Investigation of bend and shear waves in a geometrically exact elastic rod model

The propagation of bend and shear waves through an elastic rod is investigated in the framework of geometrically exact nonlinear elasticity. The model allows for shear, extension/compression, bend and twist thus enabling the study of the dynamics of all types of elastic deformations. Numerical and analytical solutions demonstrate that the propagation of planar bend or shear disturbances of finite wavelength require bend, shear and extension/compression waves. For the case of an intrinsically straight, twisted rod an exact large amplitude solution for bend-shear waves is found. In this case, the rod has a helical shape and it rotates in a clockwise or counterclockwise direction. An exact analytical solution for the large amplitude shear oscillation of a straight, untwisted rod is derived. The same type of oscillation for a twisted rod is demonstrated to exist numerically. For the case of an intrinsically straight, untwisted rod, asymptotic theory predicts that the amplitude of the extension/compression wave is proportional to the square of the amplitude of the bend or shear wave and the wavelength of the extension/compression wave is one-half the wavelength of the bend or shear wave. The propagation of planar disturbances along an intrinsically straight, twisted rod is investigated numerically and compared to an all-atom molecular dynamics simulation of DNA. The simulations are in good qualitative agreement and indicate that the chemical structure of DNA supports elastic wave propagation of the type obtained from the rod model