Nonlinear Bloch waves and balance between hardening and softening dispersion

Abstract

The introduction of nonlinearity alters the dispersion of elastic waves in solid media. In this paper, we present an analytical formulation for the treatment of finite-strain Bloch waves in one-dimensional phononic crystals consisting of layers with alternating material properties. Considering longitudinal waves and ignoring lateral effects, the exact nonlinear dispersion relation in each homogeneous layer is first obtained and subsequently used within the transfer matrix method to derive an approximate nonlinear dispersion relation for the overall periodic medium. The result is an amplitude-dependent elastic band structure that upon verification by numerical simulations is accurate for up to an amplitude-to-unit-cell length ratio of one-eighth. The derived dispersion relation allows us to interpret the formation of spatial invariance in the wave profile as a balance between hardening and softening effects in the dispersion that emerge due to the nonlinearity and the periodicity, respectively. For example, for a wave amplitude of the order of one-eighth of the unit-cell size in a demonstrative structure, the two effects are practically in balance for wavelengths as small as roughly three times the unit-cell size.

1. Introduction

(a). Phononic materials

Phononic materials are elastic materials with prescribed phonon wave propagation properties. While the term ‘phonon’ is formally used in the physical sciences to describe vibration states in condensed matter at the atomic scale, in the present context, we use it to broadly describe elastic wave propagation modes. Like crystalline materials, a phononic material has local intrinsic properties and is therefore mathematically treated as a medium that is spatially extended to infinity. Compared to a homogeneous and geometrically uniform elastic continuum, a phononic material exhibits rich and unique dynamical properties due to the presence of some form of non-homogeneity and/or non-uniformity in either an ordered or disordered manner. In the ordered case, phononic materials are constructed from a repeated array of identical unit cells which enables the calculation of the elastic band structure for a given topological configuration. This direct exposure, and access, to the inherent dynamical properties of phononic materials has vigorously chartered a new direction in materials physics, at a multitude of scales, and has already begun to impact numerous applications ranging from vibration control [1,2], through subwavelength sound focusing [3,4] and cloaking [5,6], to reducing the thermal conductivity of semiconductors [7–9] and stabilizing a wall-bounded fluid flow [10]. A discussion of applications and references are provided in recent review articles [11–13] and books [14–17], and special journal issues on the topic assemble some of the latest advances in the field [18–24].

(b). Nonlinear elastic wave dispersion

The majority of theoretical investigations of wave motion in elastic solids are based on linear analysis, that is, linear constitutive laws and linear strain–displacement relationships are assumed (see [25,26] and references therein). The incorporation of nonlinear effects, however, gives rise to a broader range of physical phenomena including amplitude-dependent elastic wave motion [27–31]. The effects of nonlinearity have also been studied for other types of waves such as water waves [32,33] and electrostatic and electromagnetic waves in plasmas [34,35].

Finite-strain waves in elastic solids is a subset among the broader class of nonlinear waves. From a mathematical perspective, a formal treatment of finite strain requires the incorporation of a nonlinear strain tensor in setting up the governing equations of motion. Regardless of the type of nonlinearity, a common analysis framework has been one in which the dispersion is viewed to arise linearly, e.g. due to the presence of a microstructure or geometrical constraints, and that such dispersion may be balanced with nonlinear effects to allow for the generation of nonlinear travelling waves of fixed spatial profile such as shock waves and solitons [30,31]. In contrast to this dispersion/nonlinearity balancing framework where the focus is on finding these special types of waves and characterizing the amplitude-dependence, or wavenumber-dependence, of their speeds, it has recently been shown that nonlinearity in itself may cause dispersion without the need for a linear dispersive mechanism [36,37]. This perspective provides a motivation to derive dispersion relations that inherently embody the effects of the nonlinearities on the dispersion, i.e. amplitude-dependent relations for general wave motion that encompass both the speed (or frequency) and the wavenumber.

Extending to periodic media, nonlinear dispersion relations appear in various contexts, for example, electronic waves in metals and semiconductors [38] and electromagnetic waves in photonic crystals [39,40]. In the context of nonlinear phononic materials, there are several studies that follow the premise of Bloch wave propagation analysis. These include investigations on systems exhibiting material nonlinearity, analysed using the method of multiple scales [41–43], perturbation analysis [44–47], the harmonic balance method [48,49] and the transfer matrix (TM) method in conjunction with a perturbation technique [50]. The effects of nonlinearity on the dynamics of periodic materials have also been explored in the context of atomic-scale models incorporating anharmonic potentials; see, for example, a recent paper focusing on phonon transport [51]. Concerning finite-strain dispersion in a layered elastic medium, this was recently investigated by Andrianov et al. [40] via a homogenization approach whereby the periodic unit cell was first homogenized as a linear medium and subsequently a finite-strain dispersion relation was derived for the averaged medium. This approach therefore does not retain the periodic character in the derived dispersion relation. On the experimental track, numerous studies have been conducted on nonlinear wave phenomena particularly in periodic granular chains, e.g. [52,53]. It is evident that the effects of nonlinearity in phononic/granular materials could be used to enrich the design of devices in numerous engineering applications, such as for shock mitigation [54], tunable wave filtering [45], focusing [55] and rectification [56]. A more recent study experimentally investigated nonlinear vibrational waves in periodic strings [57]. The field of nonlinear elastic wave propagation continues to grow and branch into new directions, such as, for example, supratransmission in dissipative periodic structures [58] and the utilization of large-amplitude waves to create topologically protected edge states in two-dimensional lattices [59].

