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Manufacturing, characterization, and macromechanical modeling of short flax/hemp fiber-hybrid reinforced polypropylene 亚麻/大麻短纤维混合增强聚丙烯的制造、表征和宏观机械建模
Q3 Engineering Pub Date : 2024-04-13 DOI: 10.1016/j.finmec.2024.100269
Bilel Miled , Slim Kammoun , Imane Belyamani , Laurent Cauret

The present work focuses on the manufacturing, mechanical characterization, and modeling of hybrid Flax/Hemp/Polypropylene (PP) composites under dry conditions. Geometric parameters of the fibers were measured both before and after the melt processing. The results indicated that the processed fibers, especially the hybrid ones, had a narrowed length distribution with an average fiber length of 0.48 mm for hybrids compared to 0.62 mm for non-hybrids. The hybrid composites were mechanically characterized using basic and loading-unloading tensile tests with various fiber combinations and weight percentages. The study also examined the effect of strain rate and cutting angle on the behaviors of the composites. The results demonstrated the significance of fiber orientation as the primary factor in explaining mechanical variations. The tensile strength and Young's Modulus of FH30 composites (PP + 15 wt% flax + 15 wt% hemp) increased by about 24.1 % and 10.9 %, respectively, when the plates were cut into dumbbell shapes at a 90° angle to the injection molding flow direction, compared to a cut at 0° The study also showed that the tensile strength is directly proportional to the mass fraction of reinforcements, with an increase in tensile strength by 7.9 % for 0° cut angle specimens between FH10 (PP + 5 wt% flax + 5 wt% hemp) and FH30 bio-composites, while the uniform strain is inversely proportional to the mass fraction of reinforcements, evidenced by a reduction in uniform strain of 30 % between FH10 and FH30 samples. Morphological observations revealed the presence of bands indicating the propagation of micro-cracks, as well as debonding or cohesive failure. Finally, a Perzyna-type elasto-viscoplastic constitutive model was applied to accurately predict the overall mechanical response of a hybrid composite material. Specifically, the model successfully captured the tension deformation behavior of hybrid natural short-fiber thermoplastic composites, including the elastic stage, yield stress, and nonlinear hardening.

本研究的重点是亚麻/大麻/聚丙烯(PP)混合复合材料在干燥条件下的制造、机械特性分析和建模。在熔融加工前后都测量了纤维的几何参数。结果表明,加工后的纤维,尤其是杂交纤维的长度分布较窄,杂交纤维的平均长度为 0.48 毫米,而非杂交纤维的平均长度为 0.62 毫米。混合复合材料的机械性能采用了基本拉伸试验和加载-卸载拉伸试验,并采用了不同的纤维组合和重量百分比。研究还考察了应变率和切削角对复合材料行为的影响。结果表明,纤维取向是解释力学变化的主要因素。与 0° 切割相比,将板材切割成与注塑流动方向成 90° 角的哑铃状时,FH30 复合材料(聚丙烯 + 15 wt% 亚麻 + 15 wt% 大麻)的拉伸强度和杨氏模量分别提高了约 24.1% 和 10.9%。而均匀应变与增强材料的质量分数成反比,FH10 和 FH30 样品的均匀应变降低了 30%。形态学观察显示,存在表明微裂纹扩展的带状现象,以及脱粘或内聚失效。最后,应用 Perzyna 型弹塑性-粘塑性构成模型准确预测了混合复合材料的整体机械响应。具体来说,该模型成功捕捉了混合天然短纤维热塑性复合材料的拉伸变形行为,包括弹性阶段、屈服应力和非线性硬化。
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引用次数: 0
High velocity steel spheres impact and damage analysis of graphene doped T700 composite shielding panels 石墨烯掺杂 T700 复合屏蔽板的高速钢球冲击和损伤分析
Q3 Engineering Pub Date : 2024-04-04 DOI: 10.1016/j.finmec.2024.100268
Nelson Matos , Virginia Infante , Manuel Gomes , Mario Ribeiro , José Pedro Sousa , Filipe Ribeiro , Ricardo Rocha

