arXiv subjects

Tomáš Roubíček

Publications and source records attributed to Tomáš Roubíček.

At least 19 recordsLinked to original sources

arXiv2026

The Stefan problem for complete melting of finitely strained solids into viscoelastic fluids

The compressible fluid-solid interaction (FSI) with a thermomechanical phase transition is formulated at large strains within the Eulerian frame. For the deviatoric part, the Jeffreys (also called anti-Zener) rheology with an additional viscosity is adopted. The core philosophy governing the mechanical solid-liquid transition is that the viscous (or viscoplastic) response is temperature-dependent and may fully degenerate to a viscoelastic fluid during thawing, so that there is no elastic response on the shear distortion. This behavior enables the free flow of the fluid, its subsequent freezing into a new configuration, and potential re-melting back into a fluid, allowing such cycles to repeat indefinitely. The classical Stefan problem, associated with the latent heat of the first-order (thawing-freezing) phase transition, is augmented by incorporating kinetic overheating and undercooling. The analysis by a time discretization with an appropriate truncation is applied to a higher-gradient modification of the original formulation, utilizing the concept of multipolar nonsimple continua.

math.AP↗

arXiv2026

Staggered time discretization in finitely-strained heterogeneous visco-elastodynamics with damage or diffusion in the Eulerian frame

The semi-implicit (partly decoupled, also called staggered or fraction-step) time discretization is applied to compressible nonlinear dynamical models of viscoelastic solids in the Eulerian description, i.e.\ in the actual deforming configuration, formulated fully in terms of rates. The Kelvin-Voigt rheology and also, in the deviatoric part, the Jeffreys rheology are considered. The numerical stability and, considering the Stokes-type viscosity multipolar of the 2nd-grade, also convergence towards weak solutions are proved in three-dimensional situations, exploiting the convexity of the kinetic energy when written in terms of linear momentum. No (poly)convexity of the stored energy is required and some enhancements (specifically towards damage and diffusion models) are briefly outlined, too.

math.NA↗

arXiv2026

A few notes about viscoplastic rheologies

The rigorous tools of convex analysis are used to examine various serial and parallel combinations of linear viscosity and perfect plasticity. Nonlinear viscosities are also considered. The general aim is to synthesize a single convex ``viscoplastic'' dissipation potential from the potentials of particular viscous or plastic elements. Rigorous serial-viscosity models are then compared with empirical models based on harmonic means, which are commonly used for various geomaterials.

math-ph↗

arXiv2025

Time discretization in convected linearized thermo-visco-elastodynamics at large displacements

The fully-implicit time discretization (i.e. the backward Euler formula) is applied to compressible nonlinear dynamical models of thermo-viscoelastic solids in the Eulerian description, i.e. in the actual deforming configuration, formulated in terms of rates. The Kelvin-Voigt rheology or also, in the deviatoric part, the Jeffreys rheology (which covers creep or plasticity) are considered, using the additive Green-Naghdi decomposition of total strain into the elastic and the inelastic strains formulated in terms of (objective) rates exploiting the Zaremba-Jaumann time derivative. A linearized convective model at large displacements is considered, focusing on the case where the internal energy additively splits the (convex) mechanical and the thermal parts.A fully implicit time-discrete scheme is devised. Considering the multipolar 2nd-grade viscosity, the numerical stability and convergence towards weak solutions are proven by exploiting, in particular, the convexity of the kinetic energy when written in terms of linear momentum instead of velocity and by estimating the temperature gradient from the entropy-like inequality.

math.NA↗

arXiv2025

Thermo-elastodynamics of finitely-strained multipolar viscous solids with an energy-controlled stress

The thermodynamical model of viscoelastic deformable solids at finite strains with Kelvin-Voigt rheology with a higher-order viscosity (using the concept of multipolar materials) is formulated in a fully Eulerian way in rates. Assumptions used in this paper allow for a physically justified free energy leading to non-negative entropy that satisfies the 3rd law of thermodynamics, i.e. entropy vanishes at zero temperature, and energy-controlled stress. This last attribute is used advantageously to prove the existence and a certain regularity of weak solutions by a simplified Faedo-Galerkin semi-discretization, based on estimates obtained from the total-energy and the mechanical-energy balances. Some examples that model neo-Hookean-type materials are presented, too.

math.AP↗

arXiv2024

Time discretization in visco-elastodynamics at large displacements and strains in the Eulerian frame

The fully-implicit time discretization (i.e. the backward Euler formula) is applied to compressible nonlinear dynamical models of viscoelastic solids in the Eulerian description, i.e. in the actual deforming configuration. The Kelvin-Voigt rheology or also, in the deviatoric part, the Jeffreys rheology are considered. Both a linearized convective model at large displacements with a convex stored energy and the fully nonlinear large strain variant with a (possibly generalized) polyconvex stored energy are considered. The time-discrete suitably regularized schemes are devised for both cases. The numerical stability and, considering the multipolar 2nd-grade viscosity, also convergence towards weak solutions are proved, exploiting the convexity of the kinetic energy when written in terms of linear momentum instead of velocity. In the fully nonlinear case, the examples of neo-Hookean and Mooney-Rivlin materials are presented. A comparison with models of viscoelastic barotropic fluids is also made.

