DOI: https://doi.org/10.20998/2079-0775.2018.7.11

БАЗОВІ ПІДХОДИ ПРИ ДОСЛІДЖЕННІ РЕАКЦІЇ ВОЛОКОННИХ МАТЕРІАЛІВ НА ЗОВНІШНЄ НАВАНТАЖЕННЯ

Mykola Tkachuk

Анотація


У роботі наведено аналіз сучасних підходів, методів та моделей, що застосовуються при дослідженні  реакції волоконних матеріалів на зовнішнє навантаження. Ставиться задача визначити властивості композиційних матеріалів у складі об’єктів військової техніки як результат моделювання статистичних наборів мікрочастин у процесі взаємодії між собою. При цьому залучаються методи статистичного опису випадкових мікроструктур для урахування властивостей елементів, орієнтації та довжини волокон, амплітуди та кривизни нерівностей. Як напрямок подальших досліджень  пропонується розвиток варіаційних методів мінімуму повної усередненої енергії для визначення механічної поведінки мікроструктури та обчислення гомогенізованих властивостей матеріалів за скінченних деформацій

Ключові слова


випадкові мікроструктури; волоконні матеріали; статистична механіка; гомогенізація

Повний текст:

PDF

Посилання


Treloar L.R.G. The Physics of Rubber Elasticity. 3rd edition. Oxford: Clarendon Press, 1975. 322 р.

Lodish H. Molecular Cell Biology. Cambridge: W.H. Freeman & Company, 2000. – 973 р.

Boal D.H. Mechanics of the cell. Cambridge: Cambridge University Press, 2002. – 406 p.

Holzapfel G.A., Ogden R.W. Mechanics of biological tissue. New York: Springer Science & Business Media, 2006. – 135 p.

Schmoller K.M., Lieleg O., Bausch A.R. Internal stress in kinetically trapped actin bundle networks. Soft Matter, 2008, vol. 4(12), рр. 2365–2367.

Schmoller K. M., Fernandez P., Arevalo R. C., Blair D. L., Bausch A. R. Cyclic hardening in bundled actin networks. Nature communications, 2010, vol. 1, рр. 134.

Lang N. R., Münster S., Metzner C. [et al.] Estimating the 3D pore size distribution of biopolymer networks from directionally biased data. Biophysical journal, 2013, vol. 105(9), рр. 1967–1975.

Wen Q., Basu A., Winer J. P. [et. al.] Local and global deformations in a strain-stiffening fibrin gel. New Journal of Physics, 2007, vol. 9(11), рр. 428.

Basu A., Wen Q., Mao X. [et. al.] Nonaffine displacements in flexible polymer networks. Macromolecules, 2011, vol.44(6). рр. 1671–1679.

Stein A. M., Vader D. A., Jawerth L. M. [et. al.] An algorithm for extracting the network geometry of three-dimensional collagen gels. Journal of microscopy, 2008, vol. 232(3), рр. 463–475.

Ponti A., Machacek M., Gupton S.L. [et. al.] Two distinct actin networks drive the protrusion of migrating cells. Science, 2004, vol. 305(5691), рр. 1782–1786.

Hearle J.W.S., Thwaites J.J. Mechanics of flexible fibre assemblies. NATO advanced study institutes series. Applied sciences. Sijthoff & Noordhoff, 1980, рр. 293–310.

Picu R.C. Mechanics of random fiber networks-a review. Soft Matter, 2011. vol. 7, рр. 6768–6785.

Gibson L.J., Ashby M.F., Harley B.A. Cellular Materials in Nature and Medicine. Cambridge: Cambridge University Press, 2010. 309 p.

Pai C. L., Boyce M. C., Rutledge G. C. On the importance of fiber curvature to the elastic moduli of electrospun nonwoven fiber meshes. Polymer, 2011, vol. 52(26), рр. 6126–6133.

Yu B., Zhao X., Zeng Y., Qi D. The influence of process parameters on needle punched nonwovens investigated using image analysis. RSC Advances, 2017, vol.7(9), рр. 5183–5188.

Martínez-Hergueta F., Ridruejo A., González C., LLorca, J. Deformation and energy dissipation mechanisms of needle-punched nonwoven fabrics: A multiscale experimental analysis. International Journal of Solids and Structures, 2015, vol. 64, рр.120–131.

