Effect of the fiber-matrix bond on the toughness of soft,short-fiber composites

Introduction

Fiber-reinforced composites can provide an excellent combination of low mass and superior mechanical properties, including high stiffness and toughness. While many composites, including carbon-fiber epoxy composites, are already widely used in industry, there are still numerous open research questions, particularly for fiber-reinforced soft composites (FRSCs) with stiffnesses on the order of kPa and MPa. Emerging applications such as stretchable electronics^1–3 and soft robotics^4–7 require tough, yet soft, materials, similar to the properties of soft biological materials such as skin. These application demands have led to a number of strategies for developing tough, soft materials, including hydrogels and elastomers. These include, for example, tough alginate-polyacrylamide hydrogels with periodic long thermoplastic fibers networks, ² random steel wool networks, ⁸ woven glass fabric composites with high strength and toughness,^9,10 and soft composites with topological interlocking between the soft matrix and a macroscale structure.^11,12

In order for composites to achieve useful mechanical properties, stress must be transferred between fibers and matrix. Shear lag models have long been used to describe this in composites with finite fiber length ¹³ and infinite fiber length, ¹⁴ providing full elasticity solutions around the fibers. Modeling of failure processes in fiber-reinforced composites has been limited to unidirectional composites with continuous fibers. Budiansky, Hutchinson, and Evans studied matrix fracture in ceramic composites without fiber fracture, characterizing the effect of frictional forces and interfacial toughness. ¹⁵ Later work allowed fibers to fracture, and showed that fiber pullout can contribute significantly to toughness during fracture.^16–19 The contribution to toughness arising from fiber pullout has also been experimentally characterized for carbon fiber-reinforced composites with stiff matrices (i.e., epoxy, ceramic).^20,21 In conventional fiber composites, other mechanisms, such as fiber fracture, fiber-matrix debonding, and matrix fracture, contribute little to the toughness of the composites. ²² Though there are many studies on fiber-reinforced polymer composites, they have mainly focused on composites with stiff matrices. Recently, a study has applied composite mechanics to FRSCs with continuous fibers. ²³ A key difference of soft composites is that the elastic contrast between fiber and matrix is much larger than for conventional composites. This results in a large load transfer length, reducing the stress concentration when a fiber breaks. Energy stored in the fibers before failure is large and dissipated through this process. ²³ Another difference with the failure of FRSCs is that matrix deformation contributes significantly to the toughness of the composites, which is generally neglected in the traditional composites literature, since matrices such as epoxy and ceramics are brittle and fail at low strain. In addition to this mechanism, which, since it is a direct result of the microstructure of the composites, is an intrinsic toughening mechanism, some fiber-reinforced composites can toughen via extrinsic mechanisms such as fiber bridging. Soft materials can also be toughened extrinsically via mechanical dissipation, a mechanism for materials that undergo hysteresis during loading and unloading (e.g., tough hydrogels with sacrificial networks). ²⁴ As the crack propagates, the material around the crack experiences unloading. The energy that drives crack growth under such conditions is the intrinsic energy required to generate new surfaces and the energy dissipated during loading and unloading. ²⁵ A previous study has characterized the effect of mechanical dissipation on the toughness of double-network hydrogels. ²⁶

