Extreme stiffness tunability through the excitation of nonlinear defect modes

Rapid Communication

Extreme stiffness tunability through the excitation of nonlinear defect modes

M. Serra-Garcia, J. Lydon, and C. Daraio

Phys. Rev. E 93, 010901(R) – Published 12 January 2016

Abstract

The incremental stiffness characterizes the variation of a material's force response to a small deformation change. In lattices with noninteracting vibrational modes, the excitation of localized states does not have any effect on material properties, such as the incremental stiffness. We report that, in nonlinear lattices, driving a defect mode introduces changes in the static force-displacement relation of the material. By varying the defect excitation frequency and amplitude, the incremental stiffness can be tuned continuously to arbitrarily large positive or negative values. Furthermore, the defect excitation parameters also determine the displacement region at which the force-displacement relation is being tuned. We demonstrate this phenomenon experimentally in a compressed array of spheres tuning its incremental stiffness from a finite positive value to zero and continuously down to negative infinity.

Received 2 February 2015
Revised 7 December 2015

DOI:https://doi.org/10.1103/PhysRevE.93.010901

Physics Subject Headings (PhySH)

Classical mechanics Defects

Chaos & nonlinear dynamics

Statistical Physics & Thermodynamics

Authors & Affiliations

M. Serra-Garcia^1,*, J. Lydon¹, and C. Daraio^1,2

¹Department of Mechanical and Process Engineering, Swiss Federal Institute of Technology (ETH), Zurich 8092, Switzerland
²Engineering and Applied Science, California Institute of Technology, Pasadena, California 91125, USA

^*sermarc@ethz.ch

Article Text (Subscription Required)

Click to Expand

Supplemental Material (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 93, Iss. 1 — January 2016

Reuse & Permissions

Access Options

Article Available via CHORUS

Download Accepted Manuscript

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Tuning the stiffness through dynamic expansion. (a) Schematic of the tunable stiffness mechanism illustrated in a one-dimensional granular chain. The diagram shows the response of the lattice to a prescribed boundary displacement. During this displacement the defect is subject to a harmonic excitation at a fixed frequency and amplitude, as a consequence, it vibrates with an amplitude $A$ . As the lattice is compressed (green arrow), the defect mode is detuned from the excitation signal (red arrows). This results in a negative incremental stiffness due to dynamic contraction of the defect mode. (b) Changes in the driving frequency and amplitude of the excitation determine the incremental stiffness, and (c) the strain point at which the stiffness is being modified. The curves are offset for clarity.
Reuse & Permissions
Figure 2
Response of the nonlinear defect mode. (a) Theoretical defect mode (blue) and acoustic band (green) frequencies dependence on the prescribed displacement. Experimental measurements are plotted as red dots with the four curves in panel (b) marked with black crosses. (b) Normalized experimental velocity of the defect mode as a function of displacement of the lattice. Curves correspond to excitation frequencies of 10 (blue, solid line), 10.5 (green, dashed line), 11 (red, dashed-dotted line), and 11.5 kHz (cyan, dotted line). The frequencies in panel (a) are obtained by fitting these curves using a Lorentzian function. (c) Experimental velocity of the defect mode $v_{d}$ , measured at the site next to the defect particle for drive amplitudes of 4.2 (blue, solid line), 9.8 (green, dashed line), and 15.4 nm (red, dotted line) all at 10.5 kHz. (d) Numerical results corresponding to (c) for defects driven at 20, 50, and 80 nm, respectively. Our discrete particle model (see the Supplemental Material [21]) qualitatively reproduces the experimental results but is unable to make precise quantitative predictions, and this could be due to the fact that our model neglects experimental factors, such as internal particle and actuator resonances, as well as the nonlinear friction between the particles and the rods.
Reuse & Permissions
Figure 3
Experimental tuning of the incremental stiffness. Force-displacement curves for excitation amplitudes of (a) 5.9 nm, (b) 6.4 nm, (c) 7.54 nm, and (d) 10.9 nm. Shown below are the defect mode velocities [proportional to the mode amplitude $A (x)$ ] as a function of the overall displacement $x$ of the lattice. In panel (d), the system discontinuously transitions between two oscillation branches. This introduces a region of completely vertical slope in the force-displacement curve. The curves have been measured at an increasing displacement rate of 0.53 nm s.
Reuse & Permissions
Figure 4
Theoretical investigation. (a) Map relating the excitation parameters with the modified incremental stiffness and displacement point. Each point position on the map corresponds to tuning the stiffness to the value on the $Y$ axis at the displacement point indicated by the $X$ axis. Each dotted red line defines a set of tuned stiffness states that are accomplished by the same excitation frequency. The solid blue lines represent sets of tuned stiffness that are attained by the same excitation amplitude. The intersection between red lines and blue lines determines the excitation frequency and amplitude required to achieve the stiffness labeled by the $Y$ axis at the displacement labeled by the $X$ axis. (b) Zero frequency band gap obtained by choosing excitation parameters corresponding to zero stiffness for the lattice. The blue and green lines show the force transmitted with the defect drive on and off, respectively. When the defect excitation is turned off, the lattice acts as a linear spring for small deformations around the prescribed displacement value; when the defect excitation is turned on, there is a band gap centered at zero frequency. The dotted red line shows the band gap edge frequency $f_{c}$ . (c) Force-displacement relationships of the system when it is driven above the bifurcation amplitude. The presence of a tunable hysteresis allows the system to be used as a tunable damper. (d) Analytical force-displacement relation for a lattice of particles with the nonlinear interaction force law $F (δ) = A δ^{0.5}$ (see the Supplemental Material for details on the parameters used [21]) with a defect excitation frequency of 13.5 KHz and amplitudes 0.72 N (blue, solid line), 0.74 N (green, dashed line), 0.76 N (red, dashed-dotted line), and 0.78 N (cyan, dotted line). For this potential exciting the defect mode results in an arbitrarily large positive stiffness.
Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics