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

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Abstract

This paper studies the internal relationship between peridynamics (PD) models, which address to the Poisson's ratio limitation in original bond-based PD theory by considering tangential bond force or tangential stiffness. Firstly, a non-ordinary state-based PD model is formulated based on a micro-potential considering a more general three-body interaction potential, which can also remove the constraint on values of Poisson’s ratio. Then, a special three-body interaction potential, called novel three-body interaction potential, is adopted in the proposed PD model stated above; it can be found that the resulting PD model can be equivalent to that considering bond stretch and rotation published earlier under assumption of small deformation conditions. This discovery in turn reveals the macroscopic mechanical mechanism of the novel three-body interaction potential in resisting shear deformation. Moreover, the result not only provides a physical explanation for the latter to some extent, but also is helpful to reveal the internal relationships between those PD models considering tangential bond force directly. In addition, another special example of PD model considering distribution of the intensity of long range forces is also given within the proposed PD modeling framework. Its performance is computationally demonstrated by a dynamic fracture problem. These results show that formulating PD models within state-based PD framework by assuming reasonable micro-potential can be proper and flexible in practical engineering problems.

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Acknowledgements

We thank Dr. Pan Li of China Academy of Engineering Physics for useful discussions. The supports from China Academy of Engineering Physics Institute of Systems Engineering are gratefully acknowledged. The authors would like to thank the editor and reviewer for valuable comments and suggestions.

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Authors and Affiliations

  1. Institute of Systems Engineering, China Academy of Engineering Physics, Mianyang, 621999, Sichuan, China

    Xiaolong Li & Zhiming Hao

  2. Shock and Vibration of Engineering Materials and Structures Key Laboratory of Sichuan Province, Mianyang, 621999, Sichuan, China

    Zhiming Hao

Authors
  1. Xiaolong Li
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  2. Zhiming Hao
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Correspondence to Zhiming Hao.

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Appendices

Appendix A Proof for Eq. (24)

Proof

By using Eq. (17), (19), we arrive at

$$\begin{array}{l}\mathbf{e}=\left(\mathbf{u}\left({\mathbf{X}}_{j}\right)-\mathbf{u}\left({\mathbf{X}}_{i}\right)\right)\cdot {\varvec{\upchi}}\otimes{\varvec{\upchi}}=\frac{{{\varvec{\upxi}}}_{ij}\cdot \left({\varvec{\upvarepsilon}}+{\varvec{\upomega}}\right)\cdot {{\varvec{\upxi}}}_{ij}}{\left|{{\varvec{\upxi}}}_{ij}\right|}{\varvec{\upchi}}\\\;\;\; =\frac{{{\varvec{\upxi}}}_{ij}}{\left|{{\varvec{\upxi}}}_{ij}\right|}\cdot \left({\varvec{\upvarepsilon}}+{\varvec{\upomega}}\right)\cdot \frac{{{\varvec{\upxi}}}_{ij}}{\left|{{\varvec{\upxi}}}_{ij}\right|}\left|{{\varvec{\upxi}}}_{ij}\right|{\varvec{\upchi}}=\frac{{{\varvec{\upxi}}}_{ij}}{\left|{{\varvec{\upxi}}}_{ij}\right|}\cdot {\varvec{\upvarepsilon}}\cdot \frac{{{\varvec{\upxi}}}_{ij}}{\left|{{\varvec{\upxi}}}_{ij}\right|}\left|{{\varvec{\upxi}}}_{ij}\right|{\varvec{\upchi}}\approx s\left|{{\varvec{\upxi}}}_{ij}\right|{\varvec{\upchi}}.\end{array}$$

Then,

$$\overline{\mathbf{e}}\approx s{\varvec{\upchi} },$$

with \(s\approx {\varvec{\upchi}}\cdot {\varvec{\upvarepsilon}}\cdot {\varvec{\upchi}}\) under small deformation and \({\varvec{\upchi}}\cdot {\varvec{\upomega}}\cdot {\varvec{\upchi}}\)=0. Therefore, \({s}^{2}=\overline{\mathbf{e} }\cdot \overline{\mathbf{e} }\) in Eq. (24) is satisfied.

Appendix B Proof for Eq. (52)

Proof

By Eq. (23)

