A thermodynamically consistent mixed-mode cohesive zone model for fracture of self-healing materials

Sigmoidal strength recovery curves were obtained from experiments on self-healing ceramic materials, asphalt concrete, microcapsule-based self-healing materials, and hydrogels.

“For example, sigmoidal strength recovery curves were obtained from experiments conducted on self-healing ceramic materials (Ozaki et al., 2016(Ozaki et al., , 2020)), asphalt concrete (García, 2012;Qiu et al., 2012), microcapsule-based self-healing materials (Zwaag, 2008), and hydrogels (Wang et al., 2017;Yu et al., 2018).”

• cites Ozaki et al., 2016 sourcePartly supported

  Section 3.2 compares healing-time-dependent strength recovery in self-healing ceramics with experimental results reported by Osada et al. (2009). Our copy does not describe those experimental curves as sigmoidal, so it supports the experimental strength-recovery portion of the claim but not the specified curve shape. The other materials are attributed to other citations in the sentence. (LLM, conf=0.81)

  Experimental — accuracy not yet measured
• cites García, 2012;Qiu et al., 2012Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites Zwaag, 2008 sourceNot checked — no full text of the source

The sigmoid-like evolution of recovery ratios is consistent with experimental observations obtained for various self-healing materials.

“which is consistent with experimental observations obtained for various self-healing materials.”

• cites Ozaki et al., 2016 sourcePartly supported

  Section 3.2 (page unavailable) reports a qualitative agreement between modeled strength recovery over healing time and experimental results for alumina/SiC nanocomposites. It does not describe the observed recovery as sigmoid-like or establish that pattern across various self-healing materials. (LLM, conf=0.82)

  Experimental — accuracy not yet measured
• cites Ozaki et al., 2020 sourcePartly supported

  The 'Comparison with experimental results' section (page unavailable) reports that strength recovery increased with healing time in the FEA and in an experiment on self-healing ceramics. It does not describe sigmoid-like recovery ratios or establish that this pattern was observed across various self-healing materials. (LLM, conf=0.84)

  Experimental — accuracy not yet measured
• cites García, 2012;Qiu et al., 2012Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Experimental load-displacement curves for the SBSmas specimens were not reported in Qiu et al. (2012).

“Experimental load-displacement curves for the SBSmas specimens were not reported inQiu et al. (2012).”

• cites Qiu et al. (2012) sourceSupported

The model can predict recovery of strength and fracture toughness as functions of time and temperature.

“The results demonstrate that the model can predict recovery of strength and fracture toughness as functions of time and temperature.”

Mixed-mode bending simulations demonstrate how mode-dependent healing kinetics and local mixed-mode history govern spatially non-uniform damage recovery and the resulting structural response.

“Finally, we conduct mixed-mode bending simulations to demonstrate how mode-dependent healing kinetics and local mixed-mode history govern spatially non-uniform damage recovery and the resulting structural response.”

Single-element tests under mode-I and mixed-mode conditions show that the model predicts damage and healing evolutions that align with theoretical expectations.

“We conduct single-element tests under mode-I and mixed-mode conditions and show that the model predicts damage and healing evolutions that align with theoretical expectations.”

The model captures the experimental strength recovery trends across all temperatures.

“As shown in the figure, our model captures the experimental strength recovery trends across all temperatures.”

• AssumptionThe authors state this as an assumption: The bulk material behaves elastically in the numerical models.

In non-proportional mixed-mode loading, the traction-separation curves reveal asymmetric recovery in normal and tangential directions, reflecting the time lag in rest periods between modes.

“The traction-separation curves in Figures 8d-e reveal asymmetric recovery in the normal and tangential directions, reflecting the time lag in rest periods between modes.”

The peak cohesive strength recovery is more significantly affected by ξ_n, with greater separation between curves, especially for ξ_n ≥ 1.5.

“In contrast, Figure 6c shows that the peak cohesive strength recovery (σ h max /σ max ) is more significantly affected, with greater separation between the curves, especially for ξ n ≥ 1.5.”

The model accurately captures the experimental load-displacement response of PBmas specimens during initial fracture, both in pre-peak and post-peak regions.

“These results show that our model accurately captures the experimental load-displacement response, both in the pre-peak and post-peak regions.”

