Rigid-plastic Finite Element Method Research Articles

Finite element analysis of deformation behaviour of metals during equal channel multi-angular pressing

In the present paper, the plastic deformation behaviour of materials during equal channel multi-angular pressing (ECMAP) of routes A and C, with a theoretical total strain of ≈2 upon a single pass, was studied using the two dimensional rigid-plastic finite element method. The ECMAP process is described in terms of the two stages based on the die filling procedure. Numerical simulation for annealed pure copper showed that the first corner gap that occurred during stage I was not formed during stage II owing to the back pressure developed due to the second intersection deformation. The back pressure effect results in the local severe shear at the bottom of the ECMAP sample for the processing routes A and C. While the overall shear deformation is more intensive in the route A sample and is lessened in the reverse direction in the route C sample due to the difference in the shear direction of the first and the second shear, the effective strain along the width direction for the route A processed workpiece is more homogeneous than that of the route C processed workpiece. The results suggest the possibility of obtaining homogeneous deformation resulting in isotropic samples by controlling the deforming route.

An Investigation of the Process Parameters for Calibre Rolling Using a Complex Analytical Model

Calibre rolling is a complex 3D boundary problem. Several analytical methods were applied to analyse this topic and some progress was achieved. However, it was difficult to include both analytical precision and CPU time saving in the calibre rolling analysis. In this paper, the simplicity of the 3D slab method and the precision of 3D rigid-plastic finite element method were combined to develop a complex analytical model. This model can rapidly simulate calibre rolling and the results obtained match well with the experimental ones. A complex analytical model can be used for discussing the influence of each process parameter on calibre rolling, which can assist the efficient process design of calibre rolling and find the optimal combination of each process parameter.

Theory and experimental research on the process of rolling shape slab

The rolling process of shape slab is simulated numerically by the rigid–plastic finite element method (RPFEM). The constraint relationship between velocity components of nodal points on the side sloping interface is discussed. The spread at side edges, the metal filled degree at ribs, separating force and rolling torque are obtained and found to be in good agreement with experimental values and the distributions of equivalent strain rate and metal flow are also calculated.

레이디얼-전방압출 공정의 성형특성에 관한 연구

This study is concerned with the analysis of the forming characteristics of radial-forward extrusion. Angle between radial and forward extrusion, gap height, and friction factor are considered as important design factors to affect forming characteristics in radial-forward extrusion. The rigid-plastic finite element method is adopted to analyze the effects of design factors on forming loads. The incremental rates of loads are nearly constant except the deformation zone from radial to forward extrusion. The smaller angle induces lesser force increment, therefore forming load increases as the angle increases. Maximum load also increases as gap-height decreases and friction factor increases.

Analysis of the non-steady state vertical–horizontal rolling process in roughing trains by the three-dimensional finite element method

The vertical–horizontal rolling process is often used to accomplish width reduction so as to provide a synchronising operation between the continuous slab casting and hot rolling processes. Numerical simulation of the non-steady state deformation behaviour around the head and tail ends during this process is made by the full three-dimensional rigid–plastic finite element method. An explanation is provided in the theory for the ‘thin element technique’ at the inlet surface of velocity discontinuity. To deal with the interpolation of friction within a surface of an element contacting partly with the roll, a new term, so-called ‘pseudo shape function’, is presented and a related new equation formula is deduced. The calculated shape of the slab edge, the separating force and the rolling torque are consistent with those measured experimental ones for the model material lead.

Analysis of stress distribution in caliber rolling by the energy method using finite-element division

The authors previously proposed an energy method using finite-element division and applied it to the analysis of the flow of material. In this work, the authors develop this method further to obtain the stress distribution in caliber rolling. Recently, Tomita obtained the stress distribution using the grid method so that the equilibrium condition may be satisfied approximately, whereas Mori evaluated the accuracy of the result obtained by the rigid-plastic finite-element method, by considering the equivalent nodal force which is assumed to be in equilibrium. Hence, the authors combined these methods and applied them to determine the stress distribution. The following results were obtained. First, in the flat rolling of bars, the roll force agrees well with the theoretical equation proposed by Saito. Second, in square-diamond rolling and round-oval rolling, the roll force depends on the ratio of the indirectly reduced area in the cross-section to the area of the cross-section. Finally, in square-oval rolling, the roll force agrees with the theoretical equation proposed by Saito when the projected material-roll contact length is replaced by the mean projected material-roll contact length in the width direction.

Analysis and design of hydroforming processes by the rigid–plastic finite element method

A three-dimensional finite element program, HydroForm-3D, based on the rigid–plastic model for the analysis and design of hydroforming process was developed. Several typical hydroforming processes such as tee extrusion, cross-extrusion, the hydroforming process combined with the pre-bent process and subframe are analyzed by using this numerical program. The hydraulic pressure force is applied to the normal direction of the tube workpiece by integrating the pressure with respect to each element’s surface area. The predicted simulation results provide information on the actual design. This numerical approach to the hydroforming process will be useful to the development and application for a wide range of automobile structural components.

