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Wave dispersion under finite deformation

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.