Analysis of the Properties of a Nonlinear Model for Shear Flow of Thixotropic Media Taking into Account the Mutual Influence of Structural Evolution and Deformation
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Abstract
Abstract A systematic analytical study was conducted on the mathematical properties of a previously proposed prototype of a nonlinear Maxwell-type constitutive equation for describing the shear flow of thixotropic substances (viscous liquid polymers, viscoelastic melts, concentrated solutions, pastes, emulsions). The equation takes into account the mutual influence of deformation and structural evolution (the kinetics of intermolecular bond formation and breaking) on viscosity and shear modulus and the effect of deformation on this kinetics. In the uniaxial case, the constitutive equation is governed by a nondecreasing material function and six positive parameters. The equation is reduced to a set of two nonlinear autonomous differential equations for the stress and the crosslinking degree. It is proved that the equilibrium point of this set is unique. The dependences of the point coordinates on all material parameters and on the shear rate for an arbitrary nondecreasing material function are investigated in general form, and all the dependences are proved to be monotonic. Equations for the flow and viscosity curves are derived and investigated. It is proved that the model leads to an increasing shear rate dependence of the equilibrium stress and to a decreasing apparent viscosity curve, which reflect the typical properties of the experimental flow curves of pseudoplastic materials. Using six arbitrary governing material parameters and the governing material function, we analytically study the phase portrait of the nonlinear set of two differential equations, to which the model is reduced, for dimensionless stress and crosslinking degree near its only equilibrium point. It is proved that the equilibrium point is always stable and can be of three types only: a stable node, a degenerate node, or a stable focus. The existence criteria for each type are found in the form of explicit constraints on the material function, model parameters, and shear rate. A stable focus indicates the nonmonotonicity of the set solutions and the existence of deformation modes with damped oscillations of stress and crosslinking degree upon reaching steady-state values. The influence of the material parameters and material function on the type of equilibrium point and on the behavior of the model integral curves is analyzed.
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OpenAlex topics: Advanced Mathematical Modeling in Engineering · Geotechnical and Geomechanical Engineering · Elasticity and Wave Propagation
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References (60)
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1. Maxwell, J.С., On the Dynamical Theory of Gases, Philos. Trans. Roy. Soc. Lond., 1867, vol. CLVII, pp. 49–88.10.1098/rstl.1867.0004 ↗
2. Bingham, E.C., Fluidity and Plasticity, New York, 1922.
3. Oldroyd, J.G., Non-Newtonian Effects in Steady Motion of Some Idealised Elastico-Viscous Liquids, Proc. Roy. Soc. Lond. A, 1958, vol. 245, pp. 278–297.10.1098/rspa.1958.0083 ↗
4. Reiner, M., Rheology, Encyclopedia of Physics. Vol. 6, Berlin–Heidelberg: Springer, 1958, pp. 434–550.10.1007/978-3-642-45887-3_4 ↗
5. Rehbinder, P.A., Surface Phenomena in Dispersed Systems. Colloidal Chemistry. Selected Works, Moscow: Nauka, 1978.
6. Coleman, B.D., Makrovitz, A., Noll, W., Viscometric Flows of Non-Newtonian Fluids. Theory and Experiment, Berlin: Springer, 1966.10.1007/978-3-642-88655-3 ↗
7. Frenkel, Ya.I., The Kinetic Theory of Fluids, Leningrad: Nauka, 1975.
8. Vinogradov, G.V. and Malkin, A.Ya., Rheology of Polymers, Moscow: Khimiya, 1977.
9. Bibik, E.E., Rheology of Dispersed Systems, Leningrad: Izd-vo Leningr. Univ., 1981.
10. Bartenev, G.M. and Zelenev, Yu.V., Physics and Mechanics of Polymers, Moscow: Vyssh. Shkola, 1983.
11. Larson, R.G., Constitutive Equations for Polymer Melts and Solutions, Boston: Butterworth, 1988.10.1016/B978-0-409-90119-1.50012-9 ↗
