This paper reports a cellular automata (CA) study of the transient dynamic responses of anti-plane shear Lamb’s problems on random fields (RFs) with fractal and Hurst effects. Both Cauchy and Dagum random field models are employed to capture the combined effects of spatial randomness in both mass density and stiffness tensor fields. First, with a dyadic representation, we formulate a second-rank anti-plane stiffness tensor random field (TRF) model with full anisotropy. Its statistical, fractal, and Hurst properties are investigated, leading to introduction of a so-called MOSP model of TRF. Then, we generalize the CA approach to incorporate the inhomogeneity in mass density as well as stiffness fields. Through parametric studies for both Cauchy and Dagum TRFs, the sensitivity of wave propagation on random fields is assessed for a wide range of fractal and Hurst parameters. In general, the mean response amplitude is lowered by the presence of randomness, and the Hurst parameter (especially, for \(\beta < 0.5\)) is found to have a stronger influence than the fractal dimension on the response. The results are compared with two simpler random fields: (1) randomness is present only in the mass density field; (2) randomness is present in the mass density field and in a locally isotropic stiffness tensor field. Overall, the results show that a second-rank anti-plane stiffness TRF with full anisotropy leads to the strongest fluctuation in displacement responses followed by a locally isotropic RF model.

本文报告了具有分形和赫斯特效应的随机场 (RF) 上反平面剪切兰姆问题的瞬态动态响应的元胞自动机 (CA) 研究。Cauchy 和 Dagum 随机场模型都用于捕捉质量密度和刚度张量场中空间随机性的综合影响。首先，通过二元表示，我们制定了具有完全各向异性的二阶反平面刚度张量随机场（TRF）模型。研究了它的统计、分形和 Hurst 属性，从而引入了所谓的MOSP 模型TRF。然后，我们将 CA 方法推广到包含质量密度和刚度场的不均匀性。通过对 Cauchy 和 Dagum TRF 的参数研究，评估了波在随机场上传播的灵敏度，以获得各种分形和 Hurst 参数。通常，平均响应幅度因随机性和 Hurst 参数（特别是对于\(\beta < 0.5\)) 被发现比分形维数对响应的影响更大。结果与两个更简单的随机场进行了比较：（1）随机性仅存在于质量密度场中；(2) 随机性存在于质量密度场和局部各向同性刚度张量场中。总体而言，结果表明，具有完全各向异性的二阶反平面刚度 TRF 导致位移响应的最强波动，其次是局部各向同性 RF 模型。