Modelling and simulation of coupled fluid transport and time-dependent fracture in fibre-reinforced hydrogel composites. (English) Zbl 1507.76010
Summary: Physicochemical and mechanical stimuli can trigger fluid transport and inelastic solid deformations in the fracture process of anisotropic hydrogel composites. The underlying mechanisms for the time-dependent deformation and fracture behaviour of hydrogel composites remain elusive and poorly understood. The present work develops an anisotropic poro-visco-hyperelastic-damage model based on the theory of porous media (TPM) to analyse the relationship between the coupled time-dependent behaviours, i.e.visco-hyperelasticity and fluid transport, and the fracture behaviour of hydrogel composites. The visco-hyperelasticity of the polymer networks resulting from the breaking and reforming kinetics of physical bonds is described by introducing internal variables based on the multiplicative decomposition of the deformation gradient tensor into elastic and inelastic parts. The fluid transport through the porous polymer networks is governed by Darcy’s law. A continuum damage model is proposed to describe the mechanical degradation of the anisotropic hydrogel composites, e.g., the cross-link unzipping and chain scission in polymer networks as well as the damage and breaking of embedded nanofibres. An integral-type nonlocal averaging algorithm is employed in the proposed damage model to eliminate the mesh dependence of numerical simulations. Furthermore, an operator split integration algorithm and an exponential mapping algorithm are applied to integrate evolution equations for internal variables and stresses. The resulting exponential tensor function is computed via spectral decompositions. The consistent tangent operators for an implicit finite element procedure are derived in detail in this contribution. Three representative numerical examples are used to validate the proposed model and investigate the mechanisms of the time-dependent fracture behaviour of anisotropic hydrogel composites. The computational results demonstrate that the proposed model can capture the fracture behaviour of hydrogel composites with different fibre orientations.
MSC:
| 76A10 | Viscoelastic fluids |
Keywords:
poro-visco-hyperelasticity; delayed fracture; nonlocal damage modelling; anisotropic hydrogel composites; crack propagationReferences:
| [1] | Kuo, C. K.; Ma, P. X., Ionically crosslinked alginate hydrogels as scaffolds for tissue engineering: Part 1. Structure, gelation rate and mechanical properties, Biomaterials, 22, 6, 511-521 (2001) |
| [2] | Griffin, D. R.; Archang, M. M.; Kuan, C.-h.; Weaver, W. M.; Weinstein, J. S.; Feng, A. C.; Ruccia, A.; Sideris, E.; Ragkousis, V.; Koh, J.; Plikus, M. V.; Di Carlo, D.; Segura, T.; Scumpia, P. O., Activating an adaptive immune response from a hydrogel scaffold imparts regenerative wound healing, Nature Mater. (2020) |
| [3] | Qiu, Y.; Park, K., Environment-sensitive hydrogels for drug delivery, Adv. Drug Deliv. Rev., 53, 3, 321-339 (2001) |
| [4] | Sun, Z.; Wang, L.; Jiang, X.; Bai, L.; Wang, W.; Chen, H.; Yang, L.; Yang, H.; Wei, D., Self-healing, sensitive and antifreezing biomass nanocomposite hydrogels based on hydroxypropyl guar gum and application in flexible sensors, Int. J. Biol. Macromol., 155, 1569-1577 (2020) |
| [5] | Guo, M.; Wu, Y.; Xue, S.; Xia, Y.; Yang, X.; Dzenis, Y.; Li, Z.; Lei, W.; Smith, A. T.; Sun, L., A highly stretchable, ultra-tough, remarkably tolerant, and robust self-healing glycerol-hydrogel for a dual-responsive soft actuator, J. Mater. Chem. A, 7, 45, 25969-25977 (2019) |
