Non-equilibrium thermodynamics modelling of the stress-strain relationship in soft two-phase elastic-viscoelastic materials
-
Pavlos S. Stephanou
,
Panayiotis Vafeas
und
Vlasis G. Mavrantzas
Abstract
In “soft–soft nanocomposites” based on film formation of latexes with structured particles, the combination of particle structure and interparticle crosslinking leads to materials that behave as nonlinear viscoelastic fluids at small strains and as highly elastic networks at larger strains. Similarly, in studies of flow-induced crystallization in polymers, a two-phase model is often invoked in which a soft viscoelastic component is coupled with a rigid semi-crystalline phase providing stiffness. In the present work, we use the framework of non-equilibrium thermodynamics (NET) to develop stress-strain relationships for such two-phase systems characterized by a viscoelastic and an elastic component by making use of two conformation tensors: the first describes the microstructure of the viscoelastic phase while the second is related to the elastic Finger strain tensor quantifying the deformation of the elastic phase due to strain and is responsible for strain-hardening. The final transport equations are formulated in the context of the generalized bracket formalism of NET and can describe the rheological behavior and mechanical response of a large variety of soft materials ranging from rubbers to artificial tissues.
-
Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
-
Research funding: No funding.
-
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
References
[1] I. Benedek, Pressure-Sensitive Adhesives and Applications, Boca Raton, CRC Press, 2004.10.1201/9780203021163Suche in Google Scholar
[2] A. Lindner, B. Lestriez, S. Mariot, et al.., “Adhesive and rheological properties of lightly crosslinked model acrylic networks,” J. Adhes., vol. 82, pp. 267–310, 2006. https://doi.org/10.1080/00218460600646594.Suche in Google Scholar
[3] M. D. Amaral, A. Roos, J. M. Asua, and C. Creton, “Assessing the effect of latex particle size and distribution on the rheological and adhesive properties of model waterborne acrylic pressure-sensitive adhesives films,” J. Colloid Interface Sci., vol. 281, pp. 325–338, 2005. https://doi.org/10.1016/j.jcis.2004.08.142.Suche in Google Scholar PubMed
[4] G. Raos and B. Zappone, “Polymer adhesion: seeking new solutions for an old problem,” Macromolecules, vol. 54, pp. 10617–10644, 2021. https://doi.org/10.1021/acs.macromol.1c01182.Suche in Google Scholar
[5] R. Messler, Joining of Materials and Structures: From Pragmatic Process to Enabling Technology, Butterworth-Heinemann, 2004. https://www.elsevier.com/books/joining-of-materials-and-structures/messler/978-0-7506-7757-8.10.1016/B978-075067757-8/50014-2Suche in Google Scholar
[6] F. Deplace, M. A. Rabjohns, T. Yamaguchi, et al.., “Deformation and adhesion of a periodic soft-soft nanocomposite designed with structured polymer colloid particles,” Soft Matter, vol. 5, pp. 1440–1447, 2009. https://doi.org/10.1039/b815292f.Suche in Google Scholar
[7] J. A. Kulkarni and A. N. Beris, “A model for the necking phenomenon in high-speed fiber spinning based on flow-induced crystallization,” J. Rheol., vol. 42, pp. 971–994, 1998. https://doi.org/10.1122/1.550913.Suche in Google Scholar
[8] A. I. Leonov, “Nonequilibrium thermodynamics and rheology of viscoelastic polymer media,” Rheol. Acta, vol. 15, pp. 85–98, 1976. https://doi.org/10.1007/bf01517499.Suche in Google Scholar