$$\begin{array}{l}\mathbf{r}\cdot \mathbf{r}=\left(\frac{\left(\mathbf{I}-{\varvec{\upchi}}\otimes{\varvec{\upchi}}\right)\cdot {\varvec{\upvarepsilon}}\cdot {{\varvec{\upxi}}}_{ij}}{\left|{{\varvec{\upxi}}}_{ij}\right|}\right)\cdot \left(\frac{\left(\mathbf{I}-{\varvec{\upchi}}\otimes{\varvec{\upchi}}\right)\cdot {\varvec{\upvarepsilon}}\cdot {{\varvec{\upxi}}}_{ij}}{\left|{{\varvec{\upxi}}}_{ij}\right|}\right)\\\qquad =\left(\frac{\mathbf{I}\cdot {\varvec{\upvarepsilon}}\cdot {{\varvec{\upxi}}}_{ij}-{\varvec{\upchi}}\left({\varvec{\upchi}}\cdot {\varvec{\upvarepsilon}}\cdot {{\varvec{\upxi}}}_{ij}\right)}{\left|{{\varvec{\upxi}}}_{ij}\right|}\right)\cdot \left(\frac{\mathbf{I}\cdot {\varvec{\upvarepsilon}}\cdot {{\varvec{\upxi}}}_{ij}-{\varvec{\upchi}}\left({\varvec{\upchi}}\cdot {\varvec{\upvarepsilon}}\cdot {{\varvec{\upxi}}}_{ij}\right)}{\left|{{\varvec{\upxi}}}_{ij}\right|}\right)\\\qquad =\left(\frac{{\varvec{\upvarepsilon}}\cdot {{\varvec{\upxi}}}_{ij}-{\varvec{\upchi}}\left({\varvec{\upchi}}\cdot {\varvec{\upvarepsilon}}\cdot {{\varvec{\upxi}}}_{ij}\right)}{\left|{{\varvec{\upxi}}}_{ij}\right|}\right)\cdot \left(\frac{{\varvec{\upvarepsilon}}\cdot {{\varvec{\upxi}}}_{ij}-{\varvec{\upchi}}\left({\varvec{\upchi}}\cdot {\varvec{\upvarepsilon}}\cdot {{\varvec{\upxi}}}_{ij}\right)}{\left|{{\varvec{\upxi}}}_{ij}\right|}\right)\\\qquad =\left({\varvec{\upvarepsilon}}\cdot {\varvec{\upchi}}-{\varvec{\upchi}}\left({\varvec{\upchi}}\cdot {\varvec{\upvarepsilon}}\cdot {\varvec{\upchi}}\right)\right)\cdot \left({\varvec{\upvarepsilon}}\cdot {\varvec{\upchi}}-{\varvec{\upchi}}\left({\varvec{\upchi}}\cdot {\varvec{\upvarepsilon}}\cdot {\varvec{\upchi}}\right)\right),\end{array}$$

under small deformation conditions, note that \(s\approx {\varvec{\upchi}}\cdot {\varvec{\upvarepsilon}}\cdot {\varvec{\upchi}}\), then

$$\begin{array}{l}\mathbf{r}\cdot \mathbf{r}=\left({\varvec{\upvarepsilon}}\cdot {\varvec{\upchi}}-{\varvec{\upchi}}s\right)\cdot \left({\varvec{\upvarepsilon}}\cdot {\varvec{\upchi}}-{\varvec{\upchi}}s\right)\\\qquad =\left({\varvec{\upvarepsilon}}\cdot {\varvec{\upchi}}\right)\cdot \left({\varvec{\upvarepsilon}}\cdot {\varvec{\upchi}}\right)-s{\varvec{\upchi}}\cdot \left({\varvec{\upvarepsilon}}\cdot {\varvec{\upchi}}\right)-\left({\varvec{\upvarepsilon}}\cdot {\varvec{\upchi}}\right)\cdot \left(s{\varvec{\upchi}}\right)+s{\varvec{\upchi}}\cdot \left({\varvec{\upchi}}s\right)\\\qquad =\left({\varvec{\upvarepsilon}}\cdot {\varvec{\upchi}}\right)\cdot \left({\varvec{\upvarepsilon}}\cdot {\varvec{\upchi}}\right)- s{\varvec{\upchi}}\cdot {\varvec{\upvarepsilon}}\cdot {\varvec{\upchi}}-\left({\varvec{\upchi}}\cdot {{\varvec{\upvarepsilon}}}^{\mathrm{T}}\right)\cdot \left(s{\varvec{\upchi}}\right)+{s}^{2}\\\qquad =\left({\varvec{\upvarepsilon}}\cdot {\varvec{\upchi}}\right)\cdot \left({\varvec{\upvarepsilon}}\cdot {\varvec{\upchi}}\right)- s{\varvec{\upchi}}\cdot {\varvec{\upvarepsilon}}\cdot {\varvec{\upchi}}-s\left({\varvec{\upchi}}\cdot {\varvec{\upvarepsilon}}\right)\cdot {\varvec{\upchi}}+{s}^{2}\\\qquad =\left({\varvec{\upvarepsilon}}\cdot {\varvec{\upchi}}\right)\cdot \left({\varvec{\upvarepsilon}}\cdot {\varvec{\upchi}}\right)-{s}^{2},\end{array}$$

which implies \(\mathbf{r}\cdot \mathbf{r}+{s}^{2}=\left({\varvec{\upvarepsilon}}\cdot {\varvec{\upchi}}\right)\cdot \left({\varvec{\upvarepsilon}}\cdot {\varvec{\upchi}}\right)\), i.e., Eq. (52) holds.

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Li, X., Hao, Z. Thoughts on the Non-ordinary Peridynamics Model Based on Three-body Potential. J Peridyn Nonlocal Model 4, 398–419 (2022). https://doi.org/10.1007/s42102-022-00084-3

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