| [6] | Choe, A.; Yeom, J.; Shanker, R.; Kim, M. P.; Kang, S.; Ko, H., Stretchable and wearable colorimetric patches based on thermoresponsive plasmonic microgels embedded in a hydrogel film, NPG Asia Mater., 10, 9, 912-922 (2018) |
| [7] | Liu, T.; Zhu, C.; Wu, W.; Liao, K.-N.; Gong, X.; Sun, Q.; Li, R. K., Facilely prepared layer-by-layer graphene membrane-based pressure sensor with high sensitivity and stability for smart wearable devices, J. Mater. Sci. Technol., 45, 241-247 (2020) |
| [8] | Shiblee, M. N.I.; Ahmed, K.; Kawakami, M.; Furukawa, H., 4D printing of shape-memory hydrogels for soft-robotic functions, Adv. Mater. Technol., 4, 8, Article 1900071 pp. (2019) |
| [9] | Wehner, M.; Truby, R. L.; Fitzgerald, D. J.; Mosadegh, B.; Whitesides, G. M.; Lewis, J. A.; Wood, R. J., An integrated design and fabrication strategy for entirely soft, autonomous robots, Nature, 536, 7617, 451-455 (2016) |
| [10] | Yang, F.; Zhao, J.; Koshut, W. J.; Watt, J.; Riboh, J. C.; Gall, K.; Wiley, B. J., A synthetic hydrogel composite with the mechanical behavior and durability of cartilage, Adv. Funct. Mater., 30, 36, Article 2003451 pp. (2020) |
| [11] | Yang, J.; Shao, C.; Meng, L., Strain rate-dependent viscoelasticity and fracture mechanics of cellulose nanofibril composite hydrogels, Langmuir, 35, 32, 10542-10550 (2019) |
| [12] | Niu, J.; Wang, J.; Dai, X.; Shao, Z.; Huang, X., Dual physically crosslinked healable polyacrylamide/cellulose nanofibers nanocomposite hydrogels with excellent mechanical properties, Carbohydr. Polymers, 193, 73-81 (2018) |
| [13] | Murai, J.; Nakajima, T.; Matsuda, T.; Tsunoda, K.; Nonoyama, T.; Kurokawa, T.; Gong, J. P., Tough double network elastomers reinforced by the amorphous cellulose network, Polymer, 178, Article 121686 pp. (2019) |
| [14] | Du, W.; Zhang, J.; Zhao, Z.; Zhang, X., Preparation of novel temperature-responsive double-network hydrogel reinforced with aramid nanofibers, Compos. Commun., 22, Article 100438 pp. (2020) |
| [15] | Martin, N.; Youssef, G., Dynamic properties of hydrogels and fiber-reinforced hydrogels, J. Mech. Behav. Biomed. Mater., 85, 194-200 (2018) |
| [16] | Tonsomboon, K.; Butcher, A. L.; Oyen, M. L., Strong and tough nanofibrous hydrogel composites based on biomimetic principles, Mater. Sci. Eng. C, 72, 220-227 (2017) |
| [17] | Xiang, C.; Wang, Z.; Yang, C.; Yao, X.; Wang, Y.; Suo, Z., Stretchable and fatigue-resistant materials, Mater. Today, 34, 7-16 (2020) |
| [18] | Karobi, S. N.; Sun, T. L.; Kurokawa, T.; Luo, F.; Nakajima, T.; Nonoyama, T.; Gong, J. P., Creep behavior and delayed fracture of tough polyampholyte hydrogels by tensile test, Macromolecules, 49, 15, 5630-5636 (2016) |
| [19] | Bonn, D.; Kellay, H.; Prochnow, M.; Ben-Djemiaa, K.; Meunier, J., Delayed fracture of an inhomogeneous soft solid, Science, 280, 5361, 265-267 (1998) |
| [20] | Skrzeszewska, P. J.; Sprakel, J.; de Wolf, F. A.; Fokkink, R.; Cohen Stuart, M. A.; van der Gucht, J., Fracture and self-healing in a well-defined self-assembled polymer network, Macromolecules, 43, 7, 3542-3548 (2010) |
| [21] | Tang, J.; Li, J.; Vlassak, J. J.; Suo, Z., Fatigue fracture of hydrogels, Extreme Mech. Lett., 10, 24-31 (2017) |
| [22] | Pizzocolo, F.; Huyghe, J.; Ito, K., Mode I crack propagation in hydrogels is step wise, Eng. Fract. Mech., 97, 72-79 (2013) |
| [23] | Yu, Y.; Bouklas, N.; Landis, C. M.; Huang, R., Poroelastic effects on the time-and rate-dependent fracture of polymer gels, J. Appl. Mech., 87, 3 (2020) |
| [24] | Yu, Y.; Landis, C. M.; Huang, R., Steady-state crack growth in polymer gels: a linear poroelastic analysis, J. Mech. Phys. Solids, 118, 15-39 (2018) |