Composite materials are increasingly being used in aircraft components such as fuselage panels which are subject to collision with debris during take-off/landing or in flight. This article provides an analysis of the impact resistance capability of graphene doped T700 (T700G) multilayer composite, when subjected to steel projectile impacts. Test coupons with 300×300 mm were impacted by steel spherical projectiles, with impact energies ranging from 34 J to 338 J, using a compressed air operated cannon. Test data, such as impact and absorbed energy were determined through impact and residual projectile velocity, acquired using ballistic chronograph and two high speed cameras. Damage assessment was performed using non-destructive techniques such as Ultrasonic testing before and after impact test. Through modelling and simulation, using ABAQUS/Explicit, and recording the change of projectile velocity, the energy loss of projectile is calculated, and the impact resistance is judged together with damage extent. T700G has similar impact resistance to AL2024, underperforming when impacted by larger projectiles and presenting an overall higher damaged area.

复合材料正越来越多地应用于飞机部件中,例如在起飞/着陆或飞行过程中会受到碎片碰撞的机身面板。本文分析了掺杂石墨烯的 T700(T700G)多层复合材料在受到钢弹丸冲击时的抗冲击能力。使用压缩空气操作的加农炮,300×300 毫米的测试样片受到钢制球形弹丸的冲击,冲击能量从 34 焦耳到 338 焦耳不等。测试数据,如冲击能量和吸收能量,是通过使用弹道计时器和两台高速摄像机获取的冲击速度和射弹残余速度确定的。在冲击试验前后,使用超声波测试等非破坏性技术进行了损伤评估。通过使用 ABAQUS/Explicit 进行建模和模拟,记录弹丸速度的变化,计算出弹丸的能量损失,并结合损伤程度判断其抗冲击性。T700G 的抗冲击性与 AL2024 相似,但在受到较大弹丸冲击时表现不佳,且总体损坏面积较大。
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引用次数: 0
Modelling and optimization of hybrid Kevlar/glass fabric reinforced polymer composites for low-velocity impact resistant applications 用于低速抗冲击应用的凯夫拉尔/玻璃纤维织物增强聚合物混合复合材料的建模和优化
Q3 Engineering Pub Date : 2024-03-30 DOI: 10.1016/j.finmec.2024.100267
T. Jambhulkar, R.K. Sahu

Kevlar and glass-based fabric-reinforced polymer composites possess excellent impact resistance, corrosion and high specific strength properties which makes them suitable candidature for the fabrication of industrial helmets, riot shields, automobile parts etc. The polymer composite components are prone to impact failure, which restricts their wide commercial usage. In this work, to enhance the impact damage resistance of polymer composite laminates, a hybridized material model consisting of glass and Kevlar fabric was employed. The hybridization of Kevlar ‘K’ with glass ‘G’ fabric [K/K/G/G/K] results in a cost reduction of 32 % without any significant variation in performance when compared to non-hybrid Kevlar fabric-based composite laminate [K/K/K/K/K]. In comparison to plain glass fabric laminates [G/G/G/G/G], the [K/K/G/G/K] configuration exhibits a significant 16.68 % improvement in energy absorption. This improvement is rooted in a parametric study and optimization process employing the Preference Selection Index (PSI) technique to determine the optimum stacking sequence of fabrics within the composite laminate. In addition, it evaluates the projectile limit velocity necessary to prevent failure and assesses the total deformation for an optimized stacked [K/K/G/G/K] composite, taking into account various projectile shapes conforming to low-velocity impact applications. The results of this study provide valuable insights into the properties and capabilities of the hybrid composite laminate and may pave the way for the development of more effective materials for impact protection in various applications.

凯夫拉纤维和玻璃纤维基织物增强聚合物复合材料具有优异的抗冲击性、耐腐蚀性和高比强度特性,因此适合用于制造工业头盔、防暴盾牌和汽车零件等。聚合物复合材料部件易受冲击而失效,这限制了其广泛的商业用途。在这项工作中,为了增强聚合物复合材料层压板的抗冲击破坏能力,采用了一种由玻璃和 Kevlar 纤维组成的杂化材料模型。与非混合型凯夫拉纤维复合层压板 [K/K/K/K/K] 相比,凯夫拉纤维 "K "与玻璃纤维 "G "纤维[K/K/G/G/K]的混合使成本降低了 32%,而性能却没有明显变化。与普通玻璃纤维层压板[G/G/G/G/G]相比,[K/K/G/G/K]配置的能量吸收能力显著提高了 16.68%。这一改进源于参数研究和优化过程,该过程采用了偏好选择指数(PSI)技术,以确定复合层压板中织物的最佳堆叠顺序。此外,考虑到符合低速冲击应用的各种射弹形状,该研究还评估了防止失效所需的射弹极限速度,并评估了优化堆叠 [K/K/G/G/K] 复合材料的总变形量。这项研究的结果为了解混合复合材料层压板的特性和能力提供了宝贵的见解,并可能为在各种应用中开发更有效的冲击防护材料铺平道路。
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引用次数: 0
Assessing lateral torsional buckling of stepped steel I beams using finite element method 用有限元法评估阶梯钢工字梁的侧向扭转屈曲
Q3 Engineering Pub Date : 2024-03-21 DOI: 10.1016/j.finmec.2024.100266
Amin H. Almasri , Israa Jabur