Extrusion-based 3D printing allows one to manufacture short, fiber-reinforced composites with controlled, spatially-varying fiber alignment, making use of shear-induced alignment of fillers during extrusion.^27–29 Using direct ink writing (DIW), an extrusion-based 3D printing approach, ³⁰ we fabricate composites with short glass fiber (GF)-reinforced Polydimethylsiloxane (PDMS) and investigate the source of toughness in such composites. In prior work, we have developed a modified shear lag model that accurately predicts the tensile behavior of GF-PDMS composites as a function of the fiber length distribution and the critical length required for shear stress transfer. ³¹ This model helps explain an observed instability in the tensile behavior at low volume fractions, as well as a ductile-to-brittle transition that occurs as a function of volume fraction. In this work, we investigate the effect of the strength of the fiber-matrix bond (specifically the critical shear stress between the fibers and the matrix) on the toughness of the composites. The strength of the fiber-matrix bond is controlled by treating the fibers with sulfuric acid prior to 3D printing. The acid treatment promotes additional hydroxyl groups on the fibers, as shown in Figure 1(a). After mixing the fibers with a PDMS ink, hydroxyl groups on glass fibers form Si-O-Si bonds with methyl groups on PDMS,^32,33 as shown schematically in Figure 1(b) and (c). Acid-treatment of the fibers improves the strength of the fiber-matrix bond, and thereby the tensile properties and fracture properties of the composite. Previous work has shown that PDMS-GF composites exhibit a hysteresis during loading and unloading, resulting from fiber-matrix debonding and subsequent fiber-matrix interaction. ³¹ Hence, we can expect mechanical dissipation to also contribute to the toughness of PDMS-GF composites, as described in Figure 1(d) and (e). This failure mechanism can also be observed in SEM images of the fracture surface, as shown in Figure 1(f). Individual fibers protrude from the matrix after complete fracture, associated with fiber-matrix debonding and subsequent fiber pullout. During initial loading, the fibers debond from the matrix irreversibly in a small region ahead of the crack tip, as shown in Figure 1(d). When the crack propagates through this region, the material in this area unloads, leaving permanent plastic stretch and dissipated energy (the area under the loading-unloading curve, as shown in Figure 1(e)). Here, we quantify the toughening mechanisms of PDMS-GF composites, including both intrinsic and extrinsic mechanisms.OPEN IN VIEWER

Materials and methods

Enhancement of tensile and fracture properties

In order to quantify the influence of the strength of the fiber-matrix bond on the mechanical properties of the composites, we measure the properties of the composites with and without treated fibers. The treated composites use glass fibers that have been treated with sulfuric acid (0.5 mol/L) to promote hydroxyl group termination on the glass surfaces. We 3D print thin samples with both untreated fibers and acid-treated fibers at 5 and 10 vol.%. First, we uniaxially load conventional tensile specimens in the direction of the fibers (details in methods) to measure the stress-stretch response of the composites (Figure 2(a)). For composites with 5 vol.% glass fibers, the acid treatment doubles the ultimate strength of the composites, but significantly decreases the ductility and the stretch of rupture (as shown in Figure 2(c)). There is still some softening in the stress-stretch response (at a stretch of approximately 1.25) due to fiber-matrix debonding. ³¹ For composites with 10 vol.% glass fibers, the yield stress increases approximately 20% while no significant change in stretch of rupture is observed after acid treatment. The unchanged stretch of rupture suggests that the failure of the composites at higher volume fractions is limited by other mechanisms, such as matrix fracture. This may result from the reduction in the mean distance between fibers at higher volume fractions, resulting in higher stress concentrations in the matrix. Importantly, for both high and low volume fractions, the initial stiffness of the composite remains unchanged, suggesting that the acid treatment of the fibers does not change the physical properties of the fibers (Figure 2(b)). In our prior work we developed a modified shear lag model to describe the tensile responses of similar composites using physical parameters such as the length of the fibers L₀, the radius of the fibers r, and the critical shear stress between fibers and matrix τ _c

σ = ( λ − 1 ) [ f E f ∫ 0 ∞ [ ( 1 − P f ) S 0 + P f S t ] P L d L + ( 1 − f ) E m ]

(1)

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Figure 2.

(a) The stress-stretch response for unnotched samples with various fiber treatments (dashed lines are fits of the modified shear lag model of Ref. ³¹ using all the same parameters other than critical shear stress τ _c ); (b) initial modulus E and (c) stretch of rupture for unnotched samples; (d) stress-stretch response for notched samples with various fiber treatments; (e) measured toughness and (f) critical stretch for various fiber treatments.

