Rao-Nakra Sandwich Beam Model

• Rao-Nakra type sandwich beam is a composite beam model with alternating stiff elastic layers and a thin viscoelastic core that enables precise control of vibrations.
• The model employs Hamilton’s principle and coupled PDEs to capture axial, bending, and shear dynamics under the thin compliant layer approximation.
• It underpins analysis of stability, controllability, and decay phenomena in smart structures with applications in vibration suppression, noise reduction, and structural health monitoring.

A Rao-Nakra type sandwich beam is a mathematical and physical model for multilayer composite beams consisting of alternating stiff elastic layers and thin compliant viscoelastic (shear) cores, characteristically derived under the "thin compliant layer" approximation. Its formal structure enables precise modeling of vibration, controllability, and stabilization phenomena in smart structures, notably including piezoelectric actuation and shear damping. The model originated in the works of Rao and Nakra (1974) and has become a primary framework for rigorous analysis and control of smart sandwich beams and plates in structural dynamics.

1. Structural Model and Rao-Nakra Assumptions

The canonical geometry is a three-layer beam (length $L$, total thickness $h$):

• Bottom layer: stiff elastic ("host"), $0 < z < z_1$
• Middle layer: thin viscoelastic core, $z_1 < z < z_2$
• Top layer: stiff elastic (including possible piezoelectric), $z_2 < z < z_3 = h$

Denote $u^i(x,t)$ as the longitudinal (axial) displacement of the $i$th layer's mid-surface and $w(x,t)$ as the common transverse deflection. For each layer, introduce (for $i=1,2,3$):

• $\psi^i(x,t) = (u^i - u^{i-1})/h_i$ (layer rotation-like measure)
• $v^i = (u^i + u^{i-1})/2$ (axial average)
• $\phi^i = \psi^i + w_x$ (total shear angle)

The stiff face layers do not exhibit shear: $\psi^1 = \psi^3 = -w_x$, $\phi^1 = \phi^3 = 0$. Only the compliant core $\phi^2\neq 0$ provides nontrivial shear behavior.

The essential "Rao–Nakra thin-compliant-layer" limit is achieved by sending the core's density and axial stiffness ($\rho_2, \alpha_2$) to zero, while retaining (i) its shear modulus $G_2$ and thickness $h_2$, (ii) the transverse kinetic energy $\dot w^2$. This justifies neglecting core normal stress transfer while ensuring shear interaction is maintained (Ozer, 2016, Ozer, 2015, Ozer, 2015, Ozer, 2017, Bautista et al., 20 Dec 2025).

2. Governing Equations: Variational Formulation and PDE System

Starting from Hamilton’s principle with kinetic, potential, and piezoelectric energies, together with external mechanical and electrical work, the state equations are derived under the above limit. For a basic three-layer model (stiff – thin compliant – stiff), the resulting coupled PDEs are:

$\begin{cases} m\,\ddot w - K_1\,\ddot w_{xx} + K_2\,w_{xxxx} - H G_2 (\phi^2)_x = 0, \[2mm] \rho_1h_1\,\ddot v^1 - \alpha^1h_1\,v^1_{xx} - G_2\,\phi^2 = 0, \[2mm] \rho_3h_3\,\ddot v^3 - \alpha^3h_3\,v^3_{xx} + G_2\,\phi^2 -\gamma h_3 (\varphi^1)_x = 0, \[2mm] -\xi (\phi^1)_{xx} + \phi^1 - \gamma v^3_x = \frac{\sigma_s(t)}{h_3}, \[2mm] \phi^2 = \frac{1}{h_2}(-v^1 + v^3 + H w_x), \end{cases}$

where $w, v^1, v^3$ are as above, $\phi^2$ is the only nonvanishing shear angle, $H = (h_1 + 2h_2 + h_3)/2$, and $m = \rho_1h_1 + \rho_3 h_3$. Piezoelectric actuation (if present) appears via electrical terms and boundary controls, depending on the physical assumptions (Ozer, 2016, Ozer, 2015). Under the electrostatic limit, the electrical field is potential-driven, giving rise to quadratic-through-thickness potentials and algebraic coupling for the voltage or charge source.

For a multilayer ($2m+1$) beam, the equations generalize to a Rayleigh (or Euler–Bernoulli) type transverse equation and $m+1$ longitudinal wave equations coupled via the compliant layer shears:

$\begin{cases} m\,\ddot w - \alpha \partial_{xx}\ddot w + K\,\partial_{xxxx}w - N^T h_E(G_E\,\psi_E+\widetilde G_E\,\dot\psi_E)' = 0, \[2mm] h_O p_O \ddot y_O - h_O E_O y_O'' + B^T (G_E\,\psi_E+\widetilde G_E \dot\psi_E) = 0, \[2mm] B\,y_O = h_E \psi_E - h_E N w', \end{cases}$

where $y_O$ collects axial displacements of stiff layers and $\psi_E$ the core shears (Ozer et al., 2014, Ozer et al., 2013).

3. Boundary Conditions and Actuation Mechanisms

Boundary conditions are typically clamped (zero displacement and rotation) at $x=0$, with free, controlled, or collocated feedback (boundary moment, force, or electrical input) at $x=L$:

• Axial feedback: $$\alpha^i h_i v^i_x(L) + \gamma h_i \phi^1(L) = g^i(t)$$
• Transverse moment/force: $$K_2 w_{xx}(L) = -M(t), \quad K_1 \ddot w_x(L) - K_2 w_{xxx}(L) + G_2H \phi^2(L) = g(t)$$
• Piezoelectric actuation (charge/voltage): $\sigma_s(t)$ prescribed as boundary input or via Dirichlet electrical conditions (Ozer, 2016, Ozer, 2015, Ozer, 2015).