[9] A. K. Doufas, A. J. McHugh, and C. Miller, “Simulation of melt spinning including flow-induced crystallization part I. Model development and predictions,” J. Non Newtonian Fluid Mech., vol. 92, pp. 27–66, 2000. https://doi.org/10.1016/s0377-0257(00)00088-4.Suche in Google Scholar
[10] A. N. Beris and B. J. Edwards, Thermodynamics of Flowing Systems: With Internal Microstructure, New York, Oxford University Press, 1994.10.1093/oso/9780195076943.001.0001Suche in Google Scholar
[11] H. C. Öttinger, Beyond Equilibrium Thermodynamics, Hoboken, New Jersey, John Wiley and Sons, 2005.10.1002/0471727903Suche in Google Scholar
[12] H. C. Öttinger and M. Grmela, “Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism,” Phys. Rev. E: Stat. Phys. Plasmas, Fluids, Relat. Interdiscip. Top., vol. 56, pp. 6633–6655, 1997. https://doi.org/10.1103/physreve.56.6633.Suche in Google Scholar
[13] M. Grmela and H. C. Öttinger, “Dynamics and thermodynamics of complex fluids. I. Development of a general formalism,” Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., vol. 56, pp. 6620–6632, 1997. https://doi.org/10.1103/physreve.56.6620.Suche in Google Scholar
[14] B. J. Edwards, M. Dressler, M. Grmela, and A. Ait-Kadi, “Rheological models with microstructural constraints,” Rheol. Acta, vol. 42, pp. 64–72, 2003. https://doi.org/10.1007/s00397-002-0256-9.Suche in Google Scholar
[15] B. J. Edwards and M. Dressler, “A rheological model with constant approximate volume for immiscible blends of ellipsoidal droplets,” Rheol. Acta, vol. 42, pp. 326–337, 2003. https://doi.org/10.1007/s00397-002-0288-1.Suche in Google Scholar
[16] M. Dressler and B. J. Edwards, “The influence of matrix viscoelasticity on the rheology of polymer blends,” Rheol. Acta, vol. 43, pp. 257–282, 2004. https://doi.org/10.1007/s00397-003-0341-8.Suche in Google Scholar
[17] M. Dressler, B. J. Edwards, and E. J. Windhab, “An examination of droplet deformation and break-up between concentrically rotating cylinders,” J. Non Newtonian Fluid Mech., vol. 152, pp. 86–100, 2008. https://doi.org/10.1016/j.jnnfm.2007.10.022.Suche in Google Scholar
[18] M. Grmela, A. Ammar, F. Chinesta, and G. Maîtrejean, “A mesoscopic rheological model of moderately concentrated colloids,” J. Non Newtonian Fluid Mech., vol. 212, pp. 1–12, 2014. https://doi.org/10.1016/j.jnnfm.2014.07.005.Suche in Google Scholar
[19] P. M. Mwasame, N. J. Wagner, and A. N. Beris, “On the macroscopic modelling of dilute emulsions under flow,” J. Fluid Mech., vol. 831, pp. 433–473, 2017. https://doi.org/10.1017/jfm.2017.578.Suche in Google Scholar
[20] P. S. Stephanou, I. C. Tsimouri, and V. G. Mavrantzas, “Flow-induced orientation and stretching of entangled polymers in the framework of nonequilibrium thermodynamics,” Macromolecules, vol. 49, pp. 3161–3173, 2016. https://doi.org/10.1021/acs.macromol.5b02805.Suche in Google Scholar
[21] P. S. Stephanou, C. Baig, and V. G. Mavrantzas, “A generalized differential constitutive equation for polymer melts based on principles of nonequilibrium thermodynamics,” J. Rheol., vol. 53, pp. 309–337, 2009. https://doi.org/10.1122/1.3059429.Suche in Google Scholar
[22] P. S. Stephanou, I. C. Tsimouri, and V. G. Mavrantzas, “Simple, accurate and user-friendly differential constitutive model for the rheology of entangled polymer melts and solutions from nonequilibrium thermodynamics,” Materials, vol. 13, p. 2867, 2020. https://doi.org/10.3390/ma13122867.Suche in Google Scholar PubMed PubMed Central
