Taghizadeh Kianoosh, Shrivastava Rohit Kumar, Luding Stefan
Multi-Scale Mechanics, Faculty of Engineering Technology, MESA+, University of Twente, 7522NB Enschede, The Netherlands.
Institute of Applied Mechanics (CE), SC SimTech, University of Stuttgart, 70569 Stuttgart, Germany.
Materials (Basel). 2021 Apr 6;14(7):1815. doi: 10.3390/ma14071815.
Energy transfer is one of the essentials of mechanical wave propagation (along with momentum transport). Here, it is studied in disordered one-dimensional model systems mimicking force-chains in real systems. The pre-stressed random masses (other types of disorder lead to qualitatively similar behavior) interact through (linearized) Hertzian repulsive forces, which allows solving the deterministic problem analytically. The main goal, a simpler, faster stochastic model for energy propagation, is presented in the second part, after the basic equations are re-visited and the phenomenology of pulse propagation in disordered granular chains is reviewed. First, the propagation of energy in space is studied. With increasing disorder (quantified by the standard deviation of the random mass distribution), the attenuation of pulsed signals increases, transiting from ballistic propagation (in ordered systems) towards diffusive-like characteristics, due to energy localization at the source. Second, the evolution of energy in time by transfer across wavenumbers is examined, using the standing wave initial conditions of all wavenumbers. Again, the decay of energy (both the rate and amount) increases with disorder, as well as with the wavenumber. The dispersive ballistic transport in ordered systems transits to low-pass filtering, due to disorder, where localization of energy occurs at the lowest masses in the chain. Instead of dealing with the too many degrees of freedom or only with the lowest of all the many eigenmodes of the system, we propose a stochastic master equation approach with reduced complexity, where all frequencies/energies are grouped into bands. The mean field stochastic model, the matrix of energy-transfer probabilities between bands, is calibrated from the deterministic analytical solutions by ensemble averaging various band-to-band transfer situations for short times, as well as considering the basis energy levels (decaying with the wavenumber increasing) that are not transferred. Finally, the propagation of energy in the wavenumber space at transient times validates the stochastic model, suggesting applications in wave analysis for non-destructive testing, underground resource exploration, etc.
能量转移是机械波传播的基本要素之一（与动量传输一起）。在此，我们在模拟实际系统中力链的无序一维模型系统中对其进行研究。预应力随机质量（其他类型的无序会导致定性相似的行为）通过（线性化的）赫兹排斥力相互作用，这使得可以解析地求解确定性问题。在重新审视基本方程并回顾无序颗粒链中脉冲传播的现象学之后，第二部分提出了主要目标，即一个更简单、更快的能量传播随机模型。首先，研究能量在空间中的传播。随着无序程度的增加（由随机质量分布的标准差量化），脉冲信号的衰减增加，从弹道传播（在有序系统中）过渡到类似扩散的特性，这是由于能量在源处的局域化。其次，利用所有波数的驻波初始条件，研究能量通过波数转移在时间上的演化。同样，能量的衰减（速率和量）随着无序程度以及波数的增加而增加。由于无序，有序系统中的色散弹道传输转变为低通滤波，其中能量在链中质量最低处发生局域化。我们不是处理系统中过多的自由度或仅处理众多本征模中最低的本征模，而是提出一种具有降低复杂度的随机主方程方法，其中所有频率/能量被分组为频段。平均场随机模型，即频段之间能量转移概率的矩阵，通过对短时间内各种频段间转移情况进行系综平均，并考虑未转移的基能级（随波数增加而衰减），从确定性解析解中校准。最后，瞬态时刻能量在波数空间中的传播验证了随机模型，表明其在无损检测、地下资源勘探等波分析中的应用。
