A new analytical theory for earing generated from anisotropic plasticity

Commercial canmaking processes include drawing, redrawing and several ironing operations. It is experimentally observed that during the drawing and redrawing processes earing develops, but during the ironing processes earing is reduced. It is essential to understand the earing mechanism during drawing and ironing for an advanced material modeling. A new analytical approach that relates the earing profile to r-value and yield stress directionalities is presented in this work. The analytical formula is based on the exact integration of the logarithmic strain. The derivation is for a cylindrical cup under the plane stress condition based on rigid perfect plasticity while force equilibrium is not considered. The earing profile is obtained solely from anisotropic plastic properties in simple tension. The earing mechanism is explained from the present theory with explicit formulae. It has been proved that earing is the combination of the contributions from r-value and yield stress directionalities. From a directionality (y-axis) vs. angle from the rolling (x-axis) plot, the earing profile is generated to be a scaled mirror image of the r-value directionality with respect to 90° (x = 90) and also a scaled mirror image of the yield stress directionality with respect to the reference yield stress (y = 1). Three different materials (Al-5% Mg alloy, AA 2090-T3 and AA 3104 RPDT control coil) are considered for verification purposes. This approach provides a fundamental basis for understanding the earing mechanism. In practice, the present theory is also very useful for the prediction of the earing profile of a drawn and iron cup and its related convolute cut-edge design for an earless cup.     

Formability of TWIP (twinning induced plasticity) automotive sheets

The main objective of this work is to experimentally and numerically evaluate the macro-performance of the automotive TWIP (twinning induced plasticity) sheet in conjunction with formability. In order to characterize the mechanical properties, the simple tension and compression tests were performed for anisotropic properties, while the strain rate test was carried out to evaluate strain rate sensitivity. The forming limit diagram was measured and incorporated into the simulation program, while the theoretical prediction of the diffuse and localized necking was also carried out utilizing Hill’s and the M-K theories as well as Dorn’s and Swift’s diffuse theories. Note that the generalized criteria of Hill’s, Dorn’s and Swift’s theories were derived for general anisotropic sheets as well in this work. For numerical simulations, the anisotropic yield functions Yld2000-2d and Hill48 as well as the isotropic Mises yield function were selectively applied along with the isotropic hardening law. Formability verification was performed, utilizing Yld2000-2d, for the hemispherical dome stretching, notch and simple tension tests with specimens selectively prepared by milling and punching, while anisotropic properties were verified through the three point bending and cylindrical cup drawing tests, comparing the performance of the three yield functions.     

Prediction of six or eight ears in a drawn cup based on a new anisotropic yield function

In this work, the recently proposed anisotropic yield function, Yld2004-18p [Barlat, F., Aretz, H., Yoon, J.W., Karabin, M.E., Brem, J.C., Dick, R.E., 2005. Linear transformation based anisotropic yield function, Int. J. Plasticity 21, 1009], is implemented in a finite element (FE) code for application to the cup drawing simulation of a circular blank sheet. A short review of the Yld2004-18p relevant features is provided and the stress integration scheme for its implementation in FE codes is described. The simulation of the drawing process is conducted for an aluminum alloy sheet sample (AA2090-T3). The predicted and experimental cup height profiles (earing profiles) with six ears are shown to be in excellent agreement. Additional simulations on a ficticious material are performed in order to show that the yield function Yld2004-18p can lead to the prediction of cups with eight ears. In order to achieve these results, a sufficient number of input data are required to calculate the yield function coefficients. Finally, a simplified analytical approach that relates the earing profile to the r-value directionality is also presented in this paper. It is shown that this approach can be very useful as a first approximation of the earing profile of drawn cups.     

