How tip geometry controls fracture in ductile polymer glasses and brittle elastomers

Abstract

Based on spatially-temporally resolved polarized optical microscopy (str-POM) measurements, we studied the fracture behavior of ductile and brittle glassy polymers as well as highly crosslinked rubbers to draw the following conclusions: (1) there is no tip plasticity below a threshold load in ductile plastics such as polyethylene terephthalate. (2) In ductile polymer glasses, before tip yielding at a common tip stress, the remote load scales with notch length a as a^−1/2, in agreement with the Inglis solution. (3) A finite stress saturation zone is observed in elastomers at loading levels even well below fatigue threshold due to significant crack tip blunting. (4) When the thickness is small enough for the plane stress condition to prevail at crack tip, in double-edge notch tension (DENT) for both ductile glassy polymers and rubbers that is characterized by ligament length l, nominal strain in the ligament is defined by ε_lig = X/l, where X is tensile displacement; tensile force F increases linearly with X independent of l; tip stress increases linearly with the far-field σ_lig (∼ε_lig). By demonstrating stress concentration at the crack tip in DENT in elastic materials and characterizing crack propagation in ductile polymers, the present study fills the missing gap in our understanding of fracture behavior in a wider range of polymeric materials. The acquired knowledge may be useful to guide specific design for packaging materials.

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1. Introduction

After several decades of investigations on polymer fracture,^1–6 there has been significant progress in elucidating the origin of Griffith's scaling in brittle plastics^7,8 and the hidden mechanisms controlling dynamic fracture of elastomers.^9–11 Specifically, applications of the spatially and temporally resolved polarized light microscopy (str-POM) method, as a stress-based approach¹² to fracture mechanics of polymeric materials, have revealed how stress buildup at a notch tip as a function of remote load. The Inglis¹³–Creager–Paris¹⁴ solution inspired approach has provided fresh insights^7,8,15,16 into polymer fracture and led to the conclusion that a lab-prepared notch in polymeric materials is not sharp and can be characterized by a finite “Inglis” radius. The existence of a sizable stress-saturation zone (SSZ) in polymeric materials⁷ allowed the estimation of the tip stress. According to str-POM observations⁷ brittle plastics show no evidence of tip yielding associated with the emergence of a plastic zone for PMMA at room temperature. Crack growth in elastomers reflects bond dissociation in backbones of load-bearing strands on a certain time scale;⁹ high crack propagation speed requires high far-field stress, not because of any higher viscoelastic energy dissipation but because only a higher degree of network stretching can shorten the elastomer's lifetime, making the crack front advance faster. A recent re-visit¹¹ to delayed rupture¹⁷ further elucidated the nature of elastomeric failure and established the phenomenological foundation.

There are still several outstanding topics in the field of polymer fracture. We addressed them in the present study by applying the str-POM method for a range of polymeric materials, from ductile and brittle glassy polymers to elastomers. We confirmed using ductile PET that no plastic zone forms at the notch tip below a threshold load. This feature can be understood in terms of Inglis–Creager–Paris (ICP) solutions and reveals the non-Irwin nature of SSZ. Before tip yielding in ductile plastics, we verified the ICP description: the tip stress is proportional to the stress intensity factor K = σa^1/2. In other words, remote load σ ∼ a^−1/2 produces a common tip stress for different notch lengths. Our str-POM measurements further reveal the existence of a finite (≥40 µm) stress saturation zone (SSZ) ahead of the notch tip in a well crosslinked elastomer in a broad range of applied load, both below the fatigue threshold and during crack growth, implying that even fatigue during cyclic load should involve high chain extension and scission on the length scale of SSZ well beyond a monolayer length scale. Additional str-POM observations demonstrate the stress field in double-edge-notch tension (DENT) in linear elastic limit and show for both plastics and elastomers that (a) ligament length l defines an effective strain ε = X/l in terms of displacement X and (b) stress intensification at the crack tip is linearly proportional to ε.

