Skip to main content
SpringerLink
Log in

Families of Stress–Strain, Relaxation and Creep Curves Generated by a Nonlinear Model for Thixotropic Viscoelastic-Plastic Media Accounting for Structure Evolution Part 2. Relaxation and Stress-Strain Curves

  • Published:
Mechanics of Composite Materials Aims and scope

A systematic analytical study of the mathematical properties of the previously constructed nonlinear model for shear flow of thixotropic viscoelastic-plastic media is continued. For arbitrary six material parameters and an (increasing) material function that control the model, the basic properties of the families of stress-strain curves at constant strain rates and relaxation curves generated by the model, and the features of the evolution of the structuredness under these types of loading are analytically studied. The dependences of these curves on time, shear rate, initial strain and initial structuredness of the material, as well as on the material parameters and function of the model, are studied. Several indicators of the applicability of the model are found which are convenient to check with experimental data. It was examined what effects typical for viscoelastic-plastic media can be described by the model and what unusual effects (unusual properties) are generated by a change in structuredness in comparison with typical stress-strain curves and relaxation curves of structurally stable materials. In particular, it has been proved that stress-strain curves can be both increasing functions and can have decreasing sections resembling a “yield tooth” or damped oscillations, that all stress-strain curves (SSCs) possess horizontal asymptotes (steady flow stress), monotonically dependent on shear rate, and flow stress increases with shear rate growth, that the instantaneous shear modulus, on the contrary, depends on the initial structuredness, but does not depend on shear rate. Under certain restrictions on the material parameters, the model is also capable of providing a bilinear form of stress-strain curves, which is typical for an ideal elastoplastic model, but with strain rate sensitivity. It has been established that the family of stress-strain curves does not have to be increasing either in initial structuredness or in shear rate: in a certain range of shear rates, in which the equilibrium position is a “mature” focus and pronounced oscillations of stress-strain curves are observed, it is possible to intertwine stress-strain curves with different shear rates. It is proved that for any material parameters and functions, all stress relaxation curves decrease and have a common zero asymptote as time tends to infinity. The analysis proved the ability of the model to describe behavior of not only liquid-like viscoelastoplastic media, but also solid-like (thickening, hardening, hardened) media: creep, relaxation, recovery, a number of typical properties of experimental relaxation curves, creep and stress-strain curves, strain rate and strain hardening, flow under constant stress and so on.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.
Fig. 7.
Fig. 8.

Similar content being viewed by others

References

  1. A. V. Khokhlov and V. V. Gulin, “Families of stress-strain, relaxation, and creep curves generated by a nonlinear model for thixotropic viscoelastic-plastic media accounting for structure evolution. Part 1. The model, its basic properties, integral curves, and phase portraits,” Mech. Compos. Mater., 60, No. 1, 49-66 (2024). https://doi.org/10.1007/s11029-024-10174-6

    Article  Google Scholar 

  2. A. M. Stolin and A. V. Khokhlov, “Nonlinear model of shear flow of thixotropic viscoelastoplastic continua taking into account the evolution of the structure and its analysis,” Moscow University Mechanics Bulletin, 77, No. 5, 127-135 (2022). https://doi.org/10.3103/S0027133022050065.

    Article  Google Scholar 

  3. A. V. Khokhlov, “Equilibrium point and phase portrait of a model for flow of tixotropic media accounting for structure evolution,” Moscow University Mechanics Bulletin, 78, No. 4, 91-101 (2023). https://doi.org/10.3103/S0027133023040039.

    Article  Google Scholar 

  4. A. V. Khokhlov and V. V. Gulin, “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,” Physical Mesomech., 26, No. 6, 621-642 (2023). https://doi.org/10.1134/S1029959923060036.

    Article  Google Scholar 

  5. A. V. Khokhlov, “Analysis of properties of ramp stress relaxation curves produced by the Rabotnov non-linear hereditary theory,” Mech. Compos. Mater., 54, No. 4, 473-486 (2018). https://doi.org/10.1007/s11029-018-9757-1.

    Article  CAS  Google Scholar 

  6. A. V. Khokhlov, “Properties of the set of strain diagrams produced by Rabotnov nonlinear equation for rheonomous materials,” Mech. Solids, 54 No. 3, 384-399 (2019). https://doi.org/10.3103/S002565441902002X.

    Article  Google Scholar 

  7. A. V. Khokhlov, “Applicability indicators and identification techniques for a nonlinear Maxwell-type elastoviscoplastic model using loading-unloading curves,” Mech. Compos. Mater., 55, No. 2 195-210 (2019). https://doi.org/10.1007/s11029-019-09809-w.

    Article  CAS  Google Scholar 

  8. A. V. Khokhlov, “Analysis of the bulk creep influence on stress-strain curves under tensile loadings at constant rates and on Poisson’s ratio evolution based on the linear viscoelasticity theory,” Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 23, No. 4 671-704 (2019). https://doi.org/10.14498/vsgtu1710.

