[Paper Review] Fundamental principles for generalized Willis metamaterials

René Pernas-Salomón, Gal Shmuel|arXiv (Cornell University)|Aug 11, 2020

Acoustic Wave Phenomena Research98 references4 citations

TL;DR

This paper establishes fundamental physical constraints—passivity, reciprocity, and causality—on generalized Willis metamaterials, which exhibit momentum coupling not only to mechanical strain and velocity but also to electric fields via piezoelectricity. The authors derive mathematical restrictions on the effective material properties, enabling validation of theoretical and experimental results and setting bounds on maximal device responses in wave manipulation applications.

ABSTRACT

Metamaterials whose momentum is constitutively coupled with their strain show promise in wave manipulation for engineering purposes and are called Willis materials. They were discovered using an effective medium theory which shows that their response is non-local in space and time. Recently, we generalized this theory to account for piezoelectricity and demonstrated that the effective momentum can depend constitutively on the electric field, thereby enlarging the design space for metamaterials. Here, we develop the mathematical restrictions on the effective properties of such generalized Willis materials, owing to passivity, reciprocity, and causality. Establishing these restrictions is of fundamental significance, as they test the validity of theoretical and experimental results and applicational importance since they provide elementary bounds for the maximal response that potential devices may achieve.

Motivation & Objective

• To establish rigorous mathematical restrictions on the effective properties of generalized Willis metamaterials based on fundamental physical principles.
• To extend prior work on standard Willis materials to include piezoelectric coupling, where momentum couples with electric fields and velocity with electric displacement.
• To ensure theoretical and experimental results for such metamaterials are physically admissible by enforcing passivity, reciprocity, and causality.
• To provide quantitative bounds on the maximum achievable response in wave manipulation devices using these materials.
• To generalize reciprocity analysis beyond the long-wavelength limit, offering new insights even for standard Willis materials.

Proposed method

• Formulate the generalized Willis equations for piezoelectric composites, incorporating coupling between mechanical momentum/velocity and electric fields.
• Apply the principles of passivity, reciprocity, and causality to derive constraints on the effective material tensors in the frequency domain.
• Use Fourier transforms and symmetry analysis to derive relations between the complex-valued material response functions, such as $\check{\mathbf{W}}(\bm{\kappa},\omega) = -\check{\mathbf{W}}^{*}(-\bm{\kappa},\omega)$.
• Introduce a modified homogenization formulation inspired by prior elastic theory, ensuring consistency with reciprocity and energy conservation.
• Analyze the symmetry properties of the Green’s function and response kernels to derive time-reversal and reciprocity conditions.
• Derive the condition $\tilde{\mathbf{W}}^{\dagger}(\mathbf{x},\mathbf{X}) = -\tilde{\mathbf{W}}^{\mathsf{T}*}(\mathbf{X},\mathbf{x})$ for statistically homogeneous media, leading to the final reciprocity relation in the Fourier domain.

Experimental results

Research questions

• RQ1 What mathematical constraints do passivity, reciprocity, and causality impose on the effective properties of generalized Willis metamaterials with piezoelectric coupling?
• RQ2 How do the coupling terms between mechanical momentum and electric field affect the symmetry and admissibility of the material response?
• RQ3 Can the reciprocity condition be generalized beyond the long-wavelength limit for these materials, and what new constraints does this yield?
• RQ4 How do the derived constraints enable validation of experimental or numerical results in metamaterial design?
• RQ5 What are the fundamental bounds on the maximum achievable response in wave manipulation devices based on these generalized Willis materials?

Key findings

• The paper derives the fundamental reciprocity relation $\check{\mathbf{W}}(\bm{\kappa},\omega) = -\check{\mathbf{W}}^{*}(-\bm{\kappa},\omega)$, which generalizes prior results to non-long-wavelength regimes.
• Passivity imposes that the imaginary part of the effective material response must be positive semi-definite, ensuring energy dissipation does not violate thermodynamic principles.
• Reciprocity leads to a symmetric relation between the response functions under time-reversal and spatial inversion, formalized via $\tilde{\mathbf{W}}^{\dagger}(\mathbf{x},\mathbf{X}) = -\tilde{\mathbf{W}}^{\mathsf{T}*}(\mathbf{X},\mathbf{x})$.
• Causality is enforced through the Kramers–Kronig relations, which link the real and imaginary parts of the frequency-domain response, ensuring no acausal behavior.
• The derived constraints provide a framework to test the physical admissibility of both theoretical models and experimental data for generalized Willis materials.
• The results establish that the maximal response in wave control devices is fundamentally bounded by these physical principles, offering design limits for optimal performance.