Simultaneous Measurement of Fiber-Matrix Interface Debonding and Tunneling Using a Dual-Vision Experimental Setup
- Open Access
- 16.09.2024
- Research paper
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Abstract
Introduction
Fiber-matrix debonding is one of the most consequential damage modes in fiber composites as it is a precursor for several other types of failure. This form of damage initiates at relatively low stresses at the interface between a fiber and its surrounding matrix. At higher stresses, the debonded area propagates circumferentially around the interface, eventually kinking out into the matrix. The kinked-out microcracks from neighboring fibers coalesce and form larger cracks that expand rapidly through the ply thickness in a direction perpendicular to the direction of the remote tensile load. These ply-thick cracks are referred to as transverse cracks and are known as one of the leading causes of structural failure in cross-ply composites [1, 2]. Transverse cracks in fiber composites affect the load-bearing capacity by reducing the stiffness, also deteriorating physical properties (e.g., thermal expansion coefficient and thermal conductivity) of the material [3]. More importantly, transverse cracks can serve as nucleation sites for other types of damage including but not limited to delamination and fiber breakage. Therefore, the implications of fiber-matrix debonding in fiber composites are significant.
Fiber-matrix debonding and its influence on other damage types in composites have been investigated for decades. An overwhelming majority of research studies on these topics have been computational and modeling-based [4‐8]. The reason for the extensive use of computational methods in this area perhaps stems from the extreme length scales associated with the nature of this type of damage. Despite the dominance of computational studies in this area, experimental investigations have also been conducted in the past decade to provide insights into the mechanics of the fiber-matrix interface to verify the numerous computational studies while also attempting to reveal the fiber-to-fiber interactions that affect the coalescence of debonded interfaces and their resultant matrix microcracks. Most if not all the experimental research studies on the aforementioned topics have been designed and conducted with image-based methods. For instance, the pioneering work of Zhang et al. [9] investigated the fiber-matrix interface debonding process by optical microscopy, leveraging pre-fragmented single glass fibers in an epoxy matrix. Supplemented by modeling studies [10], the in-situ measurements of Zhang et al. showed unstable debonding growth mechanisms that were independent of the fiber surface treatment. The effects of fiber surface treatment were reported to cause different critical stress values at the debonding initiation. Similar single-fiber optical characterizations were performed by Koyanagi et al. [11]. Experimental results obtained in this work were used in conjunction with finite element simulations to quantify the normal and shear components of the interface strength in a transversely loaded single-fiber system. Despite proving the potential effectiveness of optical measurements in single-fiber studies, the results reported in the aforementioned references relied heavily on qualitative assessments rather than quantitative data analyses.
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Quantitative evaluation of the fiber-matrix debonding mechanism was performed in the work of Martinyuk et al. [12] wherein unprecedented data on the fiber-matrix debonding and crack tunneling (i.e., propagation of the debonded interface along the fiber axis) mechanisms in single fiber composite samples were reported. This study used single-fiber samples loaded inside an X-ray computed tomography instrument. Despite being unique in terms of the design of the experiment and the promising results obtained, such in-situ test protocols are difficult to achieve and often deemed impractical due to the potential effects of the penetrating X-ray on the material and also the stepwise nature of the loading that hinders continuous observation and data extraction. In-situ characterization of the mechanics and failure at the fiber-matrix interface has also been performed by testing single and multi-fiber samples inside scanning electron microscopes (SEM). Combined with digital image correlation (DIC) [13‐15] the latter approaches have enabled a relatively accurate assessment of the interfacial damage with minimal adverse effects on the material, especially when carried out after the application of a protective DIC-suitable speckle pattern, like in Montgomery et al. [16] and Koohbor et al. [17]. However, the issues associated with the intermittent nature of testing inside an SEM chamber remain unresolved, that is, the loading must stop while an image is rasterized.
