# J-integral as driving force parameter for elastic-plastic materials

The *configurational forces concept* has allowed us also to shed new light on fundamental problems in fracture mechanics that have been unsolved for decades. One of these problems was the application of the *J*-integral concept for elastic-plastic materials. The conventional *J*-integral relies on the so-called “deformation theory of plasticity”, i.e. elastic-plastic materials are treated as if they were nonlinear elastic. This theory is not applicable in cases of non-proportional loading, i.e. if unloading processes appear in elastic-plastic materials. However, such unloading processes are inevitable during crack extension or during cyclic loading of a structure and, therefore, the conventional *J*-integral cannot be used in these cases. Moreover, the conventional *J*-integral does not characterize the crack driving force in elastic-plastic materials.

Simha et al. (2008) derived a new type of *J*-integral, called *J*^{ep}, which overcomes all these limitations. This has led to a new basis for the application of the *J*-integral in elastic-plastic materials. In recent papers, we have demonstrated that the *J*-integral for elastic-plastic materials *J*^{ep} is very useful for characterizing the crack growth rate in fatigue for cases where linear elastic fracture mechanics and the stress intensity range Δ*K* are not applicable, e.g. in low-cycle fatigue or for the propagation of short fatigue cracks.

Extension of plastic zone in a fracture mechanics specimen (left), distribution of configurational forces (right). Bulk configurational forces are induced in a homogeneous elastic-plastic material; their magnitude is proportional to the stress and the gradient of the plastic strain. Due to the bulk configurational forces, the *J*-integral *J*^{ep} becomes path dependent.

**Important publications**

N.K. Simha, F.D. Fischer, G.X. Shan, C.R. Chen, O. Kolednik, *J*-integral and crack driving force in elastic-plastic materials. *Journal of the Mechanics and Physics of Solids* **56** (2008) 2876-2895.

F.D. Fischer, N.K. Simha, J. Predan, R. Schöngrundner, O. Kolednik, On configurational forces at boundaries in fracture mechanics. *International Journal of Fracture ***174** (2012) 61-74.

O. Kolednik, R. Schöngrundner, F.D. Fischer, A new view on *J*-integrals in elastic‒plastic materials, *International Journal of Fracture* **187** (2014) 77–107.

W. Ochensberger, O. Kolednik, A new basis for the application of the *J*-integral for cyclically loaded cracks in elastic‒plastic materials. *International Journal of Fracture* **189** (2014) 77–101.

W. Ochensberger, O. Kolednik, Physically appropriate characterization of fatigue crack propagation rate in elastic–plastic materials using the *J*-integral concept. *International Journal of Fracture* **192** (2015) 25–45.

Part of this research has been funded by the Austrian COMET Competence Center Programme via the COMET K2 Center for Materials, Processing and Product Engineering in Leoben.