Modeling of softsphere normal collisions with characteristic of coefficient of restitution dependent on impact velocity

This letter presents a theoretical model of the normal (head-on) collisions between two soft spheres for predicting the experimental characteristic of the coefficient of restitution dependent on impact velocity. After the contact force law between the contacted spheres during a collision is phenomenologically formulated in terms of the compression or overlap displacement under consideration of an elastic—plastic loading and a plastic unloading subprocesses, the coefficient of restitution is gained by the dynamic equation of the contact process once an initial impact velocity is input. It is found that the theoretical predictions of the coefficient of restitution varying with the impact velocity are well in agreement with the existing experimental characteristics which are fitted by the explicit formula.     

粘着摩擦系数的分形几何研究

计及作用于接触斑点上的切向力，通过比较作用于接触斑点上的法向弹性载荷与法向塑性载荷，确定了区分弹性接触与塑性接触区域的临界接触斑点面积．总的粘着摩擦系数被表示为弹性接触区与塑性接触区的粘着摩擦系数的组合．假设屈服压力及局部粘着摩擦系数不依赖于接触斑点且等于塑性接触区中的平均值，则总的粘着摩擦系数可用简单的表达式描述．分形几何参数及归一接触面积对于粘着摩擦系数的效应已通过算例表明，研究中，分别考虑了忽略与计及接触斑点的微粒间的相互作用，两种情况的结果完全不同．     

Elastic–plastic behavior of a semicircular frame being pressed against a rigid plane

As a simplified structural model, a semicircular frame is used to study the crashworthiness behavior of an aircraft fuselage. The quasi-static large elastic-plastic deformation of a semicircular frame in the process of its being pressed against a rigid ground is analyzed. First, based on the linear elastic assumption, the quasi-static large deformation contact process of the frame can be divided into three phases, i.e., point contact, line contact and post-buckling. By means of a shooting method, the relations between the displacement and contact force as well as the distribution of bending moment in the three phases are obtained. Then, by assuming an elastic, perfectly-plastic moment-curvature relationship for the semi-circular frame, the contact process is analyzed in detail to reveal the plastic collapse mechanism, the traveling of plastic hinge and the force-displacement relationship. In order to verify the analysis, a preliminary experiment was conducted, in which two types of half rings with clamped ends were pressed by a rigid plate. In addition, a numerical simulation is also conducted by employing ABAQUS to analyze both rectangular cross-sectional beam and I-beam. Finally, the theoretical predictions are compared with the experimental results and numerical solutions, showing that the elastic-plastic analysis can predict the contact process very well.     

基于微凸体接触压力分布的粗糙表面法向接触建模

提出一种考虑微凸体弹塑性接触变形影响的粗糙表面法向接触力学模型。采用有限元模拟微凸体弹塑性接触过程,分析不同塑性屈服条件对微凸体接触载荷和实际接触面积的影响。再根据微凸体接触面上压力分布的变化规律,将微凸体的接触状态分为完全弹性接触阶段、弹塑性接触阶段、完全塑性接触阶段。分析接触面压力变化规律对微凸体法向接触载荷-变形的影响,再利用GW模型的数理统计分析的方法得到粗糙表面的法向接触载荷。将本文提出的模型与完全弹性模型、CEB模型、ZMC模型、KE模型、JG模型进行对比,并且研究了塑性指数对粗糙面接触载荷-平均高度距离的影响。结果表明,本文提出的模型能够更好地描述微凸体法向接触载荷与接触变形的变化趋势,模型预测粗糙表面法向载荷与ZMC、KE模型具有较好的一致性;粗糙面接触载荷随着平均接触距离增加而减少,随着塑性指数的增加,不同模型预测的法向接触载荷差异逐渐增大。     

Relationship between the elastic–plastic interface radius and internal pressure of thick-walled cylinders using the Lambert W function

It is necessary to determine the elastic–plastic interface radius before calculating the stresses in elastic-perfectly plastic thick-walled cylinders under internal pressure. The relationship between the radius and internal pressure is expressed by a transcendental equation in the existing literature. In this paper, we show that the radius can be explicitly expressed as a function of the internal pressure in terms of the Lambert W function, and it is very simple to use this expression to determine the values of the radius with the help of computer algebra systems.     