Hou X., Acar M., Silberschmidt V.V. 2D finite element analysis of thermally bonded nonwoven materials: Continuous and discontinuous models. Computational Materials Science, 2009, vol. 46(3), рр. 700–707.

Ridruejo A., González C., LLorca J. Micromechanisms of deformation and fracture of polypropylene nonwoven fabrics. International Journal of Solids and Structures, 2011, vol. 48(1), рр. 153–162.

Tyan Y.C., Liao J.D., Klauser R. [et. al.] Assessment and characterization of degradation effect for the varied degrees of ultra-violet radiation onto the collagen-bonded polypropylene non-woven fabric surfaces. Biomaterials, 2002, vol. 23(1), рр. 65–76.

Haverhals L.M., Reichert W.M., De Long H.C., Trulove P.C. Natural fiber welding. Macromolecular Materials and Engineering, 2010, vol. 295(5), рр. 425–430.

Li H., Zhu C., Xue J., Ke Q., Xia Y. Enhancing the Mechanical Properties of Electrospun Nanofiber Mats through Controllable Welding at the Cross Points. Macromolecular Rapid Communications, 2017, vol. 38(9).

Gandhi A., Asija N., Gaur K.K. [et. al.] Ultrasound assisted cyclic solid-state foaming for fabricating ultra-low density porous acrylonitrile–butadiene–styrene foams. Materials Letters, 2013, vol. 94, рр. 76–78.

Murr L.E., Gaytan S.M., Medina F. [et. al.] Next-generation biomedical implants using additive manufacturing of complex, cellular and functional mesh arrays. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 2010, vol. 368(1917), рр. 1999–2032.

Martínez-Hergueta F., Ridruej, A., Gonzále, C., Llorca J. Numerical simulation of the ballistic response of needle-punched nonwoven fabrics. International Journal of Solids and Structures, 2017, vol. 106, рр. 56–67.

Martínez-Hergueta F., Ridruejo A., González C., LLorca J. Ballistic performance of hybrid nonwoven/woven polyethylene fabric shields. International Journal of Impact Engineering, 2018, vol. 111, рр. 55–65.

Sun J.Y., Zhao X., Illeperuma W.R. [et. al.] Highly stretchable and tough hydrogels. Nature, 2012, vol. 489(7414), рр. 133–136.

Ethiraj G., Miehe C. Multiplicative magneto-elasticity of magnetosensitive polymers incorporating micromechanically-based network kernels. International Journal of Engineering Science, 2016, vol. 102, рр. 93–119.

Winkler R. Deformation of semiflexible chains. J. Chem. Phys, 2003, vol. 118, рр. 2919–2928.

Huisman E., Storm C., Barkema G. Monte Carlo study of multiply crosslinked semiflexible polymer networks. Phys. Rev. E, 2008, vol. 78, рр. 051801(11).

van Dillen T., Onck P.R., van der Giessen E. Models for stiffening in cross-linked biopolymer networks: A comparative study. J. Mech. Phys. Solids, 2008, vol. 56(6), рр. 2240–2264.

Onck P., Koeman T., T. van Dillen, E. van der Giessen Alternative explanation of stiffening in cross-linked semiflexible networks. Phys. Rev. Lett., 2005, vol. 95, рр. 178102(4).

Huisman E., Lubensky T. Internal stresses, normal modes, and nonaffinity in threedimensional biopolymer networks. Phys. Rev. Lett., 2011, vol. 106(8), рр. 088301(4).

Blundell J., Terentjev E. The influence of disorder on deformations in semiflexible networks. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, 2011, vol. 467, рр. 2330–2349.

Kuhn W., Grün F. Beziehungen zwischen elastischen Konstanten und Dehnungsdoppelbrechung hochelastischer Stoffe. Colloid. Polym. Sci., 1942, vol. 101(3), рр. 248–271.

Cox HL The elasticity and strength of paper and other fibrous materials. British journal of applied physics, 1952, vol. 3(3), рр. 72.

Storm C., Pastore F.C., MacKintosh T.C. [et. al.] Nonlinear elasticity in biological gels. Nature, 2005. vol. 435(7039), рр. 191–194.

