Theory and Modern Applications
Table 1 List of adopted integral kernels and the corresponding elastic potential (the function \(\mu (s,t)\) has been omitted from the model corresponding to (3.14) because it is negligible in the current context); the models in (3.15) and (3.16) are not in the table because of readability
From: Bond-based peridynamics, a survey prospecting nonlocal theories of fluid-dynamics
Equation | Φ(ξ,η) | Equation | f(ξ,η) |
|---|---|---|---|
|  | \(\frac{c}{2}(|\boldsymbol{\xi }+\boldsymbol{\eta }|-|\boldsymbol{\xi }|)^{2} \) | (3.1) | c(|ξ + η| − |ξ|)n |
(3.2) | \(\alpha (|\boldsymbol{\xi }|) (|\boldsymbol{\xi }+\boldsymbol{\eta }|^{2}-|\boldsymbol{\xi }|^{2} )^{2} \), | (3.3) | \(a(|\boldsymbol{\xi }|) (|\boldsymbol{\xi }+\boldsymbol{\eta }|^{2}-|\boldsymbol{\xi }|^{2} )(\boldsymbol{\xi }+\boldsymbol{\eta }) \) |
| Â | \(c(\boldsymbol{\xi },\delta ) \frac{(|\boldsymbol{\xi }+\boldsymbol{\eta }|-|\boldsymbol{\xi }|)^{2}}{2|\boldsymbol{\xi }|} \) | (3.14) | \(c(\boldsymbol{\xi },\delta ) \frac{|\boldsymbol{\xi }+\boldsymbol{\eta }|-|\boldsymbol{\xi }|}{|\boldsymbol{\xi }|}\boldsymbol{n} \) |
| Â | \(c(\boldsymbol{\xi },\delta ) \frac{(|\boldsymbol{\xi }+\boldsymbol{\eta }|-|\boldsymbol{\xi }|)^{2}}{ 2|\boldsymbol{\xi }|^{2}} \) | (3.17) | \(c(\boldsymbol{\xi },\delta ) \frac{|\boldsymbol{\xi }+\boldsymbol{\eta }|-|\boldsymbol{\xi }|}{ |\boldsymbol{\xi }|^{2}}\boldsymbol{n} \) |
| Â | \(\frac{C(\boldsymbol{\xi })}{r+1}(\boldsymbol{\xi }+\boldsymbol{\eta })^{r+1} \) | (3.19) | \(C(\boldsymbol{\xi })(\boldsymbol{\xi }+\boldsymbol{\eta })^{r} \) |
(3.20) | \(\kappa \frac{|\boldsymbol{\xi }+\boldsymbol{\eta }|^{p}}{|\boldsymbol{\xi }|^{N+\alpha p}}+\Psi (\boldsymbol{\xi },\boldsymbol{\eta }) \) | (3.22) | \(\kappa p \frac{|\boldsymbol{\eta}+\boldsymbol{\xi }|^{p-2} }{|\boldsymbol{\xi}|^{N+\alpha p}}(\boldsymbol{\eta +\xi})+\psi (\boldsymbol{\xi}, \boldsymbol{\eta}) \) |