Temporally stabilized peridynamics methods for shocks in solids

Abstract

The computational methods for the shocks modeling would face two major challenges: (1) the severe damage with large deformations and (2) the intermittent waves. Peridynamics (PD) takes the integral form of its governing equation and shows exceeding advantages in modeling large deformation and severe damage. On the other hand, the propagation of intermittent wave within the PD based numerical system often experiences oscillatory instability. It can be attributed to the instability in time domain and the zero energy mode. Aiming for addressing such issues, the temporally stabilized PD methods are proposed in the present work. The stabilization force component is introduced and the general framework of stabilized PD methods is established. The formulation of the corresponding force state is proposed based on the features of the spurious oscillations. The case studies indicate that the stabilized PD methods are capable of effectively suppressing the nonphysical oscillations and are well-suited for the bond-based as well as the state-based PD formulations.

Appendix A: Von Neumann stability analysis for stabilized PD

Here, the von Neumann stability analysis for stabilized PD is presented to show the effectiveness of the Method I. We conduct the analysis in Sect. 3.1 again, and the Eq. (25) turns to be:

$$\begin{aligned} \begin{aligned} \rho \ddot{u}^n(x)=\,&A\int ^{\delta }_{-\delta }C(\xi )[u^n(x+\xi )-u^n(x)] \\&+D(\xi ) [\frac{u^{n+1}(x+\xi )-u^{n-1}(x+\xi )}{2\Delta t}\\&-\frac{u^{n+1}(x)-u^{n-1}(x)}{2\Delta t}]\text {d}\xi \end{aligned} \end{aligned}$$

(73)

where $\ddot{u}^n(x)=[u^{n+1}(x)-2u^n(x)+u^{n-1}(x)]/{\Delta t}^2$, $C(\xi )=c/\left\| {\xi }\right\|$ and $D(\xi )=b \alpha _{\text{ p }}/\left\| {\xi }\right\|$ in 1-D. Similarily, let

$$\begin{aligned} u^n(x)={\lambda }^n \exp (\kappa x\sqrt{-1}) \end{aligned}$$

(74)

where $\kappa$ is a positive real number and $\lambda$ is a complex number. Then Eq. (73) can be written as:

$$\begin{aligned} \begin{aligned} \frac{\rho }{{\Delta t}^2}(\lambda -2+\lambda ^{-1})=\,&A\int ^{\delta }_{0}2C(\xi )[\cos (\kappa \xi )-1]\\&+\frac{D(\xi )}{\Delta t}\lambda [\cos (\kappa \xi )-1]\\&-\frac{D(\xi )}{\Delta t}\lambda ^{-1} [\cos (\kappa \xi )-1]\text {d}\xi \end{aligned} \end{aligned}$$

(75)

Actually, $D(\xi )$ can be regarded as $\beta C(\xi )$ and with the definition in Eq. (28), we obtain

$$\begin{aligned} \begin{aligned}&\left( 1+\frac{\beta }{\Delta t}\frac{H_\kappa \Delta {t}^2}{\rho }\right) \lambda -2\left( 1 - \frac{{H_\kappa \Delta t}^2}{\rho } \right) \\&\quad +\left( 1-\frac{\beta }{\Delta t}\frac{H_\kappa \Delta {t}^2}{\rho }\right) \lambda ^{-1}=0 \end{aligned} \end{aligned}$$

(76)

where $\beta =b \alpha _{\text{ p }}/c$. Finally, $\lambda$ is given by:

$$\begin{aligned} \lambda =\frac{\left( 1-\frac{H_\kappa \Delta {t}^2}{\rho }\right) \pm \sqrt{(1-\frac{H_\kappa \Delta {t}^2}{\rho })^2-(1-\theta ^2)}}{1+\theta } \end{aligned}$$

(77)

and $\theta =\frac{\beta H_\kappa \Delta t}{\rho } \ge 0$.

Based on Eq. (77), $| \lambda |$ of Method I is in the following form:

$$\begin{aligned} | \lambda | ^2=\frac{\left( 1-\frac{H_\kappa \Delta {t}^2}{\rho }\right) ^2 + 1-\theta ^2 - \left( 1-\frac{H_\kappa \Delta {t}^2}{\rho }\right) ^2 }{(1+\theta )^2}=\frac{1-\theta }{1+\theta } \end{aligned}$$

(78)

Obviously, $| \lambda |\le 1$ and $| \lambda |=1$ only when $\theta =0$.