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Nucleation and propagation of fracture in viscoelastic elastomers: A complete phase-field theory

T0 review · 4 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read One phase-field theory now predicts both nucleation and propagation of fracture in viscoelastic elastomers.

desk verdict A genuine first: a phase-field theory for viscoelastic elastomers that pairs a strength surface with the equilibrium-energy Griffith condition, but the load-bearing δ^ε coefficient is imported rather than derived and the validation is partly circular. read the letter →

arxiv 2506.16479 v1 pith:BZZGRCTD submitted 2025-06-19 cond-mat.soft math-phmath.MP

Farhad Kamarei , Evan Breedlove , Oscar Lopez-Pamies This is my paper · ORCID
classification cond-mat.softmath-phmath.MP MSC 74R1074D1074S05
keywords viscoelasticelastomersphase-fieldfracturenucleationstrengthsurfaceGriffithenergyfiniteviscoelasticityquasistaticloadinginternalvariables
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper seeks to establish a complete macroscopic phase-field theory that predicts where and when cracks nucleate and how they propagate in viscoelastic elastomers under slow, quasistatic loads. It argues from experiments that any such theory needs three ingredients: finite viscoelasticity, a strength surface defined in stress-deformation space, and a fracture energy. The central claim is that nucleation is governed by this strength surface while propagation and growth from large pre-existing cracks follow a Griffith condition in which only the equilibrium part of the stored elastic energy competes with the fracture energy. Simulations of three classic experiments—rubber-band uniaxial tension, pure-shear crack nucleation, and trousers tearing—match published data, which the authors present as evidence that the framework is complete. The theory reduces to solving two nonlinear PDEs coupled with a nonlinear ODE for the deformation field, a tensorial internal variable, and the phase field.

What carries the argument

The core object is the phase field z(X,t), an order parameter taking z=1 in intact material and z=0 in fractured material, coupled to an internal variable C_v that stores the viscous part of the deformation. The load-bearing identity is the reduced Griffith criticality condition -∂W^Eq/∂Γ0 = G_c, which says that crack growth is driven only by the equilibrium stored energy W^Eq competing with the fracture energy G_c. The strength of the elastomer enters through a Drucker-Prager-type strength surface F(S,F)=0, whose violation under uniform stress activates a driving force c_e inside the phase-field equation. These ingredients together generate the governing system (29)-(31): balance of linear momentum, the internal-variable evolution equation, and the phase-field equation with inequalities expressing irreversibility and the bounds 0≤z≤1.

What would settle it

Run pure-shear crack-nucleation tests on the same elastomer over stretch rates spanning at least six decades; if the critical stretch at which the pre-existing crack starts to grow changes by substantially more than experimental scatter, the theory's claim that only the equilibrium stored energy enters the Griffith condition is wrong.

Watch

Extended reading notes

Core claim

The paper claims that fracture in viscoelastic elastomers can be described by a single phase-field model in which the deformation field y(X,t), a tensorial internal variable C_v(X,t) tracking viscous relaxation, and a phase field z(X,t) evolve together under quasistatic loading. The key physical assertion is that the Griffith energy competition for crack growth involves only the equilibrium portion of the stored elastic energy—the energy that would remain if the material were held at fixed deformation until all viscous relaxation finished—and not the total stored or dissipated energy. Nucleation under spatially uniform stress is instead assigned to a strength surface F(S,F)=0 in stress-deformation space, which enters the phase-field equation through a constitutive driving force c_e. The full system, equations (29)-(31), is shown to reproduce experimental measurements of nucleation in the bulk, nucleation from large pre-existing cracks, and steady crack propagation, supporting the claim that the theory is complete for quasistatic loading.

Load-bearing premise

The theory assumes that the strength surface of a viscoelastic elastomer can be written as a single hypersurface in stress-deformation space with no explicit dependence on loading history or rate.