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 project supported by the National Natural Science Foundation of China (19832020) and Provincial Natural Science Foundation of Jilin, China (200000519)     

Macro-performance evaluation of friction stir welded automotive tailor-welded blank sheets: Part II – Formability

Formability of automotive friction stir welded TWB (tailor-welded blank) sheets was experimentally and numerically investigated in this work for four automotive sheets, aluminum alloy 6111-T4, 5083-H18, 5083-O and DP590 steel sheets, each having one or two different thicknesses. In particular, formability in three applications including the simple tension test with various weld line directions, hemisphere dome stretching and cylindrical cup drawing tests was evaluated. For numerical simulations, mechanical properties previously characterized in a joint paper (Chung et al., 2010) were utilized. To represent the mechanical properties, the non-quadratic orthogonal anisotropic yield function, Yld2000-2d, was utilized along with the (full) isotropic hardening law, while the anisotropy of the weld zone was ignored for simplicity.     

Applications of planar anisotropic yield criteria to porous sheet metal forming simulations

Planar anisotropic yield functions, with rounded vertexes especially near the equal-biaxial direction of the corresponding yield loci, appropriate for some structure metals are employed for the matrix surrounding voids in the present study. The widely adopted Hill anisotropic yield functions are also implemented into the matrix for comparisons. Mechanisms of the void growth, void nucleation, and void coalescence are simultaneously considered here. Effects of the yield function of the corresponding matrix on the sheet metal under two typical sheet forming operations, a hemispherical punch stretching operation and a cup drawing operation, are investigated via a finite element analysis. Thickness strains in various orientations of the sheet are then evaluated. Simulation results show that the yield function of the corresponding matrix plays important roles on the strain distribution and the strain localization as well. Early localization would be found for the sheet with relatively small initial void volume fraction in two operations. Yield functions of the matrix rather influence the earing phenomenon under the cup drawing procedure even similar displacement profiles of the outer boundary could be observed.     

Finite element modeling of plastic anisotropy induced by texture and strain-path change

Consideration of plastic anisotropy is essential in accurate simulations of metal forming processes. In this study, finite element (FE) simulations have been performed to predict the plastic anisotropy of sheet metals using a texture- and microstructure-based constitutive model. The effect of crystallographic texture is incorporated through the use of an anisotropic plastic potential in strain-rate space, which gives the shape of the yield locus. The effect of dislocation is captured by use of a hardening model with four internal variables, which characterize the position and the size of the yield locus. Two applications are presented to evaluate the accuracy and the efficiency of the model: a cup drawing test and a two-stage pseudo-orthogonal sequential test (biaxial stretching in hydraulic bulging followed by uniaxial tension), using an interstitial-free steel sheet. The experimental results of earing behavior in the cup drawing test, maximum pressure and strain distribution in bulging, and transient hardening in the sequential test are compared against the FE predictions. It is shown that the current model is capable of predicting the plastic anisotropy induced by both the texture and the strain-path change. The relative significance of texture and strain-path change in the predictions is discussed.     

On the use of homogeneous polynomials to develop anisotropic yield functions with applications to sheet forming

This paper investigates the capabilities of several non-quadratic polynomial yield functions to model the plastic anisotropy of orthotropic sheet metal (plane stress). Fourth, sixth and eighth-order homogeneous polynomials are considered. For the computation of the coefficients of the fourth-order polynomial an improved set of analytic formulas is proposed. For sixth and eighth-order polynomials the identification uses optimization. Simple constraints on the optimization process are shown to lead to real-valued convex functions. A general method to extend the above plane stress criteria to full 3D stress states is also suggested. Besides their simplicity in formulation, it is found that polynomial yield functions are capable to model a wide range of anisotropic plastic properties (e.g., the Numisheet’93 mild steel, AA2008-T4, AA2090-T3). The yield functions have then been implemented into a commercial finite element code as constitutive subroutines. The deep drawing of square (Numisheet’93) and cylindrical (AA2090-T3) cups have been simulated. In both cases excellent agreement with experimental data is obtained. In particular, it is shown that non-quadratic polynomial yield functions can simulate cylindrical cups with six or eight ears. We close with a discussion on earing and further examples.     

