European Journal of Mechanics A-Solids最新文献

英文中文

Rolling-induced flexural vibration 滚动引起的弯曲振动

IF 4.2 2区工程技术 Q1 MECHANICS

European Journal of Mechanics A-Solids

Pub Date : 2026-07-01 Epub Date: 2026-02-02 DOI: 10.1016/j.euromechsol.2026.106039

Chen Xuan

Flexural vibration can be triggered by rolling, an unusual way that has never been studied. Being a planar motion triggered from another motion, it integrates structural dynamics, flexural vibration and contact mechanics. I coin the term “rolling-induced flexural vibration” for this new problem, which is valuable not only to the fundamental theory of structural dynamics but also to various rolling mechanical systems like rolling soft robots. The developed wheels-on-a-string rolling beam that extends the Euler–Bernoulli theory overcomes numerical challenges in contact mechanics by removing inequalities in boundary conditions. An analysis of energy transport and dissipation highlights an unusual rolling beam system. Comparisons are conducted among 3D contact finite element analysis, the rolling beam model and a few analytic solutions. Despite ignoring certain aspect ratio dependence in 3D contact finite elements, the rolling beam model serves as a tool for assessing the efficiency and quality of 3D contact solutions to determine whether switching to alternative contact solvers is necessary. The computational efficiency of the rolling beam model can be up to hundreds of thousands of times higher than that of 3D contact finite elements.

弯曲振动可以通过滚动触发，这是一种从未被研究过的不寻常的方式。它是由另一个运动触发的平面运动，集结构动力学、弯曲振动和接触力学于一体。对于这一新问题，笔者提出了“滚动诱导弯曲振动”一词，这不仅对结构动力学的基础理论有价值，而且对滚动软机器人等各种滚动机械系统也有价值。所开发的弦上轮滚动梁扩展了欧拉-伯努利理论，消除了边界条件中的不等式，克服了接触力学中的数值难题。对一种不寻常的滚动梁系统进行了能量输运和耗散分析。对三维接触有限元分析、滚动梁模型和几种解析解进行了比较。尽管在三维接触有限元中忽略了某些纵横比依赖，但滚动梁模型可以作为评估三维接触解的效率和质量的工具，以确定是否需要切换到其他接触解。滚动梁模型的计算效率比三维接触有限元计算效率高数十万倍。

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引用次数: 0

Accuracy-enhanced modeling of 2024 aluminium alloy in electrically-assisted forming based on direct through-thickness strain measurement 基于直接贯穿厚度应变测量的2024铝合金电辅助成形精度增强建模

IF 4.2 2区工程技术 Q1 MECHANICS

European Journal of Mechanics A-Solids

Pub Date : 2026-05-01 Epub Date: 2025-12-11 DOI: 10.1016/j.euromechsol.2025.105996

Kaiyu Zhu , Hang Yuan , Xiaojie Hu , Hongrui Dong , Xiaoqiang Li , Bing Pan

Electrically-assisted forming enhances the formability of aluminum alloys while maintaining post-forming mechanical properties. However, the multi-physics coupling loading in electrically-assisted forming poses challenges for modeling the anisotropic behavior of aluminum alloy. Conventional associated flow rules typically require complex high-order yield models and customized tooling. In contrast, the non-associated flow rule combined with low-order models has demonstrated comparable accuracy with significantly reduced complexity. Encouraged by its successful application in room-temperature forming, this work proposes integrating the non-associated flow rule with the classical low-order Hill48 yield model and adopting the Grosman hardening model to characterize the anisotropic behavior of 2024-O aluminum alloy under electrically-assisted forming. Both models require accurate determinations of through-thickness strain (ε_zz). Conventional indirect methods often rely on experimentally measured in-plane strain and constant volume assumptions that are unreliable in electrically-assisted forming. To address this challenge, the recently developed mirror-assisted MV-DIC (multi-view digital image correlation) technique was adopted for direct measurement of ε_zz and parameter calibration, which eliminates the reliance on volume conservation assumptions. Electrically-assisted uniaxial tensile tests and anisotropic constitutive modeling were conducted on specimens under five distinct loading conditions. Directly and indirectly measured plastic potential function parameters revealed average relative differences of 6.2 %–14.6 %. The effects of temperature and strain rate on both model parameters and deformation behaviors were also investigated. The experimental and simulation results of shear specimens validated the accuracy of the established model and the superiority of the proposed parameter calibration method.