Editorial extensions

If this is right

  • If the theory is correct, a single set of field equations can predict both crack nucleation and crack growth in viscoelastic elastomers under arbitrary quasistatic loads, with no need to prescribe a crack path in advance.
  • The apparent rate dependence of tearing energy emerges naturally from viscous dissipation during loading, while the critical stretch at which a large crack starts to grow in pure shear is predicted to be essentially rate independent.
  • Fracture nucleation from small flaws, notches, or other nonuniform stress states is predicted to be a mediation between the strength surface and the equilibrium-energy Griffith condition, so the theory interpolates between the two classical limiting criteria.
  • All material inputs—viscoelastic parameters, uniaxial and hydrostatic strength, and fracture energy—can in principle be measured from standard experiments such as uniaxial tension, poker-chip tests, pure-shear tests, and trousers tests.
  • The numerical scheme based on non-conforming finite elements in space and explicit Runge-Kutta time stepping provides a practical route to simulate finite-deformation, near-incompressible elastomer fracture at large strains.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural next test would be to use the theory to predict the critical flaw size at which nucleation switches from strength-dominated to Griffith-dominated behavior, and to compare that prediction with the measured fracto-cohesive length scale in materials such as VHB 4905.
  • Because the strength surface is assumed history independent, the theory is likely to break down at very high loading rates or under cyclic loading; a plausible extension would make the strength surface depend on a rate-sensitive internal variable rather than on time explicitly.
  • The same phase-field structure could be adapted to fatigue by introducing a second internal variable that progressively degrades either the strength surface or the fracture energy with accumulated cycles, although the paper explicitly leaves non-monotonic loading for future work.
  • If the equilibrium-energy Griffith condition holds generally, then measurements of the critical tearing energy at different rates can be used to infer the equilibrium stored energy of an elastomer, offering a new macroscopic route to test constitutive viscoelastic models.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper develops a macroscopic phase-field theory of fracture nucleation and propagation in viscoelastic elastomers under quasistatic loading. It extends the Kumar-Francfort-Lopez-Pamies phase-field approach by coupling finite viscoelasticity (a two-potential model with internal variable C_v), a Drucker-Prager-type strength surface in stress-deformation space, and a Griffith criticality condition in which only the equilibrium free energy competes with the fracture energy. The governing equations are (29)-(31), and a numerical scheme based on Crouzeix-Raviart finite elements and explicit Runge-Kutta time stepping is presented. The theory is illustrated by simulations of uniaxial ring tests on polyurethane (Mueller 1968), pure-shear tests on VHB 4905 (Pharr et al. 2012), and trousers tests on SBR (Greensmith and Thomas 1955).

Significance. If the theory is correct, it would be a substantial contribution: it is, to my knowledge, the first phase-field framework for elastomers that simultaneously accounts for strength and viscoelasticity-related toughness in a way consistent with the equilibrium-energy Griffith condition, and it offers a complete numerical methodology. The manuscript is careful in assembling the experimental evidence, and the numerical implementation appears robust. The main caveats are that the central calibration coefficient delta^epsilon is imported from an elastic-brittle derivation without proof, and that parts of the validation reuse parameters fitted to the same experiments, making some comparisons descriptive rather than predictive. These issues do not invalidate the framework, but they need to be addressed before the paper can be judged as establishing a complete and predictive theory.