Blundell J., Terentjev E. Affine model of stress stiffening in semiflexible filament networks. ArXiv, 2008, 0808.4088: 1–11.

Arruda E.M., Boyce M.C. A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. Journal of the Mechanics and Physics of Solids, 1993, vol. 41, рр. 389–412.

Kuhl E., Garikipati E.M., Arruda K. [et. al.] Remodeling of biological tissue: Mechanically induced reorientation of a transversely isotropic chain network. J. Mech. Phys. Solids, 2005, vol. 53(7), рр. 1552–1573.

Kuhl E., Menzel A., Garikipati K. On the convexity of transversely isotropic chain network models. Philos. Mag., 2006, vol. 86(21–22), рр. 3241–3258.

Palmer J.S., Boyce M.C. Constitutive modeling of the stressstrain behavior of F-actin filament networks. Acta Biomater, 2008, vol. 4(3), рр. 597–612.

Head D., Levine A., MacKintosh F. Deformation of cross-linked semiflexible polymer networks. Phys. Rev. Lett., 2003, vol. 91,108102(4).

Kroon M. A constitutive model for strain-crystallising Rubber-like materials. Mech. Mater., 2010, vol. 42(9), рр. 873–885.

Miehe C., Göktepe S., Lulei F. A micro-macro approach to rubber-like materials – Part I: the non-affine micro-sphere model of rubber elasticity. J. Mech. Phys. Solids, 2004, vol. 52, рр. 2617–2660.

Băzant Z.P., B.H. Oh Efficient numerical integration on the surface of a sphere. Z. Angew. Math. Mech., 1986, vol. 66, рр. 37–49.

Marckmann G., Verron E. Comparison of hyperelastic models for rubber-like materials. Rubber Chem. Technol., 2006, vol. 79, рр.835–858.

Treloar LRG Stress-strain data for vulcanised rubber under various types of deformation. Trans. Faraday Soc., 1944, vol. 40. рр.59–70.

Heussinger C., Schaefer B., Frey E. Nonaffine rubber elasticity for stiff polymer networks. Phys. Rev. E., 2007, vol. 76. – 031906(12).

James Hubert M. Guth Eugene Theory of the elastic properties of rubber. The Journal of Chemical Physics, 1943. vol. 11(10). рр.455–481.

Wang Ming Chen, Eugene Guth Statistical theory of networks of non-Gaussian flexible chains. The Journal of Chemical Physics, 1952, vol. 20(7). рр.1144–1157.

Ehret AE, Itskov M., Schmid H.Numerical integration on the sphere and its effect on the material symmetry of constitutive equations: а comparative study. International journal for numerical methods in engineering, 2010, vol. 81(2). рр. 189–206.

Verron Erwan Questioning numerical integration methods for microsphere (and microplane) constitutive equations. Mechanics of Materials, 2015, vol. 89. рр.216–228.

Deam R.T., Sam F. Edwards The theory of rubber elasticity. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 1976, vol. 280(1296). рр.317–353.

Heinrich G., Kaliske M. Theoretical and numerical formulation of a molecular based constitutive tube-model of rubber elasticity. Computational and Theoretical Polymer Science, 1997, vol. 7(3). рр.227–241.

Miehe C., Göktepe S. A micro–macro approach to rubber-like materials. Part II: the micro-sphere model of finite rubber viscoelasticity. Journal of the Mechanics and Physics of Solids, 2005, vol. 53(10), рр.2231–2258.

Göktepe S., Miehe C. A micro–macro approach to rubber-like materials. Part III: The micro-sphere model of anisotropic Mullins-type damage. Journal of the Mechanics and Physics of Solids, 2005, vol. 53(10), рр. 2259–2283.

Linder C., Tkachuk M., Miehe C. A micromechanically motivated diffusion-based transient network model and its incorporation into finite rubber viscoelasticity. Journal of the Mechanics and Physics of Solids, 2011, vol. 59(10), рр. 2134–2156.

