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A coupled rate-dependent/rate-independent system for adhesive contact in Kirchhoff-Love plates

Giovanna Bonfanti, Elisa Davoli, Riccarda Rossi

TL;DR

The paper develops a dimension-reduction analysis for a coupled rate-dependent/rate-independent adhesive-contact model in visco-elastodynamic Kirchhoff-Love plates, introducing $SE$ solutions as a robust weak framework. It proves the existence of Balanced $SE$ solutions in 3D and analyzes two limiting regimes: (i) vanishing viscosity and thickness leading to a purely rate-independent elastic plate, and (ii) retained damping with vanishing thickness yielding a viscoelastic plate with a nontrivially coupled in-plane/out-of-plane adhesive contact. In the first limit, the plate behaves elastically with a reduced in-plane elasticity tensor; in the second, the limit retains viscoelastic effects and adhesion coupling, but the energy-balance property is not guaranteed. The analysis employs space-time rescalings, compactness arguments, recovery-sequence constructions, and a reduced operator framework to handle coupling across dimensions, providing a rigorous bridge between 3D visco-elastodynamics and Kirchhoff-Love plate models with adhesion.

Abstract

We perform a dimension reduction analysis for a coupled rate-dependent/rate-independent adhesive-contact model in the setting of visco-elastodynamic plates. We work with a weak solvability notion inspired by the theory of (purely) rate-independent processes, and accordingly term the related solutions `Semistable Energetic'. For Semistable Energetic solutions, the momentum balance holds in a variational sense, whereas the flow rule for the adhesion parameter is replaced by a semi-stability condition coupled with an energy-dissipation inequality. Prior to addressing the dimension reduction analysis, we show that Semistable Energetic solutions to the three-dimensional damped adhesive contact model converge, as the viscosity term tends to zero, to three-dimensional Semistable Energetic solutions for the undamped corresponding system. We then perform a dimension reduction analysis, both in the case of a vanishing viscosity tensor, and in the complementary setting in which the damping is assumed to go to infinity as the thickness of the plate tends to zero. In both regimes, the presence of adhesive contact yields a nontrivial coupling of the in-plane and out-of-plane contributions. In the vanishing-viscosity case we additionally confine the analysis to the case in which also inertia is neglected: in the vanishing-thickness limit we thus obtain purely rate-independent evolution for the adhesive contact phenomenon, still formulated in terms of the Semistable Energetic solution concept. In the second, undamped scenario, inertia is instead encompassed, thus the limiting evolution retains a mixed rate-dependent/rate-independent character, and is again given in terms of an energy-dissipation inequality and a semistability condition.

A coupled rate-dependent/rate-independent system for adhesive contact in Kirchhoff-Love plates

TL;DR

The paper develops a dimension-reduction analysis for a coupled rate-dependent/rate-independent adhesive-contact model in visco-elastodynamic Kirchhoff-Love plates, introducing solutions as a robust weak framework. It proves the existence of Balanced solutions in 3D and analyzes two limiting regimes: (i) vanishing viscosity and thickness leading to a purely rate-independent elastic plate, and (ii) retained damping with vanishing thickness yielding a viscoelastic plate with a nontrivially coupled in-plane/out-of-plane adhesive contact. In the first limit, the plate behaves elastically with a reduced in-plane elasticity tensor; in the second, the limit retains viscoelastic effects and adhesion coupling, but the energy-balance property is not guaranteed. The analysis employs space-time rescalings, compactness arguments, recovery-sequence constructions, and a reduced operator framework to handle coupling across dimensions, providing a rigorous bridge between 3D visco-elastodynamics and Kirchhoff-Love plate models with adhesion.

Abstract

We perform a dimension reduction analysis for a coupled rate-dependent/rate-independent adhesive-contact model in the setting of visco-elastodynamic plates. We work with a weak solvability notion inspired by the theory of (purely) rate-independent processes, and accordingly term the related solutions `Semistable Energetic'. For Semistable Energetic solutions, the momentum balance holds in a variational sense, whereas the flow rule for the adhesion parameter is replaced by a semi-stability condition coupled with an energy-dissipation inequality. Prior to addressing the dimension reduction analysis, we show that Semistable Energetic solutions to the three-dimensional damped adhesive contact model converge, as the viscosity term tends to zero, to three-dimensional Semistable Energetic solutions for the undamped corresponding system. We then perform a dimension reduction analysis, both in the case of a vanishing viscosity tensor, and in the complementary setting in which the damping is assumed to go to infinity as the thickness of the plate tends to zero. In both regimes, the presence of adhesive contact yields a nontrivial coupling of the in-plane and out-of-plane contributions. In the vanishing-viscosity case we additionally confine the analysis to the case in which also inertia is neglected: in the vanishing-thickness limit we thus obtain purely rate-independent evolution for the adhesive contact phenomenon, still formulated in terms of the Semistable Energetic solution concept. In the second, undamped scenario, inertia is instead encompassed, thus the limiting evolution retains a mixed rate-dependent/rate-independent character, and is again given in terms of an energy-dissipation inequality and a semistability condition.
Paper Structure (21 sections, 13 theorems, 242 equations, 1 figure)

This paper contains 21 sections, 13 theorems, 242 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Semistable Energetic solutions for the damped and undamped 3D adhesive contact systems
  3. Setup.
  4. $\mathrm{SE}$ solutions for the damped adhesive contact system
  5. Dissipation potentials and driving energy functional for the damped system.
  6. From the damped to the undamped system
  7. The dimension reduction
  8. Geometric setup.
  9. Energetic setup for the adhesive systems.
  10. The rescaled problems
  11. Adhesive contact system for $(\mathsf{u}^{\varepsilon},\mathsf{z}^{\varepsilon})$.
  12. $\vartriangleright$ Semistability condition \ref{['reduced-semistab-eps']}:
  13. $\vartriangleright$ Energy-dissipation balance \ref{['EDB-eps']}:
  14. Our dimension reduction results
  15. Our first convergence result: removing the damping
  16. ...and 6 more sections

Key Result

Theorem 2.12

RosThoBRI-INERTIA Assume ass-domain and assdata. Let $\varrho\geq 0$. Then, there exist Balanced $\mathrm{SE}$ solutions to the damped adhesive contact system satisfying the initial conditions where the initial data $({u}_0 ,\dot{u}_0 ,z_0)$ fulfill the semistability condition semistab-z at $t=0$, i.e.

Figures (1)

  • Figure 1: The set $\Omega=\Omega_+\cup \Gamma_{\!{\rm C}}\cup \Omega_-$

Theorems & Definitions (46)

  • Remark 2.2
  • Remark 2.3: Square and square root of fourth order tensors
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6: Semistable Energetic solution
  • Definition 2.7: Balanced Semistable Energetic solution
  • Remark 2.8: Time-dependent Dirichlet conditions
  • Remark 2.9: Reformulation of the semistability condition
  • Remark 2.10: 'Explicit' energy-dissipation balance
  • Remark 2.11: Rewriting the work of the external forces
  • ...and 36 more