C D

is horizontal (redrawn from [341], with permission from Elsevier).

The interfacial parameters (

$\tau _d$ τ d

and/or

$G_{ic}$ G i c

) and the interfacial frictional stress in the debonded surfaces,

$\tau _f$ τ f

, can be determined ‘indirectly’ by

$F_{{\rm max}}$ F m a x

values. This is feasible, provided that the peak forces were recorded versus

$l_{{\rm emb}}$ l e m b

over an adequately wide interval of

$l_{{\rm emb}}$ l e m b

(e.g. 0 < 

$l_{emb}$ l e m b

 ≤ 1 mm with increments of 5

${\rm \mu m}$ μ m

[332,333]). Analytical models have been proposed that relate

$F_{{\rm max}}$ F m a x

to

$l_{{\rm emb}}$ l e m b

within stress-based [332] or energy-based approaches [333]. Despite providing more accurate re-calculation of the interfacial properties from old microbond (or pull-out) tests that merely contain

$F_{{\rm max}}\lpar {l_{{\rm emb}}} \rpar$ F m a x ( l e m b )

, this indirect approach may be inaccurate [279]. This imprecision stems from underestimating the

$\tau _f$ τ f

values, particularly when for individual specimens:

$\tau _f\ll \tau _d$ τ f ≪ τ d

, and accordingly marginally overestimated

$\tau _d$ τ d

values. The results acquired by force–displacement curves of individual specimen (not for a specimen set as for the indirect method) are more precise, and therefore, the indirect method is exclusively recommended when

$F_d$ F d

and

$F_b$ F b

are unachievable [341].

Since

$F_d$ F d

is unaltered by the interfacial friction, it is the main experimentally obtained value whereby the

$\tau _{ult}$ τ u l t

can be directly deduced (as in [324]). However, for some fibre–matrix systems,

$F_d$ F d

, emerging as a ‘kink’ in the force-displacement curve, cannot be reliably measured since their curves yield a barely perceptible kink(s). The kink emergence requires a very stiff specimen and test set-up. An alternative method was proposed for quantifying the local IFSS [341] and

$G_{ic}$ G i c

[327] in pull-out and microbond tests. This method uses

$F_{{\rm max}}$ F m a x

and

$F_b$ F b

, obviating the need for

$F_d$ F d

. Mostly on the force-displacement curves, the transition point from debonding to post-debonding is barely perceptible. Using three different methods of data reduction schemes, Zhandarov and Mäder [341] demonstrated that the results of this new method are very similar to the traditional one. Thus it can be recommended as an alternative technique when the

$F_d$ F d

is indistinguishable. The ‘alternative’ method was extended to an energy-based approach to estimate the

$G_{ic}$ G i c

, and the

$\tau _f$ τ f

[327].

The

$\tau _f$ τ f

(segment CD in Figure 27), for a pull-out test can be obtained via

$$\tau _f = \displaystyle{{F_b} \over {2\pi r_f\lpar {l_{{\rm emb}}-l_b} \rpar }}$$ τ f = F b 2 π r f ( l e m b − l b )

(23)

and for the microbond test (

$l_{{\rm emb}}\gg l_b$ l e m b ≫ l b

), via

$$\tau _f = \displaystyle{{F_b} \over {2\pi r_fl_{emb}}}$$ τ f = F b 2 π r f l e m b

(24)

The typical values of

$\tau _f$ τ f

are <4 MPa for E-glass-PP systems and <18 MPa for CF-PA6,6 systems [341].

Zhandarov et al. [332] assessed the correlation between the

$F_{{\rm max}}$ F m a x

and the

$l_{{\rm emb}}$ l e m b

to separately determine the interfacial adhesion and friction. The stress-based portrayal of theoretical curves of

$F_{{\rm max}}\lpar {l_{{\rm emb}}} \rpar$ F m a x ( l e m b )

, accounting for the local IFSS,

$\tau _d$ τ d

, and the interfacial friction,

$\tau _f$ τ f

, is [309,332]:

$$ F_{{\rm max}}\lpar {l_{{\rm emb}}} \rpar = \left\{{\matrix{ {\displaystyle{{2\pi r_f} \over \beta }\left[{\tau_d\tanh \lpar {\beta l_{{\rm emb}}} \rpar -\tau_T\tanh \lpar {\beta l_{{\rm emb}}} \rpar \tanh \left({\displaystyle{{\beta l_{{\rm emb}}} \over 2}} \right)} \right]} \cr {\displaystyle{{2\pi r_f} \over \beta }\left[{\tau_d\displaystyle{u \over {\sqrt {\lpar {u^2 + 1} \rpar } }}-\tau_T\left({1-\displaystyle{1 \over {\sqrt {\lpar {u^2 + 1} \rpar } }}} \right)+ \tau_f\left[{\beta l_{{\rm emb}}-\ln \left({u + \sqrt {\lpar {u^2 + 1} \rpar } } \right)} \right]} \right]} \cr } } \right.$$ F m a x ( l e m b ) = { 2 π r f β [ τ d tanh ⁡ ( β l e m b ) − τ T tanh ⁡ ( β l e m b ) tanh ⁡ ( β l e m b 2 ) ] 2 π r f β [ τ d u ( u 2 + 1 ) − τ T ( 1 − 1 ( u 2 + 1 ) ) + τ f [ β l e m b − ln ⁡ ( u + ( u 2 + 1 ) ) ] ]

