【新文速递】2023年12月22日固体力学SCI期刊最新文章

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

International Journal of Solids and Structures

A model-based simulation framework for coupled acoustics, elastodynamics, and damage with application to nano-pulse lithotripsy

Yangyuanchen Liu, Pei Zhong, Oscar Lopez-Pamies, John E. Dolbow

doi:10.1016/j.ijsolstr.2023.112626

基于模型的声学、弹性力学和损伤耦合模拟框架，应用于纳米脉冲碎石术

We develop a model for solid objects surrounded by a fluid that accounts for the possibility of acoustic pressures giving rise to damage on the surface of the solid. The propagation of an acoustic pressure in the fluid domain is modeled by the acoustic wave equation. On the other hand, the response of the solid is described by linear elastodynamics coupled with a gradient damage model, one that is based on a cohesive-type phase-field description of fracture. The interaction between the acoustic pressure and the deformation and damage of the solid are represented by transmission conditions at the fluid-solid interface. The resulting governing equations are discretized using a finite-element/finite-difference method that pays particular attention to the spatial and temporal scales that need to be resolved. Results from model-based simulations are provided for a benchmark problem as well as for recent experiments in nano-pulse lithotripsy. A parametric study is performed to illustrate how damage develops in response to the driving force (magnitude and location of the acoustic source) as a function of the fracture resistance of the solid. The results are shown to be qualitatively consistent with experimental observations for the location and size of the damage fields on the solid surface. A study of limiting cases also suggests that both the threshold for damage and the critical fracture energy are important to consider in order to capture the transition from damage initiation to complete localization. A low-cycle fatigue model is proposed that degrades the fracture resistance of the solid as a function of accumulated tensile strain energy, and it is shown to be capable of capturing damage localization in simulations of multi-pulse nano-pulse lithotripsy.

我们为被流体包围的固体物体建立了一个模型，该模型考虑了声压对固体表面造成破坏的可能性。声压在流体域中的传播由声波方程建模。另一方面，固体的响应则由线性弹性动力学和梯度损伤模型（基于断裂的内聚型相场描述）来描述。声压与固体变形和损伤之间的相互作用由流体-固体界面上的传输条件表示。使用有限元/有限差分法对所得到的控制方程进行离散化，该方法特别关注需要解决的空间和时间尺度问题。基于模型的模拟结果适用于基准问题以及最近的纳米脉冲碎石实验。进行了一项参数研究，以说明损伤是如何随着驱动力（声源的大小和位置）的变化而发展的，是固体断裂阻力的函数。研究结果表明，固体表面损伤场的位置和大小与实验观察结果在性质上是一致的。对极限情况的研究还表明，为了捕捉从损伤开始到完全局部化的过渡，损伤阈值和临界断裂能量都是需要考虑的重要因素。我们提出了一个低循环疲劳模型，该模型可使固体的抗断裂性能退化为累积拉伸应变能的函数，并证明它能够在多脉冲纳米脉冲碎石模拟中捕捉损伤局部化。

Mechanics of Materials

Identification of thermo-viscoplastic behavior for AA6061 under in-plane biaxial loadings

J. Liang, D. Guines, L. Leotoing

doi:10.1016/j.mechmat.2023.104898

确定 AA6061 在平面双轴载荷作用下的热粘塑性行为

This work aims to investigate the thermo-visco-plastic behavior of an aluminum alloy (AA6061) sheet metal submitted to in-plane biaxial loadings under warm conditions. Biaxial tensile tests are performed in a temperature range from room temperature to 160 °C, and in a strain rate range from quasi-static to the so-called “intermediate” strain rate (up to few s−1). The specimen shape used in this study has been previously defined and validated by the authors to identify the viscoplastic hardening models of metallic alloys at large strains. A specific device leading to a uniform temperature in the sample is associated with a dynamic biaxial traction bench to carry out the temperature and strain rate dependent characterizations. From these experiments, both the experimental forces measured on each axis of the in-plane biaxial specimen and the principal strains at the center of the specimen are obtained. These experimental data are then used in an inverse analysis loop, based on a finite element model of the biaxial test, to calibrate the parameters of a thermo-viscoplastic strain hardening model by minimizing the difference between the experimental and numerical principal strains at the center point of the specimen. Finally, it is shown that the identified hardening model well predicts the flow stress of AA6061 under different temperatures and strain rates for a strain level of up to 40%.