(c). Overview

In this paper, we present a theoretical treatment of elastic wave motion in phononic materials in the presence of nonlinearity, specifically the type arising from finite elastic strain. We consider phononic crystals, which is a class of phononic materials in which the prime dispersion mechanism is Bragg scattering.¹ For ease of exposition, we focus on a one-dimensional layered material model admitting only longitudinal displacements (which may also be viewed as a model for a periodic thin rod). Since the TM method provides the backbone of our approach, we first briefly overview it, in conjunction with Bloch's theorem [61], for the exact analysis of simple one-dimensional linear phononic crystals (§2). We then review the treatment of geometric nonlinearity, i.e. finite strain, in the context of a homogeneous medium (§3a). In §3b, we merge the previous derivations, that is, we allow the finite-strain dispersion relation for a homogeneous medium to represent the motion characteristics in a single layer of a periodically layered one-dimensional phononic crystal and subsequently incorporate this relation into the TM formalism to generate the band structure. The developed formulation is then used to investigate the effects of geometric nonlinearity on the elastic band structure and Bloch mode shapes as a function of the wave amplitude. While the finite-strain dispersion within each layer is exact, the nonlinear dispersion relation we obtain for the overall one-dimensional phononic crystal represents an approximate solution. In §4, we verify the derived dispersion relation using brute-force space–time simulations followed by a Fourier transformation into the wavenumber-frequency domain. The simulations are also used to determine the upper limit of wave amplitude per unit-cell length for which the theory is accurate.

We conclude our investigation in §5 with an analysis, and an emerging interpretation, that is of consequence to the general field of nonlinear wave propagation. With our developed analytical formulation in place, we separately analyse the effect of nonlinearity in ‘hardening’ the dispersion relation at an increasing rate as the wavenumber increases, and the effect of the periodicity in linearly causing the opposite effect of ‘softening’ the dispersion relation, also at an increasing rate as the wavelength decreases. We then show that by combining both effects simultaneously, we can select the wave amplitude (and material properties) such that a neutral state is obtained up to a certain wavenumber. This selection gives us a balance between the linear and nonlinear alterations to the dispersion relation to yield a spatially invariant wave profile.

2. Wave propagation in one-dimensional linear phononic crystals

Bloch's theorem [61] provides the underlying mathematical framework for obtaining the elastic band structure (i.e. dispersion curves) for a phononic crystal. There are several approaches for applying the theorem to a unit cell modelled as a continuum. In this work, we use the TM method, which is described briefly below (for a background on the method and further details, see [62–64]).

We begin our dynamic analysis of a phononic crystal with the statement of the equation of motion. As mentioned earlier, we restrict ourselves to a simple one-dimensional model, i.e. a thin rod with the thickness being much smaller than the propagating wavelength, for which the equation of motion is

$${(E{u}_{,x})}_{,x}=\rho u_{,tt},$$ 2.1

where x, t, u = u(x, t), E = E(x) and ρ = ρ(x) denote the position, time, displacement, material Young's modulus and material density, respectively. For simplicity, we assume that the cross-sectional area of the rod is non-varying with x.

Equation (2.1) may be used to study the propagation of elastic waves in various one-dimensional media. In particular, we consider a homogeneous, linearly elastic one-dimensional rod of infinite extent (having no boundaries at which waves may reflect), and apply a plane wave solution of the form

$$u(x,t)=B{\mathrm{e}}^{\mathrm{i}(\kappa x-\omega t)},$$ 2.2

where B is the wave amplitude, κ is the wavenumber, ω is the temporal frequency of the travelling wave, and $\mathrm{i}=\sqrt{-1}$. Here, we make another assumption about the cross-sectional dimensions of the rod, namely, that they are much smaller than the wavelengths of all waves considered in the analysis. Substituting equation (2.2) into equation (2.1) provides the linear dispersion relation

$$E{\kappa }^2=\rho {\omega }^2.$$ 2.3

This approach may also be applied to heterogeneous media provided the heterogeneity is periodic. In this case, equation (2.2) specializes to represent Bloch's theorem (where B = B(x, κ)), and it suffices to analyse only a single unit cell representing the unique segment that is repeated to generate the periodic medium and to apply periodic boundary conditions to this segment. In figure 1, we present a simple bi-material model of a one-dimensional phononic crystal in the form of a layered periodic rod (where the unit cell is enclosed in a dashed box). The spatial lattice spacing of the one-dimensional phononic crystal is denoted by the constant a. The same analysis may also be applied to a unit cell with a stepwise varying cross-sectional area, but this case will not be considered here.

Figure 1.

Continuous model of a one-dimensional two-phased phononic crystal viewed as a periodic thin rod. (Online version in colour.)

For an arbitrary homogeneous layer j in the unit cell, the associated material properties, which are constant, are denoted as E^( j) and ρ^( j). The longitudinal velocity in layer j is therefore $c^{(\hspace{0.17em}j)}=\sqrt{E^{(\hspace{0.17em}j)}/{\rho }^{(\hspace{0.17em}j)}}$. The layer is bordered by layer j − 1 on the left and layer j + 1 on the right. Denoting the thickness of an arbitrary layer by d^( j), the cell length is $a=\sum _{j=1}^n{d}^{(\hspace{0.17em}j)}$ for a unit cell with n layers. Following this notation, the solution to equation (2.1) is formed from the superposition of forward (transmitted) and backward (reflected) travelling waves with a harmonic time dependence

$$u(x,t)=[B_{+}^{(\hspace{0.17em}j)}\hspace{0.17em}{\mathrm{e}}^{\mathrm{i}{\kappa }^{(\hspace{0.17em}j)}x}+B_{-}^{(\hspace{0.17em}j)}\hspace{0.17em}{\mathrm{e}}^{-\mathrm{i}{\kappa }^{(\hspace{0.17em}j)}x}]\hspace{0.17em}{\mathrm{e}}^{-\mathrm{i}\omega t},$$ 2.4

