Continuum-kinematics-inspired peridynamics
Continuum-kinematics-inspired peridynamics (CPD) is a formulation that is supposed to deliver more freedom in specifying material parameters. The internal force density is calculated as the sum of three types of point interactions which are one-, two- and three-neighbor interactions [JMS19]:
\[\boldsymbol{b}^{\mathrm{int},i} = \boldsymbol{b}_1^{\mathrm{int},i} + \boldsymbol{b}_2^{\mathrm{int},i} + \boldsymbol{b}_3^{\mathrm{int},i} \; .\]
| Size | Symbol | Unit |
|---|---|---|
| Internal force density | $\boldsymbol{b}^{\mathrm{int},i}$ | $\left[\frac{\mathrm{kg}}{\mathrm{m}^2\mathrm{s}^2}\right]$ |
| Force density shares due to one-, two- & three-neighbor interactions | $\boldsymbol{b}_1^{\mathrm{int},i}$ , $\boldsymbol{b}_2^{\mathrm{int},i}$ , $\boldsymbol{b}_3^{\mathrm{int},i}$ | $\left[\frac{\mathrm{kg}}{\mathrm{m}^2\mathrm{s}^2}\right]$ |
One-neighbor interactions
One-neighbor interactions in CPD correspond to the bonds in bond-based peridynamics, but there is a slightly different way to calculate the internal forces.
First, the neighborhood volume is determined:
\[V_\mathcal{H}^i = \beta^i \, \frac {4} {3} \, \pi \, \delta^3 \; .\]
Here $\beta^i\in [0,1]$ is a factor for the completeness of the neighborhood that takes incomplete point families at the surface into account (see figure 1).
Now, the effective one-neighbor volume can be calculated
\[V_1^i = \frac{V_\mathcal{H}^i}{N_1^i}\]
with the number of interactions $N_1^i$.
The internal force density is determined by
\[ \boldsymbol{b}_1^{{\mathrm{int}},i} = \int_{\mathcal{H}_1^i} C_1 \left( \frac{l^{ij}}{L^{ij}} - 1 \right) \frac{\boldsymbol{\Delta x}^{ij}}{l^{ij}} \; \mathrm{d} V_1^i\]
with the parameters:
| Size | Symbol | Unit |
|---|---|---|
| Neighborhood volume | $V_\mathcal{H}^i$ | $[\mathrm{m}^3]$ |
| Neighborhood completeness | $\beta^i\in [0,1]$ | $[-]$ |
| Effective one-neighbor volume | $V_1^i$ | $[\mathrm{m}^3]$ |
| Number of one-neighbor interactions | $N_1^i$ | $[-]$ |
| Material constant | $C_1$ | $[\frac{\mathrm{kg}}{\mathrm{m}^5\mathrm{s}^2}]$ |
| Relative length measures | $L^{ij}$, $l^{ij}$ | $[\mathrm{m}]$ |
Two-neighbor interactions
For two-neighbor interactions, the deformation of the area spanned by point $i$ and two of its neighbors $j$ and $k$ is analyzed to calculate the internal force density. For this, relative area measures are defined:
\[ A^{ijk}=\left| \boldsymbol{\Delta X}^{ij} \times \boldsymbol{\Delta X}^{ik} \right| \; , \qquad a^{ijk}=\left| \boldsymbol{\Delta x}^{ij} \times \boldsymbol{\Delta x}^{ik} \right| \; , \qquad \boldsymbol{a}^{ijk}= \boldsymbol{\Delta x}^{ij} \times \boldsymbol{\Delta x}^{ik} \; .\]
Other sizes needed to identify the force density are the material constant $C_2$ and the effective two-neighbor volume
\[ V_2^i = \frac{\left(V_\mathcal{H}^i\right)^2}{N_2^i}\]
with the number of interactions $N_2$.
The internal force density induced by two-neighbor interactions is
\[ \boldsymbol{b}_{2}^{\mathrm{int}, \, i} = 2 \, C_2 \int_{\mathcal{H}_2^i} \left( \frac{a^{ijk}}{A^{ijk}} - 1 \right) \frac{\boldsymbol{\Delta x}^{ik} \times \boldsymbol{a}^{ijk}}{a^{ijk}} \; \mathrm{d} V_2^i \; .\]
| Size | Symbol | Unit |
|---|---|---|
| Relative area measures | $A^{ijk}$, $a^{ijk}$, $\boldsymbol{a}^{ijk}$ | $[\mathrm{m}^2]$ |
| Effective two-neighbor volume | $V_2^i$ | $[\mathrm{m}^6]$ |
| Number of two-neighbor interactions | $N_2^i$ | $[-]$ |
| Material constant | $C_2$ | $[\frac{\mathrm{kg}}{\mathrm{m}^9\mathrm{s}^2}]$ |
Three-neighbor interactions
Three-neighbor interactions regard the volume defined by the bond vectors between point $i$ and its three neighbors $j$, $k$ and $l$:
\[V^{ijkl} = \left(\boldsymbol{\Delta X}^{ij} \times \boldsymbol{\Delta X}^{ik}\right) \cdot \boldsymbol{\Delta X}^{il} \;,\qquad v^{ijkl} = \left(\boldsymbol{\Delta x}^{ij} \times \boldsymbol{\Delta x}^{ik}\right) \cdot \boldsymbol{\Delta x}^{il} \;.\]
Additionally, the effective three-neighbor volume
\[ V_3^i = \frac{ \left(V_\mathcal{H}^i\right)^3}{N_3^i} \; .\]
is defined. For the internal force density of three-neighbor interactions, the equation
\[\boldsymbol{b}_{3}^{\mathrm{int}, \, i} = 3 \, C_3 \int_{\mathcal{H}_3^i} \left( \frac{\left|{v^{ijkl}}\right|}{\left|{V^{ijkl}}\right|} - 1 \right) \frac{\left(\boldsymbol{\Delta x}^{ik} \times \boldsymbol{\Delta x}^{il}\right) v^{ijkl}}{\left|{v^{ijkl}}\right|} \; \mathrm{d} V_3^i\]
with the material constant $C_3$ is used.
| Size | Symbol | Unit |
|---|---|---|
| Relative volume measures | $V^{ijkl}$, $v^{ijkl}$ | $[\mathrm{m}^3]$ |
| Effective three-neighbor volume | $V_3^i$ | $[\mathrm{m}^9]$ |
| Number of three-neighbor interactions | $N_3^i$ | $[-]$ |
| Material constant | $C_3$ | $[\frac{\mathrm{kg}}{\mathrm{m}^{13}\mathrm{s}^2}]$ |