Rigid viscoplastic finite element analysis of the gas-pressure constrained bulging of superplastic circular sheets into cone disk shape dies

This paper describes the development of a rigid viscoplastic finite element formulation for analysing the gas-pressure constrained bulging processes of superplastic circular sheets in cone disk shape dies. In this formulation, the effects of strain hardening and strain-rate sensitivity of materials are included, and the boundary friction condition is introduced into the formulation in the form of friction functional. The finiteelement model based on the membrane theory is developed, and then applied to simulate superplastic constrained bulging processes. The solutions by the rigid viscoplastic finite element method are compared with existing experimental data. The influences of the geometrical parameters of the dies, the friction factor in the friction functional, the strain hardening and the strain-rate sensitivity on the inhomogeneity of thickness distribution are studied in detail.     

Wave propagation in an elastic/viscoplastic hollow sphere

An analysis of the dynamic problem of a hollow sphere is presented. The medium is assumed elastic/viscoplastic, satisfying Mises condition, isotropic hardening and viscoplastic incompressibility. The cavity is subjected to a radially applied and continuously maintained impact load. The analysis for viscoplastic waves is based on the method of characteristics and a generalized form of Malvern's theory for strain-rate dependent materials. A bilinear shear stress-shear strain curve is considered and calculations are carried out on the IBM 7094 computer. The paper examines the influence of the important parameters governing spherical wave propagation on the response of the sphere; these parameters include the level of strain hardening, the viscosity coefficient, Poisson's ratio and the geometry of the sphere. It is shown that the degree of strain hardening has a significant effect on the amplitude of the oscillations with the period unaffected. The viscosity coefficient becomes more influential as the wall thickness of the sphere increases.     

球压痕试验法评价金属材料的强度

通过单次压痕试验与有限元模拟相结合的方法,结合反向分析方法与模拟退火粒子群算法,从获得的载荷-深度曲线加载部分提取材料的塑性参数,基于Ludwig硬化模型预测了不同金属材料的强度,并与单轴拉伸试验结果进行对比。结果表明:模拟得到的载荷-深度曲线与试验得到的几乎重合,二者的相对误差小于0.5%,说明模拟退火粒子群算法可有效地从压痕载荷-深度曲线中提取出金属材料的塑性参数;基于Ludwig硬化模型,利用反向分析方法从压痕载荷-深度曲线中提取的真应力-真塑性应变曲线不是唯一的,但从真应力-真塑性应变曲线计算得到的强度具有明显的收敛趋势;采用压痕试验得到不同金属材料的强度均接近于由拉伸试验得到的,屈服强度与抗拉强度的最大相对误差分别为5.9%,4.3%,说明采用压痕试验法可以准确地评价金属材料的强度。     

Material characterization of Inconel 718 from free bulging test at high temperature

Macroscopic superplastic behavior of metallic or non metallic materials is usually represented by the strain-rate sensitivity, and it can be determined by tensile tests in uniaxial stress state and bulging tests in multi axial stress state, which is the actual hot forming process. And macroscopic behavior of Non-SPF grade materials could be described in a similar way as that of superplastic materials, including strain hardening, cavity and so on. In this study, the material characterization of non-SPF grade Inconel 718 has been carried out to determine the material parameters for flow stress throughout free bulging test under constant temperature. The measured height of bulged plate during the test was used for estimation of strain-rate sensitivity, strain-hardening index and cavity volume fraction with the help of numerical analysis. The bulged height obtained from the simulation showed good agreement with the experimental findings. The effects of strain-hardening and cavity volume fraction factor for flow stress were also compared.     

Hardening laws, surface roughness and biaxial tensile limit strains of sheet aluminium alloys