2. Materials and methods

2.1. Materials and sample preparation

The bisphenol-A polycarbonate (PC) and polystyrene (PS) used in this study were sourced from Grainger and PS had thicknesses of B = 1.5 mm and 0.77 mm and PC had a thickness of 0.7 mm. The polyethylene terephthalate (PET) used in this study was obtained from Auriga Polymers and has a thickness of 0.2 mm.

Highly crosslinked butadiene rubber (BR) samples were prepared using 1 wt% dicumyl peroxide (DCP) as the curing agent. A 100 g batch of neat BR was first masticated on a two-roll mill for 30 min to ensure homogeneity. Subsequently, 1 wt% DCP was added to the matrix and thoroughly mixed. The resulting compound was then preheated at 140 °C for 5 min and compression-molded at 150 °C for 1 h to achieve fully isotropic crosslinked specimens with various thicknesses.

Single edge notches were primarily introduced into the ribbon shaped samples by gently hammering a razor into the edge, resulting in generation of a natural crack, akin to a method employed by Berry,^18,19 as done in the case of all PC and PET specimens.

Double-edge notch geometries were introduced into the ribbon-shaped specimens using a two-step process. First, a coarse notch was machined using a saw to remove the bulk of the material. This was subsequently sharpened into a natural crack by gently tapping a razor blade into the notch root; this method was employed to minimize plastic deformation and damage at the crack tip. Following preparation, the integrity of the crack tips was verified using cross-polarized light to detect any localized damage zones or residual stresses. If any such damage was observed, the specimens were annealed at a temperature near their glass transition temperature (T_g) to promote healing prior to mechanical testing.

2.2. Testing instruments

Tensile tests were conducted using a universal testing machine (Instron 5969). The deformation process was recorded with a 4K camera (HY-6110 Micro Camera) equipped with an Arducam zoom lens (5–100 mm) to enable high-resolution video capture of the specimen during loading.

Entangled transparent polymers are known to exhibit stress-induced birefringence under deformation. The applied deformation generates anisotropy in the polymer network, leading to a change in birefringence Δn. To quantify this effect, photoelastic measurements were performed simultaneously with the tensile tests. Photoelasticity is an experimental technique that relates the optical response of a material (birefringence) to its internal stress state during deformation. In the present setup, polarizers were sourced from polarization.com and arranged in a standard transmission configuration. A sodium vapor lamp with a wavelength λ of 589 nm was used as a monochromatic light source to illuminate the specimen. The transmitted light, after passing through the polarizer–specimen–analyzer assembly, was recorded using a 4K camera, allowing the evolution of stress-induced birefringence to be correlated with the mechanical response obtained from the tensile tests.

3. Experimental results

3.1. Griffith scaling in ductile glassy polymers in SEN

Whether amorphous polymers are ductile or brittle is often judged according to their macroscopic stress–strain response. Such a distinction, however, does not describe how failure is initiated in the presence of a crack. While ductile polymers can undergo plastic deformation under sufficiently high loads, tip yielding can be avoided at low loads. In the absence of plastic deformation at the crack tip, we can examine whether ICP solutions are valid in the linear elasticity limit.

We applied the str-POM method to characterize the stress state at the crack tip in the SEN configuration, specified by a notch of length a in a specimen of width W. As long as a/W < 0.2, stress intensity factor K shows negligible dependence²⁰ on W, so that the Griffith criterion applies. The system under study is a polycarbonate with thickness B = 0.7 mm, which is sufficiently thin to avoid⁸ being notch brittle. Fig. 1a shows nominal stress vs. strain curves with a = 2, 4, 8, and 20 mm, respectively. All specimens exhibit a ductile response, with the notch tip first reaching a yield point, followed by localized necking without brittle failure.

Fig. 1 (a) Nominal stress–strain curves of PC SEN specimens with notch lengths a = 2, 4, 8, and 20 mm, obtained at a crosshead speed V = 2 mm min⁻¹, where L₀ = 2W for each sample. (b) Local stress distribution ahead of the notch tip plotted against r^−1/2 at a common tip birefringence order N_tip = 14, involving different far field stress values (25, 16, 10, and 7 MPa), just prior to the onset of tip yielding, where the two images show the same birefringence order for two different notch sizes. (c) Nominal far-field stress σ at N_tip = 14 and at the peak remote load (maximum nominal stress in (a)) plotted against a.

