Skip to main content
SpringerLink
Log in

A monolithic model for phase-field fracture and waves in solid–fluid media towards earthquakes

  • Original Paper
  • Published:
International Journal of Fracture Aims and scope Submit manuscript

Abstract

Coupling of rupture processes in solids with waves also propagating in fluids is a prominent phenomenon arising during tectonic earthquakes. It is executed here in a single ‘monolithic’ model which can asymptotically capture both damageable solids (rocks) and (visco-)elastic fluids (outer core or oceans). Both ruptures on pre-existing lithospheric faults and a birth of new faults in compact rocks are covered by this model, together with emission and propagation of seismic waves, including, e.g., reflection of S-waves and refraction of P-waves on the solid–fluid interfaces. A robust, energy conserving, and convergent staggered FEM discretisation is devised. Using a rather simplified variant of such models for rupture, three computational experiments documenting the applicability of this approach are presented. Some extensions of the model towards more realistic geophysical modelling are outlined, too.

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
Fig. 9

Similar content being viewed by others

References

  • Ambrosio L, Tortorelli VM (1992) Approximation of free discontinuity problems. Boll Unione Mat Italiana 6–B:105–123

    Google Scholar 

  • Bažant Z, Jirásek M (1996) Softening-induced dynamic localization instability: seismic damage in frames. J Eng Mech 122:1149–1158

    Article  Google Scholar 

  • Bedford A (1985) Hamilton’s principle in continuum mechanics. Pitman, Boston

    Google Scholar 

  • Ben-Zion Y (1998) Properties of seismic fault zone waves and their utility for imaging low-velocity structures. J Geophys Res 103:12567–12585

    Article  Google Scholar 

  • Ben-Zion Y (2001) Dynamic ruptures in recent models of earthquake faults. J Mech Phys Solids 49:2209–2244

    Article  Google Scholar 

  • Ben-Zion Y, Ampuero JP (2009) Seismic radiation from regions sustaining material damage. Geophys J Int 178:1351–1356

    Article  Google Scholar 

  • Boger DV (1977) A highly elastic constant-viscosity fluid. J Non Newton Fluid Mech 3:87–91

    Article  Google Scholar 

  • Bourdin B, Larsen C, Richardson C (2011) A time-discrete model for dynamic fracture based on crack regularization. Int J Fract 10:133–143

    Article  Google Scholar 

  • Bourdin B, Marigo JJ, Maurini C, Sicsic P (2014) Morphogenesis and propagation of complex cracks induced by thermal shocks. Phys Rev Lett 112:014301

    Article  Google Scholar 

  • Choi E, Tan E, Lavier L, Calo V (2013) DynEarthSol2D: an efficient unstructured finite element method to study long-term tectonic deformation. J Geophys Res Solid Earth 118:2429–2444

    Article  Google Scholar 

  • Courant R, Friedrichs K, Lewy H (1928) Über die partiellen Differenzengleichungen der mathematischen Physik. Math Annalen 100:32–74

    Article  Google Scholar 

  • Focardi M (2001) On the variational approximation of free-discontinuity problems in the vectorial case. Math Models Methods Appl Sci 11:663–684

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Griffith A (1921) The phenomena of rupture and flow in solids. Philos Trans R Soc Lond A 221:163–198

    Article  Google Scholar 

  • Halphen B, Nguyen Q (1975) Sur les matériaux standards généralisés. J Mécanique 14:39–63

    Google Scholar 

  • Hamiel Y, Lyakhovsky V, Agnon A (2004) Coupled evolution of damage and porosity in poroelastic media: theory and applications to deformation of porous rocks. Geophys J Int 156:701–713

    Article  Google Scholar 

  • Hamilton W (1834) On a general method in dynamics, part II. Philos Trans R Soc 247–308

  • Harris RA et al (2009) The SCEC/USGS dynamic earthquake rupture code verification exercise. Seismol Res Lett 80:119–126. https://doi.org/10.1785/gssrl.80.1.119

    Article  Google Scholar 

  • Hofacker M, Miehe C (2012) Continuum phase field modeling of dynamic fracture: variational principles and staggered FE implementation. Int J Fract 178:113–129

    Article  Google Scholar 

  • Huang Y, Ampuero JP, Helmberger DV (2014) Earthquake ruptures modulated by waves in damaged fault zones. J Geophys Res Solid Earth B9:3133–3154

    Article  Google Scholar 

  • Kaneko Y, Lapusta N, Ampuero JP (2008) Spectral element modeling of spontaneous earthquake rupture on rate and state faults: effect of velocity-strengthening friction at shallow depths. J Geophys Res 113:B09317. https://doi.org/10.1029/2007JB005553

    Google Scholar 

  • Käser M, Dumbser M (2006) An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes I: the two-dimensional isotropic case with external source terms. Geophys J Int 166:855–877

    Article  Google Scholar 

  • Käser M, Dumbser M, de la Puente J, Igel H (2007) An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes III: viscoelastic attenuation. Geophys J Int 168:224–242