  9. E. C. Bingham, Fluidity and Plasticity. McGraw-Hill, N. Y. (1922).

  10. J. G. Oldroyd, “Non Newtonian effects in steady motion of some idealised elastico-viscous liquids,” Proc. Roy. Soc. London. Ser. A., 245, 278-297 (1958).

    Article  Google Scholar 

  11. M. Reiner, Rheology, in: Encyclopedia of Physics, 6, SpringerBerlin-Heidelberg (1958), 434-550.

  12. B. D. Coleman, A. Makrovitz, and W. Noll, Viscometric Flows of Non-Newtonian Fluids. Theory and Experiment, Springer, Berlin-Heidelberg-New York (1966).

  13. G. V. Vinogradov and A. Ya Malkin, Polymer Rheology, [in Russian], Khimiya Publ., Moscow (1977).

  14. R. G. Larson, Constitutive Equations for Polymer Melts and Solutions, Butterworth, Boston (1988).

    Google Scholar 

  15. Leonov A. I. and A. N. Prokunin, Non-Linear Phenomena in Flows of Viscoelastic Polymer Fluids, Chapman and Hall, London (1994).

    Book  Google Scholar 

  16. C. Macosko, Rheology: Principles, Measurements and Applications, VCH, N.Y. (1994).

  17. C. L. Rohn, Analytical Polymer Rheology, Hanser Publishers, Munich (1995).

    Google Scholar 

  18. R. R. Huilgol, and N. Phan-Thien, Fluid Mechanics of Viscoelasticity, Elsevier, Amsterdam (1997).

    Google Scholar 

  19. R. G. Larson, Structure and Rheology of Complex Fluids, Oxford Press, New York (1999).

    Google Scholar 

  20. R. K. Gupta, Polymer and Composite Rheology. Marcel Dekker, N. Y. (2000).

  21. R. I. Tanner, Engineering Rheology, Oxford University Press, Oxford (2000).

    Book  Google Scholar 

  22. H. Yamaguchi, Engineering Fluid Mechanics (Fluid Mechanics and Its Applications). Springer, (2008).

  23. W. W. Graessley, Polymeric Liquids and Networks: Dynamics and Rheology, Garland Science, London (2008).

    Google Scholar 

  24. M. M. Denn, Polymer Melt Processing. Cambridge University Press, Cambridge (2008).

    Book  Google Scholar 

  25. M. Kamal, A. Isayef, and S. Liu, Injection Molding Fundamentals and Applications. Hanser, Munich (2009).

    Book  Google Scholar 

  26. J. L. Leblanc, Filled Polymers, CRC Press, Boca Raton (2010).

    Google Scholar 

  27. A. Y. Malkin and A. I. Isayev, Rheology: Conceptions, Methods, Applications (2nd Ed.). ChemTec Publishing, Toronto (2012).

    Google Scholar 

  28. V. N. Pokrovskii, The Mesoscopic Theory of Polymer Dynamics, Springer (2010).

  29. A. I. Leonov, “Constitutive equations for viscoelastic liquids: formulation, analysis and comparison with data,” Rheology Series, 8, 519-575 (1999).

    Article  CAS  Google Scholar 

  30. J. J. Stickel and R. L. Powell, “Fluid mechanics and rheology of dense suspensions,” Annual Review of Fluid Mech., 37, 129-149 (2005).

    Article  Google Scholar 

  31. S. Mueller, E. W. Llewellin, and H. M. Mader, “The rheology of suspensions of solid particles,” Proc. R. Soc. A, 466, No. 2116, 1201-1228 (2010).

    Article  CAS  Google Scholar 

  32. T. Divoux, M. A. Fardin, S. Manneville, and S. Lerouge, “Shear banding of complex fluids,” Annual Review of Fluid Mech., 48, 81-103 (2016).

    Article  Google Scholar 

  33. J. F. Brady and J. F. Morris, “Microstructure of strongly sheared suspensions and its impact on rheology and diffusion,” J. Fluid Mech., 348, 103-139 (1997).

    Article  CAS  Google Scholar 

  34. C. L. Tucker and P. Moldenaers, “Microstructural evolution in polymer blends,” Annu. Rev. Fluid Mech., 34, 177-210 (2002).

    Article  Google Scholar 

  35. A. Y. Malkin and V. G. Kulichikhin, “Structure and rheology of highly concentrated emulsions: a modern look,” Russian Chemical Reviews, 84, No. 8, 803-825 (2015).

    Article  CAS  Google Scholar 

  36. V. G. Kulichikhin and A. Y. Malkin, “The role of structure in polymer rheology: review,” Polymers, 14, 1-34 (2022). https://doi.org/10.3390/polym14061262.

    Article  CAS  Google Scholar 

  37. S. S. Datta, A. M. Ardekani, P. E. Arratia, et al., “Perspectives on viscoelastic flow instabilities and elastic turbulence,” Physical Review Fluids, 7, 1-80 (2022). https://doi.org/10.1103/PhysRevFluids.7.080701

    Article  Google Scholar 

  38. D. Fraggedakis, Y. Dimakopoulos, and J. Tsamopoulos, “Yielding the yield stress analysis: A thorough comparison of recently proposed elasto-visco-plastic (EVP) fluid models,” J. Non-Newtonian Fluid Mech., 236, 104-122 (2016).