Our str-POM measurements enable us to follow stress concentration at the notch tip in these four samples. For example, as shown in Fig. 1(b) at a common birefringence order of N_tip = 14 at the notch tip, just before the onset of tip yielding and plastic deformation, a finite zone (r_ss = 100 µm) of stress saturation (SSZ) is present. In other words, the stress building at the notch tip follows r^−1/2 scaling up to SSZ; the tip stress σ_tip depends on the far-field σ as σ_tip ∼ K = σa^1/2 so that σ ∼ a^−1/2 produces a common tip stress, as shown in Fig. 1c, which is consistent with Inglis solution¹³ for an elliptical hole. The dependence of the peak nominal stress on a is much weaker, as shown by squares in Fig. 1c.

We have demonstrated with ductile PC that it is only linear elastic before tip yielding. The sizable SSZ is not Irwin's plastic zone. According to the Creager–Paris solution,¹⁴ the tip stress is linearly proportional to the far-field load and below the yield stress below a threshold load. We support this conclusion with a second ductile glassy polymer, i.e., polyethylene terephthalate (PET). Fig. 2a shows three photos from str-POM observations, showing that there exists a wide range of far-field stress within which no tip yielding occurs. For example, the Irwin–Dugdale plastic zone cannot be observed at and below σ = 26 MPa (a = 2 mm), as explained by the photos of identical birefringence patterns before and after application of σ = 26 MPa. There is no residual birefringence observed after unloading from σ = 26 MPa. On the other hand, we cannot rule out that the plastic zone does not exist below the spatial resolution of our str-POM observation. Fig. 2b–d are three equivalent plots, showing the near-tip stress field in terms of birefringence at 27 MPa for a = 2 mm, along with three other SEN samples that show the same fringe order of N_tip = 25 at the various levels of far-field stress and different notch lengths. We noted that plateauing and converging features are exactly those encompassed by the Creager–Paris solution. Like PC, there exists a stress saturation zone, a bit larger than that of PC, ca. 160 µm. Each of the three Fig. 2b–d indicates the same conclusion: the stress converges to a common value of 55 MPa.

Fig. 2 (a) Nominal stress–strain curves showing consistent ductile behavior in PET SEN specimens. The str-POM frames illustrate loading to N_tip = 25 and subsequent unloading back to the original tip pattern with no residual birefringence, indicating that the applied load is below the onset of tip yielding. (b)–(d) Local tip stress (from birefringence) plotted against r^−1/2 on linear and logarithmic scales, at a fixed tip stress corresponding to the isochromatic fringe order N_tip = 25, showing a finite stress-saturation zone in (b) and (c) and convergence to 55 MPa in (d). (e) Nominal far-field stress at N_tip = 25 and at the peak remote load (maximum nominal stress in (a)) plotted against a^−1/2.

Like PC, the far-field stress σ in PET required to produce this common tip stress scales linearly with a^−1/2, as shown in Fig. 2e. Moreover, the peak nominal stress scales with a in a weaker manner. In both Fig. 1a and 2a, the peak stress corresponds to the onset of stable ductile crack propagation.

3.2. Finite stress saturation zone in elastomers in SEN

Our str-POM observations^7,8 including the present ones in Fig. 1 and 2 indicate that crack tips in plastics are not sharp, leading to the existence of a sizable stress saturation zone (SSZ) in the SEN setting. Creager–Paris solution¹⁴ explains that the origin of SSZ is the existence of a finite tip radius. More discussion is provided in Section 4 on the relationship between SSZ and the C-P solution.