The roles of hardening laws and surface roughness have been assessed in the prediction of biaxial tensile limit strains of two A1 alloy sheet materials with different strain hardening characteristics namely AA6111-T4 and AA5754-O, utilizing the surface roughness model proposed by Parmar, Mellor and Chakrabarty [6]. In the work of Parmar et al., the predictions of limit strains were based on the Marciniak—Kuczynski inhomogeneity analysis and utilized the commonly used power hardening law (Hollomon equation) to describe the stress—strain behavior of the material. In the present work, (i) the suitability of a Voce hardening law, (ii) the effect of surface roughness parameters and (iii) the effect of grain size parameters on the prediction of biaxial limit strains has been studied. The biaxial limit strains based on Voce equation were obtained by modifying the set of equations of Parmar et al. and utilizing the experimentally measured surface roughness in 3-D, grain size parameters and stress—strain curves from uniaxial tensile and hydraulic bulge tests for the two A1 sheet materials. The predictions from Voce and Hollomon equations have been compared with the experimental forming limits determined by hemispherical punch stretching of gridded blanks. The discrepancy between predictions from Holloman equation and experiments is small for the low strain hardening AA6111-T4 material but is quite significant for the high strain hardening AA5754-O material. Further, the predictions are also strongly dependent upon the measure of surface roughness and the grain size utilized in the calculations. The results indicate good predictions of limit strains for the two alloys when (i) stress—strain data from tensile or hydraulic bulge tests are fitted to a Voce equation and (ii) half of the maximum peaks-to-valley height and grain thickness are utilized as a measure of surface roughness and grain size respectively. The results are discussed in the context of the characteristics of the hardening laws, assumptions of surface roughness model and surface and grain characteristics of the alloys studied.     

Deformation analysis of plastic strain rate hardening material using stress function under plane strain condition

The two-dimensional plane strain equation of plastic flow in accordance with the Levy-Mises constitutive relation is expressed in terms of stress functions of complex variables. Expressions for the stress, strain rate and velocity are derived for plastic flow in a non linear viscous medium assuming the stress function in the form of both the summation and product of conjugate stress functions. The plastic states are derived, also using a mixed mode solution expressed in terms of non-seperable, independent conjugate complex variables.The analysis of the block indentation associated with a nonlinear viscous (strain rate hardening) material under plane strain condition using the product form solution is performed. The effect of the variation in the strain-rate hardening exponent on the contact stress is investigated. The predicted behavior of the vertical component of the contact stress suggests the possibility of the development of a specially instrumented plane strain block indentation tests, for the rapid determination of the strain-rate sensitivity of the real material. The vertical contact stress and strain-rate obtained from the product of the complex conjugate stress function are compared with those obtained from the summation form of the complex conjugate stream function.     

An experimental investigation into load relaxation in aluminium (HE30TB) and mild steel (EN1A)

Short-time room-temperature tensile load relaxation tests were conducted on aluminium alloy (HE30TB) and mild steel (EN1A) specimens using a closed-loop, electrohydraulic servo-controlled testing machine under strain control by means of an extensometer mounted directly on the parallel section of the tested specimens. The relaxation periods (usually 60 sec.) were interruptions at chosen points in constant strain rate tensile loadings. The loading strain rates in the different tests varied from 5 × 10⁻⁴ to 10⁻² sec. The transient relaxation behaviour was investigated for the purpose of testing the applicability of the most widely assumed viscoplastic constitutive models. This was achieved by comparing the plastic strain rate just after the beginning of load relaxation at constant total strain to the plastic strain rate during the tensile loading just before the start of the relaxation interval. All common viscoplastic theories predict that the plastic strain rate ratio should be unity. The experimental results for both materials indicate, however, that the plastic strain rate ratio varies from almost zero for relaxation periods early in the loading, to a maximum of around 0·2 for some relaxation periods beginning at relatively high loads and strains. This agrees with previously reported results on pure aluminium, which is not very rate sensitive, but the results for the more rate-sensitive mild steel may be surprising. Only if the actual relaxation rate drops by a factor of about 100 in 0.2 sec, could the findings of this experimental programme be reconciled with predictions of the usual viscoplastic theories. The experimental programme also included constant strain-rate tests at several rates and jump tests, in which the rate was switched back and forth between 10⁻⁴ and 10⁻²/sec.     

On the measurement of friction coefficient utilizing the ring compression test

The main objective of this research was to investigate whether generalized friction calibration curves, as recommended in the literature for use with ring compression tests, are applicable to all types of materials and test conditions. Specifically, the effects of material properties, strain-rate sensitivity, and “barreling” on the behavior of friction calibration curves were investigated. To this end, a series of ring compression tests were conducted in order to determine the magnitude of the friction coefficient, μ, as well as the corresponding calibration curves for two types of modeling materials, white and black Plasticine. The experiments were first conducted using the Physical Modeling Technique (PMT) and then simulated via an elastic–plastic finite element code (ABAQUS). In contrast to the results available in the literature, where the same friction calibration curves are recommended for all types of materials and test conditions, the results of this investigation showed that friction calibration curves are indeed affected by the material properties and test conditions and every material possesses its own distinctive friction calibration curve.     