The use of steel beams with variable sections along their length is getting more attention in recent years. These variable sections can vary gradually along the beam or change abruptly as a drop or a step. The current study investigates the lateral torsional buckling (LTB) of stepped steel beams with flanged I section subjected to uniformly distributed pressure on the top flange. The studied parameters are the step height, the step length, and the boundary conditions of the beam. The finite element method is used to carry out a linear buckling analysis of the beams’ models. The results show significant degradation in beams LTB strength due to the jump of the beam's section. Reducing the simply supported beam depth by 25 % reduces LTB strength by about 40–50 %, while reducing the fixed beam depth by 25 % reduces LTB strength by about 30–40 %. Further reduction in depth showed no significant additional reduction in simply supported beams strength, while the fixed beams showed another 30–40 % reduction in strength with another 25 % reduction in beam total depth. The step made the compressive stress transfer between the two flange parts through the web.

近年来,使用沿长度方向具有可变截面的钢梁越来越受到关注。这些可变截面可以沿梁逐渐变化,也可以突然变化,如下降或阶梯。本研究探讨了带法兰 I 截面的阶梯钢梁在上翼缘承受均匀分布压力时的横向扭转屈曲(LTB)。研究参数包括台阶高度、台阶长度和梁的边界条件。采用有限元法对梁的模型进行线性屈曲分析。结果表明,由于梁的截面跃变,梁的 LTB 强度明显降低。将简单支撑梁深度减少 25%,LTB 强度降低约 40-50%,而将固定梁深度减少 25%,LTB 强度降低约 30-40%。进一步减小深度后,简支梁的强度没有明显的额外降低,而固定梁的强度在梁总深度再减小 25% 后又降低了 30-40%。阶梯使两个翼缘部分之间的压应力通过腹板传递。
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引用次数: 0
Exploring the acoustic activity in brittle materials in terms of the position of the acoustic sources and the power of the acoustic signals - Part II: Applications 从声源位置和声信号功率的角度探索脆性材料中的声学活动 - 第二部分:应用
Q3 Engineering Pub Date : 2024-03-11 DOI: 10.1016/j.finmec.2024.100265
Dimos Triantis , Ilias Stavrakas , Ermioni D. Pasiou , Stavros K. Kourkoulis

The present study is the continuation of a previously published one (Part I), in which the D- (three-dimensional distance between the sources of any two successive acoustic events) and P-functions (the rate of the energy of the acoustic signals) were introduced for the analysis of the acoustic activity developed in marble specimens under uniaxial tension. The concepts introduced in Part I and the analysis procedure analytically described there, are here employed to analyze the acoustic emissions data gathered from additional experimental protocols in order to support and validate the conclusions drawn in Part I. In this direction, the acoustic activity generated in marble and concrete (either plain or reinforced with short polyolefin fibers) specimens under uniaxial compression or three-point bending, respectively, is studied in the direction of detecting characteristics of the temporal evolution of the D- and P-functions that could provide early hints warning about upcoming fracture. The evolution of the D- and P-functions is considered in juxtaposition to that of the F-function, which is in fact a convenient and flexible mean for the quantification of the average frequency of generation of the acoustic events. The conclusions drawn here are in excellent agreement with the respective ones drawn in Part I, suggesting that the D- and P-functions can indeed be a valuable tool in hands of engineers working in the field of Structural Health Monitoring of structures made of brittle building materials.

