基于ZWT方程的线黏弹性球面波分析

针对时间尺度为1~100 s的爆炸冲击加载，忽略低频Maxwell体的松弛效应，同时为简化分析，忽略非线性弹簧效应项，推导了三维应力状态下的线黏弹性ZWT本构关系。基于球面波的基本动力学方程，结合ZWT线黏弹性本构关系，得到了以位移u表征的三阶波动方程。利用该方程分析固体介质中线黏弹性球面波传播过程中的吸收和弥散现象，得知:高频球面波的衰减因子趋于常数，相速趋于高频下的纵波速度；低频球面波的衰减因子和圆频率的平方成正比，其相速趋于低频下的纵波速度；低频球面波的纵波波速低于高频球面波的纵波波速，两者的比值和介质的泊松比、弹性模量及Maxwell体弹性模量相关。

Analysis of linear visco-elastic spherical waves based on ZWT constitutive equation

For the explosion and shock loading in the 1-100 μs time scale, the linear visco-elastic ZWT constitutive equation under a three-dimensional stress state was derived by ignoring the relaxation effect of the low-frequency Maxwell element and the nonlinear spring element. Based on the basic kinetic equation of the spherical wave and the linear visco-elastic ZWT constitutive equation, the third-order wave equation depicted by the displacement u was obtained. The absorption and dispersion phenomena of the spherical wave propagation in the visco-elastic solid were analyzed. Conclusions are as follows: the attenuation factor of the high-frequency spherical wave is inclined to constant, and its phase velocity to that of high-frequency longitudinal wave; the attenuation factor of the low-frequency spherical wave is in direct proportion to the square ω² of the circular frequency, and its  phase velocity is inclined to that of the low frequency longitudinal wave; the phase velocity of the low frequency spherical wave is less than that of the high-frequency spherical wave, and the ratio between them is related to the Poisson ratio as well as the elastic modules of the ZWT and Maxwell elements.