【新文速递】2024年11月7日固体力学SCI期刊最新文章

 

今日更新：International Journal of Solids and Structures 1 篇，Journal of the Mechanics and Physics of Solids 1 篇，International Journal of Plasticity 2 篇，Thin-Walled Structures 7 篇

International Journal of Solids and Structures

Investigation on dynamic impact behavior of bighorn sheep’s horn

Emre Palta, Howie Fang, Qian Wang, Zheng Li

doi:10.1016/j.ijsolstr.2024.113133

大角羊角的动态碰撞特性研究

The horn of the bighorn sheep is composed of keratin-based biological material that has a tubule-lamella structure, which gives it anisotropic hardening properties under impact loading. This paper aims to investigate the energy dissipation mechanisms inherent in bighorn sheep horns by developing a numerical material model that accounts for the horn’s anisotropic features and strain-rate effects. To this end, a transversely isotropic constitutive model, which includes both anisotropic hardening and strain-rate effects, was formulated to accurately predict the mechanical behavior of bighorn sheep horns. Material characterization was conducted through uniaxial compression tests that were conducted under quasi-static and dynamic conditions. The developed constitutive model was implemented into LS-Dyna via a user-defined material subroutine and was validated against empirical data. The validated numerical model was used to investigate the horn’s mechanical responses under dynamic loading conditions. The paper focused on impact energy dissipation mechanisms, including energy absorption and transition, stress distribution, and displacement wave propagation. The insights gained from this paper are expected to significantly contribute to the development of novel artificial materials with enhanced energy absorption and impact mitigation properties.

大角羊的角是由角蛋白为基础的生物材料，具有管片状结构，使其在冲击载荷下具有各向异性硬化特性。本文旨在通过建立考虑大角羊角各向异性特征和应变率效应的数值材料模型，研究大角羊角固有的能量耗散机制。为此，建立了考虑各向异性硬化和应变率效应的横向各向同性本构模型，以准确预测大角羊角的力学行为。通过准静态和动态条件下的单轴压缩试验进行材料表征。通过用户自定义的材料子程序在LS-Dyna中实现了所开发的本构模型，并与经验数据进行了验证。利用验证后的数值模型，研究了动载荷条件下变幅杆的力学响应。重点研究了冲击能量耗散机制，包括能量吸收和转移、应力分布和位移波传播。从本文中获得的见解预计将大大有助于开发具有增强能量吸收和减轻冲击性能的新型人工材料。

Journal of the Mechanics and Physics of Solids

Damage-induced energy dissipation in artificial soft tissues

W.K. Sun, B.B. Yin, K.M. Liew

doi:10.1016/j.jmps.2024.105933

人造软组织损伤引起的能量耗散

A systematic understanding of the toughening and self-healing mechanisms of artificial soft tissues is crucial for advancing their robust application in biomedical engineering. However, current models predominantly possess a phenomenological nature, often devoid of micromechanical intricacies and quantitative correlation between microstructure damage and macroscopic energy dissipation. To bridge this gap, we propose a novel energy dissipation mechanism-motivated network model that incorporates three unique physical ingredients with sound theoretical basis for the first time. These innovated features include the bond percolation-mediated network density and stiffness, the damage-induced energy dissipation and stress softening, and the entropic elasticity for the highly stretchable second network. The validity of this model was examined by implementing it within a meshfree peridynamic framework for artificial soft tissues subjected to simple tension and pure shear tests. We quantitatively correlated the dissipation with the network damage to reveal the effects of network density, the breaking stretch dispersion and the stiffness ratio. Our findings highlighted that the inhomogeneity and dispersion of materials properties play significant roles in the controllable progressive damage and dissipation, thereby offering valuable guidance for designing tougher artificial soft tissues. By reactivating the failed network, we further successfully captured the self-healing behaviors of artificial soft tissues. Our work provides an inspiring modeling framework for studying toughening mechanisms of artificial soft tissues.