Analyses of plane-strain upsetting by the rigid–plastic domain-boundary element method

In this paper, the rigid–plastic domain-boundary element method is formulated. This method does not need iterative calculations in any computing step. Therefore, the rigid–plastic domain-boundary element method possesses a merit in that there is no possibility of divergence of the iterative calculations and the computing costs are less than those of the rigid–plastic finite element methods which are a kind of iterative solution. On the other hand, the rigid–plastic domain-boundary element method is formulated with mixed variables which are nodal velocities and derivatives of nodal velocity. Therefore, nodal velocities and derivatives of nodal velocity can be calculated with the same precision, whilst the rigid–plastic finite element methods are formulated with single variables which are nodal velocities. Therefore, the rigid–plastic finite element methods cannot meet the need for compatibility of the nodal velocity’s derivative, and nodal velocities and derivatives of nodal velocity cannot be calculated with the same precision for these methods. In this paper a plane-strain metal forming (cold upsetting) problem is analyzed by the rigid–plastic domain-boundary element method. Contours of effective strain rate, effective strain and shear stress, etc. are obtained successfully.

The process sequence design of a power-assisted steering part

Conventional and new forging processes of a power-steering worm blank are analyzed by the rigid–plastic finite element method. The conventional process contains three stages such as indentation, extrusion and upsetting, and was designed by a forming equipment expert. Process conditions such as reduction in area, extrusion angle and upsetting ratio are considered to prevent internal or geometrical defects. The results of simulation of the conventional forging process are summarized in terms of deformation patterns, load–stroke relationships and die pressures for each forming operation. Based on the simulation results of the current three-stage process, a power-steering worm blank forging process for improving the conventional process sequence is designed. The die pressures and forming loads of the proposed process are within the limit value, which was set by experts and the proposed process is found to be suitable for manufacturing the power-steering worm blank.

The forming characteristics of radial–backward extrusion

The rigid–plastic finite element method is employed for the quantitative analysis of the radial extrusion process combined with backward extrusion. Various variables such as gap size, die corner radius and frictional conditions are adopted as design or process parameters for analysis. The emphasis is focused on the analysis of metal flow into a can and flange. Due to various die geometry and process conditions, the metal flow into a can and flange shows different patterns and its characteristics are well summarized quantitatively in this paper. Experimental data are compared with the simulation results in terms of forming load and the volume ratio of flange to can to verify the usefulness of simulation work. The comparison between experiment and simulation gives good agreement. Extensive simulation work leads to the quantitative relationships between the process conditions, volume ratio of the flange to can, and the size of the flange or can. As expected, the volume ratio of the flange to can increases as the gap size and/or die corner radius increase. However, it is found that the friction condition has little influence on the forming load and the deformation pattern. Under a certain condition of die geometry, can size turns out to be smaller than that of the initial billet. The work presented in this paper could be a good reference for process design in radial–backward extrusion.

The influence of die geometry on the radial extrusion processes

Three variants of radial extrusion are analyzed by simulation work. Case I involves forcing a cylindrical billet against a flat die. In case II, the upper punch forces a billet against a stationary punch recessed in the lower die. In case III, both the upper and lower punches move together toward the center of the billet. Major process parameters are identified as the relative gap height and the die corner radius in constant relative deformation. The simulation work is performed by the rigid–plastic finite element method. The validity of the modeling and simulation work for this process is verified by comparison with experimental data in terms of forming load. The simulation results show good agreement with the experimental data. Based on the simulation results, the different process parameters are related to the different material flow. Due to various die motions, a certain pattern in the material flow is shown in each deformation case. The die geometry has a significant influence on the material flow into the flange gap.

The forming characteristics of radial–forward extrusion

In this paper, the forming of a central hub by radial–forward extrusion is analyzed quantitatively by the rigid–plastic finite element method. Design factors such as mandrel diameter, die corner radius and friction factor are applied in the simulation. The influence of these design factors on the maximum load during the forming operation and the pressure exerted on the tooling such as the punch and the mandrel is investigated and summarized in terms of the maximum forming force. The simulation results are compared with experimental data from a reference to verify the usefulness of the simulation. Based on the simulation results, it is concluded that the maximum forming load increases as the tube diameter and/or die corner radius increase. Furthermore, the mandrel diameter among the design factors has the greatest effect on the forming load during the radial–forward extrusion process. However, the die corner radius has a slight influence on the punch load as is easily expected. The work presented in this paper might be used for basic data in the design of the radial–forward extrusion process.

3D rigid–plastic FEM numerical simulation on tube spinning

A finite element model suited to the nature of the process of tube spinning has been built and the deformation field of the process has been simulated by the 3D rigid–plastic FEM (finite element method) in this paper. The distributions of stress and strain rate of the deformation field are obtained. The results of the simulation can well explain the enlarged diameter, the reduced diameter, the reflection of the generatrices and wave, etc., and agree well with the experimental results.

