Willison Amplitude in Dynamic Media

• Willison Amplitude is a quantitative measure of nonlocal, asymmetric coupling between momentum and strain (or pressure and velocity) in dynamically homogenized materials.
• It is derived using Floquet theory and matrix-logarithm techniques, with perturbative expansions isolating the off-diagonal Willis coupling tensor.
• Its nonzero value indicates modified dispersion relations and asymmetric scattering, providing insights into the behavior of engineered microstructured media.

The Willison amplitude, or more generally the amplitude arising from Willis coupling, describes the quantitative measure of non-classical, nonlocal coupling between momentum and strain (in elasticity) or pressure and velocity (in acoustics) in dynamically homogenized heterogeneous media. Willis materials generalize standard constitutive relations by introducing additional coupling terms parameterized by the Willis coupling tensor or vector, fundamentally altering wave propagation, scattering, and effective medium behavior in systems with subwavelength structure and asymmetry.

1. Willis Constitutive Framework and Coupling Amplitude

Willis constitutive equations extend classical elasticity and acoustics by introducing frequency- and wavenumber-dependent coupling between momentum and strain. In the frequency–wavenumber domain $(\omega,k)$, for a homogeneous Willis elastic medium, the local constitutive equations are

$$\begin{align*} \sigma_{ij} &= c^{eff}_{ijkl}(\omega,k)\, e_{kl} + S_{ijr}(\omega,k)\,(-i\omega)\,u_r, \ p_q &= S_{klq}(\omega,k)\,e_{kl} + \rho^{eff}_{qr}(\omega,k)\,(-i\omega)\,u_r, \end{align*}$$

where

• $e_{kl}=\frac{1}{2}(i\,k_l\,u_k + i\,k_k\,u_l)$ is the strain,
• $u_r$ is displacement,
• $\sigma_{ij}$ is stress,
• $p_q$ is the generalized momentum density,
• $c^{eff}_{ijkl}$ is the effective stiffness tensor,
• $\rho^{eff}_{qr}$ is the anisotropic inertia tensor,
• $S_{ijr}=S_{jir}$ is the Willis coupling tensor.

The equilibrium condition $ik_j\,\sigma_{ij} = (-i\omega)\,p_i$ closes the system. The Willis amplitude specifically refers to the physical magnitude and functional form of the coupling tensor $$\begin{align*} \sigma_{ij} &= c^{eff}_{ijkl}(\omega,k)\, e_{kl} + S_{ijr}(\omega,k)\,(-i\omega)\,u_r, \ p_q &= S_{klq}(\omega,k)\,e_{kl} + \rho^{eff}_{qr}(\omega,k)\,(-i\omega)\,u_r, \end{align*}$$0, which modifies standard dynamic equations and introduces new physical effects in wave propagation and scattering (Shuvalov et al., 2010).

2. Derivation and Calculation of the Willis Amplitude

For periodically layered elastic media, the dynamic effective Willis constitutive parameters are derived from the Floquet–logarithm of the period propagator $$\begin{align*} \sigma_{ij} &= c^{eff}_{ijkl}(\omega,k)\, e_{kl} + S_{ijr}(\omega,k)\,(-i\omega)\,u_r, \ p_q &= S_{klq}(\omega,k)\,e_{kl} + \rho^{eff}_{qr}(\omega,k)\,(-i\omega)\,u_r, \end{align*}$$1. The central result is

$$\begin{align*} \sigma_{ij} &= c^{eff}_{ijkl}(\omega,k)\, e_{kl} + S_{ijr}(\omega,k)\,(-i\omega)\,u_r, \ p_q &= S_{klq}(\omega,k)\,e_{kl} + \rho^{eff}_{qr}(\omega,k)\,(-i\omega)\,u_r, \end{align*}$$2

