# A new basis for the application of the \(J\)-integral for cyclically loaded cracks in elastic–plastic materials

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## Abstract

Fatigue crack propagation is by far the most important failure mechanism. Often cracks under low-cycle fatigue conditions and, especially, short fatigue cracks cannot be treated with the conventional stress intensity range \(\Delta K\)-concept, since linear elastic fracture mechanics is not valid. For such cases, Dowling and Begley (ASTM STP 590:82–103, 1976) proposed to use the experimental cyclic \(J\)-integral \(\Delta J^{\exp }\) for the assessment of the fatigue crack growth rate. However, severe doubts exist concerning the application of \(\Delta J^{\exp }\). The reason is that, like the conventional \(J\)-integral, \(\Delta J^{\exp }\) presumes deformation theory of plasticity and, therefore, problems appear due to the strongly non-proportional loading conditions during cyclic loading. The theory of configurational forces enables the derivation of the \(J\)-integral independent of the constitutive relations of the material. The \(J\)-integral for incremental theory of plasticity, \(J^{\mathrm{ep}}\), has the physical meaning of a true driving force term and is potentially applicable for the description of cyclically loaded cracks, however, it is path dependent. The current paper aims to investigate the application of \(J^{\mathrm{ep}}\) for the assessment of the crack driving force in cyclically loaded elastic–plastic materials. The properties of \(J^{\mathrm{ep}}\) are worked out for a stationary crack in a compact tension specimen under cyclic Mode I loading and large-scale yielding conditions. Different load ratios, between pure tension- and tension–compression loading, are considered. The results provide a new basis for the application of the \(J\)-integral concept for cyclic loading conditions in cases where linear elastic fracture mechanics is not applicable. It is shown that the application of the experimental cyclic \(J\)-integral \(\Delta J^{\exp }\) is physically appropriate, if certain conditions are observed.

## Keywords

Configurational force concept Crack driving force Cyclic \(J\)-integral Incremental theory of plasticity Low-cycle fatigue## Notes

### Acknowledgments

The authors would like to express their gratitude to Prof. R. Pippan and Prof. F.D. Fischer for helpful discussions. Financial support by the Austrian Federal Government and the Styrian Provincial Government within the research activities of the K2 Competence Center on “Integrated Research in Materials, Processing and Product Engineering”, under the frame of the Austrian COMET Competence Center Programme, is gratefully acknowledged (strategic projects A4.11-WP4 and A4.20-WP3).

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