From geometric fractals to physical fractals: Concepts, models, and methods of fractional-order mechanics
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摘要: 自Newton和Leibniz时代以来, 连续介质力学中的微元体概念一直发挥着基础性作用. 当研究视域从传统工程材料拓展至生命体时, 由于生命体的多尺度、多层级特征, 微元体这一经典概念面临新的挑战. 本文聚焦于分形与分数阶微积分的交叉融合, 分析了生物多级结构中分形雪花与物理分形空间的抽象过程, 回顾了分数阶力学的探索历程. 本文从两个经典问题——即“Mandelbrot之问”与“L’Hospital之问”——出发, 展示了如下探索路径: (a) 从几何分形到物理分形, (b) 从运动学到动力学, (c) 从分形维数到分数阶次. 具体内容包括: (a) 从生物体中抽象出的分形雪花及其生长运动学, (b) 从生命体中抽象出的分数阶算子及其理论建构; (c) 分数阶算子理论的应用. 在此基础上, 本文梳理了微元体概念从经典单元、代表性体积单元RVE、点阵单胞、力学超材料单胞到分形胞元的演化脉络, 探讨了分数阶次的多重来源与分形−分数阶的时空关联性. 本文认为, 物理分形空间中的力学呈现非整数阶、非有理、非初等与非局部化的特征, 这为拓展力学的新疆界提供了全新的思路.Abstract: Since the era of Newton and Leibniz, the concept of the material element in continuum mechanics has played a fundamental role. When the research perspective expands from traditional engineering materials to living organisms, the classic concept of the material element faces new challenges due to the multi-scale and multi-level characteristics of living organisms. This article focuses on the intersection of fractals and fractional calculus, analyzes the abstraction process of fractal snowflakes and physical fractal spaces from biological multi-level structures, and reviews the exploration process of fractional-order mechanics. This article starts from two classic problems, namely the "Mandelbrot Question" and the "L’Hospital Question", and presents the following exploration paths: (a) from geometric fractals to physical fractals, (b) from kinematics to dynamics, and (c) from fractal dimension to fractional order. The specific content includes: (a) fractal snowflakes abstracted from living organisms and their growth kinematics, (b) fractional-order operators abstracted from living organisms and their theoretical construction; (c) applications of fractional-order operator theory. On this basis, this article delineates the evolutionary trajectory of the material element concept: from the classical element, the representative volume element (RVE), the lattice unit cell, and the mechanical metamaterial unit cell to the fractal cell, and discusses the multiple origins of fractional orders and the spatiotemporal correlation between fractals and fractional calculus. This article concludes that mechanics in the physical fractal spaces exhibits features that are non-integer-order, non-rational, non-elementary, and non-local, offering novel perspectives for broadening the boundaries of mechanics.
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图 3 常见的分形几何花样 (周扬杰 等 2024). (a) 谢尔宾斯基三角形, (b) 谢尔宾斯基地毯, (c) 勾股树, (d) 迭代树, (e) 科赫雪花曲线, (f) 三维谢尔宾斯基三角形, (g) 谢尔宾斯基海绵, (h) 莱维C形曲线
图 4 从生物多级结构中抽象出的分形结构 (Fan et al. 2008, He et al. 2009). (a) 人类头发纤维, (b) 羊毛纤维的多级结构, (c)(3,9 + 2) 双重分形结构, (d)(9 + 2,3) 双重分形结构
图 5 超级分形雪花 (Yin et al. 2010b). (a) 人类的小肠绒毛, (b) 千姿百态的雪花, (c) 密排型超级分形雪花, (d) 弥散型超级分形雪花
图 6 比例生长运动模式 (Yang et al. 2015). (a) 弥散型分形雪花的比例生长运动过程, (b) Sierpinski垫片的比例生长运动过程
图 10 生长路径的多样性 (杨帆 2014). (a) 由交错循环路径生成的Cantor集, (b) 由顺序串联路径生成的Cantor集, (c) 细胞有丝分裂, (d) 树木的生长过程
图 11 分数阶微积分的应用. (a) 岩石分数阶黏弹塑性损伤蠕变模型研究 (虎积元 等 2024), (b) 电力系统混沌振荡的分数阶自适应滑模控制 (孙政章 2024), (c) 基于RBF神经网络的分数阶虚拟同步机控制策略 (张赟宁 等 2025), (d) 分数阶双稳系统随机共振现象研究及FPGA实现 (竺佐 等 2022)
图 12 物理分形空间上的分数阶生物力学 (Guo et al. 2019, Peng et al. 2021, Jian et al. 2026a). (a) 基于光滑神经纤维抽象的阶梯型胞元, (b) 基于主动脉和小动脉血管段抽象的无穷级微弹性腔物理结构模型和对应的比拟电路, (c) 基于密质骨纤维抽象的阶梯型胞元, (d) 基于肌腱/韧带原纤维抽象的阶梯型胞元
图 13 无穷级粘弹性纤维链 (Yin et al. 2022b)
图 14 物理分形、分形胞元和分形元件 (Yin et al. 2022b)
图 15 物理分形的应用场景 (Yin et al. 2022b). (a) 生物力学建模 (以从肌腱中抽象出的“纤维链/组织凝胶”复合材料为例), (b) 神经信号传导, (c) 隧道噪声控制, (d) 页岩油渗流
图 17 材料力学基本分析单元的演化 (冯西桥 2017, 杨鑫 等 2021, 何玉龙 等 2026, 渐徵墨 等 2026). (a) 连续介质力学微元体, (b) 复合材料力学RVE单元, (c) 点阵材料与超材料单胞, (d) 物理分形空间中的力学元件网络: 分形胞元
图 18 物理分形空间中元件分形树的对称性演化 (Jian et al. 2024b). (a) 基于含棘突神经纤维抽象的分形树胞元, (b) 基于神经元树突抽象的分形树胞元, (c) 基于密质骨纤维抽象的分形树胞元
图 19 分数阶粘弹性模型高效数值方法的核心思想. (a) 算子空间中的不动点迭代: 分形阶梯模型的深度递推等价于算子空间中Möbius变换的不动点迭代; 四周为深度$ N=1,\;2,\;3,\;4 $的有限深度阶梯电路, 对应迭代序列 $ {{T}}^{\left(1\right)},\;{{T}}^{\left(2\right)},\;{{T}}^{\left(3\right)},\;{{T}}^{\left(4\right)} $ (红色圆点), 该序列沿蓝色螺旋收敛至压缩不动点$ {\widehat{T}}^{\left(\mathrm{\infty }\right)} $ (金色星号), (b) 数值验证: 不同深度阶梯模型的单位阶跃响应随深度增加逐渐逼近极限情形的响应曲线, 直观展示了有限深度模型向无限深度精确响应的收敛; 基于该框架, 截断迭代即可得到高效的 Prony级数近似, (c) 分数阶积分算子$ {s}^{-1/2} $的数值算法对比: 左图为$ {L}^{2} $误差随计算时间的变化, 右图为误差随单个积分点静态内存的变化
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