Jeong Whan Yoon

A pressure-sensitive yield criterion under a non-associated flow rule for sheet metal forming

Abstract Spitzig and Richmond [Acta Metall. 32 (1984) 457] proposed that plastic yielding of both polycrystalline and single crystals of steel and aluminum alloys shows a significant sensitivity to hydrostatic pressure. They further showed that under the associated flow rule, this pressure sensitivity leads to a plastic dilatancy, i.e. permanent volume change, that is at least an order of magnitude larger than observed. Indeed, the plastic dilatancy for most materials is on the order of the measurement error and must be zero in the absence of phase change and significant void nucleation during plastic deformation. A non-associated flow rule based on a pressure sensitive yield criterion with isotropic hardening is proposed in this paper that is consistent with the Spitzig and Richmond data and analysis. The significance of this work is that the model distorts the shape of the yield function in tension and compression, fully accounting for the strength differential effect (SDE). This capability is important because the SDE is sometimes described through kinematic hardening models using only pressure insensitive yield criteria.

Elasto-plastic finite element method based on incremental deformation theory and continuum based shell elements for planar anisotropic sheet materials

Abstract An implicit approach for the incremental analysis of planar anisotropic sheet forming processes is developed based on the incremental deformation theory. The incremental deformation theory based on the minimum plastic work path enables convenient decoupling of deformation and rotation by the polar decomposition. The mathematical description of a constitutive law for the incremental deformation theory is obtained from the flow theory along the minimum plastic work path. The resulting constitutive law is then incorporated in an elasto-plastic finite element analysis code. In the elasto-plastic formulation, continuum based resultant (CBR) shell element is employed. The CBR shell allows large rotation and large membrane/bending strain. The laminar coordinate system is taken to coincide with planar anisotropic material axes. Then, planar anisotropic axes during deformation are updated using a newly developed algorithm based on the polar decomposition. An iterative solving method based on the incremental deformation theory is also developed for an accurate and stable stress integration. The planar anisotropy is incorporated into the formulation for sheet forming by introducing non-quadratic Barlats yield function. For verification purposes, two examples have been simulated and compared with experimental results. The good verification results show that the present elasto-plastic formulation for planar anisotropic sheet materials can provide a good theoretical basis for more extended analyses of sheet forming processes.

Earing predictions based on asymmetric nonquadratic yield function

A nonquadratic yield function (Yld96; Barlat, F., Maeda, Y., Chung, K., Yanagawa, M., Brem, J.C., Hayashida, Y., Lege, D.J. Matsui, K., Murtha, S.J., Hattori, S., Becker, R.C., Makosey, S., 1997. Yield function development for aluminium alloy sheet. J. Mech. Phys. Solids, 45, 1727) which simultaneously accounts for the anisotropy of uniaxial yield stresses and r values was newly implemented in a finite element code. Yield surface shapes, yield stress and r-value directionalities of Yld96 were investigated and compared with those of the previous yield function, Yld91 (Barlat, F., Lege, D.J., Brem, J.C. 1991a. A six-component yield function for anistropic metals. Int. J. Plasticity, 7, 693). Complete formulations for Yld96 implementation and the calculation of coefficients were also discussed for the convenient use of Yld96. A 2090-T3 aluminum alloy sheet sample was modeled and earing formation during a cup drawing test was simulated using the FEM code. The results of earing and thickness strain profiles were compared with the results obtained with Yld91. Investigations were further carried out with a translated yield surface to account for the strength differential effect observed in this material. Computation results with the translated yield surface were in very good agreement with experimental results. It was shown that the yield surface shape and translation have a significant influence on the prediction of the cup height profile during the drawing of a circular blank.

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.

Finite element method for sheet forming based on an anisotropic strain-rate potential and the convected coordinate system

Abstract A variational formulation and the associated finite element (FE) equations have been derived for general three-dimensional deformation of a planar anisotropic rigid-plastic sheet metal which obeys the strain-rate potential proposed by Barlat et al. [Int. J. Plasticity 9 , 1(1993)] . By using the natural convected coordinate system, the effect of geometric change and the rotation of planar anisotropic axes were efficiently considered. In order to check the validity of the present formulation, a cylindrical cup deep drawing test was modeled for a 2008-T4 aluminum alloy sheet sample. Eating simulations were performed and planar anisotropic material properties were experimentally determined. Even though quantitative agreement was not fully achieved, reasonably good agreement was found between the FE simulation and the experiment in thickness strain distribution and caring. No numerical difficulty due to planar anisotropy was encountered, and the computational procedure was found to be very stable, requiring only moderate computational time. The results have shown that the present formulation for planar anisotropic deformation can provide a good basis for the analysis of sheet metal forming processes for planar anisotropic materials, especially for aluminum alloy sheets.

