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

Tomáš Roubíček

doi:10.3934/dcdss.2017044

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2017, Volume 10, Issue 4: 867-893. Doi: 10.3934/dcdss.2017044

This issue Previous Article A Harnack type inequality and a maximum principle for an elliptic-parabolic and forward-backward parabolic De Giorgi class Next Article Large solutions of parabolic logistic equation with spatial and temporal degeneracies

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

Tomáš Roubíček

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

Received: March 31, 2016

Revised: October 31, 2016

Published: July 2017

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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Abstract

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.

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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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