where κ^( j) = ω/c^( j) is the layer wavenumber. The spatial components of equation (2.4) may be written along with those of the stress

$$\sigma =E{u}_{,x},$$ 2.5

in compact form as

$$\left[\begin{array}{c}u(x)\\ \sigma (x)\end{array}\right]=\left[\begin{array}{cc}1& 1\\ \mathrm{i}{Z}^{(j)}& -\mathrm{i}{Z}^{(j)}\end{array}\right]\hspace{0.17em}\left[\begin{array}{c}{B}_{+}^{(j)}{\mathrm{e}}^{\mathrm{i}{\kappa }^{(j)}x}\\ B_{-}^{(j)}{\mathrm{e}}^{-\mathrm{i}{\kappa }^{(j)}x}\end{array}\right]={\mathbf{H}}_j\left[\begin{array}{c}{B}_{+}^{(j)}{\mathrm{e}}^{\mathrm{i}{\kappa }^{(j)}x}\\ B_{-}^{(j)}{\mathrm{e}}^{-\mathrm{i}{\kappa }^{(j)}x}\end{array}\right],$$ 2.6

where Z^( j) = E^( j)κ^( j). There are two conditions that must be satisfied at the layer interfaces: (1) the continuity of displacement and (2) the continuity of stress. This allows for the substitution of the relation x^( j)_R = x^( j)_L + d^( j) (where x^( j)_R and x^( j)_L denote the position of the right and left boundary, respectively, of layer j) into equation (2.6) and thus relating the displacement and stress at x^( j)_L to those at x^( j)_R. Subsequently, by setting x = x^( j)_L in equation (2.6), we get

$$\left[\begin{array}{c}u(x_{\mathrm{R}}^{(j)})\\ \sigma (x_{\mathrm{R}}^{(j)})\end{array}\right]={\mathbf{H}}_j{\mathbf{D}}_j{\mathbf{H}}_j^{-1}\hspace{0.17em}\left[\begin{array}{c}u(x_{\mathrm{L}}^{(j)})\\ \sigma (x_{\mathrm{L}}^{(j)})\end{array}\right]={\mathbf{T}}_j\left[\begin{array}{c}u(x_{\mathrm{L}}^{(j)})\\ \sigma (x_{\mathrm{L}}^{(j)})\end{array}\right],$$ 2.7

where

$${\mathbf{D}}_j=\left[\begin{array}{cc}{\mathrm{e}}^{\mathrm{i}{\kappa }^{(\hspace{0.17em}j)}{d}^{(\hspace{0.17em}j)}}& 0\\ 0& {\mathrm{e}}^{-\mathrm{i}{\kappa }^{(\hspace{0.17em}j)}{d}^{(\hspace{0.17em}j)}}\end{array}\right],$$ 2.8

and T_j, the TM for layer j, has the expanded form

$${\mathbf{T}}_j=\left[\begin{array}{cc}\cos ({\kappa }^{(\hspace{0.17em}j)}{d}^{(\hspace{0.17em}j)})& \left(\frac{1}{Z^{(\hspace{0.17em}j)}}\right)\sin ({\kappa }^{(\hspace{0.17em}j)}{d}^{(\hspace{0.17em}j)})\\ -Z^{(\hspace{0.17em}j)}\sin ({\kappa }^{(\hspace{0.17em}j)}{d}^{(\hspace{0.17em}j)})& \cos ({\kappa }^{(\hspace{0.17em}j)}{d}^{(\hspace{0.17em}j)})\end{array}\right].$$ 2.9

As previously stated, equation (2.7) relates the displacement and stress at x^( j)_L to those at x^( j)_R of the same layer j. However, since the construction of the TM is valid for any layer and x^( j)_L≡x^( j−1)_R, the result in equation (2.7) can be extended recursively across several layers. In the interest of brevity, let $\mathbf{y}(.)={[\mathbf{u}(.)\hspace{0.17em}\sigma (.)]}^{\mathrm{T}}$, thus,

$$\mathbf{y}(x_{\mathrm{R}}^n)={\mathbf{T}}_n{\mathbf{T}}_{n-1}\dots {\mathbf{T}}_1\mathbf{y}(x_{\mathrm{L}}^1)=\mathbf{T}\mathbf{y}(x_{\mathrm{L}}^1).$$ 2.10

Ultimately, the displacement and stress at the left end of the first layer (x = x¹_L) in a unit cell are related to those at the right boundary of the nth layer (x = xⁿ_R) by the cumulative TM, T.

Now we turn to Bloch's theorem, which states that the time harmonic response at a given point in a unit cell is the same as that of the corresponding point in an adjacent unit cell except for a phase difference of e^iκa. This relation is given by f(x + a) = e^iκaf(x), which when applied to the states of displacement and stress across a unit cell gives

$$\mathbf{y}(x_{\mathrm{R}}^n)={\mathrm{e}}^{\mathrm{i}\kappa a}\mathbf{y}(x_{\mathrm{L}}^1).$$ 2.11

Combining equations (2.10) and (2.11) yields the eigenvalue problem

$$[\mathbf{T}-\mathbf{I}\gamma ]\mathbf{y}(x_{\mathrm{L}}^1)=\mathbf{0},$$ 2.12

where γ = e^iκa. The solution of equation (2.12), which appears in complex conjugate pairs, provides the dispersion relation κ = κ(ω) for the one-dimensional phononic crystal. Real-valued wavenumbers, calculated from γ using equation (2.13), support propagating wave modes, whereas imaginary wavenumbers, extracted from γ using equation (2.14), represent spatially attenuating modes:

$${\kappa }_{\mathrm{R}}=\frac{1}{a}\hspace{0.17em}\mathrm{Re}[\frac{1}{\mathrm{i}}\hspace{0.17em}\ln \gamma ]$$ 2.13

and

$${\kappa }_{\mathrm{I}}=\frac{1}{a}\hspace{0.17em}\mathrm{Im}[\frac{1}{\mathrm{i}}\hspace{0.17em}\ln \gamma ].$$ 2.14

