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Modeling of Rayleigh-type waves in a viscoelastic piezoelectric fiber-reinforced composite layered Structure

This paper investigates the propagation of Rayleigh-type waves in a viscoelastic piezoelectric fiber-reinforced composite layer bonded to an isotropic elastic half-space via an imperfect interface, deriving an exact dispersion relation to analyze how fiber volume fraction, constituent viscoelasticity, and interfacial compliance influence wave phase velocity, attenuation, and electromechanical coupling.

Original authors: Bikram Dholey, Kshitish Ch. Mistri, Amrita Das, Gopal Chandra Shit

Published 2026-10-30

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CC BY 4.0

Original authors: Bikram Dholey, Kshitish Ch. Mistri, Amrita Das, Gopal Chandra Shit

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where the materials building our bridges, aircraft, and smart devices can not only bear heavy loads but also "feel" stress and send electrical signals in response. This is the promise of fiber-reinforced composites, materials made by embedding strong, stiff fibers into a softer, flexible glue-like matrix. When these fibers are made of piezoelectric ceramic—a substance that generates electricity when squeezed—the result is a multifunctional material capable of sensing its own health or even powering small sensors. However, real-world materials are never perfectly rigid; they have a slight internal friction, like a sponge that slowly squishes back after being pressed, which causes them to lose energy as heat. This energy loss, known as viscoelasticity, becomes critical when these materials are used to guide waves, such as the ripples that travel along the surface of the Earth during an earthquake or the ultrasonic pulses used to inspect a plane wing for cracks.

For decades, scientists have studied how these waves move through perfect, solid layers. But in reality, the bond between a composite layer and the metal or concrete it sits on is rarely perfect. It is often slightly loose or "slippery," acting like a weak spring rather than a rigid weld. Furthermore, the internal friction of both the stiff fibers and the soft glue can vary depending on how fast the wave is moving. Until now, no one had built a complete mathematical picture of how surface waves behave when they travel through a composite layer that is both internally "squishy" and imperfectly glued to a solid base. This gap in knowledge makes it difficult to design reliable sensors or to accurately interpret the signals they send.

In a new study, researchers have constructed a detailed model to fill this gap, focusing on a specific type of surface wave called a Rayleigh wave. These waves are unique because they travel along the surface of a material, with their motion dying out as they go deeper, much like the ripples on a pond that fade as they move away from the shore. The team modeled a layer of composite material, made of piezoelectric fibers embedded in an epoxy glue, sitting on top of a solid aluminum block. Crucially, they treated both the fibers and the glue as materials that lose energy, and they modeled the connection between the layer and the block as a set of tiny, flexible springs that allow the two surfaces to slide or separate slightly. This setup mimics the imperfect bonds found in real engineering structures.

The researchers used a powerful computational approach to solve the complex equations governing this system. Instead of making simplifying guesses about how small the energy loss might be, they solved the full, exact equations to find the speed at which the waves travel and how quickly they fade away. They discovered that the amount of fiber in the composite acts as a master dial for the wave's behavior. When the layer contains a high percentage of fibers, the wave travels faster and retains its energy better, moving more like a wave on a stiff board. When the layer has fewer fibers, the wave slows down and loses energy more quickly, behaving more like a wave on a soft, dampening surface.

Perhaps the most surprising finding concerns the source of the energy loss. The study showed that the internal friction of the glue matrix is the primary driver of how much the wave fades away. Changing how "sticky" or lossy the glue is had a dramatic effect on the wave's attenuation, yet it barely changed the wave's speed. In contrast, the internal friction of the stiff fibers mattered very little unless the layer was almost entirely made of fibers. This suggests that for most practical applications, the quality of the glue is far more important for signal clarity than the quality of the fibers themselves.

The imperfect bond between the layer and the base also played a decisive role. The researchers found that a loose or "slippery" connection significantly slowed down the wave and reduced the amount of energy lost to the material. This happens because the flexible springs at the interface absorb some of the motion, preventing it from being transferred into the lossy composite layer. The study also revealed a distinct "dip" in the speed of the wave packet as it travels, a phenomenon known as an Airy phase. This dip is highly sensitive to both the amount of fiber in the material and the tightness of the bond. If the bond is weak, this dip becomes deeper and occurs at different speeds, acting as a clear fingerprint of the connection's quality.

These results provide a new, precise toolkit for engineers and scientists. By understanding how the wave speed and fading change with the fiber content and the bond quality, it becomes possible to use these waves as a diagnostic tool. One could send a wave along a structure and, by measuring how fast it arrives and how much it has weakened, determine if the internal fibers are properly distributed or if the glue bond has degraded. The study confirms that while the internal friction of the glue controls how much signal is lost, the stiffness of the fibers and the tightness of the bond control how fast the signal travels. This separation of effects allows for a more accurate assessment of structural health, ensuring that the smart materials of the future can be monitored with greater reliability.

Technical Summary: Modeling of Rayleigh-type waves in a viscoelastic piezoelectric fiber-reinforced composite layered Structure

Problem Statement
This study investigates the propagation of Rayleigh-type surface waves in a viscoelastic piezoelectric fiber-reinforced composite (VRFRC) layer bonded to an isotropic elastic half-space via a mechanically imperfect interface. While piezoelectric fiber-reinforced composites (PFRCs) are widely used for structural health monitoring and sensing due to their electromechanical coupling, existing models often neglect two critical physical realities: (1) the intrinsic viscoelastic dissipation present in both the polymeric matrix and the piezoelectric fibers, and (2) the mechanical compliance of the interface (imperfect bonding) between the composite layer and the substrate. The authors address the gap in literature regarding the combined influence of bulk viscoelasticity in both constituents and interfacial imperfections on the dispersion and attenuation of surface waves.