Phenomenological study of parabolic and spherical indentation of elastic-ideally plastic material

A phenomenological study of parabolic and spherical indentation of elastic ideally plastic materials was carried out by using precise results of finite elements calculations. The study shows that no “pseudo-Hertzian” regime occurs during spherical indentation. As soon as the yield stress of the indented material is exceeded, a deviation from the, purely elastic Hertzian contact behaviour is found. Two elastic–plastic regimes and two plastic regimes are observed for materials of very large Young modulus to Yield stress ratio, E/σ_y. The first elastic–plastic regime corresponds to a strong evolution of the indented plastic zone. The first plastic regime corresponds to the commonly called “fully plastic regime”, in which the average indentation pressure is constant and equal to about three times the yield stress of the indented material. In this regime, the contact depth to penetration depth ratio tends toward a constant value, i.e. h_c/h = 1.47. h_c/h is only constant for very low values of yield strain (σ_y/E lower than 5 × 10^?6) when aE¹/Rσ_y is higher than 10,000. The second plastic regime corresponds to a decrease in the average indentation pressure and to a steeper increase in the pile-up. For materials with very large E/σ_y ratio, the second plastic regime appears when the value of the non-dimensional contact radius a/R is lower than 0.01. In the case of spherical and parabolic indentation, results show that the first plastic regime exists only for elastic-ideally plastic materials having an E/σ_y ratio higher than approximately 2.000.     

位移幅值对铜镁合金微动磨损行为的影响

在研制的多功能微动磨损试验机上,开展了不同位移幅值下铜镁合金微动磨损试验,以研究位移幅值对铜镁合金微动磨损行为的影响. 微动过程中记录摩擦系数曲线与F_t-D-N曲线,利用光学显微镜(OM)、扫描电镜(SEM)、能量色散X射线光谱仪(EDS)及三维形貌仪对损伤区域进行了微观分析. 结果显示:随着位移幅值的增加,铜镁合金微动运行状态由部分滑移进入完全滑移,未发现混合滑移状态;部分滑移区中呈现由弹性变形协调逐渐向塑性变形协调转变的趋势. 磨损体积随位移幅值的增加而增加,在完全滑移区中体积损失非常严重. 在弹性变形协调的部分滑移状态下,接触表面损伤轻微,而由塑性变形协调的部分滑移状态下,接触中心出现较大切应力,疲劳裂纹扩展至接触表面导致材料剥落,接触边缘有磨粒磨损和氧化磨损的痕迹. 在完全滑移状态下,接触表面损伤主要为疲劳剥层,磨粒磨损和氧化磨损.      

考虑粗糙结合面弹塑性接触的黏滑摩擦建模

提出一种同时考虑粗糙面上微凸体弹性变形和塑性接触的切向黏滑摩擦建模方法。采用Hertz弹性理论和Mindlin解描述弹性接触微凸体的切向载荷和相对变形的关系;采用AF(Abbott-Firstone)塑性理论和Fujimoto模型描述塑性接触微凸体切向载荷和相对变形的关系。再利用GW(Greenwood-Williamson)模型统计分析方法建立粗糙表面切向载荷和相对变形之间的关系。将模型与仅考虑微凸体弹性接触情况的模型进行对比,并研究了不同塑性指数对切向载荷和相对变形关系的影响。结果表明：与完全弹性接触模型相比,本文模型引入了塑性接触理论,能够更好地描述粗糙表面切向载荷和相对变形关系,并且考虑不同接触条件下弹性变形微凸体和塑性变形微凸体对切向接触载荷的贡献,在微滑移阶段,主要由弹性接触变形影响,而在进入宏观滑移阶段之后,切向行为主要由塑性变形影响。界面切向载荷由黏着和滑移接触作用共同决定,随着切向变形的增加,滑移接触力逐渐增加,而黏着接触力先增加后减少,反映了界面由微滑移逐渐向宏滑移演化的过程。随着塑性指数的增加,粗糙面上发生塑性接触的微凸体数目逐渐增加,切向黏滑行为主要受到塑性接触特征的控制。     