Next we tested long thin strips of the composites with a pre-cut crack oriented perpendicular with the direction of fiber alignment (see schematic in Figure 1(d)). The stress-stretch plots for these notched specimens are shown in Figure 2(d). The acid treatment results in a larger critical stretch λ _c and a higher toughness (Figure 2(d)). The acid treatment of the fibers improves the toughness by nearly 100% and 33% for specimens with 5 vol.% and 10 vol.%, respectively. The increase may be lower for composites with 10 vol.%, since higher volume fractions may cause the fracture behavior to become dominated by the matrix properties. Certain soft materials have been found to be extremely flaw insensitive, meaning that cracks below a certain length do not lead to premature material failure. These materials rupture at the same stretch at which the pristine material does. ³⁴ The PDMS-GF composites with 10 vol.% (with both types of fibers) show analogous insensitivity to flaws. The critical stretch λ _c = 1.30 (as shown in Figure 2(f)) is very close to the rupture stretch (λ _R = 1.32) of unnotched samples. This suggests that this composite is flaw-insensitive for cracks below 20 mm. Moreover, Chen et al. define a parameter called the length of flaw sensitivity r = G/W _f as the ratio between toughness G and work of fracture W _f (strain energy density at stretch of rupture). By applying this principle, we find the length of flaw sensitivity for unidirectional composites with 10 vol.% is also 20 mm, which is very close to what we observed in the notched samples with 20 mm pre-cut cracks.

Intrinsic toughening

In this section, we employ the shear lag model and our experimental data to find the origin of the observed enhancement to toughness in soft, fiber-reinforced composites. The toughness of fiber-reinforced composites is often attributed to four failure mechanisms: matrix deformation, fiber breakage, interfacial debonding, and fiber matrix sliding. ²² For our soft PDMS-GF composites, matrix deformation plays an important role in the toughness due to the ductility of PDMS (a few thousands of J/m²), ³⁵ while the contributions to toughness arising from the breakage of fibers (glass) and the interfacial debonding between the PDMS and glass fibers are low (a few tens of J/m²).^22,36 The main contribution to toughness, however, is fiber pullout. After the fiber debonds from the matrix, the fibers initiate sliding along the matrix with further deformation. During fiber pullout, debonded surfaces transfer shear stress at the critical shear stress τ _c , resulting in a large amount of work being done before crack propagation. The enhancement in toughness attributed to fiber pullout can be calculated as (following Ref. ²² )

G p = 4 f ( L 0 / r ) 2 r τ c

(2)

where L₀, r and f are known physical properties of the fibers (length, radius, and volume fraction, respectively) and τ _c is the parameter that can be extracted from our shear lag model by fitting the model to the uniaxial response of the composites. As shown in Figure 2(a), the critical shear stress between the fiber and matrix increased from 4 MPa to 8 MPa as a result of the acid-treatment of the fibers. Using the parameters characterized in prior work: ³¹ L₀ = 216 μm, r = 15 μm, and the toughness from fiber pullout (5 vol.% composite) is G _p = 2500 J/m² and G _p = 5000 J/m² for untreated and acid-treated fibers, respectively. In addition to fiber pullout, the matrix also contributes to the toughness. The PDMS used for 3D printing (with rheological modifier) has toughness of G _m = 1000 J/m². ³⁵ By adding both toughness contributions, we computed the intrinsic toughness of the composites G₀. The toughness values calculated from fiber pullout and intrinsic toughness are reported in Table 1.

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Table 1.

Parameters for calculating toughness of PDMS-GF composites.

	τ c (MPa)	G p (J/m2)	G0 (J/m2)(model)	G0 (J/m2)(experiments)	G (J/m2)
5 vol. % untreated	4	2500	3500	1500	5500
5 vol. % acid-treated	8	5000	6000	5000	13000

Extrinsic toughening

The intrinsic toughness calculated in Table 1 is generally lower than that measured from experiments. This discrepancy may arise due to the additional toughening occurring via extrinsic mechanisms. The extrinsic toughening of PDMS-GF composites is attributed to energy dissipated during crack propagation.^25,26 The overall toughness is the combination of intrinsic toughness G₀ and toughness from dissipation G _D

G = G 0 + G D

(3)

Material immediately ahead of the crack tip undergoes hysteretic loading as the crack grows and the material returns to the unloaded state. During the loading and unloading process, the material dissipates energy which is quantified as W _D . Following prior work that quantified the effect of dissipation on toughness, ²⁶ the effect of the extrinsic toughening can be calculated as