• AssumptionThe authors state this as an assumption: The bulk material behaves elastically in the numerical models.

The model predicts progressive recovery in both stiffness and strength during refracture for various rest times.

“Figure 9d shows the corresponding load-displacement curves during refracture for various rest times, clearly illustrating that our model predicts progressive recovery in both stiffness and strength.”

• AssumptionThe authors state this as an assumption: The bulk material behaves elastically in the numerical models.

In non-proportional mixed-mode loading, the healing variables h_n and h_t initiate growth only during periods when ∆n, ∆n_dot, ∆t, and ∆t_dot are simultaneously equal to zero.

“As shown in Figure 8b, the healing variables h n and h t initiate growth only during periods when ∆ n , ∆n , ∆ t , and ∆t are simultaneously equal to zero, reinforcing the importance of accounting for synchronized rest to enable healing.”

In proportional mixed-mode loading, the effective damage parameters remain constant between C and D, for which ∆n ≤ κ_h_n and ∆t ≤ κ_h_t.

“Note that the effective damage parameters remain constant between C and D, for which ∆ n ≤ κ h n and ∆ t ≤ κ h t .”

In the pure mode-I opening test, the healing variable remains equal to zero up until point B because the healing mechanism has not been triggered yet.

“The healing variable (Figure 3b) remains equal to zero up until point B because the healing mechanism has not been triggered yet.”

The healing exponents (ξn = ξt) generally decrease with increasing temperature.

“The results in Table 3 also show that the healing exponents (ξ n = ξ t ) generally decrease with increasing temperature, promoting faster progression of the healing process thermal activation becomes more pronounced.”

The reloading portion of the traction-separation curve shows a gain in stiffness and strength as a result of the healing process.

“We also note that the reloading portion of the traction-separation curve (between points C and D) show a gain in stiffness and strength as a result of the healing process.”

Increasing ξ_n systematically delays the growth of the healing variable and the corresponding recovery ratios.

“increasing ξ n systematically delays the growth of the healing variable and the corresponding recovery ratios.”

Healing evolves at a faster rate as healing temperature increases.

“As expected, the results show that healing evolves at a faster rate as healing temperature increases.”

• AssumptionThe authors state this as an assumption: The bulk material behaves elastically in the numerical models.

As rest time increases, the peak load and stiffness of partially healed beams progressively recover and eventually reach the initial fracture response.

“The results show that, as rest time increases, the peak load and stiffness of the partially healed beams progressively recover and eventually reach the initial fracture response.”

• AssumptionThe authors state this as an assumption: The bulk material behaves elastically in the numerical models.

At 30°C, the model correctly predicts the slow, sublinear increase in healing strength ratio with time, which reflects limited polymer mobility at low temperatures.

“At 30 • C (Figure 10f), the model correctly predicts the slow, sublinear increase in healing strength ratio with time, which reflects the limited polymer mobility at low temperatures.”

• AssumptionThe authors state this as an assumption: The bulk material behaves elastically in the numerical models.

The fracture energy recovery ratio shows a relatively smoother transition across the range of ξ_n considered, suggesting that energy-based metrics are somewhat less sensitive to early-stage healing variations.

“In Figure 6b, the fracture energy recovery ratio (ϕ h n /ϕ n ) shows a relatively smoother transition across the range of ξ n considered in this analysis, suggesting that energy-based metrics are somewhat less sensitive to early-stage healing variations.”

In the pure mode-I opening test, during the rest period the healing variable increases, which causes the effective damage parameter to reduce.

“During the rest period (between points B and C) the healing variable increases, which causes the effective damage parameter to reduce (as expected).”

The normal healing variable and the recovery ratios of fracture energy, cohesive strength, and initial stiffness increase at a faster rate when the healing viscosity parameter χ_n increases.

“As shown by the results, the normal healing variable (h n ) as well as the recovery ratios (ϕ h n /ϕ n , σ h max /σ max , and E h n /E n ) increase at a faster rate when χ n increases.”

The PBmas specimens exhibit a faster healing response than the SBSmas specimens.

“Returning to Figure 9e, we observe that the PBmas specimens exhibit a faster healing response than the SBSmas specimens.”

