🔬 applied physics

Nonlinear interface effects in multilayered structures: vibro-acoustic modeling and experimental analysis

This paper presents a combined theoretical and experimental study demonstrating that the dynamic response of multilayered structures, such as glass-epoxy-glass beams, exhibits nonlinear behavior at imperfect interfaces, characterized by excitation-level-dependent variations in equivalent bending stiffness, through the use of a Zig-Zag formulation and laser vibrometry analysis.

Original authors: Antoine Demiquel, Kerem Ege, Emmanuel Gourdon

Published 2026-04-06

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Antoine Demiquel, Kerem Ege, Emmanuel Gourdon

Original paper licensed under CC BY 4.0 (http://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 you have a sandwich. Not a delicious one with ham and cheese, but a structural one made of two stiff slices of glass with a soft, squishy layer of glue in the middle. This is a multilayer structure, and engineers use them everywhere—from the wings of airplanes to the screens of your phone.

Usually, when we design these sandwiches, we assume the glue is perfect. We imagine the layers are stuck together so tightly that they move as one single, solid piece. But in the real world, things aren't perfect. There might be tiny air bubbles, the glue might be a bit uneven, or the bond might be slightly weak. These are called imperfect interfaces.

This paper is like a detective story about what happens when you shake this imperfect sandwich. The researchers discovered that the "glue" doesn't just sit there; it behaves strangely depending on how hard you shake it.

Here is the breakdown of their findings using some everyday analogies:

1. The "Slippery" Glue (The Problem)

Think of the layers in the sandwich as dancers holding hands.

Perfect Bond: If they hold hands tightly, they move in perfect unison. If one steps left, the other steps left immediately.
Imperfect Bond: If their grip is loose (the "imperfect interface"), they can slip a little bit. When the music gets loud (high vibration), they might slide past each other slightly before catching up.

The researchers wanted to measure exactly how much they slip and how that sliding changes the stiffness of the whole sandwich.

2. The "Smart" Model (The Theory)

To understand this, the scientists built a mathematical model. Imagine they created a virtual twin of the glass/epoxy/glass beam on a computer.

They used a "Zig-Zag" approach. Instead of treating the whole beam as one giant blob, they treated each layer individually but added "rules" for how they talk to each other at the seams.
They realized that the "glue" acts like a spring. If you push gently, the spring is stiff. But if you push hard, the spring gets squishier. This is the nonlinear part: the behavior changes based on how hard you hit it.

3. The "Magic Sigmoid" Curve (The Pattern)

When they looked at how the beam vibrated at different speeds (frequencies), they found a specific pattern.

Imagine a traffic light that slowly changes from red to green. It doesn't switch instantly; there's a smooth transition in the middle.
The stiffness of their beam did the same thing. At low frequencies, it was very stiff (Red). At high frequencies, it was softer (Green). In the middle, it transitioned smoothly.
The researchers used a mathematical shape called a Sigmoid (an "S" curve) to describe this transition perfectly. It was like finding the exact formula for how the traffic light changes color.

4. The Laser Dance Floor (The Experiment)

To test their theory, they built a real glass/epoxy/glass beam and hung it in the air.

The Shakers: They attached small buzzers (like the ones in your phone) to the ends to shake the beam.
The Eyes: They used a super-precise laser vibrometer. Think of this as a camera that doesn't take pictures of faces, but of movement. It scanned the beam thousands of times a second to see exactly how every tiny point on the beam was dancing.
The Variable: They shook the beam at different volumes (amplitudes). First, a whisper (low vibration), then a shout (high vibration).

5. The Big Discovery (The "Aha!" Moment)

Here is the surprising part: The glue got weaker as they shook it harder.

When they shook the beam gently, the "glue" held firm, and the beam acted stiff.
When they shook it hard, the layers started to slip more. The "effective stiffness" of the whole beam dropped.
It's like a group of people trying to walk in a line. If they walk slowly, they stay in a tight line. If they start running and shouting, they start to spread out and lose their formation. The "line" becomes less rigid.

The researchers managed to quantify this. They found a number (an "interface parameter") that describes how "loose" the grip is. They proved that this number changes depending on how hard you shake the structure.

Why Does This Matter?

This is a big deal for engineers.

Safety: If you design a bridge or a plane wing assuming the glue is perfect, but it actually gets "looser" when the wind blows hard, your calculations might be wrong.
Noise Control: If you want to stop a car from rattling, you need to know exactly how the layers interact when the engine is roaring.
Future Tech: The authors suggest that in the future, we could design materials where we intentionally make the glue change its stiffness. Imagine a car suspension that gets softer when you hit a bump, or a building that becomes more flexible during an earthquake, all by controlling these "imperfect" interfaces.

In short: The paper shows that "imperfect" glue isn't just a defect; it's a dynamic feature that changes how a structure behaves. By understanding this "slippery" behavior, we can build safer, smarter, and more adaptable structures.

1. Problem Statement

Multilayered structures (e.g., laminated composites, sandwich panels) are widely used in aerospace, civil engineering, and acoustics due to their favorable stiffness-to-mass ratios. However, the dynamic performance of these structures is heavily influenced by the quality of the interfaces between layers. In practice, interfaces are rarely perfect due to manufacturing defects (air bubbles, adhesive inconsistencies, partial debonding).

