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August  2017, 10(4): 867-893. doi: 10.3934/dcdss.2017044

## An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat

 Institute of Thermomechanics, Czech Academy of Sciences, Dolejškova 5, CZ-182 00 Praha 8, Czech Republic

Received  April 2016 Revised  November 2016 Published  April 2017

Fund Project: This research has been partially supported from the grants 16-03823S "Homogenization and multi-scale computational modelling of flow and nonlinear interactions in porous smart structures" and 14-15264S "Experimentally justified multiscale modelling of shape memory alloys" of Czech Science Foundation, and from the institutional support RVO:61388998 (ČR).

The model of brittle cracks in elastic solids at small strains is approximated by the Ambrosio-Tortorelli functional and then extended into evolution situation to an evolutionary system, involving viscoelasticity, inertia, heat transfer, and coupling with Cahn-Hilliard-type diffusion of a fluid due to Fick's or Darcy's laws. Damage resulting from the approximated crack model is considered rate independent. The fractional-step Crank-Nicolson-type time discretisation is devised to decouple the system in a way so that the energy is conserved even in the discrete scheme. The numerical stability of such a scheme is shown, and also convergence towards suitably defined weak solutions. Various generalizations involving plasticity, healing in damage, or phase transformation are mentioned, too.

Citation: Tomáš Roubíček. An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 867-893. doi: 10.3934/dcdss.2017044
##### References:

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##### References:
Summary of the basic notation used through this paper
 $u$ displacements $e(u)=\frac12(\nabla u)^\top+\frac12\nabla u$ small strain tensor $v$ velocity $M$ Biot modulus $z$ damage scalar variable $\beta$ Biot coefficient $c$ concentration $\kappa$ coefficient for the ratio Fick/Darcy flow $\theta$ temperature $\varkappa$ capillarity coefficient $\vartheta$ heat content $a$ energy released per unit volume by damage $\sigma$ stress $\psi=\varphi+\phi$ free energy $\mu$ chemical potential $\varphi, \phi$ chemo-mechanical and thermal energies $\mathbb{C}$ elastic-moduli tensor $\mathfrak{u}$ internal energy $\mathbb{D}$ viscous-moduli tensor $c_{_{\rm E}}$ equilibrium concentration $\mathbb{M}$ the mobility matrix $g$ bulk force (gravity) $\mathbb{K}$ the heat-conductivity matrix $f$ traction force $c_{\rm v}$ heat capacity $h_{_{\rm{B}}}$ prescribed boundary heat flux $\varrho$ mass density $j_{_{\rm{B}}}$ prescribed boundary diffusant flux $r$ heat-production rate $\varepsilon >0$ a fixed regularization parameter $\mathfrak{s}$ entropy $\tau>0$ a time step for discretisation
 $u$ displacements $e(u)=\frac12(\nabla u)^\top+\frac12\nabla u$ small strain tensor $v$ velocity $M$ Biot modulus $z$ damage scalar variable $\beta$ Biot coefficient $c$ concentration $\kappa$ coefficient for the ratio Fick/Darcy flow $\theta$ temperature $\varkappa$ capillarity coefficient $\vartheta$ heat content $a$ energy released per unit volume by damage $\sigma$ stress $\psi=\varphi+\phi$ free energy $\mu$ chemical potential $\varphi, \phi$ chemo-mechanical and thermal energies $\mathbb{C}$ elastic-moduli tensor $\mathfrak{u}$ internal energy $\mathbb{D}$ viscous-moduli tensor $c_{_{\rm E}}$ equilibrium concentration $\mathbb{M}$ the mobility matrix $g$ bulk force (gravity) $\mathbb{K}$ the heat-conductivity matrix $f$ traction force $c_{\rm v}$ heat capacity $h_{_{\rm{B}}}$ prescribed boundary heat flux $\varrho$ mass density $j_{_{\rm{B}}}$ prescribed boundary diffusant flux $r$ heat-production rate $\varepsilon >0$ a fixed regularization parameter $\mathfrak{s}$ entropy $\tau>0$ a time step for discretisation
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