New insights into the bond-based and ordinary state-based models in Peridynamics

Twenty years in since its introduction, Peridynamics established by Silling provides an attractive theory for modeling deformation and fracturing in solid materials. The paper aims to revisit the well-established bond-based Peridynamic (BPD) and ordinary state-based Peridynamic (OSPD) models and attempts to provide new insights into them. To this aim, a novel decomposition of the Peridynamic bond length change is developed, and the bond length change components are used to define a new micro-potential function. The macroscopic nonlocal elastic strain energy density (NESED) is computed through the integral of the micro-potential over the horizon. The constitutive relation is obtained by the Frechet derivative of the NESED with respect to the deformation state. The NESEDs and constitutive bond force densities are discussed with specific influence functions. The main foundings of the present work are as follows: (1) An explicit expression of bond micro-potential is proposed, and the NESED obtained by the integral of the micro-potential over the horizon can retrieve the ad hoc NESED in the OSPD model. It could pave the way for the development of micro-potential based bond failure criterion for the OSPD model. (2) It is proven that the constitutive relation in the OSPD model can be recovered by the present constitutive relation with specific influence functions in the case where the deformation is small. (3) As the Poisson ratio is 1/3 in plane stress condition or 1/4 in plane strain condition, the present Peridynamic model with specific influence functions is reduced to the BPD model in the case of small deformation, and thus the BPD model can be considered as a reduced form of an OSPD model. Moreover, numerical investigations on a single edge-notched plate under mode II dominated loading and three-point bending tests are conducted to illustrate the descriptive and predictive capabilities of the present Peridynamic model.