这项工作旨在研究铝合金（AA6061）板材在温暖条件下承受平面双轴载荷的热粘塑性行为。双轴拉伸试验的温度范围从室温到 160 °C，应变速率范围从准静态到所谓的 "中间 "应变速率（高达几 s-1）。本研究中使用的试样形状由作者先前定义并验证，用于确定金属合金在大应变下的粘塑性硬化模型。一个能使试样温度均匀的特定装置与一个动态双轴牵引台相连接，以进行温度和应变速率相关特性分析。通过这些实验，可以获得在平面双轴试样的每个轴上测量到的实验力和试样中心的主应变。然后，基于双轴测试的有限元模型，将这些实验数据用于反分析循环，通过最小化试样中心点的实验主应变和数值主应变之间的差异，校准热粘塑性应变硬化模型的参数。结果表明，所确定的硬化模型可以很好地预测 AA6061 在不同温度和应变率下的流动应力，应变水平最高可达 40%。

International Journal of Plasticity

A coupled phase-field and crystal plasticity model for understanding shock-induced phase transition of iron

Songlin Yao, Jidong Yu, Xiaoyang Pei, Yinan Cui, Hao Zhang, Hui Peng, Yi Li, Qiang Wu

doi:10.1016/j.ijplas.2023.103860

用于理解冲击诱导铁相变的耦合相场和晶体塑性模型

In the current work, we propose a chemical-potential-based phase-field model coupled with dynamic crystal plasticity based on a unified energy framework with the aim of understanding the shock-induced α–ϵ–α phase transition (PT) of iron. The plasticity is coupled with the PT through controlling the shear strain energy and the entropy rise during the dynamic deformation. In contrast with previous models, the PT pressure in this model is not a constant but obeys the Gaussian distribution with the applied stress, which is in accordance with the hysteresis effect observed in quasi-static experiments. Moreover, the contribution of multivariants to PT can be distinguished based on reaction-pathway theory. The proposed model quantitatively reproduces the split of the three-wave structure and the “loop” features of the plasticity wave and the phase transition wave, which agree well with the shock loading experiments of polycrystalline and monocrystalline iron, and cannot be well captured by previous models. Furthermore, many new insights in shock wave physics are gained. PT kinetics is found to be influenced by plasticity via the hysteresis effect and the newly generated stress wave induced by the dynamic deformation. First, the rarefaction wave induced by plastic deformation, as well as that by the growth of the child phase, behind the shock front reduces the PT plateau in the wave profile. Second, the plasticity controls the PT driving force and influences the PT hysteresis effect behind the shock front. As yield stress increases, less strain energy is plastically dissipated, and more strain energy drives the PT behind the shock front more efficiently, which results in a sharper slope of the P2 wave.

在本工作中，我们基于统一的能量框架，提出了一种基于化学势的相场模型，该模型与动态晶体塑性相耦合，旨在理解冲击诱导的铁α-ϵ-α相变（PT）。塑性通过控制动态变形过程中的剪切应变能和熵升与 PT 相耦合。与以前的模型不同，该模型中的 PT 压力不是一个常数，而是随外加应力服从高斯分布，这与准静态实验中观察到的滞后效应相一致。此外，基于反应途径理论，可以区分多变量对 PT 的贡献。所提出的模型定量地再现了三波结构的分裂以及塑性波和相变波的 "循环 "特征，这与多晶铁和单晶铁的冲击加载实验十分吻合，是以往模型所不能很好捕捉的。此外，还获得了许多有关冲击波物理学的新见解。研究发现，PT 动力学受到塑性的影响，塑性通过滞后效应和动态变形引起的新产生的应力波产生影响。首先，塑性变形引起的稀释波和子相增长引起的稀释波在冲击波前沿后方减少了波剖面中的 PT 高原。其次，塑性控制 PT 驱动力，并影响冲击前沿后的 PT 滞后效应。随着屈服应力的增加，塑性耗散的应变能减少，更多的应变能更有效地驱动冲击前沿后的 PT，从而导致 P2 波的斜率变大。

Thin-Walled Structures

Investigation on damage behavior of composite T-shaped stiffened panels under compression after multi-point impact considering impact positions