major comments (4)
  1. [Section 3.5.4, Eq. (26)] The coefficient delta^epsilon in Eq. (26) controls both the divergence term in the phase-field equation (31) and, through Eqs. (27)-(28), the magnitude of the driving force; it is the mechanism by which crack growth from large pre-existing cracks is forced to satisfy the Griffith condition (7). The paper does not derive this coefficient for the viscoelastic setting. Section 3.5.4 states that the prescription is analogous to that of Kamarei et al. (2024) and therefore the same formula applies. That inference is not immediate: the invariants I_1 and I_2 entering c_e through Eq. (21) contain the non-equilibrium stress through psi_NEq and C_v, so the asymptotic expansion that produced delta^epsilon in the elastic-brittle case must be re-examined for finite viscoelasticity. If Eq. (26) is not the correct asymptotic calibration, the predicted critical crack-growth stretch will deviate from -dW^Eq/dGamma_0 = G_c, and the central claim fails at its most consequential point. The manuscript should either supply a derivation or demonstrate by epsilon-convergence studies that the sharp-interface limit is recovered. Section 5.2 uses a single regularization length epsilon = 0.5 mm and Section 5.3 is only qualitative, so neither supplies that evidence.
  2. [Section 5.2 and Eq. (7)] The VHB comparison in Section 5.2 does not independently test the crack-growth mechanism. The material constants in Table 4 and the fracture energy G_c = 634 N/m are taken from Shrimali and Lopez-Pamies (2023b), where they were used to explain the same Pharr et al. (2012) experiments; the strength constants in Table 5 are stipulated as being consistent with experiments. More importantly, Eq. (26) is constructed so that growth from large cracks obeys Eq. (7), and the predicted critical stretch Lambda_c = 3.63 is the value already imposed by Eq. (7). The agreement in Fig. 13 therefore confirms internal consistency and the numerical implementation, but it is not an independent validation that the theory predicts nucleation from pre-existing cracks. A test using independently measured strength and toughness, in a different geometry or loading history, would be needed.
  3. [Section 5.1, Tables 2 and 3] For the Mueller ring tests, the viscoelastic constants in Table 2 are fitted to the slowest and fastest stress-stretch curves, and the strength constants in Table 3 are fitted to the critical stress-stretch pairs for all ten stretch rates. Consequently, Fig. 9 demonstrates that the framework can describe the data set used for calibration, but it does not provide predictive evidence in the sense claimed in the abstract and Section 6. The authors should either calibrate on a subset of rates and validate on the remaining ones, or temper the predictive claim for this example.
  4. [Section 2.2.1, Eq. (2)] The strength surface is assumed to be independent of loading history, so that it can be written as F(S,F) = 0. The manuscript cites experimental support to a first degree of approximation, but this is a central assumption for the nucleation mechanism, since c_e in Eqs. (27)-(28) contains no explicit time dependence. The available data are mostly uniaxial and equi-biaxial tests; no direct evidence is provided for arbitrary multiaxial histories or for non-monotonic loading paths. The authors should state this assumption more prominently and, if possible, test the predicted nucleation under a relaxation or non-monotonic history where rate effects would be distinguishable from the stress-deformation correlation.
minor comments (4)
  1. [References] There are typographical errors in the reference list: 'Jounal of Applied Mechanics' in Chockalingam (2025) and 'Journal of the Mechancis and Physics of Solids' in Breedlove et al. (2024) should be corrected.
  2. [Remark 10] The h-correction formula for delta^epsilon would benefit from explicit parentheses; as typeset, '1 + 3/8 h/epsilon' is ambiguous and could be misread as 1 + 3/(8h/epsilon) rather than 1 + (3/8)(h/epsilon).
  3. [Table 5] Table 5 lists a2 = 0 but omits b2, although equation (5) contains b2. The authors should state that b2 is unused when a2 = 0, or add the value for completeness.
  4. [Section 5.3] The trousers comparison is explicitly qualitative, but the text around Fig. 16(b) says the theory 'predicts accurately' the propagation described by Eq. (8). Since no quantitative comparison with the Greensmith and Thomas data is made, the wording should be softened to 'reproduces qualitatively' to avoid overclaiming.

Circularity Check

3 steps flagged · score 6.0 of 10

The nucleation and propagation criteria are calibrated to their own target limits: c_e is constructed to reproduce the input strength surface, and δ^ε is imported from prior elastic work to enforce Griffith; the three validation benchmarks fall in those calibrated limits.

