International Journal of Solids and Structures最新文献_第3页

Computational study of nanoindentation on an elastoplastic solid with an interface parallel to the indentation direction 界面平行于压痕方向的弹塑性固体上纳米压痕的计算研究

IF 3.8 3区工程技术 Q1 MECHANICS

International Journal of Solids and Structures

Pub Date : 2026-01-20 DOI: 10.1016/j.ijsolstr.2026.113857

Takashi Akatsu , Wataru Kubota , Yutaka Shinoda , Fumihiro Wakai

Nanoindentation has been developed to map the local mechanical properties of heterogeneous solids. Although the local properties at the indentation site are required for mapping, the map resolution worsens because of the deformation of other components surrounding the indentation, such as the grains encompassing an indentation in heterogeneous polycrystals. To quantify the transitional change in a P-h curve with an increase in h, where P is the indentation load and h is the indentation penetration depth, this study proposes weight functions. These functions aim to quantify point-sharp nanoindentations on a circular column embedded in a matrix, which has an interface parallel to the indentation direction. The mechanical properties of the column can be accurately evaluated by extrapolating the weight functions to an indentation depth of zero, which improves the map resolution. When comparing the effects of the parallel and vertical interfaces, the impact of the vertical interface on the P-h curve is more significant than that of the parallel interface. This suggests that the correlation between the heterogeneity of local mechanical properties and the microstructure observed at the surface is controversial.

纳米压痕已被用于绘制非均质固体的局部力学特性。虽然在压痕位置的局部性质是测绘所必需的，但由于压痕周围的其他成分的变形，例如在非均质多晶中包围压痕的晶粒，地图分辨率会下降。为了量化P-h曲线随h增加的过渡变化，其中P为压痕载荷，h为压痕穿透深度，本研究提出了权函数。这些函数旨在量化嵌入在矩阵中的圆形柱上的尖锐纳米压痕，该矩阵具有平行于压痕方向的界面。通过将权重函数外推到压痕深度为零，可以准确地评估柱的力学性能，从而提高了地图分辨率。在比较平行界面和垂直界面的影响时，垂直界面对P-h曲线的影响比平行界面的影响更显著。这表明，局部力学性能的异质性与表面观察到的微观结构之间的关系是有争议的。

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引用次数: 0

Phase-field fracture modeling of polycrystalline quasicrystals subjected to hydrogen embrittlement 氢脆作用下多晶准晶相场断裂模拟

IF 3.8 3区工程技术 Q1 MECHANICS

International Journal of Solids and Structures

Pub Date : 2026-01-18 DOI: 10.1016/j.ijsolstr.2026.113853

Hongzhao Li , Raj Kiran , Yu Tan , Haidong Fan , Qingyuan Wang , Peidong Li

Quasicrystals (QCs) have demonstrated excellent hydrogen storage capabilities, making them promising candidates for hydrogen-related applications. However, exposure to hydrogen can significantly affect their mechanical integrity. In this study, a phase-field fracture model (PFM) is proposed to predict fracture behaviors of polycrystalline QCs in a hydrogen-containing environment. The damage-field and interface-field variables are introduced to regularize the cracks and interfaces (or grain boundaries) in QCs, respectively. An equivalent critical energy release rate is introduced to characterize the influence of the interfaces (or grain boundaries) on hydrogen-assisted crack propagation. The present model is numerically implemented in Comsol Multiphysics based on the Weak Form PDE module. Several numerical examples are conducted to validate the ability of the proposed model to predict the fracture behaviors of QCs and to analyze the influence of hydrogen concentration on the fracture mechanism of QCs. Numerical results indicate that the hydrogen concentration and interfaces (or grain boundaries) significantly influence the crack propagation paths and the peak force and failure displacement in the fracture test of QCs. The existence of hydrogen decreases the critical fracture load and promotes the crack propagation in QCs significantly. The developed phase-field framework and numerical implementation approach provide a convenient tool for predicting fracture failure and assessing the service reliability of polycrystalline QCs in complex hydrogen-containing environments.

