1 School of Mathematics , Cardiff University , Senghennydd Road, Cardiff CF24 4AG , UK.
2 Mathematical Institute , University of Oxford , Woodstock Road, Oxford OX2 6GG , UK.
Philos Trans A Math Phys Eng Sci. 2019 May 6;377(2144):20180068. doi: 10.1098/rsta.2018.0068.
The problem of the Rivlin cube is to determine the stability of all homogeneous equilibria of an isotropic incompressible hyperelastic body under equitriaxial dead loads. Here, we consider the stochastic version of this problem where the elastic parameters are random variables following standard probability laws. Uncertainties in these parameters may arise, for example, from inherent data variation between different batches of homogeneous samples, or from different experimental tests. As for the deterministic elastic problem, we consider the following questions: what are the likely equilibria and how does their stability depend on the material constitutive law? In addition, for the stochastic model, the problem is to derive the probability distribution of deformations, given the variability of the parameters. This article is part of the theme issue 'Rivlin's legacy in continuum mechanics and applied mathematics'.
里夫林方块问题是确定各向同性不可压缩超弹性体在等轴死载下所有均匀平衡态的稳定性。在这里，我们考虑这个问题的随机版本，其中弹性参数是遵循标准概率定律的随机变量。这些参数的不确定性可能来自于不同批次均匀样本之间的固有数据变化，或者来自于不同的实验测试。对于确定性弹性问题，我们考虑以下问题：可能的平衡是什么，它们的稳定性如何取决于材料本构定律？此外，对于随机模型，问题是给定参数的可变性，推导出变形的概率分布。本文是主题为“里夫林在连续力学和应用数学中的遗产”的一部分。
