Pressure-shear waves in 6061-T6 aluminum and alpha-titanium

Thepressure-shear plate impact technique is used to study material behavior at high rates of deformation. In this technique, plastic waves of combined pressure and shear stresses are produced by impact of parallel plates skewed relative to their direction of approach. Commercially pure alpha-titanium and 6061-T6 aluminum are tested under a variety of pressure and shear tractions by using different combinations of impact velocities and angles of inclination. A laser interferometer system is used to monitor simultaneously the normal and transverse components of motion of a point at the rear surface of the target plate. The experimental results are compared with numerical solutions based on an elastic/viscoplastic model of the material. Both isotropic and kinematic strain hardening models are used in the computations. The results indicate that unlike the normal velocity profiles, the transverse velocity profiles are sensitive to the dynamic plastic response and, thus, can be used to study material behavior at high strain rates. For the materials tested the results suggest that the flow stress required for plastic straining increases markedly with increasing strain rate at strain rates above 10⁴s^?1. Hydrostatic pressure of the order that exists in the tests (up to 2 GPa) does not affect the plastic flow in 6061-T6 aluminum and appears to have at most a minor effect on the deformation of the titanium.     

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.     

受轴向冲击薄壁圆管的几何畸变相似律研究

对于受轴向冲击载荷作用的薄壁圆管动态响应的相似律问题,由于圆管的薄壁特性导致厚度无法与高度和半径按相同的比例进行结构缩放,从而产生模型的几何畸变,此时传统的相似律已无法描述原型与畸变模型之间的动态响应规律。基于薄壁圆管轴向冲击问题的控制方程,通过能量守恒和量纲分析,推导了考虑几何畸变条件下轴向冲击载荷作用的理想弹塑性薄壁圆管动态响应的相似律。通过在给定应变与应变率区间上建立比例模型预测的流动屈服应力与原型流动屈服应力的最佳逼近关系,将几何畸变相似律进一步推广至包含应变率和应变硬化的材料。通过数值方法验证了提出的几何畸变模型相似律的适用性。分析结果表明,提出的考虑厚度畸变的受轴向冲击薄壁圆管的相似律可用于预测原型结构的冲击动态响应,并显著降低比例模型与原型结构平均载荷和能量的偏差。     

Statistical aspects of the identification of material parameters for elasto-plastic models

Summary The presented method to identify material parameters for inelastic deformation laws is based on the numerical analysis of inhomogeneous stress and strain fields received from suitable experiments. Tensile and bending tests were carried out to obtain elastic and hardening parameters. The deformation law for small elasto-plastic strains is presented as a system of nonlinear differential and algebraic equations (DAE) consisting of the stress–strain relation, evolution equations for the internal variables and the yield condition. Different rules for the evolution equations of isotropic, kinematic and distorsional hardening are proposed. The DAE are discretized using an implicit Euler method, and the resulting system of nonlinear algebraic equations is solved using the Newton method. Deterministic optimization procedures are preferred to identify material parameters from a least-squares functional of numerical and measured comparative quantities. The gradient of the objective function was calculated using a semianalytical sensitivity analysis. Due to measurement errors, the optimal sets of material parameters are non unique. The approximate estimation of confidence regions and the calculation of correlation coefficients is presented. The results of several optimization processes for material parameters of elasto-plastic deformation laws show a good agreement between measured and calculated values, but they show also problems which may occur if systematic errors will not be recognized and deleted. Received 30 September 1999; accepted for publication 8 March 2000     

应变率有关理论中Symonds模型的工程应用

讨论了在结构碰撞数值计算中遇到的应变率效应问题。进行碰撞过程模拟计算时,分别将静态弹塑性理论和应变率有关理论中的Ｓｙｍｏｎｄｓ模型引入材料的本构关系。     

摩擦材料在无围压和加围压条件下的形变比较

研究铜基粉末冶金摩擦材料在无围压和加围压条件下的动态力学特性及其形变特征。试验在Hopkinson压杆上完成,通过试验发现,材料在无围压、应变率低于1000-1s时,有应变率强化效应;在更高应变率下,材料有损伤软化效应。微观分析可以看到试件上大量的平行滑移裂纹导致了材料破坏,形变是滑移伴随少量的孪晶;裂缝中有大量的碳纤维组织,这些纤维状组织对裂纹的扩展起抑制作用。而材料在加围压下的性能得到了很大的改善,同一应变率下最大应力、屈服极限和动态杨氏模量均有提高。在高应变率时,无论是加围压还是无围压,破碎的硬质颗粒都是长条状的。由此得出推论,球状、细化的硬质颗粒,可提高基体的强度。     

Void growth and coalescence in ductile solids with stage III and stage IV strain hardening