  1. self definitional [Section 3.5.2 and Section 5.1.1-5.1.3]
    "The coefficients β ε 0 (I1), β ε 1 (I1), β ε 2 (I1) in the driving force (20) must be selected so that ... the evolution equation (19)1 for the phase field is satisfied when the given strength surface F(S(t), F(t)) = 0 ... is first violated along the given loading path in the limit as the regularization length ε↘0."

    The driving force c_e is the nucleation mechanism of the theory, and it is explicitly constructed so that, under spatially uniform stress and deformation, the phase-field equation triggers precisely when the input strength surface F(S,F)=0 is violated; Eqs. (24)-(25) enforce this identity at finite ε for uniaxial and hydrostatic loading. Section 5.1 then fits the uniaxial strength function sts(I1) of that same surface to Mueller's measured (S_c, Λ_c) failure data for all ten stretch rates, and reports the simulated (S_c, Λ_c) as a prediction. Because the gauge section stress is uniform and the stochastic perturbation is mild, the simulation returns the fitted strength surface by construction. The agreement in Fig.

  2. self citation load bearing [Section 3.5.4, Eq. (26)]
    "Given that the prescription (20) for c e (X,t), as well as the coefficients (25), are analogous to those utilized by Kamarei et al. (2024) for purely elastic elastomers, the same type of formula derived for δ ε by these authors applies here. We thus set δ ε = [(s ts (I1) + (1+2√3)s hs)/((8+3√3)s hs)] (3G c/(16ψ Eq ts (I1)ε)) + 2/5."

    δ^ε is the coefficient that controls large-crack nucleation and propagation in the phase-field equation (31), appearing as the multiplier of εG_c in the divergence term and entering c_e through (27)-(28). No derivation of this coefficient is given for finite viscoelasticity; instead the paper transfers the purely elastic formula from Kamarei et al. (2024) by analogy and self-citation. The non-equilibrium stress enters the invariants I_1, I_2 through Eq. (21), so the elastic asymptotic analysis does not automatically carry over. The subsequent agreement with the sharp Griffith condition (7) in Sections 5.2 and 5.3 is thus the designed consequence of an unverified, self-cited calibration rather than an independent prediction.

1 more flagged steps
  1. fitted input called prediction [Sections 5.2.1 and 5.2.3]
    "Their analysis showed that the viscoelastic model (14)-(15) with the materials constants listed in Table 4 provides a reasonably accurate description of the experimentally measured viscoelastic response of VHB 4905. Their analysis also showed that the fracture energy of this elastomer is about G c = 634 N/m. ..."

    The VHB benchmark is the large pre-existing crack limit, i.e., exactly the Griffith limit that the paper identified as input in Section 2.2.2 and built into the model via the imported δ^ε formula. The fracture energy G_c = 634 N/m and the viscoelastic constants are taken from the same authors' prior analysis of these same Pharr experiments, and δ^ε was imported to make the model follow the Griffith condition (7). Therefore the reported Λ_c ≈ 3.63, independent of loading rate, is the value already obtained from the sharp Griffith condition in Shrimali and Lopez-Pamies (2023b); the phase-field simulation confirms internal consistency with the built-in calibration rather than providing a first-principles prediction.

full rationale

The paper is not wholly circular: it constructs a genuinely new phase-field synthesis for viscoelastic elastomers, with a nontrivial c_e structure, a numerical implementation, and an interpolation claim for non-uniform stress states. However, the claimed validations are largely consistency checks in the two asymptotic limits that were used as inputs. The strength surface is an input, c_e is explicitly selected so that uniform-stress nucleation coincides with that surface, and the Mueller comparison uses strength parameters fitted to the very failure data being 'predicted.' Likewise, the Griffith condition (7) is an input drawn from self-cited prior work, δ^ε is imported from the authors' elastic theory without a viscoelastic derivation, and the VHB and trousers simulations recover the same Griffith behavior that the model was calibrated to reproduce. The genuinely predictive regime of the theory—fracture under non-uniform stress with finite flaws between the strength and Griffith limits—is not tested against experiments, and no ε-convergence check is provided for Eq. (26) in viscoelasticity. These factors make the central validation claims partially circular, though the framework retains independent content. Score 6.

