1 Introduction
2 Kinematics in peridynamics
3 Governing equations
Linear momentum balance | |
PD | \(\displaystyle \rho _0 \text {D}_t {\varvec{v}} = \int _{{\mathcal {H}}_0} {\varvec{p}}_0^{{{\,\mathrm{ {}^{\vert } }\,}}} \, \text {d}V^{{{\,\mathrm{ {}^{\vert } }\,}}} + {\varvec{\bar{{b}}}}_0^\text {ext} \quad \text {subject to} \quad \displaystyle \int _{{\mathcal {B}}_0} \int _{{\mathcal {H}}_0} {\varvec{p}}_0^{{{\,\mathrm{ {}^{\vert } }\,}}} \, \text {d}V^{{{\,\mathrm{ {}^{\vert } }\,}}} \, \text {d}V = \int _{\partial {\mathcal {B}}_0} {\varvec{t}}_0^\text {ext} \, \text {d}A\) |
CCM | \(\rho _0 \text {D}_t {\varvec{v}} = \text {Div} {\varvec{P}} + {{\varvec{\bar{b}}}_0^\text {ext}} \quad \text {subject to} \quad {\varvec{P}} \cdot {\varvec{N}} = {\varvec{t}}_0^\text {ext} \) |
Angular momentum balance | |
PD | \(\displaystyle \int _{{\mathcal {H}}_0} \varvec{\xi }^{{{\,\mathrm{ {}^{\vert } }\,}}} \times {\varvec{p}}_0^{{{\,\mathrm{ {}^{\vert } }\,}}} \, \text {d} V^{{{\,\mathrm{ {}^{\vert } }\,}}} = {\varvec{0}}\) |
CCM | \(\varvec{\varepsilon } : \left[ {\varvec{F}} \cdot {\varvec{P}}^\text {t} \right] = {\varvec{0}}\) |
Energy balance | |
PD | \(\displaystyle \int _{{\mathcal {H}}_0} \rho _0^{{{\,\mathrm{ {}^{\vert } }\,}}} \text {D}_t u^{{{\,\mathrm{ {}^{\vert } }\,}}} \, \text {d}V^{{{\,\mathrm{ {}^{\vert } }\,}}} = - \displaystyle \int _{{\mathcal {H}}_0} q_0^{{{\,\mathrm{ {}^{\vert } }\,}}} \, \text {d}V^{{{\,\mathrm{ {}^{\vert } }\,}}} + \bar{{\mathscr {R}}}^{ext}_0 + \int _{{\mathcal {H}}_0} {\varvec{p}}_0^{{{\,\mathrm{ {}^{\vert } }\,}}} \cdot \text {D}_t \varvec{\xi }^{{{\,\mathrm{ {}^{\vert } }\,}}} \, \text {d}V^{{{\,\mathrm{ {}^{\vert } }\,}}}\) |
\( \text {subject to} \quad \displaystyle \int _{{\mathcal {B}}_0} \int _{{\mathcal {H}}_0} q_0^{{{\,\mathrm{ {}^{\vert } }\,}}} \, \text {d}V^{{{\,\mathrm{ {}^{\vert } }\,}}} \, \text {d}V = \int _{\partial {\mathcal {B}}_0} {\mathscr {Q}}_0^\text {ext} \, \text {d}A \) | |
CCM | \(\rho _0 \text {D}_t u = - \text {Div} {\varvec{Q}} + \bar{{\mathscr {R}}}^{ext}_0 + {\varvec{P}} : \text {D}_t {\varvec{F}} \quad \text {subject to} \quad {\varvec{Q}} \cdot {\varvec{N}} = {\mathscr {Q}}_0^\text {ext} \) |
Entropy balance | |
PD | \(T \displaystyle \int _{{\mathcal {H}}_0} \rho _0^{{{\,\mathrm{ {}^{\vert } }\,}}} \text {D}_t s^{{{\,\mathrm{ {}^{\vert } }\,}}} \, \text {d}V^{{{\,\mathrm{ {}^{\vert } }\,}}} = - \displaystyle \int _{{\mathcal {H}}_0} q_0^{{{\,\mathrm{ {}^{\vert } }\,}}} \, \text {d}V^{{{\,\mathrm{ {}^{\vert } }\,}}} + \bar{{\mathscr {R}}}^{ext}_0 +T {\mathscr {S}}_0 + {\mathscr {D}}_0 \) |
\(\text {subject to} \quad \displaystyle \displaystyle \int _{{\mathcal {B}}_0} \int _{{\mathcal {H}}_0} q_0^{{{\,\mathrm{ {}^{\vert } }\,}}} \, \text {d}V^{{{\,\mathrm{ {}^{\vert } }\,}}} \, \text {d}V = \int _{\partial {\mathcal {B}}_0} {\mathscr {Q}}_0^\text {ext} \, \text {d}A \) | |
CCM | \(T \rho _0 \text {D}_t s = - \text {Div}{\varvec{Q}} + \bar{{\mathscr {R}}}^{ext}_0 +T{\mathscr {S}}_0 + {\mathscr {D}}_0 \quad \text {subject to} \quad {\varvec{Q}} \cdot {\varvec{N}} = {\mathscr {Q}}_0^\text {ext} \) |
Dissipation inequality | |
PD | \(\displaystyle {\mathscr {D}}_0 = \int _{{\mathcal {H}}_0} {\varvec{p}}_0^{{{\,\mathrm{ {}^{\vert } }\,}}} \cdot \text {D}_t \varvec{\xi }^{{{\,\mathrm{ {}^{\vert } }\,}}} \, \text {d}V^{{{\,\mathrm{ {}^{\vert } }\,}}} - \int _{{\mathcal {H}}_0} \rho _0^{{{\,\mathrm{ {}^{\vert } }\,}}} \text {D}_t \psi ^{{{\,\mathrm{ {}^{\vert } }\,}}} \, \text {d}V^{{{\,\mathrm{ {}^{\vert } }\,}}} - T{\mathscr {S}}_0 \ge 0 \) |
CCM | \({\mathscr {D}}_0 = {\varvec{P}} : \text {D}_t {\varvec{F}} - \mathrm {\rho _0 \text {D}_t \Psi - T{\mathscr {S}}_0 \ge 0 }\) |