在保持成形后力学性能的同时，电辅助成形提高了铝合金的成形性。然而，电辅助成形过程中的多物理场耦合载荷对铝合金各向异性行为的建模提出了挑战。传统的关联流规则通常需要复杂的高阶屈服模型和定制的工具。相比之下，与低阶模型相结合的非关联流规则显示出相当的准确性，同时显着降低了复杂性。受其在室温成形中的成功应用鼓舞，本文提出将非关联流动规律与经典的低阶Hill48屈服模型相结合，采用Grosman硬化模型来表征2024-O铝合金在电辅助成形中的各向异性行为。这两种模型都需要精确地确定贯穿厚度应变（εzz）。传统的间接方法通常依赖于实验测量的平面内应变和恒体积假设，这些假设在电辅助成形中是不可靠的。为了解决这一挑战，采用了最近发展的镜面辅助MV-DIC（多视图数字图像相关）技术来直接测量εzz和参数校准，从而消除了对体积守恒假设的依赖。在五种不同加载条件下，对试件进行了电辅助单轴拉伸试验和各向异性本构建模。直接和间接测量的塑性势函数参数平均相对差异为6.2% ~ 14.6%。研究了温度和应变速率对模型参数和变形行为的影响。剪切试件的试验和仿真结果验证了所建立模型的准确性和所提出参数校正方法的优越性。

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引用次数: 0

A thermodynamically consistent and strongly coupled chemo–thermo–hygral framework for early-age concrete 早期混凝土的热动力学一致性和强耦合的化学-热-水框架

IF 4.2 2区工程技术 Q1 MECHANICS

European Journal of Mechanics A-Solids

Pub Date : 2026-05-01 Epub Date: 2025-12-10 DOI: 10.1016/j.euromechsol.2025.105992

Jian Ding , Xin Wang , Xinyi Ren , Haim Waisman , Zhishen Wu

Concrete is widely used for its high strength and durability. At early ages, hydration, heat transfer, and moisture transport are the main causes of stress. These processes may lead to nonuniform deformation, high tensile stress, and early cracking, which make prediction difficult. Previous chemo–thermo–hygral models often depended on empirical relations or weak coupling assumptions and lacked thermodynamic consistency. As a result, their theoretical rigor and applicability were limited. In this study, a new free energy density function is proposed within a thermodynamically consistent framework, and a strongly coupled chemo–thermo–hygral model for early-age concrete is developed. The governing equations are derived strictly from thermodynamic principles to describe the spatial and temporal evolution of hydration, temperature, and relative humidity. The model is validated by two-dimensional and three-dimensional simulations and experimental data. The results show that the model can accurately capture the evolution of temperature–humidity coupling and has good predictive ability. A parameter analysis shows that the evolution of relative humidity is sensitive and complex, which highlights its key role in the early-age behavior of concrete. The proposed framework overcomes the limitations of existing models and provides a unified and solid basis for the study of coupled reactive porous media, with important implications for structural design, crack control, and durability improvement.

混凝土因其高强度和耐久性而被广泛使用。在早期，水合作用、热量传递和水分输送是应力的主要原因。这些过程可能导致不均匀变形、高拉伸应力和早期开裂，使预测变得困难。以往的化学-热-水模型往往依赖于经验关系或弱耦合假设，缺乏热力学一致性。因此，它们的理论严谨性和适用性受到限制。在本研究中，在热力学一致的框架内提出了一个新的自由能密度函数，并建立了早期混凝土的强耦合化学-热-水模型。控制方程严格根据热力学原理推导，用以描述水化、温度和相对湿度的时空演化。通过二维和三维仿真及实验数据对模型进行了验证。结果表明，该模型能较准确地捕捉温湿度耦合的演变过程，具有较好的预测能力。参数分析表明，相对湿度的演变是敏感和复杂的，突出了其在混凝土早期行为中的关键作用。提出的框架克服了现有模型的局限性，为耦合反应性多孔介质的研究提供了统一、坚实的基础，对结构设计、裂缝控制和耐久性提高具有重要意义。

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引用次数: 0

Quasi-static shape control of structures with topologically complex equilibrium sets 拓扑复杂平衡集结构的准静态形状控制

IF 4.2 2区工程技术 Q1 MECHANICS

European Journal of Mechanics A-Solids

Pub Date : 2026-05-01 Epub Date: 2025-12-22 DOI: 10.1016/j.euromechsol.2025.106000

