European Journal of Mechanics A-Solids最新文献_第4页

A DEM-based study of the mechanical behavior and energy dissipation of metal rubber with structural nonuniformity 结构不均匀金属橡胶力学性能及能量耗散的dem研究

IF 4.2 2区工程技术 Q1 MECHANICS

European Journal of Mechanics A-Solids

Pub Date : 2026-01-06 DOI: 10.1016/j.euromechsol.2026.106016

Yuedan Li , Chunhui Zhou , Zihao Huang , Xianjie Shi , Qiang Wan , Qinwei Wang , Peng Pi , Zhiying Ren

Metal rubber (MR) is widely used for vibration attenuation and noise reduction in harsh environments. However, the internal nonuniformity of MR leads to a more complex spatial distribution mechanism of energy dissipation, while the corresponding influence mechanism on the dynamic behavior remains unclear. Hence a Discrete Element Method (DEM) numerical model is developed together with a contact energy-conversion model for dissipation to investigate the nonuniformity of wire-to-wire contact turns in MR. An effective DEM-based nonuniformity nonlinear dynamics model (DEM-NND) is proposed to capture the dynamic hysteretic mechanical response across different specimen sizes. The results show pronounced Probability Distribution of Low Energy Dissipation (PDLED) and Probability Distribution of High Energy Dissipation (PDHED) within MR, evidencing strong nonuniformity in both the contact and dissipation fields. Specimen thickness markedly alters the volumetric fractions of PDLED and PDHED, thereby modulating stiffness and energy-dissipation performance. The proposed model agrees well with dynamic loading experiments in terms of hysteresis-loop morphology and dissipation capacity. Experimentally, reducing MR thickness from 15 mm to 5 mm decreased the PDHED volume fraction from 33.89 % to 19.61 %, accompanied by a 35.89 % reduction in the loss factor (η). This study elucidates how nonuniformity governs the macroscopic mechanical performance of MR and provides a theoretical basis and quantitative reference for structural design.

金属橡胶（MR）广泛应用于恶劣环境下的减振降噪。然而，MR内部的非均匀性导致能量耗散的空间分布机制更为复杂，而其对动力行为的影响机制尚不清楚。为此，建立了离散元法（DEM）数值模型和接触能量转换耗散模型来研究磁流变过程中导线与导线接触匝数的非均匀性，并提出了一种有效的基于DEM的非均匀性非线性动力学模型（DEM- nnd）来捕捉不同试件尺寸的动态滞后力学响应。结果表明：磁流变场内存在明显的低能量耗散概率分布（PDLED）和高能量耗散概率分布（PDHED），在接触场和耗散场均表现出较强的非均匀性。试件厚度显著改变了pded和PDHED的体积分数，从而调节了刚度和耗能性能。该模型在滞回环形态和耗散能力方面与动加载实验结果吻合较好。实验结果表明，将MR厚度从15 mm降低到5 mm， pdhd体积分数从33.89%降低到19.61%，损失系数（η）降低35.89%。本研究阐明了非均匀性对MR宏观力学性能的影响，为结构设计提供了理论依据和定量参考。

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引用次数: 0

Transient dynamic response of honeycomb non-pneumatic tires to obstacle impact under varying conditions 不同条件下蜂窝状非充气轮胎对障碍物冲击的瞬态动态响应

IF 4.2 2区工程技术 Q1 MECHANICS

European Journal of Mechanics A-Solids

Pub Date : 2026-01-02 DOI: 10.1016/j.euromechsol.2025.106012

Zhou Zheng , Fei Wang , Haichao Zhou , Darshan Dorugade , Jian Qu , Ran Guo , Chengdong Wang , Mingdi Wang , Shengbin Zhao , Chen Liang

Most existing studies typically identify only one vibration mode from resonance peaks in response spectra, which limits the interpretation of the dynamic behaviour of rolling tires. This study aims to associate dominant resonance peaks with specific vibration modes by comparing their natural frequencies with those of loaded stationary tires or pneumatic counterparts, and by analysing their dependence on selected operating and design parameters. A 3D finite element (FE) model was built for a non-pneumatic tire (NPT) with honeycomb-structured spokes to investigate its spindle and contact response characteristics. Model validity was confirmed by comparing predicted load-deflection response with experimental results from a prototype tire. Transient dynamic simulations were conducted for the loaded rotating NPT impacting obstacles of varying radii under different operating and design parameters. The results show that the dominant response peaks correspond to the hop, torsion, oval-diagonal, diametric, and oval vibration modes. Amplitudes of these peaks are significantly affected by operating parameters, whereas their frequencies remain largely unchanged. Hop and torsion frequencies are particularly sensitive to design parameters related to the cell and shear band, while the diametric mode is only slightly affected. These findings improve understanding of NPT modal behaviour and provide a basis for structural optimization and performance enhancement.

