The frame-indifferent viscoelasticity in Kelvin-Voigt rheology at large strains is formulated in the reference configuration (i.e., using the Lagrangian approach) considering also the possible self-contact in the actual deformed configuration. Using the concept of 2nd-grade nonsimple materials, existence of certain weak solutions which are a.e. injective is shown by converging an approximate solution obtained by the implicit time discretisation.
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Antman, S.S.: Physically unacceptable viscous stresses. Z. Angew. Math. Phys. 49, 980–988 (1998)
Antman, S.S.: Nonlinear Problems of Elasticity, 2nd edn. Springer, New York (2005)
Batra, R.C.: Thermodynamics of non-simple elastic materials. J. Elast. 6, 451–456 (1976)
Ciarlet, P.G., Nečas, J.: Injectivity and self-contact in nonlinear elasticity. Arch. Ration. Mech. Anal. 97, 171–188 (1987)
Fried, E., Gurtin, M.E.: Tractions, balances, and boundary conditions for nonsimple materials with application to liquid flow at small-length scales. Arch. Ration. Mech. Anal. 182, 513–554 (2006)
Healey, T.J., Krömer, S.: Injective weak solutions in second-gradient nonlinear elasticity. ESAIM Control Optim. Calc. Var. 15, 863–871 (2009)
Kružík, M., Roubíček, T.: Mathematical Methods in Continuum Mechanics of Solids. Springer, Switzerland (2019)
Mielke, A., Ortner, C., Sengül, Y.: An approach to nonlinear viscoelasticity via metric gradient flows. SIAM J. Math. Anal. 46, 1317–1347 (2013)
Mielke, A., Rossi, R., Savaré, G.: Global existence results for viscoplasticity at finite strain. Arch. Ration. Mech. Anal. 227, 423–475 (2018)
Mielke, A., Roubíček, T.: Rate-Independent Systems – Theory and Application. Springer, New York (2015)
Mielke, A., Roubíček, T.: Thermoviscoelasticity in Kelvin-Voigt rheology at large strains. Arch. Ration. Mech. Anal. 238, 1–45 (2020).
Mielke, A., Roubíček, T.: Rate-independent elastoplasticity at finite strains and its numerical approximation. Math. Models Methods Appl. Sci. 26, 2203–2236 (2016)
Neff, P.: On Korn’s first inequality with non-constant coefficients. Proc. R. Soc. Edinb. 132A, 221–243 (2002)
Neff, P.: Finite multiplicative plasticity for small elastic strains with linear balance equations and grain boundary relaxation. Contin. Mech. Thermodyn. 15, 161–195 (2003)
Neff, P.: Local existence and uniqueness for quasistatic finite plasticity with grain boundary relaxation. Q. Appl. Math. 63, 88–116 (2005)
Palmer, A.Z., Healey, T.J.: Injectivity and self-contact in second-gradient nonlinear elasticity. Calc. Var. 56, 114 (2017)
Podio-Guidugli, P.: Contact interactions, stress, and material symmetry, for nonsimple elastic materials. Theor. Appl. Mech. 28–29, 261–276 (2002)
Pompe, W.: Korn’s First Inequality with variable coefficients and its generalization. Comment. Math. Univ. Carol. 44, 57–70 (2003)
Roubíček, T.: Nonlinear Partial Differential Equations with Applications, 2nd edn. Birkhäuser, Basel (2013)
Schuricht, F.: Variational approach to contact problems in nonlinear elasticity. Calc. Var. 15, 433–449 (2002)
Šilhavý, M.: Phase transitions in non-simple bodies. Arch. Ration. Mech. Anal. 88, 135–161 (1985)
Toupin, R.A.: Elastic materials with couple stresses. Arch. Ration. Mech. Anal. 11, 385–414 (1962)
Toupin, R.A.: Theories of elasticity with couple stress. Arch. Ration. Mech. Anal. 17, 85–112 (1964)
Triantafyllidis, N., Aifantis, E.C.: A gradient approach to localization of deformation. I. Hyperelastic materials. J. Elast. 16, 225–237 (1986)
Tvedt, B.: Quasilinear equations of viscoelasticity of strain-rate type. Arch. Ration. Mech. Anal. 189, 237–281 (2008)
This research has been supported by the Czech Science Foundation through the grants 17-04301S (in particular concerning dissipative evolutionary systems), 19-29646L (especially pertaining large strains in materials science) and 19-04956S (in particular concerning nonlinear behavior of structures). Also the institutional support RVO:61388998 is acknowledged.
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Krömer, S., Roubíček, T. Quasistatic Viscoelasticity with Self-Contact at Large Strains. J Elast 142, 433–445 (2020). https://doi.org/10.1007/s10659-020-09801-9
- Kelvin-Voigt material
- Frame indifference
- Implicit time discretisation
- Lagrangian description
Mathematics Subject Classification