A simulation of three-dimensional metal rolling processes by rigid–plastic finite element method

Using different frictional shear stress models, mesh division and number of elements in the deformation zone, this paper focuses on analysing the influence of friction variation on the convergence and results of simulation such as rolling pressure, forward slip and spread by 3D rigid–plastic FEM. The effects of mesh division and the number of elements on the precision, stability and convergence of the simulation are also discussed. This investigation shows that the frictional shear stress model with a variation in the deformation zone can provide satisfactory results that are in good agreement with experimental values, and are also more accurate than results from other methods. Suitable mesh division can improve the precision and effectiveness of the simulation. The results of the simulation are discussed for front and back tensions.

Finite element simulation of the cold forging process having a floating die

In this paper, a finite element technique of analyzing a forging process having a floating die is presented and its numerical characteristics are investigated in detail. The rigid-plastic finite element method is employed together with a search scheme of obtaining the proper floating die velocity that makes the floating die load to be less than the user-specified tolerance. An application example of an axisymmetric three-stage cold forging process having a floating die is given.

Implementation of backward tracing scheme of the FEM to blank design in sheet metal forming

Blank design is a critical part for net-shape sheet forming. This study applies the backward tracing scheme of the rigid–plastic finite element method to the three-dimensional blank design of sheet metal forming. Originally, the backward tracing scheme was developed for preform design in bulk forming, and applied to several forming processes successfully. Its key concept is to trace backward from the final desirable configuration to an intermediate preform or initial blank. A program for initial blank design in sheet forming which contains the capabilities of forward loading simulation and backward tracing simulation has been developed and proven in effectiveness by applying it to a square cup stamping process. In the blank design of square cup stamping, the backward tracing program can produce an appropriate blank configuration which forms a sound net-shape cup product without requiring machining after forming. The blank design has been confirmed by a series of experiments, since the application of the backward tracing scheme was the first trial in the blank design of sheet metal forming.

Slope Stability Analysis with Rigid-Plastic Finite Element Method

Rigid plastic finite element method was formulated for slope stability assessment by introducing the shear strength reduction coefficient to define the factor of safety used in slope stability. The framework of the proposed method was clarified in comparison with the limit equilibrium method based on slice method. The limit equilibrium method was shown to coincide with the upper bound theory in limit analysis and the process of minimizing the factor of safety for possible slip line was clarified to be important from the viewpoint of limit analysis. Rigid plastic FEM analysis has the advantages of (1) no need to assume the failure mode prior to analysis, (2) mathematical minimization of factor of safety, (3) use of constitutive equation and (4) taking account of dilatancy property of geo-material. The factor of safety was examined with simple slope for various conditions. It shows the good agreement with conventional methods and the applicability of proposed method was clarified. Case studies shows non-linear slip lines were well obtained corresponding to geometry and geology of slopes in the analysis. In some cases the sliding mass included several shearing zones inside and this fact suggests the existence of complicated failure mode not to be covered in the conventional methods.

Parallelization of Finite Element Analysis Using Broadcast on Workstation Cluster

In large-scale numerical simulations based on a rigid-plastic finite element method, most of the processing time is used in solving linear equations called stiffness equations. Therefore, the method of parallel processing is introduced to speed up the process of solving stiffness equations. The conjugate gradient method is adopted for solving the equations and is parallelized on a cluster of workstations. In the present paper, we present a new cluster of workstations which uses broadcasting to transmit data from a master processor to all slave processors, instead of unicasting. Broadcasting can transmit data to all slaves by one transmission regardless of the number of slaves. Therefore, the cluster of workstations with broadcasting can reduce communication time sharply compared with sequential data transmission by conventional unicasting. We achieved more than twelvefold increase in speed for solving the stiffness equations with twenty slave processors.

Optimization of Hot Free Forging Condition for the Uniformity of Forged Shape by Three Dimensional Rigid-plastic Finite Element Analysis.

Focusing on the rough forging stage of the hot free forging process, the influence of the operational conditions on the occurrence of concave defects was first numerically analyzed using a three dimensional rigid-plastic finite element method. The validity of the analysis was examined by laboratory experiments using lead billets and the analytical results are in good agreement with experimental results. As a result, the mechanism through which the concave defects appear was clarified, and it was revealed that by applying only a small curvature to a plane surface of an anvil, the concave defects is enormously improved. Moreover, it was found that the new anvil geometry makes it possible to increase the productivity.

A study on the corner filling in the drawing of a rectangular rod from a round bar

An important characteristic of the shaped drawing process, unlike the wire drawing process, is the corner filling which influences the dimensional accuracy of the product. In this study, therefore, in order to investigate the effect of process variables, such as the reduction in area, the semi-die angle, and the rectangular ratio to the corner filling, the drawings of a rectangular rod from a round bar have been simulated by using the three-dimensional rigid-plastic finite element method (FEM) and an artificial neural network has been introduced to reduce the number of simulations. To verify the results of the study, the experimental investigations were also carried out on real industrial products. According to the results, in the case of the irregular-shaped drawing process the main process variable on the corner filling is the combination of the semi-die angle but for the regular-shaped drawing process, the reduction in area has a significant effect on the corner filling.