Zhengwei Meng, Linhai Huang, Pengfei Wang, Wei Zhang, Jin Sun, Ming Zhao, Zhaoxin Yun, Xiaoling Ai, Nianhong Li

doi:10.1016/j.tws.2023.111514

考虑撞击位置的多点撞击后复合 T 型加劲板受压破坏行为研究

Considering the randomness and non-uniqueness of the impact damage distribution of composite structures, both experimental and numerical simulation methods are employed to systematically explore the multi-point impact damage and compression-after-impact damage behavior of composite T-shaped stiffened panels under three types of impact positions. Non-destructive testing, and numerical simulation methods are used to examine the morphology of multi-point impact damage. The compression failure process of the impacted specimens and the intact specimens (reference group) is investigated by the electrometric method. In addition, the impact failure mechanism and compression failure mechanism is further studied by combining with the numerical simulation method. The results demonstrate that impact position B (the edge of each stiffener flange), the type of impact damage is more complex, in addition to fiber/matrix damage, it contains skin/stiffener debonding damage/layer fracture damage. Especially layer fracture damage, which significantly affects the compression damage behavior of structures.

考虑到复合材料结构冲击损伤分布的随机性和非唯一性，采用实验和数值模拟两种方法系统地探讨了复合材料 T 型加劲板在三种冲击位置下的多点冲击损伤和冲击后压缩损伤行为。采用无损检测和数值模拟方法研究了多点冲击损伤的形态。采用电测法研究了受冲击试样和完好试样（参照组）的压缩破坏过程。此外，还结合数值模拟方法进一步研究了冲击破坏机理和压缩破坏机理。结果表明，冲击位置 B（各加劲件翼缘边缘）的冲击破坏类型较为复杂，除纤维/基体破坏外，还包含表皮/加劲件脱粘破坏/层断裂破坏。尤其是层状断裂损伤，会对结构的压缩损伤行为产生重大影响。

A fast Chebyshev spectral approach for vibroacoustic behavior analysis of heavy fluid-loaded baffled rectangular plates with general boundary conditions

Tiantong Zhao, Tiangui Ye, Yuehua Chen, Guoyong Jin, Yukun Chen, Zhigang Liu

doi:10.1016/j.tws.2023.111518

用于分析具有一般边界条件的重流体载荷障板振动声学行为的快速切比雪夫频谱方法

A computationally efficient Chebyshev spectral approach is proposed to solve the vibroacoustic response of heavy fluid-loaded baffled rectangular plates. The governing equations for the displacements of motion in rectangular plates and the sound pressure in the Helmholtz integral are formulated using high-order, first-class Chebyshev polynomial expansions. This formulation is combined with Gauss-Chebyshev-Lobatto sampling. The quadruple integral encountered in solving the work done by the heavy fluid on the plate is reformulated into the configuration of a tensor product. The approach significantly reduces the computation time for solving the acoustic equations, bringing the vibration response of fluid-loaded plates closer to that of vacuum plates and limiting it to just a few seconds. The elastic boundary conditions of the rectangular plates are simulated using linear and rotational springs. Predictions of vibroacoustic behavior, including plate velocity, sound pressure, sound power, and radiation efficiency, are validated against literature results. The accuracy and efficiency of the Chebyshev spectral approach for the vibroacoustic coupling systems are demonstrated by the presence of exemplary agreements. Furthermore, this study investigates the effect of boundary conditions, geometric characteristics, and damping variables on the vibroacoustic behavior of rectangular plates.

本文提出了一种计算高效的切比雪夫频谱方法，用于求解重型流体负载障板矩形板的振动声学响应。矩形板运动位移和亥姆霍兹积分声压的控制方程采用高阶、一流的切比雪夫多项式展开来计算。该公式与高斯-切比雪夫-洛巴图采样相结合。在求解重流体对板做功时遇到的四重积分被重新表述为张量乘积的配置。这种方法大大减少了求解声学方程的计算时间，使流体负载板的振动响应更接近真空板的振动响应，并将其限制在几秒钟内。利用线性弹簧和旋转弹簧模拟了矩形板的弹性边界条件。振动声学行为的预测，包括板速度、声压、声功率和辐射效率，都与文献结果进行了验证。切比雪夫频谱方法在振声耦合系统中的准确性和效率通过示范协议的存在得到了证明。此外，本研究还探讨了边界条件、几何特征和阻尼变量对矩形板振动声学行为的影响。

