21. Mayumi K, Marcellan A, Ducouret G, Creton C, Narita T (2013) Stress-strain relationship of
highly stretchable dual cross-link gels: separability of strain and time effect. ACS Macro Lett
2:1065–1068
22. Mayumi K, Guo J, Narita T, Hui CY, Creton C (2016) Fracture of dual crosslink gels with
permanent and transient crosslinks. Extrem Mech Lett 6:52–59
23. Sun TL et al (2013) Physical hydrogels composed of polyampholytes demonstrate high
toughness and viscoelasticity. Nat Mater 12:932–937
24. Sun TL et al (2017) Bulk energy dissipation mechanism for the fracture of tough and selfhealing hydrogels. Macromolecules 50:2923–2931
25. Long R, Mayumi K, Creton C, Narita T, Hui CY (2014) Time dependent behavior of a dual
cross-link self-healing gel: theory and experiments. Macromolecules 47:7243–7250
26. Long R, Mayumi K, Creton C, Narita T, Hui C-Y (2015) Rheology of a dual crosslink selfhealing gel: theory and measurement using parallel-plate torsional rheometry. J Rheol
59:643–665
27. Grindy SC et al (2015) Control of hierarchical polymer mechanics with bioinspired metalcoordination dynamics. Nat Mater 14:1210–1216
28. Grindy SC, Lenz M, Holten-Andersen N (2016) Engineering elasticity and relaxation time in
metal-coordinate cross-linked hydrogels. Macromolecules 49:8306–8312
29. Tang S, Habicht A, Li S, Seiffert S, Olsen BD (2016) Self-diffusion of associating star-shaped
polymers. Macromolecules 49:5599–5608
30. Guo J, Long R, Mayumi K, Hui C-Y (2016) Mechanics of a dual cross-link gel with dynamic
bonds: steady state kinetics and large deformation effects. Macromolecules 49:3497–3507
31. Tanaka F, Edwards SF (1992) Viscoelastic properties of physically cross-linked networks.
Transient network theory. Macromolecules 25:1516–1523
32. Vernerey FJ, Long R, Brighenti R (2017) A statistically-based continuum theory for polymers
with transient networks. J Mech Phys Solids 107:1–20
33. Coleman BD, Gurtin ME (1967) Thermodynamics with internal state variables. J Chem Phys
47:597–613
34. Coleman BD, Noll W (1963) The thermodynamics of elastic materials with heat conduction and
viscosity. Arch Ration Mech Anal 13:167–178
35. Green MS, Tobolsky AV (1946) A new approach to the theory of relaxing polymeric media.
J Chem Phys 14:80–92
36. Vernerey FJ (2018) Transient response of nonlinear polymer networks: a kinetic theory. J Mech
Phys Solids 115:230–247
37. Guo J et al (2018) Fracture mechanics of a self-healing hydrogel with covalent and physical
crosslinks: a numerical study. J Mech Phys Solids 120:79–95
38. Guo J, Hui CY, Liu M, Zehnder AT (2019) The stress field near the tip of a plane stress crack in
a gel consisting of chemical and physical cross-links. Proc R Soc A Math Phys Eng Sci
475:20180863
39. Hui CY, Guo J, Liu M, Zehnder A (2019) Finite strain theory of a Mode III crack in a rate
dependent gel consisting of chemical and physical cross-links. Int J Fract 215:77–89
40. Shen T, Long R, Vernerey F (2019) Computational modeling of the large deformation and flow
of viscoelastic polymers. Comput Mech 63:725–745
41. Long KN, Dunn ML, Jerry Qi H (2010) Mechanics of soft active materials with phase
evolution. Int J Plast 26:603–616
42. Rajagopal KR, Wineman AS (1992) A constitutive equation for nonlinear solids which undergo
deformation induced microstructural changes. Int J Plast 8:385–395
43. Lavoie SR, Long R, Tang T (2016) A rate-dependent damage model for elastomers at large
strain. Extrem Mech Lett 8:114–124
44. Lavoie SR, Millereau P, Creton C, Long R, Tang T (2019) A continuum model for progressive
damage in tough multinetwork elastomers. J Mech Phys Solids 125:523–549
45. Vernerey FJ, Brighenti R, Long R, Shen T (2018) Statistical damage mechanics of polymer
networks. Macromolecules 51:6609–6622
46. Long R, Qi HJ, Dunn ML (2013) Modeling the mechanics of covalently adaptable polymer
networks with temperature-dependent bond exchange reactions. Soft Matter 9:4083