• AssumptionThe authors state this as an assumption: The bulk material behaves elastically in the numerical models.

The evolution of the recovery ratios of all quantities considered increases in a sigmoid-like manner.

“The results also show that the evolution of the recovery ratios of all quantities considered herein increase in a sigmoid-like manner”

At 40°C, the calibrated healing exponent is slightly lower than that at 50°C.

“A minor deviation from this trend is observed at 40 • C, where the calibrated exponent is slightly lower than that at 50 • C.”

The model accurately reproduces key experimental trends in healing strength ratio H_P for PBmas and SBSmas samples.

“These results show that our model accurately reproduces key experimental trends.”

• AssumptionThe authors state this as an assumption: The bulk material behaves elastically in the numerical models.

The initial stiffness recovery ratio reflects a strong dependency on ξ_n, particularly for shorter rest times, where higher exponents yield much flatter recovery curves.

“Similarly, the initial stiffness recovery ratio (E h n /E n ) in Figure 6d reflects a strong dependency on ξ n , particularly for shorter rest times, where higher exponents yield much flatter recovery curves.”

An exception to the trend arises at 30°C, where the calibrated healing viscosity parameter is anomalously high.

“An exception to this trend arises at 30 • C, where the calibrated healing viscosity parameter is anomalously high.”

• AssumptionThe authors state this as an assumption: The bulk material behaves elastically in the numerical models.

The extent of the evolution of the loading trajectory depends on the location of the element within the interface.

“The extent of this evolution depends on the location of the element within the interface.”

• AssumptionThe authors state this as an assumption: For the MMB simulations, the bulk material is linear elastic with Young's modulus E = 122 GPa and Poisson's ratio ν = 0.25.

The calibrated parameters yield recovery curves that are in good agreement with the experimental trends.

“Despite these minor deviations, the calibrated parameters yield recovery curves that are in good agreement with the experimental trends.”

In MMB element B, the relative displacement between lower and upper faces is initially dominated by tangential sliding, while normal opening becomes increasingly significant as deformation progresses.

“The relative displacement between the lower and upper faces of element B is initially dominated by tangential sliding, while normal opening becomes increasingly significant as deformation progresses.”

• AssumptionThe authors state this as an assumption: For the MMB simulations, the bulk material is linear elastic with Young's modulus E = 122 GPa and Poisson's ratio ν = 0.25.

During the rest period, the effective crack opening histories decrease according to Equation (27), driving the cohesive state toward the origin of the (κh_n, κh_t) plane and toward lower effective-damage states.

“During the rest period, the effective crack opening histories decrease according to Equation ( 27), driving the cohesive state toward the origin of the (κ h n , κ h t ) plane and, consequently, toward lower effective-damage states.”

• AssumptionThe authors state this as an assumption: For the MMB simulations, the bulk material is linear elastic with Young's modulus E = 122 GPa and Poisson's ratio ν = 0.25.

With ξn = ξt, when χn/χt = 1, the normal and tangential histories decrease proportionally, producing straight healing trajectories toward the origin.

“As observed in Figures 13a,b, with ξ n = ξ t , when χ n /χ t = 1, the normal and tangential histories decrease proportionally, producing the straight healing trajectories toward the origin.”

• AssumptionThe authors state this as an assumption: For the MMB simulations, the bulk material is linear elastic with Young's modulus E = 122 GPa and Poisson's ratio ν = 0.25.

The model's numerical predictions closely match the experimental values reported by Jud and Kausch (1979) for PMMA compact tension tests.

“As shown, our numerical predictions (solid line) closely match the experimental values (dots) reported by Jud and Kausch (1979)”

• cites Jud and Kausch (1979) sourceNot checked — no full text of the source

• AssumptionThe authors state this as an assumption: For the PMMA compact tension simulations, the bulk material is linear elastic with Young's modulus E = 5 GPa and Poisson's ratio ν = 0.3.

For both elements A and B, the loading trajectory initially develops in the κh_t direction and subsequently turns toward increasing κh_n, reflecting the evolution of the local mode mixity as the crack advances.

“For both elements, the loading trajectory initially develops in the κ h t direction and subsequently turns toward increasing κ h n , reflecting the evolution of the local mode mixity as the crack advances.”

