Bond-based peridynamics
The initial version of peridynamics is the bond-based (BB) formulation. [Sil00]
Here, a pairwise force function $\boldsymbol{f}$ is defined and calculated for each bond of two material points, which depends on the strain of the bond and is aligned in its direction:
\[ \boldsymbol{f} = c \, \varepsilon^{ij} \, \boldsymbol{n} \; .\]
Here the micro-modulus constant [SA05]
\[c = \frac{18 \, \kappa}{\pi \, \delta^4} \;\]
and the strain of the bond [SB05]
\[\varepsilon^{ij} = \frac{l^{ij}-L^{ij}}{L^{ij}}\]
with bond lengths $L^{ij} =\left|\boldsymbol{\Delta X}^{ij}\right|$ and $l^{ij} =\left|\boldsymbol{\Delta x}^{ij}\right|$ are used.
The direction vector
\[\boldsymbol{n} = \frac{\boldsymbol{\Delta x}^{ij}}{l^{ij}}\]
is oriented in the direction of the bond.
To get the resulting body forces, now the force function is integrated over the whole body:
\[\boldsymbol{b}^{\mathrm{int},i} = \boldsymbol{b}^{\mathrm{int}} (\boldsymbol{X} ^ {i} , t) = \int_{\mathcal{H}_i} \boldsymbol{f} \; \mathrm{d}V^j \; .\]
| Size | Symbol | Unit |
|---|---|---|
| Pairwise force function | $\boldsymbol{f}$ | $\left[\frac{\mathrm{kg}}{\mathrm{m}^5\mathrm{s}^2}\right]$ |
| Micro-modulus constant [SA05] | $c$ | $\left[\frac{\mathrm{kg}}{\mathrm{m}^5\mathrm{s}^2}\right]$ |
| Bond strain | $\varepsilon^{ij}$ | $[-]$ |
| Bond in $\mathcal{B}_0$ | $\boldsymbol{\Delta X}^{ij}$ | $[\mathrm{m}]$ |
| Bond in $\mathcal{B}_t$ | $\boldsymbol{\Delta x}^{ij}$ | $[\mathrm{m}]$ |
| Bond length in $\mathcal{B}_0$ | $L^{ij}$ | $[\mathrm{m}]$ |
| Bond length in $\mathcal{B}_t$ | $l^{ij}$ | $[\mathrm{m}]$ |
| Direction vector | $\boldsymbol{n}$ | $[-]$ |
| Volume of point $j$ | $V^j$ | $\left[\mathrm{m}^3\right]$ |
| Internal force density | $\boldsymbol{b}^{\mathrm{int},i}$ | $\left[\frac{\mathrm{kg}}{\mathrm{m}^2\mathrm{s}^2}\right]$ |