基于双折线抗力模型的空爆荷载梁式构件振动位移研究

为研究双折线抗力模型对空爆荷载梁式构件振动位移的影响,提出了柔性、刚性两类梁式构件正向弹塑性振动及回弹阶段弹塑性振动的分析法。应用等效单自由度法建立了各阶段振动方程并依据不同的初始条件推导出了各阶段的理论解。采用此理论解和代表性塑性强化系数,开展了双折线抗力模型中不同塑性强化程度对两类梁式构件正向弹塑性振动及回弹阶段弹塑性振动位移的典型工况验证。研究结果表明:基于双折线抗力模型位移理论解的适用范围更广;随着双折线抗力模型塑性强化系数的增大,两类梁式构件的最大弹塑性位移、残余变形均逐渐减小,且残余变形降低程度高于最大弹塑性位移;塑性强化系数增大到一定程度,梁式构件回弹阶段将出现塑性振动位移,进一步降低残余变形,无塑性回弹位移的理想弹塑性抗力模型会高估空爆荷载下梁式构件的残余变形。     

A theoretical model of collision between soft-spheres with Hertz elastic loading and nonlinear plastic unloading

This paper presents a theoretical model on the normal (head-on) collision between soft-spheres on the basis of elastic loading of the Hertz contact for compression process and a nonlinear plastic unloading for restitution one, in which the parameters all are determined in terms of the material and geometric ones of the spheres, and the behaviors of perfect elastic, inelastic, and perfect plastic collisions appeared in the classical mechanics are fully described once a value of coefficient of restitution is specified in the region of 0 ≤ ε ≤ 1. After an empirical formula of the coefficient of restitution dependent on the impact velocity is suggested to fit the existing experimental measurements by means of the least square method, the predictions of the dependency and the collision duration are in well quantitative agreement with their experimental measurements. It is found that the measurable quantities are dependent on both the impact velocity and the parameters of spheres. Following this model, finally, an approach to determine the spring coefficient in the linear viscoelastic model of the collision is also displayed. These results obtained here will be significantly beneficial for the applications where a collision model is requested in the simulations of relevant grain flows and impact dynamics etc..     

Delamination of an elastic film from an elastic–plastic substrate during adhesive contact loading and unloading

Adhesive contact between a rigid sphere and an elastic film on an elastic–perfectly plastic substrate was examined in the context of finite element simulation results. Surface adhesion was modeled by nonlinear springs obeying a force-displacement relationship governed by the Lennard–Jones potential. A bilinear cohesive zone law with prescribed cohesive strength and work of adhesion was used to simulate crack initiation and growth at the film/substrate interface. It is shown that the unloading response consists of five sequential stages: elastic recovery, interface damage (crack) initiation, damage evolution (delamination), film elastic bending, and abrupt surface separation (jump-out), with plastic deformation in the substrate occurring only during damage initiation. Substrate plasticity produces partial closure of the cohesive zone upon full unloading (jump-out), residual tensile stresses at the front of the crack tip, and irreversible downward bending of the elastic film. Finite element simulations illustrate the effects of minimum surface separation (i.e., maximum compressive surface force), work of adhesion and cohesive strength of the film/substrate interface, substrate yield strength, and initial crack size on the evolution of the surface force, residual deflection of the elastic film, film-substrate separation (debonding), crack-tip opening displacement, and contact instabilities (jump-in and jump-out) during a full load–unload cycle. The results of this study provide insight into the interdependence of contact instabilities and interfacial damage (cracking) encountered in layered media during adhesive contact loading and unloading.     

Dynamic coupling of fluid and structural mechanics for simulating particle motion and interaction in high speed compressible gas particle laden flow