Spatial representation of evolving anisotropy at large strains

A phenomenological model for evolving anisotropy at large strains is presented. The model is formulated using spatial quantities and the anisotropic properties of the material is modeled by including structural variables. Evolution of anisotropy is accounted for by introducing substructural deformation gradients which are linear maps similar to the usual deformation gradient. The evolution of the substructural deformation gradients is governed by the substructural plastic velocity gradients in a manner similar to that for the continuum. Certain topics related to the numerical implementation are discussed and a simple integration scheme for the local constitutive equations is developed. To demonstrate the capabilities of the model it is implemented into a finite element code. Two numerical examples are considered: deformation of uniform plate with circular hole and the drawing of a cup. In the two examples it is assumed that initial cubic material symmetry applies to both the elastic and plastic behavior. To be specific, a polyconvex Helmholtz free energy function together with a yield function of quadratic type is adopted.     

A non-associated constitutive model with mixed iso-kinematic hardening for finite element simulation of sheet metal forming

In this paper an anisotropic material model based on non-associated flow rule and mixed isotropic–kinematic hardening was developed and implemented into a user-defined material (UMAT) subroutine for the commercial finite element code ABAQUS. Both yield function and plastic potential were defined in the form of Hill’s [Hill, R., 1948. A theory of the yielding and plastic flow of anisotropic metals. Proc. R. Soc. Lond. A 193, 281–297] quadratic anisotropic function, where the coefficients for the yield function were determined from the yield stresses in different material orientations, and those of the plastic potential were determined from the r-values in different directions. Isotropic hardening follows a nonlinear behavior, generally in the power law form for most grades of steel and the exponential law form for aluminum alloys. Also, a kinematic hardening law was implemented to account for cyclic loading effects. The evolution of the backstress tensor was modeled based on the nonlinear kinematic hardening theory (Armstrong–Frederick formulation). Computational plasticity equations were then formulated by using a return-mapping algorithm to integrate the stress over each time increment. Either explicit or implicit time integration schemes can be used for this model. Finally, the implemented material model was utilized to simulate two sheet metal forming processes: the cup drawing of AA2090-T3, and the springback of the channel drawing of two sheet materials (DP600 and AA6022-T43). Experimental cyclic shear tests were carried out in order to determine the cyclic stress–strain behavior and the Bauschinger ratio. The in-plane anisotropy (r-value and yield stress directionalities) of these sheet materials was also compared with the results of numerical simulations using the non-associated model. These results showed that this non-associated, mixed hardening model significantly improves the prediction of earing in the cup drawing process and the prediction of springback in the sidewall of drawn channel sections, even when a simple quadratic constitutive model is used.     

An elasto-plastic constitutive model with plastic strain rate potentials for anisotropic cubic metals

An elasto-plastic constitutive model with the plastic strain rate potential was developed for finite element analysis. In the model, isotropic-kinematic hardening was incorporated under the plane stress condition for anisotropic sheet cubic metal forming analysis. The formulation is general enough for any homogeneous plastic strain rate potential (with the first-order homogeneous effective strain rate) but the plastic strain rate potential Srp2004-18p was considered here. Attention was focused on the development of the elasto-plastic transition criterion and the effective stress update algorithm. Also, to assure the quadratic convergence rate in Newton’s method, the elasto-plastic tangent modulus was analytically derived. Accuracy and convergence of the stress update algorithm were assessed by the iso-error maps, whereas stability of the algorithm was confirmed by analytical procedure. Validations were performed for the examples of the circular cup drawing, 2D draw-bending and unconstrained cylindrical bending tests, utilizing aluminum sheet alloys.     

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.     

