Quasistatic Viscoelasticity with Self-Contact at Large Strains
- PDF / 825,092 Bytes
- 13 Pages / 439.37 x 666.142 pts Page_size
- 86 Downloads / 181 Views
Quasistatic Viscoelasticity with Self-Contact at Large Strains Stefan Krömer1
· Tomáš Roubíˇcek2,3
Received: 3 April 2019 / Accepted: 8 July 2019 / Published online: 26 November 2020 © Springer Nature B.V. 2020
Abstract 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. Keywords Kelvin-Voigt material · Frame indifference · Non-selfinterpenetration · Implicit time discretisation · Lagrangian description · Pullback Mathematics Subject Classification 35K86 · 35Q74 · 74A30 · 74B20 · 74M15
1 Introduction Nonlinear elasticity and viscoelasticity is a vital part of the continuum mechanics of solids and still faces many open fundamental problems even after intensive scrutiny within past many decades. One of such problem is the possibility of non-physical self-interpenetration and analytically supported methods to prevent it. The problem is difficult because of an interaction of two configurations: the reference one (ultimately needed for analysis of problems in solid mechanics at large strains) and the actual one (ultimately need for determination the possible time-varying self-contact boundary region). So far, besides merely static situations, only rate-independent evolution of some internal variables based on (not always very realistic) concept of instantaneous global minimization and energetic solutions, possessing a good variational structure and thus allowing incorporation of the Ciarlet-Neˇcas condition [4], has been treated in [10]. In the viscoelasticity, one
B S. Krömer 1
Institute of Information Theory and Automation, Czech Acad. Sci., Pod vodárenskou vˇeží 4, 182 08 Praha 8, Czech Republic
2
Mathematical Institute, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic
3
Institute of Thermomechanics, Czech Acad. Sci., Dolejškova 5, 182 08 Praha 8, Czech Republic
434
S. Krömer, T. Roubíˇcek
cannot rely purely on a variational structure but should rather work in terms of partial differential equations. As emphasized in [8, 9], “the theory of viscoelasticity at finite strain is notoriously difficult and” that time it seemed “that the present mathematical tools are not sufficient to provide sufficiently strong solutions in the multidimensional, truly geometrically invariant case”. Since then, the quasistatic viscoelasticity has been treated in [11] and in the dynamical variant in [7, Sect. 9.3], but without globally ruling out self-interpenetration. In the case of self-contact, instead of differential equations, it is natural to describe static critical points by variational inequalities. This was developed for a purely static situation in [16] for non-simple materials involving a higher order term in
Data Loading...
