Results 1  10
of
15
QuasiStatic Crack Propagation by Griffith’s Criterion
, 2008
"... We consider the propagation of a crack in a brittle material along a prescribed crack path and define a quasistatic evolution by means of stationary points of the free energy. We show that this evolution satisfies Griffith’s criterion in a suitable form which takes into account both stable and unst ..."
Abstract

Cited by 12 (1 self)
 Add to MetaCart
We consider the propagation of a crack in a brittle material along a prescribed crack path and define a quasistatic evolution by means of stationary points of the free energy. We show that this evolution satisfies Griffith’s criterion in a suitable form which takes into account both stable and unstable propagation, as well as an energy balance formula which accounts for dissipation in the unstable regime. If the load is monotonically increasing this solution is explicit and almost everywhere unique. For more general loads we construct a solution via time discretization. Finally, we consider a finite element discretization of the problem and prove convergence of the discrete solutions.
BV SOLUTIONS AND VISCOSITY APPROXIMATIONS OF RATEINDEPENDENT SYSTEMS
"... Abstract. In the nonconvex case solutions of rateindependent systems may develop jumps as afunctionoftime. Tomodelsuchjumps,weadoptthephilosophythatrateindependenceshould be considered as limit of systems with smaller and smaller viscosity. For the finitedimensional case we study the vanishingvis ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
Abstract. In the nonconvex case solutions of rateindependent systems may develop jumps as afunctionoftime. Tomodelsuchjumps,weadoptthephilosophythatrateindependenceshould be considered as limit of systems with smaller and smaller viscosity. For the finitedimensional case we study the vanishingviscosity limit of doubly nonlinear equations given in terms of a differentiable energy functional and a dissipation potential which is a viscous regularization of agivenrateindependentdissipationpotential. The resulting definition of ‘BV solutions ’ involves, in a nontrivial way, both the rateindependent and the viscous dissipation potential, which play a crucial role in the description of the associated jump trajectories. We shall prove a general convergence result for the timecontinuous and for the timediscretized viscous approximations and establish various properties of the limiting BV solutions. In particular, we shall provide a careful description of the jumps and compare the new notion of solutions with the related concepts of energetic and local solutions to rateindependent systems. AMS Subject Classification: 49Q20,58E99. 1.
Crack growth in polyconvex materials
, 2008
"... We discuss a model for crack propagation in an elastic body, where the crack path is described apriori. In particular, we develop in the framework of finitestrain elasticity a rateindependent model for crack evolution which is based on the Griffith fracture criterion. Due to the nonuniqueness of ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
We discuss a model for crack propagation in an elastic body, where the crack path is described apriori. In particular, we develop in the framework of finitestrain elasticity a rateindependent model for crack evolution which is based on the Griffith fracture criterion. Due to the nonuniqueness of minimizing deformations, the energyrelease rate is no longer continuous with respect to time and the position of the crack tip. Thus, the model is formulated in terms of the Clarke differential of the energy, generalizing the classical crack evolution models for elasticity with strictly convex energies. We prove the existence of solutions for our model and also the existence of special solutions, where only certain extremal points of the Clarke differential are allowed.
ENERGY RELEASE RATE AND STRESS INTENSITY FACTOR IN ANTIPLANE ELASTICITY
"... Abstract. In the setting of antiplane linearized elasticity, we show the existence of the stress intensity factor and its relation with the energy release rate when the crack path is a C 1,1 curve. Finally, we show that the energy release rate is continuous with respect to the Hausdorff convergence ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
Abstract. In the setting of antiplane linearized elasticity, we show the existence of the stress intensity factor and its relation with the energy release rate when the crack path is a C 1,1 curve. Finally, we show that the energy release rate is continuous with respect to the Hausdorff convergence in a class of admissible cracks.
INTERNATIONAL SCHOOL FOR ADVANCED STUDIES Functional Analysis and Applications Sector Curriculum in Applied Mathematics
"... Quasistaticevolution problems with nonconvex energies: ..."
From ratedependent to quasistatic brittle crack propagation
, 2009
"... Abstract. On the base of many experimental results, e.g. [18], [19], [21], [12], the object of our analysis is a ratedependent model for the propagation of a crack in brittle materials. Our goal is a mathematical study of the evolution equation in the geometries of the ’Single Edge Notch Tension’ ( ..."
Abstract
 Add to MetaCart
Abstract. On the base of many experimental results, e.g. [18], [19], [21], [12], the object of our analysis is a ratedependent model for the propagation of a crack in brittle materials. Our goal is a mathematical study of the evolution equation in the geometries of the ’Single Edge Notch Tension’ (SENT) and of the ’Compact Tension ’ (ASTMCT). Besides existence and uniqueness, emphasis is placed on the regularity of the evolution making reference to the ’velocity gap’. The transition to the quasistatic regime of Griffith’s model is obtained by time rescaling, proving convergence of the rescaled evolutions and of their energies. Further, the discontinuities of the quasistatic propagation are characterized in terms of unstable branches of evolution in real time frame. The results are illustrated by a couple of numerical examples in the above mentioned geometries. AMS Subject Classification. 74R10 1
Digital Object Identifier (DOI) 10.1007/s0020501104609 From Discrete ViscoElasticity to Continuum RateIndependent Plasticity: Rigorous Results
"... We show that continuum models for ideal plasticity can be obtained as a rigorous mathematical limit starting from a discrete microscopic model describing a viscoelastic crystal lattice with quenched disorder. The constitutive structure changes as a result of two concurrent limiting procedures: the v ..."
Abstract
 Add to MetaCart
We show that continuum models for ideal plasticity can be obtained as a rigorous mathematical limit starting from a discrete microscopic model describing a viscoelastic crystal lattice with quenched disorder. The constitutive structure changes as a result of two concurrent limiting procedures: the vanishingviscosity limit and the discretetocontinuum limit. In the course of these limits a nonconvex elastic problem transforms into a convex elastic problem while the quadratic ratedependent dissipation of viscoelastic lattice transforms into a singular rateindependent dissipation of an ideally plastic solid. In order to emphasize our ideas we employ in our proofs the simplest prototypical system mimicking the phenomenology of transformational plasticity in shapememory alloys. The approach, however, is sufficiently general that it can be used for similar reductions in the cases of more general plasticity and damage models. 1.
pp. X–XX LOCAL MINIMALITY AND CRACK PREDICTION IN QUASISTATIC GRIFFITH FRACTURE EVOLUTION
"... Abstract. The mathematical analysis developed for energy minimizing fracture evolutions has been difficult to extend to locally minimizing evolutions. The reasons for this difficulty are not obvious, and our goal in this paper is to describe in some detail what precisely the issues are and why the p ..."
Abstract
 Add to MetaCart
Abstract. The mathematical analysis developed for energy minimizing fracture evolutions has been difficult to extend to locally minimizing evolutions. The reasons for this difficulty are not obvious, and our goal in this paper is to describe in some detail what precisely the issues are and why the previous analysis in fact cannot be extended to the most natural models based on local minimality. We also indicate how the previous methods can be modified for the analysis of models based on a recent definition of stability that is a bit stronger than local minimality.