3. Treatment of nonlinearity

We now provide a theoretical treatment of finite-strain dispersion; first we review the prerequisite problem of a one-dimensional homogeneous medium, and follow with the derivation for a one-dimensional phononic crystal. In the homogeneous medium problem, the approach is exact for a stable travelling wave regardless of the amplitude of the wave, i.e. strong nonlinearities are treated exactly as long as the wave has not reached a state of bifurcation. In the subsequent derivation of the phononic crystal dispersion curves, the accuracy decreases with increasing wave amplitude.

(a). Finite-strain waves in one-dimensional homogeneous media

The equation of motion and finite-strain dispersion relation is reviewed here for one-dimensional plane wave motion in a bulk homogeneous medium without consideration of lateral effects. In principle, this problem is equivalent to that of a slender rod. In the derivations, all terms in the nonlinear strain tensor are retained and no high-order terms emerging from the differentiations are subsequently neglected. These derivations are found in [36,60], but for completeness, we repeat them here. The reader is also referred to [36] for a verification of the theoretical approach by means of a comparison with a standard finite-strain numerical simulation considering instantaneous wave dispersion.

(i). Equation of motion

Introducing u as the elastic longitudinal displacement, the exact complete Green–Lagrange strain field in our one-dimensional model is given by

$$ϵ=\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+\frac{1}{2}{\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }x}\right)}^2,$$ 3.1

where the first and second terms on the r.h.s. represent the linear and nonlinear parts, respectively.

Using Hamilton's principle, we write the equation of motion under longitudinal stress as

$${\int }_0^t(\delta T-\delta U^e)\hspace{0.17em}\mathrm{d}t=0,$$ 3.2

where T and U^e denote kinetic and elastic potential energies, respectively. We note that no external nonconservative forces are permitted because of our interest in the free wave propagation problem. Furthermore, the effects of lateral inertia are neglected. The variation of kinetic energy is obtained using integration by parts and is given as

$$\delta T=-\rho A{\int }_0^l(u_{,tt}\delta u)\hspace{0.17em}\mathrm{d}x,$$ 3.3

where l denotes the length of a portion of the one-dimensional medium. Similarly, the variation of elastic potential energy is written as

$$\delta U^e={\int }_0^l{\int }_A(\sigma \delta ϵ)\hspace{0.17em}\mathrm{d}A\hspace{0.17em}\mathrm{d}x,$$ 3.4

where σ is the longitudinal stress and A is the cross-sectional area. We choose to base our analysis on the second Piola–Kirchhoff stress and model the stress–strain relationship by Hooke's Law, σ = Eϵ. Using equation (3.4), and with the aid of integration by parts, we can now write the variation of elastic potential energy as

$$\delta U^e={\int }_0^l\{\frac{1}{2}EAh(h^2-1)\delta u^{\mathrm{\prime }}\}\mathrm{d}x,$$ 3.5

where u^′ = du/dx = u_,x, and h is an agent variable defined as

$$h=1+u^{\mathrm{\prime }}.$$ 3.6

Substitution of equations (3.5) and (3.3) into equation (3.2) produces the exact finite-strain equation of motion as

$$\rho A{u}_{,tt}=\frac{1}{2}EA(3h^2-1)u^{\mathrm{\prime }\mathrm{\prime }}.$$ 3.7

If the longitudinal deformation is infinitesimal, then u^′ is small and from equation (3.6), h ≈ 1. Substitution of h = 1 into equation (3.7) leads to

$$\rho A{u}_{,tt}=EA{u}^{\mathrm{\prime }\mathrm{\prime }},$$ 3.8

which is the equation of motion describing infinitesimal longitudinal deformation.

(ii). Dispersion relation

Using equation (3.6), we rewrite equation (3.7) as

$$u_{,tt}-c^2{u}^{\mathrm{\prime }\mathrm{\prime }}=\frac{1}{2}{[c^2[3{(u^{\mathrm{\prime }})}^2+{(u^{\mathrm{\prime }})}^3]]}^{\mathrm{\prime }},$$ 3.9

where $c=\sqrt{E/\rho }$. Differentiation of equation (3.9) with respect to x gives

$${(u_{,tt})}^{\mathrm{\prime }}-c^2{u}^{(3)}=\frac{1}{2}{[c^2[3{(u^{\mathrm{\prime }})}^2+{(u^{\mathrm{\prime }})}^3]]}^{\mathrm{\prime }\mathrm{\prime }}.$$ 3.10

Defining $\overline{u}=u^{\mathrm{\prime }}$ gives

$${\overline{u}}_{,tt}-c^2{\overline{u}}_{,xx}=\frac{1}{2}{[c^2[3{\overline{u}}^2+{\overline{u}}^3]]}_{,xx},$$ 3.11

and introducing a phase variable z = κx + ω_fint, where κ represents the wavenumber for a unidirectional wave and ω_fin represents the wave frequency under finite strain, equation (3.11) becomes

$${\omega }_{\mathrm{fin}}^2{\overline{u}}_{,zz}-c^2{\kappa }^2{\overline{u}}_{,zz}=\frac{1}{2}{\kappa }^2{[c^2[3{\overline{u}}^2+{\overline{u}}^3]]}_{,zz}.$$ 3.12

Integrating equation (3.12) twice leads to

$$({\omega }_{\mathrm{fin}}^2-c^2{\kappa }^2)\overline{u}-\frac{c^2{\kappa }^2}{2}[3{\overline{u}}^2+{\overline{u}}^3]=0,$$ 3.13

where the non-zero constants of integration (in the form of polynomials in z) represent secular terms which we have set equal to zero to ensure that all waves remain bounded. Selecting the positive root of equation (3.13) we get

$$\overline{u}(z)=\frac{-3+\sqrt{1+8{\omega }_{\mathrm{fin}}^2/c^2{\kappa }^2}}{2}.$$ 3.14

Since $\overline{u}=u_{,x}$, we recognize that $\overline{u}=\kappa u_{,z}$ and therefore equation (3.14) represents a first-order ordinary differential equation with z and u as the independent and dependent variables, respectively.