[23] M. Rajabian, C. Dubois, and M. Grmela, “Suspensions of semiflexible fibers in polymeric fluids: rheology and thermodynamics,” Rheol. Acta, vol. 44, pp. 521–535, 2005. https://doi.org/10.1007/s00397-005-0434-7.Suche in Google Scholar
[24] H. Eslami, M. Grmela, and M. Bousmina, “Linear and nonlinear rheology of polymer/layered silicate nanocomposites,” J. Rheol., vol. 54, pp. 539–562, 2010. https://doi.org/10.1122/1.3372720.Suche in Google Scholar
[25] P. S. Stephanou, V. G. Mavrantzas, and G. C. Georgiou, “Continuum model for the phase behavior, microstructure, and rheology of unentangled polymer nanocomposite melts,” Macromolecules, vol. 47, pp. 4493–4513, 2014. https://doi.org/10.1021/ma500415w.Suche in Google Scholar
[26] P. S. Stephanou, “How the flow affects the phase behaviour and microstructure of polymer nanocomposites,” J. Chem. Phys., vol. 142, p. 064901, 2015. https://doi.org/10.1063/1.4907363.Suche in Google Scholar PubMed
[27] I. C. Tsimouri, P. S. Stephanou, and V. G. Mavrantzas, “A constitutive rheological model for agglomerating blood derived from nonequilibrium thermodynamics,” Phys. Fluids, vol. 30, p. 030710, 2018. https://doi.org/10.1063/1.5016913.Suche in Google Scholar
[28] P. S. Stephanou, “A constitutive hemorheological model addressing both the deformability and aggregation of red blood cells,” Phys. Fluids, vol. 32, p. 103103, 2020. https://doi.org/10.1063/5.0022493.Suche in Google Scholar
[29] P. S. Stephanou and I. C. Tsimouri, “A constitutive hemorheological model addressing the deformability of red blood cells in Ringer solutions,” Soft Matter, vol. 16, pp. 7585–7597, 2020. https://doi.org/10.1039/d0sm00974a.Suche in Google Scholar PubMed
[30] P. S. Stephanou, “Erratum: A constitutive hemorheological model addressing both the deformability and aggregation of red blood cells,” Phys. Fluids, vol. 33, p. 039901, 2021.10.1063/5.0043721Suche in Google Scholar
[31] P. S. Stephanou, “On the consistent modeling of shear-thickening polymer solutions,” Phys. Fluids, vol. 33, p. 063107, 2021. https://doi.org/10.1063/5.0053604.Suche in Google Scholar
[32] L. Treloar, The physics of Rubber Elasticity, 3rd ed. Oxford, Clarendon Press, 2005.Suche in Google Scholar
[33] R. B. Bird, R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids. Volume 1. Fluid Mechanics, 2nd ed. Hoboken, New Jersey, Wiley-Interscience, 1987.Suche in Google Scholar
[34] H. Giesekus, “A simple constitutive equation for polymer fluids based on the concept of deformation-dependent tensorial mobility,” J. Non Newtonian Fluid Mech., vol. 11, pp. 69–109, 1982. https://doi.org/10.1016/0377-0257(82)85016-7.Suche in Google Scholar
[35] R. G. Larson, Constitutive Equations for Polymer Melts and Solutions, 1st ed. Oxford, UK, Butterworth-Heinemann, 1988.10.1016/B978-0-409-90119-1.50006-3Suche in Google Scholar
[36] N. P. Thien and R. I. Tanner, “A new constitutive equation derived from network theory,” J. Non Newtonian Fluid Mech., vol. 2, pp. 353–365, 1977. https://doi.org/10.1016/0377-0257(77)80021-9.Suche in Google Scholar
[37] N. Phan-Thien, “A nonlinear network viscoelastic model,” J. Rheol., vol. 22, pp. 259–283, 1978. https://doi.org/10.1122/1.549481.Suche in Google Scholar