12. Uriev, N.B., Physico-Chemical Foundations of the Technology of Dispersed Systems and Materials, Moscow, 1988.
13. Leonov, A.I. and Prokunin, A.N., Non-Linear Phenomena in Flows of Viscoelastic Polymer Fluids, London: Chapman and Hall, 1994.10.1007/978-94-011-1258-1 ↗
14. Macosko, C., Rheology: Principles, Measurements and Applications, New York: VCH, 1994.
15. Schramm, G., A Practical Approach to Rheology and Rheometry, Karlsruhe: Gebrueder Haake GmbH, 1994.
16. Rohn, C.L., Analytical Polymer Rheology, Munich: Hanser Publishers, 1995.
17. Huilgol, R.R. and Phan-Thien, N., Fluid Mechanics of Viscoelasticity, Amsterdam: Elsevier, 1997.
18. Larson, R.G., Structure and Rheology of Complex Fluids, New York: Oxford Press, 1999.
19. Gupta, R.K., Polymer and Composite Rheology, New York: Marcel Dekker, 2000.10.1201/9781482273700 ↗
20. Tanner, R.I., Engineering Rheology, Oxford: Oxford University Press, 2000.10.1093/oso/9780198564737.001.0001 ↗
21. Yamaguchi, H., Engineering Fluid Mechanics (Fluid Mechanics and Its Applications), New York: Springer, 2008.
22. Malkin, A.Y., Isayev, A.I., Rheology: Conceptions, Methods, Applications, Toronto: Chem. Tech. Publ., 2012.10.1016/B978-1-895198-49-2.50011-6 ↗
23. Pokrovskii, V.N., The Mesoscopic Theory of Polymer Dynamics, Springer, 2010.10.1007/978-90-481-2231-8 ↗
24. Garifullin, F.A., Macromolecules and Rheological Equations, Kazan: Izd-vo KGTU, 2008.
25. Altukhov, Yu.A., Gusev, A.S., Pyshnograi, G.V., and Koshelev, K.B., Introduction to the Mesoscopic Theory of Liquid Polymer Systems, Barnaul: AltGPA, 2012.
26. Stolin, A.M., Malkin, A.Ya., and Merzhanov, A.G., Non-Isothermal Processes and Methods of Investigation in the Chemistry and Mechanics of Polymers, Russ. Chem. Rev., 1979, vol. 48, no. 8, pp. 781–797.10.1070/RC1979v048n08ABEH002412 ↗
27. Prokunin, A.N., Nonlinear Maxwell-Type Constitutive Equations for Describing the Motion of Polymer Fluids, Prikl. Matem. Mekh., 1984, vol. 48, no. 6, pp. 957–965.
28. Leonov, A.I., Constitutive Equations for Viscoelastic Liquids: Formulation, Analysis and Comparison with Data, Rheology, 1999, vol. 8, pp. 519–575.
29. Stickel, J.J. and Powell, R.L., Fluid Mechanics and Rheology of Dense Suspensions, Annu. Rev. Fluid Mech., 2005, vol. 37, pp. 129–149.10.1146/annurev.fluid.36.050802.122132 ↗
30. Mueller, S., Llewellin, E.W., and Mader, H.M., The Rheology of Suspensions of Solid Particles, Proc. R. Soc. A, 2010, vol. 466, no. 2116, pp. 1201–1228.10.1098/rspa.2009.0445 ↗
31. Malkin, A.Ya. and Patlazhan, S.A., Wall Slip for Complex Liquids—Phenomenon and Its Causes, Adv. Colloid Interface Sci., 2018, vol. 257, pp. 42–57.10.1016/j.cis.2018.05.008 ↗
32. Stolin, A.M., Khudyaev, S.I., and Buchatsky, L.M., Theory of Viscosity Superanomaly of Structured Systems, Dokl. Acad. Nauk SSSR, 1978, vol. 243, no. 26, pp. 430–433.
33. Stolin, A.M. and Irzhak, V.I., Inhomogeneous Flow Regimes in Spinning, Vysokomolekul. Soedin. B, 1993, vol. 35, no. 7, pp. 902–904.
34. Belyaeva, N.A., Stolin, A.M., and Stelmakh, L.S., Modes of Solid-Phase Extrusion of Viscoelastic Structured Systems, Inzh. Fizika, 2009, no. 1, pp. 10–16.