现有的研究大多只从响应谱的共振峰中识别出一种振动模式，这限制了对滚动轮胎动力特性的解释。本研究旨在通过将其固有频率与加载固定轮胎或气动轮胎的固有频率进行比较，并通过分析其对选定的操作和设计参数的依赖，将主要共振峰与特定振动模式联系起来。建立了蜂窝状辐条非充气轮胎（NPT）的三维有限元模型，研究了其主轴和接触响应特性。通过将预测的载荷-挠度响应与原型轮胎的试验结果进行比较，验证了模型的有效性。对不同工作参数和设计参数下加载的不同半径旋转NPT碰撞障碍物进行了瞬态动力学仿真。结果表明，结构的主导响应峰分别对应于跳跃、扭转、椭圆-对角线、直径和椭圆振动模式。这些峰值的振幅受到工作参数的显著影响，而它们的频率基本保持不变。跳频和扭转频率对与单元和剪切带相关的设计参数特别敏感，而直径模态仅受轻微影响。这些发现提高了对NPT模态行为的理解，并为结构优化和性能提高提供了基础。

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引用次数: 0

Impact of hygro-mechanical loading on crack initiation and propagation in orthotropic media: A numerical modeling approach 正交各向异性介质中水-力载荷对裂纹萌生和扩展的影响：数值模拟方法

IF 4.2 2区工程技术 Q1 MECHANICS

European Journal of Mechanics A-Solids

Pub Date : 2026-01-02 DOI: 10.1016/j.euromechsol.2025.106011

Loic Chrislin Nguedjio , Rostand Moutou Pitti , Jérôme Afoutou , Frédéric Dubois , Benoit Blaysat , Pierre Kisito Talla , Naman Recho

Moisture content gradients play a crucial role in the mechanics of wood, significantly affecting its properties and overall behavior. Among the adverse effects induced by moisture content variations, crack initiation and propagation remain one of the major challenges in wooden structures. This paper presents an energy-based fracture mechanics approach that relies on the computation of invariant integrals to evaluate fracture parameters in orthotropic materials subjected to moisture content gradients. A new concept of crack initiation is introduced by an algorithm based on the bisection method, which leads to automatic crack propagation under hygro-mechanical loading. The fracture criteria are derived from the energy release rate through a novel formulation of the invariant integral A

θ

in a moisture-coupled environment. The path-independence of this newly proposed integral is verified via finite element simulations under mixed-mode loading using the Mixed-Mode Crack Growth (MMCG) specimen. Both analytical and numerical formulations account for the combined effects of mechanical and moisture-induced pressure on the crack lips. The influence of these coupled pressures on crack initiation and propagation in wood under variable moisture content conditions is introduced and discussed. For all fracture modes, the initiation and propagation criteria are obtained through a moisture-driven propagation algorithm, which triggers automatic crack growth when a critical moisture content threshold is reached. The results demonstrate that the developed analytical tool, implemented within a finite element framework provides a reliable determination of the fracture parameters of wood under moisture content variations.

