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A new view on J-integrals in elastic–plastic materials

Abstract

It is well known that the application of the conventional \(J\)-integral is connected with severe restrictions when it is applied for elastic–plastic materials. The first restriction is that the \(J\)-integral can be used only, if the conditions of proportional loading are fulfilled, e.g. no unloading processes should occur in the material. The second restriction is that, even if this condition is fulfilled, the \(J\)-integral does not describe the crack driving force, but only the intensity of the crack tip field. Using the configurational force concept, Simha et al. (J Mech Phys Solids 56:2876–2895, 2008), have derived a \(J\)-integral, \(J^{\mathrm{ep}} \), which overcomes these restrictions: \(J^{\mathrm{ep}} \) is able to quantify the crack driving force in elastic–plastic materials in accordance with incremental theory of plasticity and it can be applied also in cases of non-proportionality, e.g. for a growing crack. The current paper deals with the characteristic properties of this new \(J\)-integral, \(J^{\mathrm{ep}}\), and works out the main differences to the conventional \(J\)-integral. In order to do this, numerical studies are performed to calculate the distribution of the configurational forces in a cyclically loaded tensile specimen and in fracture mechanics specimens. For the latter case contained, uncontained, and general yielding conditions are considered. The path dependence of \(J^{\mathrm{ep}} \) is determined for both a stationary and a growing crack. Much effort is spent in the investigation of the path dependence of \(J^{\mathrm{ep}} \) very close to the crack tip. Several numerical parameters are varied in order to separate numerical and physical effects and to deduce the magnitudes of the crack driving force for stationary and growing cracks. Interpretation of the numerical results leads to a new, completed picture of the \(J\)-integral in elastic–plastic materials where \(J^{\mathrm{ep}} \) and the conventional \(J\)-integral complement each other. This new view allows us also to shed new light on a long-term problem, which has been called the “paradox of elastic–plastic fracture mechanics”.

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Notes

  1. It might be helpful to note that during the translation of the defects, the material points of the specimen remain (almost) at the same positions. (They do remain at the same positions in the load-free reference configuration.) If the crack tip moves, the bonding between the material points in front of the current crack tip, above and below the crack plane, is cut. If the plastic zone translates with the crack tip, the region, within which the material is deformed plastically, is shifted.

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Acknowledgments

O.K. and F.D.F. gratefully acknowledge valuable discussions with Dr. N.K. Simha. 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-WP2 and A4.20-WP1).

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Kolednik, O., Schöngrundner, R. & Fischer, F.D. A new view on J-integrals in elastic–plastic materials. Int J Fract 187, 77–107 (2014). https://doi.org/10.1007/s10704-013-9920-6

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  • DOI: https://doi.org/10.1007/s10704-013-9920-6

Keywords

  • Configurational forces
  • Crack driving force
  • Crack growth
  • Elastic–plastic fracture mechanics
  • J-integral
  • Incremental theory of plasticity