math.AP↗

arXiv2024

A general thermodynamical model for finitely-strained continuum with inelasticity and diffusion, its GENERIC derivation in Eulerian formulation, and some application

A thermodynamically consistent visco-elastodynamical model at finite strains is derived that also allows for inelasticity (like plasticity or creep), thermal coupling, and poroelasticity with diffusion. The theory is developed in the Eulerian framework and is shown to be consistent with the thermodynamic framework given by General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC). For the latter we use that the transport terms are given in terms of Lie derivatives. Application is illustrated by two examples, namely volumetric phase transitions with dehydration in rocks and martensitic phase transitions in shape-memory alloys. A strategy towards a rigorous mathematical analysis is only very briefly outlined.

math-ph↗

arXiv2023

Some gradient theories in linear visco-elastodynamics towards dispersion and attenuation of waves in relation to large-strain models

Various spatial-gradient extensions of standard viscoelastic rheologies of the Kelvin-Voigt, Maxwell's, and Jeffreys' types are analyzed in linear one-dimensional situations as far as the propagation of waves and their dispersion and attenuation. These gradient extensions are then presented in the large-strain nonlinear variants where they are sometimes used rather for purely analytical reasons either in the Lagrangian or the Eulerian formulations without realizing this wave-propagation context.The interconnection between these two modeling aspects is thus revealed in particular selected cases.

physics.flu-dyn↗

arXiv2023

Thermomechanics of ferri-antiferromagnetic phase transition in finitely-strained rocks towards paleomagnetism

The thermodynamic model of visco-elastic deformable magnetic materials at finite strains is formulated in a fully Eulerian way in rates with the aim to describe thermoremanent paleomagnetism in crustal rocks. The Landau theory applied to a ferro-to-para-magnetic phase transition, the gradient theory for magnetization (leading to exchange energy) with general mechanically dependent coefficient, hysteresis in magnetization evolution by Gilbert equation involving objective corotational time derivative of magnetization, and demagnetizing field are considered in the model. The Jeffreys viscoelastic rheology is used with temperature-dependent creep to model solidification or melting transition. The model complies with energy conservation and the Clausius-Duhem entropy inequality.

physics.geo-ph↗

arXiv2023

Thermodynamics of viscoelastic solids, its Eulerian formulation, and existence of weak solutions

The thermodynamical model of visco-elastic deformable solids at finite strains is formulated in a fully Eulerian way in rates. Also effects of thermal expansion or buoyancy due to evolving mass density in a gravity field are covered. The Kelvin-Voigt rheology with a higher-order viscosity (exploiting the concept of multipolar materials) is used, allowing for physically relevant frame-indifferent stored energies and for local invertibility of deformation. The model complies with energy conservation and Clausius-Duhem entropy inequality. Existence and a certain regularity of weak solutions is proved by a Faedo-Galerkin semi-discretization and a suitable regularization. Subtle physical limitations of the model are illustrated on thermally expanding neo-Hookean materials or materials with phase transitions.

math.AP↗

arXiv2023

Inhomogeneous finitely-strained thermoplasticity with hardening by an Eulerian approach

A standard elasto-plasto-dynamic model at finite strains based on the Lie-Liu-Kröner multiplicative decomposition, formulated in rates, is here enhanced to cope with spatially inhomogeneous materials by using the reference (called also return) mapping. Also an isotropic hardening can be involved. Consistent thermodynamics is formulated, allowing for both the free and the dissipation energies temperature dependent. The model complies with the energy balance and entropy inequality. A multipolar Stokes-like viscosity and plastic rate gradient are used to allow for a rigorous analysis towards existence of weak solutions by a semi-Galerkin approximation.

math.AP↗

arXiv2023

Interaction of finitely-strained viscoelastic multipolar solids and fluids by an Eulerian approach

A mechanical interaction of compressible viscoelastic fluids with viscoelastic solids in Kelvin-Voigt rheology using the concept of higher-order (so-called 2nd-grade multipolar) viscosity is investigated in a quasistatic variant. The no-slip contact between fluid and solid is considered and the Eulerian-frame return-mapping technique is used for both the fluid and the solid models, which allows for a "monolithic" formulation of this fluid-structure interaction problem. Existence and a certain regularity of weak solutions is proved by a Schauder fixed-point argument combined with a suitable regularization.

math.AP↗

arXiv2023

Landau theory for ferro-paramagnetic phase transition in finitely-strained viscoelastic magnets

