考虑横向惯性效应非线性粘弹性杆的应变波

Strain Waves in a Nonlinear Viscoelastic Rod under the Consideration of Transverse Inertia

基于D′Alembert原理和虚功原理推导了考虑横向惯性效应下三次材料非线性粘弹性杆的基本控制方程,在有限小振幅的前提下利用多重时间尺度法及行波法,导出了反应应变畸变的MKdV-Burgers方程,在此基础上进一步推广应用双曲正切函数法得到了应变MKdV-Burgers方程激波形式的精确解,结果表明:该杆在软非线性材料的条件下可能存在激波,其传播速度与阻尼系数的平方成正比,与色散系数成反比;波宽则与阻尼成反比,与色散系数成正比,并且该种波的传播速度低于线弹性波速.

The basic longitudinal wave equation was derived by D′Alembert principle and virtual work principle for a cubic nonlinear Keilven-Voigt viscoelastic rod under the consideration of transverse inertia.Multiple scale method and traveling wave theory were applied to get the strain governing equation for small-but-finite waves.Furthermore,the exact Shock wave solutions of MKdV-Burgers equation for nonlinear viscoelastic rod were also obtained by hyperbolic tanh-function method.It is found that shock wave may exist in a rod made of soft nonlinear material,whose traveling speed is proportional to the square of damp coefficient and inversely proportional to dispersion coefficient.On the other hand,the wave width is inversely proportional to the damp coefficient and proportional to dispersion coefficient.After all the speed of the traveling wave given above is less than that of linear elastic case.