Operator-derived micropolar peridynamics

Abstract

In this article, we develop a peridynamic model for the micropolar elastic solids called operator-derived micropolar peridynamics (OMPD). Two types of OMPD models, namely the OMPD model-I based on the peridynamic differential operator and model-II based on the peridynamic operator method with second-order vector derivative, are obtained. By using the first-order Taylor series expansion (TSE), model-I can recover the previously proposed non-ordinary state-based micropolar peridynamics. However, the OMPD Model-I of any order TSE still suffers from zero-energy mode instability, while the proposed OMPD model-II is free of zero-energy mode and thus produces the correct micropolar elasticity response. The OMPD model-II produces an ordinary-like state-based micropolar peridynamic model with a symmetric horizon, which degenerates to the ordinary state-based peridynamics by vanishing the Cosserat shear modulus and micro-rotations. Furthermore, a novel bond-based micropolar peridynamics is derived with some moduli restrictions. Such bond-based micropolar peridynamics considers shear deformability and average micro-rotational effect in the shear bonds, and inherits the couple force generated by shear bond force from the OMPD model-II, which is essential to be consistent with Eringen’s micropolar elastic theory. We show that the novel bond-based micropolar peridynamics relaxes Poisson’s ratio fixation to a rigorous range from -1 to 1/4. Several numerical examples are provided to validate the models’ capacity in modeling micropolar solids and the crack propagation behavior.