Li Jun, Ostoja-Starzewski Martin
Department of Mechanical Engineering, University of Massachusetts, Dartmouth, MA 02747-2300, USA.
Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA.
Philos Trans A Math Phys Eng Sci. 2020 May 29;378(2172):20190288. doi: 10.1098/rsta.2019.0288. Epub 2020 May 11.
This article advances continuum-type mechanics of porous media having a generally anisotropic, product-like fractal geometry. Relying on a fractal derivative, the approach leads to global balance laws in terms of fractal integrals based on product measures and, then, converting them to integer-order integrals in conventional (Euclidean) space. Proposed is a new that is frame invariant, has no bias with respect to the coordinate origin and captures the differences between two fractal media having the same fractal dimension but different density distributions. A continuum localization procedure then allows the development of local balance laws of fractal media: conservation of mass, microinertia, linear momentum, angular momentum and energy, as well as the second law of thermodynamics. The product measure formulation, together with the angular momentum balance, directly leads to a generally asymmetric Cauchy stress and, hence, to a micropolar (rather than classical) mechanics of fractal media. The resulting micropolar model allowing for conservative and/or dissipative effects is applied to diffusion in fractal thermoelastic media. First, a mechanical formulation of Fick's Law in fractal media is given. Then, a complete system of equations governing displacement, microrotation, temperature and concentration fields is developed. As a special case, an isothermal model is worked out. This article is part of the theme issue 'Advanced materials modelling via fractional calculus: challenges and perspectives'.
本文提出了具有一般各向异性、类乘积分形几何的多孔介质的连续介质型力学。该方法基于分形导数，得出基于乘积测度的分形积分形式的全局平衡定律，然后将其转换为常规（欧几里得）空间中的整数阶积分。提出了一种新的[此处原文缺失具体内容]，它是框架不变的，对坐标原点无偏倚，并能捕捉具有相同分形维数但密度分布不同的两种分形介质之间的差异。连续介质局部化过程进而允许发展分形介质的局部平衡定律：质量守恒、微惯性、线性动量、角动量和能量守恒，以及热力学第二定律。乘积测度公式与角动量平衡一起，直接导致一般不对称的柯西应力，从而导致分形介质的微极（而非经典）力学。所得的允许保守和/或耗散效应的微极模型应用于分形热弹性介质中的扩散。首先，给出了分形介质中菲克定律的力学表述。然后，建立了一个完整的方程组，用于描述位移、微旋转、温度和浓度场。作为一个特殊情况，给出了一个等温模型。本文是“通过分数阶微积分进行先进材料建模：挑战与展望”主题问题的一部分。
