Two-Dimensional Formulation for J2 Plasticity in Ordinary State-Based Peridynamics

Two-dimensional formulation for J2 plasticity in ordinary state-based peridynamics

Farzaneh Mousavi, Siavash Jafrazadeh, Jiangming Zhao, Florin Bobaru

Abstract:

We present a new two-dimensional elastoplastic formulation for ordinary state-based peridynamics. Two-dimensional models, when they can be applied, offer substantial advantages in terms of efficiency compared with 3D models. Existing 2D elastoplastic peridynamic models [1], [2] have either significant limitations or use assumptions which are not consistent with the corresponding classical J2 plasticity theory. Having a correct decomposition of force and extension states is one of the important steps in deriving a PD elastoplastic model. We propose a general decomposition of 2D force and extension states into isotropic and deviatoric parts for ordinary state-based elastoplastic solids under plane stress and in plane strain conditions. We show that these decompositions are consistent with the classical decompositions for stresses and strains. Our formulation can be used in problems with small or large rotations, and small strains. We propose two types of 2D rate-independent yield functions equivalent to J2 plasticity with associated flow rule: one based on deviatoric strain energy density and the other based on deviatoric force state. We verify our model against results for the classical formulation computed with finite elements in Abaqus. To reduce/eliminate the peridynamic surface effect, we use a version of the fictitious nodes [3] method capable of handling arbitrary geometries and does not have any limitation in applying at the corner of the geometry which was problematic using previous version of fictitious nodes method [4]. Also, we notice in existing ordinary state-based elastic-perfectly plastic models, the plastic region expands unrealistically as the applied displacements continue to increase to larger values. However, our elastoplastic formulation remains valid for arbitrary rotations, and small strains.

Acknowledgements: This work was supported in part by the National Science Foundation under Grant No. 1953346 and by a Nebraska System Science award from the Nebraska Research Initiative.

References

[1]        E. Madenci and S. Oterkus, “Ordinary state-based peridynamics for plastic deformation according to von Mises yield criteria with isotropic hardening,” J. Mech. Phys. Solids, vol. 86, pp. 192–219, 2016.

[2]        A. Zang and O. Stephansson, Stress field of the Earth’s crust. Springer Science & Business Media, 2009.

[3]        J. Zhao, S. Jafarzadeh, Z. Chen and F. Bobaru, " An algorithm for imposing local boundary conditions in peridynamic models on arbitrary domain", (to be submitted)

[4]        M. Erdogan and E. Oterkus. "Peridynamic theory." In Peridynamic Theory and Its Applications, pp. 19-43. Springer, New York, NY, 2014.