Peridynamics beam plasticity theory: Yield surface for general cross-sectional geometry

Abstract

This paper presents a general way of deriving the yield surface for peridynamics (PD) Simo-Reissner beam plasticity theory from the three-dimensional (3D) von Mises yield surface and formulates a procedure to determine the plastic multiplier and elastoplastic tangent modulus. Beam plasticity problems are usually solved by employing the upper and lower bound theorems of plasticity and postulating interaction relations. However, it is well known that deriving exact expression for interaction equation is considerably difficult for a general beam cross-section and loading. To overcome this roadblock, the idea here is to express the yield surface in terms of cross-sectional coordinates by employing Simo-Reissner hypothesis on deformation field, expand the terms using Taylor series up to quadratic order, and analytically integrate over the cross-sectional area. This results in reduced form of the yield surface equation which involves beam cross-sectional properties such as area and moment of inertia. A return mapping algorithm is proposed to determine the plastic multiplier and the elastoplastic tangent modulus. This novel method does not require ad-hoc interaction relations derived from the upper and lower bound theorems and is applicable to any general beam cross-section. Numerical simulations include validation of displacement contour obtained from the proposed technique and the finite element method, and prediction of plastic deformation of cantilever beam, two-dimensional (2D) and 3D portal frames, 2D and 3D truss-frames, single unit octet lattice structure, 3×3×3 octet lattice structure, and compression-torsion lattice structures under quasi-static loading which demonstrate the potential of our approach.