Cohesive zone-type delamination in visco-elasticity.

*(English)*Zbl 1366.35190Summary: We study a model for the rate-independent evolution of cohesive zone delamination in a visco-elastic solid, also exposed to dynamics effects. The main feature of this model, inspired by M. Ortiz and A. Pandolfi [Int. J. Numer. Methods Eng. 44, No. 9, 1267–1282 (1999; Zbl 0932.74067)], is that the surface energy related to the crack opening depends on the history of the crack separation between the two sides of the crack path, and allows for different responses upon loading and unloading.

Due to the presence of multivalued and unbounded operators featuring non-penetration and the ‘memory’-constraint in the strong formulation of the problem, we prove existence of a weaker notion of solution, known as semistable energetic solution, pioneered in [T. Roubíček, Math. Methods Appl. Sci. 32, No. 7, 825–862 (2009; Zbl 1239.35158)] and refined in [R. Rossi and the first author, SIAM J. Math. Anal. 49, No. 2, 1419–1494 (2017; Zbl 1368.35009)].

Due to the presence of multivalued and unbounded operators featuring non-penetration and the ‘memory’-constraint in the strong formulation of the problem, we prove existence of a weaker notion of solution, known as semistable energetic solution, pioneered in [T. Roubíček, Math. Methods Appl. Sci. 32, No. 7, 825–862 (2009; Zbl 1239.35158)] and refined in [R. Rossi and the first author, SIAM J. Math. Anal. 49, No. 2, 1419–1494 (2017; Zbl 1368.35009)].

##### MSC:

35Q74 | PDEs in connection with mechanics of deformable solids |

35A15 | Variational methods applied to PDEs |

74H20 | Existence of solutions of dynamical problems in solid mechanics |

74C10 | Small-strain, rate-dependent theories of plasticity (including theories of viscoplasticity) |

49J53 | Set-valued and variational analysis |

49J45 | Methods involving semicontinuity and convergence; relaxation |

74C05 | Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials) |

##### Keywords:

cohesive zone delamination; weak formulation; rate-independent processes; semistable energetic solutions; non-smooth constraint; gradient systems; dynamics; irreversibility
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\textit{M. Thomas} and \textit{C. Zanini}, Discrete Contin. Dyn. Syst., Ser. S 10, No. 6, 1487--1517 (2017; Zbl 1366.35190)

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