In many configurations, mechanical (velocity) feedback is applied at the boundary, e.g.,

$$g^1(t) = -s_1 \dot v^1(L,t),\quad \sigma_s(t) = -s_3 \dot v^3(L,t),\quad M(t) = k_1 \dot w_x(L,t),\quad g(t) = k_2 \dot w(L,t).$$

Feedback law choice is critical for stability, controllability, and performance (Ozer, 2016, Ozer, 2015).

4. Mathematical Properties: Well-Posedness, Stability, and Decay

The Rao-Nakra sandwich beam yields a well-posed evolution in an energy phase space, typically constructed via the natural energetic inner product:

$$\| (v^1, v^3, w, \dot v^1, \dot v^3, \dot w) \|_H^2 = \int_0^L \{\rho_1 h_1 |\dot v^1|^2 + \rho_3 h_3 |\dot v^3|^2 + m |\dot w|^2 + \text{(elastic and shear terms)}\} dx$$

• The system is formulated abstractly as $$\dot \Phi(t) + \mathcal A \Phi(t) + \mathcal B \Phi(t) = 0$$, with the generator $\mathcal A$ encoding spatial operators and $\mathcal B$ the boundary-damping operators (Ozer, 2016, Ozer, 2015, Bautista et al., 20 Dec 2025).
• Under collocated mechanical feedback, the semigroup generated by $\mathcal A + \mathcal B$ is uniformly exponentially stable: there exist constants $M, \omega > 0$ such that

$$\| e^{(\mathcal A + \mathcal B)t} \Phi_0 \|_H \leq M e^{-\omega t} \| \Phi_0 \|_H$$

for all initial data (Ozer, 2016, Ozer, 2017, Ozer, 2015, Bautista et al., 20 Dec 2025).

Technical proofs involve demonstrating maximal dissipativity of $\mathcal A$, unique continuation (no imaginary axis eigenmodes), compactness of coupling (so the system is a compact perturbation of a decoupled, exponentially stable generator), and then application of Triggiani's compact-perturbation theorem (Ozer et al., 2013, Ozer, 2016, Ozer, 2017).

Polynomial decay estimates (energy $E(t) \le C (1+t)^{-\beta}$ for $\beta > 0$) are obtained for beams under frictional or fractional (memory-type) damping, with explicit quantification of decay rate in terms of physical and fractional parameters (Ammari et al., 2024, Guesmia, 2022, Akil et al., 2021).

5. Boundary Controllability and Observability

Boundary controllability is established for multilayer Rao–Nakra beams by coupling the Rayleigh (bending) and wave (axial) branches via the viscoelastic shears. Observability inequalities are proved for the adjoint (backward-in-time) system using multiplier arguments and unique continuation. The Hilbert Uniqueness Method (HUM) then guarantees null controllability:

• For the undamped system, exact boundary controllability holds for any minimal actuation time $\tau$ determined by wave/beam speed, with controls $M(t)$ (moment) and $g_O(t)$ (axial forces) in $L^2(0,T)$ (Ozer et al., 2014).
• For damped systems, exact controllability persists if damping is sufficiently small (Ozer et al., 2014).
• The control time threshold and minimal control regularity are sharp; higher damping levels can degrade controllability.
• Boundary inputs required for controllability do not scale with the number of layers (Ozer et al., 2014, Bautista et al., 20 Dec 2025).

6. Generalizations, Extensions, and Computational Aspects

• Piezoelectric and Electromagnetic Coupling: Extensions to active constrained layer (ACL) models include piezoelectric layers actuated by voltage, charge, or current, with magnetic effects retained or neglected according to the Maxwell closure (fully dynamic, quasi-static, or electrostatic). Full variational derivations yield Rao-Nakra–type systems incorporating additional electric variables and coupling (Ozer, 2015, Ozer, 2015, Ozer, 2017).
• Fractional and Memory Damping: The introduction of spatially distributed fractional-order (e.g., Caputo) or memory-type damping leads to augmented system formulations and polynomial decay rates, with numerical methods (energy-conserving finite-difference, β-Newmark) providing validation (Ammari et al., 2024).
• Stabilization via Partial or Indirect Damping: Analysis of strong and polynomial stability for beams with partial or indirect viscous damping reveals sharp necessary and sufficient conditions on the damping locations and material asymmetries for stability and decay (Akil et al., 2021, Guesmia, 2022).
• Boundary-delayed Feedback: Dynamic boundary conditions, including time-varying delay and spatially dependent weights, are rigorously treated using Kato semigroup theory and Lyapunov functional construction; exponential stability and null-controllability are preserved under suitable feedback forms (Bautista et al., 20 Dec 2025).

7. Significance and Applications

The Rao-Nakra type sandwich beam model is fundamental in the analysis and control of smart structures containing viscoelastic layers, especially for applications in vibration suppression, noise reduction, and active structural health monitoring. The model’s compact block-coupled PDE structure is particularly amenable to semigroup, spectral, and control-theoretic analysis, facilitating rigorous proofs of stability, robust controllability, and time-optimal control laws suitable for engineering design (Ozer, 2016, Ozer, 2015, Bautista et al., 20 Dec 2025, Ozer et al., 2013, Ozer, 2017).

The analytical framework is robust to multilayer generalization, piezoelectric/electromagnetic actuation, fractional damping, strong/partial damping, and realistic boundary actuation constraints. These features confirm the Rao–Nakra model's central role in the modern mathematical theory of composite beam and plate structures.