    Article  Google Scholar 

  • Khoei A, Vahab M, Hirmand M (2016) Modeling the interaction between fluid-driven fracture and natural fault using an enriched-FEM technique. Int J Fract 197:1–24

    Article  Google Scholar 

  • Komatitsch D, Tromp J (2002a) Spectral-element simulations of global seismic wave propagation—I. Validation. Geophys J Int 149:390–412

    Article  Google Scholar 

  • Komatitsch D, Tromp J (2002b) Spectral-element simulations of global seismic wave propagation—II. Three-dimensional models, oceans, rotation and self-gravitation. Geophys J Int 150:303–318

    Article  Google Scholar 

  • Kružík M, Roubíček T (2019) Mathematical methods in continuum mechanics of solids. Springer, Basel

    Google Scholar 

  • Larsen C, Ortner C, Süli E (2010) Existence of solution to a regularized model of dynamic fracture. Math Models Methods Appl Sci 20:1021–1048

    Article  Google Scholar 

  • Lay T, Wallace TC (1995) Modern global seismology. Academic Press, San Diego

    Google Scholar 

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

    Article  Google Scholar 

  • Lyakhovsky V, Ben-Zion Y (2014b) Damage-breakage rheology model and solid-granular transition near brittle instability. J Mech Phys Solids 64:184–197

    Article  Google Scholar 

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

    Google Scholar 

  • Lyakhovsky V, Hamiel Y, Ampuero JP, Ben-Zion Y (2009) Non-linear damage rheology and wave resonance in rocks. Geophys J Int 178:910–920

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Mielke A, Roubíček T (2015) Rate-independent systems—theory and application. Springer, New York

    Book  Google Scholar 

  • Pelties C, de la Puente J, Ampuero JP, Brietzke GB, Käser M (2012) Three-dimensional dynamic rupture simulation with a high-order discontinuous Galerkin method on unstructured tetrahedral meshes. J Geophys Res 117:B02309. https://doi.org/10.1029/2011JB008857

    Article  Google Scholar 

  • Roubíček T (2014) A note about the rate-and-state-dependent friction model in a thermodynamical framework of the Biot-type equation. Geophys J Int 199:286–295

    Article  Google Scholar 

  • Roubíček T (2017a) An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat. Discret Contin Dynam Syst S 10:867–893

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Roubíček T (2018) Seismic waves and earthquakes in a global monolithic model. Contin Mech Thermodynam 30:709–729

    Article  Google Scholar 

  • Roubíček T (2019) Models of dynamic damage and phase-field fracture, and their various time discretisations. In: Rodrigues JE, Hinttermüller M (eds) Topics in applied analysis and optimisation, CIM Series in Math. Sci. Springer, Berlin (in print). (Preprint arXiv:1906.04110)

  • Roubíček T, Panagiotopoulos CG (2017) Energy-conserving time-discretisation of abstract dynamical problems with applications in continuum mechanics of solids. Numer Funct Anal Optim 38:1143–1172

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Roubíček T, Panagiotopoulos C, Tsogka C (2019) Explicit time-discretisation of elastodynamics with some inelastic processes at small strains. Preprint arXiv:1903.11654

  • Tago J, Cruz-Atienza VM, Virieux J, Etienne V, Sánchez-Sesma FJ (2012) A 3D hp-adaptive discontinuous Galerkin method for modeling earthquake dynamics. J Geophys Res 117:B09312. https://doi.org/10.1029/2012JB009313

    Article  Google Scholar 

  • Tosi N, Čadek O, Martinec Z (2009) Subducted slabs and lateral viscosity variations: effects on the long-wavelength geoid. Geophys J Int 179:813–826

    Article  Google Scholar 

Download references

Acknowledgements

The support from the Grants 17-04301S (as far as dissipative evolutionary aspects concerns) and 19-04956S (as far as modelling of dynamic and nonlinear behaviour concerns) of the Czech Sci. Foundation and VEGA 1/0078/16 of the Ministry of Education, Science, Research and Sport of the Slovak Republic, and the institutional support RVO:61388998 (ČR) are acknowledged.

Author information

Authors and Affiliations

  1. Institute of Thermomechanics, Czech Academy of Sciences, Dolejškova 5, 182 00, Praha 8, Czech Republic

    Tomáš Roubíček

  2. Mathematical Institute, Charles University, Sokolovská 83, 186 75, Praha 8, Czech Republic

    Tomáš Roubíček

  3. Civil Engineering Faculty, Technical University of Košice, Vysokoškolská 4, 042 00, Kosice, Slovakia

    Roman Vodička

Authors
  1. Tomáš Roubíček
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Roman Vodička
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tomáš Roubíček.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Roubíček, T., Vodička, R. A monolithic model for phase-field fracture and waves in solid–fluid media towards earthquakes. Int J Fract 219, 135–152 (2019). https://doi.org/10.1007/s10704-019-00386-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10704-019-00386-6

Keywords

Mathematics Subject Classification

Search

Navigation