具有记忆的简单材料中的一维冲击波

Y. B. Fu, N. H. Scott
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引用次数: 16

摘要

本文用两种近似方法推导了平面膨胀激波在简单记忆材料中的渐近演化规律。第一种方法是奇异曲面理论与微扰方法的结合。导出了激波振幅和伴随的二阶不连续振幅的两个耦合一阶常微分方程系统。假设激波幅值很小,但伴随的二阶不连续可以被认为是有限的,也可以被认为与激波幅值一样小。第一种情况对应于所施加载荷的持续时间与粘性松弛时间相比较小的情况,我们表明两个不连续点的演化行为受到材料非线性的强烈影响。然而,第二种情况对应于持续时间与粘性松弛时间相当的情况,我们能够证明进化行为是由粘弹性线性理论预测的。在这两种情况下,在允许粘性松弛时间趋于无穷时,都得到了相应的弹性结果。第二种近似方法是应用于调制单波理论的激波拟合方法,它本身是一种基于小振幅有限速率假设的近似,与上面讨论的第一种情况等效。这两种近似方法在其共同的有效范围内产生相同的演化规律。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
One-dimensional shock waves in simple materials with memory
Asymptotic evolution laws for plane dilatational shock waves travelling in simple materials with memory are derived in this paper by using two approximation methods. The first method is a combination of singular surface theory and perturbation methods. A system of two coupled first-order ordinary differential equations is derived for the shock amplitude and the amplitude of the accompanying second-order discontinuity. The shock amplitude is assumed to be small, but the accompanying second-order discontinuity may be taken either to be finite or to be small with the shock amplitude. The first case corresponds to the situation in which the duration time of the applied load is small compared with the viscous relaxation time and we show that the evolutionary behaviour of the two discontinuities is strongly affected by material nonlinearity. The second case, however, corresponds to the situation in which the duration time is comparable with the viscous relaxation time and we are able to show that the evolutionary behaviour is as predicted by the linear theory of viscoelasticity. In both cases the corresponding elastic results are obtained on allowing the viscous relaxation time to tend to infinity. The second approximation method is the shock-fitting method applied to a modulated simple wave theory, which is itself an approximation based on a small-amplitude finite-rate assumption equivalent to the first case discussed above. The two approximation methods are shown to yield the same evolution laws within their common range of validity.
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