Assumptions & free parameters 6 free parameters · 8 assumptions · 0 invented entities

The central claim rests on a large set of material parameters that are fit to experiments for each elastomer, and on domain assumptions about the rate-independence of strength, the equilibrium-energy Griffith condition, and the adequacy of the chosen constitutive forms. No new physical entities are introduced: the phase field z is a regularized order parameter, C_v is a standard internal variable, and the strength surface F(S,F)=0 is a calibrated mathematical construct. The theory is therefore a synthesis of existing ingredients plus a new coupling, rather than a derivation from independent first principles.

free parameters (6)
  • Viscoelastic model constants (Eq. (13)-(15)): mu1, mu2, kappa, alpha1, alpha2, nu1, nu2, beta1, beta2, eta0, eta_inf… = Mueller: Table 2; VHB 4905: Table 4; trousers: Table 6
    Fifteen material constants fit to stress-stretch or relaxation data for each elastomer; the Mueller fit uses only the slowest and fastest stretch rates.
  • Uniaxial strength model constants a0, a1, b1, a2, b2, c0, c1 (Eq. (5)) = Mueller: Table 3; VHB: Table 5; trousers: Table 7
    Fit to uniaxial tensile strength data; for VHB the values are only consistent with two experiments, not measured directly.
  • Hydrostatic strength s_hs = Mueller: 3 MPa; VHB: 500 kPa; trousers: 2 MPa
    Estimated from poker-chip or related hydrostatic experiments; a single scalar constant in the strength surface (3).
  • Fracture energy G_c = Mueller: 41 N/m; VHB: 634 N/m; trousers: 200 N/m
    Fracture energy is a material input measured or estimated from pure-shear or tearing tests.
  • Regularization length epsilon = Mueller: 0.02 mm; VHB: 0.5 mm; trousers: 0.25 mm
    Chosen smaller than material length scales; the paper states results are insensitive to smaller values but does not show a systematic convergence study.
  • Stochastic perturbation of strength parameters a1, a2 = Plus or minus 5 percent variation in subregions of size 5*epsilon for Mueller simulations
    Introduced to break symmetry and match the experimental observation that fracture occurs in only one gauge section.
assumptions (8)
  • domain assumption Balance of linear momentum and balance of configurational forces govern deformation and fracture (Section 3.4).
    The governing equations (17)-(19) are derived from these two balance principles, following Kumar et al. (2018a).
  • domain assumption The strength of viscoelastic elastomers is independent of loading history to first approximation, so F(S,F)=0 (Section 2.2.1, Eq. (2)).
    This is the key premise allowing a time-independent strength surface; the paper states experiments support this only as a first approximation.
  • domain assumption Fracture nucleation from large cracks and propagation follow the Griffith competition between only the equilibrium stored energy and fracture energy, -dW^Eq/dGamma0 = G_c (Section 2.2.2, Eq. (7)).
    This is imported from Shrimali and Lopez-Pamies (2023b), a self-cited paper, and is built into the phase-field equation (31).
  • ad hoc to paper The specific Drucker-Prager-type strength surface (3) with the uniaxial strength formula (5) adequately represents the strength of the elastomers considered.
    The surface is chosen for simplicity and because it fits uniaxial, biaxial, and hydrostatic data; the paper acknowledges more data is needed to validate it.
  • domain assumption The two-potential viscoelastic model (11)-(12), with the specific form (13), describes the finite deformation viscoelasticity of the elastomers.
    This constitutive class is standard and the specific form has been used by the authors before, but it is an input assumption for the fracture theory.
  • domain assumption Quasistatic loading: inertial effects are negligible (Section 1).
    The theory targets quasistatic loading, so fast crack dynamics with inertia are outside its scope.
  • domain assumption No other dissipation mechanisms are active: strain-induced crystallization, Mullins effect, and healing are negligible or absent (Section 1).
    The paper restricts attention to unfilled elastomers or those with small fillers, and lists these omitted mechanisms as future work.
  • domain assumption Viscous dissipation is isochoric, det(C_v)=1 (Section 3.2.1).
    This is a constitutive constraint on the internal variable and is enforced in the evolution equation (12).

