Wave propagation in continuous bimodular media

Oscillating Nonlinear Acoustic Waves in a Mooney–Rivlin Rod

“…Considering the formation of shock wave fronts in solid mechanics, it has been found that in a 1D bi-modular elastic rod, a shock wave front arises when a slowly moving wave pulse is overtaken and absorbed by a faster moving pulse; see [15][16][17][18][19][20][21]. Within the considered one-dimensional case, a bi-modular material is defined by a step-wise dependence of the elastic modulus E(ε) on strain ε [22,23], with a one-dimensional infinitesimal strain derived by the linear Cauchy strain-displacement relation:…”

“…where u is the infinitesimal displacement; x is the spatial coordinate; and t is the time. It has recently been found [23] that in contrast to fluid dynamics where the shock wave fronts propagate with supersonic velocities, the shock wave front in a 1D elastic bi-modular rod propagates with an intermediate velocity v s satisfying the condition.…”

“…Herein, x j are the spatial grid nodes and t n are the nodes along the time-axis. It should also be noted that Lax-Wendroff scheme, along with the Godunov scheme, is recommended for solving nonlinear hyperbolic differential equations with discontinuities in coefficients [40] and evaluating discontinuities in stress and strain fields [40]; see also [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] where this scheme was used in constructing solutions with the appearance and propagation of shock wave fronts.…”

Elastic waves in media with bimodular nonlinearity taking into account the effects of reflection from shock fronts

“…Quite a lot of studies have been devoted to the study of nonlinear propagation of longitudinal elastic waves in bimodular solids without dispersion (and without taking into account reflection from gaps) [8][9][10][11][12][13][14][15][16][17]. In such media, the nonlinear propagation mode occurs only for heteropolar waves, and unipolar single pulses propagate linearly, with constant but different velocities depending on their polarity.…”

Упругие Волны В Средах С Разномодульной Нелинейностью С Учетом Эффектов Отражения От Ударных Фронтов

“…Изучению нелинейного распространения продольных упругих волн в разномодульных твердых телах без дисперсии (и без учета отражения от разрывов) посвящено довольно много работ [8][9][10][11][12][13][14][15][16][17]. В таких средах нелинейный режим распространения имеет место только для разнополярных волн, а однополярные одиночные импульсы распространяются линейно, с постоянными, но различными скоростями, зависящими от их полярности.…”