Dynamic breaking of axial symmetry of acoustic waves in crystals as the origin of nonlinear elasticity and chaos: Analytical model and MD simulations

TL;DR

This study proposes a Chain of Springs and Masses (CSM) framework to interpret molecular dynamics of oriented FCC crystals under surface shear, revealing a dynamical force perpendicular to the applied load that scales as the square of the pressure and breaks axial symmetry for transverse waves. By deriving an interlayer potential with a dominant axially symmetric term plus a threefold-angular component, the authors connect the observed nonlinear coupling and Z-motions to a 3D extension of the Hénon–Heiles potential, uncovering rich nonlinear and potentially chaotic dynamics. Phase-space analyses and driven-oscillator models show how energy transfers between directions and how angular dependence follows a cos(3α) pattern, with clear implications for nonlinear elasticity and wave propagation in bulk crystals. The work also links these findings to Christoffel equations and suggests experimental avenues to verify dynamic symmetry breaking, highlighting material-dependent variations in the nonlinear coupling strength and the potential for broader impact on dislocation dynamics and acoustic metamaterials.

Abstract

A Chain of Springs and Masses (CSM) model is used in the interpretation of molecular dynamics (MD) simulations of movement of atoms in orientated FCC crystals. A force of dynamic origin is found that is perpendicular to the direction of the external shear pressure. It is proportional to the square of the applied pressure; It causes breaking of axial symmetry for propagation of transverse acoustic waves. It leads to a non-linear elastic response of crystals and to chaotic patterns in the motion of atoms. We provide an analytical derivation of an effective atomistic 3D potential for interaction between crystallographic layers. The potential is found to possess a component that has an anharmonic threefold axial symmetry around one direction. It reduces to the H{é}non-Heinen potential in a 2D cross-section, leading to mathematically rich, complex dynamic features. Results of simulation predict displacements of atoms that are inconsistent with the static theory of elasticity that may have been overlooked in experiments.

Figures (15)

• Figure 1: Comparison of average displacement of layers in the X-direction (figure a) and in the Z-direction (b), when the shear pressure of 100 MPa is applied at the surface in the X-direction. Labels indicate the number of layer.
• Figure 2: Comparison of average velocity of layers in the X- and Z-directions, $V_X$ and $V_Z$, under the same simulation conditions and for the same layers as in figure \ref{['fig:XYZL00A2']}. Velocities are time derivatives of displacement curves shown in figure \ref{['fig:XYZL00A2']}.
• Figure 3: Comparison of average forces acting on layers in the X- and Z-directions, under the same simulation conditions and for the same layers as in figures \ref{['fig:XYZL00A2']} and \ref{['fig:XYZL00D2']}. Notice the difference in vertical scale in a) and b).
• Figure 4: Phase-space portrait of $V_z(V_x)$ for three values of pressure, as shown in the legend, when pressure is applied at an angle $\ang{0}$ with respect to the X-axis. a) shows dependencies for layer 1 and b) for layer 2). $V_{x0}$ scales up linearly with pressure; therefore, its value changes here by 6000 times, while $V_{z0}$ changes as $P^2$; hence, it changes here $36\cdot 10^6$ times.
• Figure 5: Phase space dependencies $V_{z}(V_{x})$ for layer 50 (time is a hidden parameter there). For a pressure of 1000 MPa a number of curves are shown for a set of angle values, where angles are the direction of the applied pressure at the surface. For pressure applied at an angle of $\ang{0}$ with respect to the X-axis, there are also curves drawn for 2000, 3000 and 4000 MPa.