Methodology
The authors develop an analytical electromechanical framework based on the following key methodological steps:

Constitutive Modeling: Both the PZT-5A fibers and the epoxy matrix are modeled as viscoelastic solids using Kelvin–Voigt elements. The fibers are treated as viscoelastic piezoelectric solids, while the matrix is a viscoelastic dielectric. The constitutive equations are formulated in the stress–charge form with complex, frequency-dependent constants derived via the elastic–viscoelastic correspondence principle.
Homogenization: The effective properties of the VRFRC are determined using the strength-of-materials/rule-of-mixtures (SM/RM) relations. Complex constituent constants are substituted into these relations to yield complex, frequency-dependent effective constants for the homogenized medium. The model ensures passivity (energy dissipation rather than generation) by verifying that the loss matrix is positive semi-definite.
Boundary Value Problem: The system consists of a VRFRC layer ( $0 \le z \le h$ ) on an isotropic elastic half-space ( $z \ge h$ , modeled as aluminum). The upper surface is traction-free and electrically open. The interface at $z=h$ is modeled using a linear-slip (spring) model, allowing for displacement discontinuities proportional to the traction via normal ( $K_n$ ) and tangential ( $K_t$ ) stiffness parameters. The substrate is grounded.
Dispersion Analysis: The governing equations of motion and Gauss's law are solved for time-harmonic fields. An exact $8 \times 8$ complex secular equation is derived by satisfying boundary conditions at the free surface and the imperfect interface. This equation is solved numerically for the complex wavenumber $k$ (yielding phase velocity $c$ and normalized attenuation $\delta$ ) without employing small-loss approximations.
Validation: The numerical implementation is validated against several limiting cases, including the Rayleigh wave of the substrate, the surface wave of a VRFRC half-space, the decoupled elastic limit, and the antisymmetric Lamb mode ( $A_0$ ) of a detached plate.

Key Contributions and Results
The study provides a systematic analysis of how fiber volume fraction ( $V_f$ ), viscous relaxation times, and interfacial compliance affect wave propagation:

Effective Properties: The effective complex constants are not simple volume averages; they depend on the coupling mechanism (iso-strain vs. iso-stress). The storage parts of the constants increase with $V_f$ , while the loss parts are dominated by the matrix relaxation times at lower $V_f$ but shift toward fiber properties at high $V_f$ .
Fiber Volume Fraction ( $V_f$ ): $V_f$ primarily controls dispersion. Higher $V_f$ increases layer stiffness and piezoelectric stiffening, resulting in higher phase velocities at short wavelengths and a higher group velocity minimum. Conversely, higher $V_f$ reduces attenuation in the dispersive range because the stiffer fibers store more energy relative to the lossy matrix.
Viscous Relaxation Times:
- Matrix: The viscous relaxation times of the matrix (specifically $\eta_{33}^c$ and $\eta_{44}^c$ ) govern the attenuation significantly but have a negligible effect on phase and group velocities (changes $< 1.2\%$ ). $\eta_{33}^c$ dominates in the dispersive range, while $\eta_{44}^c$ affects short wavelengths.
- Fibers: Fiber viscoelasticity is negligible at low fiber content but becomes significant for fiber-rich layers ( $V_f = 0.9$ ), where neglecting it leads to an underestimation of attenuation.
Imperfect Interface: Interfacial compliance, particularly the normal compliance ( $\xi_n$ ), lowers the phase velocity and attenuation in the dispersive range. For stiff, weakly bonded layers, the phase velocity can dip below the half-space surface-wave speed, creating a minimum. The group velocity minimum (Airy phase) is highly sensitive to bond quality, shifting to lower wavenumbers and decreasing in magnitude as compliance increases.
Group Velocity: The group velocity exhibits a pronounced minimum (Airy phase) that serves as a sensitive indicator of both fiber content and bond quality. This minimum occurs where the wave motion transitions from being substrate-dominated to layer-dominated.

Significance and Claims
The paper claims that its exact $8 \times 8$ complex dispersion relation, solved without small-loss approximations, provides a rigorous framework for understanding surface waves in multifunctional piezoelectric composites. The study highlights that:

Passivity is Critical: Proper formulation of dielectric relaxation is necessary to ensure the medium is passive (dissipative), avoiding unphysical energy generation.
Interfacial Compliance is Non-Negligible: Ignoring imperfect bonding leads to erroneous estimates of surface-wave properties, particularly when wavelengths are comparable to interfacial dimensions.
Diagnostic Potential: The sensitivity of the group velocity minimum and attenuation to fiber volume fraction and bond quality suggests that these parameters can be used for the non-destructive assessment of bonded piezoelectric composite layers in structural health monitoring and ultrasonic sensing applications.

The authors conclude that their results offer guidelines for designing and assessing bonded piezoelectric composite layers, emphasizing the need to treat both constituents as viscoelastic and to account for interfacial compliance in wave-based sensing applications.

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