Springback prediction for sheet metal forming process using a 3D hybrid membrane/shell method

To reduce the computational time of finite element analyses for sheet forming, a 3D hybrid membrane/shell method has been developed and applied to study the springback of anisotropic sheet metals. In the hybrid method, the bending strains and stresses were calculated as post-processing, considering the incremental change of the sheet geometry obtained from the membrane finite element analysis beforehand. To calculate the springback, a shell finite element model was used to unload the sheet. For verification purposes, the hybrid method was applied for a 2036-T4 aluminum alloy square blank formed into a cylindrical cup, in which stretching is dominant. Also, as a bending-dominant problem, unconstraint cylindrical bending of a 6111-T4 aluminum alloy sheet was considered. The predicted springback showed good agreement with experiments for both cases.

The effect of plastic anisotropy on compressive instability in sheet metal forming

Abstract The wrinkling behavior of a thin sheet with perfect geometry is associated with compressive instability. The compressive instability is influenced by many factors such as stress state, mechanical properties of the sheet material, geometry of the body, contact conditions and plastic anisotropy. The analysis of compressive instability in a plastically deforming body is difficult considering all the factors because the effects of the factors are very complex and the instability behavior may show a wide variation for a small deviation of the factors. In this study, the bifurcation theory is introduced for the finite element analysis of puckering initiation and growth of a thin sheet with perfect geometry. All the above mentioned factors are conveniently considered by the finite-element method. The instability limit is found by the incremental analysis and the post-bifurcation behavior is analyzed by introducing the branching scheme proposed by Riks. The finite-element formulation is based on the incremental deformation theory and elastic–plastic material modeling. The finite-element analysis is carried out using the continuum-based resultant shell elements considering the anisotropy of the sheet metal. In order to investigate the effect of plastic anisotropy on the compressive instability, a square plate that is subjected to compression in one direction and tension in the other direction is analyzed by the above-mentioned finite-element analysis. The critical stress ratios above which buckling does not take place are found for various plastic anisotropic modeling methods and discussed. Finally, the effect of plastic anisotropy on the puckering behavior in the spherical cup deep drawing process is investigated. From the results of the finite-element analysis, it is shown that puckering behavior of sheet metal is largely affected by plastic anisotropy.

Investigation into wrinkling behavior in the elliptical cup deep drawing process by finite element analysis using bifurcation theory

Abstract The initiation and growth of wrinkles in sheet metal forming processes are influenced by many factors such as the stress state, the mechanical properties of the sheet material, the geometry of the body, and the contact conditions. It is difficult to analyze wrinkling initiation and growth considering these factors, because the effects of the factors are very complex and the wrinkling behavior may show a wide variation for small deviation of the factors. In this study, bifurcation theory is introduced for the finite element analysis of wrinkling initiation and growth. All the above mentioned factors are conveniently considered by the finite element method. The wrinkling initiation is determined by checking the determinant of the stiffness matrix at each iteration and the wrinkling behavior is analyzed by successive iteration with the perturbed guess along the eigenvector. The finite element formulation is based on the incremental deformation theory and elastic–plastic material modeling. The finite element analysis is carried out using continuum-based resultant shell elements. The initiation and growth of wrinkling in the elliptical cup deep drawing process are analyzed by the proposed algorithm. The effect of the aspect ratio of a punch on the wrinkling behavior in the elliptical cup deep drawing process is investigated.

Influence of initial back stress on the earing prediction of drawn cups for planar anisotropic aluminum sheets

Abstract Anisotropy is closely related to the formability of sheet metal and should be considered carefully for more realistic analysis of actual sheet metal forming operations. In order to better describe anisotropic plastic properties of aluminum alloy sheets, a planar anisotropic yield function which accounts for the anisotropy of uniaxial yield stresses and strain rate ratios simultaneously was proposed recently. This yield function was used in the finite element simulations of cup drawing tests for an aluminum alloy 2008-T4. Isotropic hardening with a fixed initial back stress based on experimental tensile and compressive test results was assumed in the simulation. The computation results were in very good agreement with the experimental results. It was shown that the initial back stress as well as the yield surface shape have a large influence on the prediction of the cup height profile.

Incorporation of sheet-forming effects in crash simulations using ideal forming theory and hybrid membrane and shell method

In order to achieve reliable but cost-effective crash simulations of stamped parts, sheet-forming process effects were incorporated in simulations using the ideal forming theory mixed with the three-dimensional hybrid membrane and shell method, while the subsequent crash simulations were carried out using a dynamic explicit finite element code. Example solutions performed for forming and crash simulations of I- and S-shaped rails verified that the proposed approach is cost effective without sacrificing accuracy. The method required a significantly small amount of additional computation time, less than 3% for the specific examples, to incorporate sheet-forming effects into crash simulations. As for the constitutive equation, the combined isotropic-kinematic hardening law and the nonquadratic anisotropic yield stress potential as well as its conjugate strain-rate potential were used to describe the anisotropy of AA6111-T4 aluminum alloy sheets.