Wave dispersion under finite deformation

Abstract

We derive exact dispersion relations for axial and flexural elastic wave motion in a rod and a beam under finite deformation. For axial motion we consider a simple rod model, and for flexural motion we employ the Euler-Bernoulli kinematic hypothesis and consider both a conventional transverse motion model and an inextensional planar motion model. The underlying formulation uses the Cauchy stress and the Green-Lagrange strain. For all models, we consider linear constitutive relations in order to isolate the effect of finite motion. The proposed methodology, however, is applicable to problems that also exhibit material nonlinearity. For the rod model, we obtain the exact analytical explicit solution of the derived finite-deformation dispersion relation, and compare it with data obtained via numerical simulation of nonlinear wave motion in a finite rod. For the beam model, we obtain a solution by numerical root finding. The results allow us to quantify the deviation in the dispersion curves when exact large deformation is considered compared to following the assumption of infinitesimal deformation. We show that incorporation of finite deformation following the chosen definitions of stress and strain raises the frequency branches for both axial and flexural waves above the nominal values associated with linear motion. For the beam problem, only the inextensional planar motion model provides an accurate description of the finite-deformation response for both static deflection and wave dispersion. Our findings, which represent the first derivation of finite-strain, amplitude-dependent dispersion relations for any type of elastic media, draw attention to 1) the tangible effect of finite deformation on wave dispersion and consequently on the speed of sound in an elastic medium and 2) the importance of incorporating longitudinal-transverse motion coupling in nonlinear analysis of thin structures.Comment: Accepted for publication in Wave Motio

Dispersion characteristics of a nonlinear elastic metamaterial

“…Capturing this property within the dispersion relation provides a general and fundamental description of the nonlinear wave propagation characteristics. Abedinnasab and Hussein 36 derived exact dispersion relations for axial and flexural elastic wave motion in homogeneous rods and beams under finite strain.…”

“…The nonlinearity considered arises from large elastic deformation in the rod, whereas the metamaterial behavior is associated with the dynamics of the local resonators. Our model is based on embedding the exact finite-strain dispersion relation for a 1D homogeneous rod 36 for a periodic elastic medium. 43 This approach has been applied earlier to a PC rod to examine the interplay between nonlinearity and periodicity; 42 here we apply it to a MM rod.…”

“…This analysis places emphasis on the role of the resonator parameters. We then review the treatment of geometric nonlinearity, i.e., finite strain, in the context of a homogeneous medium following the theory proposed by Abedinnasab and Hussein 36 (Section III A). In Section III B, we combine the previous derivations.…”

“…The reader is referred to Ref. [36] for more details as well as a validation of the theoretical approach by means of a comparison with a standard finite-strain numerical simulation of a corresponding 1D model with finite dimensions.…”

“…Frequency dispersion curves for a 1D homogeneous elastic medium 36. The finite-strain dispersion relation is based on Eq.…”