系统地了解人造软组织的增韧和自愈机制对于推进其在生物医学工程中的强大应用至关重要。然而，目前的模型主要具有现象学性质，往往缺乏微观力学的复杂性和微观结构损伤与宏观能量耗散之间的定量关联。为了弥补这一差距，我们首次提出了一种新的能量耗散机制驱动的网络模型，该模型包含了三种独特的物理成分，具有良好的理论基础。这些创新的特征包括键渗透介导的网络密度和刚度，损伤引起的能量耗散和应力软化，以及高度可拉伸的第二网络的熵弹性。通过对人工软组织进行简单拉伸和纯剪切试验的无网格周动力框架，验证了该模型的有效性。我们定量地将耗散与网络损伤进行了关联，揭示了网络密度、断裂拉伸色散和刚度比对网络耗散的影响。研究结果表明，材料性能的非均匀性和分散性在可控制的渐进损伤和耗散中起着重要作用，从而为设计更坚韧的人造软组织提供了有价值的指导。通过重新激活失败的网络，我们进一步成功地捕获了人工软组织的自愈行为。我们的工作为研究人造软组织的增韧机制提供了一个鼓舞人心的建模框架。

International Journal of Plasticity

Towards extraordinary strength-ductility synergy in pure Mg via dislocation transmutation

Liuyong He, Jiang Zheng, Mengning Xu, Tianjiao Li, Dongdi Yin, Bin Jiang, Fusheng Pan, Hao Zhou

doi:10.1016/j.ijplas.2024.104160

通过位错嬗变在纯Mg中实现非凡的强度-延展性协同作用

Navigating the strength-ductility trade-off has been a persistent challenge in Mg alloys. Here, we address this issue through a novel multiple-direction pre-deformation at room temperature that introduces a high density of <c+a> dislocations into pure Mg via dislocation transmutation. This approach achieves a remarkable enhancement in the strength-ductility synergy, increasing the yield strength from 87.6 MPa to 156.6 MPa and improving elongation to failure from 7.7% to 17.6%. In general, introducing a high-density <c+a> dislocations in Mg alloys have been difficult due to the high CRSS at room temperature. Our findings reveal that extension twinning can act as a “dislocation converter,” transforming basal dislocations in the matrix into <c+a> dislocations within twins. Intensive basal dislocations were induced in pure Mg through pre-tension and subsequently transformed into <c+a> dislocations via extension twinning during compression. This process led to a substantial number of <c+a> dislocations and I1 stacking faults, contributing to the enhanced strength. The high density of <c+a> dislocations, combined with I1 stacking faults and a reduced c/a ratio within twins, enhances the activity of pyramidal <c+a> slip, thereby significantly improving ductility. This dislocation transmutation strategy offers a promising way for producing strength-ductility synergy in Mg alloys.

在强度和延展性之间进行权衡一直是镁合金面临的挑战。在这里，我们通过一种在室温下的新型多方向预变形来解决这个问题，该变形通过位错嬗变将高密度的<c+a>位错引入纯Mg。该方法显著增强了强度-塑性协同效应，使屈服强度从87.6 MPa提高到156.6 MPa，破坏伸长率从7.7%提高到17.6%。一般来说，由于室温下的高CRSS，在镁合金中引入高密度<c+a>位错是困难的。我们的研究结果表明，延伸孪晶可以作为“位错转换器”，将基体中的基底位错转化为孪晶中的<c+a>位错。在纯Mg中，通过预拉伸诱导了密集的基底位错，随后在压缩过程中通过扩展孪晶转化为<c+a>位错。这一过程导致大量的<c+a>位错和I1层错，有助于提高强度。高密度的<c+a>位错，结合I1层错和孪晶内减小的c/a比，增强了锥体<c+a>滑移的活动性，从而显著提高了塑性。这种位错嬗变策略为镁合金强度-塑性协同效应的产生提供了一条有希望的途径。

Exceptional tensile ductility and strength of a BCC structure CLAM steel with lamellar grains at 77 kelvin

Jinhua Zhou, Jing Wang, Jungang Ren, Robert O. Ritchie, Zuncheng Wang, Yuchao Wu, Zhufeng He, Xin Wang, Ying Fu, Yifu Jiang, Lin Wang, Xiaowei Yin

doi:10.1016/j.ijplas.2024.104161

具有77开尔文层状晶粒的BCC结构CLAM钢的优异拉伸延展性和强度

The low-temperature tensile brittleness of body-centered cubic (BCC) metals and alloys can seriously compromise their service applications. In this study, we prepared a BCC structured China low activation martensitic steel (CLAM) steel with lamellar grains by regulating the rolling and heat-treatment processes, successfully reversing the decreasing trend of ductility in the steel with decrease in temperature. Compared with current face-centered cubic (FCC) structural steels and high-entropy alloys, the lamellar grained CLAM steel exhibits an excellent synergy of strength and ductility at 77K, but with lower raw material costs. The superior low temperature ductility of the lamellar grained steel can be attributed to an increase in grain strength at low temperatures which promotes the propagation of layered tearing cracks; this in turn leads to a significant increase in the necking area of the steel, thereby compensating for the decrease in ductility. We conclude that our lamellar grain structures can be utilized to significantly enhance the low-temperature tensile ductility of BCC metals and alloys, thereby expanding their service range to cryogenic temperatures.

