Ordinary state-based Peridynamics

For an isotropic Peridynamic solid and small deformations we can define $\underline{x}=|\underline{\mathbf{X}}|$ and $\underline{y}=|\underline{\mathbf{Y}}|$ and $\underline{e}=\underline{y}-\underline{x}=|\boldsymbol{\eta}|$

\[\underline{y}-\underline{x}\neq|\boldsymbol{\eta}|\]

for the general case

The force density scalar state can be defined as $\underline{t}=|\underline{\mathbf{T}}|$

The weighted volume is

\[m_V = \int_{\mathcal{H}} \underline{\omega}\langle \boldsymbol{\xi}\rangle \underline{x} \underline{x} dV\]

The dilatation is given as

\[\theta = \frac{3}{m_V} = \int_{\mathcal{H}}\underline{\omega}\langle \boldsymbol{\xi}\rangle \underline{x} \underline{e}\langle \boldsymbol{\xi}\rangle dV\]

\[\underline{t} = \frac{\omega\langle \boldsymbol{\xi}\rangle }{m_v}\left[3K \theta \underline{x} + 15G \underline{e}^d \right]\]

with the decomposition in the devatoring and isotropic part of the strain

\[\underline{e}^d\langle \boldsymbol{\xi}\rangle = \epsilon_{ij}^d\xi_i\frac{x_j}{|\boldsymbol{\xi}|}\]

and

\[\underline{e}^i\langle \boldsymbol{\xi}\rangle = \epsilon_{ij}^i\xi_i\frac{x_j}{|\boldsymbol{\xi}|}\]

The force density can be determined as

\[\underline{\mathbf{T}}=\underline{t}\frac{\underline{\mathbf{Y}}}{|\underline{\mathbf{Y}}|}\]

For plane stress and plane strain the equations are taken form [1].

TODO