| [25] | Hong, W.; Zhao, X.; Zhou, J.; Suo, Z., A theory of coupled diffusion and large deformation in polymeric gels, J. Mech. Phys. Solids, 56, 5, 1779-1793 (2008) · Zbl 1162.74313 |
| [26] | Mao, Y.; Anand, L., A theory for fracture of polymeric gels, J. Mech. Phys. Solids, 115, 30-53 (2018) · Zbl 1441.74023 |
| [27] | Ding, J.; Remmers, J. J.; Leszczynski, S.; Huyghe, J. M., Swelling driven crack propagation in large deformation in ionized hydrogel, J. Appl. Mech., 85, 2 (2018) |
| [28] | Pan, Y.; Zhong, Z., A nonlinear constitutive model of unidirectional natural fiber reinforced composites considering moisture absorption, J. Mech. Phys. Solids, 69, 132-142 (2014) · Zbl 1328.74026 |
| [29] | Cheng, J.; Jia, Z.; Li, T., A constitutive model of microfiber reinforced anisotropic hydrogels: With applications to wood-based hydrogels, J. Mech. Phys. Solids, 138, Article 103893 pp. (2020) |
| [30] | Bosnjak, N.; Wang, S.; Han, D.; Lee, H.; Chester, S. A., Modeling of fiber-reinforced polymeric gels, Mech. Res. Commun., 96, 7-18 (2019) |
| [31] | Liu, S.; Stupp, S. I.; De La Cruz, M. O., Anisotropic contraction of fiber-reinforced hydrogels, Soft Matter, 14, 37, 7731-7739 (2018) |
| [32] | Flory, P. J.; Rehner, J., Statistical mechanics of cross-linked polymer networks I. Rubberlike elasticity, J. Chem. Phys., 11, 11, 512-520 (1943) |
| [33] | Long, R.; Hui, C.-Y., Fracture toughness of hydrogels: measurement and interpretation, Soft Matter, 12, 39, 8069-8086 (2016) |
| [34] | Mao, Y.; Lin, S.; Zhao, X.; Anand, L., A large deformation viscoelastic model for double-network hydrogels, J. Mech. Phys. Solids, 100, 103-130 (2017) |
| [35] | Long, R.; Mayumi, K.; Creton, C.; Narita, T.; Hui, C.-Y., Time dependent behavior of a dual cross-link self-healing gel: Theory and experiments, Macromolecules, 47, 20, 7243-7250 (2014) |
| [36] | Guo, J.; Long, R.; Mayumi, K.; Hui, C.-Y., Mechanics of a dual cross-link gel with dynamic bonds: steady state kinetics and large deformation effects, Macromolecules, 49, 9, 3497-3507 (2016) |
| [37] | Guo, J.; Liu, M.; Zehnder, A. T.; Zhao, J.; Narita, T.; Creton, C.; Hui, C. Y., Fracture mechanics of a self-healing hydrogel with covalent and physical crosslinks: A numerical study, J. Mech. Phys. Solids, 120, 79-95 (2018) |
| [38] | Shen, T.; Vernerey, F. J., Rate-dependent fracture in transient networks, J. Mech. Phys. Solids, Article 104028 pp. (2020) |
| [39] | Lin, J.; Zheng, S. Y.; Xiao, R.; Yin, J.; Wu, Z. L.; Zheng, Q.; Qian, J., Constitutive behaviors of tough physical hydrogels with dynamic metal-coordinated bonds, J. Mech. Phys. Solids, Article 103935 pp. (2020) |
| [40] | Chen, Y.; Lorentz, E.; Besson, J., Crack initiation and propagation in small-scale yielding using a nonlocal GTN model, Int. J. Plast., 130, Article 102701 pp. (2020) |
| [41] | Duddu, R.; Waisman, H., A nonlocal continuum damage mechanics approach to simulation of creep fracture in ice sheets, Comput. Mech., 51, 6, 961-974 (2013) · Zbl 1366.86005 |
| [42] | Yun, K.; Wang, Z.; Ronald, S.; Pak, Y., An advanced continuum damage mechanics model for predicting the crack progress process based on the consideration of the influence of crack direction under quasi-static load, Int. J. Mech. Sci., 130, 487-496 (2017) |
| [43] | Gao, Z.; Zhang, L.; Yu, W., A nonlocal continuum damage model for brittle fracture, Eng. Fract. Mech., 189, 481-500 (2018) |
| [44] | Li, X.; Gao, W.; Liu, W., A mesh objective continuum damage model for quasi-brittle crack modelling and finite element implementation, Int. J. Damage Mech., 28, 9, 1299-1322 (2019) |