A large SSZ¹⁵ in prenotched elastomers also has implications. Using a highly crosslinked polybutadiene, we applied the str-POM method to quantify the stress intensification at the notch tip. Stretching at V = 3 mm min⁻¹ (L₀ = 30 mm) with cut size a = 3.4 mm produces a stress response as shown in Fig. 3a and allows us¹⁶ to measure crack speed v_c on a continuous basis at various far-field loads. Fig. 3b shows the existence of SSZ from σ = 0.03 to 0.12 MPa based on the birefringence images in Fig. 3c. The size of SSZ only increases. Fig. 3d presents measured values of v_c at different values of σ in open circles above 0.1 MPa when crack growth becomes visible. See Movie S1 in the SI. This disclosed relationship between v_c and σ permits us to indicate extrapolated values of v_c at 0.03, 0.05 and 0.07 MPa (e.g., 10⁻⁸ mm s⁻¹ at 0.05 MPa) and to suggest that these loading levels are well below a fatigue threshold G₀. Typical cyclic loading usually produces crack growth much higher than 10⁻⁸ mm s⁻¹. At these loading levels a sizable SSZ is already present. At far-field stress equal to 0.12 MPa, crack growth is significant as shown in Fig. 3d, with r_ss ∼ 110 µm, implying that energy release involves a large scale far greater than the monolayer scale. Since r_ss is also sizable well below the fatigue threshold, as shown in Fig. 3b, cyclic crack growth during fatigue tests should also involve energy release on the length scale of r_ss. Our analysis suggested¹⁰ that crack growth under applied load is due to chain network breakdown at the crack tip via chain scission on a time scale characterized by v_c and r_ss.

Fig. 3 (a) Nominal stress vs. strain curve for BR1phr in SEN with length, width and thickness given by 30, 15 and 1.5 mm, and a = 3.4 mm, produced with V = 3 mm min⁻¹ and L₀ = 30 mm. (b) Simultaneous str-POM measurements of local birefringence distribution with respect to the distance r to the at the notch tip, plotted against r^−1/2 at four levels of far field loads: σ₀ = 0.03, 0.05, 0.07 and 0.12 MPa. (c) Four images at the four respective loads. (d) Crack speed v_c from video recording (Movie S1) of the same test described in (a)–(c), where three filled circles show the extrapolated values of v_c at 0.03, 0.05 and 0.07 MPa, respectively, along with the actual reading of v_c in open circles.

3.3. Scaling behavior in terms of nominal and local stresses in DENT

While str-POM measurements^7–11 have been carried out to uncover physics in polymer fracture in SEN and pure shear, fracture in DENT configuration remains unexplored by the str-POM method. Below, it is shown that str-POM measurements provide fresh insight into the nature of various scaling behavior uncovered by far-field measurements.

3.3.1. Effective strain. To investigate fracture characteristics of ductile polymers and discuss the framework of the essential work of fracture (EWF), we considered the DENT configuration. This geometry is distinct from SEN, typically used for brittle materials; it is designed to operate in the limit where the ligament length (l) is significantly smaller than the specimen width (D). In this regime (l ≪ D), the ligament length becomes the sole characteristic length scale governing the deformation. Unlike geometries where fracture is dictated by crack length, the mechanical response of a ductile DENT specimen prior to global yielding or brittle fracture behaves analogously to a linear spring. In other words, we found the emergent tensile force to scale linearly with the applied displacement, effectively independent of l.

Unlike brittle polymers, in the DENT of ductile glassy polymers, the tensile force also continues to increase beyond the point of tip yielding. Using a bisphenol-A polycarbonate (PC) sheet with thickness B = 0.77 mm and a polyethylene terephthalate (PET) sheet with B = 0.2 mm, Fig. 4a and b respectively illustrate that the tensile force F depends primarily on the displacement X in a linear fashion, independent of l up to the maximum force, at which point the entire ligament has undergone necking. Below, we discussed the implications using dimensional analysis.

Fig. 4 Mechanical responses of DENT specimens in terms of tensile force vs. displacement X for (a) PC (B = 0.77 mm) and (b) PET (B = 0.2 mm) at different ligament lengths. The inset (a) illustrates the DENT configuration.