本研究是之前发表的研究报告(第一部分)的继续,其中引入了 D 函数(任意两个连续声波事件的声源之间的三维距离)和 P 函数(声波信号的能量率),用于分析大理石试样在单轴拉伸条件下产生的声波活动。在这方面,研究了大理石和混凝土(普通或用短聚烯烃纤维加固)试样在单轴压缩或三点弯曲下分别产生的声学活动,目的是检测 D 和 P 函数的时间演变特征,以便为即将发生的断裂提供早期预警。D 函数和 P 函数的演变与 F 函数的演变并列考虑,而 F 函数实际上是量化声学事件平均发生频率的一种方便灵活的方法。这里得出的结论与第一部分得出的结论非常一致,表明 D 函数和 P 函数确实可以成为从事脆性建筑材料结构健康监测领域工作的工程师手中的宝贵工具。
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引用次数: 0
Free vibration and static analysis of sandwich composite plate with auxetic core and GPLRC facing sheets in hygrothermal environment 带辅助磁芯和 GPLRC 面板的三明治复合板在湿热环境中的自由振动和静态分析
Q3 Engineering Pub Date : 2024-03-04 DOI: 10.1016/j.finmec.2024.100264
Nima Namazinia, Akbar Alibeigloo, Mahsa Karimiasl

The main objective of this study is to investigate the vibration and static analysis of a sandwich composite plate in the hygrothermal environment. The plate consists of three layers such as an Aluminum auxetic core and Graphene platelet-reinforced composite (GPLRC) facing sheets. Based on the Halpin-Tsai theory and Gibson's model, the macro mechanical properties of facing sheets and auxetic core are derived. Via Reddy's third-order shear deformation theory (TSDT), the equations of motion of the sandwich plate are obtained and solved by both Navier and the generalized differential quadrature method (GDQM). The current formulation is validated by comparing the numerical results with those that are reported in the literature. Finally, the influence of different parameters such as the inclined angle of auxetic cells, thickness to the length, and core to total thickness on the natural frequencies, deflection and stresses of the sandwich plate, is examined. Even though some studies have been conducted to investigate the bending or vibration behavior of sandwich structures with negative Poisson's ratio (NPR) cores, the bending and vibration behavior of such structures in the hygrothermal environment is remained unexplored. Additionally, the analysis of stress component variations along the thickness of the plate was a novel feature that has not been addressed in prior studies on the similar structure.

本研究的主要目的是研究夹层复合板在湿热环境下的振动和静态分析。该板材由三层组成,如铝辅助芯材和石墨烯血小板增强复合材料(GPLRC)面片。根据 Halpin-Tsai 理论和 Gibson 模型,推导出了面片和辅助磁芯的宏观力学性能。通过 Reddy 的三阶剪切变形理论 (TSDT),得到了夹层板的运动方程,并用纳维法和广义微分正交法 (GDQM) 进行了求解。通过将数值结果与文献报道的结果进行比较,验证了当前的公式。最后,研究了不同参数对夹层板固有频率、挠度和应力的影响,如辅助单元的倾斜角、厚度与长度的比例、夹芯与总厚度的比例。尽管已有一些研究对具有负泊松比(NPR)芯材的夹层结构的弯曲或振动行为进行了调查,但仍未对此类结构在湿热环境下的弯曲和振动行为进行探讨。此外,对沿板厚的应力分量变化进行分析是一个新颖的特点,在之前对类似结构的研究中尚未涉及。
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引用次数: 0
Probabilistic analysis of FRP efficacy in seismic risk reduction 降低地震风险的 FRP 效能概率分析
Q3 Engineering Pub Date : 2024-02-03 DOI: 10.1016/j.finmec.2024.100259
Adil Ziraoui , Benaissa Kissi , Hassan Aaya

Engineers are tasked with the challenging task of evaluating the performance and analyzing the risk of systems in the context of performance-based seismic design. All sources of random uncertainty must be taken into account during the design phase in order to complete this assignment. The performance limit states for a structure must be defined using appropriate procedures that take into consideration the system characteristics describing the structure, the soil, and the loads applied to the structural reactions. The main objective of this study is to conduct an in-depth analysis, both linear and non-linear (Pushover), of seismic vulnerability for a reinforced concrete (RC) structure. This aims to probabilistically evaluate the effectiveness of composite materials, particularly those reinforced with glass and carbon fibers, in reducing seismic risk when used to reinforce structural columns. The outcomes of this study will provide valuable insights into the efficacy of FRP reinforcements in enhancing seismic resistance, regardless of the analytical approach adopted (linear or non-linear). They reveal a seismic risk reduction of 48 % for structures equipped with glass fiber-reinforced columns and 67 % for those with carbon fiber-reinforced columns.

