A Priori Inequalities between Energy Release Rate and Energy Concentration for 3D Quasistatic Brittle Fracture Propagation

Abstract

We study the properties of absolute minimal and equilibrium states of generalized Mumford—Shah functionals, with applications to models of quasistatic brittle fracture propagation. The main results, theorems 7.3, 8.4 and 9.1, concern a priori inequalities between energy release rate and energy concentration for 3D cracks with complex shapes, seen as outer measures living on the crack edge.

Keywords

3D brittle fracture energy methods Mumford—Shah functional

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References

Mumford, D. , and Shah, J. Optimal approximation by piecewise smooth functions and associated variational problems. Communications on Pure and Applied Mathematics, XLII, 5 ( 1989).

De Giorgi, Ennio ; Ambrosio, Luigi , New functionals in the calculus of variations . Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 82 ( 1988), no. 2, 199-210 (1989)

Ambrosio, L. Variational problems in SBV and image segmentation. Acta Applicandae Mathematicæ, 17, 1-40 (1989).

Ambrosio, L. Existence theory for a new class of variational problems. Archive for Rational Mechanics and Analysis, 111, 291-322 ( 1990).

Francfort, G. , and Marigo, J.-J. Stable damage evolution in a brittle continuous medium. The European Journal of Mechanics A/Solids, 12, 149-189 (1993).

Francfort, G. , and Marigo, J.-J. Revisiting brittle fracture as an energy minimization problem. Journal of the Mechanics and Physics of Solids, 46, 1319-1342 (1998).

Mielke, A. Evolution in rate-independent systems (Ch. 6). In Dafermos, C , and Feireisl, E. (eds.) Handbook of Differential Equations, Evolutionary Equations, Vol. 2, pp. 461-559, Elsevier B.V., Amsterdam, 2005.

Dal Maso, G. , Francfort, G. , and Toader, R. Quasistatic crack growth in nonlinear elasticity . Archive for Rational Mechanics and Analysis, 176, 165-225 (2005).

Buliga, M. Variational Formulations in Brittle Fracture Mechanics, PhD Thesis, Institute of Mathematics of the Romanian Academy, 1997

10.

Buliga, M. Energy concentration and brittle crack propagation . Journal of Elasticity, 52(3), 201-238 (1999).

11.

Buliga, M. Brittle crack propagation based on an optimal energy balance. Revue Roumaine de Mathématique Pures et Appliquées, 45(2), 201-209 ( 2001).

12.

Bellettini, G. , Coscia, A. , and Dal Maso, G. Compactness and lower semicontinuity properties in SBD(Ω). Mathematische Zeitschrift, 228(2), 337-351 (1998).