The Variational Approach to Fracture

ISBN-13: 9781402063947 / Angielski / Twarda / 2008 / 164 str.

Blaise Bourdin; Gilles A. Francfort; Jean-Jacques Marigo

Opis

One of the goals of the Journal of Elasticity: The Physical and Ma- ematical Science of Solids is to identify and to bring to the attention of the research community in the physical and mathematical sciences extensive expositions which contain creative ideas, new approaches and currentdevelopmentsinmodellingthebehaviourofmaterials. Fracture has enjoyed a long and fruitful evolution in engineering, but only in - cent years has this area been considered seriously by the mathematical science community. In particular, while the age-old Gri?th criterion is inherently energy based, treating fracture strictly from the point of view of variational calculus using ideas of minimization and accounting for the singular nature of the fracture ?elds and the various ways that fracture can initiate, is relatively new and fresh. The variational theory of fracture is now in its formative stages of development and is far from complete, but several fundamental and important advances have been made. The energy-based approach described herein establishes a consistent groundwork setting in both theory and computation. While itisphysicallybased, thedevelopmentismathematicalinnatureandit carefully exposes the special considerations that logically arise rega- ing the very de?nition of a crack and the assignment of energy to its existence. The fundamental idea of brittle fracture due to Gri?th plays a major role in this development, as does the additional dissipative feature of cohesiveness at crack surfaces, as introduced by Barenblatt. Thefollowinginvited, expositoryarticlebyB. Bourdin, G. Francfort and J. -J. Marigo represents a masterful and extensive glimpse into the fundamentalvariationalstructureoffractu

Spis treści

1 Introduction; 2 Going variational; 2.1 Griffith’s theory; 2.2 The 1-homogeneous case – A variational equivalence; 2.3 Smoothness – The soft belly of Griffith’s formulation; 2.4 The non 1-homogeneous case – A discrete variational evolution; 2.5 Functional framework – A weak variational evolution; 2.6 Cohesiveness and the variational evolution; 3 Stationarity versus local or global minimality – A comparison; 3.1 1d traction; 3.1.1 The Griffith case – Soft device; 3.1.2 The Griffith case – Hard device; 3.1.3 Cohesive case – Soft device; 3.1.4 Cohesive case – Hard device; 3.2 A tearing experiment; 4 Initiation; 4.1 Initiation – The Griffith case; 4.1.1 Initiation – The Griffith case – Global minimality; 4.1.2 Initiation – The Griffith case – Local minimality; 4.2 Initiation – The cohesive case; 4.2.1 Initiation – The cohesive 1d case – Stationarity; 4.2.2 Initiation – The cohesive 3d case – Stationarity; 4.2.3 Initiation – The cohesive case – Global minimality; 5 Irreversibility; 5.1 Irreversibility – The Griffith case – Well-posedness of the variational evolution; 5.1.1 Irreversibility – The Griffith case – Discrete evolution; 5.1.2 Irreversibility – The Griffith case – Global minimality in the limit; 5.1.3 Irreversibility – The Griffith case – Energy balance in the limit; 5.1.4 Irreversibility – The Griffith case – The time-continuous evolution; 5.2 Irreversibility – The cohesive case; 6 Path; 7 Griffith vs. Barenblatt; 8 Numerics and Griffith; 8.1 Numerical approximation of the energy; 8.1.1 The first time step; 8.1.2 Quasi-static evolution; 8.2 Minimization algorithm; 8.2.1 The alternate minimization algorithm; 8.2.2 The backtracking algorithm; 8.3 Numerical experiments; 8.3.1 The 1D traction (hard device); 8.3.2 The Tearing experiment; 8.3.3 Revisiting the 2D traction experiment on a fiber reinforced matrix; 9 Fatigue; 9.1 Peeling Evolution; 9.2 The limitfatigue law when d tends to 0; 9.3 A variational formulation for fatigue; 9.3.1 Peeling revisited; 9.3.2 Generalization; Appendix; Glossary; References.