体心立方（BCC）金属和合金的低温拉伸脆性严重影响其使用。本研究通过调整轧制工艺和热处理工艺，制备了具有层状晶粒的BCC组织中国低活化马氏体钢（CLAM），成功扭转了该钢随温度降低而塑性下降的趋势。与现有的面心立方（FCC）结构钢和高熵合金相比，层状晶状CLAM钢在77K时表现出良好的强度和延展性协同作用，但原材料成本较低。层状晶钢具有较好的低温延展性，主要是由于在低温条件下晶粒强度的提高促进了层状撕裂裂纹的扩展；这反过来又导致钢的颈缩面积显著增加，从而补偿了延展性的降低。我们的结论是，我们的层状晶粒结构可以用来显著提高BCC金属和合金的低温拉伸延展性，从而扩大它们的使用范围到低温。

Thin-Walled Structures

Web Shear Buckling of Steel-Concrete Composite Girders – Advanced Numerical Analysis

Mehmed Numanović, Markus Knobloch

doi:10.1016/j.tws.2024.112671

钢-混凝土组合梁腹板剪切屈曲的高级数值分析

Load-bearing capacity of plate girders, often used in design of bridges and high-rise buildings, is limited by the shear capacity of connected slender plate elements subjected to shear buckling. To quantify this, experimental investigations on five large-scale steel and steel-concrete composite plate girders loaded solely in shear, with a minimal influence of bending moments, have been conducted and evaluated. In this paper, the phenomenon of web shear buckling is investigated within the numerical analysis using the ABAQUS Software. An advanced numerical model has been developed and results validated against existing experimental findings. One of the focal points of this study represents the methodology of developing such a comprehensive numerical model, implementation of suitable analysis procedures, material models, boundary conditions, finite elements and interactions, in order to correctly replicate the observed response in the tests. In addition, case studies tackling the influence of web slenderness, aspect ratio, initial imperfections, shear connection and concrete classes on the structural-mechanical behavior of steel-concrete composite girders in shear as well as the applicability and suitability of the existing analytical model are also presented and analyzed.

通常用于桥梁和高层建筑设计的板梁，其承载能力受到受剪切屈曲作用的细长板单元的剪切能力的限制。为了量化这一点，对五个大型钢-钢-混凝土组合板梁进行了试验研究，仅受剪切作用，弯矩影响最小，并进行了评估。本文利用ABAQUS软件对腹板剪切屈曲现象进行了数值分析。建立了一个先进的数值模型，并与现有的实验结果进行了验证。本研究的重点之一是开发这样一个全面的数值模型的方法，实施适当的分析程序、材料模型、边界条件、有限元和相互作用，以便正确地复 制试验中观察到的响应。此外，还对腹板长细比、纵横比、初始缺陷、受剪连接和混凝土等级对钢-混凝土组合梁受剪结构力学性能的影响以及现有分析模型的适用性和适宜性进行了实例分析。

Shock mitigation and failure mechanism of copper foam/paraffin phase change reinforced composites

Jingjing Song, Yuliang Lin, Minzu Liang, Wen Liang, Jiakai Guo, Yuwu Zhang

doi:10.1016/j.tws.2024.112673

泡沫铜/石蜡相变增强复合材料的减震破坏机理

Copper foam/paraffin phase change reinforced composites (CPPC) were fabricated using vacuum immersion technology to address the pressing need for phase change reinforced composite applications. Experiments were conducted to explore the influence of strain rate and relative density of the matrix material on the mechanical properties of the CPPC under both quasi-static and dynamic conditions. A 3D-Voronoi model of the CPPC was developed with randomly varying relative density, based on real porous metal foam and utilizing graphical parametric design tools. The mechanical behavior of the CPPC under impact loading was studied, focusing on deformation, energy absorption, and damage mechanisms. Comparison and analysis of stress-strain curves and deformation modes were performed using experimental and modeling data. The shear failure modes of CPPC under quasi-static compression include 'X-shaped fracture,' 'blocky spalling,' or '45° parallel fracture,' depending on the relative density of the copper foam matrix. The addition of paraffin effectively improved the energy-absorbing properties of copper foam. As the relative density of the copper foam matrix increased, the enhancement in energy absorption became more pronounced, while the improvement in modulus and yield strength decreased. The composite exhibited an 83% increase in specific energy absorption compared to copper foam alone, with the paraffin filler absorbing 69% of the total energy during impact loading. The CPPC acted as a mechanical filter through stress wave reflection and transmission attenuation. The investigation into the shock mitigation and failure mechanisms of CPPC could offer valuable insights for the design of functional composites. Furthermore, it could inspire the creation of impact-resistant and heat dissipation structures for electronic devices.

