Revisiting the peridynamic motion equation due to characterization of boundary conditions

Abstract

The peridynamic motion equation was investigated once again. The origin of incompatibility between boundary conditions and peridynamics was analyzed. In order to eliminate this incompatibility, we proposed a new peridynamic motion equation in which the effects of boundary traction and boundary displacement constraint were introduced. The new peridynamic motion equation is invariant under the transformations of rigid translation and rotation. Meanwhile, it also satisfies the requirements of total linear and angular momentum equilibrium. By this motion equation, three kinds of boundary value problems containing the displacement boundary condition, the traction boundary condition and mixed boundary condition are characterized in peridynamics. As examples, we calculated static tension and longitudinal vibration of a finite rod. The acquired solutions exhibit obvious nonlocal features, and the vibration has the dispersion similar to one dimensional atom chain vibration.

Appendices

Appendix A

With Eq. (11), we have

$$\begin{aligned}&{} \mathbf y (\mathbf x ,t)\times \left\{ \int _{\partial \Omega }[K(\mathbf x ,\mathbf x ')\mathbf p (\mathbf x ',t)+ L(\mathbf x ,\mathbf x ')\mathbf y (\mathbf x ',t)]\mathrm {d}a(\mathbf x ')\right. \nonumber \\&\qquad \left. +\,\int _{\Omega }{} \mathbf f (\mathbf x ,\mathbf x ',t)\mathrm {d}v(\mathbf x ')+\mathbf b (\mathbf x ,t)\right\} \nonumber \\&\quad = \mathbf y (\mathbf x ,t)\times \rho (\mathbf x )\ddot{\mathbf{u }}(\mathbf x ,t), \end{aligned}$$

(A1)

where $\mathbf y (\mathbf x ,t)=\mathbf x +\mathbf u (\mathbf x ,t)$. Integrating Eq. (A1) over $\Omega$ leads to

$$\begin{aligned}&\int _{\Omega }{} \mathbf y (\mathbf x ,t)\times \int _{\partial \Omega }[K(\mathbf x ,\mathbf x ')\mathbf p (\mathbf x ',t)\nonumber \\&\qquad +\, L(\mathbf x ,\mathbf x ')\mathbf y (\mathbf x ',t)]\mathrm {d}a(\mathbf x ')\mathrm {d}v(\mathbf x )\nonumber \\&\qquad +\,\int _{\Omega }{} \mathbf y (\mathbf x ,t)\times \int _{\Omega }{} \mathbf f (\mathbf x ,\mathbf x ',t)\mathrm {d}v(\mathbf x ') \mathrm {d}v(\mathbf x )\nonumber \\&\qquad +\,\int _{\Omega }{} \mathbf y (\mathbf x ,t)\times \mathbf b (\mathbf x ,t)\mathrm {d}v(\mathbf x )\nonumber \\&\quad = \int _{\Omega }{} \mathbf y (\mathbf x ,t)\times \rho (\mathbf x )\ddot{\mathbf{u }}(\mathbf x ,t)\mathrm {d}v(\mathbf x ). \end{aligned}$$

(A2)

In Eq. (A2), the first term can be rewritten as

$$\begin{aligned}&\int _{\Omega }{} \mathbf y (\mathbf x ,t)\times \int _{\partial \Omega } [K(\mathbf x ,\mathbf x ')\mathbf p (\mathbf x ',t)\nonumber \\&\qquad +\, L(\mathbf x ,\mathbf x ')\mathbf y (\mathbf x ',t)]\mathrm {d}a(\mathbf x ')\mathrm {d}v(\mathbf x )\nonumber \\&\quad = \int _{\partial \Omega }\int _{\Omega }[\mathbf y (\mathbf x ,t)-\mathbf y (\mathbf x ',t)]\times [K(\mathbf x ,\mathbf x ')\mathbf p (\mathbf x ',t)\nonumber \\&\qquad +\,L(\mathbf x ,\mathbf x ')\mathbf y (\mathbf x ',t)] \mathrm {d}v(\mathbf x )\mathrm {d}a(\mathbf x ')\nonumber \\&\qquad +\,\int _{\partial \Omega }{} \mathbf y (\mathbf x ,t)\times \mathbf p (\mathbf x ,t)\mathrm {d}a(\mathbf x ). \end{aligned}$$