where $$\begin{align*} \sigma_{ij} &= c^{eff}_{ijkl}(\omega,k)\, e_{kl} + S_{ijr}(\omega,k)\,(-i\omega)\,u_r, \ p_q &= S_{klq}(\omega,k)\,e_{kl} + \rho^{eff}_{qr}(\omega,k)\,(-i\omega)\,u_r, \end{align*}$$3 is partitioned into $$\begin{align*} \sigma_{ij} &= c^{eff}_{ijkl}(\omega,k)\, e_{kl} + S_{ijr}(\omega,k)\,(-i\omega)\,u_r, \ p_q &= S_{klq}(\omega,k)\,e_{kl} + \rho^{eff}_{qr}(\omega,k)\,(-i\omega)\,u_r, \end{align*}$$4 blocks: $$\begin{align*} \sigma_{ij} &= c^{eff}_{ijkl}(\omega,k)\, e_{kl} + S_{ijr}(\omega,k)\,(-i\omega)\,u_r, \ p_q &= S_{klq}(\omega,k)\,e_{kl} + \rho^{eff}_{qr}(\omega,k)\,(-i\omega)\,u_r, \end{align*}$$5 The Willis coupling is extracted from these off-diagonal blocks:

• The "normal" coupling ($$\begin{align*} \sigma_{ij} &= c^{eff}_{ijkl}(\omega,k)\, e_{kl} + S_{ijr}(\omega,k)\,(-i\omega)\,u_r, \ p_q &= S_{klq}(\omega,k)\,e_{kl} + \rho^{eff}_{qr}(\omega,k)\,(-i\omega)\,u_r, \end{align*}$$6) at $$\begin{align*} \sigma_{ij} &= c^{eff}_{ijkl}(\omega,k)\, e_{kl} + S_{ijr}(\omega,k)\,(-i\omega)\,u_r, \ p_q &= S_{klq}(\omega,k)\,e_{kl} + \rho^{eff}_{qr}(\omega,k)\,(-i\omega)\,u_r, \end{align*}$$7 is

$$\begin{align*} \sigma_{ij} &= c^{eff}_{ijkl}(\omega,k)\, e_{kl} + S_{ijr}(\omega,k)\,(-i\omega)\,u_r, \ p_q &= S_{klq}(\omega,k)\,e_{kl} + \rho^{eff}_{qr}(\omega,k)\,(-i\omega)\,u_r, \end{align*}$$8

• Additional components arise from $$\begin{align*} \sigma_{ij} &= c^{eff}_{ijkl}(\omega,k)\, e_{kl} + S_{ijr}(\omega,k)\,(-i\omega)\,u_r, \ p_q &= S_{klq}(\omega,k)\,e_{kl} + \rho^{eff}_{qr}(\omega,k)\,(-i\omega)\,u_r, \end{align*}$$9-derivatives of $e_{kl}=\frac{1}{2}(i\,k_l\,u_k + i\,k_k\,u_l)$0 (Shuvalov et al., 2010).

At low frequency/long wavelength, Magnus expansion yields

$e_{kl}=\frac{1}{2}(i\,k_l\,u_k + i\,k_k\,u_l)$1

with $e_{kl}=\frac{1}{2}(i\,k_l\,u_k + i\,k_k\,u_l)$2 a double-commutator integral involving microscale compliance and density. The expansion converges rapidly for $e_{kl}=\frac{1}{2}(i\,k_l\,u_k + i\,k_k\,u_l)$3, where $e_{kl}=\frac{1}{2}(i\,k_l\,u_k + i\,k_k\,u_l)$4, $e_{kl}=\frac{1}{2}(i\,k_l\,u_k + i\,k_k\,u_l)$5 (Shuvalov et al., 2010).

3. Physical Interpretation and Implications

Nonzero Willis amplitude signifies that time-varying strain produces momentum density (and vice versa) beyond the usual mass-density response. This results in

• Modified dispersion relations, leading to frequency-dependent wave speeds and impedance,
• Non-reciprocal or asymmetric scattering behaviors,
• Enhanced relevance in media with strong heterogeneity or lack of symmetry.