In recent years, image-based characterizations based on optical DIC have been introduced as an alternative and utilized successfully to provide a continuous stream of data (i.e., no need for intermittent loading-imaging process), minimizing the potential effects of X-ray and e-beam irradiation on the material properties, and no requirement for customized speckle pattern application to enable measurements at reduced length-scales. For example, the series of optical-DIC studies performed by Tabiai et al. [18‐20] on macro-fiber components have highlighted the promise of optical full-field measurements in revealing the fundamental mechanisms that govern the fiber-matrix interface failure. The full-field measurements performed by Tabiai et al. [18‐20] on a wide range of fiber materials (e.g., carbon fiber, PTFE, PLA, carbon steel, galvanized steel) and fiber diameters (from 7.5 to ~ 700 μm in diameter) provided extremely useful information that allowed comparison of interface strength in various fiber-matrix systems. More importantly, quantitative comparisons of the kinematic fields in the vicinity of a single fiber and a bundle of fibers in Tabiai et al. [18] suggested that a fiber bundle can be reasonably assumed to behave like an isolated single fiber in terms of fundamental fiber-matrix debonding initiation and propagation mechanisms. However, the experimental methods discussed in the aforementioned reports were still incapable of providing quantitative information on crack tunneling and correlations between in-plane and out-of-plane debonding mechanisms. The application of optical microscopy on in-situ real-time measurement of interface debonding in short fiber composites was also discussed in a recent publication by Nikforooz et al. [21]. The methodology proposed by Nikforooz et al. enabled quantitative comparison of different fiber sizings in mixed-mode debonding growth in E-glass/epoxy systems. Recently, the present authors also introduced systematic coupled experimental-computational approaches that utilized full-field strain measurements in macro-fiber systems to uncover some of the fundamental mechanisms that control the fiber-matrix interface debonding initiation and growth, and the effect of adjacent fibers on the fiber-matrix interface failure in glass fiber-epoxy systems [22, 23]. Although proven extremely effective, these papers only focused on in-plane deformation kinematics without taking into account the out-of-plane deformations in the fiber-matrix vicinity and the crack tunneling effect along the fiber-matrix interface. The companion modeling studies suggested that the in-plane data can indeed be used as a first-order approximation of the fiber-matrix interface fracture properties, but more realistic characterizations require considering the actual 3D configuration including the contribution of out-of-plane fields as well [24, 25].
The present work takes the above-discussed experimental [22, 23] investigation of fiber-matrix interface to the next step by introducing a novel method that utilizes a simple yet effective computer vision approach to enable the simultaneous characterization of surface and through-thickness mechanics and failure in single macro fiber model composites. The presented approach is unique because it, first, sheds light on the out-of-plane deformation at the fiber-matrix interface, a concept that has been missing from almost all the preceding body of experimental-based literature on the topic. Second, the new dual-vision method used in this work enables a continuous collection of data to correlate remote stress and strains with their corresponding local crack propagation parameters, both on the surface (in-plane debonding) and through-thickness (crack tunneling). The merit of such measurements is realized and discussed through experimental data sets as input to inverse-identification computational studies that describe the cohesion and separation of the bi-material interface in fiber composites. The inverse methods presented herein also discuss the shortcomings of an ‘experimental only’ approach and provide practical strategies to address image-based measurements.
Experimental
Materials and Sample Preparation
Single macro fiber samples were fabricated by embedding a 1 mm diameter quartz glass rod (hereafter referred to as glass macro fiber) in a clear room-curable epoxy resin (105/209 epoxy, West System, MI, US). The resin was mixed and poured inside a silicone rubber mold whose dimensions were in accordance with ASTM D638. The glass macro fiber (McMaster-Carr, USA) was immediately inserted into and held in place inside the resin using a 3D-printed jig [22]. Before insertion, the glass macro fiber was cleaned with isopropanol and treated with a silane glass treatment solution (AP115 Silane Glass Treatment, 3 M, MN, USA). This solution creates a hydrophobic surface that leads to stronger bonds with the epoxy resin.
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The sample was cured at room temperature inside a chemical fume hood for 48 h. After curing, the clear epoxy resin embedding the single glass macro fiber at its center was extracted from the silicone mold and held for an additional 12 h at room temperature to ensure all surfaces were fully cured. The extra length of the macro fiber protruding out of the rear side of the sample was trimmed by a diamond glass cutter about 1 mm above the cured epoxy surface and then further polished (with fine sandpaper) to ensure the back side of the sample was as flat as possible. The side surfaces of the sample were lightly polished using extra fine sandpapers to ensure the surfaces were free of scratches. A highly polished side surface was also crucial for the side-view measurements of crack tunneling, discussed in the following. The front surface of the sample was also lightly polished to ensure that the cross-section of the glass macro fiber was flush with the epoxy on the camera-facing side of the tensile samples.
Digital image correlation (DIC) was used to characterize the full-field strain distribution on and in the vicinity of the macro fiber on the front surface of the tensile samples. To enable optical DIC (see Mechanical Tests and Image‑Based Measurements for details), a fine black-and-white pattern was applied. The polished front surface of the tensile sample was first coated with a thin layer of matte white paint. Immediately after the application of the white substrate coating, the sample was subjected to an aerosol cloud of black paint particles inside a closed container. The immediate application of the black paint particles ensured their adhesion to the surface of the damp white paint, resulting in a sufficiently fine pattern suitable for optical DIC at sub-millimeter scales. The quality assessment of this patterning technique has been reported by Girard et al. [23]. To ensure that the pattern application would not affect the side-view (crack tunneling) characterizations, all sample sides except the front surface were masked with painter tape. Figure 1 shows the tensile sample dimensions and a representative speckle pattern used for digital image correlation analyses in this work. The speckle-patterned surfaces were stored for another two days inside a chemical fume hood to ensure that the coating was fully dried and brittle enough to crack upon the initiation of fiber-matrix interface debonding. Note that the potential effect of sample edges on the deformation and failure of the 1 mm diameter was studied previously and shown to be negligible [23]. The slightly off-centered and asymmetric placement of the macro fiber was a challenge encountered in this work and our previous articles [22, 23]. The displacement of the fiber is likely due to the small volumetric variations that occurred during the curing of the epoxy resin to chemical shrinkage. The exact location of the fiber center was determined optically and used as input to the FE analyses to ensure realistic replication of experimental measurements by the model.
