Cup Drawing Process Research Articles

Formulation of Contact Problems in Sheet Metal Forming Simulation Using Local Interpolation for Tool Surfaces

Treatment of contact between a sheet and tools is one of the most difficult problems to deal with in finite-element simulations of the sheet metal forming processes. In order to obtain more accurate tool models without increasing the number of elements, this paper describes new formulations for contact problems using local interpolation proposed by Nagata for tool surfaces. A contact search algorithm between the sheet nodes and the interpolated tool surfaces and the consistent tangent stiffness matrix for the sliding nodes were developed. The developed algorithms were introduced into the static-explicit elastoplastic finite-element method code STAMP3D. Simulations of a square cup drawing process with a very coarsely discretized punch model were carried out. The simulated results showed that the proposed algorithms gave the proper deformation process, thus demonstrating the validity of the proposed formulations.

An improved model for predicting limiting drawing ratio

During the material selection and the tool design stage for a new component the knowledge of formability of the material under consideration is required. Limiting drawing ratio is one such measure. In the present work the model of Liu [D.K. Liu, The limiting drawing ratio for plastic instability of the cup-drawing process, J. Mater. Process. Technol. 86 (1999) 168–176] in which LDR is a function of material properties, coefficient of friction (taken into account in an approximate manner) and the die arc radius, is improved. The present model considers the bending and the unbending around the die arc radius. The friction effect is considered more realistically by taking into account the blank holding force (BHF). Since in the drawing operation the BHF should be just enough to prevent wrinkling, this is calculated using the expression of Siebel and Beisswanger [K. Lange, Handbook of Metal Forming, McGraw-Hill, 1985, p. 20.10]. The prediction from the present model is in good agreement with the experimental results and is an improvement over that predicted from the existing model of Liu [D.K. Liu, The limiting drawing ratio for plastic instability of the cup-drawing process, J. Mater. Process. Technol. 86 (1999) 168–176]. The effect of various material properties, blank holding force, friction coefficient and the die arc radius have been predicted in the comprehensive parametric studies.

Integration of RP and explicit dynamic FEM for the visualization of the sheet metal forming process

This paper develops a FORTRAN program to convert the explicit dynamic finite element method (FEM)-simulated deformed sheet to the stereolithography (STL) format used in the rapid prototyping (RP) apparatus. Such integration of the RP/FEM can rapidly produce a visualized 3D physical part of the sheet deformation state. Three cases – cylindrical drawing, bore expanding and square cup drawing processes, simulated by explicit dynamic FEM – were investigated to verify the integration system. The wrinkled flange in the cylindrical drawing process, the circle hole expansion in the bore expanding process, and the square cup in the square cup drawing were successfully predicted by explicit dynamic FEM, and the rapid prototyping 3D physical parts also showed good visualization of the deformed sheet for the above three cases. It proves that the integration system of RP/FEM will be able to supply a useful method for the visualization of the 3D physical part in the sheet metal forming process.

Plane stress yield function for aluminum alloy sheets—part II: FE formulation and its implementation

A recently proposed plane stress yield function [Yld2000-2d; Int. J. Plasticity 19 (2003) 1297—part I of this work] that well describes the anisotropic behavior of aluminum alloy sheets was implemented in a finite element code. A short review of the Yld2000-2d relevant features was provided and the complete formulation for the yield function implementation was proposed for its convenient use. Yield surface shapes, yield stress and r-value directionalities predicted with Yld2000-2d for Al–5 wt.% Mg and 6016-T4 alloy sheet samples were compared with those of previously suggested yield functions. Simulations of the cup drawing process for the Al–Mg binary sheet were performed to compute the cup height profiles (earing profiles) with different yield functions. The predicted profiles were compared to experimental data and it was shown that the simulation using Yld2000-2d led to the best agreement between theoretical and experimental results. The drawing and redrawing processes of a circular cup using the NUMISHEET'99 (Gelin, J.C., Picart, P., 1999. In: Gelin, J.C., Picart, P. (Eds.), Proceedings of NUMISHEET'99, Vol. 2. 13–17 September, Besancon, France) conference geometry and specifications for a 6111-T4 aluminum alloy sheet sample were also simulated. The predicted load-punch displacement curves and sheet thickness profiles along different radial directions of the cup were shown to be in excellent agreement with experimental data.