Identification of material parameters in constitutive model for sheet metals from cyclic bending tests

This paper deals with the identification of material parameters in a constitutive model for sheet metals using the bending moment versus curvature diagrams obtained by cyclic bending tests. The model can describe the cyclic strain hardening by the isotropic hardening and the Bauschinger effect by the kinematic hardening. An optimization technique based on the iterative multipoint approximation concept was used for the identification of the material parameters. This paper describes the experimentation, the fundamentals and the technique of the identification problem, and the verification of this approach.     

Cyclic loading of beams based on the Prager and Frederick–Armstrong kinematic hardening models

The kinematic hardening theory of plasticity based on the Prager and Frederick–Armstrong models are used to evaluate the cyclic loading behavior of a beam under the axial, bending, and thermal loads. The beam material is assumed to follow non-linear strain hardening property. The material's strain hardening curves in tension and compression are assumed to be both identical for the isotropic material and different for the anisotropic material. A numerical iterative method is used to calculate the stresses and plastic strains in the beam due to cyclic loadings. The results of the analysis are checked with the known experimental tests. It is concluded that the Prager kinematic hardening theory under deformation controlled conditions, excluding creep, results into reversed plasticity. The load controlled cyclic loading under the Prager kinematic hardening model with isotropy assumption results into reversed plasticity. Under anisotropy assumption of tension/compression curve, this model predicts ratcheting. On the other hand, the Frederick–Armstrong model predicts ratcheting behavior of the beam under load controlled cyclic loading with non-zero mean load. This model predicts reversed plasticity under the load controlled cyclic loading with zero mean load, and deformation controlled cyclic loading.     

Containment of explosions in cylindrical shells

An experimental and theoretical investigation of the high explosive containment ability of circular cylindrical shells is presented. Approximate expressions for final circumferential strain or radial displacement as a function of cylinder length are developed in terms of elementary functions, based upon an assumed rigid-plastic material relation for the containment vessel. The material relation includes strain hardening and approximate strain-rate sensitivity. The expressions are presented in a form convenient for containment design purposes and are shown to be in reasonable agreement with experimental results for several container materials and radius-to-thickness ratios.     

The influence of strain hardening of polymers on the piling-up phenomenon in scratch tests: Experiments and numerical modelling

The aim of this study was to relate the scratching behaviour of polymers to their mechanical properties. A thermosetting resin (CR39) and a thermoplastic polymer (PMMA) were studied using a microscratch tester allowing in situ observation of the contact area. These two polymers exhibit different elastic and viscoplastic properties, the main difference being the large ability of CR39 to strain harden, whereas PMMA softens. A spherical indenter was used to vary the level of deformation imposed on the samples. The response was initially elastic, then viscoelastic and finally mainly viscoplastic with increasing penetration of the indenter into the material. The two polymers displayed the same response for small levels of deformation, while at larger strains PMMA showed more pronounced plastic behaviour. The origin of this difference in behaviour was investigated by means of a three dimensional finite element analysis. The rheology of PMMA and CR39 was simplified and modelled by assuming linear elastic behaviour and a viscoplastic law taking into account their strain hardening capacity at high strains. Strain hardening was found to be a key factor to correctly model the material flow around the indenter. The response of the polymers was governed by the ratio between the plastic and elastic strains involved in the deformation in the contact region. In first approximation, the representative strain was imposed mainly by the geometry of the indenter, while the elastic deformation was controlled by the mechanical properties of the material, a larger strain hardening leading to a greater elastic deformation and a lower plastic strain thus a better scratch resistance of the specimen.     

Mechanical characterization at intermediate strain rates for rate effects on an epoxy syntactic foam

An intermediate strain-rate mechanical testing technique was developed through proper modifications of a hydraulically driven loading frame (MTS 810) and a split Hopkinson pressure bar (SHPB). The modified MTS and SHPB were used to obtain valid stress–strain data for an epoxy syntactic foam at intermediate strain rates in the order from 10⁻¹ to 10² s⁻¹. Additionally, lower and higher strain-rate characterization of the foam material was conducted, such that the compressive stress–strain data of the syntactic epoxy foam were obtained at strain rates from 0.005 to 2150 s⁻¹ without any gap in the intermediate strain-rate range. The syntactic epoxy foam exhibited nonlinear strain-rate dependency of failure strength.     

GH2132高温高应变率下力学性能分析与Johnson-Cook本构模型的建立

对铁基高温合金GH2132进行了准静态压缩试验和分离式霍普金森压杆(SHPB)试验,获得了该材料在不同温度和应变率下的应力应变曲线,分析了其力学行为.GH2132在准静态压缩过程中出现加工硬化且没有明显的屈服阶段.在SHPB试验中,GH2132有明显的温度软化效应,当应变率在4000～8000 s-1之间时表现出应变率...     

