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A convective model for poro-elastodynamics with damage and fluid flow towards Earth lithosphere modelling

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Abstract

Devised towards geophysical applications for various processes in the lithosphere or the crust, a model of poro-elastodynamics with inelastic strains and other internal variables like damage (aging) and porosity as well as with diffusion of water is formulated fully in the Eulerian setting. Concepts of gradient of the total strain rate as well as the additive splitting of the total strain rate are used while eliminating the displacement from the formulation. It relies on that the elastic strain is small while only the inelastic and the total strains can be large. The energetics behind this model is derived and used for analysis as far as the existence of global weak energy-conserving solutions concerns. By this way, the model of Lyakhovsky et al. (Appl Geophys 171:3099–3123, 2014; J Mech Phys Solids 59:1752–1776, 2011) is completed to make it mechanically consistent and amenable for analysis.

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References

  1. Ben-Zion, Y.: Collective behavior of earthquakes and faults: continuum-discrete transitions, progressive evolutionary changes, and different dynamic regimes. Rev. Geophys. 46, RG4006 (2008)

  2. Benešová, B., Forster, J., Liu, C., Schlömerkemper, A.: Existence of weak solutions to an evolutionary model for magnetoelasticity. SIAM J. Math. Anal. 50, 1200–1236 (2018)

    Article  MathSciNet  Google Scholar 

  3. Billen, M.I., Hirth, G.: Rheologic controls on slab dynamics. Geochem. Geophys. Geosyst. 8, Q08012 (2007)

    Article  ADS  Google Scholar 

  4. Biot, M.A.: General theory of three-dimensional consolidation. J. Appl. Phys. 12, 155–164 (1941)

    Article  ADS  Google Scholar 

  5. Brenner, H.: Kinematics of volume transport. Physica A 349, 11–59 (2005)

    Article  ADS  Google Scholar 

  6. Brenner, H.: Fluid mechanics revisited. Physica A 349, 190–224 (2006)

    Article  ADS  Google Scholar 

  7. Burczak, J., Málek, J., Minakowski, P.: Stress-diffusive regularization of non-dissipative rate-type materials. Disc. Cont. Dyn. Syst.-S 10, 1233–1256 (2017)

  8. Ericksen, J.L.: Liquid crystals with variable degree of orientation. Arch. Ration. Mech. Anal. 113, 97–120 (1991)

    Article  MathSciNet  Google Scholar 

  9. Finzi, Y., Muhlhaus, H., Gross, L., Kamirbekyan, A.: Shear band formation in numerical simulations applying a continuum damage rheology model. Pure Appl. Geophys. 170, 13–25 (2013)

    Article  ADS  Google Scholar 

  10. Green, A., Naghdi, P.: A general theory of an elastic–plastic continuum. Arch. Ration. Mech. Anal. 18, 251–281 (1965)

    Article  MathSciNet  Google Scholar 

  11. Hashiguchi, K., Yamakawa, Y.: Introduction to Finite Strain Theory for Continuum Elasto-Plasticity. J. Wiley, Chichester (2013)

    Google Scholar 

  12. Haupt, P.: Continuum Mechanics and Theory of Materials, 2nd edn. Springer, Berlin (2002)

    Book  Google Scholar 

  13. Hill, R.: A general theory of uniqueness and stability in elastic–plastic solids. J. Mech. Phys. Solids 6, 236–249 (1948)

    Article  ADS  Google Scholar 

  14. Jiao, Y., Fish, J.: Is an additive decomposition of a rate of deformation and objective stress rates passé? Comput. Methods Appl. Mech. Eng. 327, 196–225 (2017)

  15. Korteweg, D.J.: Sur la forme que prennent les équations du mouvement des fuides si lón tient compte des forces capillaires causées par des variations de densité considérables mais continues et sur la théorie de la capillarité dans l’hypothèse d’une variation continue de la densité. Arch. Néerl. Sci. Exactes Nat. 6, 1–24 (1901)

  16. Kröner, E.: Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen. Arch. Ration. Mech. Anal. 4, 273–334 (1960)

    Article  MathSciNet  Google Scholar 

  17. Kružík, M., Roubíček, T.: Mathematical Methods in Continuum Mechanics of Solids. Sringer, Cham (2019)

    Book  Google Scholar 

  18. Lee, E., Liu, D.: Finite-strain elastic–plastic theory with application to plain-wave analysis. J. Appl. Phys. 38, 19–27 (1967)

    Article  ADS  Google Scholar 

  19. Lin, F.-H., Liu, C., Zhang, P.: On hydrodynamics of viscoelastic fluids. Commun. Pure Appl. Math. 58, 1437–1471 (2005)

    Article  MathSciNet  Google Scholar 

  20. Lyakhovsky, V., Ben-Zion, Y.: A continuum damage-breakage faulting model and solid-granular transitions. Pure Appl. Geophys. 171, 3099–3123 (2014)

    Article  ADS  Google Scholar 

  21. Lyakhovsky, V., Hamiel, Y.: Damage evolution and fluid flow in poroelastic rock. Izvestiya Phys. Solid Earth 43, 13–23 (2007)

    Article  ADS  Google Scholar 

  22. Lyakhovsky, V., Hamiel, Y., Ben-Zion, Y.: A non-local visco-elastic damage model and dynamic fracturing. J. Mech. Phys. Solids 59, 1752–1776 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  23. Lyakhovsky, V., Myasnikov, V.P.: On the behavior of elastic cracked solid. Phys. Solid Earth 10, 71–75 (1984)

