Non-ordinary state based Peridynamics
Correspondence Peridynamics
The correspondence formulation is a non-ordinary state-based formulation provided by [14]. It has the goal to apply classical models to Peridynamics.
The non-local deformation gradient is defined as
\[\mathbf{F}=\int_{\mathcal{H}}\underline{\omega}\langle \boldsymbol{\xi}\rangle\underline{\mathbf{Y}}\langle \boldsymbol{\xi}\rangle\otimes\underline{\mathbf{X}}\langle \boldsymbol{\xi}\rangle dV \cdot \mathbf{K}^{-1}\]
with the positive definite shape tensor as
\[\mathbf{K}=\int_{\mathcal{H}}\underline{\omega}\langle \boldsymbol{\xi}\rangle\underline{\mathbf{X}}\langle \boldsymbol{\xi}\rangle\otimes\underline{\mathbf{X}}\langle \boldsymbol{\xi}\rangle dV\]
In numerical applications if bonds break the shape tensor is positive semi definite. $\det\mathbf{K}=0$ can occur and the inversion of the shape tensor won't work.
Based on this definition strain measures can be created to calculate the Cauchy stresses
\[\boldsymbol{\sigma} = f(\mathbf{F}, t, T, ...)\]
To get the force densities the First-Piola Kirchhoff stress tensor has to be calculated by
\[\mathbf{P} = \text{det}(\mathbf{F})\boldsymbol{\sigma}\mathbf{F}^{-T}\]
and finaly the force density vector can be determined as
\[\underline{\mathbf{T}} = \underline{\omega}\langle \boldsymbol{\xi}\rangle\mathbf{P}\mathbf{K}^{-1}\mathbf{\xi}\]
The 2D plane strain or plane stress models are represented in the Cauchy stresses by assuming that the strain in the third direction are zero or the stresses.
Zero-energy modes
For correspondence models, the so called zero-energy modes could occur [15]. These modes are non-physical and lead to unstable or unreasonable solutions. Several stabilization methods were published to overcome this problem [16], [17], [18], [19],[20],[21].
A promising approach implemented as global control in PeriLab was published by Wan et al. in 2019 [22]. Instead of a bond-based stabilization method proposed by Silling [23] Wan et al. developed a state-based stabilization method. As positive side effect this method stabilizes the solution for anisotropic material as well. The corrected force density state $\underline{\mathbf{T}}^C$ with suppression of the zero-energy mode is:
\[\underline{\mathbf{T}}^C=\underline{\mathbf{T}}+\underline{\mathbf{T}}^S.\]
Following Wan et al. \cite{WanJ2019} the suppression force density state $\underline{\mathbf{T}}^S$ is:
\[\underline{\mathbf{T}}^S\langle \boldsymbol{\xi}\rangle = \underline{\omega}\langle\boldsymbol{\xi}\rangle\mathbf{C}_1\underline{\mathbf{z}}.\]
with $\underline{\mathbf{z}}$ as the non-uniform deformation state
\[\underline{\mathbf{z}}\langle \boldsymbol{\xi}\rangle= \underline{\mathbf{Y}}\langle\boldsymbol{\xi} \rangle-\mathbf{F}\boldsymbol{\xi}\]
caused by the zero-energy mode. If the approximated non-local deformation gradient $\mathbf{F}$ exactly maps each undeformed bond to the deformed configuration no zero-energy mode occur. In that case the non-uniform deformation state is zero and the corrected force density state $\underline{\mathbf{T}}^C$ is equal to the force density state $\underline{\mathbf{T}}$. The second order tensor $\mathbf{C}_1$ is given as
\[\mathbf{C}_1=\mathbf{C}\cdot\cdot\mathbf{K}^{-1},\]
utilizing the elasticity tensor.