Hydraulic-pressure-enhanced cup-drawing processes—an appraisal

Hydraulic-pressure-enhanced deep drawing techniques generally increase the depth-to-diameter ratio possible in cup drawing. Draw ratios of up to 3.5 have been achieved by some of these processes. Phenomena being used in these processes include a peripheral push to aid the draw force, hydraulic pressure tightly holding the wall of the cup against the punch and fluid lubrication of the die radius and the flange of the cup being drawn. Various aspects including the principles underlying these are presented and discussed in this paper.

Construction of sliding constraint surfaces and initial guess shapes for intermediate steps in multi-step finite element inverse analysis

Multi-step inverse analysis has been developed for more accurate results while one-step analysis can affirm qualitative deformation behavior. An inverse finite element approach has more capability to design the optimum blank shape from the desired final shape with small amount of computation time and effort. In forming simulation with the inverse method, multi-step inverse analysis is difficult to apply because sliding constraint surfaces and initial guesses of intermediate steps described by complex die and punch set are hard to generate with present mapping schemes. In this paper, intermediate sliding constraint surfaces for multi-step finite element inverse analysis are automatically constructed from the die–punch set at each step with the given stroke. The initial guess of intermediate steps is constructed by a direct mesh mapping scheme that is suggested to generate an initial guess on the constructed sliding constraint surface. Construction of sliding constraint surfaces and initial guesses for intermediate steps is applied to a multi-step rectangular cup drawing process and an S-rail forming process. The numerical example of the multi-step rectangular cup drawing process demonstrates that the proposed mapping scheme is applicable to construct sliding constraint surfaces and initial guesses for analysis of a multi-step drawing process with large aspect ratio. The analysis result of the S-rail forming process is compared with that of a direct finite element analysis to evaluate the effectiveness of the mapping scheme proposed.

Design sensitivity analysis of sheet metal forming processes with a direct differentiation method

A design sensitivity analysis scheme is proposed for an elasto-plastic finite element analysis with explicit time integration using a direct differentiation method (DDM). The direct differentiation scheme proposed deals with large deformation, an elasto-plastic constitutive relation considering the planar anisotropy, shell elements with reduced integration and complicated contact between the sheet and the dies. The design sensitivities with respect to the process parameter are calculated with direct analytical differentiation of the governing equation. The present DDM result is compared with the result obtained from a finite difference method (FDM) in sheet metal forming processes such as a cylindrical cup drawing process and a U draw-bending process for the verification of its reliability, accuracy and versatility. DDM results show good agreement with FDM results. Results fully demonstrate that sensitivity calculation with FDM is sometimes very dangerous since it could lead to a fatal pitfall and selection of the sensitivity scheme should be carefully done for specific problems. The analysis shows that the sensitivity of state variables provides useful information not only for the strain control but also for the inspection of defect initiation such as fracture or wrinkle. The result demonstrates that the sensitivity calculation scheme proposed is applicable to analysis and design of complicated sheet metal forming.

Development of a one point quadrature shell element for nonlinear applications with contact and anisotropy

A general purpose shell element for nonlinear applications including sheet metal forming simulation is developed based on reduced integration with one point quadrature. The developed shell element has five degrees of freedom and four nodes. It covers flexible warping behavior without artificial warping correction. A physical stabilization scheme with the assumed natural strain method is employed to derive a strain field that can be decomposed into the sum of a constant and a linear term with respect to the natural coordinates. The rigid body projection is introduced to treat rigid body rotations effectively. The shell element incorporates elasto-plastic planar anisotropic material models based on the incremental deformation theory. Linear and nonlinear patch tests are performed and the results are compared with analytical or previously reported results. Simulations that include impact and deformable body contact are performed to show the robustness of the contact algorithm. Finally, to demonstrate the capability of handling anisotropic materials, the developed shell element is used for the circular cup drawing process simulation in order to predict the earing profile of Al 2008-T4 alloy sheet.

Modeling the Bauschinger effect for sheet metals, part II: applications

The modified Chaboche and ANK model proposed in Part I are implemented into second order accurate user material routines for ABAQUS/Explicit and /Standard. A criterion to detect a load reversal for ANK model is also proposed. They are applied to investigate the role of Bauschinger effect in the simulation of sheet metal forming and springback prediction under multi-cyclic loading conditions. Reverse cup drawing and draw-bead tests are simulated and results are compared with available experimental data. The ANK model predicted the maximum punch forces more accurately with a correct trend for the reverse cup drawing process. The draw-bead test simulation results with the ANK model show encouragingly good comparisons with measured data. An accurate modeling of the Bauschinger effect appears to be more important when multiple cycles of bending and reverse bending loading conditions are expected.