Eszter Fehér , András Á. Sipos , Péter L. Várkonyi

Inspired by biological systems, we introduce a general framework for quasi-static shape control of load-bearing structures under slowly varying external actions or requirements. In this setting, shape control aims to traverse the stable sub-manifolds of the equilibrium set to meet some predefined requirements or optimization criteria. This paper explores the implications of large shape changes and high compliance, such as the emergence of unstable equilibria and equilibrium sets with non-trivial topology. We identify various adaptivity scenarios, ranging from inverse kinematics to optimization and path planning problems, and discuss the role of time-dependent loads and requirements. The applicability of the proposed concepts is demonstrated through the example of a soft arch that is susceptible to snap-through behavior.

受生物系统的启发，我们介绍了在缓慢变化的外部作用或要求下，承载结构的准静态形状控制的一般框架。在这种情况下，形状控制的目标是遍历平衡集的稳定子流形，以满足一些预定义的要求或优化准则。本文探讨了大形状变化和高顺应性的含义，如非平凡拓扑的不稳定平衡和平衡集的出现。我们确定了各种适应性场景，从逆运动学到优化和路径规划问题，并讨论了时间相关负载和要求的作用。所提出的概念的适用性是通过一个软拱的例子来证明的，该软拱易受弹跳行为的影响。

引用次数: 0

Topology optimization of soft robots with local enhancement 基于局部增强的软体机器人拓扑优化

IF 4.2 2区工程技术 Q1 MECHANICS

European Journal of Mechanics A-Solids

Pub Date : 2026-05-01 Epub Date: 2025-12-24 DOI: 10.1016/j.euromechsol.2025.106006

Shikeng Zhao, Jianhua Xiang, Yongfeng Zheng, Jiale Huang

The demand for pneumatic soft robots is rapidly increasing for addressing repetitive and complex tasks in engineering. However, traditional soft robots mainly use solid materials without fully considering the material Poisson's ratio's impact on the structural performance. Additionally, local enhanced composites can enhance structural mechanical strength. This study firstly presents the comprehensive study on topology optimization of layered soft robots, using local enhanced composites. An ideal natural pressure field was constructed via fluid loads mathematical model, the density optimization method with penalization and Heaviside filter, were combined to make the structural boundaries clear, and the homogenization method is used to obtain the macroscopic elastic matrices of local enhanced composites. Moreover, design variables of soft robots were updated by using the moving asymptote method, to solve the optimization problem with non-monotonic sensitivity. This paper investigates topology optimization of layered soft robots under various Poisson's ratios, enhanced matrix ratios and enhancement Young's moduli. Computational results indicate that for local enhanced composites, a suitable enhanced matrix ratio can increase soft robots' output displacement, composites with 100 % enhanced matrix ratio have approximately 20 % lower strain energy than those with 0 % ratio.

为了解决工程中重复和复杂的任务，对气动软机器人的需求正在迅速增加。然而，传统的软体机器人主要采用固体材料，没有充分考虑材料泊松比对结构性能的影响。此外，局部增强复合材料可以提高结构的机械强度。本研究首次对基于局部增强复合材料的层状柔性机器人拓扑优化进行了全面研究。通过流体载荷数学模型构建理想的自然压力场，结合惩罚密度优化方法和Heaviside滤波方法清晰结构边界，采用均质化方法得到局部增强复合材料的宏观弹性矩阵。利用运动渐近线法更新软机器人的设计变量，解决了具有非单调灵敏度的优化问题。研究了不同泊松比、增强矩阵比和增强杨氏模下层状软机器人的拓扑优化问题。计算结果表明，对于局部增强复合材料，适当的增强矩阵比例可以增加软机器人的输出位移，100%增强矩阵比例的复合材料应变能比0%增强矩阵比例的复合材料应变能降低约20%。

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引用次数: 0

Simple shear in incompressible transversely isotropic hyperelasticity: experiments versus generalized linear models 不可压缩横向各向同性超弹性中的简单剪切：实验与广义线性模型

IF 4.2 2区工程技术 Q1 MECHANICS

European Journal of Mechanics A-Solids

Pub Date : 2026-05-01 Epub Date: 2025-12-29 DOI: 10.1016/j.euromechsol.2025.106010

