Abstract
We study the time evolution of non-linear viscoelastic solids in the presence of inertia and (self-)contact. For this problem we prove the existence of weak solutions for arbitrary times and initial data, thereby solving an open problem in the field. Our construction directly includes the physically correct, measure-valued contact forces and thus obeys conservation of momentum and an energy balance. In particular, we prove an independently useful compactness result for contact forces.
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Notes
In the absence of rigid bodies and point-masses this turns out to also be a sufficient assumption. For more details see the discussion in Sect. 6.
As a necessary step towards the proof, we also improve their result to give a quantization of the contact force as a measure, which in [13] was only characterized as part of a distribution. We thus in fact solve both of their open problems (See Remark 5.1). In particular, we believe that the more detailed treatment of convergence of contact forces used for this might be of independent interest.
To reduce the distinction between cases, it is best to not think of \(\Omega \) as the domain, but of \(\mathbb {R}^n \setminus \Omega \) as a fixed, rigid obstacle. Thus \(n_\Omega \) is the interior normal of that obstacle, in the same way \(n_\eta \) is the interior normal of the movable solids.
Note that this implies that (3.5) characterizes the interior of \(T_{\eta }(\mathcal {E})\). Additionally, for a sufficiently regular \(\eta \), it is not hard to prove that (iv) is in fact equivalent to the other three conditions. However, this proof involves splitting \(\varphi \) into its precise normal and tangential components and is thus not easily transferred to the general situation.
As before, \(\eta ^{-1}\) denotes the preimage of \(\eta (t,\cdot )\).
To avoid confusion, we note that since we use interior normals, some signs and inequalities have to be turned around.
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Acknowledgements
The authors acknowledge the support of the the Primus research programme of Charles University under Grant No. PRIMUS/19/SCI/01. The research of A.Č. and M.K. was partly funded by the ERC-CZ Grant LL2105. A.Č. further acknowledges the support of Charles University, project GA UK No. 393421. The work of G.G. and M.K. was partially supported by the Charles University research program No. UNCE/SCI/023 and by the Czech Science Foundation (GAČR) under Grant No. GJ19-11707Y.
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Communicated by Andrea Mondino.
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Češík, A., Gravina, G. & Kampschulte, M. Inertial evolution of non-linear viscoelastic solids in the face of (self-)collision. Calc. Var. 63, 55 (2024). https://doi.org/10.1007/s00526-023-02648-7
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DOI: https://doi.org/10.1007/s00526-023-02648-7