35. Kuznetsova, Yu.L. and Skul’skiy, O.I., Effect of Different Flows on the Shear Banding of a Liquid with a Non-Monotonic Flow Curve, J. Appl. Math. Tech. Phys., 2019, vol. 60, no. 1, pp. 22–30. https://doi.org/10.1134/S002189441901004810.1134/S0021894419010048 ↗
36. Brady, J.F. and Morris, J.F., Microstructure of Strongly Sheared Suspensions and Its Impact on Rheology and Diffusion, J. Fluid Mech., 1997, vol. 348, pp. 103–139.10.1017/S0022112097006320 ↗
37. Tucker, C.L. and Moldenaers, P., Microstructural Evolution in Polymer Blends, Annu. Rev. Fluid Mech., 2002, vol. 34, pp. 177–210.10.1146/annurev.fluid.34.082301.144051 ↗
38. Malkin, A.Ya. and Kulichikhin, V.G., Structure and Rheology of Highly Concentrated Emulsions: A Modern Look, Russ. Chem. Rev., 2015, vol. 84, no. 8, pp. 803–825.10.1070/RCR4499 ↗
39. Padmanabhan, K.A., Vasin, R.A., and Enikeev, F.U., Superplastic Flow: Phenomenology and Mechanics, Berlin: Springer, 2001.10.1007/978-3-662-04367-7 ↗
40. Fraggedakis, D., Dimakopoulos, Y., and Tsamopoulos, J., Yielding the Yield Stress Analysis: A Thorough Comparison of Recently Proposed Elasto-Visco-Plastic (EVP) Fluid Models, J. Non-Newtonian Fluid Mech., 2016, vol. 236, pp. 104–122.10.1016/j.jnnfm.2016.09.001 ↗
41. Eglit, M.E., Yakubenko, A.E., and Zaiko, Yu.S., Mathematical Modeling of Slope Flows Taking into Account the Non-Newtonian Properties of the Fluid, Trudy Matem. Instituta Steklova, 2018, vol. 300, pp. 229–239.10.1134/S0081543818010194 ↗
42. Pyshnograi, G., Merzlikina, D., Filip, P., and Pivokonsky, R., Mesoscopic Single and Multi-Mode Rheological Models for Polymeric Melts Viscometric Flows Description, WSEAS Trans. Heat Mass Transfer, 2018, vol. 13, pp. 49–65.
43. Varchanis, S., Makrigiorgos, G., Moschopoulos, P., Dimakopoulos, Y., and Tsamopoulos, J., Modeling the Rheology of Thixotropic Elasto-Visco-Plastic Materials, J. Rheology, 2019, vol. 63, no. 4, pp. 609–639.10.1122/1.5049136 ↗
44. Khokhlov, А.V., Properties of a Nonlinear Viscoelastoplastic Model of Maxwell Type with Two Material Functions, Moscow Univ. Mech. Bull., 2016, vol. 71, no. 6, pp. 132–136. https://doi.org/10.3103/S002713301606002910.3103/S0027133016060029 ↗
45. Khokhlov, A.V., Long-Term Strength Curves Generated by the Nonlinear Maxwell-Type Model for Viscoelastoplastic Materials and the Linear Damage Rule under Step Loading, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2016, no. 3, pp. 524–543. http://dx.doi.org/10.14498/vsgtu151210.14498/vsgtu1512 ↗
46. Khokhlov, A.V., The Nonlinear Maxwell-Type Model for Viscoelastoplastic Materials: Simulation of Temperature Influence on Creep, Relaxation and Strain-Stress Curves, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2017, vol. 21, no. 1, pp. 160–179. https://doi.org/10.14498/vsgtu152410.14498/vsgtu1524 ↗
47. Khokhlov, A.V., A Nonlinear Maxwell-Type Model for Rheonomic Materials: Stability under Symmetric Cyclic Loadings, Moscow Univ. Mech. Bull., 2018, vol. 73, no. 2, pp. 39–42. https://doi.org/10.3103/S002713301802003610.3103/S0027133018020036 ↗
48. Khokhlov, A.V., Applicability Indicators and Identification Techniques for a Nonlinear Maxwell-Type Model Using Multi-Step Creep Curves, Vestn. MGTU Baumana. Ser. Estestv. Nauki, 2018, no. 6, pp. 92–112. https://doi.org/10.18698/1812-3368-2018-6-92-11210.18698/1812-3368-2018-6-92-112 ↗