| [45] | de Borst, R.; Verhoosel, C. V., Gradient damage vs phase-field approaches for fracture: Similarities and differences, Comput. Methods Appl. Mech. Engrg., 312, 78-94 (2016) · Zbl 1439.74347 |
| [46] | Steinke, C.; Zreid, I.; Kaliske, M., On the relation between phase-field crack approximation and gradient damage modelling., Comput. Mech., 59, 5 (2017) |
| [47] | Mandal, T. K.; Nguyen, V. P.; Heidarpour, A., Phase field and gradient enhanced damage models for quasi-brittle failure: A numerical comparative study, Eng. Fract. Mech., 207, 48-67 (2019) |
| [48] | Bui, T. Q.; Hu, X., A review of phase-field models, fundamentals and their applications to composite laminates, Eng. Fract. Mech., Article 107705 pp. (2021) |
| [49] | Ehlers, W., Foundations of multiphasic and porous materials, (Porous Media (2002), Springer), 3-86 · Zbl 1062.76050 |
| [50] | Markert, B., A biphasic continuum approach for viscoelastic high-porosity foams: comprehensive theory, numerics, and application, Arch. Comput. Methods Eng., 15, 4, 371-446 (2008) · Zbl 1170.74325 |
| [51] | De Boer, R., Trends in Continuum Mechanics of Porous Media, Vol. 18 (2006), Springer Science & Business Media |
| [52] | Ehlers, W.; Markert, B., A macroscopic finite strain model for cellular polymers, Int. J. Plast., 19, 7, 961-976 (2003) · Zbl 1090.74527 |
| [53] | Markert, B., A constitutive approach to 3-d nonlinear fluid flow through finite deformable porous continua, Transp. Porous Media, 70, 3, 427 (2007) |
| [54] | Ehlers, W.; Karajan, N.; Markert, B., An extended biphasic model for charged hydrated tissues with application to the intervertebral disc, Biomech. Model. Mechanobiol., 8, 3, 233 (2009) |
| [55] | Gültekin, O.; Dal, H.; Holzapfel, G. A., Numerical aspects of anisotropic failure in soft biological tissues favor energy-based criteria: A rate-dependent anisotropic crack phase-field model, Comput. Methods Appl. Mech. Engrg., 331, 23-52 (2018) · Zbl 1439.74199 |
| [56] | Denli, F. A.; Gültekin, O.; Holzapfel, G. A.; Dal, H., A phase-field model for fracture of unidirectional fiber-reinforced polymer matrix composites, Comput. Mech., 1-18 (2020) · Zbl 1462.74137 |
| [57] | Gültekin, O.; Hager, S. P.; Dal, H.; Holzapfel, G. A., Computational modeling of progressive damage and rupture in fibrous biological tissues: application to aortic dissection, Biomech. Model. Mechanobiol., 18, 6, 1607-1628 (2019) |
| [58] | Ma, S.; Scheider, I.; Bargmann, S., Anisotropic constitutive model incorporating multiple damage mechanisms for multiscale simulation of dental enamel, J. Mech. Behav. Biomed. Mater., 62, 515-533 (2016) |
| [59] | Ehlers, W.; Luo, C., A phase-field approach embedded in the theory of porous media for the description of dynamic hydraulic fracturing, Comput. Methods Appl. Mech. Engrg., 315, 348-368 (2017) · Zbl 1439.74107 |
| [60] | Ehlers, W.; Luo, C., A phase-field approach embedded in the theory of porous media for the description of dynamic hydraulic fracturing, Part II: the crack-opening indicator, Comput. Methods Appl. Mech. Engrg., 341, 429-442 (2018) · Zbl 1440.74123 |
| [61] | Ehlers, W.; Markert, B., A linear viscoelastic biphasic model for soft tissues based on the theory of porous media, J. Biomech. Eng., 123, 5, 418-424 (2001) |
| [62] | Ehlers, W.; Markert, B., On the viscoelastic behaviour of fluid-saturated porous materials, Granul. Matter, 2, 3, 153-161 (2000) |
| [63] | Reese, S.; Govindjee, S., A theory of finite viscoelasticity and numerical aspects, Int. J. Solids Struct., 35, 26-27, 3455-3482 (1998) · Zbl 0918.73028 |