含水率梯度在木材力学中起着至关重要的作用，显著影响其性能和整体行为。在含水率变化引起的不利影响中，裂纹的萌生和扩展仍然是木结构的主要挑战之一。本文提出了一种基于能量的断裂力学方法，该方法依赖于不变积分的计算来评估受含水量梯度影响的正交各向异性材料的断裂参数。提出了一种基于等分法的裂纹萌生新概念，从而实现了水力学载荷下裂纹的自动扩展。在湿耦合环境下，通过一种新的不变积分a θ公式，从能量释放率推导出断裂判据。通过混合模式裂纹扩展（MMCG）试件的混合加载有限元模拟验证了该积分的路径无关性。解析公式和数值公式都考虑了机械压力和水致压力对裂纹唇的综合影响。介绍并讨论了变含水率条件下这些耦合压力对木材裂纹萌生和扩展的影响。对于所有断裂模式，通过水分驱动扩展算法获得起裂和扩展准则，当达到临界含水率阈值时触发裂纹自动扩展。结果表明，所开发的分析工具在有限元框架内实现，可以可靠地确定木材在含水率变化下的断裂参数。

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引用次数: 0

Temperature-dependent identification and active vibration control under time-varying loads for composite laminates 时变载荷下复合材料层合板的温度相关辨识与主动振动控制

IF 4.2 2区工程技术 Q1 MECHANICS

European Journal of Mechanics A-Solids

Pub Date : 2026-01-02 DOI: 10.1016/j.euromechsol.2025.106014

Hui Zhang , Yu Zhang , Wei Sun , Haitao Luo , Hongwei Ma , Kunpeng Xu

To address the difficulties regarding mechanical property fluctuations and vibration control of carbon fiber reinforced composite (CFRC) structures in environments with coupled high temperatures and time-varying loads, this study centers on identifying the temperature dependence of the elastic and damping parameters of CFRC and devising vibration control approaches. Firstly, experimental tests are conducted to acquire the natural frequencies and modal damping ratios of CFRC plates at various temperatures. Then, a dynamic model of the CFRC plate that takes temperature dependence into account is built, and via inverse identification, the nonlinear patterns of how the elastic modulus and loss factor change with temperature are uncovered. Subsequently, by introducing the concept of fractional-order differentiation, a novel fractional-order linear extended state observer-sliding mode control (FOLESO-SMC) strategy is proposed. This method enhances the dynamic capturing capability of the system, enabling precise estimation of the states of the system under time-varying load disturbances and rapid response to compensate for the vibrations of the system. Finally, the generality of the temperature dependence parameters of CFRC is fully validated. Through both simulation calculations and experimental tests, it is verified that FOLESO-SMC can effectively cope with the dual influences of the thermal environment and complex time-varying loads, thus achieving efficient vibration suppression. This study addresses the research gap concerning the active vibration control applied to CFRC structures under the coupled effect of thermal environments and time-varying loads, laying a foundation for the intelligent vibration control of such structures in extreme service scenarios by providing theoretical support and technical safeguards.

针对碳纤维增强复合材料（CFRC）结构在高温和时变载荷耦合环境下力学性能波动和振动控制的困难，研究了CFRC弹性和阻尼参数的温度依赖性，并设计了振动控制方法。首先进行了试验测试，获得了不同温度下CFRC板的固有频率和模态阻尼比。在此基础上，建立了考虑温度依赖的CFRC板动力模型，通过反辨识，揭示了弹性模量和损耗因子随温度变化的非线性规律。随后，通过引入分数阶微分的概念，提出了一种分数阶线性扩展状态观测器-滑模控制（folesco - smc）策略。该方法增强了系统的动态捕获能力，能够准确估计系统在时变负载扰动下的状态，并能快速响应以补偿系统的振动。最后，充分验证了CFRC温度相关参数的通用性。通过仿真计算和实验测试，验证了FOLESO-SMC能够有效应对热环境和复杂时变载荷的双重影响，从而实现高效的振动抑制。本研究填补了热环境与时变载荷耦合作用下CFRC结构振动主动控制的研究空白，为极端使用场景下CFRC结构振动智能控制提供理论支持和技术保障。

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引用次数: 0

Damage detection in natural fiber composites using vibrational analysis and a curvature-based indicator 基于振动分析和曲率指示器的天然纤维复合材料损伤检测

IF 4.2 2区工程技术 Q1 MECHANICS

European Journal of Mechanics A-Solids

Pub Date : 2025-12-31 DOI: 10.1016/j.euromechsol.2025.106013

Ons Lahbib, Nadia Massé, Ali El Hafidi, Philippe Leclaire

Natural fiber-reinforced composite materials offer a sustainable and eco-friendly alternative to traditional composites in various sectors. For these materials, it is essential to develop new methods for characterizing and detecting internal defects and damages. This study aims to detect and locate damage in natural fiber-reinforced composite materials, specifically flax/epoxy beams and plates, which exhibit lower elastic moduli and higher damping compared to traditional composites. A new indicator based on vibrational modal curvature shapes and taking into account damping factor was employed to identify invisible low-energy impact damage and delamination. Experimental vibration tests on cantilever flax fiber composite beams validated the method's effectiveness. The results show that the method can detect and localize hidden damage, with damage in flax/epoxy composites remaining generally more confined, whereas in carbon/epoxy composites it tends to spread over a larger area.