In this work, structural finite element analyses of particles moving and interacting within high speed compressible flow are directly coupled to computational fluid dynamics and heat transfer analyses to provide more detailed and improved simulations of particle laden flow under these operating conditions. For a given solid material model, stresses and displacements throughout the solid body are determined with the particle–particle contact following an element to element local spring force model and local fluid induced forces directly calculated from the finite volume flow solution. Plasticity and particle deformation common in such a flow regime can be incorporated in a more rigorous manner than typical discrete element models where structural conditions are not directly modeled. Using the developed techniques, simulations of normal collisions between two 1 mm radius particles with initial particle velocities of 50–150 m/s are conducted with different levels of pressure driven gas flow moving normal to the initial particle motion for elastic and elastic–plastic with strain hardening based solid material models. In this manner, the relationships between the collision velocity, the material behavior models, and the fluid flow and the particle motion and deformation can be investigated. The elastic–plastic material behavior results in post collision velocities 16–50% of their pre-collision values while the elastic-based particle collisions nearly regained their initial velocity upon rebound. The elastic–plastic material models produce contact forces less than half of those for elastic collisions, longer contact times, and greater particle deformation. Fluid flow forces affect the particle motion even at high collision speeds regardless of the solid material behavior model. With the elastic models, the collision force varied little with the strength of the gas flow driver. For the elastic–plastic models, the larger particle deformation and the resulting increasingly asymmetric loading lead to growing differences in the collision force magnitudes and directions as the gas flow strength increased. The coupled finite volume flow and finite element structural analyses provide a capability to capture the interdependencies between the interaction of the particles, the particle deformation, the fluid flow and the particle motion.     

机械结合面切向接触阻尼计算模型

针对两粗糙表面在法向力和切向力共同作用下相互接触时结合面切向阻尼的问题进行了研究。首先,根据KE模型对单个微凸体在弹性、弹塑性、塑性变形阶段的切向接触行为进行了分析,获得了微凸体在三个变形阶段的黏滑特性;然后,基于GW统计模型建立了一种在微凸体法向弹性、弹塑性和塑性变形机制基础上,考虑微凸体黏滑摩擦行为的机械结合面切向接触阻尼统计模型;最后,分别讨论了机械结合面的法向预载荷、切向激振频率和切向动态位移幅值对机械结合面切向阻尼的影响。研究表明:结合面切向接触阻尼系数随着结合面法向载荷的增大而增大,随着切向激振频率和切向动态位移幅值的增大而减小;在高频率、大幅值下,结合面切向接触阻尼系数几乎与动态位移幅值和激振频率无关。为了验证模型的准确性,构建了动态切向力作用下的结合面切向阻尼试验,其试验结果与理论仿真变化规律与量级基本一致,从而证明了本文所提出的切向阻尼模型的有效性。      

多体系统碰撞动力学中接触力模型的研究进展

接触碰撞行为作为大自然与多体系统中的常见现象,其接触力模型对于多体系统的碰撞行为机理研究与性能预测至关重要.静态弹塑性接触模型与考虑能量耗散的连续接触力模型是研究接触碰撞行为的两类不同方法,在多体系统碰撞动力学中存在诸多共性与差异.本文分别从上述两类接触模型的发展历程入手,详细介绍了两类模型的区别与联系.首先,根据阻尼项分母中是否含有初始碰撞速度将连续接触力模型分为黏性接触力模型与迟滞接触力模型,讨论了能量指数与Hertz接触刚度之间的关系,阐述了现有连续接触力模型在计算弹塑性材料接触碰撞行为时存在的问题.其次,着重介绍了分段连续的准静态弹塑性接触力模型(可连续从完全弹性转换到完全塑性接触阶段),分析了利用此类弹塑性接触力模型计算碰撞行为的技术特点.同时,以恢复系数为桥梁和借助线性化的弹塑性接触刚度,避免了Hertz刚度对弹塑性接触刚度的计算误差,根据碰撞前后多体系统的能量与动能守恒推导了弹塑性接触模型等效的迟滞阻尼因子.探索了连续接触力模型与准静态弹塑性接触力模型之间的内在联系,数值计算结果定量说明了人为阻尼项代表的能量耗散与弹塑性接触力模型中加卸载路径代表的能量耗散具有等效性.另外...     

基于Iwan模型的接合面切向响应建模

针对工程中常见预紧力作用下的搭接接头,研究其在小幅切向位移激励时的切向位移响应问题,为此提出一种新的基于实际表面形貌和材料性能参数的滑移力密度分布函数.应用该分布函数得到搭接接头切向响应本构模型,并获得单位加载周期内的迟滞曲线和能量耗散值,通过与已出版的实验结果相对比,发现得到的模拟值与实验结果吻合,证明该模型的合理性.在此基础上利用该分布函数研究了接合面切向位移与切向力、切向接触刚度及能量耗散之间的关系,结果表明:建立的模型能很好地描述接合面间切向力与切向位移之间的关系,临界滑移力函数开始迅速上升,到达最大值后迅速收敛到零;切线力与切向位移之间表现出非线性特性,随着切向位移的增大,切向接触刚度表现出"软化"现象;初始切向刚度与法向载荷、粗糙度参数及塑性指数有关,对于确定的接触表面,法向力越大,初始切向刚度越大;初始切向刚度同样也随着塑性指数的增大而增大.     

