Well-posedness of the governing equations for a quasi-linear viscoelastic model with pressure-dependent moduli in which both stress and strain appear linearly

• Published in: Zeitschrift für angewandte Mathematik und Physik 2024-02, Vol.75 (1), Article 22
• Main Authors: Itou, Hiromichi, Kovtunenko, Victor A., Rajagopal, Kumbakonam R.
• Format: Article
• Language: English
• Subjects: Constitutive relationships; Engineering; Exact solutions; Mathematical analysis; Mathematical Methods in Physics; Pressure dependence; Theoretical and Applied Mechanics; Viscoelasticity; Well posed problems
• ISSN: 0044-2275; 1420-9039
• DOI: 10.1007/s00033-023-02160-0

The response of a body described by a quasi-linear viscoelastic constitutive relation, whose material moduli depend on the mechanical pressure (that is one-third the trace of stress) is studied. The constitutive relation stems from a class of implicit relations between the histories of the stress and the relative deformation gradient. A-priori thresholding is enforced through the pressure that ensures that the displacement gradient remains small. The resulting mixed variational problem consists of an evolutionary equation with the Volterra convolution operator; this equation is studied for well-posedness within the theory of maximal monotone graphs. For isotropic extension or compression, a semi-analytic solution of the quasi-linear viscoelastic problem is constructed under stress control. The equations are studied numerically with respect to monotone loading both with and without thresholding. In the example, the thresholding procedure ensures that the solution does not blow-up in finite time.