天然纤维增强复合材料在各个领域为传统复合材料提供了可持续和环保的替代品。对于这些材料，开发表征和检测内部缺陷和损伤的新方法至关重要。本研究旨在检测和定位天然纤维增强复合材料的损伤，特别是亚麻/环氧树脂梁和板，与传统复合材料相比，它具有更低的弹性模量和更高的阻尼。提出了一种基于振动模态曲率形状并考虑阻尼因素的新指标，用于识别不可见的低能冲击损伤和分层。对悬臂式亚麻纤维复合梁的振动试验验证了该方法的有效性。结果表明，该方法能够有效地检测和定位潜在损伤，其中亚麻/环氧复合材料的损伤范围较窄，而碳/环氧复合材料的损伤范围较广。

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 : 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

Study on data-driven inverse identification of structural parameters for large-scale structures 大型结构参数数据驱动反演方法研究

IF 4.2 2区工程技术 Q1 MECHANICS

European Journal of Mechanics A-Solids

Pub Date : 2025-12-29 DOI: 10.1016/j.euromechsol.2025.106009

Jie Luo , Yiwen Li , Guangyan Liu , Kai Zhang

Accurate identification of structural parameters from deformation fields remains a pivotal objective in theoretical research and engineering practice. As inverse-problem methodologies evolve, engineering applications now require parameter identification at unprecedented spatial scales and levels of fidelity. Conventional approaches, however, encounter substantial limitations when applied to large-scale structures, primarily because of the high dimensionality, nonlinearity, and heterogeneity inherent in full-field deformation data. Inspired by multi-scale decomposition principles, this study introduces a segmentation–assembly–optimization (SAO) framework that systematically reduces the complexity of large-scale inverse problems. A lightweight convolutional neural network (CNN) is trained to map local displacement fields onto their corresponding structural parameters; these local estimates are subsequently assembled and refined by a mechanics-driven optimization procedure to reconstruct the global parameter distribution. Comprehensive numerical experiments demonstrate that the proposed framework achieves accuracies exceeding 98 % within the region of interest (ROI) for large-scale structures with intricate geometries, and maintains robust reconstruction accuracy (>90 % under 1 % noise), whereas the standalone CNN performance degrades significantly. The SAO framework thereby overcomes scale-dependent constraints and delivers a reliable, data-driven solution for high-resolution structural identification.

从变形场中准确识别结构参数一直是理论研究和工程实践的关键目标。随着反问题方法的发展，工程应用现在需要在前所未有的空间尺度和保真度水平上进行参数识别。然而，传统的方法在应用于大规模结构时遇到了很大的限制，主要是因为全场变形数据的高维性、非线性和非均质性。受多尺度分解原理的启发，本研究引入了一个分段-装配-优化（SAO）框架，系统地降低了大规模逆问题的复杂性。训练轻量级卷积神经网络（CNN）将局部位移场映射到相应的结构参数上；这些局部估计随后通过力学驱动的优化程序进行组合和改进，以重建全局参数分布。综合数值实验表明，对于具有复杂几何形状的大型结构，所提出的框架在感兴趣区域（ROI）内的精度超过98%，并且在1%噪声下保持了稳健的重建精度（> 90%），而单独的CNN性能明显下降。因此，SAO框架克服了规模相关的限制，为高分辨率结构识别提供了可靠的、数据驱动的解决方案。

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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 : 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

A finite-strain chemo-electro-mechanical model for gel polymer electrolytes with dynamic ion exchange between fluid and polymer phases 流体与聚合物相动态离子交换凝胶聚合物电解质的有限应变化学-机电模型

