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

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)

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  • 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.

    Mathematics Subject Classification: Primary: 65K15, 65P99, 74F10, 74H15; Secondary: 35Q74, 37N15, 74J99, 74R20, 76S05, 80A17.


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  • Table 1.  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
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