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

TL;DR

This paper develops an exact operator formulation for the dynamic interaction between a finite circular elastic plate and an elastic half-space, extending classical infinite-plate analyses to finite radii with explicit matrix representations.

Contribution

It introduces a novel operator approach using a truncated Lamb kernel and Bessel basis to accurately model finite-radius plate-half-space interactions.

Findings

Reproduces finite-radius experimental results for small R

Approaches the infinite-radius limit as R increases

Provides explicit matrix elements involving Cauchy principal values

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 \[ \mathscr{M}(\omega) = \chi_{[0,R]} \, T(\omega)\, \chi_{[0,R]}, \] where $T(\omega)$ is the Hankel multiplier involving the Rayleigh denominator $\Omega(\xi,\omega)$. Truncation destroys the diagonal structure of $T(\omega)$ and introduces real-axis singularities associated with the Rayleigh pole, in addition to square-root branch points at $\xi = k_T$ and $\xi = k_L$. We represent the action of $\mathscr{M}(\omega)$ on a finite-disk Bessel basis $\{ \phi_n(r) = A_{1,n} J_0(\lambda_n r) + A_{2,n} I_0(\lambda_n r)\},$ 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.