针对相变增强复合材料应用的迫切需要，采用真空浸泡技术制备了泡沫铜/石蜡相变增强复合材料（CPPC）。在准静态和动态两种条件下，研究了应变速率和基体材料相对密度对CPPC力学性能的影响。基于真实多孔金属泡沫，利用图形参数化设计工具，建立了相对密度随机变化的CPPC 3D-Voronoi模型。研究了CPPC在冲击载荷作用下的力学行为，重点研究了CPPC的变形、能量吸收和损伤机理。利用实验数据和模型数据对应力-应变曲线和变形模式进行了对比分析。根据泡沫铜基体的相对密度，CPPC在准静态压缩下的剪切破坏模式可分为“x形断裂”、“块状剥落”或“45°平行断裂”。石蜡的加入有效地改善了泡沫铜的吸能性能。随着泡沫铜基体相对密度的增加，吸能增强更加明显，而模量和屈服强度的提高幅度减小。与单独的泡沫铜相比，该复合材料的比能量吸收增加了83%，其中石蜡填料在冲击加载时吸收了总能量的69%。CPPC通过应力波反射和透射衰减起到机械滤光器的作用。对CPPC减震失效机理的研究可为功能复合材料的设计提供有价值的见解。此外，它可以激发电子设备的抗冲击和散热结构的创造。

A comprehensive study of beam modal functions in the free vibration analysis of cylindrical shells: critical examination on the applicability to the clamped-free boundary condition

Ganghui Xu, Changsheng Zhu

doi:10.1016/j.tws.2024.112674

圆柱壳自由振动分析中梁模态函数的综合研究：对无夹紧边界条件适用性的关键检验

Over the past few decades, approximate methods that can provide solutions of sufficient accuracy have received considerable attention in the free vibration analysis of cylindrical shells, where a great deal of studies adopted the beam modal functions as the trial functions for the axial mode shapes of cylindrical shells. Nevertheless, most studies were restricted to the application of single term beam modal function and failed to simulate elastic boundary conditions of cylindrical shells, while the accuracy of the corresponding methods has recently sparked significant controversy, especially for cylindrical shells under the clamped-free boundary condition. This paper presents a comparative study of three forms of beam modal functions in the free vibration analysis of cylindrical shells, one of which is proposed for the first time to simulate elastic boundary conditions of cylindrical shells. A unified model is developed using the general Rayleigh-Ritz method, incorporating the breathing modes with circumferential orders being zero, and four types of commonly used thin shell theories, namely the Donnell, Reissner, Love, and Sanders theories. From both perspectives of natural frequencies and mode shapes, numerical results are validated by comparison with those existing in the literature and those calculated from the finite element method (FEM). The results not only clarify the distinction of different forms of beam modal functions used in the Rayleigh-Ritz method, but also provide explanations for the controversy raised in recent studies. Furthermore, the unified formulations can be extended to vibration analysis of various forms of shell structures, and can also be helpful to the vibration analysis of beams and plates with elastic boundary conditions.

近几十年来，能够提供足够精度解的近似方法在圆柱壳的自由振动分析中受到了相当大的重视，其中大量的研究采用梁模态函数作为圆柱壳轴向模态振型的试函数。然而，大多数研究仅限于应用单项梁模态函数，未能模拟圆柱壳的弹性边界条件，而相应方法的准确性近年来引起了很大的争议，特别是对无夹固边界条件下的圆柱壳。本文对圆柱壳自由振动分析中的三种梁模态函数形式进行了比较研究，首次提出了一种用于模拟圆柱壳弹性边界条件的梁模态函数形式。采用一般的瑞利-里兹方法，将周向阶数为零的呼吸模式与Donnell、Reissner、Love和Sanders四种常用的薄壳理论相结合，建立了统一的模型。从固有频率和模态振型两方面对数值结果进行了验证，并与已有文献和有限元法计算结果进行了比较。研究结果不仅澄清了Rayleigh-Ritz方法中不同形式梁模态函数的区别，而且对近年来研究中出现的争议提供了解释。统一的公式可以推广到各种形式的壳结构的振动分析中，也可以用于具有弹性边界条件的梁和板的振动分析。

