Free Longitudinal Vibrations of Bimodular Beams: A Comparative Study

Abstract

In this paper, we consider the problem of the free longitudinal vibrations of a beam made of a bimodular material, i.e. an elastic material whose in-tension Young's modulus is a fraction of that under compression. After recalling the exact solutions for an infinite beam and for a beam with fixed ends calculated via the characteristics method, we apply high-resolution methods based on the finite-element approach to solve the nonlinear equation of the motion. In particular, we compare the exact solutions with the numerical solutions calculated using the collocation and least-squares method developed in the present study, the space-time element method, as well as total variation diminishing (TVD) and Newmark methods.

Wave propagation in a semi-infinite heteromodular elastic bar subjected to a harmonic loading

“…(10), and boundary condition at the free end of the bar. The Hugoniot conditions (15) and (16) at all discontinuous wave fronts are also satisfied exactly.…”

“…To do this we check Eq. (16). First of all, it is easy to see that u u 0 before the shock wave, since the shock wave propagates faster thanthe sound velocity in the domain before the shock.…”

“…Lucchesi and Pagni [15] considered the Riemann problem and proved that a unique solution always exists except for the special case of no-tension materials. Recently, Kharenko et al [16] considered free longitudinal vibration of both infinite and finite heteromodular bar using high-resolution methods based on the finite-element approach.…”

On Combining Different Acceleration Techniques at the Iterative Solution of PDEs by the Method of Collocations and Least Residuals

“…В [39,40] различные варианты метода КНН применены для расчета напряженного состояния пластин из изотропных и анизо-тропных композиционных материалов. В [41] метод КНН приложен к решению за-дачи о продольных колебаниях бимодулярных балок, в [42] к решению уравнения Бюргерса и уравнения Кортевега-де Вриза-Бюргерса, в [27,43] к трехмерному моделированию лазерной сварки металлических пластин на адаптивных сетках в областях с криволинейной поверхностью.…”

Wave propagation through an elastically asymmetric architected material

“…To handle better this artefact, an adapted integration technique could be used. In [46] the authors suggested a collocation and least-squares (CLS) method, which over-performs other alternative integration methods in terms of accuracy and in simplicity of use. It is worth mentioning here that such a strongly non-linear hyperbolic problem with inherent shock waves and other discontinuities has to be treated using general energetic principles and so-called entropy conditions formulated in [47], notably the choice between possible solution has to follow the principle of the growth of entropy of particles crossing the shock front.…”