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Pith. "Pith review of Nucleation and propagation of fracture in viscoelastic elastomers: A complete phase-field theory." pith.science (2026). https://pith.science/paper/BZZGRCTD

@misc{pith2026250616479,
  author       = {Pith},
  title        = {Pith review of: Nucleation and propagation of fracture in viscoelastic elastomers: A complete phase-field theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BZZGRCTD}},
  note         = {Machine review of arXiv:2506.16479}
}
read the original abstract

This paper presents a macroscopic theory, alongside its numerical implementation, aimed at describing, explaining, and predicting the nucleation and propagation of fracture in viscoelastic materials subjected to quasistatic loading conditions. The focus is on polymers, in particular, on elastomers. To this end, the starting point of this work is devoted to summarizing the large body of experimental results on how elastomers deform, nucleate cracks, and propagate cracks when subjected to mechanical loads. When viewed collectively, the experiments make it plain that there are three basic ingredients that any attempt at a complete macroscopic theory of fracture in elastomers ought to account for: i) the viscoelasticity of the elastomer; ii) its strength; and iii) its fracture energy. A theory is then introduced that accounts for all these three basic ingredients by extending the phase-field theory initiated by Kumar, Francfort, and Lopez-Pamies (J. Mech. Phys. Solids 112 (2018), 523--551) for elastic brittle materials to seamlessly incorporate viscous energy dissipation by deformation, a generalized strength surface that is a hypersurface in stress-deformation space (and not just in stress space as for elastic brittle materials), and the pertinent Griffith criticality condition for materials that dissipate energy not just by the creation of surface but also by deformation, in this case, by viscous deformation (Shrimali and Lopez-Pamies (2023) Extreme Mech. Lett. 58, 101944). From an applications point of view, the proposed theory amounts to solving an initial-boundary-value problem comprised of two nonlinear PDEs coupled with a nonlinear ODE for the deformation field, a tensorial internal variable, and the phase field. A robust scheme is presented to generate solutions for these equations.

Figures

Figures reproduced from arXiv: 2506.16479 by the authors.