Fig. 1
a Schematic of the tensile test sample with a glass macro fiber at its center. b camera view of the macro fiber and its surrounding epoxy showing the speckle pattern. c A zoomed-in view of the macro fiber and the symmetric debonding at its poles after failure
Mechanical Tests and Image-Based Measurements
The speckle-patterned samples were subjected to uniaxial tensile tests performed at a constant crosshead speed of 1 mm.min−1 until fracture. The mechanical test frame used herein was a table-top Shimadzu AGS-X equipped with a 10 kN load cell capacity. The force-displacement data acquisition rate was synced with the imaging rate of 1 Hz to ensure the global stress values corresponding to each image were recorded. Three sets of mechanical tests and DIC analyses were performed in this study, as described in the following.
1.
The mechanical response of the base epoxy was characterized in-house. To this end, neat epoxy samples were prepared following the process elaborated in Materials and Sample Preparation but without the inclusion of the glass macro fiber. The in-plane longitudinal, εyy, and transverse, εxx, strain components were measured by DIC and used to determine the true stress-strain behavior and the Poisson’s ratio of the epoxy in this work. The procedure to characterize the tensile response of the neat epoxy is excluded here for brevity but is described in detail in the authors’ previous publication [22].
2.
A stereo DIC analysis was conducted to characterize the deformation response in single-fiber samples emphasizing out-of-plane deformations in the fiber vicinity. As elaborated in the forthcoming sections, slight out-of-plane deformations occur at the fiber-matrix interface due to a combination of the Poisson effect and sharp property gradients across the interface. Although insignificant, these out-of-plane deformations contribute to the stress triaxiality at the fiber-matrix interface and are, therefore, essential for the interpretation of crack tunneling mechanisms and FE model calibration. As shown schematically in Fig. 2, the stereo DIC setup herein consisted of two five-mega-pixel cameras each equipped with a 100 mm macro lens (Tokina AT-X) facing the front surface of the tensile sample and oriented at an approximately 20° stereo angle. The imaging system was calibrated using back-lit miniature calibration targets supplied by Correlated Solutions, Inc (SC, USA).
3.
The same cameras used for stereo DIC measurements were repurposed to create a dual-vision system for the simultaneous characterization of in-plane fiber-matrix debonding (and its corresponding strain fields) and debond crack tunneling along the fiber, referred hereafter to as front-view and side-view measurements, respectively (Fig. 2b).
Fig. 2
Schematic of the two image-based measurement methods used in this work: a stereo-DIC for surface measurements, b dual-vision system for simultaneous characterization of surface deformation and crack tunneling through thickness
In all the experiments mentioned above, the same speckle patterning process was employed. Depending on the utilized DIC mode, the images captured during mechanical tests were analyzed either in Vic-3D for stereo DIC, or in Vic-2D for front-view dual-vision analyses. (Vic-2D/3D are commercial image correlation software packages by Correlation Solutions, Inc., SC, USA). The main image correlation parameters used here are listed in Table 1. The dual-vision tests were performed on at least five single fiber samples. The results presented and discussed in Sect. 3.2 and thereafter are representative of a best-case scenario, reported after ensuring consistency between the trends and values among all independent measurements.
Table 1
The DIC parameters and the corresponding measurement noise values
Imaging system | ||
|---|---|---|
Image Correlation Parameter | Stereo DIC | Front-view 2D DIC |
Subset size, Lsub (µm) | 101.2 | 142.8 |
Step size, Lst (µm) | 2.9 | 5.7 |
Strain window size, Lwindow (data point) | 5 | 5 |
Virtual strain gauge size*, LVSG (µm) | 112.8 | 165.6 |
Strain noise floor | 0.485 × 10−3 | 1.571 × 10−3 |
Strain bias | 0.125 × 10−3 | 0.212 × 10−3 |
In-plane displacement noise floor (µm) | 0.62 | 0.14 |
In-plane displacement bias (µm) | 5.34 | 3.59 |
Out-of-plane displacement noise floor (µm) | 1.96 | - |
Image scale (µm/pixel) | 2.89 | 5.71 |
Matching Criterion | Zero-normalized squared differences | |
The subset, step, and strain window sizes were selected to ensure consistency between the virtual strain gauge sizes between the two measurement scenarios (i.e., between Stereo DIC and Front-view 2D DIC measurement approaches). The strain noise floor and bias corresponding to each imaging system were characterized using baseline measurement protocols elaborated in [26, 27].