Miehe C., Joel Méndez Diez, S. Göktepe, Lisa-Marie Schänzel Coupled thermoviscoplasticity of glassy polymers in the logarithmic strain space based on the free volume theory. International Journal of Solids and Structures, 2011, vol. 48(13), рр. 1799–1817

Miehe C., Göktepe S., Méndez Diez J. Finite viscoplasticity of amorphous glassy polymers in the logarithmic strain space. International Journal of Solids and Structures, 2009, vol. 46(1), рр.181–202.

Kuhl E., Menzel A., Garikipati K. On the convexity of transversely isotropic chain network models. Philosophical Magazine, 2006, vol. 86(21-22), рр.3241–3258.

Kuhl E., Garikipati K., Arruda E.M., Grosh K. Remodeling of biological tissue: mechanically induced reorientation of a transversely isotropic chain network. Journal of the Mechanics and Physics of Solids, 2005, vol. 53(7), рр.1552–1573.

Alastrué V., Martinez M.A., Doblaré M., Menzel A. Anisotropic microsphere-based finite elasticity applied to blood vessel modelling. Journal of the Mechanics and Physics of Solids, 2009, vol. 57(1), рр. 178–203

Menzel A. Waffenschmidt T. A microsphere-based remodelling formulation for anisotropic biological tissues. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 2009, vol. 367(1902), рр. 3499–3523.

Waffenschmidt T. Modelling and simulation of adaptation and degradation in anisotropic biological tissues. Ph.D. Thesis. – 2013.

Holzapfel G.A., Unterberger M. J., Ogden R. W. An affine continuum mechanical model for cross-linked F-actin networks with compliant linker proteins. Journal of the mechanical behavior of biomedical materials, 2014, vol. 38, рр.78–90.

Van Oosterwyck H. José F. Rodríguez, M. Doblaré, José M. García Aznar An affine micro-sphere-based constitutive model, accounting for junctional sliding, can capture F-actin network mechanics. Computer methods in biomechanics and biomedical engineering, 2013, vol. 16(9), рр.1002–1012.

Guilie Joachim, Thien-Nga Lê, Patrick Le Tallec Microsphere model for straininduced crystallization in rubber. In Proceedings of the 8th conference on constitutive models in rubbers. Taylor & Francis, 2013, рр. 467–472.

Guilie Joachim, Thien-Nga Le, Patrick Le Tallec Micro-sphere model for straininduced crystallisation and three-dimensional applications. Journal of the Mechanics and Physics of Solids, 2015, vol. 81, рр.58–74.

Le Tallec P. polymer modelling: from macroscopic hyperelasticity to strain induced crystallisation. 11th. World Congress on Computational Mechanics (WC CM XI) 5th. European Conference on Computational Mechanics (ECCM V) 6th. European Conference on Computational Fluid Dyn amics (ECFD VI) July 20–25, 2014, Barcelona, Spain

Mistry Sunny J., Sanjay Govindjee A micro-mechanically based continuum model for strain-induced crystallization in natural rubber. International Journal of Solids and Structures, 2014, vol. 51(2), рр.530–539.

Rastak R., Linder C. A non-affine micro-macro approach to strain-crystallizing rubber-like materials. Journal of the Mechanics and Physics of Solids, 2018. vol. 111. рр.67–99.

Thylander S., Menzel A., Ristinmaa M. A non-affine electroviscoelastic microsphere model for dielectric elastomers: Application to VHB 4910 based actuators. Journal of Intelligent Material Systems and Structures, 2016, рр. 1045389X16651157.

Rey T., Chagnon G., Favier D., Le J.-B.Cam Hyperelasticity with rate-independent microsphere hysteresis model for rubberlike materials. Computational Materials Science, 2014, vol. 90, рр.89–98.

Bergström J.S., Boyce M.C. Constitutive modeling of the large strain time-dependent behavior of elastomers. Journal of the Mechanics and Physics of Solids, 1998, vol. 46(5), рр.931–954.

Dal Hüsnü, Michael Kaliske Bergström–Boyce model for nonlinear finite rubber viscoelasticity: theoretical aspects and algorithmic treatment for the FE method. Computational Mechanics, 2009, vol.44(6), рр. 809–823.

Sáez P., Alastrué V., Peña E., Doblaré M., Martínez M.A. Anisotropic microsphere-based approach to damage in soft fibered tissue. Biomechanics and modeling in mechanobiology, 2012, vol. 11(5), рр. 595–608.