准晶体（qc）已经证明了出色的储氢能力，使其成为氢相关应用的有希望的候选者。然而，暴露在氢气中会显著影响其机械完整性。本研究提出了一种相场断裂模型（PFM）来预测多晶qc在含氢环境中的断裂行为。引入损伤场和界面场变量分别对裂纹和界面（或晶界）进行正则化。引入等效临界能量释放率来表征界面（或晶界）对氢辅助裂纹扩展的影响。基于弱形式PDE模块，在Comsol Multiphysics中对该模型进行了数值实现。通过数值算例验证了该模型对碳纤维断裂行为的预测能力，并分析了氢浓度对碳纤维断裂机理的影响。数值结果表明，氢浓度和界面（或晶界）对裂纹扩展路径、峰值力和破坏位移有显著影响。氢的存在显著降低了临界断裂载荷，促进了裂纹扩展。所开发的相场框架和数值实现方法为复杂含氢环境下多晶qc的断裂失效预测和使用可靠性评估提供了方便的工具。

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引用次数: 0

Data-Driven-Design (D3) of multi-material systems: A novel framework and its application to viscoelastic metamaterials 多材料系统的数据驱动设计（D3）：一个新的框架及其在粘弹性超材料中的应用

IF 3.8 3区工程技术 Q1 MECHANICS

International Journal of Solids and Structures

Pub Date : 2026-01-17 DOI: 10.1016/j.ijsolstr.2026.113844

Rayehe Karimi Mahabadi, Hossein Salahshoor

While multimaterial additive manufacturing enables finely programmed heterogeneity, there remains no robust and objective-driven framework to assign materials across complex architectures under practical constraints. We introduce Data-Driven-Design (

D^{3}

) as a robust computational framework for multi-material lattice design, optimized with respect to a prescribed performance objective. The framework relies on representing physical constraints, material data, and design objectives as sets in a phase space and formulating the material selection problem as a distance minimization problem among the encoded sets. We showcase the approach in multi-material design of viscoelastic lattices provided with measurements of complex moduli as a function of frequency with the design objective of maximizing dissipation. For our numerical experiments, we import dynamic viscoelasticity measurement for twenty five different materials from literature, and show that multi-material designs can match or outperform the dissipation obtained from homogeneous designs made of the most dissipative material among the data registry. In a finite lattice example, we show that

D^{3}

design yields a mechanical dissipation with 300% increase compared to best homogeneous design from a limited collection of materials. Beyond viscoelastic lattices, the

D^{3}

framework generalizes naturally to multi-physics and multi-objective metastructure design, offering a unified, data-driven approach to optimal material selection under complex constraints.

虽然多材料增材制造能够实现精细编程的异质性，但在实际限制下，仍然没有强大的、客观驱动的框架来跨复杂架构分配材料。我们介绍了数据驱动设计（D3）作为多材料晶格设计的鲁棒计算框架，针对规定的性能目标进行了优化。该框架依赖于将物理约束、材料数据和设计目标表示为相空间中的集合，并将材料选择问题表述为编码集合之间的距离最小化问题。我们展示了粘弹性晶格的多材料设计方法，提供了复模量作为频率函数的测量，设计目标是最大化耗散。在我们的数值实验中，我们从文献中引入了25种不同材料的动态粘弹性测量，并表明多材料设计可以匹配或优于由数据注册表中耗散最大的材料制成的均匀设计所获得的耗散。在一个有限晶格的例子中，我们表明D3设计产生的机械耗散与有限材料的最佳均匀设计相比增加了300%。除了粘弹性网格之外，D3框架还可以自然地推广到多物理场和多目标元结构设计，为复杂约束下的最佳材料选择提供统一的数据驱动方法。

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引用次数: 0

Mechanical dissipation in random fiberglass networks under small strains 随机玻璃纤维网在小应变下的力学耗散

IF 3.8 3区工程技术 Q1 MECHANICS

International Journal of Solids and Structures

Pub Date : 2026-01-16 DOI: 10.1016/j.ijsolstr.2026.113854

Gregoire Markey , Etienne Barthel , Catalin R. Picu

Glass wool is used extensively for thermal and acoustic insulation in buildings, appliances, automotive and aerospace applications. In this work we study the response of fiberglass networks crosslinked using polymeric binders to small amplitude oscillatory excitation in the low acoustic frequency range. Two types of binders are used, with low and high damping, and a range of fiber densities and binder volume fractions are considered. We report the scaling of storage and loss moduli of the network with these parameters. Further, we evaluate the loss factor and find that it is independent of the fiber mass density but is power law related to the binder volume fraction. We develop a model that provides a mechanistic basis for the experimental results reported and may guide material design.