G = G 0 1 − α h max

(4)

h = W D / W

(5)

Experimentally, we first measure the hysteresis ratio as a function of the pre-applied stretch by conducting cyclic tests, with increasing stretch applied with each cycle. In Figure 3(a), we plot the initial stiffness of the loading portion of each cycle as a function of the strain energy density W (λ _p ), normalized by the total work of fracture W _f (area under the stress-stretch curve for unnotched specimens). As the material is stretched to higher amplitudes, there is a drop in the initial stiffness during reloading. This indicates that damage has occurred, analogous to the Mullins effect. The initial stiffness degrades less for composites with acid-treated fibers (50% degradation for composites with acid-treated fibers, vs 70% degradation for composites with untreated fibers). The hysteresis ratio h is calculated for each cycle and plotted versus normalized strain energy density in Figure 3(b). Composites with untreated fibers dissipate a significant amount of energy even at very low stretch, up to 40% at an applied stretch of λ _p = 1.10. In contrast, the composites with acid-treated fibers show little dissipation at small stretch, dissipating 40% of the energy only at an applied stretch of λ _p = 1.20. As the applied stretch increases, the hysteresis ratio eventually plateaus, with a maximum at h_max. To quantify the maximum dissipation, we fit the experimental results to an exponential function, which we previously used to model damage in the PDMS-GF composites during monotonic loading ³¹

h = h max [ 1 − exp − W ( λ p ) β ]

(6)

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Figure 3.

(a) Residual modulus during reloading versus normalized strain energy density for a given pre-stretch λ _p . (b) The hysteresis ratio from cyclic loading tests.

Next, we measure the toughness of the composites that have been pre-stretched. The experimental steps (i to iv) are illustrated in Figure 4(a). A pristine specimen with no pre-cut crack is stretched to λ _p (step i and ii). After unloading from the pre-stretch (step iii), a 20 mm crack is added using a razor blade. The notched sample is then loaded to failure (step iv), allowing extraction of the critical stretch at which crack propagation is initiated. This experimental method has been used previously to measure the effect of dissipation, e.g., for hydrogels. ²⁶ For composites with untreated fibers, we pre-stretch three individual samples to λ _p = 1.2, 1.35, 1.5. For composites with acid-treated fibers, we pre-stretch three individual samples to λ _p = 1.125, 1.25, 1.375. We report the toughness of the composites as a function of normalized strain energy density in Figure 4(b). We also report the toughness of pristine samples, corresponding to the data points for which the pre-stretch strain energy density is zero. For composites with untreated fibers, the toughness drops rapidly and stabilizes with the latter two pre-stretch values. This indicates that the intrinsic toughness G₀ is indeed being measured. In contrast, the toughness of the composites with acid-treated fibers drops linearly as the pre-stretch strain energy increases. At pre-stretch λ _p = 1.375 (which is very close to 1.44, the stretch of rupture for unnotched samples), the composites with acid-treated fibers are still tougher than the composites with untreated fibers without prior loading.OPEN IN VIEWER

The intrinsic toughness G₀ of both composites is estimated via a micro-mechanical model according to equations (2) and (3). It can also be determined experimentally using equation (4) by measuring h_max (from the repeated cyclical loading test of Figure 3(b)) and toughness of the composites without pre-stretch. This estimate is shown in Figure 4(b) as dashed lines for 5 vol.% untreated and acid-treated composites with α = 1, showing a good agreement with the estimate of intrinsic toughness reported in Table 1. This suggests that the energy dissipated during loading and unloading of materials contributes fully to the enhancement of toughness. Also, by extrapolating the experimental toughness with various pre-stretch (Figure 4(b)), one can find that the intrinsic toughness obtained from equation (4) is very close to the toughness if W (λ _p )/W _f = 1. Hence, equation (4) can be used to extrapolate the intrinsic toughness, which is reported in Table 1. Comparing the experimental value with the micromechanical model, a small discrepancy is found for composites with untreated fibers. They agree well for composites with acid-treated fibers. This supports the validity of the micromechanical model, which indicates that fiber-matrix pullout significantly increases the toughness, in some cases by more than a factor of four. By improving the critical shear stress between fiber and matrix, the intrinsic toughness of the composites can be greatly improved while the energy dissipation is maintained. We also point out that this model assumes extensive fiber-matrix sliding, which means that the composites need to be quite stretchable. For the 10% vol. composites, the model would overestimate the toughness due to the reduced amount of sliding.