Results and Discussion
Mechanical Properties of the Neat Epoxy
Figure 3 shows the stress-strain curve of the neat epoxy used in this work. The examined room-curable epoxy shows slight ductility and a true failure strain of about 0.03. The Young’s modulus, Poisson’s ratio, and tensile strength of the epoxy were characterized as 3.04 ± 0.11 GPa, 0.41 ± 0.03, and 59.5 ± 2.0 MPa respectively. These values were determined as the average of five independent measurements to account for experimental uncertainty and variability. Glass macro fibers used in this study were not tested in-house. However, the elastic modulus of the fibers was provided by the supplier as 72 GPa.
Fig. 3
True stress-strain curve of the epoxy. Scatter bars indicate experimental variability (standard deviation) among five measurements
Out-of-Plane Deformation
Figure 4 shows the out-of-plane deformation data collected at various stages of deformation (i.e., different global stresses). Out-of-plane deformation values extracted over two orthogonal 3 mm line markers are shown in Fig. 4(a), (b), indicating larger inward deformations (contraction) of the matrix in the x-direction, i.e., perpendicular to the loading direction.
Fig. 4
Variation of out-of-plane displacement, uz, with respect to remote tensile stress, \(\:{\sigma\:}^{\infty\:}\), along orthogonal directions in the vicinity of the glass macro fiber in (a) x-direction, b y-direction. c Contour maps showing the full-field distribution of uz at increasing stresses
The asymmetry observed on the left and right-hand sides of the macro fiber could be due to the slight misalignment of the fiber inside the matrix and/or asymmetry in the loading. Stronger out-of-plane deformation gradients are also observed along the same direction, as also indicated by the contour maps in Fig. 4(c). Despite the strong gradients developed in the fiber vicinity, the region of influence of such three-dimensional deformation fields remains limited to only an area slightly larger than one fiber diameter. In other words, the out-of-plane deformation gradients decay rapidly outside an area roughly three times the size of the fiber. Nevertheless, the highly localized out-of-plane shear strains caused by the steep out-of-plane deformation gradients across the fiber-matrix interface are a major contributor to the debonding initiation process. The effect of such local shear-dominated zones is more consequential at the top and bottom poles of the fiber since these areas are also subjected to large local tensile stress fields. However, the reason why the out-of-plane deformations are seemingly less intense in these regions (i.e., at the top and bottom poles) is due to debonding, which leads to partial strain release in the epoxy areas adjacent to the top and bottom interface regions, as discussed in detail in the forthcoming sections. In contrast, the out-of-plane displacement data extracted from the horizontal line marker (Fig. 4a) indicate sharper variations across the interface since the complete debonding at the left and right poles of the interface is expected to occur at much higher remote stresses.
Although limited by the relatively low resolution of the stereo DIC measurements, the out-of-plane deformation characterization in this work verifies the recent computational findings which point to the necessity of three-dimensional models for accurate analysis of fiber-matrix interface mechanics [24]. As such, the modeling approach presented in later sections herein has been intentionally designed based on three-dimensional geometries that replicate the experimental configurations discussed above.
In-Plane Deformation and Interface Debonding Characterization
In-plane deformation and strain fields were obtained using the front-view camera shown earlier in Fig. 2(b). The in-plane strain maps are illustrated in Fig. 5. The longitudinal in-plane strain contour maps, εyy, show the evolution of narrow high-strain bands at the top and bottom poles of the macro fiber at remote tensile stresses between 8.5 and 14 MPa. These high-strain regions mark the evolution and growth of the interface debonding as they intensify in magnitude and propagate circumferentially around the fiber at higher stresses. At remote tensile stresses approaching 40 MPa, complete separation between the fiber and its surrounding matrix is observed. The complete separation of macro fiber from the matrix activates strain localization in the matrix indicated by large strain bands formed in the shape of a cross surrounding the fiber. Consistent with the εyy strain fields, longitudinal and shear strain components also show gradual changes at stress levels lower than 40 MPa, followed by rapid changes as the remote stress reaches the ultimate tensile strength of the sample, i.e., 41 MPa.
A noteworthy observation is the evolution of negative (compressive) transverse strains at the left and right points of the fiber-matrix interface. The compressive strains are developed due to the Poisson’s behavior and impose a counteracting effect that partially restrains the opening and/or shearing of the interface. This restraint enables a further increase of out-of-plane deformation in these regions with minimal damage evolution, which reaffirms the presence of larger out-of-plane deformations along the x-direction as shown in Fig. 4.
