Mathematical modeling of sintering during powder forming processes

This paper describes a study of densification induced by local capillary forces during compaction of powder based materials. A coupled sinter-compaction model with an internal state parameter was proposed. An internal state parameter was assumed as the sintering stresses on contact areas between powder particles. The mechanical model describing the plastic deformation during the P/M forging of a preform is based on the plasticity theory of porous metals. The numerical investigation of P/M forming processes is based on the rigid-plastic finite element model. A finite element program taking into account the sintering effect during P/M forming is created. A numerical example is considered.     

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.     

Simulation of microscopic shrinkage behaviour in sintering of powder compact

A method for simulating microscopic shrinkage behaviour of powder particles in sintering of a compact is proposed on the basis of the granular element method. In this method, the powder particles are modelled as many circular elements undergoing viscoplastic deformation due to surface tension during the sintering, and the microscopic shrinkage is calculated by equilibrating forces acting on the elements. The variation in shape of necks between the elements during the sintering is taken into consideration. Plane-strain shrinkage in sintering is calculated under regular and irregular dispositions of powder particles. In the regular disposition of powders having the same diameter, the obtained shrinkage behaviour is compared with the experimental one using glass rods and the calculated one by the viscoplastic finite-element simulation respectively. It is shown from the simulation of irregular disposition that the densification due to the sintering is accelerated by mixing powders having different diameters.     

Isostatic compaction of bimodal powder mixtures and composites

We study the effect of using a mixture of particles of two different sizes for the compaction of spherical metallic powders by discrete element method simulations (DEM). The study focuses on large size ratios (4 and 8) for which a wide range of volume fractions of large particles is investigated (from 0% to 80% volume). The bimodality of the powder is shown to affect all stages of powder compaction. The relative density of the powder when no significant load is applied (tap density) is a function of both the size ratio and the volume fraction of large particles. Furthermore, we show that bimodal compacts exhibit a stiffer response during isostatic plastic compaction compared to the monomodal case. The important practical application of a mixture of soft deformable particles (metallic) with hard particles (ceramic) is also studied. Following compaction, the unloading of the compact and the resulting tensile strength (green strength) are also shown to depend on the bimodality of the compact.     

Analysis of the forming process of sintered powder metals by a rigid-plastic finite-element method

A finite-element method for analyzing plastic deformation of sintered powder metals is developed. To give a theoretical basis, a variational principle for rigid-plastic deformation of porous metals is derived. The effect of the change in density during plastic deformation due to expansion or contraction of the internal pores is accommodated to the rigid-plastic finite-element method. As an example of analysis, plane-strain continuous rolling of sintered powder plates is treated. The distributions of density, strain, strain rate and stress in the plate and the pressure distribution over the roll surface are calculated. It is shown that the deformation behaviour of sintered powder metals is significantly influenced by the initial density.     

不同压制工艺对粉末冶金制品性能影响的有限元模拟

利用MSC/MARC有限元分析软件对金属粉末压制过程进行模拟分析.采用基于更新拉格朗日方法的热-力耦合分析不同工艺条件(温度、摩擦条件和压制方式等)对压坯性能的影响规律,同时对压制过程中的力学特征(压制力、脱模力、侧压力和应力分布等)进行分析.结果显示,摩擦条件是影响压坯密度大小及分布的关键因素,通过采用双向压制方法可以有效地改善单向压制压坯密度分布不均、差值较大的现象.温度的提高有利于提高粉末颗粒的塑性变形能力,但效果不明显.另外,由于温度影响润滑剂润滑性能,因此在制定压制工艺时需考虑温度对压坯性能的影响.此外温度的提高、摩擦的降低均有利于降低压制力,提高压坯密度均匀性,改善压坯应力集中现象.     

Numerical simulation of sheet metal stamping by using ODF data

An elasto-polycrystalline plastic finite element method is developed to simulate the deformation behavior and corresponding crystallographic texture evolution of sheet metals in stamping processes. A rate-independent crystal plasticity model and a degenerated shell element are introduced into the explicit finite element formulation. A successive integration method by which the shear rates on the slip systems can be calculated without prior determination of active/non-active state of the slip systems, as generally done in rate-dependent models, is employed to calculate the plastic strain rate of a crystal during the deformation. Representative crystal orientations are extracted from the orientation distribution function (ODF) data and assigned to the integration points of elements according to the distributional characteristic of crystal orientations. The macroscopic stress of a polycrystalline aggregate is evaluated by a volume-averaged response of the representative crystals. Two drawing processes are simulated, one with a cylindrical punch and the other with a square punch. Good agreements between the calculated results and experimental ones are obtained.     