A finite element analysis of the flange earrings of strong anisotropic sheet metals in deep-drawing processes

Flange earrings of strong anisotropic sheet metals in deep-drawing process are numerically analyzed by the elastic-plastic large deformation finite element formulation based on a discrete Kirchhoff triangle plate shell element model. A Barlat-Lian anisotropic yield function and a quasi-flow corner theory are used in the present formulation. The numerical results are compared with the experimental ones of cylindrical cup drawing process. The focus of the present researches is on the numerical analysis and the constraining scheme of the flange earring of circular sheets with strong anisotropy in square cup drawing process.

The numerical analysis of anisotropic sheet metals in deep-drawing processes

Flange earing of strong anisotropic sheet metals in the deep-drawing process is analyzed numerically by the elastic–plastic large deformation finite element formulation based on the discrete Kirchhoff triangle plate shell element model. A Barlat–Lian anisotropy yield function and a quasi-flow corner theory are introduced in the present formulation. The numerical results are compared with the experimental results of the cylindrical cup drawing process. The focus of the present research is on the numerical analysis and the constraining scheme of the flange earing of circular sheets with strong anisotropy in the square cup drawing process.

Design modification in a multi-stage rectangular cup drawing process with a large aspect ratio by an elasto-plastic finite element analysis

Finite element analysis of multi-stage deep drawing processes is carried out for the process design of rectangular cup drawing with a large aspect ratio. The analysis incorporates shell elements for an elasto-plastic finite element method with an explicit time integration scheme using LS-DYNA3D. Simulation is performed for thorough investigation of failures such as tearing and wrinkling during the forming process. The analysis reveals that the difference of the drawing ratio within the cross-section produces non-uniform metal flow to cause wrinkling and severe extension. The irregular contact condition between the blank and the die also induces non-uniform metal flow. This paper identifies such an unfavorable mechanism in rectangular cup drawing and proposes a modification guideline in the design of the process and the tool shape. The analysis result confirms that the modified design not only improves the quality of a deep-drawn product but also reduces the possibility of failure.

Tool temperature control to increase the deep drawability of aluminum 1050 sheet

A feasibility study on the tool temperature control to increase the deep drawability of Al-1050 sheet is performed. The conventional deep drawing process is limited to a certain limit drawing ratio (LDR) beyond which failure will ensue. The purpose of this study is to examine the possibilities of relaxing the above limitation through the tool temperature control, aiming towards a process with an increased LDR. The idea which may lead to this goal is strengthening the punch-nose radius part by cold punch which has frequently been potential failure area in cup drawing process, while heating the remainder of the blank to reduce the stress on the cup sidewalls. Over the ranges of conditions investigated, the deep drawability of Al-1050 is found to be strongly sensitive to the temperature of the die and punch. The experimental implementation shows that the tool temperature control is very effective way to promote deep drawability of Al-1050.

Analysis of the hydromechanical deep drawing of cylindrical cups

The hydromechanical deep drawing process of aluminum cups and mild steel cups is analyzed experimentally and numerically. The effects of the process parameters on the final product quality are discussed. A working zone with a suitable maximum chamber pressure is obtained from the experimental results. The explicit finite element method is used with Hill’s transversely anisotropic material model for the numerical analysis. The numerical results are compared with those obtained in the experiments, process defects of local thinning are predicted and the thickness variations are discussed.

Acoustic emission monitoring of sheet metal forming: characterization of the transducer, the work material and the process

A systematic investigation of acoustic emission (AE) in cup-drawing was conducted. The AE transducer was characterized by subjecting it to known excitations both directly and through a plastically-deforming material. The AE activity in well-controlled tensile tests was characterized using root-mean-square (RMS) AE measurement. A model of RMS AE based on dislocation generation, motion and annihilation was formulated. Model parameter values were determined using tensile-test results. Finite element calculations of the strain distributions in the cup-drawing process using experimentally determined friction boundary conditions were performed. The calculated strain and strain rate were used in the RMS AE model to predict AE activity in cup-drawing. The RMS of AE signals was measured in cup-drawing, and predicted and measured results were compared. Transducer and signal transmission path characteristics have significant effects on measured AE signals. If only correlations between process variables and RMS AE is sought, the combined transducer, signal path and AE effects can be used. The RMS AE model developed using experimental tensile-test results was capable of predicting RMS AE measurements in cup-drawing. The large AE activity in the initial stages of plastic deformation in cup-drawing makes AE measurements easy and meaningful, but the small signal after this part of the process makes the monitoring of subtle changes in this part of the process doubtful.

