Thoughts on the Non-ordinary Peridynamics Model Based on Three-body Potential

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

“…according to the definition of Frechet derivative as coined in Silling et al [9]. And, the conservation of angular momentum is guaranteed by the theoretical equivalence to the ones proposed in Li and Hao [29,31] due to the same PD SED being used. It could be found that the first term on the RHS of equation ( 69) is associated with dilatation, therefore, is defined as dilatational part…”

“…form mentioned above can be obtained by analytically evaluating the integral term in equation ( 2) as [29,31]…”

“…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.…”

“…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. 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].…”

“…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 .…”