The thermodynamic model of visco-elastic deformable magnetic materials at finite strains is formulated in a fully Eulerian way in rates. The Landau theory applies for ferro-to-para-magnetic phase transition, the gradient theory (leading exchange energy) for magnetization with general mechanically dependent coefficient, hysteresis in magnetization evolution by Landau-Lifshitz-Gilbert equation involving objective corotational time derivative of magnetization, and demagnetizing field are considered in the model. The Kelvin-Voigt viscoelastic rheology with a higher-order viscosity (exploiting the concept of multipolar materials) is used, allowing for physically relevant frame-indifferent stored energies and for local invertibility of deformation. The model complies with energy conservation and Clausius-Duhem entropy inequality. Existence and a certain regularity of weak solutions is proved by a Faedo-Galerkin semi-discretization and a suitable regularization.

math.AP↗

arXiv2023

A thermodynamical model for paleomagnetism in Earth's crust

A thermodynamically consistent model for soft deformable viscoelastic magnets is formulated in actual space (Eulerian) coordinates. The possibility of a ferro-paramagnetic-type (or ferri-antiferromagnetic) transition exploiting Landau phase transition theory as well as mechanical melting or solidification is considered, being motivated and applicable to paleomagnetism (involving both thermo- and isothermal and viscous remanent magnetization) in rocks in Earth's crust and to rock-magma transition. The temperature-dependent Jeffreys rheology in the deviatoric part combined with the Kelvin-Voigt rheology in the spherical (volumetric) part is used. The energy balance and the entropy imbalance behind the model are demonstrated, and its analysis is performed by time discretization, proving existence of weak solutions.

math.AP↗

arXiv2022

The Stefan problem in a thermomechanical context with fracture and fluid flow

The classical Stefan problem, concerning mere heat-transfer during solid-liquid phase transition, is here enhanced towards mechanical effects. The Eulerian description at large displacements is used with convective and Zaremba-Jaumann corotational time derivatives, linearized by using the additive Green-Naghdi's decomposition in (objective) rates. In particular, the liquid phase is a viscoelastic fluid while creep and rupture of the solid phase is considered in the Jeffreys viscoelastic rheology exploiting the phase-field model and a concept of slightly (so-called "semi") compressible materials. The $L^1$-theory for the heat equation is adopted for the Stefan problem relaxed by allowing for kinetic superheating/supercooling effects during the solid-liquid phase transition. A rigorous proof of existence of weak solutions is provided for an incomplete melting, employinga time-discretisation approximation.

math.AP↗

arXiv2022

Qualitative study of a geodynamical rate-and-state model for elastoplastic shear flows in crustal faults

The Dieterich-Ruina rate-and-state friction model is transferred to a bulk variant and the state variable (aging) influencing the dissipation mechanism is here combined also with a damage influencing standardly the elastic response. As the aging has a separate dynamics, the overall model does not have a standard variational structure. A one-dimensional model is investigated as far as the steady-state existence, localization of the ataclastic core, and its time response, too. Computational experiments with a damage-free variant show stick-slip behavior (i.e. seismic cycles of tectonic faults) as well as stable slip under very large velocities.

math.AP↗

arXiv2022

Quasistatic hypoplasticity at large strains Eulerian

The isothermal quasistatic (i.e.\ acceleration neglected) hardening-free plasticity at large strains is considered, based on the standard multiplicative decomposition of the total strain and the isochoric plastic distortion. The Eulerian velocity-strain formulation is used. The mass density evolves too, but acts only via the force term with a given external acceleration. This rather standard model is then re-formulated in terms of rates (so-called hypoplasticity) and the plastic distortion is completely eliminated, although it can be a-posteriori re-constructed. Involving gradient theories for dissipation, existence and regularity of weak solutions is proved rather constructively by a suitable regularization combined with a Galerkin approximation. The local non-interpenetration through a blowup of stored energy when elastic-strain determinant approaches zero is enforced and exploited. The plasticity is considered rate dependent and, as a special case, also creep in Jeffreys' viscoelastic rheology in the shear is covered while the volumetric response obeys the Kelvin-Voigt rheology.

math.AP↗

arXiv2022

Visco-elastodynamics at large strains Eulerian

Isothermal visco-elastodynamics in the Kelvin-Voigt rheology is formulated in the spatial Eulerian coordinates in terms of velocity and deformation gradient. A generally nonconvex (possibly also frame-indifferent) stored energy is admitted. The model involves a nonlinear 2nd-grade nonsimple (multipolar) viscosity so that the velocity field is well regular. To simplify analytical arguments, volume variations of the solid material are assumed to be only rather small so that the mass density is constant, exploiting the concept of semi-compressible materials. Existence of weak solutions is proved by using the Galerkin method combined with a suitable regularization, using nontrivial results about transport by smooth velocity fields.

math.AP↗