Tkachuk M., Linder Ch. The maximal advance path constraint for the homogenization of materials with random network microstructure. Philosophical Magazine, 2012, vol.92(22). рр. 2779–2808. Research Program: Computational modelling of adhesive contact 5


Пристатейна бібліографія ГОСТ


  1. Treloar L.R.G. The Physics of Rubber Elasticity. 3rd edition. Oxford: Clarendon Press, 1975.  322 р.
  2. Lodish H. Molecular Cell Biology. Cambridge: W.H. Freeman & Company, 2000. – 973 р.
  3. Boal D.H. Mechanics of the cell. Cambridge: Cambridge University Press, 2002. – 406 p.
  4. Holzapfel G.A., Ogden R.W. Mechanics of biological tissue. New York: Springer Science & Business Media,  2006. – 135 p.
  5. Schmoller K.M., Lieleg O., Bausch A.R. Internal stress in kinetically trapped actin bundle networks. Soft Matter, 2008, vol. 4(12), рр. 2365–2367.
  6. Schmoller K. M., Fernandez P., Arevalo R. C., Blair D. L., Bausch A. R. Cyclic hardening in bundled actin networks. Nature communications, 2010, vol. 1, рр. 134.
  7. Lang N. R., Münster S., Metzner C. [et al.] Estimating the 3D pore size distribution of biopolymer networks from directionally biased data. Biophysical journal, 2013, vol. 105(9), рр. 1967–1975.
  8. Wen Q., Basu A., Winer J. P. [et. al.] Local and global deformations in a strain-stiffening fibrin gel. New Journal of Physics, 2007, vol. 9(11), рр. 428.
  9. Basu A., Wen Q., Mao X. [et. al.] Nonaffine displacements in flexible polymer networks. Macromolecules, 2011, vol.44(6). рр.  1671–1679.
  10. Stein A. M., Vader D. A., Jawerth L. M. [et. al.] An algorithm for extracting the network geometry of three-dimensional collagen gels. Journal of microscopy, 2008, vol. 232(3), рр. 463–475.
  11. Ponti A., Machacek M., Gupton S.L. [et. al.] Two distinct actin networks drive the protrusion of migrating cells. Science, 2004, vol. 305(5691), рр. 1782–1786.
  12. Hearle J.W.S., Thwaites J.J.  Mechanics of flexible fibre assemblies. NATO advanced study institutes series. Applied sciences. Sijthoff & Noordhoff, 1980, рр.  293–310.
  13. Picu R.C. Mechanics of random fiber networks-a review. Soft Matter, 2011. vol. 7, рр. 6768–6785.
  14. Gibson L.J., Ashby M.F., Harley B.A. Cellular Materials in Nature and Medicine. Cambridge: Cambridge University Press, 2010. 309 p.
  15. Pai C. L., Boyce M. C., Rutledge G. C.  On the importance of fiber curvature to the elastic moduli of electrospun nonwoven fiber meshes. Polymer, 2011, vol. 52(26), рр. 6126–6133.
  16. Yu B., Zhao X., Zeng Y., Qi D. The influence of process parameters on needle punched nonwovens investigated using image analysis. RSC Advances, 2017, vol.7(9), рр. 5183–5188.
  17. Martínez-Hergueta F., Ridruejo A., González C., LLorca, J. Deformation and energy dissipation mechanisms of needle-punched nonwoven fabrics: A multiscale experimental analysis. International Journal of Solids and Structures, 2015, vol. 64, рр.120–131.
  18. Hou X., Acar M., Silberschmidt V.V. 2D finite element analysis of thermally bonded nonwoven materials: Continuous and discontinuous models. Computational Materials Science, 2009, vol.  46(3), рр. 700–707.
  19. Ridruejo A., González C., LLorca J. Micromechanisms of deformation and fracture of polypropylene nonwoven fabrics. International Journal of Solids and Structures, 2011, vol.  48(1), рр. 153–162.
  20. Tyan Y.C., Liao J.D., Klauser R. [et. al.] Assessment and characterization of degradation effect for the varied degrees of ultra-violet radiation onto the collagen-bonded polypropylene non-woven fabric surfaces. Biomaterials, 2002, vol.  23(1), рр. 65–76.
  21. Haverhals L.M., Reichert W.M., De Long H.C., Trulove P.C. Natural fiber welding. Macromolecular Materials and Engineering, 2010, vol. 295(5), рр. 425–430.
