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, formulated fully in terms of rates. 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.
KružíkMRoubíčekT.Mathematical methods in continuum mechanics of solids. Cham: Springer, 2019.
2.
RoubíčekT.Some gradient theories in linear visco-elastodynamics towards dispersion and attenuation of waves in relation to large-strain models. Acta Mech2024; 235: 5187–5211.
MindlinR.Micro-structure in linear elasticity. Archive Ration Mech Anal1964; 16: 51–78.
5.
FriedEGurtinM.Tractions, balances, and boundary conditions for nonsimple materials with application to liquid flow at small-length scales. Arch Ration Mech Anal2006; 182: 513–554.
NečasJNovotnýAŠilhavýM.Global solution to the ideal compressible heat conductive multipolar fluid. Comment Math Univ Carolinae1989; 30: 551–564.
8.
NečasJRůžičkaM. Global solution to the incompressible viscous-multipolar material problem. J Elast1992; 29: 175–202.
9.
RoubíčekT.Visco-elastodynamics at large strains Eulerian. Z Angew Math Phys2022; 73: 80.
10.
GallouëtTMalteseDNovotnýA.Error estimates for the implicit MAC scheme for the compressible Navier-Stokes equations. Numer Math2019; 141: 495–567.
11.
FeireislEHošekRMalteseD, et al. Error estimates for a numerical method for the compressible Navier–Stokes system on sufficiently smooth domains. ESAIM Math Modelling Numer Anal2017; 51: 279–319.
12.
FeireislEKarperTPokornýM.Mathematical theory of compressible viscous fluids. Cham: Birkhäuser, 2016.
13.
FeireislELukáčová-MedvidováMMizerováH, et al. Numerical analysis of compressible fluid flows. Cham: Springer, 2021.
14.
KarperT.A convergent FEM-DG method for the compressible Navier-Stokes equations. Numer Math2013; 125: 441–510.
15.
ZatorskaE.Analysis of semidiscretization of the compressible Navier-Stokes equations. J Math Anal Appl2012; 386: 559–580.
16.
IiSSugiyamaKTakeuchiS, et al. An implicit full Eulerian method for the fluid-structure interaction problem. Int J Numer Meth Fluids2011; 65: 150–165.
17.
NochettoRSalgadoATomasI.The micropolar Navier-Stokes equations: a priori error analysis. Math Models Meth Appl Sci2014; 24: 1237–1264.
18.
JaumannG.Geschlossenes system physikalischer und chemischer differential-gesetze. Sitzungsber Akad Wiss Wien (IIa)1911; 120: 385–530.
19.
ZarembaS.Sur une forme perfectionées de la théorie de la relaxation. Bull Int Acad Sci Cracovie1903; 594–614.
20.
BiotM.Mechanics of incremental deformation. New York: John Wiley, 1965.
MoresiLQuenetteSLemialeV, et al. Computational approaches to studying non-linear dynamics of the crust and mantle. Phys Earth Planet Inter2007; 163: 69–82.
27.
MuhlhausHBRegenauer-LiebK.Towards a self-consistent plate mantle model that includes elasticity: simple benchmarks and application to basic modes of convection. Geophys J Int2005; 163: 788–800.
28.
PatočkaVČadekOTackleyP, et al. Stress memory effect in viscoelastic stagnant lid convection. Geophys J Int2017; 209: 1462–1475.
29.
ThielmannMKausBPopovA.Lithospheric stresses in Rayleigh-Bénard convection: effects of a free surface and a viscoelastic Maxwell rheology. Geophys J Int2015; 203: 2200–2219.
30.
GeryaT.Introduction to numerial geodynamic modelling. 2nd ed.New York: Cambridge University Press, 2019.
31.
JiaoYFishJ.Is an additive decomposition of a rate of deformation and objective stress rates passé?Comput Methods Appl Mech Eng2017; 327: 196–225.
XiaoHBruhnsOMeyersA.Strain rates and material spins. J Elast1998; 52: 1–41.
34.
MeyersASchießePBruhnsO.Some comments on objective rates of symmetric Eulerian tensors with application to Eulerian strain rates. Acta Mech2000; 139: 91–103.
35.
GreenANaghdiP.A general theory of an elastic-plastic continuum. Arch Ration Mech Anal1965; 18: 251–281.
36.
RoubíčekT.Thermodynamics of viscoelastic solids, its Eulerian formulation, and existence of weak solutions. Z Angew Math Phys2024; 75: Art.no.51.
RoubíčekT.Nonlinear partial differential equations with applications. 2nd ed.Basel: Birkhäuser, 2013.
39.
ZeidlerE. Nonlinear functional analysis and its applications. I–IV . New York: Springer, 1985-90.
40.
HuSPapageorgiouN.Handbook of multivalued analysis. Vol. I. Theory. Dordrecht: Kluver, 1997.
41.
GurtinMFriedEAnandL.The mechanics and thermodynamics of continua. New York: Cambridge University Press, 2010.
42.
MartinecZ.Principles of continuum mechanics. Basel: Birkhäuser/Springer, 2019.
43.
RubinM.Continuum mechanics with Eulerian formulations of constitutive equations. Cham: Springer, 2021.
44.
DavoliERoubíčekTStefanelliU.A note about hardening-free viscoelastic models in Maxwellian-type rheologies. Math Mech Solids2021; 26: 1483–1497.
45.
RoubíčekT.Quasistatic hypoplasticity at large strains Eulerian. J Nonlin Sci2022; 32: 45.
46.
CharrierPDacorognaBHanouzetB, et al. An existence theorem for slightly compressible materials in nonlinear elasticity. SIAM J Math Anal1988; 19: 70–85.
47.
HartmannSNeffP.Polyconvexity of generalized polynomial-type hyperelastic strain energy functions for near-incompressibility. Intl J Solids Struct2003; 40: 2767–2791.
