Dynamic buckling of elastic–plastic square tubes under axial impact—II: structural response

Dynamic elastic–plastic buckling of thin-walled square tubes is studied from the viewpoint of elastic–plastic stress wave propagation, which originates from an axial impact loading. The influence of the impact velocity and the striking mass on the development of the buckling shape is discussed when considering the transient deformation process. It is shown that the maximum load, which results from a high velocity impact load and occurs at t=0, is a function of the impact velocity and is related to the speed of the elastic–plastic stress waves propagating along the tube. The predictions for the initiation of buckling based on a numerical simulation of the axial impact of strain rate insensitive square tubes using the FE code ABAQUS show good agreement with the results from experiments on aluminium alloy tubes impacted at various initial velocities. A comparison between the buckling initiation in square tubes and geometrically equivalent circular tubes reveals differences in the response, which are attributed to the stress wave propagation phenomena and to the structural differences between the two structures.     

Rebound behaviour of spheres for plastic impacts

This paper presents a study on the rebound behaviour of spheres impacted normally against a target wall using finite element methods. The emphasis is on the prediction of the coefficient of restitution and the effects of material properties and impact velocities on the rebound behaviour of the sphere. Finite deformation during plastic impact is addressed. The finite element results show that, for impacts of small plastic deformation, the coefficient of restitution is mainly dependent on the ratio of the impact velocity V_i to the yield velocity V_y which is consistent with those predicted by the theory of impact mechanics; while for impacts of finite-plastic-deformation it is also dependent on the ratio of the representative Young's Modulus E^* to the yield stress Y. The FEA results suggest that for impacts of finite-plastic-deformation the coefficient of restitution can be approximated to be proportional to [(V_i/V_y)/(E^*/Y)]^−1/2.     

确定轴向冲击下薄壁圆柱壳第二临界速度的新方法

研究薄壁圆柱壳的动态屈曲模式，有助于构造具有高吸能率的抗冲击结构。根据轴向冲击下的薄壁圆柱壳存在使其屈曲模式由轴对称转换为非轴对称的第二临界速度，且当冲击速度大于第二临界速度时薄壁圆柱壳的屈曲模式先呈现轴对称形式，然后随着冲击响应时间逐渐由轴对称形式转化为非轴对称形式这一理论，基于有限元仿真，比较撞击系统动能的时间历程和屈曲变形的时间历程，提出了用以确定第二临界速度的能量迭代法。应用此方法设计薄壁圆柱壳的动力屈曲结构可有效地减少试验次数，降低实验成本。该方法的可行性和正确性利用落锤实验得到了验证。     

Inertia effects in axisymmetrically deformed cylindrical shells under axial impact

The axisymmetric buckling of elastic–plastic cylindrical shells subjected to axial impact are studied using a finite element analysis. This study reveals that shells, subjected to axial impact, are both velocity and mass sensitive, so that larger energies can be absorbed by a shell for high-velocity impacts when decreasing the striking mass. It is shown that the inertia characteristics of the shell, together with the material properties, determine particular patterns of the axial stress wave propagation, thus, causing either dynamic plastic or dynamic progressive buckling to develop during the initial phase of the shell response. Domains of the load parameters, where the different buckling phenomena develop, are obtained for two particular shells. Strain rate effects are also considered when discussing the energy absorbing properties of the shells.     

Instability analysis in pressurized three-layered fiber-reinforced anisotropic rubber tubes in torsion

Instability analysis of the large deformation of three-layered fiber-reinforced anisotropic rubber tubes subjected to an inner pressure combined with torsion and axial stretch is examined within the framework of finite elasticity. Based on a fiber-reinforced incompressible hyper-elastic material model, the relation between the principal stretch and the inner pressure, torsion and axial stretch is obtained. It is shown that there exists a critical thickness for the tubes. For a thin-walled tube which is thinner than the critical thickness, the tube under-goes a uniform stable inflation when the inner pressure is small. However, the tube may under-go a strikingly non-uniform deformation when the inner pressure is larger than a certain critical value. One portion of the tube may become highly distended as a bubble and it is proved that this deformation is unstable. While a thick-walled tube which is thicker than the critical thickness, always under-goes a uniform inflation. The critical thickness of the tube to take unstable deformation is given based on the deformation curve and its slope. Furthermore, the effect of axial stretch, torsion and the distribution angle or the strength of the fiber on the deformation and the stress of the tube are discussed.     