IF 4.2 2区工程技术 Q1 MECHANICS

European Journal of Mechanics A-Solids

Pub Date : 2025-12-24 DOI: 10.1016/j.euromechsol.2025.105988

Mattia Serpelloni , Alberto Salvadori , Luigi Cabras

Climate change pivots on shifting to renewable energy sources and to reliable, readily available energy storage systems; at present, lithium-ion batteries (LiBs) are the most advanced industrial technology. Great efforts towards novel materials are underway to overcome well known safety concerns in conventional liquid electrolytes. Gel polymer electrolytes (GPEs) are promising candidates. They are composed of a fluid mixture that fills the interstitial spaces in a solid polymer network. The confined liquid boosts the conductivity and improves the surface contact with electrodes. We devise a multiphysics model for GPEs, framed in the finite-strains thermo-mechanics of continua. It accounts for the electro-chemistry, transport, and mechanics of energy storage. Predictive science is achieved through simulations of the transport and chemical interactions of solvent and ions during material advection. Insightful information on the behavior of GPE during charge–discharge of (Li-ion) batteries are attained.

气候变化的关键是转向可再生能源和可靠、现成的能源储存系统；目前，锂离子电池是最先进的工业技术。人们正在努力开发新材料，以克服传统液体电解质中众所周知的安全问题。凝胶聚合物电解质（GPEs）是很有前途的候选材料。它们由流体混合物组成，填充在固体聚合物网络中的间隙空间。密闭的液体提高了电导率，并改善了与电极的表面接触。我们设计了一个gpe的多物理场模型，框架在连续体的有限应变热力学。它解释了电化学、传输和能量储存的力学。预测科学是通过模拟物质平流过程中溶剂和离子的传递和化学相互作用来实现的。获得了锂离子电池充放电过程中GPE行为的深刻信息。

引用次数: 0

Finite element analysis of non-uniform functionally graded multi-cracked Timoshenko beams using an equilibrium-based formulation 非均匀功能梯度多裂纹Timoshenko梁的有限元分析

IF 4.2 2区工程技术 Q1 MECHANICS

European Journal of Mechanics A-Solids

Pub Date : 2025-12-24 DOI: 10.1016/j.euromechsol.2025.106008

H.A.F.A. Santos , V.V. Silberschmidt

A novel finite element formulation is introduced for the static analysis of non-uniform functionally graded multi-cracked Timoshenko beams with small deformations. The cracks, assumed to remain open, are modelled using the so-called discrete spring approach, in which Dirac delta generalized functions are introduced into the bending flexibility of the beams. The formulation is derived on the basis of a complementary variational approach that involves only the elements’ shear forces and bending moments as the fundamental unknown fields. The corresponding element flexibility matrix is obtained in closed-form, with the crack contributions explicitly separated from the standard bending and shear flexibility terms. The numerical solutions produced by the formulation are strictly equilibrated, i.e., they satisfy all equilibrium conditions of the associated boundary-value problem in strong form. The effectiveness and accuracy of the formulation are numerically assessed through its application to several benchmark problems. The obtained results are analysed and compared, where possible, to exact (or reference) solutions and solutions given by the standard displacement-based finite element formulation, clearly illustrating the capability of the formulation to deliver highly accurate results for both thin and thick beams, even on meshes with only a few degrees-of-freedom.

提出了一种新的有限元公式，用于分析小变形非均匀功能梯度多裂纹Timoshenko梁的静力分析。假设裂缝保持开放，使用所谓的离散弹簧方法进行建模，其中将狄拉克三角洲广义函数引入梁的弯曲灵活性。该公式是在一个互补变分方法的基础上推导出来的，该方法只涉及单元的剪力和弯矩作为基本的未知场。相应的单元柔度矩阵以封闭形式得到，其中裂纹贡献与标准弯曲和剪切柔度项明确分离。由该公式生成的数值解是严格平衡的，即满足相关强形式边值问题的所有平衡条件。通过对几个基准问题的应用，对该公式的有效性和准确性进行了数值评价。在可能的情况下，对获得的结果进行分析和比较，以精确（或参考）解决方案和基于标准位移的有限元公式给出的解决方案，清楚地说明了该公式对薄梁和厚梁提供高精度结果的能力，即使在只有几个自由度的网格上也是如此。

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引用次数: 0