  22. Li H., Zhu C., Xue J., Ke Q., Xia Y. Enhancing the Mechanical Properties of Electrospun Nanofiber Mats through Controllable Welding at the Cross Points. Macromolecular Rapid Communications, 2017, vol. 38(9).
  23. Gandhi A., Asija N., Gaur K.K. [et. al.] Ultrasound assisted cyclic solid-state foaming for fabricating ultra-low density porous acrylonitrile–butadiene–styrene foams. Materials Letters, 2013, vol. 94, рр. 76–78.
  24. Murr L.E., Gaytan S.M., Medina F. [et. al.] Next-generation biomedical implants using additive manufacturing of complex, cellular and functional mesh arrays. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 2010, vol.  368(1917), рр. 1999–2032.
  25. Martínez-Hergueta F., Ridruej, A., Gonzále, C., Llorca J. Numerical simulation of the ballistic response of needle-punched nonwoven fabrics. International Journal of Solids and Structures, 2017, vol. 106, рр. 56–67.
  26. Martínez-Hergueta F., Ridruejo A., González C., LLorca J. Ballistic performance of hybrid nonwoven/woven polyethylene fabric shields. International Journal of Impact Engineering, 2018, vol. 111, рр. 55–65.
  27. Sun J.Y., Zhao X., Illeperuma W.R. [et. al.] Highly stretchable and tough hydrogels. Nature, 2012, vol.  489(7414), рр. 133–136.
  28. Ethiraj G., Miehe C. Multiplicative magneto-elasticity of magnetosensitive polymers incorporating micromechanically-based network kernels. International Journal of Engineering Science, 2016, vol. 102, рр. 93–119.
  29. Winkler R. Deformation of semiflexible chains. J. Chem. Phys, 2003, vol. 118, рр. 2919–2928.
  30. Huisman E., Storm C., Barkema G. Monte Carlo study of multiply crosslinked semiflexible polymer networks. Phys. Rev. E, 2008, vol. 78, рр. 051801(11).
  31. van Dillen T., Onck P.R., van der Giessen E. Models for stiffening in cross-linked biopolymer networks: A comparative study. J. Mech. Phys. Solids,  2008, vol. 56(6), рр. 2240–2264.
  32. Onck  P., Koeman T., T. van Dillen, E. van der Giessen Alternative explanation of stiffening in cross-linked semiflexible networks. Phys. Rev. Lett., 2005, vol.  95, рр. 178102(4).
  33. Huisman E., Lubensky T. Internal stresses, normal modes, and nonaffinity in threedimensional biopolymer networks. Phys. Rev. Lett., 2011, vol. 106(8), рр. 088301(4).
  34. Blundell J., Terentjev E. The influence of disorder on deformations in semiflexible networks. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, 2011, vol.  467, рр. 2330–2349.
  35. Kuhn W., Grün F. Beziehungen zwischen elastischen Konstanten und Dehnungsdoppelbrechung hochelastischer Stoffe. Colloid. Polym. Sci., 1942, vol.  101(3), рр. 248–271.
  36. Cox HL The elasticity and strength of paper and other fibrous materials. British journal of applied physics, 1952, vol. 3(3), рр.  72.
  37. Storm C., Pastore F.C., MacKintosh T.C. [et. al.] Nonlinear elasticity in biological gels. Nature, 2005. vol. 435(7039), рр. 191–194.
  38. Blundell J., Terentjev E. Affine model of stress stiffening in semiflexible filament networks. ArXiv, 2008, 0808.4088: 1–11.
  39. Arruda E.M., Boyce M.C. A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. Journal of the Mechanics and Physics of Solids, 1993, vol. 41, рр. 389–412.
  40. Kuhl E., Garikipati E.M., Arruda K. [et. al.] Remodeling of biological tissue: Mechanically induced reorientation of a transversely isotropic chain network. J. Mech. Phys. Solids, 2005, vol. 53(7), рр. 1552–1573.
  41. Kuhl E., Menzel A., Garikipati K. On the convexity of transversely isotropic chain network models. Philos. Mag., 2006, vol. 86(21–22), рр. 3241–3258.
  42. Palmer J.S., Boyce M.C. Constitutive modeling of the stressstrain behavior of F-actin filament networks. Acta Biomater, 2008, vol. 4(3), рр. 597–612.