单峰接触研究及其在分形表面接触中的应用

基于有限元方法,建立了弹塑性单峰的接触模型.粗糙峰为理想的弹塑性材料,为了考虑不同的材料特性对微凸体变形的影响,分别对9种不同的材料进行了分析.根据有限元计算结果,分析了接触面积,平均接触压力和接触力与变形干涉量之间的关系,并进行了经验公式的拟合.单峰接触所经历的4个不同的阶段,以及不同阶段之间的转化点均作了明确的表达.然后,根据分形理论,将单峰接触模型扩展到了三维的粗糙表面的接触,并提出了一个计算接触表面法向刚度的模型.通过与实验数据和以往模型的结果对比,证明本文中所提出的模型具有较高的精度.     

Hydrodynamic lubrication in fully plastic asperity contacts

A line contact inlet zone analysis is carried out for the hydrodynamic lubrication in a fully plastic asperity contact. A governing equation of the central film thickness i.e. the film thickness in the fully plastic contact area is derived. An equation predicting this film thickness is also derived. It is found that for the fully plastic contact, under relatively light loads the prediction accuracy for the central film thickness is good, while at the load heavy enough the prediction equation greatly overestimates the central film thickness and the central film thickness solved from the analytical governing equation is significantly low showing the asperity in boundary layer lubrication. For the fully plastic contact, the central film thickness is nearly half of that obtained based on the elastic contact assumption for relatively light loads or even lower for heavier loads. The hydrodynamic lubrication is found difficult to form in the fully plastic asperity contact for the carried load heavy enough or the significantly low sliding speed between the asperities. To achieve a high hydrodynamic lubrication film thickness in the fully plastic asperity contact it is recommended to employ a high sliding speed or a high fluid viscosity. However, in the fully plastic asperity contact, the potential hydrodynamic load-carrying capacity is limited and much smaller than that based on the elastic contact assumption or predicted by conventional line contact elasto-hydrodynamic lubrication theory.     

Influence of penetration depth and mechanical properties on contact radius determination for spherical indentation

Knowledge of the relationship between the penetration depth and the contact radius is required in order to determine the mechanical properties of a material starting from an instrumented indentation test. The aim of this work is to propose a new penetration depth–contact radius relationship valid for most metals which are deformed plastically by parabolic and spherical indenters. Numerical simulation results of the indentation of an elastic–plastic half-space by a frictionless rigid paraboloïd of revolution show that the contact radius–indentation depth relationship can be represented by a power law, which depends on the reduced Young’s modulus of the contact, on the strain hardening exponent and on the yield stress of the indented material. In order to use the proposed formulation for experimental spherical indentations, adaptation of the model is performed in the case of a rigid spherical indenter. Compared to the previous formulations, the model proposed in the present study for spherical indentation has the advantage of being accurate in the plastic regime for a large range of contact radii and for materials of well-developed yield stress. Lastly, a simple criterion, depending on the material mechanical properties, is proposed in order to know when piling-up appears for the spherical indentation.     

Anomalous behavior revisited: Dynamic response of elastic-plastic structures

Based on the minimum principle of acceleration in the elastic-plastic continua under finite deformation, the dynamic response of an elastic-perfectly plastic pin-ended beam subjected to rectangular impulse loading is studied with the help of a numerical approach. The calculated results once again show the anomalous behavior of the beam during its response process, which was previously found in [1]. By carefully analyzing the instantaneous distribution of the bending moment, the membrane force, the curvature and displacement during the response process, it is concluded that the interactive effect between the geometry and materials nonlinearities of the structure is the key reason for leading to the anomalous behavior. This will be helpful for clarifying some misunderstandings in explaining the problem before. Project supported by the National Natural Science Foundation of China.