(A3)

Owing to Eq. (14) or (16), Eq. (A3) reduces to

$$\begin{aligned}&\int _{\Omega }{} \mathbf y (\mathbf x ,t)\times \int _{\partial \Omega } [K(\mathbf x ,\mathbf x ')\mathbf p (\mathbf x ',t)\nonumber \\&\qquad +\, L(\mathbf x ,\mathbf x ')\mathbf u (\mathbf x ',t)]\mathrm {d}a(\mathbf x ')\mathrm {d}v(\mathbf x )\nonumber \\&\quad = \int _{\partial \Omega }{} \mathbf y (\mathbf x ,t)\times \mathbf p (\mathbf x ,t)\mathrm {d}a(\mathbf x ). \end{aligned}$$

(A4)

With Eq. (2), it is easy to give that [2]

$$\begin{aligned}&\int _{\Omega }{} \mathbf y (\mathbf x ,t)\times \int _{\Omega }{} \mathbf f (\mathbf x ,\mathbf x ',t)\mathrm {d}v(\mathbf x ') \mathrm {d}v(\mathbf x )\nonumber \\&\quad =\frac{1}{2}\int _{\Omega }\int _{H_x}[\mathbf y (\mathbf x ,t)-\mathbf y (\mathbf x ',t)]\nonumber \\&\qquad \times \, \mathbf f (\mathbf x ,\mathbf x ',t)\mathrm {d}v(\mathbf x ')\mathrm {d}v(\mathbf x ). \end{aligned}$$

(A5)

In terms of Eq. (15) or (17), Eq. (A5) reduces to

$$\begin{aligned} \int _{\Omega }{} \mathbf y (\mathbf x ,t)\times \int _{\Omega }{} \mathbf f (\mathbf x ,\mathbf x ',t)\mathrm {d}v(\mathbf x ') \mathrm {d}v(\mathbf x )=0. \end{aligned}$$

(A6)

Substituting Eqs. (A4) and (A6) into Eq. (A2) yields

$$\begin{aligned}&\int _{\partial \Omega }{} \mathbf y (\mathbf x ,t)\times \mathbf p (\mathbf x ,t)\mathrm {d}a(\mathbf x )+ \int _{\Omega }{} \mathbf y (\mathbf x ,t)\times \mathbf b (\mathbf x ,t)\mathrm {d}v(\mathbf x )\nonumber \\&\quad = \int _{\Omega }{} \mathbf y (\mathbf x ,t)\times \rho (\mathbf x )\ddot{\mathbf{u }}(\mathbf x ,t)\mathrm {d}v(\mathbf x ), \end{aligned}$$

(A7)

which shows that the equilibrium of angular momentum is satisfied for a bounded body subjected to boundary traction, boundary displacement constraint and body force.

Appendix B

With Eq. (36), Eq. (33) is written as

$$\begin{aligned} \alpha (x)p+\beta (x)y(x)+\gamma \int _{-l/2}^{l/2}y(x)\mathrm {d}x-l\gamma y(x)=0. \end{aligned}$$

(B1)

Let

$$\begin{aligned} c=\int _{-l/2}^{l/2}y(x)\mathrm {d}x{,} \end{aligned}$$

(B2)

so we have

$$\begin{aligned} \alpha (x)p+\beta (x)y(x)+\gamma c-l\gamma y(x)=0. \end{aligned}$$

(B3)

That is

$$\begin{aligned} y(x)=\frac{\gamma c+p\alpha (x)}{l\gamma -\beta (x)}. \end{aligned}$$

(B4)

Inserting Eq. (B4) in Eq. (B2) leads to Eq. (39).