State of the art ductile fracture models often rely on simple power laws to describe the strain hardening of the matrix material. Power laws do not distinguish between the two main stages of hardening observed in polycrystals, referred to as stage III and stage IV hardening, and which emerge from the evolution of the dislocation substructure. The aim of this study is to couple a physics based strain hardening law including these two stages to a micromechanics based ductile damage model. One of the main motivations is that, the stage IV constant hardening rate stage, occurring only at large strain, will be attained in most ductile failure problems if not at the overall level of deformation, at least locally around the growing voids. Furthermore, proper modelling of the stage III involving dislocation storage and recovery terms and the transition to stage IV provides a link with the underlying physical mechanisms of deformation and with the microstructure. First, in order to evaluate the effects of the stage III and stage IV hardening on void growth and coalescence, an extensive parametric study is performed on two-dimensional (2D) axisymmetric finite element (FE) unit cell calculations, using a Kocks-Mecking type hardening law. The cell calculations demonstrate that accounting for the stage IV hardening can have a profound effect on delaying void coalescence and increasing the ductility. The magnitude of the recovery term during stage III has also a significant effect on the void growth rate. Then, the Kocks-Mecking law is incorporated into the Gologanu-Leblond-Devaux (GLD) porous plasticity model supplemented by two different versions of the Thomason void coalescence criterion. The predictions of the damage model are in good agreement with the results of the FE calculations in terms of the stress-strain curves, the evolution of void shape and porosity, as well as the strain value at the onset of void coalescence.     

On a finite strain viscoplastic theory based on a new internal dissipation inequality

This work is focused on the theoretical development and numerical implementation of a viscoplastic law. According to the second law of thermodynamics a dissipation inequality described in the rotated material coordinate system is developed. Based on this dissipation inequality and the principle of maximum dissipation a finite strain viscoplastic model described also in the rotated material coordinate system is formulated. The evolution equations are expressed in terms of the material time derivatives of the rotated elastic logarithmic strain, the accumulated plastic strain and the strain-like tensor conjugate to the rotated back stress. The mathematical structure of this theory is concise and similar to that of the infinitesimal viscoplastic theory. These characteristics make the numerical implementation of this theory easy. The stress integration algorithm and the algorithmic tangent moduli for the infinitesimal theory can be applied to the numerical implementation of the present finite strain theory with a little reformulation. The complicated algorithmic formulations for most of other finite plastic laws can be therefore circumvented. In order to check the effectivity of the present finite strain theory a set of numerical examples under strict deformation conditions are presented. These numerical examples prove the excellent performance of the present viscoplastic material law at describing the finite strain elastoplastic and viscoplastic problems.     

Analysis of repeated impacts on a steel rod with visco-plastic material behavior

In machine dynamics impacts are usually common phenomena, resulting from collisions of moving bodies. Even low velocity impacts might produce high stresses in the contact region, which result in inelastic deformation. Thereby, visco-plastic materials, such as steel, show a significant increase of the yield stress with the strain rate. In machine dynamics repeated collisions occur, resulting in repeated impacts on a previously deformed contact area. Then, inelastic deformation and the resulting residual stresses produced by previous impacts have an influence on the behavior of the following impacts. Thus, the impact behavior varies with the number of impacts. This paper presents a numerical and experimental evaluation of repeated impacts with identical impact velocity up to 3 m/s, whereby the deformation history of the contact area, due to previous impacts, is included. The approach is applied to longitudinal impacts of an elastic steel sphere on a steel rod with distinct visco-plastic material behavior which is identified by Split Hopkinson Pressure Bar tests. A Finite Element analysis and experimental verification using two Laser-Doppler-Vibrometers are performed. It is shown that for an accurate impact simulation the FE model must include the visco-plastic material behavior of the steel. Further it is found that the maximal contact force, the rebound velocity and the coefficient of restitution increase with the number of impacts, while the contact duration decreases with the number of impacts. After several impacts these quantities show saturation to a constant value, indicating no significant additional inelastic deformation in the later impacts. Further, the residual stress distribution, the maximal von Mises stress distribution and the local deformation at the contact point are evaluated and a characteristic force-deformation diagram is obtained. Finally, an analysis is performed to describe the relation between maximal force and remaining crater at the contact point.     

Collisions and fractures: A predictive theory

We investigate collisions of solids which can fracture. Equations of motion and constitutive laws provide a predictive theory. Assuming the collision as instantaneous, the equations of motion are derived from the principle of virtual work introducing new interior forces which describe the very large stresses and the very large contact forces resulting from the kinematic incompatibilities. They are interior volume percussion stresses and interior surface percussions both on the unknown fractures and on the colliding surface. In order to approximate these equations, we assume solids are damageable. In this point of view, it results that velocity is continuous with respect to space but its strain rate is very large in a thin region where the material is completely damaged, so approximating a fracture. When the velocity before collision is very large, the damaged zone may be large accounting for parts of the solid completely transformed into powder. The constitutive laws result from dissipative functions satisfying the second law of thermodynamics and able to model the fracturation phenomenon at the macroscopic engineering level. Representative numerical examples confirm that the model accounts for the fracturation qualitative properties.     

