Zhang Xian, Ostoja-Starzewski Martin
Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA.
Institute for Condensed Matter Theory and Beckman Institute, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA.
Philos Trans A Math Phys Eng Sci. 2020 May 29;378(2172):20190591. doi: 10.1098/rsta.2019.0591. Epub 2020 May 11.
This paper reports the application of cellular automata to study the dynamic responses of Lamb-type problems for a tangential point load and a concentrated moment applied on the free surface of a half-plane. The medium is homogeneous, isotropic and linear elastic while having a random mass density field with fractal and Hurst characteristics. Both Cauchy and Dagum random field models are used to capture these effects. First, the cellular automata approach is tested on progressively finer meshes to verify the code against the continuum elastodynamic solution in a homogeneous continuum. Then, the sensitivity of wave propagation on random fields is assessed for a wide range of fractal and Hurst parameters. Overall, the mean response amplitude is lowered by the mass density field's randomness, while the Hurst parameter (especially, for < 0.2) is found to have a stronger influence than the fractal dimension on the response. The resulting Rayleigh wave is modified more than the pressure wave for the same random field parameters. Additionally, comparisons with previously studied Lamb-type problems under normal in-plane and anti-plane loadings are given. This article is part of the theme issue 'Advanced materials modelling via fractional calculus: challenges and perspectives'.
本文报道了应用元胞自动机研究半平面自由表面上切向点载荷和集中力矩作用下兰姆型问题的动态响应。介质是均匀、各向同性且线弹性的，同时具有具有分形和赫斯特特征的随机质量密度场。柯西和达古姆随机场模型都被用来捕捉这些效应。首先，在逐渐细化的网格上对元胞自动机方法进行测试，以验证该代码与均匀连续体中的连续弹性动力学解是否一致。然后，针对广泛的分形和赫斯特参数，评估随机场对波传播的敏感性。总体而言，质量密度场的随机性降低了平均响应幅度，同时发现赫斯特参数（特别是对于<0.2）对响应的影响比分形维数更强。对于相同的随机场参数，瑞利波的变化比压力波更大。此外，还给出了与先前研究的平面内和平面外正常载荷下兰姆型问题的比较。本文是主题为“通过分数阶微积分进行先进材料建模：挑战与展望”的一部分。