Now we return to equation (3.9) and consider for initial conditions a sinusoidal displacement field with amplitude B. This represents a fundamental harmonic signal for which we seek to characterize its dispersive behaviour. In principle, any choice of the initial velocity field is permitted. Following the change of variables that has been introduced, we impose a balance between the spatial and temporal phase which allows us to set up the problem at z = 0. Thus, we have the following restrictions on the $\overline{u}(z)$ function given in equation (3.14):

$$\overline{u}(0)=\kappa B\mathrm{and}{\overline{u}}_{,z}(0)=0.$$ 3.15

These represent initial conditions in the wave phase, z, for equation (3.12) and allow for the introduction of the wave amplitude, B, into the formulation. Applying equation (3.15) to equation (3.14) enables us to use the latter to solve for ω_fin for a given value of κ. This leads to the exact dispersion relation,

$${\omega }_{\mathrm{fin}}(\kappa ;B)=\sqrt{\frac{2+3B\kappa +{(B\kappa )}^2}{2}}\omega ,$$ 3.16

where ω is the frequency based on infinitesimal strain,

$$\omega (\kappa )=c\kappa .$$ 3.17

By taking the limit, ${\lim }_{B\to 0}{\omega }_{\mathrm{fin}}(\kappa ;B)$, in equation (3.16) we recover equation (3.17) which is the standard linear dispersion relation for a one-dimensional homogeneous elastic medium or a thin rod [65]. We note that equation (3.16) is a general nonlinear dispersion relation that is independent of the wave profile.

For demonstration, six amplitude-dependent finite-strain dispersion curves based on equation (3.16) are plotted in figure 2. These curves describe the fundamental dispersive properties that emerge due to the incorporation of finite strain. The curves demonstrate that nonlinearity by itself causes wave dispersion in an elastic medium, i.e. without the need for a linear dispersive mechanism. From a physical point of view, one may envision an initial prescribed harmonic wave being set free at some point in time. The dispersion relation of equation (3.16) describes the frequency versus wavenumber relation for this wave as it disperses in the presence of amplitude-dependent finite strain. This concept was tested numerically and validated in [36]. Superimposed in figure 2 is the dispersion curve based on infinitesimal strain, i.e. equation (3.17). It is noted that the deviation between a finite-strain curve and the infinitesimal-strain curve increases with wavenumber, and the effect of the wave amplitude on this deviation is illustrated in the figure for six finite-strain cases where the value of B is doubled from one case to the other.

Figure 2.

Frequency dispersion curves for a one-dimensional homogeneous elastic medium [36]. The finite-strain dispersion relation is based on equation (3.16); the infinitesimal strain dispersion relation is based on equation (3.17). (Online version in colour.)

(b). Finite-strain waves in one-dimensional phononic crystals

The TM method is now used to obtain a dispersion relation for a one-dimensional phononic crystal whose constituent materials are exhibiting finite-strain dispersion. The outcome is an approximate overall dispersion relation since the construction of the TM is based on a linear strain–displacement relationship (see equations (2.5) and (2.6)). While not exact, this approach provides a quantitative prediction of the effects of nonlinearity on the location and size of band gaps and the values of the group velocity across the spectrum, all as a function of wave amplitude.

As presented in §2, the TM method is applicable in either the absence or presence of nonlinearity; the distinction is made in the definition of κ^( j) in equation (2.4). For the linear problem, κ^( j) = κ^( j)(ω) = ω/c^( j) as outlined earlier. A similar relationship between the jth-layer wavenumber and the finite-strain wave frequency, ω_fin, is now developed, i.e. κ^( j) = κ^( j)(ω_fin).

First we rewrite equation (3.16) explicitly for layer j,

$${\omega }_{\mathrm{fin}}=c^{(\hspace{0.17em}j)}{\kappa }^{(\hspace{0.17em}j)}\sqrt{\frac{2+3B{\kappa }^{(\hspace{0.17em}j)}+{[B{\kappa }^{(\hspace{0.17em}j)}]}^2}{2}},$$ 3.18

which may be cast as the following fourth-order characteristic equation:

$${[{\kappa }^{(\hspace{0.17em}j)}]}^2(1+B{\kappa }^{(\hspace{0.17em}j)})(2+B{\kappa }^{(\hspace{0.17em}j)})-2\frac{{\omega }_{\mathrm{fin}}^2}{{[c^{(\hspace{0.17em}j)}]}^2}=0.$$ 3.19