164
Q. Guo and R. Long
highly stretchable dual cross-link gels: separability of strain and time effect. ACS Macro Lett
2:1065–1068
22. Mayumi K, Guo J, Narita T, Hui CY, Creton C (2016) Fracture of dual crosslink gels with
permanent and transient crosslinks. Extrem Mech Lett 6:52–59
23. Sun TL et al (2013) Physical hydrogels composed of polyampholytes demonstrate high
toughness and viscoelasticity. Nat Mater 12:932–937
24. Sun TL et al (2017) Bulk energy dissipation mechanism for the fracture of tough and selfhealing hydrogels. Macromolecules 50:2923–2931
25. Long R, Mayumi K, Creton C, Narita T, Hui CY (2014) Time dependent behavior of a dual
cross-link self-healing gel: theory and experiments. Macromolecules 47:7243–7250
26. Long R, Mayumi K, Creton C, Narita T, Hui C-Y (2015) Rheology of a dual crosslink selfhealing gel: theory and measurement using parallel-plate torsional rheometry. J Rheol
59:643–665
27. Grindy SC et al (2015) Control of hierarchical polymer mechanics with bioinspired metalcoordination dynamics. Nat Mater 14:1210–1216
28. Grindy SC, Lenz M, Holten-Andersen N (2016) Engineering elasticity and relaxation time in
metal-coordinate cross-linked hydrogels. Macromolecules 49:8306–8312
29. Tang S, Habicht A, Li S, Seiffert S, Olsen BD (2016) Self-diffusion of associating star-shaped
polymers. Macromolecules 49:5599–5608
30. Guo J, Long R, Mayumi K, Hui C-Y (2016) Mechanics of a dual cross-link gel with dynamic
bonds: steady state kinetics and large deformation effects. Macromolecules 49:3497–3507
31. Tanaka F, Edwards SF (1992) Viscoelastic properties of physically cross-linked networks.
Transient network theory. Macromolecules 25:1516–1523
32. Vernerey FJ, Long R, Brighenti R (2017) A statistically-based continuum theory for polymers
with transient networks. J Mech Phys Solids 107:1–20
33. Coleman BD, Gurtin ME (1967) Thermodynamics with internal state variables. J Chem Phys
47:597–613
34. Coleman BD, Noll W (1963) The thermodynamics of elastic materials with heat conduction and
viscosity. Arch Ration Mech Anal 13:167–178
35. Green MS, Tobolsky AV (1946) A new approach to the theory of relaxing polymeric media.
J Chem Phys 14:80–92
36. Vernerey FJ (2018) Transient response of nonlinear polymer networks: a kinetic theory. J Mech
Phys Solids 115:230–247
37. Guo J et al (2018) Fracture mechanics of a self-healing hydrogel with covalent and physical
crosslinks: a numerical study. J Mech Phys Solids 120:79–95
38. Guo J, Hui CY, Liu M, Zehnder AT (2019) The stress field near the tip of a plane stress crack in
a gel consisting of chemical and physical cross-links. Proc R Soc A Math Phys Eng Sci
475:20180863
39. Hui CY, Guo J, Liu M, Zehnder A (2019) Finite strain theory of a Mode III crack in a rate
dependent gel consisting of chemical and physical cross-links. Int J Fract 215:77–89
40. Shen T, Long R, Vernerey F (2019) Computational modeling of the large deformation and flow
of viscoelastic polymers. Comput Mech 63:725–745
41. Long KN, Dunn ML, Jerry Qi H (2010) Mechanics of soft active materials with phase
evolution. Int J Plast 26:603–616
42. Rajagopal KR, Wineman AS (1992) A constitutive equation for nonlinear solids which undergo
deformation induced microstructural changes. Int J Plast 8:385–395
43. Lavoie SR, Long R, Tang T (2016) A rate-dependent damage model for elastomers at large
strain. Extrem Mech Lett 8:114–124
44. Lavoie SR, Millereau P, Creton C, Long R, Tang T (2019) A continuum model for progressive
damage in tough multinetwork elastomers. J Mech Phys Solids 125:523–549
45. Vernerey FJ, Brighenti R, Long R, Shen T (2018) Statistical damage mechanics of polymer
networks. Macromolecules 51:6609–6622
46. Long R, Qi HJ, Dunn ML (2013) Modeling the mechanics of covalently adaptable polymer
networks with temperature-dependent bond exchange reactions. Soft Matter 9:4083
164
Q. Guo and R. Long