Separation and velocity dependence of the drag force applied to a rigid ovoid of Rankine nosed projectile penetrating an elastic–perfectly-plastic target

Yarin et al. have developed an analytical solution for normal penetration of a rigid projectile of the shape of an ovoid of Rankine into an incompressible elastic–perfectly-plastic target. Here, the closed form expressions from this solution are used to analyze separation and velocity dependence of the drag force applied to the projectile. It is shown that for penetration velocities V below a critical value V_s, the target material maintains full contact with the projectile's surface, the drag force is due solely to the resistance of plastic flow in the target and it is independent of the velocity V. For V equal to V_s the separation point jumps from the projectile's tail to a point closer to its tip. For increasing values of V the separation point moves even further towards the projectile's tip causing a strong dependence of the drag force on velocity. In contrast, results of the cavity expansion model applied to this problem indicate that separation cannot occur and that the drag force depends on velocity for all values of V. These results emphasize the importance of accurately modeling the flow field around the projectile.     

径向冲击下复合材料层合圆柱壳的动力屈曲

采用Lagrange方程导出包含横向剪切变形和转动惯量的复合材料层合圆柱壳径向脉冲屈曲控制方程;用四阶Runge-Kuta方法对方程数值求解,寻找占优屈曲模态数及对应于允许初缺陷放大值时的临界冲击速度;通过计算碳/环氧材料角铺设层合圆柱壳,讨论了横向剪切变形、壳体几何尺寸、铺层角度等因素对层合圆柱壳动力屈曲的影响.     

Ultrasound-velocity measurement of strain in metallic polycrystals

During extension tests of metallic polycrystals, in situ measurement of ultrasound velocity (USV, V _US ) was carried out for the plastically deforming metallic polycrystalline specimens. USV has been found to depend on the plastic strain or acting stress of plastic flow. The complicated form of these dependencies is discussed in the context of the concept of evolution of lattice defects. V _US is shown to be associated with the multistage nature of the plastic flow curve. This allows plastic flow stages to be recognised with a higher degree of certainty. Thus the linear stage of work hardening has been separated for Al polycrystals by the procedure of USV measurement and certain peculiarities of plastic strains localisation as associated with USV variation have been investigated. Received: 2 April 2001 / Accepted: 24 May 2001     

Foreign object damage in disks of gas-turbine-grade silicon nitrides by steel ball projectiles at ambient temperature

Foreign object damage (FOD) behavior of two commercial gas-turbine-grade silicon nitrides, AS800, SN282, was determined at ambient temperature through postimpact strength testing of target disks impacted by steel ball projectiles with a diameter of 1.59 mm in a velocity range from 115 to 440 m/s. AS800 silicon nitride exhibited a greater FOD resistance than SN282, primarily due to its greater value of fracture toughness (K _Ic). The critical impact velocity V _c for which the corresponding postimpact strength was the lowest was V _c 440, 300 m/s for AS800, SN282, respectively. A unique lower strength regime was typified for both silicon nitrides depending on impact velocity, was attributed to significant radial cracking. The damage generated by projectile impact was typically in the form of ring, radial, cone cracks with their severity, combination being dependent on impact velocity. Unlike the thick (3 mm) flexure bar target specimens used in previous studies, the thin (2 mm) disk target specimens exhibited a unique back-side radial cracking on the reverse side just beneath the impact sites at, above impact velocities of 160 m/s for SN282, 220 m/s for AS800.     

受轴向冲击圆柱壳非对称屈曲分析

轴向冲击荷载作用下薄壁圆柱壳屈曲变形研究一直受人们关注,探讨其动态响应的特征和机理,不仅可以丰富冲击屈曲研究内容,而且为提高结构的抗冲击能力提供理论基础。众多实验现象表明,圆柱壳在轴向冲击荷载作用下非对称屈曲时截面为规则几何形状,且不只三角形一种模式,也不是随机现象,而具有一定规律性。文章对环向截面屈曲耗能计算进行理论推导,通过分析对比屈曲耗能与折叠边数、圆柱壳半径的关系,根据最小耗能原理,证实变形规律存在,并分析得到,随半径增加圆柱壳非对称屈曲由三角形模式向多边形模式发展,最终有转变为轴对称变形模式的可能。     

