Nonlinear Phononic Materials

Phononic materials are engineered media that can manipulate waves propagating through them due to their synthetic and periodic architecture. However, phononic materials are mostly studied by assuming linear elasticity laws, which apply to small-amplitude waves only. Real-world materials experience high-amplitude excitations and can even exhibit enriched dynamic responses due to nonlinearity. How such high-amplitude mechanical waves interact with phononic media is at large under investigation and the primary focus of this project. In this work, we are developing a fundamental understanding of how waves evolve through nonlinear phononic materials, specifically through periodic local microstructures, e.g. rough contacts, inspired from geomaterials. The nonlinearity in our materials stems from the nonlinear mechanical deformation of the rough surfaces. We analyzed these materials through two approaches: 1) analytically by developing an equivalent discrete periodic model [Grinberg and Matlack, Wave motion, 2020] and 2) numerically by considering a continuum with discrete contacts [Patil and Matlack, Wave motion, 2021].

  • Discrete spring-mass model

We employed an equivalent discrete spring-mass model to understand the fundamental wave propagation characteristics of our nonlinear phononic material. Wave propagation was studied using a perturbation approach to obtain approximate analytical solutions and verified through full-scale numerical simulations. Our study focused on primary and secondary transmission characteristics, and showed that secondary transmission depends on the band structure of the underlying media, and also on system parameters such as the number of contacts, excitation amplitude, unit cell constant, and external precompression [Grinberg and Matlack, Wave motion, 2020].

Phononic material with periodic rough contacts and equivalent discrete model. Plots show amplitude-dependent second harmonic transmission. The related article “Nonlinear elastic wave propagation in phononic material with periodic solid-solid contact interface” appeared in Wave Motion, 102466(2020) and may be found at (
  • Continuum model with discrete nonlinearity

To understand the role of elastic deformation between nonlinear contacts, we simulated nonlinear wave propagation within a finite element framework. Our model revealed that the wave self-interaction with nonlinear contacts generates zero- and second-harmonic frequencies, and also causes self-demodulation effects that generate very low frequencies. An interesting finding of the work was that the amplitude of these nonlinearly generated frequencies depends upon the local coupling of discrete contacts with the surrounding continuum and also on the contact distribution in the continuum [Patil and Matlack, Wave motion, 2021].

The effect of local discrete-continuum coupling on the amplitudes of nonlinearly-generated frequencies. The related article “Wave self-interactions in continuum phononic materials with periodic contact nonlinearity” appeared in Wave Motion, 102763 (2021) and may be found at (

This work was partially supported by the Army Research Office under Grant Number W911NF-20-1-0250

Journal Papers:


Conference Presentations:

  • G. U. Patil*, K.H. Matlack, Tunable bandgaps and second harmonic generation of a one-dimensional nonlinear phononic material with periodic rough interfaces. The Virtual Meeting 2020 of the Society of Engineering Sciences (2020)