Model of quasi-ductile deformations that bridges the scales Dusan Krajcinovic a, * , Sreten Mastilovic b a Mechanical and Aerospace Engineering, Arizona State University, Tempe, AZ 85287-6106, USA b Framatonme Cogema Fuels, 1211 Town Center Dr., Las Vegas, NV 89144, USA Abstract This study proposes a model of the deformation of materials susceptible to microcracking that is based on the thermodynamics of irreversible processes and fracture mechanics. The study considers the process on the atomic-, micro-, meso- and macroscales that are connected by the process of homogenization. The limitations of the process are carefully spelled out. Ó 2001 Elsevier Science Ltd. All rights reserved. 1. Introduction The objective of this study is to de®ne the es- sential basis of the models of non-elastic defor- mation and failure of materials susceptible to microcracking and brittle fracture. For example, rocks, concrete, epoxies, many composites, ce- ramics, glasses, many metals, bones, etc. belong to this class of materials. The modeling of this class of non-elastic, non-linear, irreversible deformation must be based on thermodynamics and fracture mechanics. To be useful all mechanical parameters of these models must be clearly identi®ed and measurable in laboratory. In the case of failure it is necessary to identify the order parameter which quanti®es the ``distance'' from the failure to assess the residual strength of the damaged structure and minimize the probability of structural and material failures. To reach the lofty goal of this study it is im- perative to keep in mind that the material texture is a random variable on the microcrack scale. Hence, the pattern of microcracks and their mor- phology are random variables on the same scale. In other words the propagation of microcracks depends on the local ¯uctuations of the energy barriers quenched within the material and local ¯uctuations of stresses. At some point of the ho- mogenization process from the atomic to micro- scale to mesoscale to macro-scale the models should change from being statistical to become deterministic. Since the propagation of micro- cracks can be unstable and the formation of mi- crocrack clusters self-organized, the failure process on the macroscopic scale for this class of materials is typically a statistical phenomenon. Thus, the application of continuum models in estimates of the failure is limited at best. The process of homogenization [1] represents a trade during which the eciency of the engineering continuum models is reached losing, at least par- tially, the rigor and the ®ne resolution of quantum physics. This trade is reasonable only when the parameters of the state on the coarse scale are related to the same parameters of the state on the ®ner scale. This is the only strategy that will pro- vide realistic continuum models, when possible, www.elsevier.com/locate/tafmec Theoretical and Applied Fracture Mechanics 37 2001) 167±182 * Corresponding author. Tel.: +1-602-9658656; fax: +1-602- 9651384. E-mail address: dusan@asu.edu D. Krajcinovic). 0167-8442/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 8 4 4 2  0 1 ) 0 0 0 8 4 - 2