A stabilized hybrid peridynamic method compatible with constitutive models of different dimensions

Peridynamics (PD) has been increasingly widely employed to address cracking and fracture behaviors of engineering materials. However, the existing two types of PD, namely, bond based peridynamics (BPD) and state based peridynamics (SPD), have their respective limitations. The BPD is practical and capable of solving real engineering problems, but cannot work with three-dimensional (3D) constitutive models. The SPD is able to use 3D constitutive models but suffers from the so called “zero-energy mode” problem especially when the system is highly nonlinear. This paper presents a stabilized hybrid PD method (HPD) that can be effectively compatible with 1D to 3D constitutive models and does not have zero-energy mode issue for highly nonlinear problems. The proposed HPD creates bonds between each PD point and other PD points inside its horizon, and obtains the total force on the PD point by all bond forces. For 2D or 3D (or simply written as nD) constitutive models, each bond force consists of an axial force and a shear force evaluated at the midpoint of the bond, and is calculated by the nD stress tensor multiplied by the direction and area of the bond. The nD stress can be obtained by using an nD constitutive model based on the nD strain at the midpoint of the two connected PD points. The nD strain is estimated by using displacements of PD points within a new sub-horizon, which is defined by the overlap of two horizons of the connected PD points. For 1D constitutive model cases, the axial force is obtained by the axial strain and the shear force is ignored, and the HPD is degraded to BPD. Four application examples are presented to verify the proposed HPD method, i.e., static analyses for a linear cantilever beam and a nonlinear soil column using 2D material models, seismic analysis for a soil column using 3D nonlinear model and dynamic analysis of a plate with pre-crack using 1D and 3D models. The responses (e.g., displacements, stresses, strains at representative points) are analyzed and compared with those by FEM, BPD and SPD. The HPD method presented herein is demonstrated to be accurate, stable, and compatible with various material models of different dimensions to solve a wide range of complicated problems.