Dynamic Response of a Finite Circular Plate on an Elastic Half-Space Using the Truncated Lamb Kernel

TL;DR

This work provides an exact operator formulation for the dynamic interaction between a finite circular plate and an elastic half-space by introducing the truncated Lamb operator $\mathcal{M}(\omega)=\chi_{[0,R]}T(\omega)\chi_{[0,R]}$ and expanding fields in a finite-disk Bessel basis. The truncated operator yields a dense, spectrally convergent matrix that couples all plate modes, with integrals interpreted as Cauchy principal values plus Rayleigh-pole residues that capture radiation damping. The authors implement a stable numerical inversion using a four-interval Gauss-Legendre quadrature tailored to endpoint singularities and demonstrate that small-radius results reproduce known finite-radius experiments while large-radius limits converge to the infinite-radius model. The framework clarifies the distinction between finite-radius and infinite-plane assumptions, quantifies finite-radius corrections, and opens avenues for time-domain extensions and non-axisymmetric generalizations. Overall, the paper delivers a rigorous, exact operator-level treatment of finite-radius plate–half-space interaction with practical implications for structural foundations, sensors, and nondestructive testing.

Abstract

We develop an exact operator formulation for the dynamic interaction between a finite circular elastic plate and an elastic half-space. Classical analyses, beginning with Lamb's representation of the half-space response, typically assume an infinite plate and rely on diagonalization of the soil operator via the continuous Hankel transform. For a plate of finite radius $R$, however, both traction and displacement are supported only on $0 \le r \le R$, leading to the spatially truncated Lamb operator \[ \mathscr{M}(ω) = χ_{[0,R]} \, T(ω)\, χ_{[0,R]}, \] where $T(ω)$ is the Hankel multiplier involving the Rayleigh denominator $Ω(ξ,ω)$. Truncation destroys the diagonal structure of $T(ω)$ and introduces real-axis singularities associated with the Rayleigh pole, in addition to square-root branch points at $ξ= k_T$ and $ξ= k_L$. We represent the action of $\mathscr{M}(ω)$ on a finite-disk Bessel basis $\{ φ_n(r) = A_{1,n} J_0(λ_n r) + A_{2,n} I_0(λ_n r)\},$ which satisfies the free-edge boundary conditions of the plate, and derive explicit expressions for the resulting matrix elements. These involve integrals of the Lamb kernel evaluated as Cauchy principal values, with residue contributions corresponding to radiation damping in the half-space. The resulting operator matrix is dense but spectrally convergent. Its inversion yields a complete frequency-domain solution for finite-radius plates. The analysis reproduces Chen et al.'s finite-radius experiments for small $R$, approaches the infinite-radius limit as $R \to \infty$, and quantifies finite-radius corrections. To our knowledge, this is the first exact operator-level treatment of finite-radius plate-half-space interaction that retains the full nonlocal Lamb kernel.

Figures (11)

• Figure 1: Contour Path
• Figure 2: Spectral convergence of the discretized $\mathbf{\widehat{S}}$-matrix to a reference solution $\mathbf{\widehat{S}}_{\mathrm{ref}}$. $\xi_{\text{max}} = 2200, R = 0.0762$m
• Figure 3: Error over number of quadrature nodes for various values of $R$, for fixed $\omega = 1000$ and $\xi_{\text{max}} = 2800$
• Figure 4: Radial strain in a plate of radius $R=76.2$mm at the point $r=12.7$mm over time.
• Figure 5: Radial strain in a plate of radius $R=1000$mm at the point $r=12.7$mm over time.

Theorems & Definitions (22)

• Claim 1
• proof
• Claim 2
• proof
• Definition 1: $a$-Cauchy and $a$-convergence
• Definition 2: Closed form
• Claim 3
• proof
• Claim 4
• proof