Figure 1
Figure 1. Strength surface (3) with (5) and the material constants given in [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Schematic of the specimens (L = 60 mm, H = 6 mm, B = 1.5 mm, A = 10 mm) and results for the critical global stretch Λc = hc/H and the corresponding critical global stress Sc = Pc/(BL), plotted as functions of the loading rate Λ =˙ h/H˙ , at which fracture nucleates from the large pre-existing cracks in the “pure-shear” fracture tests on SBR reported by Wang et al. (2017). 9 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Schematic of the specimens (L = 101.6 mm, H = L = 101.6 mm, B = 0.7938 mm, A = 5.08 mm) and results for the critical time tc, plotted as a function of the applied global stress S0 = P/(BL), at which fracture nucleates from the large pre-existing cracks in the delayed fracture tests on a polyurethane elastomer reported by Knauss (1970). of the loading rate at which the test is carried out; see, e.g., Major and Lang (… view at source ↗
Figures from the paper (15 more)
Figure 4
Figure 4. Figure 4: Schematic of the specimens (L ≤ 100 mm, H ≤ 50 mm, B = 0.5 mm) and results for the critical global stress Sc = Pc/(BH), plotted as a function of the size A of the pre-existing crack, at which fracture nucleates from the pre-existing cracks in the single edge notch frac…
Figure 5
Figure 5. Figure 5: Schematic of the specimens (L = 150 mm, H = 40 mm, B = 1 mm, A = 50 mm) and results for the normalized critical force 2Pc/B at which the crack grows at a constant speed ˙a ≈ ˙ l/2 in the trousers tests on a SBR (vulcanizate D) reported by Greensmith and Thomas (1955). …
Figure 6
Figure 6. Figure 6: Rheological representation of the two-potential viscoelastic model (11)-(12). In the comparisons with experiments that are presented in Section 5 below, we will make use of the following specific forms for the equilibrium and non-equilibrium free energy functions in (9…
Figure 7
Figure 7. Figure 7: (a) Schematic of the ring specimens (A = 16.5 mm, B = 19 mm, R = 5 mm, L = π(A − R) = 36.1 mm) and loading in the experiments carried out by Mueller (1968) on a polyurethane elastomer; the thickness of the specimens is H = 2.5 mm. (b) Comparison between the response pr…
Figure 8
Figure 8. Figure 8: One of the FE meshes used to simulate the experiments of Mueller (1968) and the subregions over which the strength material constants are stochastically perturbed. 5.1.3. Comparisons with the experiments [PITH_FULL_IMAGE:figures/full_fig_p032_8.png]
Figure 9
Figure 9. Figure 9: compares the results predicted by the phase-field theory with the experimental data reported by Mueller (1968) for (a) the critical stress Sc and (b) the critical stretch Λc at which the specimens fracture; both results are presented as functions of the stretch rate Λ …
Figure 10
Figure 10. Figure 10: One of the FE meshes used to simulate the experiments of Pharr et al. (2012). A line has been drawn on the mesh to help visualize the location of the pre-existing crack. 5.2.3. Comparisons with the experiments [PITH_FULL_IMAGE:figures/full_fig_p034_10.png]
Figure 11
Figure 11. Figure 11: (a) compares the stress-stretch response predicted by the phase-field theory with the experi￾mental data reported by Pharr et al. (2012) for three different stretch rates, Λ = 1 ˙ .67 × 10−3 , 6.67 × 10−2 , 1.67 s−1 . The main observation from this figure is that the …
Figure 12
Figure 12. Figure 12: Contour plots of the phase field z(X, t) over the undeformed and deformed configurations of specimens stretched at Λ = 1 ˙ .67 × 10−3 , 6.67 × 10−2 , 1.67 s−1 . The results pertain to two values of the applied stretch Λ for each Λ. ˙ 0 50 100 150 200 250 10-3 10-2 10-…
Figure 13
Figure 13. Figure 13: Comparisons between the values predicted by the phase-field theory (lines) and the experimental data (circles) of Pharr et al. (2012) for (a) the critical stretch Λc and (b) the critical stress Sc at which fracture nucleates from the pre-existing crack. Both sets of r…
Figure 5
Figure 5. Figure 5: They reported results for the normalized critical force 2 [PITH_FULL_IMAGE:figures/full_fig_p036_5.png]
Figure 14
Figure 14. Figure 14: (a) Uniaxial stress-stretch response at various constant stretch rates described by the viscoelastic model (14)-(15) with the material constants listed in [PITH_FULL_IMAGE:figures/full_fig_p036_14.png]
Figure 15
Figure 15. Figure 15: One of the FE meshes used to simulate the trousers experiments. 37 [PITH_FULL_IMAGE:figures/full_fig_p037_15.png]
Figure 16
Figure 16. Figure 16: (a) presents results, plotted as functions of the applied grip-to-grip separation l, for the force P predicted by the phase-field theory for trousers tests carried out at three different constant rates of separation between the grips, ˙ l = 5 × 10−3 , 5 × 10−1 , 500 m…
Figure 17
Figure 17. Figure 17: Contour plot of the phase-field z(X, t) over the undeformed and deformed configurations at the grip-to-grip separation l = 114 mm for the specimen teared at ˙ l = 500 mm/s. 38 [PITH_FULL_IMAGE:figures/full_fig_p038_17.png]

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