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

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.

keywords: Viscoelasticity at small strains inertia damage crack AmbrosioTortorelli approximation Cahn-Hilliard equation Fick and Darcy flow fractional step splitting Crank-Nicolson formula convergence existence of weak solutions
Thermodynamics of perfect plasticity
Tomáš Roubíček
Discrete & Continuous Dynamical Systems - S 2013, 6(1): 193-214 doi: 10.3934/dcdss.2013.6.193
Viscoelastic solids in Kelvin-Voigt rheology at small strains exhibiting also stress-driven Prandtl-Reuss perfect plasticity are considered quasistatic (i.e. inertia neglected) and coupled with heat-transfer equation through dissipative heat production by viscoplastic effects and through thermal expansion and corresponding adiabatic effects. Enthalpy transformation is used and existence of a weak solution is proved by an implicit suitably regularized time discretisation.
keywords: thermodynamics thermal expansion Kelvin-Voigt rheology adiabatic effects small strains bounded-deformation-valued solution. Prandtl-Reuss plasticity
On certain convex compactifications for relaxation in evolution problems
Tomáš Roubíček
Discrete & Continuous Dynamical Systems - S 2011, 4(2): 467-482 doi: 10.3934/dcdss.2011.4.467
A general-topological construction of limits of inverse systems is applied to convex compactifications and furthermore to special convex compactifications of Lebesgue-space-valued functions parameterized by time. Application to relaxation of quasistatic evolution in phase-change-type problems is outlined.
keywords: generalized Young measures Limits of inverse systems Helly principle.
Thermomechanics of hydrogen storage in metallic hydrides: Modeling and analysis
Tomáš Roubíček Giuseppe Tomassetti
Discrete & Continuous Dynamical Systems - B 2014, 19(7): 2313-2333 doi: 10.3934/dcdsb.2014.19.2313
A thermodynamically consistent mathematical model for hydrogen adsorption in metal hydrides is proposed. Beside hydrogen diffusion, the model accounts for phase transformation accompanied by hysteresis, swelling, temperature and heat transfer, strain, and stress. We prove existence of solutions of the ensuing system of partial differential equations by a carefully-designed, semi-implicit approximation scheme. A generalization for a drift-diffusion of multi-component ionized ``gas'' is outlined, too.
keywords: anelastic strain hydrogen diffusion swelling time discretisation semiconductors weak solutions fuel cells Metal-hydrid phase transformation convergence.
Modelling of thermodynamics of martensitic transformation in shape-memory alloys
Tomáš Roubíček
Conference Publications 2007, 2007(Special): 892-902 doi: 10.3934/proc.2007.2007.892
Existence of a solution to the thermo-visco-elasto-"plastic-type" system involving also higher capillarity/viscosity terms and describing thermodynamics of activated martensitic transformation at large strains is proved by a careful successive passage to a limit in a suitably regularized Galerkin approximation.
keywords: nonconvex stored energy Thermoviscoelasticity activated processes.
A quasistatic mixed-mode delamination model
Tomáš Roubíček V. Mantič C. G. Panagiotopoulos
Discrete & Continuous Dynamical Systems - S 2013, 6(2): 591-610 doi: 10.3934/dcdss.2013.6.591
The quasistatic rate-independent evolution of delamination in the so-called mixed-mode, i.e. distinguishing opening (mode I) from shearing (mode II), devised in [45], is described in detail and rigorously analysed as far as existence of the so-called energetic solutions concerns. The model formulated at small strains uses a delamination parameter of Frémond's type combined with a concept of interface plasticity, and is associative in the sense that the dissipative force driving delamination has a potential which depends in a 1-homogeneous way only on rates of internal parameters. A sample numerical simulation documents that this model can really produce mode-mixity-sensitive delamination.
keywords: interface plasticity variational inequality simulations. Rate-independent interface fracture Rothe method energetic solution inelastic debonding convergence analysis damage

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