Fig. 5
Evolution of longitudinal (εyy), transverse (εxx), and shear (εxy) strain components in the vicinity of a single glass macro fiber at different remote tensile stresses. Tensile load is applied in the y-direction. The sample failed by complete separation at 41 MP
The full-field nature of surface strain measurements in this work allows the extraction of local strains from any region within the area of interest. Here, we have selected four representative regions located at 90° angular distances from one another in the matrix with a 200 μm offset from the interface. The offset distance was chosen to ensure that the singularity at the fiber-matrix interface (esp. at the top and bottom poles) does not challenge our interpretations, as was discussed in detail in a previous article [23]. Variations of local strains with remote stress in the four representative points are shown in Fig. 6. At tensile stresses below 8 MPa, all local strain curves follow the same increasing trend also showing similar values. At 8.5 MPa, a sudden drop is observed in the local strain data representing the top pole deformation. A similar response is captured at the bottom of the interface at 11.7 MPa. The two drops in the local strain curves are associated with the occurrence of debonding at the top and bottom poles of the interface, leading to partial strain release in the neighboring matrix regions. Also observed and discussed in previous investigations [23], the occurrence of such local strain drops is an effective indicator of the debonding initiation in the composite. Interestingly, the debonding initiation at the vertically opposite points around the interface has minimal effect on the local strain fields in the transverse directions.
Fig. 6
Variation of local longitudinal strain component, εyy, with remote tensile stress extracted from four representative locations around the fiber, shown schematically on the right. The global strain curve is included in this graph for reference
In addition to the perturbation in the strain fields, debonding initiation can also be tracked directly from the measurement of crack opening across the interface at the top and bottom poles. Figure 7 shows debonded crack openings at interface poles as functions of remote tensile stress and global strain. Opening displacements reported in this figure were measured using two 200 μm optical gauges placed at the two poles and oriented vertically, i.e., parallel to the loading direction. The 200 μm gauge at each pole was placed symmetrically about the interface, i.e., with 100 μm of its length spanning across either side of the interface. The sensitivity of the measured opening displacements to the length of the optical gauge was investigated by Girard et al. [23], where it was shown that the opening displacement was independent of the gauge sizes of up to 660 μm. Consistent with the findings in Fig. 6, the initiation of debonding is associated with a step increase in the opening displacement. The remote stresses associated with such step increases match perfectly with those at which partial strain drops were observed in Fig. 6. While confirming the occurrence of interface debonding, the difference between the critical stresses associated with the top and bottom debonding suggests the presence of significant asymmetry, possibly due to slight misalignment and/or inconsistent fiber surface characteristics in this work. Inferred to be a direct result of uncertainty associated with the experimental nature of this work, a remote tensile stress of 10 ± 1.7 MPa (mean ± range of variation of the two remote stress values corresponding to debonding initiation at the top and bottom poles) was identified as the critical remote stress for the debonding initiation in this work.
Fig. 7
Variation of debonded crack opening with remote tensile stress and global strain extracted from top and bottom fiber poles. Crack opening was measured via two 200 μm optical gauges placed at the two poles and oriented vertically, i.e., parallel to the loading direction
Debonding Propagation along the Fiber and Crack Tunneling
The experimental protocols designed herein allowed us to observe and quantify the propagation of the debonded interface along the fiber length in the samples. Such measurements were possible due to the different refractive indices of the two constituents (i.e., epoxy and glass macro fibers). The refractive indices of the two materials were provided by the suppliers as 1.46 for the glass macro fiber and 1.57 for the fully cured epoxy resin. Due to the differences between these indices, a ray of light shone from the rear side of the sample will be refracted differently when the interface is intact versus when there is a slight opening at the interface, noting that the presence of a debonded interface introduces an additional medium to the otherwise bi-material (i.e., glass-epoxy) interface. While the optics underpinnings of the above process are beyond the scope of this work, the resulting effects of the light refraction due to interface debonding were observed and used for crack tunneling characterization. The schematic illustration of a debonded interface and its corresponding crack tunneling are shown in Fig. 8(a). An exemplary raw image along with its binarized replica are also shown in Fig. 8(b), (c), respectively. The raw image in Fig. 8(b) illustrates a side-view image of the sample after the initiation and partial propagation of the debonded interface along the glass macro fiber axis. The debonded interface is clearly shown in this figure by a light wedge-like region located at the top of the fiber-matrix interface. Note that a similar wedge-like region is also partially visible on the opposite side of the sample, propagating toward the front. The placement of the backlight in this work was designed for maximum illumination of the crack tunneling initiated from the front side. Such a design enabled direct comparison and correlations between the front-view and side-view measurements. To enable a more accurate identification of the location of the crack tip, the raw image was thresholded into a binary black-and-white image, as shown in Fig. 8(c). The same process (i.e., the same thresholding approach) was performed on all images captured during the mechanical loading, a process that enabled the visual detection and tracking of the tip of the tunneled crack parallel to the fiber-matrix interface. Figure 8(d) illustrates the progression of crack tunneling along the fiber as a collage of individual binarized images at increasing remote stresses. The red triangles mark the tip of the debonded interface along the z-direction.