• AssumptionThe authors state this as an assumption: For the MMB simulations, the bulk material is linear elastic with Young's modulus E = 122 GPa and Poisson's ratio ν = 0.25.

The effective normal and tangential damage parameters are highly nonuniform over the first 35 mm ahead of the initial crack tip, reflecting the spatial variation in the loading histories accumulated along the interface.

“The effective normal and tangential damage parameters are highly nonuniform over the first 35 mm ahead of the initial crack tip, reflecting the spatial variation in the loading histories accumulated along the interface.”

• AssumptionThe authors state this as an assumption: For the MMB simulations, the bulk material is linear elastic with Young's modulus E = 122 GPa and Poisson's ratio ν = 0.25.

Once crack propagation resumes, the reloading curves soften and progressively merge with the original equilibrium path.

“Once crack propagation resumes, the reloading curves soften and progressively merge with the original equilibrium path.”

• AssumptionThe authors state this as an assumption: For the MMB simulations, the bulk material is linear elastic with Young's modulus E = 122 GPa and Poisson's ratio ν = 0.25.

As the fracture process zone approaches an element, normal opening develops rapidly, causing the local response to become increasingly dominated by mode I.

“As the fracture process zone approaches an element, normal opening develops rapidly, causing the local response to become increasingly dominated by mode I.”

• AssumptionThe authors state this as an assumption: For the MMB simulations, the bulk material is linear elastic with Young's modulus E = 122 GPa and Poisson's ratio ν = 0.25.

The relative normal and tangential healing rates control the direction of recovery in the (κh_n, κh_t) space.

“The relative normal and tangential healing rates control the direction of recovery in the (κ h n , κ h t ) space.”

• AssumptionThe authors state this as an assumption: For the MMB simulations, the bulk material is linear elastic with Young's modulus E = 122 GPa and Poisson's ratio ν = 0.25.

In fracture-healing-refracture tests on modified dogbone specimens made of asphalt mastic, the model accurately captured the recovery of peak load, quantified through the healing strength ratio H_P, across rest times spanning minutes to several days for both virgin and polymer-modified mastics.

“In fracture-healing-refracture tests on modified dogbone specimens made of asphalt mastic, the model accurately captured the recovery of peak load, quantified through the healing strength ratio H P , across rest times spanning minutes to several days for both virgin and polymer-modified mastics.”

With χt held fixed, increasing χn/χt shifts the recovery curves toward shorter rest times, reflecting the expected increase in the normal healing rate relative to the fixed tangential healing rate.

“With χ t held fixed, increasing χ n /χ t shifts the recovery curves toward shorter rest times, reflecting the expected increase in the normal healing rate relative to the fixed tangential healing rate.”

• AssumptionThe authors state this as an assumption: For the MMB simulations, the bulk material is linear elastic with Young's modulus E = 122 GPa and Poisson's ratio ν = 0.25.

The healing paths do not retrace the loading paths.

“As illustrated by the red trajectories in Figures 13a,b, the healing paths do not retrace the loading paths.”

• AssumptionThe authors state this as an assumption: For the MMB simulations, the bulk material is linear elastic with Young's modulus E = 122 GPa and Poisson's ratio ν = 0.25.

The model introduces two internal state variables representing the maximum effective crack opening displacements in the normal and tangential directions.

“Our model introduces two internal state variables representing the maximum effective crack opening displacements in the normal and tangential directions”

The cohesive-healing model provides a versatile and predictive framework for simulating fracture and healing in self-healing materials under general loading conditions.

“Together, these results demonstrate that our cohesive-healing model provides a versatile and predictive framework for simulating fracture and healing in self-healing materials under general loading conditions.”

The healing evolution equations for hn and ht are valid for general loading-unloading-reloading as well as during rest periods.

“which are valid for general loading-unloading-reloading as well as during rest periods.”

• AssumptionThe authors state this as an assumption: During rest periods, the healing variable h evolves according to a reaction-type equation ḣ = χ(1-h)^ξ [1-H(∆)].

Under pure loading, the cohesive tractions reduce to those of the original PPR model.