Finite element dynamic modeling and vibration reduction analysis of the double-lap bolted thin plate with partially attached constrained layer damping

Rongfei Zhang, Wei Sun, Haitao Luo, Hongwei Ma, Hui Zhang

doi:10.1016/j.tws.2023.111522

带部分附着约束层阻尼的双搭接螺栓薄板的有限元动态建模和减振分析

The double-lap bolted thin plate is widely used, but it is easy to produce vibration. In this paper, the method of attaching H-shaped constrained layer damping (CLD) is adopted to reduce vibration, and the finite element dynamic model of the double-lap bolted thin plate with H-shaped CLD is investigated. The innovations of the modeling method are described as follows. A composite element with four nodes and 28 degrees of freedom (DOFs) is produced by taking into account the interlaminar shear deformation of the CLD structure and employing complex modulus to quantify the damping characteristics of the viscoelastic layer. A series of transition elements are constructed to conduct the assembly of elements with inconsistent dimensions. Complex stiffness spring components and mass elements are presented to simulate the stiffness, damping, and mass characteristics of the double-lap bolt joint based on the crock-shaped stress distribution in the double-lap bolt joint and taking the bolt effect region into account. Finally, a case study is carried out, and the rationality of the finite element model of the bolted structure in solving the natural characteristics and vibration response parameters is verified by the established experimental tests. At the same time, the test and modeling results show that the H-shaped CLD reduces vibration on the double-lap bolted thin plate. Furthermore, based on the established finite element model, the influence of the attaching position of the H-shaped CLD, thicknesses of the viscoelastic layer, and the constraining layer on the vibration characteristics of the bolted structure are analyzed.

双搭接螺栓薄板应用广泛，但容易产生振动。本文采用附加 H 型约束层阻尼（CLD）的方法来减少振动，并研究了附加 H 型约束层阻尼的双搭接螺栓薄板的有限元动态模型。建模方法的创新点如下。考虑到 CLD 结构的层间剪切变形，并采用复模量量化粘弹性层的阻尼特性，生成了一个具有四个节点和 28 个自由度 (DOF) 的复合元素。我们构建了一系列过渡元件，用于组装尺寸不一致的元件。提出了复杂的刚度弹簧元件和质量元件，以模拟双圈螺栓连接处的刚度、阻尼和质量特性，其依据是双圈螺栓连接处的钩形应力分布，并考虑了螺栓效应区域。最后，进行了案例研究，并通过已建立的试验验证了螺栓结构有限元模型在求解自然特性和振动响应参数方面的合理性。同时，试验和建模结果表明，H 型 CLD 可减少双搭接螺栓薄板的振动。此外，基于已建立的有限元模型，分析了 H 型 CLD 连接位置、粘弹性层和约束层厚度对螺栓结构振动特性的影响。

Axial and radial reversibility and energy absorption behaviors of braided shape memory composite thin-walled tubes

Qin Yang, Renyi Liu, Zhiyong Li, Bohong Gu, Baozhong Sun, Wei Zhang

doi:10.1016/j.tws.2023.111524

编织形状记忆复合材料薄壁管的轴向和径向可逆性及能量吸收行为

Braided shape memory polymer composites (SMPCs), which integrate braided structures with shape memory polymers, are well-known for their superior mechanical properties, reversible deformation, and recovery capabilities. This study focused on exploring the reusable energy absorption capacity of intelligent braided SMPC tubes. The effects of braiding structure on compression modes, mechanical properties, energy absorption, shape memory performance, and repeatability under axial and radial compression have been comprehensively investigated. The results show that SMPC tubes have a shape recovery ratio exceeding 94% after four cycles. As the number of compressions increased, both the mechanical and energy absorption properties decreased, particularly during the first compression. Higher energy absorption capacity has been observed in SMPC tubes with larger braiding angles, up to 32.51 J, 2.3 times that of pure TPU tubes. These findings offer potential approaches for the development of high-performance reusable protective devices and lightweight intelligent structures.

