A two-dimensional non-ordinary state-based peridynamic model based on three-body potential for elastic brittle fracture analysis

A non-ordinary state decomposition method for three-body potential peridynamics model

“…by considering equations ( 29) and (30). Combining the equations above and Lemma 1, DW (h r ) can be approximated as…”

“…As is well known, there exists Poisson’s ratio limitation in original BB PD model due to central potential assumption. To eliminate this limitation, the authors proposed the TBP PD model [28–31] based on a micropotential w ij combined with three-body interaction potential. The expression shown in equation (2) is the so-called novel TBP defined in Li and Hao [29] (or modified TBP in Li and Hao [31]) and will be the basis of this study.…”

“…This is a more general computing scheme, by which the rotation tensor ω i , a classical local quantity, was also calculated in Madenci et al [40] to enable the PD model computationally feasible. In the present study, the classical quantities, such as stress and strain tensors, for all points will be formulated based on the corresponding deformation gradient tensor [9] reduced from deformation vector state Y true¯ as done in Li and Hao [30]. Then, a well-established plasticity quantity, von Mises stress, will be formulated in 3D and 2D cases.…”

“…In OSB-PD models above, the PD force density vectors between two points are assumed to be parallel to their relative position vector in deformed state, leading to the limitation of shear deformation for bond vectors. By comparison, three-body potential (TBP) PD model [28][29][30][31], belonging to non-ordinary state-based (NOSB) PD models such as those in the literature [9,[32][33][34][35][36][37][38][39][40] among others, can resolve this limitation due to introducing three-body interaction potential between material points, which enables the shear deformability of bond vectors. Meanwhile, as researched in Li and Hao [29], if a specific expression, provided by novel TBP, is used, this PD model will take bond shear form in the hypothesis of small deformation conditions and be equivalent to the one presented by Madenci et al [40], that is, the PD model considering bond rotation.…”

“…Meanwhile, as researched in Li and Hao [29], if a specific expression, provided by novel TBP, is used, this PD model will take bond shear form in the hypothesis of small deformation conditions and be equivalent to the one presented by Madenci et al [40], that is, the PD model considering bond rotation. However, no matter in the TBP PD model or the one considering bond rotation, the PD SED and force functions are directly expressed in terms of bond stretch and shear deformations [28][29][30][31]40]. Also, the decomposition method appearing in OSB-PD model [9,25] could not be directly used to obtain the required decomposition forms for them due to the bond shear deformability .…”