In classical homogenized media, $e_{kl}=\frac{1}{2}(i\,k_l\,u_k + i\,k_k\,u_l)$6 and inertia is isotropic. For periodic stratification, the Willis amplitude vanishes only if the unit cell profile is symmetric; asymmetry gives rise to finite Willis coupling. The Willis amplitude encodes the leading-order nonlocal correction to classical continuum models and is required to exactly match the Bloch–Floquet propagator of the microstructured medium (Shuvalov et al., 2010).

4. Willis Amplitude in Acoustic Scattering

In acoustics, the standard constitutive equations for a Willis fluid introduce a Willis coupling vector $e_{kl}=\frac{1}{2}(i\,k_l\,u_k + i\,k_k\,u_l)$7, supplementing bulk modulus $e_{kl}=\frac{1}{2}(i\,k_l\,u_k + i\,k_k\,u_l)$8 and density $e_{kl}=\frac{1}{2}(i\,k_l\,u_k + i\,k_k\,u_l)$9: $u_r$0 with $u_r$1 the pressure, $u_r$2 the volume strain, $u_r$3 the velocity, and $u_r$4 the momentum density. The nondimensional asymmetry vector $u_r$5 ($u_r$6) quantifies the strength of Willis coupling (Muhlestein et al., 2018).

For a Willis–fluid cylinder, analytic expressions for the scattering amplitude coefficients, to first order in $u_r$7, are given as: $u_r$8 where $u_r$9 is the classical (Willis-free) coefficient, and $\sigma_{ij}$0 represents the leading Willis-coupling correction, involving coupling between angular modes $\sigma_{ij}$1 and $\sigma_{ij}$2. The amplitude characterizes both altered cross-section and mode-coupling in the presence of Willis effect (Muhlestein et al., 2018).

5. Analytical Methods and Series Expansions

Willis amplitude calculations rely on perturbative expansions and matrix-logarithm techniques. The Magnus series provides a systematic expansion of the Floquet-averaged Stroh matrix in powers of the small parameter $\sigma_{ij}$3, yielding concise formulas for the coupling tensor and its frequency/wavenumber scaling. In scattering, generalized impedance methods enable higher-order corrections to the amplitude, with the impedance/admittance kernel expanded in $\sigma_{ij}$4 to control accuracy. For small $\sigma_{ij}$5, the Born approximation suffices; higher orders capture increased nonlocal and anisotropic effects (Shuvalov et al., 2010, Muhlestein et al., 2018).

6. Dependence on Microstructure and Symmetry

The Willis amplitude is identically zero for unit-cell symmetric media (due to vanishing odd Magnus terms), but increases with structural contrast and asymmetry. The leading term of $\sigma_{ij}$6 is linear in frequency, with higher-order terms scaling as $\sigma_{ij}$7. In strong-contrast or highly asymmetric periodic media, the Willis amplitude constitutes a significant correction, dominating dispersive and nonlocal response (Shuvalov et al., 2010).

A summary of the key dependencies:

Microstructural Property	Willis Amplitude ($\sigma_{ij}$8)
Unit-cell symmetry	$\sigma_{ij}$9 (all orders)
Asymmetry	$p_q$0 (leading order)
Strong contrast	Larger $p_q$1

7. Practical Computation and Range of Validity

Computation of the Willis amplitude involves:

1. Calculating the propagator $p_q$2 over one period,
2. Forming $p_q$3,
3. Extracting off-diagonal blocks to recover $p_q$4.

For scattering problems, the amplitude is extracted from expansions of the mode amplitudes $p_q$5 or the impedance kernel, with convergence and accuracy controlled by the smallness of the coupling parameter $p_q$6 or $p_q$7. For $p_q$8, first-order perturbation remains accurate, and for $p_q$9 higher-order impedance expansions provide reliable amplitude estimates (Muhlestein et al., 2018).

A plausible implication is that systematic computation of the Willison amplitude is feasible for a wide class of engineered and natural microstructured materials, provided the microstructure induces significant spatial asymmetry and the frequency regime remains within the validity of the homogenization (long-wavelength) assumption.