编织形状记忆聚合物复合材料（SMPC）是编织结构与形状记忆聚合物的集成，以其优异的机械性能、可逆变形和恢复能力而闻名。本研究的重点是探索智能编织 SMPC 管的可重复使用能量吸收能力。全面研究了编织结构对轴向和径向压缩下的压缩模式、机械性能、能量吸收、形状记忆性能和重复性的影响。结果表明，SMPC 管在四个循环后的形状恢复率超过 94%。随着压缩次数的增加，机械性能和能量吸收性能都有所下降，尤其是在第一次压缩时。在具有较大编织角的 SMPC 管中观察到了更高的能量吸收能力，高达 32.51 J，是纯 TPU 管的 2.3 倍。这些发现为开发高性能可重复使用的保护装置和轻质智能结构提供了潜在的方法。

Investigation of nonlinear magnetic stiffness based thin-layer Stacked piezoelectric generators with a force-amplification structure

Min Wang, Jiepeng Liu, Yan Peng, Zhongjie Li

doi:10.1016/j.tws.2023.111525

基于具有力放大结构的薄层叠层压电发生器的非线性磁刚度研究

This paper investigates a nonlinear piezoelectric stack energy harvester incorporating a magnetic spring (NPEH) that utilizes impact-induced frequency modulation to augment the harvested power. The proposed configuration comprises a force amplifier, a piezoelectric stack, a spring-mass system, and a limiter. We formulate a two-degree-of-freedom (2DoF) theoretical model that accounts for the nonlinear characteristics of the system and corroborate it with experimental data. We examine the effects of system parameters on the output performance of the NPEH and demonstrate that the nonlinear magnetic spring facilitates a wideband energy harvesting. For instance, when the surface magnetic field intensities of the upper and lower magnets are 156mT and 256mT, respectively, the operational bandwidth reaches 5.23Hz, which is 154% greater than that of a linear spring system. Additionally, the nonlinear magnetic spring mitigates the voltage decay during the collision period due to its hysteresis response, thereby increasing the average power. At an excitation frequency of 9 Hz and with surface field strengths of the upper and lower magnets set at 156mT and 256mT respectively, the NPEH generates an RMS power of 2.2mW (with an instantaneous power peak of 1.20W). We provide design guidelines for optimizing the energy harvesting performance based on the parametric analysis.

本文研究了一种包含磁性弹簧（NPEH）的非线性压电堆能量收集器，它利用冲击引起的频率调制来增强收集的功率。所提出的配置包括一个力放大器、一个压电叠层、一个弹簧-质量系统和一个限制器。我们制定了一个两自由度（2DoF）理论模型，该模型考虑了系统的非线性特性，并与实验数据相互印证。我们研究了系统参数对 NPEH 输出性能的影响，并证明非线性磁弹簧有助于宽带能量采集。例如，当上下磁体的表面磁场强度分别为 156mT 和 256mT 时，工作带宽达到 5.23Hz，比线性弹簧系统高出 154%。此外，非线性磁性弹簧由于其滞后响应，可减轻碰撞期间的电压衰减，从而提高平均功率。在激励频率为 9 Hz，上下磁体的表面磁场强度分别为 156mT 和 256mT 时，NPEH 产生的有效功率为 2.2mW（瞬时功率峰值为 1.20W）。我们根据参数分析提供了优化能量收集性能的设计指南。

Theoretical Model of Energy Absorption for Square Tubes under Transverse Loading with Variable Axial Constraint Condition

Junyuan Zhang, Jian Xie, Danfeng Zheng, Chang Su, Hao Zhou

doi:10.1016/j.tws.2023.111526

轴向约束条件可变的横向载荷下方形管的能量吸收理论模型

Thin-walled tubes in the actual engineering structure (e.g., autobody) are usually connected to other structure, and the boundary constraint condition at both ends of tubes are usually very complex when bending deformation occurs under transverse loading. This paper aims at the effect and mechanism of axial constraint on the bending energy absorption effect of thin-walled tube under limited and variable axial constraint condition at both ends. A typical of thin-walled tube, square tube, is taken as an example. Firstly, we compare the experiment results of square tube under transverse loading with three-point-bending condition and fully clamped condition, revealing that the axial constraint force has a significant effect on the energy absorption effect. After that, a FE model of the square tube under variable axial constraint condition at both ends is established and validated based on experiment. Based on the validated FE model, the effect of different sizes of axial constraint stiffness on the bending energy absorption of square tube are investigated, and it is found that it has a significant effect on the deformation mode and bending energy absorption. With the increase of axial constraint stiffness, it will mainly affect the latter segment of the force-displacement curve in the bending process, which increases the bending energy absorption. Finally, theoretical models are developed to predict the force-displacement curves and bending energy absorption of the square tube under three different types of variable axial constraint properties: linear axial constraint, unequal linear axial constraint and nonlinear axial constraint.