While linear models exist to describe imperfect interfaces (often using spring-layer approximations), the nonlinear behavior of these interfaces under varying excitation levels remains poorly understood and characterized. Specifically, there is a lack of experimental methodologies to identify how interface properties (and consequently, the global equivalent bending stiffness) change as a function of vibration amplitude. This nonlinearity can lead to significant shifts in natural frequencies and damping characteristics, which are critical for accurate vibro-acoustic prediction.

2. Methodology

The authors propose a combined theoretical and experimental approach to characterize these nonlinear interface effects.

A. Theoretical Modeling

Zig-Zag Formulation: The study utilizes a multilayer model based on the Guyader-Lesueur formulation. This approach assigns independent kinematic fields to each layer while enforcing interfacial coupling conditions. It reduces the number of degrees of freedom by expressing the kinematics of all layers relative to a reference layer.
Imperfect Interface Modeling: The model incorporates "jump functions" to represent displacement discontinuities at interfaces. These are modeled using a linear spring-layer approach, where shear stress is proportional to relative displacement via an interface stiffness ( $K$ ) or compliance ( $B$ ).
Nonlinear Approximation: Recognizing that real interfaces exhibit nonlinear behavior, the authors approximate the nonlinear traction law using a piecewise linear representation. They introduce a single scalar equivalent interface parameter, $B$ , assumed to be homogeneous and isotropic.
Equivalent Plate & Sigmoid Model:
- The multilayer system is reduced to an equivalent homogeneous Kirchhoff–Love plate to derive a frequency-dependent effective bending stiffness, $D(f)$ .
- An asymptotic analysis reveals that $D(f)$ transitions between a low-frequency limit ( $D_{low}$ ) and a high-frequency limit ( $D_{high}$ ).
- A sigmoid-based model (adapted from Arasan et al.) is employed to explicitly describe this frequency-dependent transition. The model is defined by a transition frequency ( $f_T$ ) and a slope factor ( $R$ ). Crucially, the high-frequency limit and the transition frequency depend on the interface parameter $B$ .

B. Experimental Setup

Sample: A three-layer beam consisting of two stiff glass skins and a soft epoxy core (DP-190 3M adhesive). The sample contains manufacturing-induced imperfections (e.g., air bubbles).
Excitation: The beam is excited by piezoelectric buzzers at both ends using linear frequency chirps (1 kHz – 50 kHz).
Measurement: A scanning laser vibrometer measures the transverse displacement field ( $u_z$ ) along the beam length.
Amplitude Variation: Seven distinct excitation levels are tested, ranging from low to high mechanical amplitudes (approx. 22 pm to 290 pm).

C. Data Processing & Identification

Wavenumber Extraction: The measured displacement fields are processed using a 2D Spatial FFT to extract the dispersion curves. The longitudinal bending wavenumber ( $k_b$ ) is isolated from other modes.
CFAT Application: The Corrected Force Analysis Technique (CFAT) is applied to the measured displacement fields to estimate the equivalent dynamic Young's modulus ( $E$ ) and bending stiffness directly from experiments, serving as a validation tool.
Parameter Identification: The experimental transition frequency ( $f_T^{exp}$ ) is extracted by fitting the sigmoid model to the dispersion data. Using the theoretical relationship between $f_T$ and the interface parameter $B$ , the value of $B$ is numerically solved for each excitation level.

3. Key Contributions

Experimental Identification of Nonlinear Interface Parameter: The paper successfully identifies an equivalent interface parameter ( $B$ ) that varies with excitation amplitude, providing a quantitative measure of interface nonlinearity.
Sigmoid-Based Modeling Framework: The authors demonstrate that a sigmoid function can accurately model the frequency-dependent evolution of bending stiffness in imperfect multilayer structures, bridging the gap between complex multilayer theories and simple equivalent plate models.
Amplitude-Dependent Stiffness Characterization: The study establishes a direct link between excitation amplitude and the softening of the structure's equivalent bending stiffness, mediated by the interface parameter.

4. Key Results

Nonlinear Evolution of $B$ : The identified interface parameter $B$ increases significantly (by a factor of approximately two) as the excitation amplitude increases. This indicates that the interface becomes "softer" or more compliant at higher vibration levels.
Frequency Shift: The increase in $B$ causes a downward shift in the transition frequency ( $f_T$ ) of the bending stiffness curve. The study observed a frequency shift of approximately $\Delta f_T \approx 3$ kHz between the lowest and highest excitation levels.
Stiffness Softening: As $B$ increases, the equivalent dynamic Young's modulus ( $E$ ) decreases in the transition and high-frequency regions, while the low-frequency stiffness remains largely unaffected (consistent with the asymptotic analysis).
Model Validation: The analytical predictions using the Guyader model with the identified $B$ values show excellent agreement with the experimental data obtained via the CFAT method, particularly in the 7–30 kHz range.

5. Significance

Predictive Accuracy: This work highlights that assuming linear, constant interface properties can lead to significant errors in predicting the dynamic response of multilayer structures, especially at higher excitation levels.
Design Implications: The ability to quantify amplitude-dependent interface behavior is crucial for the design of adaptive structures, metamaterials, and vibration control systems where performance must be predictable under varying operational loads.
Methodological Advancement: The combination of the Zig-Zag model, sigmoid-based fitting, and CFAT provides a robust, non-destructive methodology for characterizing hidden interface defects and their nonlinear behavior without requiring direct access to the interface itself.
Future Applications: The authors suggest that this approach could be extended to additive manufacturing (3D printing) to design and tune interface geometries for specific mechanical properties, or to integrate active transducers for tunable multilayer structures.