“In contrast, under pure loading, κ (i+1) n = ∆ (i+1) n and κ (i+1) t = ∆ (i+1) t , and thus the cohesive tractions given by Equation ( 14) reduce to those of the original PPR model (Park et al., 2009).”

• cites Park et al., 2009Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The model is thermodynamically consistent.

“We demonstrate that the model is thermodynamically consistent”

• Also stated in: § P r e p r i n t n o t p e e r r e v i e w e d · boosted

  “This discussion demonstrates that our proposed cohesive-healing model satisfies the Clausius-Duhem inequality, ensuring satisfaction of the second law of thermodynamics.”

• AssumptionThe authors state this as an assumption: The Helmholtz free energy can be split into a mechanical part and a chemical part, where the chemical free energy is related to the extent of the chemical reaction driving the self-healing process.

The mixed-mode extension allows the model to capture the coupled effects of normal and tangential crack opening and their influence on both damage and healing evolution.

“This extension allows the model to capture the coupled effects of normal and tangential crack opening and their influence on both damage and healing evolution.”

The approach enables the recovery of mode-I and mode-II cohesive properties.

“This approach enables the recovery of mode-I and mode-II cohesive properties”

During unloading, the damage parameters remain unchanged.

“During unloading, κ n = κ (i) n and κ (i+1) t = κ (i)(i+1) t , and thus the damage parameters remain unchanged.”

The history variables κn and κt must evolve irreversibly to ensure thermodynamic consistency.

“Given that κ n and κ t represent, respectively, the maximum normal and absolute tangential crack openings over the loading history, and the damage parameters are required to be irreversible to ensure thermodynamic consistency (Spring et al., 2016), the history variables themselves must also evolve irreversibly.”

• cites Spring et al., 2016 sourceNot checked — no full text of the source

The two internal state variables evolve according to healing kinetics activated only when crack faces are in contact and at rest.

“which evolve according to healing kinetics activated only when crack faces are in contact and at rest.”

The model is validated against experimental data from fracture-healing tests on self-healing asphalt mastics and polymethyl methacrylate (PMMA).

“We further validate the model against experimental data from fracture-healing tests on self-healing asphalt mastics and polymethyl methacrylate (PMMA).”

The Heaviside function in the healing evolution law ensures that healing occurs only when the crack faces are in contact.

“The Heaviside function ensures that healing occurs only when the crack faces are in contact (i.e., when interface wetting is possible).”

• AssumptionThe authors state this as an assumption: During rest periods, the healing variable h evolves according to a reaction-type equation ḣ = χ(1-h)^ξ [1-H(∆)].

The damage parameters consistent with the PPR model are functions of two internal history parameters, κn and κt, representing the maximum normal opening and absolute tangential opening in the history of loading.

“The damage parameters consistent with the PPR model are functions of two internal history parameters, κ n and κ t , representing the maximum normal opening and absolute tangential opening in the history of loading, respectively.”

• cites Spring et al., 2016 sourceNot checked — no full text of the source

The anomaly at 40°C likely stems from the limited number of experimental data points at 40°C, particularly within the intermediate rest time range of approximately 10 to 90 hours.

“This anomaly likely stems from the limited number of experimental data points at This preprint research paper has not been peer reviewed.Electronic copy available at: https://ssrn.com/abstract=7528848P r e p r i n t n o t p e e r r e v i e w e d 40 • C, particularly within the intermediate rest time range of approximately 10 to 90 hours.”

The model accurately tracks healing under complex, non-proportional mixed-mode paths.

“These results confirm that the model accurately tracks healing under complex, non-proportional mixed-mode paths.”

ξ_n has a dominant effect on the kinetics of recovery for fracture energy, cohesive strength, and stiffness.

“This delay is especially pronounced in Figures 6b-d, indicating that ξ n has a dominant effect on the kinetics of recovery for fracture energy, cohesive strength, and stiffness.”

The model captures the interplay between damage and healing mechanisms in mixed-mode settings with time-dependent behavior and mode coupling.

“These results demonstrate that the model captures the interplay between damage and healing mechanisms in mixed-mode settings with time-dependent behavior and mode coupling.”

The healing kinetics law can be recast in Onsager form, which is a sufficient and thermodynamically consistent form.