Die design optimization for axisymmetric hot extrusion of metal matrix composites

Strain rate sensitive materials such as Ti alloys, superplastic materials and metal matrix composites (MMCs) can be deformed only in very narrow range of strain rate. In this work, a new process design method for controlling strain rate in workpiece during hot extrusion process is proposed. In this approach, a coupled numerical approach of finite element analysis and optimization technique to optimal profiled die which yields more uniform strain rate distribution in the deforming region is applied to the hot extrusion process of MMCs. Extrusion die profiles are defined by Bezier curves, and FPS (flexible polyhedron search) method is used as optimization technique. The change of relative deviation of strain rate, the progressive development of die profiles with increase of iteration for optimization and the corresponding strain rate distributions are investigated. In addition, the die profiles by optimization scheme for different extrusion ratios are compared with those by analytical solution.     

高应变率下Cu-P/M摩擦材料正向和反向应变率效应

研究了冲击载荷下铜基粉末冶金(Cu P/M)摩擦材料不同的应变率效应。试验在分离式Hopkinson压杆 (SHPB)上完成。应变率范围为:102/s～103/s。通过试验得到了该材料的动态应力应变曲线,发现该材料在应变率 1000/s以下,表现为应变率强化效应;在应变率1000/s以上,表现为应变率弱化效应。也就是说,应变率1000/s是 该材料的临界应变率。为了与静态时的情况比较,在MTS试验机上又做了10-4/s～10-3/s应变率范围内的准静态 实验。比较动静态试验结果,发现动态时的屈服极限大于静态的;而屈服后的应变硬化率是静态大于动态的。通过 对样品进行微观组织分析,发现在压制烧结时有硬质颗粒破碎。在冲击载荷下材料内部的损伤演化形成大范围的 多源裂纹及孔洞分布群导致裂纹迅速扩展,同时伴随硬质颗粒破碎。     

Influence of material parameters on the hydrostatic bulging of a circular diaphragm

The plastic bulging of pressurized circular membranes is examined with particular attention to the effect of material parameters on the inherent inhomogeneity of the test. A rigid-viscoplastic behaviour based on flow-theory of plasticity is considered for materials with transversely isotropic properties. The numerical calculations display the effects of strain-hardening, strain-rate hardening and yield surface shape on the bulging pressure, the strain-distribution and the bulge shape. Comparisons with available experimental data are also presented.     

Micro–macro simulation of sintering process by coupling Monte Carlo and finite element methods

A micro–macro method for simulating a sintering process of ceramic powder compacts based on the Monte Carlo and finite element methods is proposed. Macroscopic non-uniform shrinkage during the sintering is calculated by the viscoplastic finite element method. In the microscopic approach using the Monte Carlo method, powder particles and pores among the particles are divided into many cells, and the growth of grains in the particles and the disappearance of pores are simulated by means of the Potts model.The shrinkage strain rate required as a materials constant in the macroscopic method is calculated by the microscopic approach. The microscopic and macroscopic approaches are coupled by exchanging microscopic and macroscopic results in each finite element step. In the Monte Carlo method, the effect of macroscopic viscoplastic deformation on the microstructural change is taken into consideration by including viscoplastic strain rate calculated by the finite element method in the disappearance frequency of pore cells. The shrinkage behaviour in the sintering of circular two-layer compacts is simulated by the proposed micro–macro method.     

Plastic deformation analysis of strain-rate sensitive materials under plane strain conditions

The two-dimensional plane strain equation of plastic flow in accordance with the Levy-Mises constitutive relation is expressed in terms of stream functions of complex variables. Expressions for the stress, strain-rate and velocity are derived, assuming the stream function in the forms of both the summation and product of conjugate flow functions, for plastic flow in a nonlinear viscous (strain-rate sensitive) medium. The plastic states are also derived using a mixed mode solution expressed in terms of non-separable, independent conjugate complex variables. Application of the summation form solution is illustrated through the block indentation problem. Calculations are made on the effect of variation of the strain-rate sensitivity exponent on the contact stress. The predicted behavior of the contact stress suggests the possibility of the development of a specially instrumented plane strain block indentation test for the rapid determination of the strain-rate sensitivity of real materials. By reducing the results of the indentation of a perfectly plastic material it is found that the contact stress is uniform and the external load is constant. The stress on the contact surface obtained using the present analysis is identical to that available from a slip line solution to the problem.