旋节线拓扑的非周期结构化材料: 力学性能预测、设计和应用

张简; 严子铭; 庄茁; 柳占立

doi:10.6052/1000-0992-26-001

旋节线拓扑的非周期结构化材料: 力学性能预测、设计和应用

doi: 10.6052/1000-0992-26-001 cstr: 32046.14.1000-0992-26-001

清华大学航天航空学院, 北京 100084

基金项目: 国家自然科学基金项目 (12525208, 12272204) 资助.

详细信息

作者简介:
柳占立 (1981-), 男, 博士, 教授, 博士生导师, 清华大学航天航空学院工程力学系. 主要从事固体动态强度与失效、物理AI加速的力学仿真和逆向设计等研究. 担任《力学与实践》副主编、多个期刊编委, 中国力学学会理事, 中国力学学会计算力学专委会副主任, 北京力学会秘书长等. 曾获中国力学学会青年科技奖、教育部自然科学奖一等奖、钱令希计算力学青年奖等奖励和荣誉. 主持国家自然科学基金青年项目 (A类) 等项目. 在《Journal of the Mechanics and Physics of Solids》、《Computer Methods in Applied Mechanics and Engineering》、《Nature Communications》等期刊发表学术论文200余篇, 出版专著、教材4部

通讯作者:
liuzhanli@tsinghua.edu.cn

中图分类号: O3
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出版历程
- 收稿日期: 2026-01-07
- 录用日期: 2026-03-03
- 网络出版日期: 2026-03-30

Spinodoid Non-periodic Architected Materials: Mechanical Performance Prediction, Design, and Applications

School of Aerospace Engineering, Tsinghua University, Beijing 100084, China

More Information

Corresponding author: liuzhanli@tsinghua.edu.cn

摘要

摘要: 结构化材料(Architected Materials)通过结构设计实现可控的超常性能, 在生物医学、航空航天、能源环境等多领域得到广泛应用. 通过在结构设计过程中引入随机非周期性, 可以解决传统周期性材料局部化失效导致的高脆性问题, 提高材料的韧性、损伤容限与缺陷不敏感性, 实现优异的综合性能. 但是, 这也同时使结构整体复杂度增加, 研究、设计成本大幅提升, 亟需发展新的理论和方法. 旋节线拓扑构成的材料(Spinodoid/Spinodal-Like Metamaterials)是一种具有代表性的非周期结构化材料, 其研究范式可以推广至多种复杂非周期结构的力学性能预测、拓扑结构设计. 本文以旋节线拓扑为代表, 介绍非周期结构化材料的建模原理、等效力学性能、设计和制造方法及典型应用, 对现有研究进行总结并提出未来的发展方向.
- 旋节线拓扑 /
- 结构化材料 /
- 非周期性 /
- 力学性能预测 /
- 逆向设计
Abstract: Through structural design, architected materials can achieve extraordinary properties and therefore have found broad applications across biomedical, aerospace, energy, and environmental domains. Incorporating non-periodicity into structural design helps mitigate the brittleness arising from localized failure in conventional periodic materials, leading to improved toughness, damage tolerance, and defect insensitivity. However, the added complexity imposes computational and manufacturing challenges, calling for the development of new theoretical frameworks and design methodologies. Spinodoid materials/Spinodal-Like materials, characterized by spinodal topology, represent a class of non-periodic architected materials, and the research paradigm established for these materials can be generalized to predict mechanical performance and guide structural design of various complex non-periodic architectures. This review focuses on spinodoid structures as a representative example, introducing their modeling principles, effective mechanical properties, design and manufacturing methods, and typical applications. We summarize the current research and propose future research directions, with the aim of charting a roadmap for non-periodic architected materials based on the advanced methods across the integrated "modeling–design–manufacturing–application" pipeline.
- Spinodoid /
- Spinodal topology /
- Architected materials /
- Mechanical performance prediction /
- Inverse design

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图 1 自然界的随机非周期拓扑结构. (a)犀鸟鸟喙(Schaedler & Carter 2016), (b)松叶横截面(Bhate 2019), (c)枫香树果实(Chiang et al. 2021), (d)人体股骨内部骨小梁(Musthafa & Walker 2024)

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图 2 非周期仿生材料. (a)仿骨小梁的随机Voronoi杆状材料(Jiao C et al. 2022), (b)仿向日葵髓部截面的双重梯度结构(Bai et al. 2024), (c)与犀鸟鸟喙内部结构高度相似的拓扑优化结构(Aage et al. 2017)

