Oscillating vector solitary waves in soft laminates

https://doi.org/10.1016/j.jmps.2020.104058Get rights and content

Abstract

Vector solitary waves are nonlinear waves of coupled polarizations that propagate with constant velocity and shape. In mechanics, they hold the potential to control locomotion, mitigate shocks and transfer information, among other functionalities. Recently, such elastic waves were numerically observed in compressible rubber-like laminates. Here, we conduct numerical experiments to characterize the possible vector solitary waves in these laminates, and expose a new type of waves whose amplitude and velocity oscillate periodically without dispersing in time. This oscillation is a manifestation of a periodic transfer of energy between the two wave polarizations, which we consider as internal mode of the solitary wave. We find that the vector solitary waves propagate faster at higher amplitudes, and determine a lower bound for their velocity. We describe a procedure for identifying which initial strains generate such vector solitary waves. This procedure also enables an additional classification between tensile and compressive solitary waves, according to the way that the axial strain changes as the waves propagate.

Introduction

Nonlinearities in wave mechanics are the source of fascinating phenomena, from instabilities (Skipetrov and Maynard, 2000), harmonic generation (Ganesh, Gonella, 2017, Khajehtourian, Hussein, Saltiel, Neshev, Fischer, Krolikowski, Arie, Kivshar, 2008), extreme energy transfer (Zhang et al., 2018) and non-reciprocal transmission (Lepri and Casati, 2011), to the formation of shocks (Chockalingam and Cohen, 2020), rogue waves (Baronio et al., 2012) and solitary waves (Silling, 2016). The latter are amplitude-dependent waves which propagate with a constant velocity and fixed shape, owing to a balance between dispersion and nonlinearity in the system (Dauxois and Peyrard, 2006). Initially studied in fluid mechanics (Boyd, 2015), quantum mechanics (Kasamatsu and Tsubota, 2006) and optics (Kivshar, Agrawal, 2003, Stegeman, Segev, 1999), solitary waves are recently gaining increased attention from the solid mechanics community (Deng, Wang, He, Tournat, Bertoldi, 2018, Deng, Wang, Tournat, Bertoldi, 2020b, Katz, Givli, 2019, Mo, Singh, Raney, Purohit, 2019, Nadkarni, Daraio, Kochmann, 2014). This interest was triggered by the quest for mechanical metamaterials—artificial composites with properties and functionalities not found in nature (Bertoldi, Vitelli, Christensen, van Hecke, 2017, Christensen, Kadic, Kraft, Wegener, 2015, Craster, Guenneau, 2012, Kadic, Milton, van Hecke, Wegener, 2019, Srivastava, 2015); and its pertinent progress in additive manufacturing techniques for creating these architectured materials (Raney and Lewis, 2015).

In addition to the mathematical significance of their analysis, the study of solitary waves in mechanical systems shows also technological potential in impact mitigation (Yasuda et al., 2019), mechanical logic gates (Raney et al., 2016), nondestructive testing (Nasrollahi et al., 2017), wave focusing (Deng et al., 2019a), and robot locomotion (Deng et al., 2020a). The framework of these works is almost exclusively of discrete models, with only few results on elastic continua. A specific class of continuum that exhibits nonlinearities is of soft materials, whose microscopic composition (Arruda and Boyce, 1993) and ability to sustain large deformations (Ogden, 1997) are the source of constitutive and geometrical nonlinearities, respectively. Thus, soft materials constitute a barely explored platform to theoretically study solitary waves, and in turn experimentally realize using current 3D printing capabilities (Bandyopadhyay, Vahabzadeh, Shivaram, Bose, 2015, Garcia, Wu, Kim, Yu, Zhu, 2019, Truby, Lewis, 2016).