(25)

where

$\beta$ β

can be equalised to

$\beta _{{\rm Nayfeh}}$ β N a y f e h

(see Equation 23A). Using the definition of IFSSapp in Equation (13),

$\tau _{{\rm app}}\lpar {l_{{\rm emb}}} \rpar$ τ a p p ( l e m b )

is achieved as [309]

$$\tau _{{\rm app}}\lpar {l_{{\rm emb}}} \rpar = \left\{{\matrix{ {\displaystyle{{2\pi r_f} \over \beta }\left[{\tau_d\displaystyle{{\tanh \lpar {\beta l_{{\rm emb}}} \rpar } \over {\lpar {\beta l_{{\rm emb}}} \rpar }}-\tau_{{\rm thermal}}\displaystyle{{\tanh \lpar {\beta l_{{\rm emb}}} \rpar } \over {\lpar {\beta l_{{\rm emb}}} \rpar }}\tanh \left({\displaystyle{{\beta l_{{\rm emb}}} \over 2}} \right)} \right]} \cr {\displaystyle{{\tau_d} \over {\beta l_{{\rm emb}}}}\displaystyle{u \over {\sqrt {\lpar {u^2 + 1} \rpar } }}-\displaystyle{{\tau_{{\rm thermal}}} \over {\beta l_{{\rm emb}}}}\left({1-\displaystyle{1 \over {\sqrt {\lpar {u^2 + 1} \rpar } }}} \right)+ \displaystyle{{\tau_f} \over {\beta l_{{\rm emb}}}}\left[{\beta l_{{\rm emb}}-\ln \left({u + \sqrt {\lpar {u^2 + 1} \rpar } } \right)} \right]} \cr } } \right.$$ τ a p p ( l e m b ) = { 2 π r f β [ τ d tanh ⁡ ( β l e m b ) ( β l e m b ) − τ t h e r m a l tanh ⁡ ( β l e m b ) ( β l e m b ) tanh ⁡ ( β l e m b 2 ) ] τ d β l e m b u ( u 2 + 1 ) − τ t h e r m a l β l e m b ( 1 − 1 ( u 2 + 1 ) ) + τ f β l e m b [ β l e m b − ln ⁡ ( u + ( u 2 + 1 ) ) ]

(26)

with

$$u = \displaystyle{{\sqrt {\tau _{{\rm thermal}}^2 + 4\tau _f\lpar {\tau_d-\tau_f} \rpar } -\tau _{{\rm thermal}}} \over {2\tau _f}}$$ u = τ t h e r m a l 2 + 4 τ f ( τ d − τ f ) − τ t h e r m a l 2 τ f

(27)

If

$\beta l_{{\rm emb}} \lt \ln \left({u + \sqrt {\lpar {u^2 + 1} \rpar } } \right)$ β l e m b < ln ⁡ ( u + ( u 2 + 1 ) )

, the debonding is catastrophic (unstable and

$\tau _f = 0$ τ f = 0

), while for

$\beta l_{{\rm emb}} \ge \ln \left({u + \sqrt {\lpar {u^2 + 1} \rpar } } \right)$ β l e m b ≥ ln ⁡ ( u + ( u 2 + 1 ) )

cases, the debond propagation is stable [309]. The effects of specimen geometry and

$l_{emb}$ l e m b

on

$u$ u

are addressed in [309]. These theoretical curves, with two fitting parameters

$\tau _d$ τ d

and

$\tau _f$ τ f

and a non-linear least-squares method [332], were compared to the experimental microbond (and pull-out) data in [309,332]. Similarly, as a sequel to their work, an energy-based analysis [333] was developed, which exhibited a better fit for larger

$l_{emb}$ l e m b

.

The Nairn analytical model for the pull-out and the microbond test [276], representing the

$G\lpar {l_d} \rpar$ G ( l d )