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图 3 材料数据库的构建及非周期结构的代表性优化设计流程. (a) ~ (d)材料数据库的力学性能分布范围, 包括(a)沿水平和竖直方向的等效杨氏模量(星形代表2939种超越经典超材料的新型材料类型)、(b)泊松比与(c, d)各向异性, 其中(c)展示了各向异性指数A^SU与E_x/E_y及ν_yx/ν_xy的关系图及投影, (d)展示了11组新型材料和经典超材料A^SU范围的浮动柱状图(Chen J X et al. 2025), (e) ~ (f)基于“频率提示 + 拓扑优化”的不规则微结构虚拟生长方法, 其中圆环颜色占比反映了不同基础单元的频率; 均匀化计算后, 设计参数与力学性能通过神经网络建立映射关系, 进而在迭代中向力学目标优化(Jia Y Q et al. 2024b)

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图 4 四种旋节线结构的建模方法. (a)不同相对密度和演化时间下, 相场演化法产生的三维实体结构(Hsieh et al. 2019), (b)不同参数模式下, 谱密度函数法获得的二维结构(Deng S G et al. 2026), (c) 3D-CRNN预测的三维实体结构, 其展现了良好的空间泛化效果(Lanzoni et al. 2024), (d)高斯随机场方法 + 水平集方法所得的四种旋节线典型构型(Deng W T et al. 2024)

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图 5 由细观角度建立宏观统计量的方法. (a)上方为旋节线元胞变形过程的拉伸能代理量(红)分布与弯曲能代理量(蓝)分布, 材料分布对两种能量的分布影响较大; 下方为实验所得(归一化)力学性能与响应代理量间的对比(Dhulipala & Portela 2025), (b)左侧为具有各向异性的正交对称TPMS结构及其弹性面, 右侧为引入边结构张量后理论预测值与数值计算值的对比(Daynes 2026)

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图 6 汇总现有代表性研究的Ashby图. (a)归一化等效模量(Hsieh et al. 2019, Izard et al. 2019, Portela et al. 2020, Anandan et al. 2025, Li K et al. 2026, Wang X H et al. 2025), (b)归一化等效强度(Hsieh et al. 2019, Liu Y J et al. 2024, Anandan et al. 2025)

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图 7 旋节线结构在单轴压缩载荷下的力学响应. (a)正交各向异性的加卸载过程(Deng W T et al. 2024), (b)分布于低密度区域的失效过程(Liu Y J et al. 2024)

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图 8 不同结构化材料的低速抗冲击性能. (a)为四种旋节线壳形式材料和一种随机TPMS材料的示意图、仿真计算结果及相应的载荷−侵彻深度曲线, (b)为周期性、随机性TPMS结构及旋节线结构的能量耗散对比, 自上至下分别为内能、损伤能量耗散、塑性能量耗散(Ejeh et al. 2024), (c)为不同晶格结构的残余速度−冲击速度曲线, 曲线与横轴的交点为极限穿透速度, EBEP结构的极限穿透速度较大, 抗穿透性能较好(Alhembar et al. 2025)

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图 9 含有梯度的生物组织与人工设计旋节线结构. (a)竹子横截面的扫描电子显微镜(SEM)图像(Mao et al. 2023), (b)压痕载荷下纵向、横向和梯度结构取向复合材料的应力分布(Liu Z Q et al. 2016, 2020), (c)通过插值函数对不同拓扑的旋节线结构单胞实现平滑过渡(Senhora et al. 2022), (d)海螺型(Conch)纳米陶瓷功能梯度材料具有高损伤容限的优势: 左上图展示了其具体结构; 右侧图呈现了壳厚为10 nm、40 nm、80 nm时, 结构经反复加卸载后的结构恢复情况; 左下图对比了不同类型结构在加卸载后的恢复率, 海螺型梯度结构无论壳层厚薄, 均表现出良好的损伤容限(Anandan et al. 2025)

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图 10 基于数据驱动的代表性先进逆向设计方法. (a)基于仿真计算小样本量数据驱动的多目标贝叶斯优化方法, 其中$ \{ {}^i({\boldsymbol{\varTheta }},{\bf{P}})\} $为描述符−性能对的初始数据集, $ {\bf{P}}({\boldsymbol{\varTheta}} )$为高斯随机过程建立的结构−性能关系; $ \{ {}^j{{\boldsymbol{\varTheta }}^*}\} $为贝叶斯优化新提出的结构描述符, $ \{ {}^j{{\bf{P}}^*}\} $为对应的性能 (Raßloff et al. 2025), (b) ~ (c)基于非线性实验数据的深度算子网络(DeepONet)代理模型, 其中(b)为正向建模与逆向设计示意图, (c)通过双光子光刻技术制备样品, 利用原位微压缩实验采集训练数据(Jin et al. 2025), (d) ~ (f)为 “建模−计算−预测−设计”一体化的GNDM方法, (d)中GNDM以一组基底材料和物理仿真算法为输入, 通过学习几何特征与性能之间的关系输出具有预期性能的微结构, (e)为面向目标性能的逆向设计流程, (f)说明无论训练样本的几何尺寸如何, GNDM都能生成填充任意空间的连续设计(Shen et al. 2026)