“We also note that the healing kinetics law shown in Equations ( 28)-( 29) can be recast in the following Onsager form, which is a sufficient and thermodynamically consistent form (Onsager, 1931a,b):”

• cites Onsager, 1931a,bNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• AssumptionThe authors state this as an assumption: The Helmholtz free energy can be split into a mechanical part and a chemical part, where the chemical free energy is related to the extent of the chemical reaction driving the self-healing process.

The sparsity of data in the intermediate rest time region limited the ability to properly calibrate the healing parameters to fit the early stages of healing at 40°C.

“The sparsity of data in this region limited our ability to properly calibrate the healing parameters to fit the early stages of healing at that temperature.”

The consistency between numerical and experimental healing strength ratios across a wide range of conditions validates the predictive capabilities of the model.

“The consistency between numerical and experimental healing strength ratios across a wide range of conditions validates the predictive capabilities of our model.”

• AssumptionThe authors state this as an assumption: The bulk material behaves elastically in the numerical models.

The proposed exact integration approach for h_n and h_t during rest periods enables the use of arbitrarily large time steps while preserving accuracy in the evolution of the healing variables.

“This integration scheme enables the use of arbitrarily large time steps during rest periods, offering a substantial gain in computational efficiency while preserving accuracy in the evolution of the healing variables.”

Smaller exponent values result in faster progression of healing, as seen in the PBmas response.

“In contrast, smaller exponent values result in faster progression of healing, as seen in the PBmas response.”

• AssumptionThe authors state this as an assumption: The bulk material behaves elastically in the numerical models.

The proposed exact integration approach offers a substantial gain in computational efficiency.

“This integration scheme enables the use of arbitrarily large time steps during rest periods, offering a substantial gain in computational efficiency while preserving accuracy in the evolution of the healing variables.”

The behavior after point D is expected because κ_h_n increases with time, leading to ḣn < 0 and ḋn > 0.

“This behavior is expected because, after point D, κ h n increases with time, leading to ḣn < 0 (as given by Equation ( 28)) and ḋn > 0 (which can be determined from Equation ( 25)).”

The SBS polymer modification may significantly reduce the ability of the material to heal under the tested conditions.

“These observations suggest that the SBS polymer modification, while beneficial for improving other mechanical properties, may significantly reduce the ability of the material to heal under the tested conditions.”

• AssumptionThe authors state this as an assumption: The bulk material behaves elastically in the numerical models.

The paper's approach to modeling self-healing materials is based on a heuristic rule for the evolution of healing variables that mimics more accurate polymer chain dynamics models at the computational expense of simplified cure kinetics models.

“Our approach to modeling self-healing materials is based on a heuristic rule for the evolution of the healing variables which mimics these more accurate polymer chain dynamics models, yet at the computational expense of simplified cure kinetics models.”

The model has capacity to represent healing behavior in applications where mixed-mode loading and partial contact are prevalent (e.g., delamination in composites).

“They also highlight its capacity to represent healing behavior in applications where mixed-mode loading and partial contact are prevalent (e.g., delamination in composites).”

The anomalously high healing viscosity parameter at 30°C can be attributed to the significantly large healing exponent (ξ_n = ξ_t) obtained at the same temperature, which effectively delays the onset of healing despite the relatively high healing viscosity value.

“This discrepancy can be attributed to the significantly large healing exponent (ξ n = ξ t ) obtained at the same temperature, which effectively delays the onset of healing despite the relatively high healing viscosity value.”

• AssumptionThe authors state this as an assumption: The bulk material behaves elastically in the numerical models.

The large healing exponent obtained at 30°C reflects the healing behavior observed at low temperatures, where molecular mobility is limited.

“The large healing exponent obtained at 30 • C reflects the healing behavior observed at low temperatures, where molecular mobility is limited.”

• AssumptionThe authors state this as an assumption: The bulk material behaves elastically in the numerical models.

Larger healing exponent values lead to a more gradual increase in recovery, consistent with trends in Figures 6.

“This is consistent with the trends observed in Figures 6, which show that larger healing exponent values lead to a more gradual increase in recovery.”

• AssumptionThe authors state this as an assumption: The bulk material behaves elastically in the numerical models.