Fig. 8
a Schematic illustration and geometric characteristics of fiber-matrix debonding and crack tunneling in single fiber model composites. Experimental characterization of crack tunneling along the glass macro fiber: b a raw image showing the tunneled crack tip at 10.4 MPa with its corresponding binarized image in (c). d Progression of crack tunneling shown in the form of a vertically stacked collage at different remote tensile stresses. Red triangles show the location of the crack tip identified by visual detection
The crack tunneling progression was quantified by measuring the crack length (difference between the current and initial coordinates of the crack tip) with respect to the remote tensile stress. Figure 9 shows the variation of the crack tunneling length, Ld, with remote tensile stress. Note that due to complicated light refractions at the left-hand side of the images, the crack length identification at stress-free and small stress conditions was impractical. Specifically, due to the free surface effect and light amplification at the front-side intersection (due to the backlight placement), binarized images at tensile stresses lower than 3 MPa appeared identical to that of 3.1 MPa. This issue challenged the visual detection of the crack tip in stress-free and small stress conditions. As such, the crack length-stress data points shown in Fig. 9 do not initiate at zero. Instead, the first data point identifiable by the abovementioned procedure was at a remote stress of approximately 3 MPa. While such experimental challenges were undesirable, the fact that the first data points were collected at stresses below the debonding initiation (see Fig. 6) makes the crack tunneling results reliable, at least for baseline characterizations and model-based inverse identification purposes discussed later.
Considering the results presented in Fig. 9, it is clearly observed that the tunneling propagation rate is variable, with zero (or near zero) propagation rates at small stresses, transitioning into faster rates when the remote stress exceeds ~ 7 MPa. This rate change has been co-plotted in Fig. 9 by including the crack propagation rate with respect to the tensile stress, dLd/dσ, where the differential was determined numerically by the forward difference method. The results shown in this figure indicate that crack tunneling initiates as early as the occurrence of the in-plane debonding. Further propagation of the debonding at the free surface of the sample provides the driving force for the faster propagation of the interface separation along the fiber axis, as denoted by the ever-increasing debonding rate values shown in Fig. 9. The crack tunneling process ends when the separated interface reaches the distal end of the sample, i.e., Ld=5 mm, at a remote stress of 12.5 MPa. Interestingly, the latter stress coincides with the instance when the debonded zones at the top and bottom of the interfaces reach one another at the left and right poles. The coalescence of the interface debonding on the surface was qualitatively observed earlier in Fig. 5 (see strain contour maps at 14.1 MPa). It should be noted that, as discussed above, the same tunneling effect initiates at the distal (rear) end of the sample and propagates in the opposite direction towards the front. While the crack tunneling in the opposite direction was not evaluated here, it is reasonable to assume that the strain energy released by the propagation of the debonding from the distal end affects and expedites the debonding on the other end. The ever-increasing rate of crack tunneling might be a consequence of the propagation on the opposite end or debonding on the opposite pole (on the same side), or both.
Fig. 9
Variation of crack tunneling length, Ld, and the propagation rate with respect to tensile stress, dLd/dσ, as functions of remote tensile stress
Another noteworthy remark associated with the measurements herein is the consistency between the trends observed in this work with those reported in Martyniuk et al. [12] obtained with X-ray microtomography. Although different in the critical remote stresses (as indeed expected due to different sample geometries, materials, and fiber sizing), the general trends observed in the in-plane normal opening and lengthwise tunneling are found to be consistent between the present study and those presented by Martyniuk et al. [12]. In both investigations, the interface debonding was shown to initiate at the free surface of the sample parallel to the direction of the applied tensile load and symmetrically at the top and bottom poles of the interface. By increasing the remote tensile stress, the debonded area propagates both in-plane (y-direction) and along the fiber axis (z-direction) at increasing propagation rates. The increasing crack propagation rate in this work suggests the presence of an unstable phase that initiates at remote stresses equivalent roughly to 30–40% of the ultimate tensile strength of the samples.
Finally, the most significant difference between the results discussed herein and those reported earlier in Ref. [23] is the absence of matrix microcracks in our investigation. The absence of matrix failure and crack kinking does not affect the accuracy of the findings. However, it is understood that the weak fiber-matrix interface strength (also evidenced by the relatively low debonding initiation stress) is the primary reason for the absence of matrix failure. While this will be addressed in our future efforts through improving the interface strength by applying different fiber sizing agents, the experimental measurements here were deemed useful enough for calibrating 3D finite element models discussed in the following.
