Frictionless Motion of Lattice Defects

Energy dissipation by fast crystalline defects takes place mainly through the resonant interaction of their cores with periodic lattice. We show that the resultant effective friction can be reduced to zero by appropriately tuned acoustic sources located on the boundary of the body. To illustrate the general idea, we consider three prototypical models describing the main types of strongly discrete defects: dislocations, cracks, and domain walls. The obtained control protocols, ensuring dissipation-free mobility of topological defects, can also be used in the design of metamaterial systems aimed at transmitting mechanical information.

Figure 1

(a) Piecewise linear stress-strain relation $\sigma =\sigma (ϵ)$; the macroscopic driving force $G^M(V)=S_2-S_1$. (b) Dispersion relation $\omega (k)$ for $\mathrm{Im}(k)=0$ (acoustic branch); $k_j$ correspond to the radiated waves in cases $K=1$ and $K=3$; see the text. Green circles correspond to ac sources behind the defect, and magenta squares ahead of the defect.

Figure 2

(a) Kinetic domain for the case $K=1$; open squares show the selected TW solutions reached numerically [34]. (b) Amplitude dependence of $G^{*}$ and $V^{*}$; insets show normalized strains $\overline{ϵ}(\eta )=ϵ(\eta )/{ϵ}_c$. Parameters: ${\sigma }_0=2$ and ${ϵ}_c=1$.

Figure 3

Steady propagation of the phase transition front obtained numerically for a finite chain with $N=1000$ starting from the initial data (3) with $V=0.5$. Normalized strain ${\overline{ϵ}}_j(t)={ϵ}_j(t)/{ϵ}_c$ is shown at three different moments of time. The inset shows a comparison with the analytical TW solution (solid line).

Figure 4

(a) Dislocation propagation driven by the constant force $\tau$; macroscopic driving force $G^M(V)=S_2-S_1$. (b) Dispersion relation for $\mathrm{Im}(k)=0$ (optical branch). The wave numbers $k_j$ define radiated waves in the cases $K=1$ and $K=3$.

Figure 5

(a) Kinetic domains for dislocations showing admissible solutions with $K=1$. There is only one damping wave in the big pink domain and more than one in the smaller ones. (b) The ac amplitude dependence of $G^{*}(V^{*})$ and $V^{*}$; insets show displacements $\overline{u}(\eta )=u(\eta )/u_c$. Parameters: ${\sigma }_0=2$ and $u_c=1$.

Figure 6

(a) Constitutive relation for a mechanical fuse and schematic representation of a crack propagating with velocity $V$ under remote load $\tau$; the strain energy jump $⟦w⟧=-S_1$. (b) Dispersion relations ${\omega }_{+}(k)$ (optical branch) and ${\omega }_{-}(k)$ (acoustic branch) for $\mathrm{Im}(k)=0$ characterizing intact and broken lattices, respectively.

Figure 7

(a) Kinetic domains for admissible solutions with $K=1$ and the ac wave coming from ahead. There is only one damping wave in the big pink domain and more than one in the smaller ones. (b) Amplitude dependence of $G^{*}(V^{*})$ and $V^{*}$. Material properties are $\gamma =1$ and $u_c=1$.