在基于性能的抗震设计中,工程师面临着评估系统性能和分析系统风险的艰巨任务。为了完成这项任务,在设计阶段必须考虑到所有随机不确定性来源。结构的性能极限状态必须使用适当的程序来定义,这些程序应考虑到描述结构、土壤和施加在结构反力上的荷载的系统特征。本研究的主要目的是对钢筋混凝土(RC)结构的地震脆弱性进行深入的线性和非线性(Pushover)分析。这样做的目的是从概率上评估复合材料,尤其是使用玻璃纤维和碳纤维增强的复合材料,在用于加固结构柱时降低地震风险的有效性。无论采用哪种分析方法(线性或非线性),这项研究的成果都将为玻璃纤维增强材料在提高抗震性方面的功效提供有价值的见解。研究结果表明,采用玻璃纤维加固柱的结构可降低 48% 的地震风险,而采用碳纤维加固柱的结构可降低 67% 的地震风险。
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引用次数: 0
A review of additive manufacturing capabilities for potential application in offshore renewable energy structures 海上可再生能源结构潜在应用的快速成型制造能力综述
Q3 Engineering Pub Date : 2024-02-01 DOI: 10.1016/j.finmec.2024.100255
Fraser O'Neill, Ali Mehmanparast

Offshore renewable energy structures are subject to harsh environments with loading from wind, wave, and tides which introduce fatigue damage in corrosive and erosive environments. An effective approach that has been found to improve mechanical and fatigue resistance of engineering structures is employment of Additive Manufacturing (AM) technology. However, little research has been conducted for implementation of AM technology in offshore renewable energy structures. This study aims to collate and critically discuss the advantages that AM technology can offer to enhance the lifespan of offshore renewable energy structures. In addition to fatigue life improvement, the potential of AM technology to enhance corrosion and erosion resistance in offshore renewable energy structures has been explored. It has been found in this study that among the existing AM techniques, Wire Arc Additive Manufacturing (WAAM) offers promising potentials for life enhancement of offshore wind turbine and tidal turbine support structures. Early research into the potential of using WAAM to create corrosion resistance coatings and components highlights many benefits achieved from this new emerging manufacturing technology, but further research is required to justify the use of the processes for commercial applications. In terms of erosion and wear resistance even less research has been conducted but initial findings show that AM has the potential to add a great level of resistance compared to the wrought material. This study presents the key advantages that AM technology offers to enhance the design life and integrity of offshore renewable energy structures as a first step towards unlocking the great potentials of AM for consideration and implementation in the energy transition roadmap.

海上可再生能源结构所处的环境十分恶劣,风、波浪和潮汐都会对其造成负荷,从而在腐蚀和侵蚀环境中产生疲劳损伤。增材制造(AM)技术是提高工程结构机械性能和抗疲劳性能的有效方法。然而,在海上可再生能源结构中应用 AM 技术的研究却很少。本研究旨在整理和批判性地讨论 AM 技术在提高海上可再生能源结构寿命方面的优势。除了提高疲劳寿命外,还探讨了 AM 技术在提高海上可再生能源结构抗腐蚀和抗侵蚀能力方面的潜力。研究发现,在现有的 AM 技术中,线弧快速成型技术(WAAM)为提高海上风力涡轮机和潮汐涡轮机支撑结构的寿命提供了巨大潜力。对使用 WAAM 制作耐腐蚀涂层和部件的潜力进行的早期研究突出显示了这一新兴制造技术所带来的诸多益处,但还需要进一步的研究来证明将该工艺用于商业应用的合理性。在抗侵蚀和抗磨损方面的研究则更少,但初步研究结果表明,与锻造材料相比,AM 有可能增加更高水平的抗磨损能力。本研究介绍了 AM 技术在提高近海可再生能源结构的设计寿命和完整性方面所具有的关键优势,这是在能源转型路线图中考虑和实施 AM 技术以释放其巨大潜力的第一步。
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引用次数: 0
Dynamic impedances of multiple strips on multi-layered transversely isotropic poroelastic soils 多层横向各向同性孔弹性土上多条带的动态阻抗
Q3 Engineering Pub Date : 2024-02-01 DOI: 10.1016/j.finmec.2024.100260
Teerapong Senjuntichai , Barami Phulsawat , Suraparb Keawsawasvong , Wichairat Kaewjuea