    Article  CAS  Google Scholar 

  39. S. Varchanis, G. Makrigiorgos, P. Moschopoulos, Y. Dimakopoulos, and J. Tsamopoulos, “Modeling the rheology of thixotropic elasto-visco-plastic materials,” J. Rheology, 63, No. 4, 609-639 (2019).

    Article  CAS  Google Scholar 

  40. A. V. Khokhlov, “Long-term strength curves generated by the nonlinear Maxwell-type model for viscoelastoplastic materials and the linear damage rule under step loading,” J. Samara State Tech. Univ., Ser. Phys. Math. Sci., No 3, 524-543 [in Russian] (2016). https://doi.org/10.14498/vsgtu1512

  41. A. V. Khokhlov, “Nonlinear Maxwell-type elastoviscoplastic model: General properties of stress relaxation curves and restrictions on the material functions,” Vestn. Mosk. Gos. Tekh. Herald of the Bauman Moscow State Tech. Univ., Nat. Sci., No. 6, 31-55 (2017) [In Russian]. https://doi.org/10.18698/1812-3368-2017-6-31-55

  42. A. V. Khokhlov, “The nonlinear Maxwell-type model for viscoelastoplastic materials: Simulation of temperature influence on creep, relaxation and strain-stress curves,” J. Samara State Tech. Univ., Ser. Phys. Math. Sci., 21, No. 1, 160-179 (2017) (in Russian). https://doi.org/10.14498/vsgtu1524.

  43. A. V. Khokhlov, “Possibility to describe the alternating and non-monotonic time dependence of Poisson’s ratio during creep using a nonlinear Maxwell-type viscoelastoplasticity model,” Russian Metallurgy, No. 10, 956-963 (2019). https://doi.org/10.1134/S0036029519100136.

    Article  Google Scholar 

  44. T. G. Nieh, J. Wadsworth and O. D. Sherby, Superplasticity in Metals and Ceramics, Cambridge Univ. Press, Cambridge (1997).

    Book  Google Scholar 

  45. K. A. Padmanabhan, R. A. Vasin and F. U. Enikeev, Superplastic Flow: Phenomenology and Mechanics, Heidelberg: Springer-Verlag, Berlin, (2001).

    Book  Google Scholar 

  46. V. M. Segal, I. J. Beyerlein, C. N. Tome, V. N. Chuvil’deev and V. I. Kopylov, Fundamentals and Engineering of Severe Plastic Deformation, Nova Science Pub. Inc., New York (2010).

  47. A. P. Zhilayev and A. I. Pshenichnyuk, Superplasticity and Grain Boundaries in Ultrafine-Grained Materials, Cambridge Intern. Sci. Publ., Cambridge (2010).

    Google Scholar 

  48. R. Z. Valiev, A. P. Zhilyaev and T. G. Langdon, Bulk Nanostructured Materials: Fundamentals and Applications, TMSWiley, Hoboken (2014).

  49. E. R. Sharifullina, A. I. Shveykin and P. V. Trusov, “Review of experimental studies on structural superplasticity: internal structure evolution of material and deformation mechanisms,” PNRPU Mechanics Bulletin, 3, 103-127 (2018). https://doi.org/10.15593/perm.mech/2018.3.11.

    Article  Google Scholar 

  50. A. G. Mochugovskiy, A. O. Mosleh, A. D. Kotov, A. V. Khokhlov, L. Y. Kaplanskaya, A.V. Mikhaylovskaya, “Microstructure evolution, constitutive modelling, and superplastic forming of experimental 6XXX-type alloys processed with different thermomechanical treatments,” Materials, 16, No. 1, 1-18 (2023). https://doi.org/10.3390/ma16010445.

    Article  CAS  Google Scholar 

Download references

Acknowledgment

The paper was done with the financial support of the Ministry of Education and Science of the Russian Federation as part of the program of the Moscow Center for Fundamental and Applied Mathematics under the agreement No. 075-15-2022-284.

Author information

Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, Russia

    A. V. Khokhlov & V. V. Gulin

  2. Moscow Center for Fundamental and Applied Mathematics, Moscow, Russia

    A. V. Khokhlov

  3. North-Eastern Federal University, Yakutsk, Russia

    A. V. Khokhlov & V. V. Gulin

Authors
  1. A. V. Khokhlov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. V. Gulin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. V. Khokhlov.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Khokhlov, A.V., Gulin, V.V. Families of Stress–Strain, Relaxation and Creep Curves Generated by a Nonlinear Model for Thixotropic Viscoelastic-Plastic Media Accounting for Structure Evolution Part 2. Relaxation and Stress-Strain Curves. Mech Compos Mater 60, 259–278 (2024). https://doi.org/10.1007/s11029-024-10197-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11029-024-10197-z

Keywords

Search

Navigation