Cold compaction of an array of cylindrical fibres

The biaxial compaction of a square array of circular cylinders is studied using slip-line field, upper bound and finite element methods. Densification is assumed to occur by plastic deformation at the contacts. It is found that contact–contact interaction has a softening effect on the local indentation pressure at each contact. The yield surfaces resulting from hydrostatic and closed-die compaction are constructed at various stages of densification: the size and shape of the yield surface depend upon the loading history and upon the relative density of the compact. The finite element predictions suggest the formation of a vertex at the loading point for the entire densification process in the case of isostatic compaction. On the other hand, a vertex at the loading point is formed only for a relative density D0.85 in the case of closed-die compaction.     

NUMERICAL SIMULATION FOR LASER BENDING OF SHEET METAL

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Cosserat continuum theory to simulate microscopic rotation of magnetic powder in applied magnetic field

Numerical method based on the Cosserat continuum theory is proposed for simulating behavior of a magnetic powder in an applied magnetic field. The Maxwell stress is induced in the magnetic powder. During powder forming process in the magnetic field, the magnetic particles are thus rotated and transferred by both mechanical and magnetic interaction. To simulate such powder behavior, we formulate a finite element equation considering Maxwell stress based on the Cosserat continuum theory of compressible plasticity. The powder behavior with magnetic alignment during compaction in magnetic field is simulated by the proposed method and the effect of couple-stress on the powder behavior is discussed.     

Hot backward extrusion comparative analyses by a combined finite element method

Deformation and temperature of hot backward extrusion are complex owing to interaction between deformation and temperature. In this paper, two- and one-way axisymmetric hot backward extrusion problems are analyzed by a combined finite element method, which consists of the volumetrically elastic and deviatorically rigid–plastic finite element method and the heat transfer finite element method. The volumetrically elastic and deviatorically rigid–plastic finite element method is different from the conventional rigid–plastic finite element methods, and has some merits in comparison with the conventional methods. Because contact surfaces between workpiece and tools of the one-way extrusion are different from those of the two-way one, the deformation and temperature of the one-way extrusion are different from those of the two-way one. Contours of effective strain rate, effective strain, temperature, effective stress and hydrostatic stress, as well as plots at different reductions for the two extrusions are obtained successfully. Differences of calculated results for the two extrusions can be clearly seen through comparative analyses.Because the bulk modulus is introduced into the volumetrically elastic and deviatorically rigid–plastic finite element method, influence of temperature on hydrostatic stress can be considered in this paper.     

Influence of the induced anisotropy in powder-based materials

Development of anisotropy, induced during processes of compaction of metal powders and plastic deformation of sintered metals is considered. A mechanical model, taking into account the induced anisotropy is proposed. The application of the finite element (FE) method for the proposed model is given and a FE code is created. Numerical simulation of compaction and of forging processes are given, showing the influence of the induced anisotropy on the relative density distribution and on the form of the end product.     

Consideration of geometric nonlinearity in rigid-plastic finite element formulation of continuum elements for large deformation

A rigid—plastic finite element formulation for the continuum elements employing the geometric nonlinearity during an incremental time step is presented. In sheet metal deformation, the displacement for each step is considerably large even though the effective strain increment is very small. For such large displacement problems, geometric nonlinearity must be considered. In the elastic—plastic finite element using continuum elements, general incremental formulations to include the geometric nonlinearity are available. However, in the conventional rigid—plastic finite element analysis using continuum, elements, the geometric nonlinearity has not been considered properly during an incremental time step. In this paper, in order to incorporate geometric nonlinearity to rigid—plastic continuum elements during a step, the convected coordinate system is introduced. To show the stability of strain distributions by the effect of geometric nonlinearity according to incremental step size, two sheet metal forming processes, stretching and deep drawing process, are analysed with various step sizes. Then the computed results using the derived equation are compared with those obtained without considering geometric nonlinearity.     

Experimental based finite element simulation of cold isostatic pressing of metal powders

The present work addresses the various ingredients required for reliable finite element simulations of cold isostatic pressing (CIP) of metal powders. A plastic constitutive model for finite deformation is presented and implemented into an explicit finite element (FE) code. The FE implementation is verified so that numerical errors (both temporal and spatial errors) are kept under control. Thereafter, uniaxial die compaction experiments are performed required for determining the material parameters in the constitutive model. Subsequently they are applied for the simulation of a “complex” CIP process. The experimental observations of the complex CIP process were used to validate the overall method by comparing the FE results (final dimensions and average relative density) to the experimental observations. The numerical results (final dimensions and relative density) are in good agreement with the experimental observations.     