Geo-materials naturally display a certain degree of anisotropy due to various effects such as deposition. Besides, they are often two-phase materials with a solid skeleton and voids filled with water, and commonly known as poroelastic materials. In the past, despite numerous studies investigating the vibrations of strip foundations, dynamic impedance functions for multiple strip footings bonded to the surface of a multi-layered transversely isotropic poroelastic half-plane have never been reported in the literature. They are first presented in this paper. All strip foundations are assumed to be rigid, fully permeable, and subjected to three types of time-harmonic loadings. The dynamic interaction problem is investigated by using an exact stiffness matrix method and a discretization technique. The flexibility equations are established by enforcing the appropriate rigid body displacement boundary conditions at each footing-layered soil interface. Numerical results for dynamic impedance functions of two-strip system are presented to illustrate the influence of various effects on dynamic responses of multiple rigid strip foundations.

由于沉积等各种影响,地质材料自然会表现出一定程度的各向异性。此外,它们通常是两相材料,具有固体骨架和充满水的空隙,通常被称为孔弹性材料。过去,尽管对带状地基振动进行了大量研究,但文献中从未报道过粘结在多层横向各向同性孔弹性半平面表面的多个带状地基的动态阻抗函数。本文首次对其进行了介绍。假定所有条形地基都是刚性的、完全透水的,并承受三种时谐荷载。本文采用精确刚度矩阵法和离散化技术研究了动态相互作用问题。通过在每个基脚-分层土壤界面强制执行适当的刚体位移边界条件,建立了柔性方程。文中给出了双条带系统动态阻抗函数的数值结果,以说明各种效应对多刚性条带地基动态响应的影响。
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引用次数: 0
Probabilistic fracture analysis of double edge cracked orthotropic laminated plates using the stochastic extended finite element method 使用随机扩展有限元法对双边缘开裂正交层压板进行概率断裂分析
Q3 Engineering Pub Date : 2024-02-01 DOI: 10.1016/j.finmec.2024.100257
Shailesh P. Palekar , Atteshamuddin S. Sayyad , Prasad M. Patare , Achchhe Lal

The current computational investigation employs the stochastic extended finite element approach, which the authors have previously developed, to investigate the probabilistic fracture response of double edge cracked orthotropic laminated composite plates under varying stress conditions. The well-known extended finite element method is used to determine the mean and coefficient of variation of stress intensity factors KI and or KII by treating the input parameters as random variables. This is done under the assumption that all of the laminated plate's layers are perfectly bonded to one another and that there is no delamination effect between the layers, the matrix, or the fibres. And it's believed that the plate has through thickness crack. A combination of input random Gaussian variables is used to model the various input factors, such as the lamination angle, the applied loads, and the crack parameters (such the crack length and location). Typical numerical results are shown to investigate the effects of varying degrees of uncertainty in the lamination angle, crack length, crack length to plate width ratio, crack positions, and applied tensile, shear, and combined (tensile and shear) stresses. An excellent agreement arises when the findings generated with the stochastic extended finite element method methodology are assessed against the results found in the published literature through Monte Carlo simulations.

当前的计算研究采用了作者之前开发的随机扩展有限元方法,以研究双边缘开裂正交层状复合板在不同应力条件下的概率断裂响应。通过将输入参数视为随机变量,使用著名的扩展有限元法确定应力强度因子 KI 和 KII 的平均值和变化系数。这种方法的假设前提是层压板的所有层之间都完全粘合,层、基体或纤维之间不存在分层效应。并且认为板材存在贯穿性裂缝。使用输入随机高斯变量组合来模拟各种输入因素,如层压角、外加载荷和裂纹参数(如裂纹长度和位置)。典型的数值结果显示了层压角、裂纹长度、裂纹长度与板宽比率、裂纹位置、施加的拉伸、剪切和组合(拉伸和剪切)应力的不同不确定程度的影响。通过蒙特卡罗模拟将随机扩展有限元方法得出的结果与已发表文献中的结果进行对比评估,结果非常一致。
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Forces in mechanics
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