实际工程结构（如车身）中的薄壁管通常与其他结构相连，在横向载荷作用下发生弯曲变形时，管两端的边界约束条件通常非常复杂。本文旨在研究在两端有限和可变轴向约束条件下，轴向约束对薄壁管弯曲能量吸收效应的影响和机理。本文以典型的薄壁方管为例。首先，我们比较了方管在三点弯曲条件和全夹紧条件下横向加载的实验结果，发现轴向约束力对能量吸收效果有显著影响。随后，根据实验结果建立并验证了两端可变轴向约束条件下方管的有限元模型。根据验证后的 FE 模型，研究了不同大小的轴向约束刚度对方管弯曲能量吸收的影响，结果发现轴向约束刚度对变形模式和弯曲能量吸收有显著影响。随着轴向约束刚度的增大，主要会影响弯曲过程中力-位移曲线的后一段，从而增加弯曲能量吸收。最后，建立了理论模型来预测方管在三种不同类型的可变轴向约束特性（线性轴向约束、不等线性轴向约束和非线性轴向约束）下的力-位移曲线和弯曲能量吸收。

Experimental investigation of liquid-tank interaction effects on full containment LNG storage tanks through shaking table tests

Wei Liu, Chang Xiao, Hao Zhou, Chenyan Wang

doi:10.1016/j.tws.2023.111527

通过振动台试验研究液罐相互作用对全密封液化天然气储罐的影响

Liquefied natural gas (LNG) storage tanks play a crucial role as primary storage facilities in modern gas supply systems. This study investigates the seismic response of full containment LNG storage tanks through shaking table tests. A 1/14 scale model structure is meticulously constructed and subjected to testing. The test comprises two distinct phases: an empty tank phase and a water-filled tank phase. The primary emphasis of this paper is the exploration of the effects of liquid–tank interaction. A novel no-contact binocular video synchronization measurement system is developed by utilizing high-speed cameras to accurately measure the liquid sloshing height. The seismic responses of both the liquid and the tank, including acceleration and hydrodynamic pressure, are meticulously measured and analyzed. Findings reveal that the filling fluid has a significant effect on the tank, particularly the steel inner tank. The liquid sloshing phenomenon influences the hydrodynamic pressures, with the magnitude of influence decreasing as the liquid depth increases. In addition, a comparison is conducted between the experimental liquid sloshing height and hydrodynamic pressure results and those obtained from liquid linear forced sloshing theory. Liquid linear forced sloshing theory fails to accurately predict liquid sloshing height for large-amplitude sloshing but provides a rough estimate of hydrodynamic pressure distribution on the tank wall. Two simplified mechanical models for liquid sloshing are proposed and compared with the experimental results. The outcomes demonstrate that the Housner method exhibits relatively large errors, while the simplified mechanical models based on liquid linear forced sloshing theory accurately describe cases with small-amplitude liquid sloshing.

液化天然气（LNG）储罐是现代天然气供应系统中的主要储存设施，起着至关重要的作用。本研究通过振动台试验研究了全封闭液化天然气储罐的地震响应。我们精心建造了一个 1/14 比例的模型结构，并对其进行了测试。测试包括两个不同的阶段：空罐阶段和充水阶段。本文的主要重点是探索液体与储罐相互作用的影响。通过利用高速摄像机精确测量液体荡动高度，开发了一种新型非接触式双目视频同步测量系统。对液体和罐体的地震响应，包括加速度和流体动力压力，进行了细致的测量和分析。研究结果表明，填充液体对罐体，尤其是钢制内罐有很大影响。液体滑动现象会影响流体动力压力，影响程度随着液体深度的增加而减小。此外，还对实验得出的液体荡高和水动力压力结果与液体线性强制荡高理论得出的结果进行了比较。液体线性强制滑动理论无法准确预测大振幅滑动时的液体滑动高度，但可以粗略估计罐壁上的水动力压力分布。本文提出了两种简化的液体荡动力学模型，并与实验结果进行了比较。结果表明，Housner 方法的误差相对较大，而基于液体线性强制滑动理论的简化机械模型则能准确描述小振幅液体滑动的情况。