多目标质量的粉末包套等通道挤扭成形致密工艺参数优化

剧烈塑性变形法—挤扭以剪切塑变为主变形方式,成形工艺复杂,影响因素很多,难以精确地建立工艺参数与成形质量之间的关系;选取优化合理的工艺参数匹配是材料尤其是粉末材料成形无破裂、塑性好、致密化程度高等成形质量的关键。以等通道横截面扭转角、螺旋角、摩擦因子、挤压速度、初始相对密度为自变量,正交试验为设计方案,对纯铝粉末烧结材料进行一道次包套等通道扭挤数值模拟,获得以等效塑性应变、静水压力、最大损伤值、相对密度数据,通过层次分析法计算多质量目标的权重,运用追踪点法和灰色系统理论的灰色关联度优化工艺参数,使设计目标值达到等效应变最大、静水压力最大、最大损伤值最小、相对密度最大。模拟和试验结果表明,运用多目标优化组合参数进行等通道挤扭成形能使纯铝粉末体材料迅速地形变累积,静水压力提高,损伤值显著地减小,致密效率高,晶粒明显细化,提高了材料综合质量。     

Collapse simulation of tubular structures using a finite element limit analysis approach and shell elements

This paper describes the collapse simulation of thin-walled tubular structures using a finite element limit analysis approach and degenerated four-node shell elements. The simulation traces the path of sequential deformation of the structure modelled by considering the strain-hardening effect, which is important for the analysis of collapse behaviour and energy absorption efficiency. The collapse analysis of some square tubes was used to verify the simulation method proposed. Numerical results are compared with experimental observations for sequential collapse loads and deformation modes, showing fairly good coincidence. The collapse analysis of an S-rail was then carried out for sequential collapse loads as well as deformation modes and its results are compared with elasto-plastic analysis results obtained from the explicit dynamic code PAM-CRASH. The energy absorption capacity was studied for a variety of rectangular cross-section aspect ratios. The results show that the energy absorption capacity increases as the height-to-width aspect ratio becomes larger. Results also demonstrate that the finite element limit analysis can predict the plastic collapse load and collapse mode of thin-walled structures efficiently and systematically. The present algorithm with a simple formulation has the advantage of stable convergence, computational efficiency and easy access to strain-hardening materials compared to the incremental rigid–plastic finite element analysis.     

Simulation of severe plastic deformation by finite element method with spatially fixed elements

A method for simulating large plastic deformation in metal forming processes is proposed. To avoid the difficulties arising from the distortion of elements in the rigid-plastic finite element method, the elements are defined with reference to an unchanging spatial grid rather than the workpiece, so that the elements are not changed by the deformation of material. The monitoring points embedded in the deforming material provide the information of the distortion of grid attached to the material and of distribution of flow stress. The sharp change of flow direction at the corner of tool is treated by the element with a singular point. Axi-symmetric backward extrusion of cans is simulated as an example of a process with severe deformation. The computed grid distortion, distribution of equivalent strain and working load are in good agreement with the experimental ones measured in extrusion of aluminium cans.     

A rigid plastic finite element perturbation method for the effective prediction of the influence of the change in parameters in the constitutive equation upon deformation behaviour

In order to predict the influence of change in parameters in the constitutive equation upon deformation behaviour, a rigid plastic finite element perturbation method is developed. The change of such field variables as velocity, pressure, forming force and so on are expressed in terms of a power series of the perturbation of the parameters. The coefficients of the series are determined by solving the rigid plastic finite element perturbation equations. Several examples are presented to illustrate the advantages of using the proposed method to solve specific problems in preference to the conventional rigid plastic finite element method.     

Modeling and estimation of deformation behavior of particle-reinforced metal–matrix composite

In order to estimate the characteristic feature of the deformation behavior of materials with a length scale, the strain gradient plasticity theories, corresponding variational principle and a finite element method are given. Then the finite element method is applied to the estimation of the mechanical characteristics of the particle reinforced metal–matrix composites modeled under plane strain conditions. The effects of the volume fraction, size and distribution pattern of the reinforcement particles on the macroscopic mechanical property of the composite are discussed. It has been clarified that the deformation resistance of the composite is substantially increased with decreasing particle size under a constant volume fraction of the reinforcement material. The main cause of the increase of the deformation resistance in the plastic range is the high strain gradient appearing in the matrix material, which increases with the reduction of the distance between particles.     

FEA for damping of structures having elastic bodies,viscoelastic bodies,porous media and gas

A numerical method is proposed to calculate damping properties for soundproof structures involving solid bodies, porous media and air in two-dimensional regions. Both effective density and bulk modulus have complex quantity to represent damped sound fields in the porous media. Particle displacements in the media are discretized using finite element method. For damped solid bodies, displacements are formulated using conventional finite elements including complex modulus of elasticity. Displacement vectors as common unknown variables are solved under coupled condition between solid bodies, porous media and gas. Further, by applying asymptotic method to complex eigenvalue problem, explicit expressions of modal loss factor for the mixed structures are derived. The proposed methods yield appropriate results for some typical problems and this method diminish computational time for large-scaled finite element models concerning the mixed structure. Moreover, it is found that damping can be coupled in the mixed structures between solid bodies, porous media and air.