Solving equation (3.19) gives

$${\kappa }_{1,2}^{(\hspace{0.17em}j)}=\frac{1}{12B}(-9+P^{(\hspace{0.17em}j)}\mp \sqrt{Q^{(\hspace{0.17em}j)}-R^{(\hspace{0.17em}j)}})$$ 3.20a

and

$${\kappa }_{3,4}^{(\hspace{0.17em}j)}=-\frac{1}{12B}(9+P^{(\hspace{0.17em}j)}\pm \sqrt{Q^{(\hspace{0.17em}j)}+R^{(\hspace{0.17em}j)}}),$$ 3.20b

where

$$\begin{array}{rl}{P}^{(\hspace{0.17em}j)}& =\sqrt{\frac{33c^{(\hspace{0.17em}j)}{A}^{(\hspace{0.17em}j)}+12(4{[c^{(\hspace{0.17em}j)}]}^2-24B^2{\omega }_{\mathrm{fin}}^2+{[A^{(\hspace{0.17em}j)}]}^2)}{c^{(\hspace{0.17em}j)}{A}^{(\hspace{0.17em}j)}}},\end{array}$$ 3.21a

$$\begin{array}{rl}{Q}^{(\hspace{0.17em}j)}& =\frac{66c^{(\hspace{0.17em}j)}{A}^{(\hspace{0.17em}j)}-48({[c^{(\hspace{0.17em}j)}]}^2-6B^2{\omega }_{\mathrm{fin}}^2)-12{[A^{(\hspace{0.17em}j)}]}^2}{c^{(\hspace{0.17em}j)}{A}^{(\hspace{0.17em}j)}},\end{array}$$ 3.21b

$$\begin{array}{rl}{R}^{(\hspace{0.17em}j)}& =\frac{54\sqrt{3c^{(\hspace{0.17em}j)}{A}^{(\hspace{0.17em}j)}}}{\sqrt{11c_0^{(\hspace{0.17em}j)}{A}^{(\hspace{0.17em}j)}+4(4{[c^{(\hspace{0.17em}j)}]}^2+{[A^{(\hspace{0.17em}j)}]}^2-24B^2{\omega }_{\mathrm{fin}}^2)}},\end{array}$$ 3.21c

$$\begin{array}{rl}\text{and}{A}^{(\hspace{0.17em}j)}& ={(-99B^2{\omega }_{\mathrm{fin}}^2{c}^{(\hspace{0.17em}j)}+8{[c^{(\hspace{0.17em}j)}]}^3+3B{\omega }_{\mathrm{fin}}\sqrt{(1536B^2+321{[c^{(\hspace{0.17em}j)}]}^2)B^2{\omega }_{\mathrm{fin}}^4-48{[c^{(\hspace{0.17em}j)}]}^4})}^{1/3}.\end{array}$$ 3.21d

At this point, equation (3.20) is substituted into the Z^( j) = E^( j)κ^( j) equations in the TM formulation presented in §2. This yields a nonlinear enriched eigenvalue problem that we may use to obtain an approximation of the finite-strain dispersion curves of a one-dimensional phononic crystal.² While the technique is not limited to small values of B/a, its accuracy reduces as the strength of the nonlinearity increases. In §5, we numerically examine the accuracy as a function of B/a.

To demonstrate the effects of nonlinearity, we consider the same geometric features as the periodic bi-material rod in figure 1 and the following ratio of material properties: c⁽²⁾/c⁽¹⁾ = 2 and ρ⁽²⁾/ρ⁽¹⁾ = 3. Furthermore, we consider a bi-layered unit cell in which d⁽²⁾ = d⁽¹⁾. The results are shown in figure 3 for a phononic crystal of size a = 1 (arbitrary units) and a value of wave amplitude of B/a = 1/8. Superimposed, for comparison, are the dispersion curves on the basis of infinitesimal strain and the corresponding dispersion curves for an equivalent statically homogenized medium for which the speed of sound is c (obtained by the standard rule of mixtures). We observe in the figure that the finite-strain dispersion curves asymptotically converge to the infinitesimal and homogenized curves at long wavelengths as expected. We also note that the finite strain causes the dispersion branches to rise and the band-gap sizes to increase significantly—an attractive trait for many applications involving sound and vibration control. This behaviour, however, is dependent on the type of nonlinearity considered.

Figure 3.

Frequency band structure for a one-dimensional phononic crystal under finite strain (obtained using equation (3.20) in conjunction with equation (2.12)). For comparison, the dispersion curves under infinitesimal strain are included. Also, corresponding dispersion curves for a statically equivalent one-dimensional homogeneous elastic medium are overlaid. The nonlinearity-induced shifting of the dispersion curves is marked at two frequencies. Points P_L and P_NL are at frequency ω/c⁽¹⁾ = 2 and lie on the first infinitesimal-strain and the first finite-strain pass-band branch, respectively. Points S_L and S_NL are at frequency ω/c⁽¹⁾ = 5 and lie on the first infinitesimal-strain and the first finite-strain stop-band branch, respectively. (Online version in colour.)

The influence of the nonlinearity on the frequency–wavenumber relation naturally impacts the spectrum of group velocities, defined as

$$c_{\mathrm{g}}=\frac{\mathrm{\partial }{\omega }_{\mathrm{fin}}(\kappa ;B)}{\mathrm{\partial }\kappa }.$$ 3.22

In figure 4, we show the amplitude-dependent relationship between the frequency and the group velocity and between the group velocity and the wavenumber. The unfolded frequency band structure is also included for correlation. Most noticeable in this figure is the significant rise in the group velocity with amplitude. A similar rise takes effect for the phase velocity as well (not shown), indicating that with finite strain, the medium's permissible wave speeds are supersonic with respect to the nominal speeds under linear, infinitesimal strain. We also note that the homogenized medium's group velocity curves under finite strain are linear and exceed the maximum group velocity values for the corresponding phononic crystal; whereas, in contrast, the maximum group velocity in the infinitesimal-strain problem overlaps with the corresponding homogenized medium's horizontal group velocity line. This disparity may be a manifestation of the linear approximation inherent in the TM method. Thus, the minimum distance between the maximum group velocity of a phononic crystal and the corresponding homogenized medium's group velocity line may be viewed as a measure of accuracy for a given value of wave amplitude B/a. This conjecture is a subject for a future investigation.

