Softening Instability: Part I—Localization Into a Planar Band

[+] Author and Article Information
Zdeněk P. Bažant

Department of Civil Engineering, Northwestern University, Evanston, IL 60208

J. Appl. Mech 55(3), 517-522 (Sep 01, 1988) (6 pages) doi:10.1115/1.3125823 History: Received March 13, 1987; Revised February 01, 1988; Online July 21, 2009


Distributed damage such as cracking in heterogeneous brittle materials may be approximately described by a strain-softening continuum. To make analytical solutions feasible, the continuum is assumed to be local but localization of softening strain into a region of vanishing volume is precluded by requiring that the softening region, assumed to be in a state of homogeneous strain, must have a certain minimum thickness which is a material property. Exact conditions of stability of an initially uniform strain field against strain localization are obtained for the case of an infinite layer in which the strain localizes into an infinite planar band. First, the problem is solved for small strain. Then a linearized incremental solution is obtained taking into account geometrical nonlinearity of strain. The stability condition is shown to depend on the ratio of the layer thickness to the softening band thickness. It is found that if this ratio is not too large compared to 1, the state of homogeneous strain may be stable well into the softening range. Part II of this study applies Eshelby’s theorem to determine the conditions of localization into ellipsoidal regions in infinite space, and also solves localization into circular or spherical regions in finite bodies.

Copyright © 1988 by ASME
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