玻璃棉广泛用于建筑、电器、汽车和航空航天领域的隔热和隔音。在本工作中，我们研究了用聚合物粘合剂交联的玻璃纤维网络在低声波范围内对小振幅振荡激励的响应。使用两种类型的粘合剂，具有低和高阻尼，并考虑了纤维密度和粘合剂体积分数的范围。我们用这些参数报告了网络的存储和损失模的缩放。进一步，我们评估了损耗因子，发现它与纤维质量密度无关，但与粘合剂体积分数有关。我们开发了一个模型，为实验结果提供了一个机制基础，并可能指导材料设计。

引用次数: 0

Reformulating a Gurson-based dynamic damage model and demonstrating improved predictive power and numerical robustness 重新制定了基于gurson的动态损伤模型，并证明了改进的预测能力和数值鲁棒性

IF 3.8 3区工程技术 Q1 MECHANICS

International Journal of Solids and Structures

Pub Date : 2026-01-16 DOI: 10.1016/j.ijsolstr.2026.113833

Thao Nguyen, Darby J. Luscher, Miles Buechler, Michael B. Prime, Saryu J. Fensin

The original Tepla (TEnsile PLAsticity) ductile damage model, based on the Gurson yield surface, has long been used to model damage evolution and material failure under dynamic loading. Unfortunately, Tepla suffered from mesh sensitivity, numerical instability, and limited predictive capability. Here, we theoretically reformulate Tepla to address these issues. We especially focus on the prediction of porosity, which is the key state variable used for modeling ductile damage, as compared to more easily measured surface velocities, which are at best an indirect measure of damage. Key model changes include separating the viscosity during volumetric void growth from underlying shear strength behavior and switching to an iterative bisection solver. The new Tepla is then calibrated on incipient spall experiments on half-hard copper and tantalum, which demonstrate its ability to simultaneously fit the model to recovered porosity distributions and measured surface velocities, a stringent test. Improved numerical behavior, such as greatly reduced mesh sensitivity, is also shown in those simulations. Finally, the new Tepla model is applied to several high-explosive loaded, sweeping wave experiments, showing the ability of the model to predict behavior on tests with significantly different loading conditions and histories than the calibration data.

基于Gurson屈服面的Tepla（拉伸塑性）韧性损伤模型早已被用来模拟材料在动载荷作用下的损伤演化和破坏。不幸的是，Tepla存在网格敏感性、数值不稳定性和有限的预测能力。在这里，我们从理论上重新制定Tepla来解决这些问题。与更容易测量的表面速度相比，我们特别关注孔隙度的预测，这是用于模拟韧性损伤的关键状态变量，而表面速度充其量只是间接测量损伤。关键的模型变化包括将体积空隙生长过程中的粘度从潜在的剪切强度行为中分离出来，并切换到迭代的等分求解器。新的Tepla随后在半硬铜和钽的初期碎屑实验中进行了校准，证明了它能够同时拟合模型，以恢复孔隙度分布和测量表面速度，这是一项严格的测试。在这些模拟中也显示了改进的数值行为，例如大大降低了网格灵敏度。最后，将新Tepla模型应用于若干高爆载扫波实验，结果表明，该模型能够预测与校准数据有显著不同加载条件和历史的试验行为。

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引用次数: 0

Semi-analytical solution of VABS-based Timoshenko Beam Model for free vibration of composite structures 基于vabs的复合结构自由振动Timoshenko梁模型半解析解

IF 3.8 3区工程技术 Q1 MECHANICS

International Journal of Solids and Structures

Pub Date : 2026-01-14 DOI: 10.1016/j.ijsolstr.2026.113852

Sichen Liu, Wenbin Yu

In practical engineering applications, slender structures often have complex geometries, such as aircraft wings and rotor blades. Slender structures with low aspect ratios were often analyzed using three-dimensional (3D) Finite Element Analysis (FEA) as beam models were widely considered inadequate. However, analyzing slender structures with complex geometries was usually impractical due to the large number of degrees of freedom (DOFs) needed. This underscored the need to evaluate whether beam models can preserve accuracy while substantially reducing computational cost. To perform this assessment, a semi-analytical solution of the Timoshenko beam model, based on Variational Asymptotic Beam Sectional Analysis (VABS) considering all possible couplings, was used to predict the natural frequencies of general composite structures in free vibration analysis. The natural frequencies obtained from the semi-analytical solution were compared against the results from 3D FEA to assess the accuracy of the VABS-based Timoshenko model. The findings indicated that, for general composite beams, the Timoshenko model provided accurate predictions of the first several modes, excluding non-beam modes, mitigating the need for resource-intensive 3D FEA.