Healing of the fiber-matrix bond

Even though the extrinsic toughening associated with the damage mechanisms contributes to a higher toughness, these inelastic processes degrade the performance of the composite under cyclic loading (Figure 4(b)). In this section, we explore the feasibility of healing the fiber matrix bond to recover lost stiffness and toughness. This is possible because the matrix (PDMS) has low hysteresis and does not have appreciable inelastic degradation. ³⁷ To study this effect, specimens are first loaded to λ _p and unloaded to zero stress. Next, they are placed in the oven at 100°C for 10, 30, 60, or 120 min. After this, we immediately test the composites either in uniaxial tension (to characterize stiffness) or in a tear test (to characterize toughness). As a control, we also leave some damaged specimens at room temperature for the same time intervals.

We plot the ratio of the residual stiffness E _r to the initial stiffness E₀ (defined in Figure 5(a)) as a function of the healing time in Figure 5(b). All composites are pre-stretched in uniaxial tension up to λ _p = 1.35. No significant recovery of stiffness is observed when specimens are left at room temperature, regardless of fiber treatment. In contrast, after 10 min in the oven, untreated and acid-treated fiber composites recover up to 80% and 85% of the initial stiffness, respectively. After 120 min, untreated and acid-treated specimens recover 92% and 99% of the initial stiffness, respectively. The nearly perfect recovery of the specimens with acid-treated fibers may be the result of the additional hydroxyl groups, as illustrated in Figure 1(a), which could be reactivated under heat to bond with the PDMS network. ³² OPEN IN VIEWER

As shown in Figure 4(b), pre-stretching the composites results in the loss of toughness. However, the healing process restores some of this. To measure this effect, we first stretch specimens with no pre-cut crack to a pre-stretch of λ _p , followed by unloading. The samples are subsequently placed in the oven for 120 min. Next, we cut a notch and test the samples until failure. As controls, some specimens are subjected to the same procedures but without exposure to elevated temperature. Figure 5 panels c and d show the measured toughness values for specimens with untreated fibers and acid-treated fibers, respectively. For composites with untreated fibers, after healing, the composites are more than 100% tougher than specimens that did not undergo healing at pre-stretch values of 1.35 and 1.50. For composites with acid-treated fibers, the original drop in toughness after pre-stretching is not as significant as that for composites with untreated fibers. After healing, the toughness recovery is not as significant in percentage. Nevertheless, composites with acid-treated fibers can recover more than 75% of the toughness after pre-stretching at large stretch (λ _p = 1.50), which is 65% larger than for composites with untreated fibers.

Conclusion

In summary, fiber-matrix debonding is the damage mechanism in the PDMS-GF composites that causes inelastic behavior such as load plateau and a cyclic loading response analogous to the Mullins effect. We found such damage in the composite provided both intrinsic and extrinsic toughening via fiber pullout and mechanical dissipation as a result of fiber-matrix interaction after debonding. We investigated the effect of the strength of the fiber-matrix bond and found that it enhances the intrinsic toughness of the composites while slightly decreasing the contribution from extrinsic toughening. However, the amount of extrinsic toughening depends on the intrinsic toughness (as shown in Equation (4)), leading to higher toughness with improved fiber-matrix bond strength. Finally, we showed that the fibers can be rebonded to the matrix after the composites are relaxed from loading through a simple thermal process. This allows the composites to regain stiffness and toughness even after large pre-stretching. Particularly, composites with acid-treated fibers can recover more stiffness and toughness during the healing process. This allows these composites to be used multiple times and still remain tough.