49. Khokhlov, А.V., Applicability Indicators and Identification Techniques for a Nonlinear Maxwell-Type Elastoviscoplastic Model Using Loading-Unloading Curves, Mech. Compos. Mater., 2019, vol. 55, no. 2, pp. 195–210. https://doi.org/10.1007/s11029-019-09809-w10.1007/s11029-019-09809-w ↗
50. Khokhlov, A.V., Possibility to Describe the Alternating and Non-Monotonic Time Dependence of Poisson’s Ratio during Creep Using a Nonlinear Maxwell-Type Viscoelastoplasticity Model, Russ. Metallurg. (Metally), 2019, no. 10, pp. 956–963. https://doi.org/10.1134/S003602951910013610.1134/S0036029519100136 ↗
51. Khokhlov, A.V., Two-Sided Estimates for the Relaxation Function of the Linear Theory of Heredity Via the Relaxation Curves during the Ramp-Deformation and the Methodology of Identification, Mech. Solids, 2018, vol. 53, no. 3, pp. 307–328. https://doi.org/10.3103/S002565441807010510.3103/S0025654418070105 ↗
52. Khokhlov, A.V., Properties of the Set of Strain Diagrams Produced by Rabotnov Nonlinear Equation for Rheonomous Materials, Mech. Solids, 2019, vol. 54, no. 3, pp. 384–399. https://doi.org/10.3103/S002565441902002X10.3103/S002565441902002X ↗
53. Stolin, A.M. and Khokhlov, A.V., Nonlinear Model of Shear Flow of Thixotropic Viscoelastoplastic Continua Taking into Account the Evolution of the Structure and Its Analysis, Moscow Univ. Mech. Bull., 2022, vol. 77, no. 5, pp. 127–135. https://doi.org/10.3103/S002713302205006510.3103/S0027133022050065 ↗
54. Segal, V.M., Beyerlein, I.J., Tome, C.N., Chuvil’deev, V.N., and Kopylov, V.I., Fundamentals and Engineering of Severe Plastic Deformation, New York: Nova Science Pub., 2010.
55. Zhilayev, A.P. and Pshenichnyuk, A.I., Superplasticity and Grain Boundaries in Ultrafine-Grained Materials, Cambridge: Cambridge Int. Sci. Publ., 2010.10.1533/9780857093837 ↗
56. Valiev, R.Z., Zhilyaev A.P., and Langdon, T.G., Bulk Nanostructured Materials: Fundamentals and Applications, Hoboken, New Jersey: TMS-Wiley, 2014.10.1002/9781118742679 ↗
57. Ovid’ko, I.A., Valiev, R.Z., and Zhu, Y.T., Review on Superior Strength and Enhanced Ductility of Metallic Nanomaterials, Progr. Mater. Sci., 2018, vol. 94, pp. 462–540.10.1016/j.pmatsci.2018.02.002 ↗
58. Sharifullina, E.R., Shveykin, A.I., and Trusov, P.V., Review of Experimental Studies on Structural Superplasticity: Internal Structure Evolution of Material and Deformation Mechanisms, PNRPU Mech. Bull., 2018, no. 3, pp. 103–127.
59. Mikhaylovskaya, A.V., Kishchik, A.A., Kotov, A.D., Rofman, O.V., and Tabachkova, N.Y., Precipitation Behavior and High Strain Rate Superplasticity in a Novel Fine-Grained Aluminum Based Alloy, Mater. Sci. Eng. A, 2019, vol. 760, pp. 37–46.10.1016/j.msea.2019.05.099 ↗
60. Mochugovskiy, A.G., Mosleh, A.O., Kotov, A.D., Khokhlov, A.V., Kaplanskaya, L.Y., and Mikhaylovskaya, A.V., Microstructure Evolution, Constitutive Modelling, and Superplastic Forming of Experimental 6XXX-Type Alloys Processed with Different Thermomechanical Treatments, Materials, 2023, vol. 16, no. 1, pp. 445-1–18. https://doi.org/10.3390/ma1601044510.3390/ma16010445 ↗