[38] A. Souvaliotis and A. N. Beris, “An extended White–Metzner viscoelastic fluid model based on an internal structural parameter,” J. Rheol., vol. 36, pp. 241–271, 1992. https://doi.org/10.1122/1.550344.Suche in Google Scholar
[39] H. Alexander, “A constitutive relation for rubber-like materials,” Int. J. Eng. Sci., vol. 6, pp. 549–563, 1968. https://doi.org/10.1016/0020-7225(68)90006-2.Suche in Google Scholar
[40] K. D. Housiadas and A. N. Beris, “Extensional behavior influence on viscoelastic turbulent channel flow,” J. Non Newtonian Fluid Mech., vol. 140, pp. 41–56, 2006. https://doi.org/10.1016/j.jnnfm.2006.03.017.Suche in Google Scholar
[41] G. Marckmann and E. Verron, “Comparison of hyperelastic models for rubber-like materials,” Rubber Chem. Technol., vol. 79, pp. 835–858, 2006. https://doi.org/10.5254/1.3547969.Suche in Google Scholar
[42] A. N. Gent, “Elastic instabilities in rubber,” Int. J. Non Lin. Mech., vol. 40, pp. 165–175, 2005. https://doi.org/10.1016/j.ijnonlinmec.2004.05.006.Suche in Google Scholar
[43] A. N. Gent and A. G. Thomas, “Forms for the stored (strain) energy function for vulcanized rubber,” J. Polym. Sci., vol. 28, pp. 625–628, 1958. https://doi.org/10.1002/pol.1958.1202811814.Suche in Google Scholar
[44] A. N. Gent, “A new constitutive relation for rubber,” Rubber Chem. Technol., vol. 69, pp. 59–61, 1996. https://doi.org/10.5254/1.3538357.Suche in Google Scholar
[45] A. N. Gent, “Elastic instabilities of inflated rubber shells,” Rubber Chem. Technol., vol. 72, pp. 263–268, 1999. https://doi.org/10.5254/1.3538799.Suche in Google Scholar
[46] T MathWorks, MATLAB (R2020b), The MathWorks Inc., 2020.Suche in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Research Articles
- Influence of drive chamber discharging process on non-linear displacer dynamics and thermodynamic processes of a fluidic-driven Gifford-McMahon cryocooler
- Stability Analysis of Double Diffusive Convection in Local Thermal Non-equilibrium Porous Medium with Internal Heat Source and Reaction Effects
- Maximum work configuration of finite potential source endoreversible non-isothermal chemical engines
- Densities and isothermal compressibilities from perturbed hard-dimer-chain equation of state: application to nanofluids
- Jeffery-Hamel flow extension and thermal analysis of Oldroyd-B nanofluid in expanding channel
- Non-equilibrium thermodynamics modelling of the stress-strain relationship in soft two-phase elastic-viscoelastic materials
- Minimum power consumption of multistage irreversible Carnot heat pumps with heat transfer law of q ∝ (ΔT) m
Artikel in diesem Heft
- Frontmatter
- Research Articles
- Influence of drive chamber discharging process on non-linear displacer dynamics and thermodynamic processes of a fluidic-driven Gifford-McMahon cryocooler
- Stability Analysis of Double Diffusive Convection in Local Thermal Non-equilibrium Porous Medium with Internal Heat Source and Reaction Effects
- Maximum work configuration of finite potential source endoreversible non-isothermal chemical engines
- Densities and isothermal compressibilities from perturbed hard-dimer-chain equation of state: application to nanofluids
- Jeffery-Hamel flow extension and thermal analysis of Oldroyd-B nanofluid in expanding channel
- Non-equilibrium thermodynamics modelling of the stress-strain relationship in soft two-phase elastic-viscoelastic materials
- Minimum power consumption of multistage irreversible Carnot heat pumps with heat transfer law of q ∝ (ΔT) m
