Journal
ADVANCES IN CALCULUS OF VARIATIONS
Volume 16, Issue 4, Pages 903-934Publisher
WALTER DE GRUYTER GMBH
DOI: 10.1515/acv-2021-0073
Keywords
Inertial balanced viscosity solutions; inertial virtual viscosity solutions; rate-independent systems; vanishing inertia and viscosity limit; minimizing movements scheme; variational methods
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The notion of inertial balanced viscosity (IBV) solution is introduced to describe rate-independent evolutionary processes. It is proved that these solutions converge in finite dimension and can be obtained via a natural extension of the minimizing movements algorithm. In the case of a nontrivial kernel, the weaker notion of inertial virtual viscosity (IVV) solution is introduced, and the analogous convergence result holds.
The notion of inertial balanced viscosity (IBV) solution to rate-independent evolutionary processes is introduced. Such solutions are characterized by an energy balance where a suitable, rate-dependent, dissipation cost is optimized at jump times. The cost is reminiscent of the limit effect of small inertial terms. Therefore, this notion proves to be a suitable one to describe the asymptotic behavior of evolutions of mechanical systems with rate-independent dissipation in the limit of vanishing inertia and viscosity. It is indeed proved, in finite dimension, that these evolutions converge to IBV solutions. If the viscosity operator is neglected, or has a nontrivial kernel, the weaker notion of inertial virtual viscosity (IVV) solutions is introduced, and the analogous convergence result holds. Again in a finite-dimensional context, it is also shown that IBV and IVV solutions can be obtained via a natural extension of the minimizing movements algorithm, where the limit effect of inertial terms is taken into account.
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