在实际工程应用中，细长结构通常具有复杂的几何形状，如飞机机翼和旋翼叶片。由于梁模型被普遍认为不充分，低纵横比细长结构通常采用三维有限元分析（FEA）进行分析。然而，由于需要大量的自由度（dof），分析具有复杂几何形状的细长结构通常是不切实际的。这强调了评估光束模型是否可以在大幅降低计算成本的同时保持精度的必要性。为了进行评估，基于变分渐近梁截面分析（VABS）的Timoshenko梁模型的半解析解考虑了所有可能的耦合，用于预测一般复合材料结构在自由振动分析中的固有频率。将半解析解得到的固有频率与三维有限元分析结果进行比较，以评估基于vabs的Timoshenko模型的准确性。研究结果表明，对于一般复合梁，Timoshenko模型提供了准确的前几个模态预测，不包括非梁模态，减少了对资源密集型三维有限元分析的需求。

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引用次数: 0

Revealing the strain and strain rate hardening of HDPE according to strain loading rate by measuring local stresses and strains 通过测量HDPE的局部应力和应变，揭示了HDPE随应变加载率的应变和应变率硬化

IF 3.8 3区工程技术 Q1 MECHANICS

International Journal of Solids and Structures

Pub Date : 2026-01-13 DOI: 10.1016/j.ijsolstr.2026.113850

Xinyue Wu , Shengwang Hao

Strain rate effects represent a critical aspect of the constitutive relationship describing the plastic flow behaviour of a ductile material owing to the heterogeneous distributions of strain and strain rate induced by necking localisation. This study quantitatively evaluated hardening by measuring the evolutions of the local true stresses and strains in high-density polyethylene (HDPE) specimens subjected to tension at different strain loading rates. The local true stress–strain curves exhibited three typical behaviours in the plastic flow stage owing to the different distances of each section from the centre of necking. The strain and strain rate hardening coefficients were estimated using statistics describing the relationships between the local true stresses, strains, and strain rates measured at different cross-sections, where local true strains were at the same values. The strain rate hardening coefficient was constant when the local true strains were moderate and decreased logarithmically according to the applied strain loading rate. Based on these results, the strain hardening coefficient was calculated, and the constitutive equation of plastic flow behaviours was discussed. The relationship of local true stress and true strain in the necked zone was divided into four stages: the early viscoelastic stage, constant strain rate hardening coefficient stage (necking forming stage), transition stage, and necking propagation stage. Different mechanical response mechanisms host the deformation behaviour in each stage. The strain rate hardening coefficient in the second stage decreases as a logarithmic dependence on the applied (average) strain rate. The dependence of local stress on local true strain in the first and second stages follows different expressions. In the third (transition) stage, the strain rate hardening coefficient increases with the increase in local strain. Finally, these results are verified against the measured true stress and strain gradients as well as the relationship between them. The results of this study provide insights into the constitutive relationship for the whole deformation process of polymers and suggest an approach for describing mechanical behaviours through measurements of local mechanical properties.

由于颈缩局部化引起的应变和应变率的不均匀分布，应变率效应是描述延性材料塑性流动行为的本构关系的一个关键方面。本研究通过测量高密度聚乙烯（HDPE）试样在不同应变加载率下拉伸时的局部真应力和应变演变来定量评价其硬化。由于各断面距离颈缩中心的距离不同，塑性流动阶段的局部真应力-应变曲线呈现出三种典型特征。应变和应变率硬化系数使用描述在不同截面上测量的局部真应力、应变和应变率之间关系的统计数据来估计，其中局部真应变处于相同的值。应变率硬化系数在局部真应变中等时保持不变，随外加应变加载率的增大呈对数递减。在此基础上，计算了应变硬化系数，讨论了塑性流变特性的本构方程。将颈区局部真应力与真应变的关系划分为早期粘弹性阶段、等应变率硬化系数阶段（颈化形成阶段）、过渡阶段和颈化扩展阶段4个阶段。不同的力学响应机制主导着每个阶段的变形行为。第二阶段的应变率硬化系数随施加的（平均）应变率呈对数依赖关系而减小。在第一阶段和第二阶段，局部应力对局部真应变的依赖表现为不同的表达式。在第三阶段（过渡阶段），应变速率硬化系数随局部应变的增大而增大。最后，根据实测的真实应力应变梯度以及它们之间的关系，验证了这些结果。本研究的结果为聚合物的整个变形过程提供了本构关系的见解，并提出了一种通过测量局部力学性能来描述力学行为的方法。

