Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Nonlinear effect of initial stress on crack propagation between similar and dissimilar orthotropic media

Author: M. A. Biot
Journal: Quart. Appl. Math. 30 (1973), 379-406
MSC: Primary 73.35
DOI: https://doi.org/10.1090/qam/400852
MathSciNet review: 400852
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Abstract: The theory of crack propagation in orthotopic media is developed by applying the theory of incremental deformations in the vicinity of a state of initial stress. This is carried out in the context of a new approach to analytical methods and a physical analysis which takes into account plastic deformation under prestress. The state of initial stress is triaxial along the directions of elastic symmetry, and the crack is parallel to these directions. An additional shear component for the initial stress is also taken into account and general conditions are derived for crack propagation, including the case of fluid injection into the crack. The analysis is first carried out for an homogeneous medium. The nonlinear influence of the initial stress appears in two ways: first, through a fundamental purely elastic effect related to the occurrence of surface instability, and second, through the influence of the initial stress on plastic behavior. The particular cases of an isotropic elastic medium with finite initial strain and an orthotropic incompressible medium are discussed. The analysis is extended to a crack between dissimilar orthotropic media with initial stress. The method of analysis leads to a number of simplifications and brings out new properties of the solutions for this type of problem. For incompressible media without initial stress, the typical oscillatory behavior disappears. Uniqueness of the solutions is also derived.

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DOI: https://doi.org/10.1090/qam/400852
Article copyright: © Copyright 1973 American Mathematical Society

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