  43. Head D., Levine A., MacKintosh F. Deformation of cross-linked semiflexible polymer networks. Phys. Rev. Lett., 2003, vol. 91,108102(4).
  44. Kroon M. A constitutive model for strain-crystallising Rubber-like materials. Mech. Mater., 2010, vol. 42(9), рр. 873–885.
  45. Miehe C., Göktepe S., Lulei F. A micro-macro approach to rubber-like materials – Part I: the non-affine micro-sphere model of rubber elasticity. J. Mech. Phys. Solids,  2004,  vol. 52, рр. 2617–2660.
  46. Băzant Z.P., B.H. Oh Efficient numerical integration on the surface of a sphere. Z. Angew. Math. Mech., 1986, vol. 66, рр. 37–49.
  47. Marckmann G., Verron E. Comparison of hyperelastic models for rubber-like materials. Rubber Chem. Technol., 2006, vol. 79, рр.835–858.
  48. Treloar LRG  Stress-strain data for vulcanised rubber under various types of deformation. Trans. Faraday Soc., 1944,  vol. 40. рр.59–70.
  49. Heussinger C., Schaefer B., Frey E. Nonaffine rubber elasticity for stiff polymer networks. Phys. Rev. E., 2007, vol. 76. – 031906(12).
  50. James Hubert M. Guth Eugene Theory of the elastic properties of rubber. The Journal of Chemical Physics, 1943. vol. 11(10). рр.455–481.
  51. Wang Ming Chen,  Eugene Guth Statistical theory of networks of non-Gaussian flexible chains. The Journal of Chemical Physics, 1952, vol. 20(7). рр.1144–1157.
  52. Ehret AE, Itskov M., Schmid H.Numerical integration on the sphere and its effect on the material symmetry of constitutive equations: а comparative study. International journal for numerical methods in engineering, 2010,  vol. 81(2). рр. 189–206.
  53. Verron Erwan  Questioning numerical integration methods for microsphere (and microplane) constitutive equations. Mechanics of Materials, 2015, vol. 89. рр.216–228.
  54. Deam R.T., Sam F. Edwards The theory of rubber elasticity. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 1976, vol. 280(1296). рр.317–353.
  55. Heinrich G., Kaliske M. Theoretical and numerical formulation of a molecular based constitutive tube-model of rubber elasticity. Computational and Theoretical Polymer Science, 1997, vol. 7(3). рр.227–241.
  56. Miehe C., Göktepe S. A micro–macro approach to rubber-like materials. Part II: the micro-sphere model of finite rubber viscoelasticity. Journal of the Mechanics and Physics of Solids, 2005, vol. 53(10), рр.2231–2258.
  57. Göktepe S., Miehe C. A micro–macro approach to rubber-like materials. Part III: The micro-sphere model of anisotropic Mullins-type damage. Journal of the Mechanics and Physics of Solids, 2005, vol. 53(10), рр. 2259–2283.
  58. Linder C., Tkachuk M., Miehe C. A micromechanically motivated diffusion-based transient network model and its incorporation into finite rubber viscoelasticity. Journal of the Mechanics and Physics of Solids, 2011, vol. 59(10), рр. 2134–2156.
  59. Miehe C., Joel Méndez Diez, S. Göktepe, Lisa-Marie Schänzel Coupled thermoviscoplasticity of glassy polymers in the logarithmic strain space based on the free volume theory. International Journal of Solids and Structures, 2011, vol. 48(13), рр. 1799–1817
  60. Miehe  C., Göktepe S., Méndez Diez J. Finite viscoplasticity of amorphous glassy polymers in the logarithmic strain space. International Journal of Solids and Structures, 2009, vol. 46(1), рр.181–202.
  61. Kuhl E., Menzel A., Garikipati K. On the convexity of transversely isotropic chain network models. Philosophical Magazine, 2006, vol. 86(21-22), рр.3241–3258.
  62. Kuhl E., Garikipati K., Arruda E.M., Grosh K. Remodeling of biological tissue: mechanically induced reorientation of a transversely isotropic chain network. Journal of the Mechanics and Physics of Solids, 2005, vol. 53(7), рр.1552–1573.
