Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fractures

By Alessandro Giacomini. 2003

164 citations recorded by the source corpus.

Abstract.We define a notion of quasistatic evolution for the elliptic approximation of the Mumford-Shah functional proposed by Ambrosio and Tortorelli. Then we prove that this regular evolution converges to a quasi static growth of brittle fractures in linearly elastic bodies.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/613f34ff9aba52fdab7dc5e0eebd87d4e4892608

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In Applied Mathematics

In Computational Theory and Mathematics

In Mechanics of Materials

Citations in the corpus

Experimental validation of a phase-field model for fracture (2017, methodology): “In order to prevent the crack from healing, additional equality (Bourdin 1999) or inequality (Giacomini 2005) constraints on the phase field evolution have been used.”
Gradient Damage Models and Their Use to Approximate Brittle Fracture (2011, unknown): “The same is true for the corresponding quasi-static evolutions ruled by a global minimality principle (Giacomini, 2005).”
An overview of the modelling of fracture by gradient damage models (2016, unknown): “The same is true for the corresponding quasi-static evolutions ruled by a global minimality principle [22].”
The Variational Approach to Fracture (2008, background): “In [57], crack growth is enforced through the monotonicity of v in time, i.”
Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments (2009, background): “Moreover, Giacomini (2005) proves that the time-discrete quasi-discrete evolution obtained by minimizing the regularized functional under the irreversibility condition on a converges to the quasi-static evolution of the brittle fracture model of Francfort and Marigo (1998).”
A time-discrete model for dynamic fracture based on crack regularization (2011, background): “In our approach, the energy corresponding to the phase-field function, v, is minimized at each discrete time (subject to irreversibility constraints as in [7, 18]), while the displacement u follows dynamics, affected by v only on a small set modeling the crack, where u is allowed to be discontinuous”
An adaptive finite element approximation of a generalized Ambrosio-Tortorelli functional (2013, unknown): “In this paper we consider the minimization of a generalised functional together with a new method for implementing crack irreversibility, which is based on a monotonicity condition specified by Giacomini [27].”
Crack nucleation in variational phase-field models of brittle fracture (2018, unknown): “…a broad class of energies, including the ones above was achieved with various degrees of reﬁnement going from static scalar elasticity to time discrete and time continuous quasi-static evolution linearized elasticity, and their ﬁnite element discretization [12, 16, 23, 46, 27, 28, 45, 25, 26, 56].”
A higher-order phase-field model for brittle fracture: Formulation and analysis within the isogeometric analysis framework (2014, methodology): “Although it has been used in numerous numerical implementations and theoretical papers (see Bellettini and Coscia [4], Bourdin [7], Bourdin, Francfort, and Marigo [9], Giacomini [20], Bourdin [8], Del Piero, Lancioni, and March [17], Lancioni and Royer-Carfagni [25], Amor, Marigo, and Maurini [2], B”
EXISTENCE OF SOLUTIONS TO A REGULARIZED MODEL OF DYNAMIC FRACTURE (2010, methodology): “An analysis of this approximation in the quasi-static setting is provided by Giacomini[ 13 ].”