Physics-informed radial basis networks for force finding of cable domes

Mingliang Zhu, Jin Wang, Jiamin Guo

doi:10.1016/j.tws.2024.112675

用于索穹顶受力测量的物理信息径向基网络

The stiffness of cable dome structures is entirely derived from the prestress in their cables and struts, making force-finding a critical step in their design. However, traditional force-finding methods are often complex to implement and have limited applicability. To address these challenges, this paper establishes a general force-finding framework for cable domes based on physics-informed radial basis networks (PIRBN), utilizing neural network techniques to achieve an efficient and reliable force-finding process. Additionally, a loss function is derived that incorporates the physical characteristics of cable domes from the perspective of structural stiffness. Case studies on three types of cable domes were conducted, and the hyperparameter tuning of the network model was simplified using the Optuna hyperparameter optimization method. The results show that PIRBN is suitable for force-finding analysis in both single and multiple prestress mode cable domes, considering the effects of external loads, and provides high computational efficiency and broad applicability.

索穹顶结构的刚度完全来源于其索和支柱的预应力，因此求力是其设计的关键步骤。然而，传统的测力方法往往是复杂的实现和有限的适用性。为了解决这些挑战，本文建立了一个基于物理信息径向基网络（PIRBN）的电缆圆顶通用力查找框架，利用神经网络技术实现高效可靠的力查找过程。此外，从结构刚度的角度，导出了一个包含索穹顶物理特性的损失函数。以三种类型的电缆穹顶为例，采用Optuna超参数优化方法简化了网络模型的超参数整定。结果表明，PIRBN在考虑外荷载影响的情况下，适用于单预应力模式和多预应力模式索穹顶受力分析，计算效率高，适用性广。

Self-healing effect on the impact-resistance of hybrid stitch toughening CFRP composites: Experimental and numerical study

Zhenzhen Zhang, Yutong Liu, Ying Tie, Yuliang Hou, Cheng Li

doi:10.1016/j.tws.2024.112635

复合缝增韧CFRP复合材料抗冲击性能的自愈效应：实验与数值研究

The self-healing effect on the impact-resistance has been investigated for hybrid stitch toughening CFRP composites using multiscale modeling. The stitches made of the healing agent, poly ethylene-co-methacrylic acid (EMAA), facilitate the repair of delamination damages via a self-healing process. The other stitches, fabricated from carbon fiber, contribute to the enhancement of interlaminar toughness. Considering the local structural features adjacent to the stitches, an equivalent fiber-embedded laminate (EFEL) cell is established to characterize the mesoscale behavior. A modified constitutive model is developed to accurately describe the deformation modes of the EFEL cell. Subsequently, a macroscale model is constructed by directly extending the EFEL cells. The self-healing of the impact-resistance is numerically explored through multiple low-velocity impact (LVI) tests. The proposed modeling approach enables a prediction error less than 8.4% and the computation time of approximately 17.3 h (1036 min), demonstrating the high accuracy and efficiency. After the self-healing process, the peak impact forces of the LVI specimens increase, while decreases in absorbed energy are observed. Moreover, the healed specimens exhibit fewer damaged elements and a smoother damaged surface compared with the unhealed ones. It demonstrates that the EMAA healing agent possesses the capability to improve the impact-resistance of hybrid stitch toughening CFRP composites.

采用多尺度模型研究了复合针状增韧CFRP复合材料的自愈效应。由愈合剂——聚乙烯-甲基丙烯酸（EMAA）制成的缝线，通过自愈过程促进分层损伤的修复。其他缝线由碳纤维制成，有助于增强层间韧性。考虑到缝线附近的局部结构特征，建立了等效的纤维嵌入层压（EFEL）单元来表征缝线的中尺度行为。提出了一种修正的本构模型，以准确地描述电光放电单元的变形模式。然后，通过直接扩展EFEL单元，构建了宏观尺度模型。通过多次低速冲击（LVI）试验，数值探讨了抗冲击的自愈性。该建模方法预测误差小于8.4%，计算时间约为17.3 h (1036 min)，具有较高的精度和效率。自愈过程后，LVI试样的峰值冲击力增大，而吸收能量减小。与未愈合的试件相比，愈合后试件的损伤元素较少，损伤表面更光滑。结果表明，EMAA愈合剂具有提高复合针状增韧CFRP复合材料抗冲击性能的能力。