Finite Element Simulation Results
The aforementioned experimental observations provide significant information concerning the debonding process from localized interface debonding to tunneling propagation. Such information is particularly relevant for establishing an experimental-numerical dialogue for further analysis of fiber-matrix interface failure modeling. Notably, it allows for the inverse identification of the fiber-matrix interface fracture properties using suitable numerical approaches such as cohesive zone models [11, 29] or the coupled criterion [30, 31]. Both approaches can be implemented in 3D FE simulations to account for the actual sample geometry and also for the stress singularity acting at the free edge, contrary to a 2D model. However, those models require the loading configuration together with the associated debonding shape to be determined in order to accurately perform the fracture property identification. Actually, from the present experimental observations, the debonding opening can be derived as a function of the debonding length, while the debonding angle and tunneling shapes have to be determined. Koyanagi et al. [11] proposed to extract the debonding angle at several locations through the thickness using side-view measurements and by identifying the crack front location. However, the authors reported that accurate determination of the debonded shape is challenging since local compressive stresses act on the fiber equator. This challenge aligns well with the DIC strain fields presented in Fig. 5. Consequently, the interface opening becomes smaller or even null when approaching the fiber equator, which makes the precise debonding front determination uncertain.
To overcome the abovementioned challenges, we developed a numerical model to assess the debonding shapes that best match the experimental observations; namely, the debonding opening at the free surface and its length along the sample thickness. For this purpose, a 3D FE model was established in the commercial FE software Abaqus to replicate the tested sample geometry and boundary conditions. Quadratic tetrahedral elements were used to discretize the model under linear elasticity and small deformation assumptions. A mesh convergence study was performed to ensure that the results were similar for a smaller mesh size. Elements with a size of 0.03 mm were adopted at the debonding front, resulting in models with around 1,500,000 degrees of freedom. Isotropic elastic properties were used for the matrix (see In‑Plane Deformation and Interface Debonding Characterization), while the manufacturer-reported properties were implemented to model the mechanical response of the glass macro fiber. The displacement boundary condition was applied at the top surface of the model consistent with the experimental configuration, and the remote stress was computed using the summation of the nodal reaction forces at the model boundary. Figure 10 shows various parameters that characterize the debonding geometry in our FE simulations.
Fig. 10
Different parameters that describe the debonding geometry in the FE simulations
Two geometric parameters, i.e., the debonding opening and length, can be extracted directly from the experimental observations. However, the other two parameters that remain unknown and are required to define the complete debonding geometry are the debonding angle at the free surface and the debonding shape between the free surface and the crack tip at the fiber pole (see Fig. 10). The main objective of the numerical approach herein is to retrieve the experimental debonding opening for fixed debonding length, angle, and shape. Therefore, the suitable numerical solution would be able to match both experimental debonding opening and length simultaneously, providing quantitative data for the debonding angle and shape. It is noteworthy that each debonding stage corresponds to a specific loading level that is replicated in the numerical model. A range of 10 debonding angles from 20° to 180° are tested for each debonding length. The several possible debonding shapes are approximated using the expression given in equation (1), where n can be varied from 2 to 6 to assess a wide range of shape options.
$$\:z\left(x\right)=\:{\left({L}_{\text{d}}^{\text{n}}\:\left({1-\left(\frac{x}{sin\:\left({\theta\:}_{\text{d}}/2\right)\:r}\right)}^{\text{n}}\right)\right)}^{\frac{1}{\text{n}}},$$
(1)
where r is the fiber radius. The determined shapes are drawn in the x-z plane and then projected onto the fiber surface. The interface nodes are unbuttoned within the surfaces defined by the debonding front shape. The debonding opening is further determined as the distance between the matrix and fiber nodes at the free edge pole of the fiber. This approach is also conducted by considering the opposite side debonding into consideration, i.e., debonding initiated at the rear side of the sample propagating in -z direction. However, this opposite-side debonding propagation appeared to have a negligible influence on the opening quantities extracted, smaller than 1%. Figure 11 shows the debonding opening observed experimentally as a function of the debonding length.
Fig. 11
Variation of the experimental debonded interface opening (in x-y plane) as a function of its length (along z-direction)
Figure 12 shows the variation of the three previous quantities, namely debonding angle, interface opening, and length obtained from the numerical model using a fixed order n = 4, for which the experimental measurements best match with the simulation results. The debonding opening increases with the debonding angle and length. Similarly, for each value of n in equation (1), a numerically determined debonding angle can be identified corresponding to a numerical interface opening that matches the experimentally measured values for a fixed crack length. The appropriate numerical debonding semi-angles are therefore superimposed on the numerical opening in Fig. 12 with a solid line and square markers.
Fig. 12
Variation of the debonding opening as a function of the debonding length and angle determined numerically. The numerical debonding parameters that correspond to the experimental debonding opening are depicted by the solid red line
Figure 13 shows the suitable numerical debonding angles for three different orders, i.e., different n values. The debonding angle starts between 55° and 60° and increases to 90°. It can be observed that the debonding shape order plays only a minor role in the obtained debonding angle with differences smaller than 15°. From a 2.25 mm debonding length, the numerical model failed to replicate the correct experimentally measured debonding openings, as the numerical results underestimated those measured experimentally. The discrepancy between numerical and experimental results here could be explained by the presence of gradual and space-wise mechanical and interface properties along the sample thickness resulting from a spatially variable degree of cure along the z-direction. If such variations in the degree of cure of the matrix exist, the sample center would be expected to exhibit softer behavior, leading to larger debonding openings obtained experimentally. Regardless, further debonding length can thus be approximated by a 90° debonding angle.
