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

  • Tomáš RoubíčekEmail author
  • Roman Vodička
Original Paper


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.


Fracture of faults Tectonic earthquake dynamics Elastic waves Elastic-fluid/solid interaction Numerical modelling 

Mathematics Subject Classification

74F10 74J10 74R20 74S05 86-08 



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.


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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Institute of ThermomechanicsCzech Academy of SciencesPraha 8Czech Republic
  2. 2.Mathematical InstituteCharles UniversityPraha 8Czech Republic
  3. 3.Civil Engineering FacultyTechnical University of KošiceKosiceSlovakia

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