History effect in fatigue crack growth under mixed-mode loading conditions

Plastic deformation within the crack tip region introduces internal stresses that modify subsequent behaviour of the crack and are at the origin of history effects in fatigue crack growth. Consequently, fatigue crack growth models should include plasticity-induced history effects. A model was developed and validated for mode I fatigue crack growth under variable amplitude loading conditions. The purpose of this study was to extend this model to mixed-mode loading conditions. Finite element analyses are commonly employed to model crack tip plasticity and were shown to give very satisfactory results. However, if millions of cycles need to be modelled to predict the fatigue behaviour of an industrial component, the finite element method becomes computationally too expensive. By employing a multiscale approach, the local results of FE computations can be brought to the global scale. This approach consists of partitioning the velocity field at the crack tip into plastic and elastic parts. Each part is partitioned into mode I and mode II components, and finally each component is the product of a reference spatial field and an intensity factor. The intensity factor of the mode I and mode II plastic parts of the velocity fields, denoted by dρ_I/dt and dρ_II/dt, allow measuring mixed-mode plasticity in the crack tip region at the global scale. Evolutions of dρ_I/dt and dρ_II/dt, generated using the FE method for various loading histories, enable the identification of an empirical cyclic elastic–plastic constitutive model for the crack tip region at the global scale. Once identified, this empirical model can be employed, with no need of additional FE computations, resulting in faster computations. With the additional hypothesis that the fatigue crack growth rate and direction can be determined from mixed-mode crack tip plasticity (dρ_I/dt and dρ_II/dt), it becomes possible to predict fatigue crack growth under I/II mixed-mode and variable amplitude loading conditions. To compare the predictions of this model with experiments, an asymmetric four point bend test system was setup. It allows applying any mixed-mode loading case from a pure mode I condition to a pure mode II. Initial experimental results showed an increase of the mode I fatigue crack growth rate after the application of a set of mode II overload cycles.     

Generalised critical free-surface flows

Nonlinear waves in a forced channel flow are considered. The forcing is due to a bottom obstruction. The study is restricted to steady flows. A weakly nonlinear analysis shows that for a given obstruction, there are two important values of the Froude number, which is the ratio of the upstream uniform velocity to the critical speed of shallow water waves, F _C>1 and F _L<1 such that: (i) when F<F _L, there is a unique downstream cnoidal wave matched with the upstream (subcritical) uniform flow; (ii) when F=F _L, the period of the cnoidal wave extends to infinity and the solution becomes a hydraulic fall (conjugate flow solution) – the flow is subcritical upstream and supercritical downstream; (iii) when F>F _C, there are two symmetric solitary waves sustained over the site of forcing, and at F=F _C the two solitary waves merge into one; (iv) when F>F _C, there is also a one-parameter family of solutions matching the upstream (supercritical) uniform flow with a cnoidal wave downstream; (v) for a particular value of F>F _C, the downstream wave can be eliminated and the solution becomes a reversed hydraulic fall (it is the same as solution (ii), except that the flow is reversed!). Flows of type (iv), including the hydraulic fall (v) as a special case, are computed here using the full Euler equations. The problem is solved numerically by a boundary-integral-equation method due to Forbes and Schwartz. It is confirmed that there is a three-parameter family of solutions with a train of waves downstream. The three parameters can be chosen as the Froude number, the obstruction size and the wavelength of the downstream waves. This three-parameter family differs from the classical two-parameter family of subcritical flows (i) but includes as a particular case the hydraulic falls (ii) or equivalently (v) computed by Forbes.     

Impact of nonlocal thin elastic plates by a cylindrical projectile

This paper is concerned with the analysis of the problem of dynamic impact of an elastic thin plate by a cylindrical projectile. The behavior of the plate material is assumed to be nonlocal elastic, and the effect of the impact is represented by a uniform velocity distribution over a circular region of the plate surface. Assuming the plate is thin, only the contributions of vertical shearing stress are considered, and the expressions of shear stress, axial displacement and the corresponding velocity components are obtained. Finally, the value of the total strain energy—crack initiation energy—for which the plastic flow will start, has been calculated and compared with experiments.     

Estimation of a crater volume formed by impact of a projectile on a metallic target

The crater volume has been an important factor in ballistics and has many influences such as material strength, initial projectile velocity, angle of incidence, and nose shape. The goal of this research is to predict the resulting crater volume of a long rod penetration based on the initial projectile velocity and mass. Mooney’s (Bull Seism Soc Am 64(2):473, 1974) displacement equations were used to calculate the elastic crater volume for a given impulse force, P, varying as a delta function in time on the surface of an isotropic, semi-infinite solid. This estimated elastic volume, V_elastic is linearly related to the experimental ballistic volume, V_experimental by an “energy dissipation factor”, k. V_elastic = kV_experimental. The energy dissipation factor “lumps” together elastic and plastic deformation mechanisms. Terminal ballistic data for a steel long-rod projectile into “semiinfinite” steel or aluminum target will be compared to the crater volume calculated through the use of k.     