Fig. 13
Numerical debonding angles that lead to similar openings for different debonding lengths
The numerically determined debonding shapes are shown on the fiber-matrix interface in Fig. 14 for an arbitrarily chosen order n = 4. The debonding propagates significantly along the thickness while the increase in debonding angle remains less pronounced. Similarly, the determined debonding shapes are finally superimposed onto the raw side images obtained experimentally. Overall good agreement is achieved at small remote stresses. The debonding shape indeed well encompasses the lighter area that is assimilated to the debonded zone. Nevertheless, larger discrepancies are obtained for larger stresses. Particularly, the numerically determined debonding areas are larger than the lighter areas observed on the side view images of the sample. The lighter zone seems to be representative of the actual fiber-matrix debonding except in areas loaded under combined shear and compression. In these zones, the presence of a debonding cannot be revealed only by the light contrast brought by the presence of a debonded interface.
Fig. 14
a Variation of debonding (crack tunneling) front as a function of remote stress based on the angles that best match the experimental opening and length, for n = 4. b Debonding front shapes for several loading stages superimposed on the optical images. c Variation of mode mixity at the fiber-matrix interface obtained analytically from Goodier’s solution [32]
The lighter zones in raw images disappear for a debonding semi-angle of 50°. Goodier’s analytical solution provides the stress fields at the interface of an embedded circular inclusion under plane strain assumption [32]. The mode mixity, \(\:\psi\:\), at the interface can thus be assessed by the arc tangent of the shear-to-normal stress ratio, which varies between 0 for pure opening to π/2 for pure shear. Figure 14(b) shows the variation of the mode mixity as a function of the debonding semi-angle. Therefore, it appears that for mode mixity values larger than one, the debonding can no longer be identified using the light contrast generated by the debonding opening along the fiber axis. The opening mode then becomes negligible in comparison to the shear mode.
Fig. 15
a Front view of the debonding angle corresponding to a fully debonded interface. b Side view of the debonding for the similar remote loading, highlighting the discrepancies between experimental and numerical determination of the debonded shape
The above conclusions can be confirmed by observing the debonding area for larger stresses, e.g., a remote stress of 40 MPa, as shown in Fig. 15. On the one hand, a fully debonded interface can be observed at the front surface (Fig. 15a) where a complete debonding angle is present. On the other hand, Fig. 15(b) shows the side view where the lighter zone can be observed. The debonding angle derived from the lighter zone does not correspond to a fully debonded area. Such observation, therefore, confirms that lighter areas observed on the side view only enable partial capturing of the actual debonding shape and its corresponding area. Consequently, the side-view optical observations of the debonding only provide a correct debonding shape at the early stages of propagation. Comparison with the front-view observations is required to confirm that the lighter areas observed on the side view correspond to the real debonding shape. More detailed investigations regarding the possible shortcomings of the present optical-based methods are currently underway. Specifically, the authors intend to take this research to the next step by comparing the results of the present optical-based analyses with tests performed inside X-ray CT instruments.
Conclusions
This work investigated the in-plane and out-of-plane debonding at the fiber-matrix interface in model single macro-fiber composite samples subjected to transverse tension. The interface failure mechanisms were first characterized by full-field measurements facilitated by optical DIC. First, the out-of-plane deformation fields were characterized by stereo-DIC, highlighting the presence of out-of-plane deformations at the fiber-matrix interface. Though negligible, the characteristic dimensions that were affected by such out-of-plane deformation fields were identified to be as large as three fiber diameters. Next, a dual-vision system was designed and utilized to enable the simultaneous characterization of in-plane debonding and its resultant through-thickness debond propagation, also referred to as crack tunneling. The dual-vision experimental setup allowed for the identification of critical strains and stresses corresponding to the initiation and propagation of debonding damage along the fiber axis. These measurements were then used to calibrate an inverse identification approach that outputs the shape of the debonded interface along the fiber-matrix interface. It was found that the experimental measurements only were insufficient for the complete identification of critical damage parameters, including in-plane debonding opening and angle as well as the length and shape of the damage along the fiber length. In particular, side-view optical measurements were incapable of revealing the damage shape due to the presence of mixed-mode fractures. However, experimentally-informed computer simulations were shown to be effective in revealing all the critical failure parameters, allowing for a comprehensive understanding of the fiber-matrix failure in transversely-loaded fiber composites.
Acknowledgements
B.K. gratefully acknowledges the financial support provided by the Advanced Materials & Manufacturing Institute at Rowan University. B.K. also acknowledges partial financial support provided by New Jersey Economic Development Authority (NJEDA) through a Wind Institute Fellowship.
Declarations
Competing Interest
The authors have no competing interests to declare that are relevant to the content of this article.
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