Safa Mathlouthi , Mohamed Trifa , Makrem Arfaoui , Luiz Carlos Da Silva Nunes

This study presents a comprehensive assessment of five transversely isotropic incompressible hyperelastic models under simple shear loading. The analysis is grounded on two independent experimental datasets involving fiber-reinforced soft materials, enabling a robust evaluation of model performance across a broad range of fiber orientations. Each model was calibrated using shear stress data and subsequently tested on its ability to predict the normal stress response, an indicator of the Poynting effect, without further parameter adjustment. Results show that while all models capture the shear behavior reasonably well, their predictive accuracy for normal stresses varies significantly with fiber alignment and dataset. Notably, one model consistently demonstrated superior performance, maintaining predictive robustness across both datasets. In contrast, other models exhibited greater sensitivity to anisotropy and deformation conditions. The findings underscore the importance of validating constitutive models against multiple stress components to reveal limitations that may be masked in standard calibration procedures. The results of this work gives practical insights into the modeling of fiber-reinforced soft materials and highlights the critical role of multi-criteria validation for selecting reliable hyperelastic formulations in anisotropic elasticity.

本研究提出了在简单剪切载荷下的五种横向各向同性不可压缩超弹性模型的综合评估。该分析基于两个独立的实验数据集，涉及纤维增强软材料，能够在广泛的纤维取向范围内对模型性能进行可靠的评估。每个模型都使用剪切应力数据进行校准，随后测试其预测正常应力响应的能力，这是Poynting效应的一个指标，无需进一步调整参数。结果表明，虽然所有模型都能很好地捕捉剪切行为，但它们对正应力的预测精度因纤维取向和数据集的不同而有很大差异。值得注意的是，一个模型始终表现出卓越的性能，在两个数据集上保持预测稳健性。相反，其他模型对各向异性和变形条件表现出更大的敏感性。研究结果强调了针对多个应力分量验证本构模型的重要性，以揭示可能在标准校准程序中被掩盖的局限性。这项工作的结果为纤维增强软材料的建模提供了实际的见解，并强调了在各向异性弹性中选择可靠的超弹性配方的多准则验证的关键作用。

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引用次数: 0

Mechanical behaviour of an optimised novel Ti6Al4V lattice structure fabricated via LPBF: “An experimental and FEA investigation” LPBF制备的新型优化Ti6Al4V晶格结构的力学行为：“实验和有限元研究”

IF 4.2 2区工程技术 Q1 MECHANICS

European Journal of Mechanics A-Solids

Pub Date : 2026-05-01 Epub Date: 2026-01-14 DOI: 10.1016/j.euromechsol.2026.106017

Kirandeep Singh , Guofang Liang , Kevin Tetsworth , Justin Cooper-White , Deniz U. Erbulut , Mingxing Zhang

Critical-sized bone defects lack the inherent capacity for self-repair and require engineered bone scaffold structures to provide mechanical stability while facilitating osteointegration. Conventional lattice architectures often fail to reconcile high strength with elevated porosity due to discontinuous geometries and stress concentrations at nodal junctions. The Hexanoid (HH) scaffold, inspired by curved surfaces and exhibiting three-dimensional periodicity, has demonstrated superior in-silico cell proliferation but remains mechanically suboptimal. This study introduces a titanium-based Modified Hexanoid (MH) scaffold, engineered to enhance structural performance while retaining porosity conductive to bone ingrowth. Scaffolds were fabricated using Ti6Al4V alloy via Laser Powder Bed Fusion (L-PBF) and mechanically benchmarked against HH, Cubic (CU) and Circular (CR) scaffold structures. Quasi-static compression testing reveals that the MH scaffold achieved an elastic modulus of ∼9 GPa, a yield strength of ∼104 MPa, and a compressive strength of ∼154 MPa, representing improvements of 24 %, 58 %, and 37 %, respectively, over the HH design. The MH design maintained a porosity of approximately 73 %, exceeding HH (∼61 %) and being comparable to CU (∼77 %) and CR (∼76 %). By combining porosity similar to that of trabecular bone with mechanical properties approaching those of cortical bone, the MH scaffold overcomes the strength-porosity trade-off, demonstrating strong potential for load-bearing orthopaedic implants.