Figure 4.

Effect of nonlinearity on the group velocity for the one-dimensional phononic crystal and the statically equivalent one-dimensional homogeneous elastic medium considered in figure 3. (a) Unfolded frequency band structure, (b) frequency versus group velocity, (c) group velocity versus wavenumber. The nonlinearity-induced shifting of the dispersion curves at frequency ω/c⁽¹⁾ = 2 is noted. (Online version in colour.)

In figure 5, we show three time snapshots of Bloch mode shapes corresponding to the pair of isofrequency pass-band points (top row) and the pair of isofrequency stop-band points (bottom row) marked in figure 3 (see [64] for a description of the procedure we use to calculate the Bloch mode shapes using the TM method). The increase in the wavelength due to the nonlinearity at a given frequency, e.g. by comparing point P_NL to P_L, is observed in figure 5 in the form of a slight stretching of the waveform. The effect of the nonlinearity on the group velocity, as indicated in figure 4, is less obvious in the mode shape diagrams. The effect of the nonlinearity on stop-band stationery waves appears to be a smoothening of the spatial profile. A strengthening of the spatial attenuation is also observed and is consistent with the prediction in figure 3 at the selected frequency.

Figure 5.

Three time snapshots of the Bloch mode shape over six unit cells corresponding to the four points P_L, P_NL, S_L and S_NL marked in figure 3. The Bloch mode shape for points P_L and P_NL are shown in (a–c), while the Bloch mode shapes for points S_L and S_NL are shown in (d–f). Dashed curves correspond to finite strain and black solid curves correspond to infinitesimal strain. Each pass-band and stop-band sets of curves are normalized with respect to the maximum displacement value of the infinitesimal strain case at time t. Upon further progression in time, the periodic character of the pass-band signal and the oscillatory nature of the stop-band signal will both become more apparent. (Online version in colour.)

4. Numerical verification

In this section, we conduct direct numerical simulations to verify the theory and determine the limits of its accuracy as the wave amplitude is increased. For this purpose and in the interest of simplicity, we consider a bi-material rod with the following material properties ratio: c⁽²⁾/c⁽¹⁾ = 3/2 and ρ⁽²⁾/ρ⁽¹⁾ = 4/9. The rod consists of 200 unit cells each consisting of two equal sized layers, and each unit cell has 25 grid points. A standard second-order finite-difference scheme is used to solve equation (3.11) with periodic boundary conditions applied. A prescribed sinusoidal displacement gradient field with an initial amplitude of κ_eB and wavenumber κ_e is applied across the length of the rod, i.e. $\overline{u}(x,t=0)={\kappa }_{e}B\sin ({\kappa }_{e}x)$. A constant time step of 10⁻³ s is used through the integrations sweep with a total time varying between 300 and 400 s depending on the value of κ_e and the requirements for numerical stability. The space–time solution is then Fourier transformed to obtain an intensity distribution in the wavenumber-frequency domain. The results are shown in figure 6 for a unit cell of size a = 10 and wave amplitude of B/a = 1/8, for a normalized excitation wavenumber equal to κ_ea = 1.3. In figure 6a,b, we plot the simulated spectrum in logarithmic scale. This allows us to highlight only the fundamental harmonic (because it has the highest intensity among the harmonics) in our comparison with the theoretical results. The existence of higher harmonics is apparent when the numerical results are plotted in linear scale as shown in figure 6c. The results for eight different values of κ_e are shown in figure 7. Superimposed, for comparison, are the dispersion curves on the basis of finite as well as infinitesimal strains. A perfect agreement between the numerical wavenumber-frequency spectrum and the analytically predicted dispersion relation from equation (3.20) is clearly observed. Furthermore, this agreement holds throughout the first and second Brillouin zones and possibly further. Figure 8 shows the same set of results for κ_ea = 3.1 and for a range of values of B/a indicating that the present technique is accurate for up to B/a = 0.125, i.e. for a wave amplitude that is one-eighth of the unit cell length. For higher relative wave amplitudes, errors of 2.8, 4.4 and 7.8% are observed for B/a = 0.15, B/a = 0.175, B/a = 0.2, respectively, at κ_ea = 3.1.

Figure 6.

Numerical verification of the proposed TM method with nonlinear enrichment as applied to the one-dimensional bi-material phononic crystal rod described in §4. (a) Finite-strain dispersion curves obtained by theory. The spectrum obtained by simulation for a normalized excitation wavenumber equal to κ_ea = 1.3 is also plotted in logarithm scale in order to highlight the fundamental harmonic. For comparison, the linear, infinitesimal-strain dispersion curves are also plotted. The simulated spectrum shown in (a) is plotted again in (b) in logarithmic scale and in (c) in linear scale. In (c), higher-order harmonics are clearly observed. (Online version in colour.)

Figure 7.

Numerical verification of the proposed TM method with nonlinear enrichment as applied to the one-dimensional bi-material phononic crystal rod described in §4. (a–h) Finite-strain dispersion curves obtained by theory, and by simulation for a range of excitation wavenumbers covering the first and second Brillouin zones. (i) A superposition of the simulations spectra overlaid on the theoretical results. For comparison, the linear, infinitesimal-strain dispersion curves are also plotted. (Online version in colour.)

Figure 8.

Examination of the limit on wave amplitude for obtaining accurate results by the proposed TM method with nonlinear enrichment. Similar to figure 7, these results are for the one-dimensional bi-material phononic crystal rod described in §4. For κ_ea = 3.1, the theoretical results break down when B/a exceeds 0.125. For comparison, the linear, infinitesimal-strain dispersion curves are also plotted. (Online version in colour.)

