This paper reviews recent theories of nonclassical, structured deformations and integral representations for their Helmholtz free energy. Energy minimizers for a body undergoing shearing at two different length scales and for a bar experiencing both smooth extension and macroscopic fractures are determined, and applications to the shearing of single crystals and to the cohesive fracture of solids are discussed. Yield, hysteresis, and the associated dissipation in two-level shears are shown to arise from instabilities at the micro level, and the dichotomy between brittle and ductile fracture is related precisely to a critical length of a bar.
Get full access to this article
View all access options for this article.
References
1.
[1] Del Piero, G. and Owen, D. R.: Structured deformations of continua. Arch. Rat. Mech. Anal., 124, 99-155 (1993).
2.
[2] Del Piero, G. and Owen, D. R.: Integral-gradient formulae for structured deformations. Arch. Rat. Mech. Anal., 131, 121-138 (1995).
3.
[3] Owen, D. R.: Disarrangements in continua and the geometry of microstructure, in Recent Advances in Elasticity, lscoelasticity, and Inelasticity, pp. 67-81, ed., K. Rajagopal, World Scientific, Singapore, 1995.
4.
[4] Choksi, R. and Fonseca, I.: Bulk and interfacial energy densities for structured deformations of continua. Arch. Rat. Mech. Anal., 138, 37-103 (1997).
5.
[5] Barenblatt, G. I.: The formation of equilibrium cracks during brittle fracture: General ideas and hypotheses. Axially symmetric cracks. Appl. Math. Mech. (PMM), 23, 622-636 (1959).
6.
[6] Noll, W. and Virga, E.: Fit regions and functions of bounded variation. Arch. Rat. Mech. Anal., 102, 1-22 (1988).
7.
[7] Evans, L. C. and Gariepy, R. F.: Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992.
8.
[8] Ziemer, W. P.: Weakly Differentiable Functions, Springer, Berlin, 1989.
9.
[9] De Giorgi, E. and Ambrosio, L.: Un nuovo tipo di funzionale del calcolo delle variazioni. Atti Accademia Nazionale Lincei, 82, 199-210 (1988).
10.
[10] Alberti, G.: A Lusin type theorem for gradients. J. Funct. Anal., 100, 110-118 (1991).
11.
[11] Ambrosio, L.: Compactness for a special case of functions of bounded variation. Bollettino Unione Matematica Italiana, 3-B7, 857-881 (1989).
12.
[12] Fonseca, I. and Muller, S.: Relaxation of quasiconvex functionals in BV (Ω; Rp) for integrands f (x, u, ▿ u). Arch. Rat. Mech. Anal., 123, 1-49 (1993).
13.
[13] Nakanishi, F.: Selected Papers, eds., T Okasaki, S. Kumagai, A. Vatari, Y. Sato, K. Tsuda, and J. Shioiri. Department of Aeronautics, University of Tokyo, 1966.
14.
[14] Grolig, G.: Plastische Deformation als Ausdruck Mikroskopischer Instabilitlit. Dissertation, Fachbereich Mechanik der Technischen Hochschule Darmstadt, 1985.
15.
[15] Hill, R.: The Mathematical Theory of Plasticity, Oxford University Press, London, 1950.
16.
[16] Taylor, G. I.: The mechanism of plastic deformation of crystals. Part I. Theoretical. Proc. Roy. Soc., A, 145,362-387 (1934).
17.
[17] Needleman, A.: On the competition between failure and instability in progressively softening solids. J. Appl. Mech., 58, 294-296 (1991).
18.
[18] Carpinteri, A.: Cusp catastrophe interpretation of fracture instability. J. Mech. Phys. Solids, 37, 567-582 (1989).
19.
[19] Truskinovsky, L.: Fracture as a phase transition, in Contemporary Research in the Mechanics and Mathematics of Materials, pp. 322-332, ed., R. C. Batra and M. F. Beatty, CIMNE, Barcelona, 1996.
20.
[20] Schreyer, H. L. and Chen, Z.: One-dimensional softening with localization. J. Appl. Mech., 53, 791-797 (1986).
21.
[21] Del Piero, G. and Owen, D.: Structured Deformations of Continua, Lecture Notes, XXII Scuola Estiva di Fisica Matematica, Ravello, September 1997, forthcoming.
