Abstract
To introduce the theory of structured deformations, a good starting point is perhaps to illustrate a situation in which such objects arise naturally, although unexpectedly, from a problem in fracture mechanics.1
Presently on leave at the Centro Linceo Interdisciplinare “Beniamino Segre”, Accademia Nazionale dei Lincei, Rome, Italy. This research was supported by the the Italian Ministry for University and Scientific Research through the Programma Cofinanziato 2002 Modelli matematici per la scienza dei materiali. I thank D.R. Owen for his accurate reading of the manuscript and for many valuable suggestions and comments.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
G. Alberti. A Lusin type theorem for gradients. Journal of Functional Analysis. 100: 110–118, 1991.
L. Ambrosio. Free discontinuity problems and special functions with bounded variation. Progress in Mathematics, 168: 15–35, 1998.
L. Ambrosio, N. Fusco and D. Pallara. Functions of Bounded Variation and Free Discontinuity Problems. Oxford Science Publications, 2000.
H. Cartan. Cours de calcul différentiel. Hermann, Paris 1967.
R. Choksi and I. Fonseca. Bulk and interfacial energy densities for structured deforma- tions of continua. Archive for Rational Mechanics and Analysis, 138: 37–103, 1997.
R. Choksi, G. Del Piero, I. Fonseca and D.R. Owen. Structured deformations as energy minimizers in models of fracture and hysteresis. Mathematics and Mechanics of Solids, 4: 321–356, 1999.
B. Dacorogna. Direct methods in the Calculus of Variations. Springer-Verlag, 1989.
G. Dal Maso and R. Toader. A model for the quasi-static growth of a brittle fracture: existence and approximation results. Archive for Rational Mechanics and Analysis, 162: 101–135, 2002.
G. Del Piero. A new function space for the mathematical theory of plasticity. In A.S. Khan and M. Tokuda, editors, Proceedings of the Second International Symposium on Plasticity and its Current Applications. Pergamon Press, 1989.
G. Del Piero. Towards a unified approach to fracture, yielding, and damage. In E. Ivan and K.Z. Markov, editors, Proceedings of the 9th International Symposium “Continuum Models and Discrete Systems”. World Scientific, Singapore, 1998.
G. Del Piero. The energy of a one-dimensional structured deformation. Mathematics and Mechanics of Solids, 6: 387–408, 2001.
G. Del Piero. A class of fit regions and a universe of shapes for continuum mechanics. Journal of Elasticity, 70: 175–195, 2003.
G. Del Piero and D.R. Owen. New concepts in the mechanics of fractured continua. Proceedings of the 10th National Congress AIMETA, Pisa 1990.
G. Del Piero and D.R. Owen. Structured deformations of continua. Archive for Rational Mechanics and Analysis, 124: 99–155, 1993.
G. Del Piero and D.R. Owen. Integral-gradient formulae for structured deformations. Archive for Rational Mechanics and Analysis, 131: 121–138, 1995.
G. Del Piero and D.R. Owen. Structured Deformations. Quaderni I.N.d.A.M., Gruppo Nazionale Fisica Matematica no. 58, Centro Stampa 2p, Firenze 2000.
G. Del Piero and L. Truskinovsky. Elastic bars with cohesive energy. In preparation.
G. Francfort and J.J. Marigo. Revisiting brittle fracture as an energy minimization problem. Journal of the Mechanics and Physics of Solids, 46: 1319–1342, 1998.
P.R. Halmos. Finite-Dimensional Vector Spaces. Springer-Verlag 1974.
A.N. Kolmogorov and S.V. Fomin. Introductory Real Analysis. Dover Publications, 1975.
W. Noll. Lectures on the foundations of Continuum Mechanics and Thermodynamics. Archive for Rational Mechanics and Analysis, 52: 62–92, 1973.
W. Noll, E.G. Virga. Fit regions and functions of bounded variation. Archive for Rational Mechanics and Analysis, 102: 1–21, 1988.
D.R. Owen. Disarrangements in continua and the geometry of microstructure. In K. Rajagopal, editor, Recent Advances in Elasticity, Viscoelasticity, and Inelasticity, pp. 67–81. World Scientific, 1995.
C. Truesdell. A First Course in Rational Continuum Mechanics, Vol.1. Academic Press, 1991.
A.I. Vol’pert and S.I. Hudjaev. Analysis of Classes of Discontinuous Functions and Equations of Mathematical Physics. Nijhoff, 1985.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Wien
About this chapter
Cite this chapter
Del Piero, G. (2004). Foundations of the Theory of Structured Deformations. In: Del Piero, G., Owen, D.R. (eds) Multiscale Modeling in Continuum Mechanics and Structured Deformations. International Centre for Mechanical Sciences, vol 447. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2770-4_4
Download citation
DOI: https://doi.org/10.1007/978-3-7091-2770-4_4
Publisher Name: Springer, Vienna
Print ISBN: 978-3-211-22425-0
Online ISBN: 978-3-7091-2770-4
eBook Packages: Springer Book Archive