To the best of our knowledge, the first numerical observation of one-dimensional solitary waves in elastic periodic continua was by LeVeque (2002b) and Bale et al. (2002), using a finite-volume method they developed, which was followed by the studies of Andrianov, Danishevskyy, Ryzhkov, Weichert, 2013, Andrianov, Danishevskyy, Ryzhkov, Weichert, 2014; Hussein and Khajehtourian (2018); LeVeque and Yong (2003); Navarro et al. (2015) and the references therein. As explained in these works, the periodicity of the media causes dispersion, whose balance with geometrical and constitutive nonlinearities forms elastic solitary waves. Only recently, this principle for generating solitary waves was tested by Ziv and Shmuel (2020) in two-dimensional continuum elastodynamics. There, the authors have developed a designated finite-volume method to simulate finite motions of soft laminates whose displacement field consists of two coupled components. By application of this method to a periodic repetition of compressible Gent (1996) layers that differ in their mass density, the authors were able to observe the formation of vector solitary waves—solitary waves with (at least) two components and two polarizations that are coupled one with the other—in this case, when the axial and transverse components of the displacement field are coupled. To the best of our knowledge, that was the first report of vector solitary waves in an elastic continuum, within the framework of nonlinear elastodynamics (Ogden, 1997). An earlier report of such waves in a discrete mechanical medium was given by Deng, Raney, Tournat, Bertoldi, 2017, Deng, Tournat, Wang, Bertoldi, 2019b, who conceived a model made of rigid squares that are connected using linear springs at their corners, thereby allowing for two coupled degrees of freedom for the squares: rotation and translation. While both the systems considered by Deng, Raney, Tournat, Bertoldi, 2017, Deng, Tournat, Wang, Bertoldi, 2019b and the system considered here are two-dimensional and nonlinear, there are fundamental differences between them, as we explain in the sequel. These differences lead to opposite velocity-amplitude relation for the vector solitary waves in each system.

Here, we carry out a comprehensive study of the vector solitary waves that were observed by Ziv and Shmuel (2020), by analyzing a large set of numerical experiments that were generated using the method developed in that work, with the following findings. First, we find that these vector solitary waves can be divided into two types which differ in their velocity and strain profile. We term the two types quasi-pressure and quasi-shear solitary waves, since in the limiting case when the heterogeneity vanishes and the strains are small, they reduce to the standard pressure and shear waves, respectively. We show that the profile of quasi-pressure vector solitary waves resembles the sech2 function, which is the solitary wave solution of the well-known KdV equation (Korteweg and de Vries, 1895). We find that while quasi-pressure solitary waves maintain an identical wave profile as they pass between adjacent unit cells, the profile and velocity of quasi-shear solitary waves change between different cells in a periodic and permanent manner. This oscillation occurs through a periodic transfer of energy between the two polarizations of the vector solitary wave, which we consider as internal mode of the solitary wave. Similar long-lived oscillations were reported before in other fields (Campbell, Schonfeld, Wingate, 1983, Kivshar, Pelinovsky, Cretegny, Peyrard, 1998, Mantsyzov, 1995, Szankowski, Trippenbach, Infeld, Rowlands, 2010), and were also reported recently in an acoustic system that is described using the continuum limit of a discrete model based on mechanical–electrical analogies (Zhang et al., 2017). Our results are the first report of oscillating vector solitary waves that are generated from the equations of continuum elastodynamics.

Next, we characterize the relation between the amplitude of the strain and the velocity of the solitary waves. This is done for a wide range of initial conditions, which were carefully chosen to generate in each experiment a single solitary wave. These experiments show that the velocity is a monotonically increasing function of the amplitude. This observation agrees with the results of Ziv and Shmuel (2020), who generated trains of solitary waves from a single set of initial conditions, and observed that the taller waves in the train propagate faster than the shorter waves. Notably, this velocity-amplitude relation is opposite to the relation for the vector solitary waves in the discrete mechanical model of Deng, Raney, Tournat, Bertoldi, 2017, Deng, Tournat, Wang, Bertoldi, 2019b, there the waves with smaller amplitudes are faster. We also determine the velocity of Bloch-Floquet waves in the laminate in the limit of low-frequency, long-wavelength linear elastodynamics (Santosa and Symes, 1991), to find they serve as a lower bound to the velocity of solitary waves in the nonlinear settings. Our results thus generalize the findings of LeVeque and Yong (2003) and Andrianov et al. (2014), who showed that one-dimensional solitary waves in nonlinear laminates are supersonic,1 similarly to the feature of solitary wave solutions to the KdV equation (Dauxois and Peyrard, 2006).