(energy release rate as a function of debonded length) is

$$\eqalign{&#38; G\lpar {l_d} \rpar =\cr &#38; \displaystyle{{r_f} \over 2}\Bigg\{{C_{33s}{\bar{\sigma }}^2 + 2D_{3s}\bar{\sigma }\Delta T + \Bigg({\displaystyle{{D_3^2 } \over {C_{33}}} + \displaystyle{{V_m{\lpar {\alpha_{\,fT}-\alpha_m} \rpar }^2} \over {V_fA_0}}} \Bigg)}\lpar {{\rm \Delta }T} \rpar^2\cr&#38; {-\Bigg[{\displaystyle{{\sigma_0} \over 2}\Bigg({\displaystyle{1 \over {E_{\,fL}}}-\displaystyle{1 \over {E_m}}} \Bigg)+ D_{3s}\Delta T} \Bigg]\hfill} \cr &#38; {\times \Bigg[{kC_T\lpar {l_d} \rpar -\Bigg({\bar{\sigma } + \displaystyle{{\lpar {1 + m} \rpar D_3{\rm \Delta }T} \over {C_{33}}}} \Bigg){\mathop C\limits^{\prime} }_T\lpar {l_d} \rpar } \Bigg]\hfill} \Bigg\}}$$ G ( l d ) = r f 2 { C 33 s σ ¯ 2 + 2 D 3 s σ ¯ Δ T + ( D 3 2 C 33 + V m ( α f T − α m ) 2 V f A 0 ) ( Δ T ) 2 − [ σ 0 2 ( 1 E f L − 1 E m ) + D 3 s Δ T ] × [ k C T ( l d ) − ( σ ¯ + ( 1 + m ) D 3 Δ T C 33 ) C ′ T ( l d ) ] }

(28)

where

$$\bar{\sigma } = \sigma -kl_d-\displaystyle{{\sigma _0E_{\,fL}} \over {V_fE_{\,fL} + V_mE_m}}$$ σ ¯ = σ − k l d − σ 0 E f L V f E f L + V m E m

(29)

with

$\sigma = F/\pi r_f^2$ σ = F / π r f 2

(tensile stress on fibre),

$\sigma _0 = \sigma V_f\lpar {1-m} \rpar$ σ 0 = σ V f ( 1 − m )

as the net axial stress and

$k = 2\tau _f/r_f$ k = 2 τ f / r f

being the frictional stress transfer rate (for frictionless case:

$k = 0$ k = 0

).

$A_0$ A 0

,

$C_{33}$ C 33

,

$C_{33s}$ C 33 s

,

$D_3$ D 3

and

$D_{3s}$ D 3 s

, dependent on the properties of the constituents, are given in the Appendix of [276]. Although

$C_{33}\approx C_{33s}$ C 33 ≈ C 33 s

and

$D_3\approx D_{3s}$ D 3 ≈ D 3 s

in many cases, it is recommended to distinguish between them [276]. For the pull-out test,

$m = 0$ m = 0

and for the microbond test,

$m = 1$ m = 1

. Using an SLM, the Liu and Nairn [329] approach for evaluating the cumulative stress transfer function,

$C_T\lpar {l_d} \rpar$ C T ( l d )

, led to:

$$C_T\lpar {l_d} \rpar = \displaystyle{1 \over {\beta _{{\rm Nayfeh}}}}\tanh \displaystyle{{\beta _{{\rm Nayfeh}}\lpar {l_{{\rm emb}}-l_d} \rpar } \over 2}$$ C T ( l d ) = 1 β N a y f e h tanh β N a y f e h ( l e m b − l d ) 2

(30)

$${C}^{\prime}_T\lpar {l_d} \rpar = -\displaystyle{1 \over 2}{\rm sec}{\rm h}^2\displaystyle{{\beta _{{\rm Nayfeh}}\lpar {l_{{\rm emb}}-l_d} \rpar } \over 2}$$ C T ′ ( l d ) = − 1 2 s e c h 2 β N a y f e h ( l e m b − l d ) 2

(31)

Considering

$G\lpar {l_d} \rpar = G_{ic} = {\rm const}.$ G ( l d ) = G i c = c o n s t .

, the applied force as a function of the debond length [333] becomes

$$F\lpar {l_d} \rpar = \pi r_f^2 \left[{-\displaystyle{{c_1\lpar {l_d} \rpar } \over {2c_2\lpar {l_d} \rpar }} + \sqrt {{\left({\displaystyle{{c_1\lpar {l_d} \rpar } \over {2c_2\lpar {l_d} \rpar }}} \right)}^2-\displaystyle{{c_0\lpar {l_d} \rpar -G_{ic}} \over {c_2\lpar {l_d} \rpar }}} } \right]$$ F ( l d ) = π r f 2 [ − c 1 ( l d ) 2 c 2 ( l d ) + ( c 1 ( l d ) 2 c 2 ( l d ) ) 2 − c 0 ( l d ) − G i c c 2 ( l d ) ]

(32)

This is the corrected version which is given in [327]. The equations of

$\tau \lpar {l_d} \rpar$ τ ( l d )

and

$G\lpar {l_d} \rpar$ G ( l d )

(variations of

$\tau$ τ

and

$G$ G

with the debonded length that build

$\; \tau$ τ

- and

$R$ R

-curves), given in [312], satisfactorily characterised the fibre–matrix interface in microbond tests [312]. Regarding the correct inclusion of

$V_f$ V f

in these calculations, it should be pointed out that

$V_f\;$ V f

depends on the

$l_{emb}$ l e m b

(not in the case of cylindrical droplets) and consequently,

$\beta$ β

,

$\tau _{{\rm thermal}}$ τ t h e r m a l

and

$u$ u

also are functions of the

$l_{emb}$ l e m b

[309]. Exclusive

$V_f$ V f

calculations for different geometries (microbond and pull-out tests) are presented in [332].