The calibrated healing parameters effectively encode the interplay between temperature-dependent molecular mobility and interfacial healing kinetics, allowing the model to reproduce healing behavior across a broad spectrum of thermal conditions.

“These results confirm that the calibrated healing parameters effectively encode the the interplay between temperature-dependent molecular mobility and interfacial healing kinetics, allowing the model to reproduce healing behavior across a broad spectrum of thermal conditions.”

The same healing-rate ratio can produce different amounts of damage recovery depending on the normal and tangential histories accumulated during loading.

“Consequently, the same healing-rate ratio can produce different amounts of damage recovery depending on the normal and tangential histories accumulated during loading.”

The model was verified under pure mode I loading.

“Model verification under pure mode I loading: (a) Prescribed normal crack opening history, ∆n, with points A-D used to denote loading, unloading, rest, and reloading phases; (b) evolution of the normal healing variable, hn; (c) evolution of the effective normal damage parameter, dn; and (d) mode-I cohesive traction, Tn, as a function of the crack opening displacement.”

For the MMB configuration, cohesive softening and progressive crack propagation produce strongly nonlinear equilibrium paths, making robust path-following essential for capturing the complete post-peak behavior.

“For this configuration, cohesive softening and progressive crack propagation produce strongly nonlinear equilibrium paths, making robust path-following essential for capturing the complete post-peak behavior.”

The model was validated against a compact tension test on a specimen made of PMMA.

“Compact tension test on specimen made of PMMA: (a)-(b) Geometry, mesh, and boundary conditions; (c) numerical load-displacement responses for refracture tests after various rest times; and (d) numerical vs. experimental comparison of fracture toughness, KIi, recovery as a function of rest time.”

The model was validated against a mixed-mode bending (MMB) test.

“Mixed-mode bending (MMB) test and cohesive-healing response: (a) specimen geometry and rigid-lever loading arrangement; (b) finite element mesh; (c) deformed configuration showing cohesive elements A and B and the local mixed-mode separation at element B; (d) load-displacement responses for different rest periods obtained for χn/χt = 1, together with beam-theory and LEFM reference solutions; (e)”

Because this tangential degradation lies closer to the steep transition of the PPR damage surface, reductions in the effective histories during rest produce a comparatively rapid decrease in tangential damage.

“Because this tangential degradation lies closer to the steep transition of the PPR damage surface, reductions in the effective histories during rest produce a comparatively rapid decrease in tangential damage.”

The two-step calibration approach enabled efficient and physically interpretable parameter identification across a range of self-healing materials and loading conditions.

“This two-step approach enabled efficient and physically interpretable parameter identification across a range of self-healing materials and loading conditions.”

This dependence follows from the nonlinear PPR effective damage surfaces, which determine how reductions in κh_n and κh_t translate into damage recovery along each healing trajectory.

“This dependence follows from the nonlinear PPR effective damage surfaces, which determine how reductions in κ h n and κ h t translate into damage recovery along each healing trajectory.”

The model considers two internal state variables representing the maximum effective crack openings in the normal and tangential directions.

“The model considers two internal state variables representing the maximum effective crack openings in the normal and tangential directions, along with a phenomenological healing kinetics law that activates only during full crack-face contact.”

• AssumptionThe authors state this as an assumption: The healing kinetics law activates only during full crack-face contact.

The subsequent increase in load as the crack propagates beyond a = L reflects the change in the global structural response associated with the advanced crack configuration.

“The subsequent increase in load as the crack propagates beyond a = L reflects the change in the global structural response associated with the advanced crack configuration.”

The model was validated against tensile tests on modified dogbone specimens made of bituminous asphalt mastics.

“Tensile test on modified dogbone specimens made of bituminous asphalt mastics: (a)-(b) Geometry, mesh, and boundary conditions; (c) numerical vs. experimental load-displacement during initial fracture test (before healing); (d) load-displacement curves from refracture tests after various healing times; and (e) numerical vs. experimental comparison of healing strength ratio, HP , versus healing”

Healing represents constitutive recovery of the cohesive state rather than reverse propagation of the fracture process.

“Healing therefore represents constitutive recovery of the cohesive state rather than reverse propagation of the fracture process.”

A key attribute of the model is its ability to handle mixed-mode healing behavior through a unified, physically interpretable framework.

