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Mathematical Problems in Engineering
Volume 2010, Article ID 963043, 20 pages
Research Article

Spectral Classification of Non-Coaxiality for Two-Dimensional Incremental Stress-Strain Response

1Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China
2Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Tongji University, Shanghai 200092, China
3Sonny Astani Department of Civil and Environmental Engineering, University of Southern California, Los Angeles, CA 90089-2531, USA

Received 31 August 2010; Accepted 30 December 2010

Academic Editor: K. R. Rajagopal

Copyright © 2010 Jiangu Qian et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The present study examines the non-coaxial aspects of incremental material behavior, and attempts to classify the incremental non-coaxiality that relates stress and strain increments. In the solid mechanics literature, non-coaxiality (NC) refers usually to incremental strains and stress states having different principal directions. Departing from conventional non-coaxiality, the analysis investigates the incremental non-coaxiality (INC) of linearized rate-type solids. This study uses the concept of deviatoric second-order work for examining the relations between stability and incremental non-coaxiality. Based on a spectral analysis of the constitutive compliance matrix, it proposes three classifications for distinguishing various degrees of incremental non-coaxiality and stability. These classifications determine the conditions for the existence of incremental coaxiality (i.e., colinearity of stress and strain increments), stability, instability, and stable-instable transition (i.e., positive, negative, or zero second-order deviatoric work). The study illustrates these classifications in the cases of generic elastic and elastoplastic constitutive models. The analysis pertains to two-dimensional cases. Additional research is required to extend the analysis from two to three dimensions.