Impulsive loading of ideal fibre-reinforced rigid-plastic beams—II Beam with end supports

The theory outlined in Part I is applied to the problem of a beam, supported at both ends, struck transversely at any point by a mass which subsequently adheres to the beam. For sufficiently long beams, the resulting moving discontinuities in slope and velocity do not propagate to the ends of the beam, and for this case the solution is obtained for the non-linear strain-hardening law used in Part I. A simpler approximate solution is derived for the case of low impact velocity and/or slight strain hardening. For shorter beams, the propagating discontinuities may undergo one or more successive reflections at the end points and at the point of initial impact. The deformation after such reflections is analysed, for linear strain-hardening, in the final two sections. Some particularly simple results are found in the case of central impact.     

爆炸荷载作用下多排钢管结构的大变形响应

采用八塑性铰变形模态,对爆炸荷载作用下多排钢管的压扁运动场进行了描述。基于虚速度原理,建立了多排钢管动力压扁模态解方程。由拉格朗日第二类方程确立初始条件,用差分法求解出各层钢管结构的传递荷载和压扁量大变形响应。计及应变率效应和材料硬化效应,计算值与试验结果相符。结果表明,采用模态法求解多排钢管的大变形动响应是可行的。     

The uniaxial tension of particulate composite materials with nonlinear interface debonding

Debonding of particle/matrix interfaces can significantly affect the macroscopic behavior of composite material. We have used a nonlinear cohesive law for particle/matrix interfaces to study interface debonding and its effect on particulate composite materials subject to uniaxial tension. The dilute solution shows that, at a fixed particle volume fraction, small particles lead to hardening behavior of the composite while large particles yield softening behavior. Interface debonding of large particles is unstable since the interface opening (and sliding) displacement(s) may have a sudden jump as the applied strain increases, which is called the catastrophic debonding. A simple estimate is given for the critical particle radius that separates the hardening and softening behavior of the composite.     

基于水平集-离散元方法的球谐函数颗粒材料缓冲性能分析

在自然环境与工业领域中,颗粒材料是一种常见的缓冲材料,其中大量形态各异的非球形颗粒表现出复杂的力学特性并应用于不同工程领域。本文采用球谐函数构造不同球面度和表面凹凸特性的非规则颗粒,通过水平集方法计算球谐函数颗粒间的接触点和碰撞力,并对冲击过程中球形和凹形颗粒的缓冲性能进行离散元分析。数值结果表明,颗粒床厚度、冲击速度和颗粒形状显著影响球谐函数颗粒材料的缓冲性能。颗粒床底部的冲击力峰值随着颗粒床厚度和表面凹凸性的增加而降低,同时冲击力峰值随着冲击速度和颗粒球面度的增加而增加。与球形颗粒相比,球谐函数颗粒具有凹凸表面和多接触点特性,这有利于冲击荷载向四周扩展并提高凹形颗粒的缓冲效果。     

Mode I crack tip field with strain gradient effects

A plane strain mode I crack tip field with strain gradient effects is investigated. A new strain gradient theory is used. An elastic-power law hardening strain gradient material is considered and two hardening laws, i. e. a separation law and an integration law are used respectively. As for the material with the separation law hardening, the angular distributions of stresses are consistent with the HRR field, which differs from the stress results^[19]; the angular distributions of couple stresses are the same as the couple stress results^[19]. For the material with the integration law hardening, the stress field and the couple stress field can not exist simultaneously, which is the same as the conclusion^[19], but for the stress dominated field, the angular distributions of stresses are consistent with the HRR field; for the couple stress dominated field, the angular distributions of couple stresses are consistent with those in Ref. [19]. However, the increase in stresses is not observed in strain gradient plasticity because the present theory is based on the rotation gradient of the deformation only, while the crack tip field of mode I is dominated by the tension gradient, which will be shown in another paper. Supported by the National Science Foundation of China (No. 19704100), Science Foundation of Chinese Academy of Sciences (Project KJ951-1-20), CAS K. C. Wong Post-doctoral Research Award Fund and the Post Doctoral Science Fund of China.     