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引用次数: 0

Optimization, additive manufacturing, and testing of bird-bone-inspired materials for aircraft wing designs 优化，增材制造和测试鸟骨启发材料的飞机机翼设计

IF 3.8 3区工程技术 Q1 MECHANICS

International Journal of Solids and Structures

Pub Date : 2026-01-12 DOI: 10.1016/j.ijsolstr.2026.113846

Sepideh Ebad Sichani, Xin Ning

Inspired by the continuous shells and graded porous interiors of natural bird bones, this study presents a framework to design, optimize, and additively manufacture bird-bone-like materials for a new class of aircraft wing designs without traditional components such as ribs, spars, and stiffeners. Additive manufacturing, including fused deposition modeling (FDM), enables the rapid fabrication of these complex bio-inspired geometries with minimal material waste but introduces significant anisotropy due to its layer-by-layer deposition process. We implemented a transversely isotropic material model with Hill’s yield criterion to capture the directional dependence of FDM-printed polylactic acid (PLA). Using the Covariance Matrix Adaptation Evolution Strategy (CMA-ES), the bird-bone-inspired materials were optimized to minimize wing mass while maximizing load-carrying capacity. This framework achieved substantial improvements in structural efficiency, with 48–54 % for wings with lattice-based internal structures and 23–37 % for foam-based internal structures compared to reference designs. Experimental validation through structural testing of 3D-printed wings showed strong agreement with numerical predictions, with differences in effective stiffness and load-carrying capacity within 1.4–3.3 % and 1.2–13.5 %, respectively, of simulated values. The results confirm the effectiveness of this integrated framework for designing lightweight, high-performance bird-bone-inspired materials for aerospace applications.

受天然鸟骨的连续壳和渐变多孔内部的启发，本研究提出了一种框架，用于设计、优化和增材制造类鸟骨材料，用于新型飞机机翼设计，而不需要传统的部件，如肋、梁和加强筋。增材制造，包括熔融沉积建模（FDM），能够以最小的材料浪费快速制造这些复杂的生物启发几何形状，但由于其逐层沉积过程，引入了显着的各向异性。我们实现了一个横向各向同性材料模型与希尔的屈服准则，以捕捉定向依赖的fdm打印的聚乳酸（PLA）。采用协方差矩阵自适应进化策略（CMA-ES）对鸟骨启发材料进行优化，使机翼质量最小化，同时使承载能力最大化。与参考设计相比，该框架在结构效率方面取得了实质性的改进，网格内部结构的机翼效率提高了48 - 54%，泡沫内部结构的机翼效率提高了23 - 37%。通过对3d打印机翼的结构测试进行的实验验证与数值预测非常吻合，有效刚度和承载能力的差异分别在模拟值的1.4 - 3.3%和1.2 - 13.5%之间。结果证实了这种集成框架在设计用于航空航天应用的轻质、高性能鸟骨材料方面的有效性。

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引用次数: 0

Using topological defects to unfold thin structures: A graph-based approach with energy-driven distortion minimization 利用拓扑缺陷展开薄结构：基于图的能量驱动畸变最小化方法