临界尺寸的骨缺损缺乏固有的自我修复能力，需要工程骨支架结构在促进骨整合的同时提供机械稳定性。由于几何形状不连续和节点处的应力集中，传统的晶格结构往往无法调和高强度和高孔隙率。受曲面启发，具有三维周期性的Hexanoid （HH）支架显示出优越的硅细胞增殖能力，但机械性能仍不理想。本研究介绍了一种基于钛的改性类己烷支架（MH），旨在提高结构性能，同时保持有利于骨长入的孔隙度。支架采用Ti6Al4V合金通过激光粉末床熔合（L-PBF）制造，并以HH、Cubic （CU）和Circular （CR）支架结构为机械基准。准静态压缩测试表明，MH支架的弹性模量为~ 9 GPa，屈服强度为~ 104 MPa，抗压强度为~ 154 MPa，分别比HH设计提高了24%，58%和37%。MH设计保持了约73%的孔隙度，超过HH(~ 61%)，与CU（~ 77%）和CR（~ 76%）相当。通过结合类似小梁骨的孔隙度和接近皮质骨的力学性能，MH支架克服了强度-孔隙度的权衡，显示出承重骨科植入物的强大潜力。

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引用次数: 0

Oblique impact behavior of bio-inspired turtle shell suture interfaces 仿生龟壳缝合界面的斜冲击行为

IF 4.2 2区工程技术 Q1 MECHANICS

European Journal of Mechanics A-Solids

Pub Date : 2026-05-01 Epub Date: 2026-01-13 DOI: 10.1016/j.euromechsol.2026.106028

Xu Zhang , Tom Allen , Shouji Zhao , Wu Yan , Qiang Fu , Zhenqing Wang

The study investigates the dynamic response of bionic suture interfaces inspired by turtle shell under oblique impact at angles of 0°, 15°, 30°, 45°, and 60°. Four geometric configurations (triangular, trapezoidal, anti-trapezoidal, and rectangular) are examined numerically. Results demonstrate that increasing the impact angle enhances frictional energy dissipation through impactor sliding, accompanied by greater energy absorption, structural deformation, and damage area. The interdigitated suture design effectively suppresses interlaminar slippage under oblique loading. When impact occurs perpendicular to the tip angle, the structural response resembles that of a conventional flat plate under normal impact, exhibiting comparatively inferior mechanical performance. Among the tested geometries, the triangular suture interface outperforms others in impact resistance. At identical impact angles, the load-bearing capacities of anti-trapezoidal, trapezoidal, and triangular interfaces exceed that of the rectangular interfaces by 23 %–37 %, 19 %–40 %, and 81 %–89 %, respectively. Notably, the triangular suture with a tip angle of 4.6° demonstrates optimal performance. This systematic study provides valuable data on the oblique impact behavior of bioinspired suture interfaces, offering fundamental guidance for the design of advanced protective composites and bioinspired artificial armor systems.

研究了以龟壳为灵感的仿生缝合界面在0°、15°、30°、45°和60°角度的斜冲击下的动态响应。四种几何构型（三角形、梯形、反梯形和矩形）进行了数值检验。结果表明：增大冲击角能增强冲击块滑动的摩擦能耗散，同时能吸收增大，结构变形增大，损伤面积增大；指间缝线设计有效地抑制了斜载荷下的层间滑移。当冲击发生在垂直于尖端角时，结构响应与传统平板在正常冲击下的响应相似，力学性能相对较差。在测试的几何形状中，三角形缝合界面的抗冲击性优于其他几何形状。在相同的冲击角度下，反梯形界面、梯形界面和三角形界面的承载能力分别比矩形界面高出23% ~ 37%、19% ~ 40%和81% ~ 89%。值得注意的是，尖端角为4.6°的三角形缝线表现出最佳的性能。该系统研究为仿生缝合界面的斜冲击行为提供了有价值的数据，为先进防护复合材料和仿生人工装甲系统的设计提供了基础指导。

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引用次数: 0

Modelling lateral spread in wire flat rolling 线材轧制横向扩展的建模

IF 4.2 2区工程技术 Q1 MECHANICS

European Journal of Mechanics A-Solids

Pub Date : 2026-05-01 Epub Date: 2026-01-12 DOI: 10.1016/j.euromechsol.2026.106027

Mozhdeh Erfanian , Carl D. Slater , Edward J. Brambley

A mathematical model for wire rolling is developed, focusing on predicting the lateral spread. This provides, for the first time, an analytic model of lateral spread without any fitting parameters. The model is derived directly from the governing equations, assuming a rigid, perfectly plastic material and exploiting the thinness of the wire (in thickness and width) relative to the roller size. Results are compared against experiments performed on stainless steel wire using