  63. Alastrué V., Martinez M.A., Doblaré M., Menzel A. Anisotropic microsphere-based finite elasticity applied to blood vessel modelling. Journal of the Mechanics and Physics of Solids, 2009, vol. 57(1), рр. 178–203
  64. Menzel A. Waffenschmidt T. A microsphere-based remodelling formulation for anisotropic biological tissues. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 2009, vol. 367(1902), рр. 3499–3523.
  65. Waffenschmidt T. Modelling and simulation of adaptation and degradation in anisotropic biological tissues. Ph.D. Thesis. – 2013.
  66. Holzapfel G.A., Unterberger M. J., Ogden R. W. An affine continuum mechanical model for cross-linked F-actin networks with compliant linker proteins.  Journal of the mechanical behavior of biomedical materials, 2014, vol. 38, рр.78–90.
  67. Van Oosterwyck H. José F. Rodríguez, M. Doblaré, José M. García Aznar An affine micro-sphere-based constitutive model, accounting for junctional sliding, can capture F-actin network mechanics. Computer methods in biomechanics and biomedical engineering, 2013, vol. 16(9), рр.1002–1012.
  68. Guilie Joachim, Thien-Nga Lê, Patrick Le Tallec Microsphere model for straininduced crystallization in rubber. In Proceedings of the 8th conference on constitutive models in rubbers. Taylor & Francis, 2013, рр. 467–472.
  69. Guilie Joachim, Thien-Nga Le, Patrick Le Tallec Micro-sphere model for straininduced crystallisation and three-dimensional applications. Journal of the Mechanics and Physics of Solids, 2015, vol. 81, рр.58–74.
  70. Le Tallec P. polymer modelling: from macroscopic hyperelasticity to strain induced crystallisation. 11th. World Congress on Computational Mechanics (WC CM XI)  5th. European Conference on Computational Mechanics (ECCM V) 6th. European Conference on Computational Fluid Dyn amics (ECFD VI) July 20–25, 2014, Barcelona, Spain
  71. Mistry Sunny J., Sanjay Govindjee A micro-mechanically based continuum model for strain-induced crystallization in natural rubber. International Journal of Solids and Structures, 2014, vol. 51(2), рр.530–539.
  72. Rastak R., Linder C. A non-affine micro-macro approach to strain-crystallizing rubber-like materials. Journal of the Mechanics and Physics of Solids, 2018. vol.  111. рр.67–99.
  73. Thylander S.,  Menzel A., Ristinmaa M. A non-affine electroviscoelastic microsphere model for dielectric elastomers: Application to VHB 4910 based actuators. Journal of Intelligent Material Systems and Structures, 2016, рр. 1045389X16651157.
  74. Rey T., Chagnon G., Favier D., Le J.-B.Cam Hyperelasticity with rate-independent microsphere hysteresis model for rubberlike materials. Computational Materials Science, 2014, vol.  90, рр.89–98.
  75. Bergström J.S., Boyce M.C. Constitutive modeling of the large strain time-dependent behavior of elastomers. Journal of the Mechanics and Physics of Solids, 1998, vol.  46(5), рр.931–954.
  76. Dal Hüsnü, Michael Kaliske Bergström–Boyce model for nonlinear finite rubber viscoelasticity: theoretical aspects and algorithmic treatment for the FE method. Computational Mechanics, 2009, vol.44(6), рр. 809–823.
  77. Sáez P., Alastrué V., Peña E., Doblaré M., Martínez M.A. Anisotropic microsphere-based approach to damage in soft fibered tissue. Biomechanics and modeling in mechanobiology, 2012, vol. 11(5), рр. 595–608.
  78. Tkachuk M., Linder  Ch.  The maximal advance path constraint for the homogenization of materials with random network microstructure. Philosophical Magazine, 2012, vol.92(22). рр. 2779–2808. Research Program: Computational modelling of adhesive contact 5

 





ISSN 2079-0775. Вісник Національного Технічного Університету «ХПІ».