IF 3.8 3区工程技术 Q1 MECHANICS

International Journal of Solids and Structures

Pub Date : 2026-01-12 DOI: 10.1016/j.ijsolstr.2026.113849

Shah Wasif Sazzad , Sanjay Dharmavaram , Luigi E. Perotti

The ability to unfold three-dimensional curved surfaces to flat templates has many applications ranging from space exploration, to communication, to mapping and image processing. In this context, we propose a new algorithm to unfold curved surfaces to a planar template that is selected under chosen design criteria. The given surface is first tessellated and potential cutlines are identified by joining the topological defects in the tessellation. These cutlines isolate regions (or unit patches) of lower Gauss curvature, which can be flattened with smaller distortion. Vice versa, regions of high Gauss curvature are driven toward the boundaries of the template, where area can be more easily added or subtracted with the same goal of minimizing distortion. Based on the determined cutlines, a graph is constructed where the nodes correspond to the unit patches and the edges to the patches’ connectivity. The edge weights are assigned based on chosen design criteria so that the graph’s minimum spanning tree determines the connections between the unit patches in the unfolded template. In this work, we consider criteria to avoid overlapping and based on the area or shape of the unfolded template, or leading to compact refolding. Each unit patch is mapped to the flat template and linked to its adjacent subunits following the minimum spanning tree. An elastic energy minimization scheme is applied to reduce distortion. The unfolding procedure can be reversed using a separate path to achieve compact refolding, which may be advantageous for transportation and storage. The proposed strategy is demonstrated in the unfolding of icosahedral shells, geodesic domes, and a paraboloid according to different design criteria. In all these examples, the limited distortion of the unfolded template with respect to the original surface is presented.

将三维曲面展开为平面模板的能力有许多应用，从空间探索、通信、映射和图像处理。在这种情况下，我们提出了一种新的算法展开曲面的平面模板，是选定的设计标准下选择。给定的表面首先被镶嵌，潜在的切线是通过加入镶嵌中的拓扑缺陷来识别的。这些切线隔离了较低高斯曲率的区域（或单位块），可以用较小的失真将其平坦化。反之亦然，高高斯曲率的区域被驱动到模板的边界，在那里可以更容易地增加或减少面积，以最小化扭曲的相同目标。基于确定的切线，构建图，其中节点对应于单元补丁，边缘对应于补丁的连通性。根据选择的设计准则分配边缘权重，以便图的最小生成树确定展开模板中单元补丁之间的连接。在这项工作中，我们考虑了避免重叠的标准，并基于展开模板的面积或形状，或导致紧凑的再折叠。每个单元补丁被映射到平面模板，并按照最小生成树链接到相邻的子单元。采用弹性能量最小化方案来减小变形。展开过程可以使用单独的路径进行反向，以实现紧凑的再折叠，这可能有利于运输和存储。根据不同的设计标准，在二十面体壳体、测地线圆顶和抛物面体的展开中证明了所提出的策略。在所有这些例子中，展示了展开模板相对于原始表面的有限畸变。

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引用次数: 0

Novel non-local frequency scale invariance for nanostructures 纳米结构的新型非局域频率尺度不变性

IF 3.8 3区工程技术 Q1 MECHANICS

International Journal of Solids and Structures

Pub Date : 2026-01-12 DOI: 10.1016/j.ijsolstr.2026.113848

Fernando Ramirez , Arturo Rodriguez-Herrera , Paul R. Heyliger

The natural vibration behavior of traction-free nanoparticles using an integral formulation of non-local elasticity theory is reported. Frequency spectra of sphere- and cubed-shaped particles composed of silicon, carbon, and germanium were computed using the finite element method. Various particle sizes, material internal lengths, and non-local weighting factors were considered. It was found that non-local frequencies are consistently lower than those obtained using local elasticity, indicating a material softening effect introduced by the non-local theory. Additionally, non-local frequencies approach those calculated using classical local elasticity as the local weighting factor increases and/or the material-internal-length to particle-size ratio decreases. Finally, the non-local frequency-radius product varies with particle size, indicating that the frequency scale invariance holding in classical elasticity is not valid. Instead, it was found that normalized non-local frequencies remain constant for a given material-internal-length to particle-size ratio, regardless of the particle size. This result introduces an alternative concept of scale invariance within the framework of non-local elasticity.

利用非局部弹性理论的积分公式研究了无牵引力纳米粒子的固有振动行为。用有限元法计算了硅、碳和锗组成的球形和立方体颗粒的频谱。考虑了不同粒度、材料内部长度和非局部加权因素。研究发现，非局部频率始终低于用局部弹性计算得到的频率，表明非局部理论引入的材料软化效应。此外，当局部权重因子增加和/或材料内部长度与粒径比减小时，非局部频率接近使用经典局部弹性计算的频率。最后，非局域频率-半径积随粒径的变化而变化，表明经典弹性力学中持有的频率尺度不变性不成立。相反，研究发现，对于给定的材料内部长度与粒径比，无论粒径大小如何，归一化的非局部频率保持不变。这一结果在非局部弹性的框架内引入了尺度不变性的另一种概念。

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引用次数: 0