Quasi-static rate-independent evolutions: characterization, existence, approximation and application to fracture mechanics

Matteo Negri

ESAIM: Control, Optimisation and Calculus of Variations (2014)

  • Volume: 20, Issue: 4, page 983-1008
  • ISSN: 1292-8119

Abstract

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We characterize quasi-static rate-independent evolutions, by means of their graph parametrization, in terms of a couple of equations: the first gives stationarity while the second provides the energy balance. An abstract existence result is given for functionals ℱ of class C1 in reflexive separable Banach spaces. We provide a couple of constructive proofs of existence which share common features with the theory of minimizing movements for gradient flows. Moreover, considering a sequence of functionals ℱn and its Γ-limit ℱ we provide, under suitable assumptions, a convergence result for the associated quasi-static evolutions. Finally, we apply this approach to a phase field model in brittle fracture.

How to cite

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Negri, Matteo. "Quasi-static rate-independent evolutions: characterization, existence, approximation and application to fracture mechanics." ESAIM: Control, Optimisation and Calculus of Variations 20.4 (2014): 983-1008. <http://eudml.org/doc/272770>.

@article{Negri2014,
abstract = {We characterize quasi-static rate-independent evolutions, by means of their graph parametrization, in terms of a couple of equations: the first gives stationarity while the second provides the energy balance. An abstract existence result is given for functionals ℱ of class C1 in reflexive separable Banach spaces. We provide a couple of constructive proofs of existence which share common features with the theory of minimizing movements for gradient flows. Moreover, considering a sequence of functionals ℱn and its Γ-limit ℱ we provide, under suitable assumptions, a convergence result for the associated quasi-static evolutions. Finally, we apply this approach to a phase field model in brittle fracture.},
author = {Negri, Matteo},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {quasi-static evolutions; phase-field; quasi-static rate-independent evolutions; graph parametrization; reflexive separable Banach spaces; -limit; brittle fracture},
language = {eng},
number = {4},
pages = {983-1008},
publisher = {EDP-Sciences},
title = {Quasi-static rate-independent evolutions: characterization, existence, approximation and application to fracture mechanics},
url = {http://eudml.org/doc/272770},
volume = {20},
year = {2014},
}

TY - JOUR
AU - Negri, Matteo
TI - Quasi-static rate-independent evolutions: characterization, existence, approximation and application to fracture mechanics
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2014
PB - EDP-Sciences
VL - 20
IS - 4
SP - 983
EP - 1008
AB - We characterize quasi-static rate-independent evolutions, by means of their graph parametrization, in terms of a couple of equations: the first gives stationarity while the second provides the energy balance. An abstract existence result is given for functionals ℱ of class C1 in reflexive separable Banach spaces. We provide a couple of constructive proofs of existence which share common features with the theory of minimizing movements for gradient flows. Moreover, considering a sequence of functionals ℱn and its Γ-limit ℱ we provide, under suitable assumptions, a convergence result for the associated quasi-static evolutions. Finally, we apply this approach to a phase field model in brittle fracture.
LA - eng
KW - quasi-static evolutions; phase-field; quasi-static rate-independent evolutions; graph parametrization; reflexive separable Banach spaces; -limit; brittle fracture
UR - http://eudml.org/doc/272770
ER -

References

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