Shear performance prediction for corrugated steel web girders based on machine-learning algorithms

Yong Liu, Wei Ji, Jieqi Li, ShiBo Liu, Wenjuan Yang

doi:10.1016/j.tws.2024.112668

基于机器学习算法的波纹钢腹板梁抗剪性能预测

This study aimed to predict the shear strength of corrugated steel web girders (CSWGs) by developing a new method based on four machine-learning (ML) algorithms, namely the support vector machine, artificial neural network, random forest, and XGBoost. Based on the acquired experimental and numerical data, a database containing 552 samples was constructed to train and test the ML models. A five-fold cross-validation approach was adopted during training to prevent model overfitting. A RandomizedSearchCV was used to optimize the hyperparameters of each model. The performance of the trained models was evaluated using four performance metrics, and the results revealed that the coefficients of determination (R2) of all ML models exceeded 0.97 when used on both training and validation sets, demonstrating the excellent performance of the ML models in predicting the shear strength of CSWGs. Additionally, the implemented ML models outperformed existing design codes and empirical formulae. The XGBoost model yielded the best prediction results with an R2 of 0.999, mean absolute error of 44.98 kN, root-mean-square error of 66.67 kN, and mean absolute percentage error of 2.1 %. By using the Shapley additive explanation to derive a visual, quantitative explanation of the XGBoost model, the yield strength, web thickness, and web height were identified as the most critical factors affecting the shear strength of CSWGs, and their average absolute Shapley values accounted for approximately 91.45 % of the total value. The ML models implemented in this study provide a promising new approach for pre-designing and verifying the stability of CSWGs.

基于支持向量机（support vector machine）、人工神经网络（artificial neural network）、随机森林（random forest）和XGBoost四种机器学习算法，对波纹钢腹板梁（CSWGs）的抗剪强度进行预测。基于获得的实验数据和数值数据，构建了包含552个样本的数据库，对ML模型进行训练和测试。在训练过程中采用五重交叉验证方法来防止模型过拟合。使用RandomizedSearchCV对每个模型的超参数进行优化。采用4个性能指标对训练模型的性能进行评价，结果表明，在训练集和验证集上使用的ML模型的决定系数（R2）均超过0.97，表明ML模型在预测cswg抗剪强度方面具有良好的性能。此外，实现的ML模型优于现有的设计规范和经验公式。XGBoost模型预测结果最佳，R2为0.999，平均绝对误差为44.98 kN，均方根误差为66.67 kN，平均绝对百分比误差为2.1%。利用Shapley加性解释对XGBoost模型进行可视化、定量的解释，确定了屈服强度、腹板厚度和腹板高度是影响cswg抗剪强度的最关键因素，其平均绝对Shapley值约占总量的91.45%。本研究中实现的ML模型为cswg的预设计和稳定性验证提供了一种有希望的新方法。

Weak form quadrature shell elements based on absolute nodal coordinate formulation

Zixuan He, Huayi Li, Hongzhi Zhong

doi:10.1016/j.tws.2024.112670

基于绝对节点坐标公式的弱形式正交壳单元

Weak form quadrature elements for moderately thick shells with arbitrary initial configurations are developed under the framework of continuum mechanics and the absolute nodal coordinate formulation (ANCF). Locking problems of shell analysis are discussed. Nonlinear analysis of various shell structures is conducted. The joint constraint equations for shells with discontinuous slopes are established. Five examples encompassing static and dynamic shell analysis, post-buckling analysis of shells, as well as analysis of shells with discontinuous mid-surface slopes are examined to assess the performance of the proposed elements. Satisfactory results are obtained, validating the efficacy of the proposed elements.

在连续介质力学和绝对节点坐标公式的框架下，建立了具有任意初始构型的中厚壳的弱形式正交单元。讨论了壳体分析中的锁定问题。对各种壳结构进行了非线性分析。建立了具有不连续斜率的壳的联合约束方程。五个例子包括静力和动力壳分析，壳的后屈曲分析，以及具有不连续中表面斜率的壳的分析，以评估所提出的单元的性能。得到了满意的结果，验证了所提出元件的有效性。