Plastic deformation behaviors and mechanical properties of rolled rings of 20CrMnTi alloy in combined radial and axial ring rolling

In this paper, a new combined radial and axial ring rolling process is proposed, in which a large increase in both the ring diameter and height can be achieved. Using the finite element (FE) method, the plastic deformation behaviors and mechanical properties of the rolled rings of 20CrMnTi alloy in combined radial and axial ring rolling are numerically investigated. It is found that under different axial rolling ratio λ_a and radial rolling ratio λ_r, there are three kinds of plastic deformation behaviors during the process. The first one is that when λ_a = 1 and λ_r > 1, the ring produces the deformation of thickness reduction and diameter expansion and its height basically remains unchanged. The second one is that when λ_a > 1 and λ_r > 1, the ring produces the deformation of thickness reduction, diameter expansion and height increase. The third one is that when λ_a > 1 and λ_r = 1, the ring produces the deformation of thickness reduction, outer diameter constancy and height increase. Owing to these plastic deformation behaviors, the axial and circumferential strain distribution of the rolled rings with the same final geometry is different. To evaluate the mechanical properties of these rolled rings, the compression and tensile tests have been carried out numerically and it is found that the rolled rings with the same final geometry may have different mechanical properties due to the different axial and circumferential strain distribution and the proposed combined radial and axial ring rolling process can thus be adopted to manufacture the rings with the same final geometry but different mechanical properties.     

Stationary Waves in a Supersonic Flow of a Two-Component Bose Gas

A stationary wave pattern occurring in a flow of a two-component Bose-Einstein condensate past an obstacle is studied. We consider the general case of unequal velocities of two superfluid components. The Landau criterium applied to the two-component system determines a certain region in the velocity space in which superfluidity may take place. Stationary waves arise out of this region, but under the additional condition that the relative velocity of the components does not exceed some critical value. Under increase of the relative velocity the spectrum of the excitations becomes complex valued and the stationary wave pattern is broken. In case of equal velocities two sets of stationary waves that correspond to the lower and the upper Bogolyubov mode can arise. If one component flows and the other is at rest only one set of waves may emerge. Two or even three interfere sets of waves may arise if the velocities approximately of equal value and the angle between the velocities is close to π/2. In two latter cases the stationary waves correspond to the lower mode and the densities of the components oscillate out-of-phase. The ratio of amplitudes of the components in the stationary waves is computed. This quantity depends on the relative velocity, is different for different sets of waves, and varies along the crests of the waves. For the cases where two or three waves interfere the density images are obtained.     

Investigation of stress corrosion cracking of low-alloy steel in water

For four low-alloy steels with a wide range of tensile strength, the dynamical processes of the nucleation and propagation of stress corrosion cracking (SCC) in water with various polarization conditions and in a 0.1 N K₂Cr₂O₇ solution were traced with an optical microscope. The results show that if the tensile strength of the steel is higher than a critical value that is different in different polarization conditions and K_IK_ISCC, the plastic zone in front of a loaded crack tip is enlarged with time i.e. the delayed plastic deformation occurs in all the environments used. The nucleation and propagation of SCR will proceed when this delayed plastic deformation develops to a critical condition. Neither anodic and cathodic polarization nor the inhibitor can change the feature of the delayed plasticity and the nucleation and propagation of SCC in water.In all the environments used, K_ISCC is increased and $\text{da}\text{dt}$ is decreased with the decrease in the strength of the steel. K_ISCC is increased and da/dt is decreased with the anodic polarization and the addition of the inhibitor, but the cathodic polarization has the opposite effect.     

Vibrations of an infinitely long flowing liquid jet—Influence of spin and axial flow

For an infinitely long liquid column the influence of axial flow velocity and spin has been investigated. The results are exhibited for axisymmetric mode m=0 and asymmetric modes m=1 and 2. A frictionless liquid shows with the increase of axial flow an increase of the frequency in flow direction and a decrease of the oscillation frequency in the opposite flow direction for axisymmetric motion. It also means that a larger surface tension, larger diameter or larger density of the liquid column exhibit the same behavior. For asymmetric motions the effect of axial velocity w₀ is the opposite. With increasing axial wave length the natural frequencies decrease. At certain axial speed magnitudes both waves move in flow direction with different magnitude. The effect of increasing spin is a decrease of natural frequencies and an instability for smaller axial wave lengths. Viscous effects show usually smaller oscillation frequencies.     

Dynamic response of an FGM cylindrical shell under moving loads

This paper presents an analytical study on the dynamic behavior of the infinitely-long, FGM cylindrical shell subjected to combined action of the axial tension, internal compressive load and ring-shaped compressive pressure with constant velocity. It is assumed that the cylindrical shell is a mixture of metal and ceramic that its properties changes as a function of the shell thickness. The problem is studied on the basis of the theory of vibrations of cylindrical shells. Derived formulas for the maximum static and dynamic displacements, dynamic factors and critical velocity for the FGM cylindrical shell subjected to moving loads. Numerical calculations have been made for fully metal, fully ceramic and FGM (Si₃N₄/SUS304) cylindrical shells. A parametric study is conducted to demonstrate the effects of the material property gradient, the radius to thickness ratio and the velocity of the moving load on the dynamic displacements and dynamic factors of the inner and ring-shaped pressures for FGM cylindrical shells.