Abstract We develop formulas to approximate the effective thermal conductivity of composite materials consisting of a matrix with nominal dimensionless conductivity K m a t = 1 containing inclusions (holes) with a conductivity K h o l e ≤ 1 laid out in fractal patterns: 2D Sierpinski carpets and 3D Menger sponges. Direct simulation of steady state heat conduction in finite stages of construction of these objects provide benchmark conductivity values. Recent theory on random walks in fractal media provides conductivity estimates for Sierpinski carpets for the case where K h o l e = 0 . Recognizing that the estimate in the fractal generators is related to the classic Rayleigh model, we obtain an expression for the conductivity in the more general case 0 ≤ K h o l e ≤ 1 . Comparison with direct simulated values shows a relative error below 6% when the conductivity contrast is high ( K h o l e ≈ 0 ) and below 1% when the contrast is low. This significantly improves the accuracy of estimates obtained from simple resistor networks, which are interpreted as upper and lower bounds. The extension of this approach to Menger sponges gives a conductivity estimator related to the classic Maxwell conductivity model. This estimator, accounting for the effects of pattern correlations, has relative errors in the same range as the carpet estimator. A conjecture for the random walk dimension in sponges is also derived. The effective conductivities for 2D and 3D patterns, with low volume fractions of the matrix and high conductivity contrast, also represent a significant improvement over the classic formulas for uncorrelated distributions of inclusions.

中文翻译：

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确定分形物体的有效电导率

摘要 我们开发了近似复合材料的有效热导率的公式，该复合材料由具有标称无量纲传导率 K mat = 1 的基体组成，其中包含传导率 K 孔 ≤ 1 的夹杂物（孔洞），以分形图案布置：2D 谢尔宾斯基地毯和 3D Menger 海绵. 在这些物体的有限构造阶段直接模拟稳态热传导可提供基准电导率值。最近关于分形介质中随机游走的理论为 K 孔 = 0 的情况提供了谢尔宾斯基地毯的电导率估计值。认识到分形生成器中的估计与经典瑞利模型相关，我们获得了更一般情况下的电导率表达式 0 ≤ K 孔 ≤ 1 。与直接模拟值的比较表明，当电导率对比度高（K 孔 ≈ 0）时，相对误差低于 6%，而当对比度低时，则低于 1%。这显着提高了从简单电阻网络获得的估计的准确性，这些电阻网络被解释为上限和下限。将此方法扩展到 Menger 海绵可提供与经典 Maxwell 电导率模型相关的电导率估算器。该估计器考虑了模式相关性的影响，与地毯估计器的相对误差范围相同。还推导出了海绵中随机游走维度的猜想。2D 和 3D 图案的有效电导率，具有低基质体积分数和高电导率对比度，