Lastly, we describe a procedure for determining which initial deformations will generate vector solitary waves. This is done using contour maps of the characteristic velocities as functions of the strain, identifying the initial strain and loading path in these maps, and employing a certain criterion regarding the gradient of the characteristic velocity along the path, as explained in detail later. These maps are also useful in identifying how the axial strain in the laminate changes during the propagation of the solitary waves. Specifically, we identify two domains in the map for quasi-pressure solitary waves, namely, a domain of solitary waves that tend to increase the axial strain in the laminate and a separate domain of solitary waves that tend to decrease the axial strain in the laminate. In accordance with this tendency, we term the latter compressive solitary waves, notwithstanding the fact that the sign of axial strain in the laminate may be positive. Similarly, we term the former tensile solitary waves, notwithstanding the fact that the sign of axial strain may be negative. Interestingly, such a separation is absent from the map for quasi-shear solitary waves, since these waves can only decrease the axial strain, as explained later.

The rest of the paper is organized as follows. Section 2 contains a description of the relevant elastodynamics problem, together with its governing vectorial equations. Section 3 first revisits solutions of the differential relations between the strain components associated with the benchmark problem of a homogeneous medium (Ziv and Shmuel, 2019). In the second part of this Sec., the computational study of the heterogeneous medium is carried out, where we also show that the solutions that are associated with the homogeneous medium serve as an estimators for the differential relations in the main problem. A recap of our main results together with comments on future work close the paper in Section 4.

Section snippets

Problem statement and governing equations

We consider an infinite periodic repetition in the X1 direction of two hyperelastic phases, namely, a and b, governed by the same strain energy density function Ψ, and different initial mass density ρL. We set X1=0 at the beginning of a certain a phase, and denote its number by n=0, such that even and odd values of n correspond to layers made of phase a and b, respectively. At the initial state, the laminate is subjected to a combination of axial and transverse displacement fields as functions

Computational study

Homogeneous benchmark problem. Before we proceed to the study of the main problem, it is advantageous to present the strain relations of the homogeneous benchmark problem. As we will show in the sequel, the reason is that they serve as estimators to the laminated case. Thus, the differential relation between the axial and shear strains in quasi-shear and quasi-pressure waves traversing homogeneous media are given by (Davison, 1966; Ziv and Shmuel 2019)quasi-pressure:ϵTϵA=η+,quasi-shear:ϵTϵA=

Summary

Ziv and Shmuel (2020) have developed a finite-volume method to solve the equations governing finite-amplitude smooth waves with two coupled components in nonlinear compressible laminates. The application of this method to simulate the response of pre-strained compressible Gent two-phase laminates has revealed the generation of vector solitary waves, whose axial and transverse polarizations are coupled. Here, we have used the method to conduct a large set of numerical experiments with different

CRediT authorship contribution statement

Ron Ziv: Conceptualization, Methodology, Software, Formal analysis, Investigation, Writing - original draft, Writing - review & editing, Visualization. Gal Shmuel: Conceptualization, Writing - original draft, Writing - review & editing, Visualization, Supervision, Funding acquisition.

Declaration of Competing Interest

I hereby declare that there is no conflict of interest.

Acknowledgments

We thank an anonymous reviewer whose constructive comments helped improving this paper. We also thank Prof. Oleg Gendelman for exposing us to the analogy with internal modes of kinks in Φ4 theory. We acknowledge the support of the Israel Science Foundation, funded by the Israel Academy of Sciences and Humanities (Grant no. 1912/15), the United States-Israel Binational Science Foundation (Grant no. 2014358), and Ministry of Science and Technology (grant no. 880011).

References (71)

  • Y.S. Kivshar et al.

    Internal modes of solitary waves

    Phys. Rev. Lett.

    (1998)
  • C. Mo et al.