    Google Scholar 

  24. Martinec, Z.: Principles of Continuum Mechanics. Birkhäuser/Springer, Cham (2019)

    Book  Google Scholar 

  25. Maugin, G.A.: The Thermomechanics of Plasticity and Fracture. Cambridge University Press, Cambridge (1992)

    Book  Google Scholar 

  26. Öttinger, H.C., Struchtrup, H., Liu, M.: Inconsistency of a dissipative contribution to the mass flux in hydrodynamics. Phys. Rev. E 80, Art.no. 056303 (2009)

  27. Podio-Guidugli, P.: Inertia and invariance. Ann. Mat. Pura Appl. 172, 103–124 (1997)

    Article  MathSciNet  Google Scholar 

  28. Prager, W.: An elementary discussion of definitions of stress rate. Q. Appl. Math. 18, 403–407 (1961)

    Article  MathSciNet  Google Scholar 

  29. Regenauer-Lieb, K., Yuen, D.A.: Modeling shear zones in geological and planetary sciences: solid- and fluid-thermal-mechanical approaches. Earth Sci. Rev. 63, 295–349 (2003)

    Article  ADS  Google Scholar 

  30. Roubíček, T.: Nonlinear Partial Differential Equations with Applications, 2nd edn. Birkhäuser, Basel (2013)

    Book  Google Scholar 

  31. Roubíček, T.: Geophysical models of heat and fluid flow in damageable poro-elastic continua. Cont. Mech. Thermodyn. 29, 625–646 (2017)

    Article  MathSciNet  Google Scholar 

  32. Roubíček, T.: Coupled time discretisation of dynamic damage models at small strains. IMA J. Numer. Anal. 40, 1772–1791 (2020)

    Article  MathSciNet  Google Scholar 

  33. Roubíček, T.: From quasi-incompressible to semi-compressible fluids. Disc. Cont. Dynam. Syst. S (2020). https://doi.org/10.3934/dcdss.2020414

  34. Roubíček, T.: The Stefan problem in a thermomechanical context with fracture and fluid flow. Preprint arXiv:2012.15248 (2020)

  35. Roubíček, T.: Thermodynamically consistent model for poroelastic rocks towards tectonic and volcanic processes and earthquakes. Geophys. J. Intl., in print. Preprint arXiv:2103.11663 (2021). https://doi.org/10.1093/gji/ggab317

  36. Roubíček, T., Stefanelli, U.: Thermodynamics of elastoplastic porous rocks at large strains towards earthquake modeling. SIAM J. Appl. Math. 78, 2597–2625 (2018)

    Article  MathSciNet  Google Scholar 

  37. Surana, K.S.: Advanced Mechanics of Continua. CRC Press, Boca Raton (2015)

    MATH  Google Scholar 

  38. Temam, R.: Sur l’approximation de la solution des équations de Navier–Stokes par la méthode des pas fractionnaires (I). Archive Ration. Mech. Anal. 32, 135–153 (1969)

  39. Temam, R.: Navier–Stokes Equations—Theory and Numerical Analysis. North-Holland, Amsterdam (1977)

    MATH  Google Scholar 

  40. Tomassetti, G.: An interpretation of Temam’s stabilization term in the quasi-incompressible Navier-Stokes system. Appl. Eng. Sci. 5, Art.no. 100028 (2021)

  41. Ván, P., Pavelka, M., Grmela, M.: Extra mass flux in fluid mechanics. J. Non-Equilib. Thermodyn. 42, 133–152 (2017)

    Article  ADS  Google Scholar 

  42. Volokh, K.Y.: An approach to elastoplasticity at large deformations. Euro. J. Mech. A Solids 39, 153–162 (2013)

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgements

The authors are deeply thankful to two anonymous referees for very careful reading of the original version and many suggested improvements and mistake corrections.

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Authors and Affiliations

  1. Mathematical Institute, Charles University, Sokolovská 83, 186 75, Prague 8, Czech Republic

    Tomáš Roubíček

  2. Institute of Thermomechanics, Czech Academy of Sciences, Dolejškova 5, 182 00, Prague 8, Czech Republic

    Tomáš Roubíček

  3. Department of Engineering, Roma Tre University, Via Vito Volterra 62, 00146, Rome, Italy

    Giuseppe Tomassetti

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  1. Tomáš Roubíček
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  2. Giuseppe Tomassetti
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Correspondence to Tomáš Roubíček.

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Communicated by Andreas Öchsner.

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This research has been partially supported from the CSF (Czech Science Foundation) Project 19-04956S, the MŠMT ČR (Ministry of Education of the Czech Rep.) Project CZ.02.1.01/0.0/0.0/15-003/0000493, and the institutional support RVO: 61388998 (ČR). This research has also been partially supported by the Italian INdAM-GNFM (Istituto Nazionale di Alta Matematica-Gruppo Nazionale per la Fisica Matematica), the Grant of Excellence Departments, MIUR-Italy (Art.1, commi 314-337, Legge 232/2016), and the Grant “Mathematics of active materials: from mechanobiology to smart devices” (PRIN 2017, prot. 2017KL4EF3) funded by the Italian MIUR.

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Roubíček, T., Tomassetti, G. A convective model for poro-elastodynamics with damage and fluid flow towards Earth lithosphere modelling. Continuum Mech. Thermodyn. 33, 2345–2361 (2021). https://doi.org/10.1007/s00161-021-01043-x

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  • DOI: https://doi.org/10.1007/s00161-021-01043-x

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