For F to depend on X but be independent of l, we infer that the effective tensile stress σ_lig in the ligament must be higher in shorter ligaments at a common value of X. Specifically,

σ_lig = F/Bl = E_ligε_lig (1)

defines the effective strain ε_lig is given by

ε_lig = X/l, (2)

so that we have

F = (E_ligB)X. (3)

Here, the slope given by E_ligB can be read from Fig. 4a and b to reveal an effective modulus, E_lig. We found E_lig = 0.43 GPa for PC, which is approximately four times smaller than the bulk Young's modulus E = 1.8 GPa. Similarly, E_lig = 0.75 GPa for PET.

3.3.2. Force–displacement linearity. According to Fig. 4a and b, the linearity between F and X holds nearly up to the force maximum. Thus, the linear scaling is observed throughout the full range of displacement, including regimes where tip yielding (TY) and ligament necking (LN) take place. Str-POM measurements can quantify stress intensification at the notch tips, which are the two edges in DENT, in terms of the birefringence fringe order (N_tip) and identify the onset of tip yielding. Moreover, str-POM observations allow us to confirm the assertion that the strain field in the ligament is given by eqn (2): At a common value of ε_lig or fixed “far-field” σ_lig, the tip birefringence should be the same, independent of l. For example, reading from Fig. 5a the first and sixth squares, at N_tip = 16, tip yielding emerges at F = 25 N (X = 0.094 mm) for l = 1 mm and F = 230 N (X = 0.64 mm) for l = 8 mm, corresponding to σ_lig = 40 MPa (l = 1 mm) and 34 MPa (l = 8 mm). The value of σ_lig is approximately half of the yield stress σ_y ≈ 70 MPa, as if the stress intensification factor is around two, independent of ligament length.

Fig. 5 (a) Tensile force vs. displacement for all ligament lengths at three characteristic states: pre-tip-yielding (PTY, N_tip = 10), tip yielding (TY, N_tip = 16), and total ligament necking (LN) at the maximum force, where the two images show that the tip birefringence orders reach the same value of N_tip = 16. (b) Tensile forces at the three states, F_PTY, F_TY, and F_LN, plotted against ligament length l, showing linear scaling. (c) Linear relationship between displacement and ligament length to confirm that each of the three states involves a common effective strain ε_lig = X/l.

Fig. 5a shows the two images of the birefringence pattern at tip yielding, along with the relationship between force and displacement for all ligament lengths at pre-tip-yielding (PTY – N_tip = 10), tip yielding (TY – N_tip = 16) and total ligament necking (LN) at the force maximum. The linearity in eqn (3) also manifests in linear relationships between tensile force F_PTY, F_TY and F_LN and ligament length l, as shown in Fig. 5b. Finally, a common strain ε_lig to produce these three special states among all specimens of different ligament length implies a third linearity, as shown in Fig. 5c.

Using the str-POM method, we explored the stress intensification in DENT to the tip stress to the remote load, which may be expressed in terms of the effective strain ε_lig given in eqn (2). To convert birefringence order N to tensile strength, we first obtained the calibration, i.e., the stress-optical relation (SOR) in Fig. 6a. Here, linearity breaks down as PC undergoes partial yielding. At the notch tips (edges of DENT), as shown in Fig. 6b, the tip birefringence linearly increases with the “far field” strain ε_lig. Given eqn (1) and that linearity in SOR in Fig. 6a is bounded by 40 MPa, the horizontal axis of Fig. 6b is proportional to fringe order N so that Fig. 6b shows N_tip = 2N, independent of l for the explored range 3 mm to 9.3 mm. In contrast, in SEN with notch length, a str-POM observation of SEN revealed^7,21 that at a common tip stress or birefringence, the far-field stress σ ∼ a^−1/2. It is remarkable that neither ligament length l nor tip curvature appears in the stress intensification in DENT. Converting to stress based on SOR in Fig. 6a and c shows the stress intensification at the tip. Here, the initial slope is two, i.e., the tip stress intensifies by a factor of two, and the nonlinear relationship between σ_tip and σ_lig arises from partial tip yielding.