The limiting drawing ratio for plastic instability of the cup-drawing process

The plastic instability of cup-drawing is usually measured by the limiting drawing ratio (LDR). A new and practically applicable equation for estimating the LDR in the cup-drawing of a cylindrical cup with a flat-nosed punch is derived using an integral technique based on the load-maximum principle for localization of the plastic flow. It is a function of the process parameters of normal anisotropy value R̄, strain-hardening exponent n, friction coefficient μ, radius of the die arc rd, half die opening r1 and yield strength σy, and can clearly explore the interaction between the process parameters and the LDR in a theoretical manner. Good agreement between calculation and experiment is obtained. It is thereby possible to better understand and control cup-drawing behavior for optimum press-drawability.

Limiting drawing ratio of the cup-drawing process with stabilization matrices approach

This paper adopts a four-node bi-linear quadrilateral element incorporating stabilization matrices to analyze the prediction of the limiting drawing ratio (LDR) in the axisymmetric cylindrical cup-drawing process. These stabilization matrices can control the ‘hour-glass’ effect occurring with the reduced integration scheme. To validate the formulation the theory and the development of a computer code, experimental dies for the cup-drawing process have been signed and tested in a sheet-forming machine. The predicted value of the punch load and thickness distribution in the simulation correlate well with the experimental results. In particular, the extended rmin technique, used to determine the punch travel in each step, is proven adequate for tracing the penetration and separation phenomena between the node and tool contact over the entire deformation history of the cup-drawing process.

Increasing the drawability of AA2014 Al-Cu by differential heat treatment

The limiting drawing ratio of an AA2014 Al-Cu alloy sheet has been increased from a value of 1.8 in uniform, annealed material to a value of 2.6 by the use of differential heat treatment. That treatment involved setting up a transient thermal gradient to give a gradient of solid solution, which then led to a large strength gradient after quenching and aging. Both the heat treatment and the cup drawing process were modeled to predict an optimum treatment. The finite-element modeling (FEM) method was used to simulate cup drawing, with a material description that included plastic anisotropy, which is known to have a significant influence on the process.

Keywords

The aim of this paper is to demonstrate the capability of the tangent additive Mori-Tanaka scheme to simulate complex processes like sheet metal forming. The strengths of such homogenization scheme are exemplified by performing numerical simulations on AISI 304L grade classified as TRIP (TRansformation Induced Plasticity) stainless steel. Some experiments are also performed to validate numerical results. The experimental data are obtained from tensile tests performed at different strain rates. During tensile tests, the evolution of the martensite volume fraction and of the temperature in the AISI 304L specimen are registered. Material parameters of the phase transformation constitutive model (which is coupled with the elastic-viscoplastic Mori-Tanaka homogenization approach) are identified using tensile experimental data. To validate the Mori-Tanaka homogenization scheme, a special focus is devoted to the comparison of the simulated responses to experimental data obtained via tensile tests. Overall stress-strain responses and volume fraction of martensite are accurately reproduced. In addition, local responses in each phase are provided. Numerical simulations of cup drawing process were also performed and a parametric study on material parameters, friction coefficient, holder force enabled to capture the evolution of the punch force and volume fraction of martensite in the blank. An isotropization strategy is also proposed which consists in using an isotropic instead of the anisotropic tangent stiffness modulus for the reference medium to evaluate the tangent polarization tensor present in the interaction law. Relevance of this isotropization is discussed.

Prediction of the maximum drawing load in the cup-drawing process of sheet metals

Based on the load-maximum principle of plastic flow, a new equation, including the process parameters of the strain-hardening exponent n, the normal anisotropy value R̄, the coefficient of friction μ, the radius of the die arc rd (the bending effect) and the thickness of the blank t0, for evaluating the maximum drawing load Pd at a particular drawing ratio in the cup-drawing of a cylindrical cup with a flat-nosed punch is derived by an elementary analysis. Compared with published experimental work, good agreement between calculation and the experiment is obtained. The new equation is simple and practically applicable, exploring the interaction between the process parameters mentioned above and the value of Pd and thereby enabling a better understanding of the cup-drawing behavior of sheet metal.