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The Theory of Materials Failure$
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Richard M. Christensen

Print publication date: 2013

Print ISBN-13: 9780199662111

Published to Oxford Scholarship Online: May 2013

DOI: 10.1093/acprof:oso/9780199662111.001.0001

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The Failure Theory for Isotropic Materials

The Failure Theory for Isotropic Materials

(p.30) 4 The Failure Theory for Isotropic Materials
The Theory of Materials Failure

Richard M. Christensen

Oxford University Press

Some essential preliminaries introduce the failure-theory problem by considering the different roles assumed by analytical forms that involve parameters as opposed to those that involve properties of the material. The first section of the derivation of the failure theory is entitled “The Organizing Principle”, and therein it is found that only two properties are allowed for the failure theory calibration and that they must be the uniaxial tensile and compressive strengths, T and C. Furthermore, it is established that the ratio T/C varies between the limits of 0 and 1, and covers the entire spectrum of materials types from the most brittle types to the most ductile types, and all those in between. Next, a polynomial expansion in the isotropic invariants of stress is used to define the representation for the elastic energy of the body, and most importantly, also used independently for the representation defining the limits of this constitutive form. This latter result is then the polynomial invariants failure criterion. Although this criterion provides the basic and guiding theoretical form, it is shown that it cannot provide the complete and all-inclusive failure description. There must be a competitive fracture criterion in certain ranges of T/C, and this fracture criterion is found to be controlled by the maximum principal stress. These two failure criteria, controlled by only T and C, cover all the types of isotropic materials where failure represents the cessation of linear elastic behavior, whether the failure event is that of brittle fracture or plastic flow.

Keywords:   failure theory, failure criteria, organizing principle, calibration, tensile strength, compressive strength, spectrum, ductile, brittle, principal stress, plastic flow

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