    Cnoidal wave propagation in an elastic metamaterial

    Phys. Rev. E

    (2019)
  • R.W. Ogden

    Non-Linear Elastic Deformations

    (1997)
  • J. Raney et al.

    Stable propagation of mechanical signals in soft media using stored elastic energy

    Proc. Natl. Acad. Sci. USA

    (2016)
  • J. Zhang et al.

    Bright and gap solitons in membrane-type acoustic metamaterials

    Phys. Rev. E

    (2017)
  • D.S. Bale et al.

    A wave propagation method for conservation laws and balance laws with spatially varying flux functions

    SIAM J. Sci. Comput.

    (2002)
  • A. Bandyopadhyay et al.

    Three-dimensional printing of biomaterials and soft materials

    MRS Bulletin

    (2015)
  • F. Baronio et al.

    Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves

    Phys. Rev. Lett.

    (2012)
  • K. Bertoldi et al.

    Flexible mechanical metamaterials

    Nat. Rev. Mater.

    (2017)
  • D.R. Bland

    Plane isentropic large displacement simple waves in a compressible elastic solid

    Z. Angew. Math. Phys. ZAMP

    (1965)
  • J.P. Boyd

    Dynamical meteorology | solitary waves

  • D.K. Campbell et al.

    Resonance structure in kink-antikink interactions in ϕ4 theory

    Phys. D

    (1983)
  • S. Chockalingam et al.

    Shear shock evolution in incompressible soft solids

    Journal of the Mechanics and Physics of Solids

    (2020)
  • A.R. Cioroianu et al.

    Normal stresses in elastic networks

    Phys. Rev. E

    (2013)
  • T. Dauxois et al.

    Physics of Solitons

    (2006)
  • L. Davison

    Propagation of plane waves of finite amplitude in elastic solids

    Journal of the Mechanics and Physics of Solids

    (1966)
  • L. Davison

    Fundamentals of Shock Wave Propagation in Solids

    (2008)
  • B. Deng et al.

    Pulse-driven robot: motion via solitary waves

    Sci. Adv.

    (2020)
  • B. Deng et al.

    Focusing and mode separation of elastic vector solitons in a 2D soft mechanical metamaterial

    Phys. Rev. Lett.

    (2019)
  • B. Deng et al.

    Anomalous collisions of elastic vector solitons in mechanical metamaterials

    Phys. Rev. Lett.

    (2019)
  • B. Deng et al.

    Metamaterials with amplitude gaps for elastic solitons

    Nat. Commun.

    (2018)
  • B. Deng et al.

    Nonlinear transition waves in free-standing bistable chains

    Journal of the Mechanics and Physics of Solids

    (2020)
  • D. Espíndola et al.

    Shear shock waves observed in the brain

    Phys. Rev. Appl.

    (2017)
  • R. Ganesh et al.

    Nonlinear waves in lattice materials: Adaptively augmented directivity and functionality enhancement by modal mixing

    J. Mech. Phys. Solids

    (2017)
  • D. Garcia et al.

    Heterogeneous materials design in additive manufacturing: Model calibration and uncertainty-guided model selection

    Additive Manufacturing

    (2019)
  • Cited by (5)

    • Vibrations and waves in soft dielectric elastomer structures

      2023, International Journal of Mechanical Sciences
      Citation Excerpt :

      Furthermore, Dai and Li [113] examined strongly nonlinear axisymmetric waves in a circular hyperelastic rod characterized by a compressible Mooney-Rivlin material model and observed the appearance of seven types of nonlinear waves, namely solitary waves of radial contraction and radial expansion, solitary shock waves of radial contraction and radial expansion, periodic waves and two types of periodic shock waves. Recently, the large-amplitude nonlinear elastic waves in soft periodic structures were investigated [114–116] and the experimental and numerical results demonstrated the existence of the vector solitary waves in these soft periodic structures. Moreover, the results revealed that the structural geometry and the initial deformation can be harnessed to tune the wave characteristics and control the selective generation of the solitary waves.

    • Wavenumber-space band clipping in nonlinear periodic structures

      2021, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
    View full text