“A key attribute of our model lies in its ability to handle mixed-mode healing behavior through a unified, physically interpretable framework.”

The evolution from predominantly mode-II toward mixed-mode kinematics is important for healing because the resulting normal and tangential loading histories determine the state from which recovery begins.

“This evolution from predominantly mode-II toward mixed-mode kinematics is important for healing because the resulting normal and tangential loading histories determine the state from which recovery begins.”

Jud and Kausch (1979) report values of the mode-I K-calibration factor Y ranging between 11 and 16 for their CT specimens.

“Jud and Kausch (1979) report values of Y ranging between 11 and 16 for their CT specimens.”

• cites Jud and Kausch (1979) sourceNot checked — no full text of the source

Spring et al. (2016) showed that the normal and tangential damage parameters, dn and dt, defined in Equation (10), are monotonically increasing functions of κn and κt.

“Spring et al. (2016) showed that the normal and tangential damage parameters, dn and dt, defined in Equation (10), are monotonically increasing functions of κn and κt.”

• cites Spring et al. (2016) sourceNot checked — no full text of the source

The effect of the healing-rate ratio is state-dependent because it governs the evolution of the healing variables from a history state determined by the preceding loading path.

“The effect of the healing-rate ratio is therefore state-dependent because it governs the evolution of the healing variables from a history state determined by the preceding loading path.”

The model was verified under proportional mixed-mode loading.

“Model verification under proportional mixed-mode loading: (a) Prescribed normal (∆n) and tangential (∆t) separation history, with points A-D used to denote loading, unloading, rest, and reloading phases; (b) evolution of healing variables, hn and ht; (c) evolution of the effective damage parameters, dn and dt; and (d)-(e) cohesive tractions, Tn and Tt, as a function of their respective separations”

Enforcing the irreversibility of dn and dt requires that κn and κt be non-decreasing over time.

“Therefore enforcing the irreversibility of dn and dt simply requires that κn and κt be non-decreasing over time.”

The model was validated against a three-point bending test on a notched beam made of bituminous asphalt mastic.

“Three-point bending test on a notched beam made of bituminous asphalt mastic: (a)-(b) Geometry, mesh, and boundary conditions (dimensions in mm); (c) prescribed vertical displacement history at beam midspan; (d) numerical load-displacement responses for refracture tests after various rest times and experimental load-displacement curve for initial fracture test (before healing); (e) numerical vs.”

The healing kinetics law is governed by four parameters, namely the healing viscosity parameters (χn and χt) and nonlinear healing exponents (ξn and ξt), which together enable the model to seamlessly transition between damage and healing under arbitrary mixed-mode loading conditions.

“The healing kinetics law is governed by four parameters, namely the healing viscosity parameters (χ n and χ t ) and nonlinear healing exponents (ξ n and ξ t ), which together enable the model to seamlessly transition between damage and healing under arbitrary mixed-mode loading conditions.”

The model was validated against three sets of experimental data.

“To demonstrate the predictive capability of our model, we validated it against three sets of experimental data.”

The paper introduced a thermodynamically consistent mixed-mode cohesive-healing model that extends the Park-Paulino-Roesler (PPR) cohesive zone model to intrinsically self-healing materials.

“In this study, we introduced a thermodynamically consistent mixed-mode cohesive-healing model that extends the Park-Paulino-Roesler (PPR) cohesive zone model to intrinsically self-healing materials.”

Varying the healing-rate ratio changes not only the rate of recovery but also the path followed by the cohesive state through the mixed-mode effective-history space.

“Thus, varying the healing-rate ratio changes not only the rate of recovery but also the path followed by the cohesive state through the mixed-mode effective-history space.”

The cohesive-healing framework accurately captures both the kinetics of and mechanical aspects of crack healing in PMMA.

“Overall, these results demonstrate that our cohesive-healing framework accurately captures both the kinetics of and mechanical aspects of crack healing in PMMA.”

The material tangent matrix derived here is general and applies to arbitrary loading conditions, including loading, unloading, reloading, and rest periods.

“The material tangent matrix derived here is general and applies to arbitrary loading conditions, including loading, unloading, reloading, and rest periods.”