| [64] | Liu, D.; Ma, S.; Stoffel, M.; Markert, B., A biphasic visco-hyperelastic damage model for articular cartilage: application to micromechanical modelling of the osteoarthritis-induced degradation behaviour, Biomech. Model. Mechanobiol., 19, 3, 1055-1077 (2019) |
| [65] | Andrade, F.; César de Sá, J.; Andrade Pires, F., A ductile damage nonlocal model of integral-type at finite strains: Formulation and numerical issues, Int. J. Damage Mech., 20, 4, 515-557 (2011) |
| [66] | Berger, L.; Bordas, R.; Kay, D.; Tavener, S., A stabilized finite element method for finite-strain three-field poroelasticity, Comput. Mech., 60, 1, 51-68 (2017) · Zbl 1386.74134 |
| [67] | Van der Voet, A., A comparison of finite element codes for the solution of biphasic poroelastic problems, Proc. Inst. Mech. Eng. Part H, 211, 2, 209 (1997) |
| [68] | Sun, J.-Y.; Zhao, X.; Illeperuma, W. R.; Chaudhuri, O.; Oh, K. H.; Mooney, D. J.; Vlassak, J. J.; Suo, Z., Highly stretchable and tough hydrogels, Nature, 489, 7414, 133-136 (2012) |
| [69] | Bluhm, J., Modelling of saturated thermo-elastic porous solids with different phase temperatures, (Porous Media (2002), Springer), 87-118 · Zbl 1064.74056 |
| [70] | Govindjee, S.; Simo, J. C., Mullins’ effect and the strain amplitude dependence of the storage modulus, Int. J. Solids Struct., 29, 14-15, 1737-1751 (1992) · Zbl 0764.73073 |
| [71] | Markert, B.; Ehlers, W.; Karajan, N., A general polyconvex strain-energy function for fiber-reinforced materials, Proc. Appl. Math. Mech., 5, 1, 245-246 (2005) · Zbl 1391.74051 |
| [72] | Simo, J. C., Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory, Comput. Methods Appl. Mech. Engrg., 99, 1, 61-112 (1992) · Zbl 0764.73089 |
| [73] | Waffenschmidt, T.; Polindara, C.; Menzel, A.; Blanco, S., A gradient-enhanced large-deformation continuum damage model for fibre-reinforced materials, Comput. Methods Appl. Mech. Engrg., 268, 801-842 (2014) · Zbl 1295.74007 |
| [74] | Tanaka, M.; Fujikawa, M.; Balzani, D.; Schröder, J., Robust numerical calculation of tangent moduli at finite strains based on complex-step derivative approximation and its application to localization analysis, Comput. Methods Appl. Mech. Engrg., 269, 454-470 (2014) · Zbl 1296.74129 |
| [75] | Ostwald, R.; Kuhl, E.; Menzel, A., On the implementation of finite deformation gradient-enhanced damage models, Comput. Mech., 64, 3, 847-877 (2019) · Zbl 1465.74142 |
| [76] | Liu, M.; Guo, J.; Hui, C. Y.; Zehnder, A., Crack tip stress based kinetic fracture model of a PVA dual-crosslink hydrogel, Extreme Mech. Lett., 29, Article 100457 pp. (2019) |
| [77] | Guo, J.; Hui, C.-Y.; Liu, M.; Zehnder, A. T., The stress field near the tip of a plane stress crack in a gel consisting of chemical and physical cross-links, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 475, 2227, Article 20180863 pp. (2019) · Zbl 1472.74188 |
| [78] | Kazimierska-Drobny, K.; El Fray, M.; Kaczmarek, M., Determination of mechanical and hydraulic properties of PVA hydrogels, Mater. Sci. Eng. C, 48, 48-54 (2015) |
| [79] | Cai, S.; Hu, Y.; Zhao, X.; Suo, Z., Poroelasticity of a covalently crosslinked alginate hydrogel under compression, J. Appl. Phys., 108, 11, Article 113514 pp. (2010) |
| [80] | Yu, Y.; Landis, C. M.; Huang, R., Steady-state crack growth in polymer gels: A linear poroelastic analysis, J. Mech. Phys. Solids, 118, 15-39 (2018) |
| [81] | Silva, P.; Crozier, S.; Veidt, M.; Pearcy, M., An experimental and finite element poroelastic creep response analysis of an intervertebral hydrogel disc model in axial compression, J. Mater. Sci. Mater. Med., 16, 7, 663-669 (2005) |