Fig. 6 (a) Stress-optical relationship measured from unnotched PC. (b) Tip (edge) birefringence as a function of effective strain ε_lig of eqn (2) for different ligament lengths. (c) Tip stress vs. “far field load” σ_lig.

The scaling behavior of PC in DENT takes place within linear elasticity. Analyzing the DENT tests presented in Fig. 4b, we confirmed that PET also displays the same linearities, as shown in Fig. 7a and b.

Fig. 7 Tensile force at ligament necking F_NL vs. ligament length l. (c) Linear relationship between displacement and ligament length to confirm that ligament necking involves a common effective strain ε_lig = X/l.

3.4. Brittle polystyrene in DENT: thickness effect

To explore the conditions for eqn (2) and (3) to hold in DENT for linear elastic materials, we examined a brittle polystyrene that undergoes fracture at room temperature within linear elasticity. The chain network breaks down via chain pullout before yielding.^21–23 In other words, at room temperature, brittle PS is incapable of undergoing plastic deformation. By changing the thickness B from 0.55 to 1.5 mm, the scaling behavior of F ∼ X, described in eqn (3) and seen from Fig. 4a and b, from which Fig. 5a–c and 7a, b arise, disappears. Fig. 8a is replaced by Fig. 8b, which shows higher force responses for higher ligament length. Thus, the effective strain defined in eqn (2) appears to be valid only when specimens are thin enough, where perhaps plane-stress deformation prevails. At the present time, to the best of our knowledge, we are unaware of any theoretical calculation that demonstrates such a thickness effect or derives eqn (1) to eqn (2).

Fig. 8 (a) Force vs. displacement curves for (a) thin PS (B = 0.55 mm) and (b) thick PS (B = 1.5 mm) specimens with various ligament lengths l. (c) Nominal stress σ_lig defined by eqn (1) at brittle fracture as a function of ligament length for both thicknesses.

At B = 0.55 mm, PS in DENT with smaller l shows the same level of mechanical resistance (in terms of tensile force F) against displacement X, like PC and PET. However, since PS can be expected to fracture at the same local tip stress, Fig. 8c explicitly confirms that fracture occurs at a common level of σ_lig. In other words, specimens with larger l are stronger and more strain tolerant, as indicated in Fig. 8a. As far as fracture conditions are concerned, the thickness effect is unremarkable, as shown by the comparison between circles and squares in Fig. 8c. In other words, PS in DENT experiences brittle fracture like the fracture observed with a notch-free specimen. On the other hand, σ_lig(fract) is well defined and shows a relatively small range of uncertainty. Thus, for flaw-tolerant brittle materials including PS, DENT is a rather interesting protocol for fracture characterization. Here we can apply the conclusion established in Fig. 8c, i.e., σ_tip = 2σ_lig, to estimate the fracture stress of PS as σ_b = 2σ_lig(fract), i.e., ca. 50 MPa, which is indeed a typical value for PS, as shown in Fig. S1 in the SI.

3.5. Elastomer in DENT

The linear relation of eqn (3) may occur for any linear elastic material if the DENT specimens are thin enough. We examined one more type of solid that may exhibit the scaling behavior in eqn (3). Highly crosslinked BR1phr follows linear elastic fracture mechanics and develops fast crack growth in a narrow range of applied far-field stress. When it is thin enough, e.g., with B = 0.7 mm, the force resistance is like that of PS, as shown in Fig. 9a. In contrast, BR1phr with B = 1.5 mm behaves like the thicker PS because Fig. 9b resembles Fig. 8b.

Fig. 9 Force vs. displacement curves for DENT specimens of crosslinked polybutadiene BR1phr for (a) thickness B = 0.7 mm and (b) B =1.5 mm at different ligament lengths. (c) Nominal ligament stress at fracture as a function of l, showing little dependence at B = 0.7 mm.

