Separation of Scales in Fracture Mechanics: From Molecular to Continuum Theory via Γ Convergence
Publication: Journal of Engineering Mechanics
Volume 130, Issue 2
Abstract
We propose a procedure to obtain a consistent, mesh-objective, continuous model starting from chains composed of discrete springs exhibiting strain softening. Observing the size-dependent response of tensile chains and the corresponding scaling law, recent results for the variational convergence of discrete functionals (Γ convergence) are used to pass from a molecular to a continuum theory. The limit model, where softening and fracture are interpreted by the dichotomy of bulk and surface energies, reproduces the same overall properties of the discrete system. In particular, fracture energy does not vanish in the limit and the discrete approximations of the resulting continuum model are mesh objective.
Get full access to this article
View all available purchase options and get full access to this article.
References
Bažant, Z.(1976). “Instability, ductility and size-effect in strain-softening concrete.” J. Eng. Mech., 102(2), 331–344.
Bažant, Z., and Belytschko, T.(1985). “Wave propagation in strain softening bar: Exact solutions.” J. Eng. Mech., 111(3), 381–389.
Bažant, Z., and Cedolin, L. (1991). Stability of structures: Elastic, inelastic, fracture and damage theories, Oxford University Press, New York.
Bažant, Z., and Chen, E. P.(1997). “Scaling of structural failure.” Appl. Mech. Rev., 50(10), 593–627.
Bažant, Z. and Planas, J. (1998). Fracture and size effect in concrete and other quasi-brittle materials, CRC, New York.
Braides, A. (2002). Γ-convergence for beginners, Oxford University Press, Oxford, U.K.
Braides, A., Dal Maso, G., and Garroni, A.(1999). “Variational formulation of softening phenomena in fracture mechanics: The one-dimensional case.” Arch. Ration. Mech. Anal., 146, 23–58.
Braides, A., and Gelli, M. S.(2002). “Continuum limits of discrete systems without convexity hypotheses.” Math. Mech. Solids, 7, 41–66.
Choksi, R., Del Piero, G., Fonseca, I., and Owen, D.(1999). “Structured deformations as energy minimizers in models of fracture and hysteresis.” Math. Mech. Solids, 4(3), 321–356.
Del Piero, G. (1999). “One-dimensional ductile-brittle transition, yielding and structured deformations.” Variations de Domaines et Frontières Libres en Mécanique des Solides, P. Argoul, M. Frèmond, and Q. S. Nguyen, eds., Kluwer, Dordrecht, The Netherlands.
Del Piero, G., and Truskinovsky, L.(1998). “A one-dimensional model for localized and distributed fractures.” J. Phys. IV, 8, 95–102.
Del Piero, G., and Truskinovsky, L.(2001). “Macro- and micro-cracking in one dimensional elasticity.” Int. J. Solids Struct., 38, 1135–1148.
Dunn, J. E., and Fosdick, R.(1980). “The morphology and stability of material phases.” Arch. Ration. Mech. Anal., 74, 1–99.
Ericksen, J. L. (1983). “Ill-posed problems in thermoelasticity theory.” System of non-linear partial differential equations, J. M. Ball, ed., Kluwer, Dordrecht, The Netherlands 71–93.
Kolmogorov, A. N., and Fomin, S. V. (1975). Introductory real analysis, Dover, New York.
Meyer, C., and Okamura, H., eds. (1986). Finite element analysis of reinforced concrete structures, ASCE, New York.
Müller, I., and Villaggio, P.(1977). “A model for an elastic plastic body.” Arch. Ration. Mech. Anal., 102, 25–46.
Prigogine, I., and Kondepudi, D. (1998). Modern thermodynamics: From heat engines to dissipative structures, Wiley, Chichester, U.K.
Rogers, R. C., and Truskinovsky, L.(1997). “Discretization and hysteresis.” Physica B, 233, 370–375.
Truskinovsky, L. (1996). “Fracture as a phase transition.” Contemporary research in the mechanics and mathematics of materials, R. C. Batra and M. F. Beatty, CIMNE, Barcelona, Spain, 322–332.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 130 • Issue 2 • February 2004
Pages: 204 - 215
Copyright
Copyright © 2004 American Society of Civil Engineers.
History
Received: Jul 15, 2002
Accepted: Dec 4, 2002
Published online: Jan 16, 2004
Published in print: Feb 2004
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.