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arXiv · 2610.18466

Dynamic viscoelastic membranes

Martin Kružík·Riccardo Voso
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Abstract

We rigorously derive a dynamic membrane theory for nonlinear viscoelastic solids with Kelvin-Voigt rheology from the corresponding three-dimensional bulk theory, in the limit of vanishing thickness. The bulk model accounts for inertia, a nonsimple elastic energy depending on the second deformation gradient, and a frame-indifferent dissipation. Orientation preservation is ensured by the energy, while self-interpenetration of the limiting membrane is prevented by a uniform averaged invertibility condition. Our analysis relies on a hyperbolic minimizing movements scheme with two time-scales, which recasts the evolution into a hierarchy of incremental, sequential, and time-delayed problems. For fixed thickness, we prove the existence of solutions to the scheme and their convergence to a weak solution of the continuous-time problem as the time-scales vanish. We then identify the membrane limit at each level of the hierarchy, proving the $Γ$-convergence of the incremental functionals and the convergence of solutions to the sequential and time-delayed problems. Since the constant in the Korn inequality is not uniform in the thickness, these limits rely on estimates for the time derivative of the Cauchy-Green tensor rather than for the velocity gradient. Finally, as the outer time-scale vanishes, the membrane time-delayed solutions converge to a measure-valued solution of the membrane continuous-time problem.

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Martin Kružík, Riccardo Voso. 2026-10-13. Dynamic viscoelastic membranes. https://arxiv.org/abs/2610.18466

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