Unlike PS, with sufficient birefringence and linear SOR for BR1phr, we can further illustrate and confirm how stress intensification at the edges of DENT takes place, driven by the far-field load. Specifically, like Fig. 6b, Fig. 10a shows that the birefringence order N_tip at the tips increases linearly with the effective far-field stress σ_lig. In contrast, with thicker BR = 1.5 mm, Fig. 10b shows that the tip birefringence becomes linearly dependent on F instead of F/l ∼ σ_lig. The birefringence images in Fig. 10a and b display the deformation of BR1phr specimens under a far-field load of 1.5 N. For the thin sample (B = 0.7 mm), higher tip birefringence shows up for l = 4 mm than for l = 7 mm because it depends on F/l rather than on F alone. In contrast, for the thick sample, the same tip birefringence order appears for l = 3 mm and l = 9 mm, indicating that the tensile force F dictates the tip birefringence regardless of ligament length.

Fig. 10 Evolution of the local notch-tip birefringence, N_tip, as a function of σ_lig for polybutadiene (BR) specimens with thickness (a) B = 0.7 mm and (b) B = 1.5 mm. Images show birefringence pattern at a fixed load of F = 1.5 N, corresponding to the two open symbols in (a).

3.6. Essential work from DENT

For ductile materials that can undergo considerable plastic deformation, Cotterell and Reddel (CR)²⁴ first proposed to investigate their mechanical responses in the DENT configuration, provided the ligament length l is smaller than twice the maximum plastic zone size r_p*. In this limit, along with l > 5B, CR envisioned that the total mechanical work W needed to drive crack growth beyond tip yielding until final macroscopic separation should comprise both linear and quadratic terms. The dominant part of W should scale quadratically with l since plastic deformation may occur in a region whose volume is quadratic in l, i.e.,

W = w_elB + w_pBl², (4)

where l < 2r_p* and l > 5B, so that

w_f = W/lB = w_e + w_pl. (5)

However, we showed that linear-elastic materials such as PS and BR1phr examined in the preceding Sections 3.4 and 3.5 also indicate a quadratic dependence of W on l². In other words, when specimens in DENT are sufficiently thin for eqn (2) and (3) to be valid, W for fracture also follows eqn (4) or eqn (5) in the absence of plasticity. For example, according to Fig. 8a and 9a, W = FX = l²(F/l)(X/l) ∼ l² because F (∼X) and X both linearly increase with l until fracture. Fig. 11a and b show w_f vs. l with a clearly defined intercept at l = 0 as w_e. The value for w_e is consistent with the reported¹⁹ toughness G_c for PS; however, w_e ∼ 0.5 kJ m⁻² is more than twice that of G_c = 0.21 kJ m⁻², which is estimated from pure shear measurements using Rivlin–Thomas formula,¹⁵ as shown in Fig. S2.

Fig. 11 Specific work of fracture w_f, plotted as a function of ligament length l, for (a) PS of thickness B = 0.55 and 1.5 mm and (b) BR1phr of thickness B = 0.7, 1 and 1.5 mm.

Finally, because of the scaling characteristics displayed in eqn (2) and (3), the essential work w_e can be evaluated from Fig. 4a and b for PC and PET, either up to the force maximum or by integrating over the “tails” associated with ligament necking, which is massive for PET and little for PC, and negligible for BR1phr and non-existent for PS. Fig. 12a and b show w_e = 5 kJ m⁻² for PC and 20 kJ m⁻² for PET, which is consistent with the literature.²⁵

Fig. 12 Specific work of fracture, w_f, plotted versus ligament length l, based on either total area under force vs. displacement curves (conventional) or area up to the maximum (peak force) for (a) PC and (b) PET.

4. Discussion

The str-POM method has allowed us to describe what happens at the notch tip and confirm the existence of SSZ that Creager–Paris solution¹⁴ predicted in the linear elasticity limit. In other words, the C-P solution predicts the existence of SSZ and elucidates the origin of SSZ that we observed in PS and BR. Since SSZ emerges in these brittle polymers as well as ductile PC and PET below the tip yielding threshold, it is not the plastic zone that Irwin proposed for metals. The absence of a notch tip yielding below a finite far-field load conversely reveals that notch tips are only of finite sharpness. The size of SSZ reveals the level of tip blunting in these materials. Although the existence of a damage or process zone⁵ has been suggested initially^26–29 for metals, the concept has been adopted⁴ for polymers, including plastics and elastomers. One of the main aims of the present study is precisely to raise a question about the past treatment of polymer fracture. Our observations suggest the SSZ emerges before fracture and thus does not coincide with the concept of damage zone. While the emergence of SSZ supports the C-P description of tip stress, beyond linear elasticity and finite spatial resolution of the str-POM method, we cannot be certain that only elastic SSZ is present at finite loads. On the other hand, there is little indication that PC, PET and well-crosslinked BR cannot be regarded as only showing linear-elastic responses before tip yielding for PC and PET and fracture for BR.