Tension of a cylindrical membrane partially stretched over a rigid cylinder

We study a contact problem with friction for a hyperelastic long thin-walled tube. One end of the tube is placed over an immovable, rough, rigid cylinder and an axial force is applied to another end. We assume the deformation of the tube is finite and axisymmetric. The tube is modeled by a semi-infinity cylindrical membrane. The axial force tends to a constant value at large distances from the inclusion. The membrane is made of an incompressible, homogeneous, isotropic elastic material. A contact between the membrane and the rigid cylinder is with a dry friction. The membrane will not slide off the cylinder only by friction and at a sufficient contact area. The friction is described by Coulomb’s law. We study a minimum length of the membrane which is in contact with the rigid cylinder and is needed to hold the membrane on the rigid cylinder. We obtain an explicit solution for the Bartenev–Khazanovich (Varga) strain–energy function and numerical results for the Mooney–Rivlin and Fung models.     

On internal dissipation inequalities and finite strain inelastic constitutive laws: Theoretical and numerical comparisons

Internal dissipation always occurs in irreversible inelastic deformation processes of materials. The internal dissipation inequalities (specific mathematical forms of the second law of thermodynamics) determine the evolution direction of inelastic processes. Based on different internal dissipation inequalities several finite strain inelastic constitutive laws have been formulated for instance by Simo [Simo, J.C., 1992. Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory. Computer Methods in Applied Mechanics and Engineering 99, 61–112]; Simo and Miehe [Simo, J.C., Miehe, C., 1992. Associative coupled thermoplasticity at finite strains: formulation, numerical analysis and implementation. Computer Methods in Applied Mechanics and Engineering 98, 41–104]; Lion [Lion, A., 1997. A physically based method to represent the thermo-mechanical behavior of elastomers. Acta Mechanica 123, 1–25]; Reese and Govindjee [Reese, S., Govindjee, S., 1998. A theory of finite viscoelasticity and numerical aspects. International Journal of Solids and Structures 35, 3455–3482]; Lin and Schomburg [Lin, R.C., Schomburg, U., 2003. A finite elastic–viscoelastic–elastoplastic material law with damage: theoretical and numerical aspects. Computer Methods in Applied Mechanics and Engineering 192, 1591–1627]; Lin and Brocks [Lin, R.C., Brocks, W., 2004. On a finite strain viscoplastic theory based on a new internal dissipation inequality. International Journal of Plasticity 20, 1281–1311]; and Lin and Brocks [Lin, R.C., Brocks, W., 2005. An extended Chaboche’s viscoplastic law at finite strains: theoretical and numerical aspects. Journal of Materials Science and Technology 21, 145–147]. These constitutive laws are consistent with the second law of thermodynamics. As the internal dissipation inequalities are described in different configurations or coordinate systems, the related constitutive laws are also formulated in the corresponding configurations or coordinate systems. Mathematically, these constitutive laws have very different formulations. Now, a question is whether the constitutive laws provide identical constitutive responses for the same inelastic constitutive problems. In the present work, four types of finite strain viscoelastic and endochronically plastic laws as well as three types of J₂-plasticity laws are formulated based on four types of dissipation inequalities. Then, they are numerically compared for several problems of homogeneous or complex finite deformations. It is demonstrated that for the same inelastic constitutive problem the stress responses are identical for deformation processes without rotations. In the simple shear deformation process with large rotation, the presented viscoelastic and endochronically plastic laws also show almost identical stress responses up to a shear strain of about 100%. The three laws of J₂-plasticity also produce the same shear stresses up to a shear strain of 100%, while different normal stresses are generated even at infinitesimal shear strains. The three J₂-plasticity laws are also compared at three complex finite deformation processes: billet upsetting, cylinder necking and channel forming. For the first two deformation processes similar constitutive responses are obtained, whereas for the third deformation process (with large global rotations) significant differences of constitutive responses can be observed.     

On the application of the additive decomposition of generalized strain measures in large strain plasticity

In the present paper two thermodynamically consistent large strain plasticity models are examined and compared in finite simple shear. The first model (A) is based on the multiplicative decomposition of the deformation gradient, while the second one (B) on the additive decomposition of generalized strain measures. Both models are applied to a rigid-plastic material described by the von Mises-type yield criterion. Since both models include neither hardening nor softening law, a constant shear stress response even for large amounts of shear is expected. Indeed, the model A exhibits the true constant shear stress behavior independent of the elastic material law. In contrast, the model B leads to a spurious shear stress increase or drop such that its applicability under finite shear deformations may be questioned.     

应变硬化薄板夹持型滚弯成形过程分析

现存文献对薄板的连续滚弯成形过程的数学模拟多限于讨论三辊轮解锥型弯板过程,且基于较多的假定和板的小变形情况。而开发有更多的功能和更高的生产效率的四辊轮连续弯板机需研究夹持型滚弯成形过程。本文将「１」的工作,理想塑性材料平面应变薄板的夹持型连续滚弯成形过程的弹塑性大变形弯曲的力学分析,推广至一般应变硬化材料的情形。得出了控制微分方程的解析解。采用固支边界条件模拟刚度很大的平持系统对板的夹持。通过对线