Matériaux, résonances et interfaces (MRI)

Résumé: Identifying a universal material constitutive law which describes the mechanical response of rubber-like solids for all deformation fields and achievable extensions, is an outstanding challenge. Here, we propose to exploit the propagation of elastic waves and demonstrate that monitoring incremental guided wave propagation in an elastomer plate undergoing uniaxial extension reveals model sensitivities that are inaccessible in the corresponding static test. We measure the dispersion relations of the three zero-order guided modes propagating parallel and perpendicular to the direction of imposed elongation. We compare them with predictions from the acoustoelastic theory, which also takes into account material rheology, using parameters extracted from fitting the uniaxial stress–strain curve across three successive elongation regimes, following the methodical procedure of (Destrade et al. 2017 Proc. R. Soc. A: Math. Phys. Eng. Sci. 473, 20160811. (doi:10.1098/rspa.2016.0811)). We evidence that our approach lifts the degeneracy between hyperelastic models with different functional forms of the so-called (Formula presented) term, which remains indistinguishable from static uniaxial tension stress–strain measurements alone. However, like their static counterpart, our dynamic measurements cannot distinguish between different generalized neo-Hookean models.

Résumé: Semi-analytical methods for modeling guided waves in structures of constant cross-section yield frequency-dependent polynomial eigenvalue problems for the wavenumbers and mode shapes. Solving these eigenvalue problems over a range of frequencies results in continuous eigencurves. Recent research has shown that eigencurves of differentiable parameter-dependent eigenvalue problems can alternatively be computed as solutions to a system of ordinary differential equations (ODEs) obtained by postulating an exponentially decaying residual of a modal solution. Starting from an approximate initial guess of the eigenvalue and eigenvector at a given frequency, the complete eigencurve is obtained using standard numerical ODE solvers. We exploit this idea to develop an efficient method for computing the dispersion curves of plate structures coupled to unbounded solid or fluid media. In these scenarios, the approach is particularly useful because the boundary conditions give rise to nonlinear terms that severely hinder the application of traditional solvers. We discuss suitable approximations of the nonlinearity for obtaining initial values, analyze computational costs and robustness of the proposed algorithm, and verify results by comparison against existing methods.