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图 11 不同尺度的旋节线材料制造方案. (a)自组装纳米壳基旋节线材料, 比例尺分别为: (ii, iii) 10 μm, (iv) 100 μm, (v) 10 μm, (vi) 5 μm (Portela et al. 2020), (b)双光子光刻和原子层沉积技术制造的微纳级壳结构(壳厚11 nm), 比例尺: 50 μm, (c) SLS技术打印的TPU壳结构(壳厚约1 mm), 元胞尺寸50 mm (Wang X H et al. 2025), (d)大尺寸水射流粉末床技术制造的镁基水泥旋节线结构, 比例尺: 10 cm (Nale et al. 2025)

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图 12 增材制造在复杂结构中产生的不同缺陷类型. (a)不锈钢颗粒部分熔化导致A处孔隙表面存在大量粘结颗粒(Liu Y J et al. 2024), (b) B处存在表面气孔, 由部分空气或金属蒸汽无法及时逸出导致, (c) C处表面存在阶梯状缺陷, 与结构表面的曲面形态有关(Wojciechowski et al. 2023), (d)支柱间连接的细丝D, 由打印过程喷嘴位置移动引起

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图 13 旋节线结构的典型应用. (a)基于逆向模型, 以弹性常数为目标的人体骨替代物设计: 从影像数据中提取骨小梁结构, 对增材制造骨样本提取力学性能(等效弹性模量曲面图); 借助逆向模型生成具有相似力学性能的旋节线结构, 并进行制造与验证(Deng W T et al. 2024), (b) ~ (d)利用旋节线铁电材料“力−热−电”响应的具体应用: (b)基于功能梯度旋节线结构设计的一体化三维力传感器及其制备样品, (c)功能梯度设计在三个方向单向冲击下的实验电压响应及有限元模拟结果, (d)将轻质铁电超材料用于埃菲尔铁塔结构健康监测的构建模块(Shi et al. 2024)

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表 1 旋节线结构的主要建模方法

建模方法	原理	结构类型	影响形态的主要参数	计算方法	优势	不足	代表文献
相场演化方法	基于Cahn-Hilliard动力学方程的相分离演化	三维壳/实体结构	相对密度$\bar \rho $ 演化时间t 相界面宽度参数δ 初始扰动场u₀	有限差分法	符合真实物理过程, 可用于自组装制备所得材料的验证.	计算速度慢、成本高; 材料以各向同性为主, 难以直观调控结构各向异性.	Hsieh et al. 2019
		二维结构	相对密度$\bar \rho $ 界面宽度、演化速率相关无量纲参数α、β	有限差分法	参数直观、简洁, 物理意义明确; 参数调控自由且便利.	无法生成三维旋节线结构; 参数需控制范围以避免数值不稳定现象.	Mandolesi et al. 2025
谱密度函数方法	基于可调谱密度函数由白噪声重构结构	二维结构	相对密度$\rho_m $ 空间频率k 各向异性指数α^mon、$\alpha _1^{{\rm{ort}}} $、$\alpha _2^{{\rm{ort}}} $ 频率旋转角γ	基于快速傅里叶变换的频域滤波与阈值化方法	参数物理意义明确, 实现了高效低维参数化; 计算速度快、具有可控的随机性和鲁棒性.	覆盖形态有限; 依赖于原始样本, 存在重构结果失真的风险.	Deng S G et al. 2026
神经网络方法	基于机器学习方法学习相分离演化过程	三维结构	神经网络超参数演化时间t	物理启发的卷积循环神经网络(CRNN)	在线阶段模拟相分离过程快速生成结构, 无需数值计算演化过程; 在时空上具备一定可泛化性.	数据集生成及离线训练所需计算成本高、具有数据集依赖性; 演化时间较长时可能产生误差积累.	Lanzoni et al. 2024
高斯随机场方法	基于大量随机波矢叠加的水平集方法	三维壳/实体结构	相对密度$\bar \rho $ GRF锥角参数 θ₁、θ₂、θ₃ 波数β	高斯随机场方法、水平集方法	结构生成速度相对较快, 计算便捷; 可自由、直观地调控结构形态与各向异性.	叠加波数量、单胞尺寸较大时计算速度较慢; GRF锥角参数与各向异性弹性矩阵缺乏直接关联.	Soyarslan et al. 2018 Kumar S et al. 2020

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