5. Balance between linear and nonlinear dispersion

In the literature, a common view is that nonlinearity of the type considered in this study tends to steepen, and subsequently narrow, a wave because large-amplitude constituent waves are able to catch up with slower low-amplitude ones; and, in contrast, dispersion causes a wave to widen its profile spatially because different constituent waves travel at different speeds [67]. When a wave maintains its spatial profile, such as the case for a soliton, it follows that this is due to a balance between nonlinearity and dispersion. Here, we view this problem from a different perspective. We consider the influence of the periodicity in altering the dispersion (which is a linear mechanism and has a softening effect) and, in parallel, the influence of the nonlinearity in also altering the dispersion (which in this case has a hardening effect).³ In figure 9a, we reproduce the results displayed in figure 3 with a focus on the first Brillouin zone. The figure illustrates the two effects when taking place separately or in combination. In the structure considered, we observe that an amplitude of B/a = 1/8 allows the two effects to be practically in balance up to approximately κ = 5π/8a, which corresponds to a wavelength as small as roughly three times the unit cell size. Such condition, in principle, brings rise to a spatially invariant wave within this range of wavelengths. This proposed interpretation that the balance is between hardening and softening dispersion (with one of these mechanisms being due to nonlinearity⁴ ) has implications for the study of solitons as it provides a new understanding of the nature of their formation.

Figure 9.

Illustration of the contrast between the effect of the periodicity on the dispersion as seen in the linear phononic crystal (see §2) versus the effect of the finite-strain nonlinearity on the dispersion which is here brought about by increasing the wave amplitude in a homogeneous medium with the same properties as a statically homogenized version of the phononic crystal (see §3a). In the nonlinear phononic crystal considered in §3b, the two opposing effects are simultaneously present and a balance may be practically realized up to a certain wavenumber. For a wave amplitude of B/a = 1/8, the two effects are approximately in balance up to κ = 5π/8a as shown in (a). The impact of this balance on the space–time displacement profile is demonstrated in (b). The A, B and C profiles are obtained, respectively, by direct numerical integration, the present TM method with nonlinear enrichment, and the standard TM method. (Online version in colour.)

The linear–nonlinear dispersion balancing phenomenon is elucidated further by plotting in figure 9b the mode shapes corresponding to points A, B and C marked in figure 9a. We observe in A that the finite strain steepens a sinusoidal wave until it eventually reaches the point of bifurcation (not shown). By contrast, in C we observe a linear wave with a spatially variant profile at each time step due to the periodically alternating material properties. In B, which corresponds to a balanced state, no wave profile steepening is observed (thus the wave is stabilized) and the profile is now spatially invariant at each time window. Wave profile invariance potentially may be used towards coherent information transfer over long distances, among other applications.

6. Conclusion

We have theoretically derived a wavenumber versus frequency finite-strain dispersion relation for Bloch wave propagation in a slender one-dimensional phononic crystal (layered periodic elastic medium with a small cross section). The effect of finite strain has been incorporated exactly at the individual homogeneous layer level. This part of the formulation is accurate for strong nonlinearity, in fact with no limit on the wave amplitude as long as the wave is stable and has not reached, for example, a state of bifurcation. Subsequently, the TM method has been applied to the unit cell to analytically provide an approximate nonlinear dispersion relation for the periodic medium. Thus, this approach represents an application of the TM method with a nonlinear enrichment (in principle, it can be implemented also for other types of waves such as nonlinear electromagnetic waves in a photonic crystal). Owing to the assumption of a linear strain–displacement gradient relation in the TM method, as well as the application of Bloch's theorem across the unit cell, the analysis becomes less accurate as the strength of the nonlinearity increases. Using brute-force finite-difference numerical simulations, we demonstrated that the technique is accurate for up to a wave amplitude of one-eighth the unit-cell length (figure 8). To our knowledge, this is the most accurate analytical method to date for wave propagation analysis in strongly nonlinear periodic media.

The dynamic behaviour revealed by equations (3.16) and (3.20) is based on our underlying assumption of Green–Lagrange strain at the level of each homogeneous layer from which the layered medium is constructed. The same technique may be applied for a model with material nonlinearity instead of, or in addition to, the geometric nonlinearity. Specific choices of material and geometric nonlinearities could in principle lead to qualitatively different dispersion behaviour. Material nonlinearities may include, for example, polynomial constitutive relations of different orders describing hardening or softening stress–strain behaviour [67]; and geometric nonlinearities may include any choice of the strain–displacement gradient relationship from among the Seth–Hill family of strain measures [68,69]. Thus it is necessary for future work considering specific materials to examine the problem experimentally to determine the most appropriate constitutive relation and strain measure guided by the theory presented in this paper.

Finally, and importantly, the results provide a quantitative prediction of the changes in the dispersion curves when finite strain and periodicity are introduced separately or in combination. The finite strain creates a hardening effect, whereas the periodicity creates a softening effect. We have shown that the wave amplitude (and the material properties) could be chosen to create an approximate balance between these two dispersive effects up to a certain wavenumber, creating a spatially invariant wave profile as illustrated in figure 9. This represents a subtle but fundamentally different interpretation of the balancing mechanism responsible for spatial invariance in a wave profile. The proposed interpretation contrasts with the common view that the balance is between nonlinear and dispersion effects.

Data accessibility

All data are included in the manuscript.

Authors' contributions

M.I.H. proposed (i) the enrichment of the transfer-matrix method with an exact nonlinear dispersion relation and (ii) the balance analysis provided in figure 9. M.I.H. and R.K. designed and performed the research and analysed the results; M.I.H. and R.K. co-wrote the paper.

Competing interests

We declare we have no competing interests.

Funding

This work has been supported by the National Science Foundation CAREER grant no. 1254931.