In elastomers that undergo purely elastic stretching at temperatures well above the glassy transition temperature T_g, there is neither plastic process nor viscous dissipation at the crack tip. For example, BR has T_g = −100 °C and does not develop a process zone. In the case of fracture of thermoplastic elastomers,³⁰ more complicated processes may take place at the notch tip. Further discussion is beyond the scope of the present study.

5. Conclusions

We have employed the str-POM method to investigate several important questions in the field of polymer fracture. The access to tip stress based on photoelastic effects shows that no sizable plastic zone exists in ductile PET below a critical load (cf. Fig. 2a). Thanks to str-POM observations, we have generalized Griffith scaling law and confirmed the Creager–Paris solution using PC and PET below tip yielding: Stress intensity factor K prescribes the tip stress so that σ ∼ a^−1/2 produces a common tip stress (cf. Fig. 1c and 2e) prior to tip yielding and plasticity at the notch tip. For elastomers in the linear elastic limit, there exists a finite stress saturation zone (e.g., r_ss ≥ 40 µm, cf. Fig. 3b) before fracture so that energy release rate G involves release of stored elastic energy over a spatial region of size r_ss, far greater than the monolayer thickness. This observation can explain why G reaches a much higher level than G₀.

So far, the single-edge notch (SEN) has been a dominant configuration for fracture mechanics investigation of various brittle materials, including plastics and elastomers. Ductile materials are not subject to Griffith's account and may be characterized by resistance curves. To quantify energy involved in ductile fracture, double-edge-notch tension (DENT) is widely used to extract the essential work w_e as a quantity “equivalent” to Griffith's toughness. Application of str-POM allows us to elucidate how stress builds up at the tip of double edges in DENT and produce insight into the origin of w_e. We found for both plastics (ductile and brittle) and elastomers (well-crosslinked) that in the limit of linear elasticity and under the condition of plane stress (or for specimens sufficiently thin) stress intensification at the notch tip is independent of ligament length (cf. Fig. 8c and 9c) and depends only on the effective strain, which is displacement X normalized by ligament length l (cf. Fig. 6b and 10a). Thus, surprisingly, prior to failure and within the explored range of ligament length l, the tensile force arising from drawing is independent of l (cf. Fig. 4a, b, 8a and 9a). Moreover, this peculiar scaling, validated using linear-elastic brittle solids such as PS and highly-crosslinked BR (cf. Fig. 12a and b), necessarily produces a term in the total mechanical work that is quadratic in l although no plastic deformation is involved.

Author contributions

Asal Y Siavoshani: quantitative birefringence measurements, data curation, formal analysis, validation, and writing – experimental section. Zehao Fan: curation of data in Fig. 3. Shi-Qing Wang: supervision, conceptualization, methodology, formal analysis, validation, and writing – drafting, reviewing and editing.

Conflicts of interest

There are no conflicts to declare.

Data availability

The data supporting this article have been included as part of the supplementary information (SI). Supporting information includes the representative stress–strain (SS) curve for the uncut polystyrene (PS) specimen to fracture, the methodology for determining the fracture energy release rate G for the butadiene rubber (BR) system, SS curves for uncut BR specimens, and Movie SI.1 showing dynamic crack propagation in the BR1phr elastomer. See DOI: https://doi.org/10.1039/d6sm00148c.

Acknowledgements

This work is supported, in part, by the Polymers program at the US National Science Foundation through Special Creativity Extension of grant DMR-2210184.

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