| [82] | Miri, A. K.; Hosseinabadi, H. G.; Cecen, B.; Hassan, S.; Zhang, Y. S., Permeability mapping of gelatin methacryloyl hydrogels, Acta Biomater., 77, 38-47 (2018) |
| [83] | Zhang, Y.; Ye, L., Improvement of permeability of poly(vinyl alcohol) hydrogel by using poly(ethylene glycol) as porogen, Polymer - Plastics Technol. Eng., 50, 8, 776-782 (2011) |
| [84] | Paradossi, G.; Cavalieri, F.; Capitani, D.; Crescenzi, V., Physicochemical characterization of chemical hydrogels based on PVA, J. Polym. Sci. B, 37, 12, 1225-1233 (1999) |
| [85] | Peppas, N., Crystallization of polyvinyl alcohol-water films by slow dehydration, Eur. Polym. J., 12, 8, 495-498 (1976) |
| [86] | Zhang, H.; Peng, H.; Li, Y.; Xu, Y.; Weng, W., Compositional- and time-dependent dissipation, recovery and fracture toughness in hydrophobically reinforced hybrid hydrogels, Polymer, 80, 130-137 (2015) |
| [87] | Shu, M.; Long, S.; Huang, Y.; Li, D.; Li, H.; Li, X., High strength and antibacterial polyelectrolyte complex CS/HS hydrogel films for wound healing, Soft Matter, 15, 38, 7686-7694 (2019) |
| [88] | Xu, H.; Xie, Y.; Zhu, E.; Liu, Y.; Shi, Z.; Xiong, C.; Yang, Q., Supertough and ultrasensitive flexible electronic skin based on nanocellulose/sulfonated carbon nanotube hydrogel films, J. Mater. Chem. A, 8, 13, 6311-6318 (2020) |
| [89] | Zheng, S. Y.; Tian, Y.; Zhang, X. N.; Du, M.; Song, Y.; Wu, Z. L.; Zheng, Q., Spin-coating-assisted fabrication of ultrathin physical hydrogel films with high toughness and fast response, Soft Matter, 14, 28, 5888-5897 (2018) |
| [90] | Smith, T. L.; Stedry, P. J., Time and temperature dependence of the ultimate properties of an SBR rubber at constant elongations, J. Appl. Phys., 31, 11, 1892-1898 (1960) |
| [91] | Wang, X.; Hong, W., Delayed fracture in gels, Soft Matter, 8, 31, 8171-8178 (2012) |
| [92] | Zheng, S.; Yu, H.; Yang, C.; Hong, W.; Zhu, F.; Qian, J.; Wu, Z.; Zheng, Q., Fracture of tough and stiff metallosupramolecular hydrogels, Mater. Today Phys., 13, Article 100202 pp. (2020) |
| [93] | Han, S.; Gemmell, S.; Helmer, K.; Grigg, P.; Wellen, J.; Hoffman, A.; Sotak, C., Changes in ADC caused by tensile loading of rabbit achilles tendon: evidence for water transport, J. Magn. Reson., 144, 2, 217-227 (2000) |
| [94] | Wellen, J.; Helmer, K.; Grigg, P.; Sotak, C., Application of porous-media theory to the investigation of water adc changes in rabbit achilles tendon caused by tensile loading, J. Magn. Reson., 170, 1, 49-55 (2004) |
| [95] | Federico, S.; Herzog, W., On the anisotropy and inhomogeneity of permeability in articular cartilage, Biomech. Model. Mechanobiol., 7, 5, 367-378 (2008) |
| [96] | Pierce, D. M.; Ricken, T.; Holzapfel, G. A., A hyperelastic biphasic fibre-reinforced model of articular cartilage considering distributed collagen fibre orientations: continuum basis, computational aspects and applications, Comput. Methods Biomech. Biomed. Eng., 16, 12, 1344-1361 (2013) |
| [97] | Simo, J. C.; Taylor, R. L., Quasi-incompressible finite elasticity in principal stretches. continuum basis and numerical algorithms, Comput. Methods Appl. Mech. Engrg., 85, 3, 273-310 (1991) · Zbl 0764.73104 |
| [98] | Itskov, M., A generalized orthotropic hyperelastic material model with application to incompressible shells, Internat. J. Numer. Methods Engrg., 50, 8, 1777-1799 (2001) · Zbl 0997.74006 |
| [99] | Holmes, D.; Loughran, J., Numerical aspects associated with the implementation of a finite strain, elasto-viscoelastic-viscoplastic constitutive theory in principal stretches, Internat. J. Numer. Methods Engrg., 83, 3, 366-402 (2010) · Zbl 1193.74149 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