100 mm

diameter rolls, demonstrating accurate predictions of lateral spread across a wide range of wire diameters (

2.96 mm

–

7.96 mm

) and reduction ratios (20%–60%), all without the need for fitting parameters. Since the model requires only seconds to compute, the model’s valid range is explored for varying roll diameter, wire diameter, and reduction ratio, and their effects on the resulting lateral spread characterised. The model can serve as a robust tool for validating FE results, guiding process design, and laying the foundation for future improved models. Matlab code to evaluate the model is provided in the supplementary material.

建立了线材轧制的数学模型，重点研究了线材横向扩展的预测。这首次提供了一个没有任何拟合参数的横向扩散的解析模型。该模型直接从控制方程中推导出来，假设材料是刚性的、完全塑性的，并利用线材的细度（厚度和宽度）相对于辊的尺寸。结果与使用直径为100mm的轧辊在不锈钢丝上进行的实验进行了比较，证明了在很宽的丝径范围内（2.96mm-7.96mm）的横向扩散和减速率（20%-60%）的准确预测，所有这些都不需要拟合参数。由于该模型只需要几秒钟的计算时间，因此该模型的有效范围是针对不同的轧辊直径、线材直径和减速比，以及它们对产生的横向扩散特性的影响进行探索的。该模型可以作为验证有限元结果、指导过程设计以及为未来改进模型奠定基础的强大工具。在补充资料中提供了对模型进行评估的Matlab代码。

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引用次数: 0

Nonlinear forced vibrations of FGM microplates: a numerical approach in the context of Mindlin's strain gradient elasticity FGM微板的非线性强迫振动：在Mindlin应变梯度弹性的背景下的数值方法

IF 4.2 2区工程技术 Q1 MECHANICS

European Journal of Mechanics A-Solids

Pub Date : 2026-05-01 Epub Date: 2026-01-08 DOI: 10.1016/j.euromechsol.2026.106023

Y. Gholami , R. Ansari , H. Rouhi

This study aims to investigate the geometrically nonlinear forced vibration behavior of micro-scale plates made of functionally graded materials (FGMs) by developing a variational numerical approach that accounts for strain gradient effects. The proposed approach, which is based on the variational differential quadrature technique, is capable of addressing the problem with arbitrary geometry (e.g. quadrilateral plate, annular sector plate, triangular plate, etc.). Besides, Mindlin's strain gradient theory is applied that leads to a formulation which encompasses the modified versions of strain gradient and couple stress theories (MSGT & MCST). An important novelty of present work is its vector-matrix presentation which can be beneficial for researchers working on numerical methods. Based on Hamilton's principle together with Mindlin's plate theory, the governing equations are derived. In the numerical results, the effects of thickness-to-material length-scale parameter on the frequency-response curves of FG plates with various shapes are analyzed. Also, comparisons are made between the predictions of MCST, MSGT as well as the classical theory. The results indicate that strain gradient terms have a pronounced influence on the nonlinear dynamic response of FGM microplates, giving rise to evident stiffening behavior and noticeable changes in the frequency–response curves. These observations highlight the importance of using higher-order continuum models to achieve reliable predictions of nonlinear forced vibration behavior at the micro-scale.

本研究旨在通过发展考虑应变梯度效应的变分数值方法来研究由功能梯度材料（fgm）制成的微尺度板的几何非线性强迫振动行为。该方法基于变分微分正交技术，能够解决任意几何形状（如四边形板、环形扇形板、三角形板等）的问题。此外，采用Mindlin的应变梯度理论，得到了一个包含应变梯度和耦合应力理论（MSGT &； MCST）修正版本的公式。当前工作的一个重要新颖之处在于它的向量矩阵表示，这对研究数值方法的研究人员是有益的。根据汉密尔顿原理和明德林的平板理论，推导了控制方程。数值结果分析了不同形状FG板的厚度-材料长度尺度参数对频率响应曲线的影响。并对MCST、MSGT和经典理论的预测结果进行了比较。结果表明，应变梯度项对FGM微孔板的非线性动态响应有显著影响，使其出现明显的加筋行为，频率响应曲线发生明显变化。这些观察结果强调了使用高阶连续体模型在微观尺度上实现非线性强迫振动行为可靠预测的重要性。

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