Analysis of blank thickening in deep drawing processes using the theory of a Cosserat generalized membrane

This paper analyzes a transient, nonlinear deep drawing process where a circular blank of a rigid-plastic material is forced by a rigid circular punch to deform into a cylindrical cup. Attention is focused on the plastic flow beneath the blank-holder. Using the Cosserat theory of a generalized membrane it is possible to obtain analytical solutions which examine the following two major effects: (a) the importance of added “rim pressure” acting on the outer edge of the blank; and (b) the importance of a controlled moveable blank-holder to allow blank thickening during the drawing process. Guided by these analytical results, a new deep drawing machine was built to exploit these effects and increase the limit drawing ratio (LDR) of the drawing process. Specifically, the LDR (in one stroke) reached the value of 3.16 compared with the value of about 2.0 in the conventional process. Moreover, the analytical prediction of the punch force versus the punch stroke is in good agreement with the experimental data and with simulations using the computer code DYTRAN.     

筒形件液压拉深过程最优压边力的研究

结合筒形件液压拉深新装置的基本工作原理，提出了筒形件拉深法兰部分切向应变更为精确的简化计算公式，进一步获得了该法兰区域板料径向及切向应力的新的数学表达式。应用能量法重新建立了筒形件拉深时法兰区板料不起皱的最小压边力的数学表达式。明确指出液压拉深时拉深中的危险断面在板料凹模圆角区和筒壁区相交处，建立了相应的拉深力的计算公式，并获得了拉深破裂临界状态时极限拉深力及径向应力表达式。推导出了拉深破裂最大压边力的公式．指出实际拉深时最佳压边力应在防皱最小压边力和拉破最大压边力之间选取，同时根据该结果也能计算液体压力与最小拉深系数的关系。计算结果表明，该液压拉深方法可大幅度提高拉深变形程度，值得在工业实际中推广应用。     

Evaluation of advanced anisotropic models with mixed hardening for general associated and non-associated flow metal plasticity

The main objective of this paper is to develop a generalized finite element formulation of stress integration method for non-quadratic yield functions and potentials with mixed nonlinear hardening under non-associated flow rule. Different approaches to analyze the anisotropic behavior of sheet materials were compared in this paper. The first model was based on a non-associated formulation with both quadratic yield and potential functions in the form of Hill’s (1948). The anisotropy coefficients in the yield and potential functions were determined from the yield stresses and r-values in different orientations, respectively. The second model was an associated non-quadratic model (Yld2000-2d) proposed by Barlat et al. (2003). The anisotropy in this model was introduced by using two linear transformations on the stress tensor. The third model was a non-quadratic non-associated model in which the yield function was defined based on Yld91 proposed by Barlat et al. (1991) and the potential function was defined based on Yld89 proposed by Barlat and Lian (1989). Anisotropy coefficients of Yld91 and Yld89 functions were determined by yield stresses and r-values, respectively. The formulations for the three models were derived for the mixed isotropic-nonlinear kinematic hardening framework that is more suitable for cyclic loadings (though it can easily be derived for pure isotropic hardening). After developing a general non-associated mixed hardening numerical stress integration algorithm based on backward-Euler method, all models were implemented in the commercial finite element code ABAQUS as user-defined material subroutines. Different sheet metal forming simulations were performed with these anisotropic models: cup drawing processes and springback of channel draw processes with different drawbead penetrations. The earing profiles and the springback results obtained from simulations with the three different models were compared with experimental results, while the computational costs were compared. Also, in-plane cyclic tension–compression tests for the extraction of the mixed hardening parameters used in the springback simulations were performed for two sheet materials.     

Electromechanical response of (3–0, 3–1) particulate,fibrous, and porous piezoelectric composites with anisotropic constituents: A model based on the homogenization method

An analytical framework based on the homogenization method has been developed to predict the effective electromechanical properties of periodic, particulate and porous, piezoelectric composites with anisotropic constituents. Expressions are provided for the effective moduli tensors of n-phase composites based on the respective strain and electric field concentration tensors. By taking into account the shape and distribution of the inclusion and by invoking a simple numerical procedure, solutions for the electromechanical properties of a general anisotropic inclusion in an anisotropic matrix are obtained. While analytical forms are provided for predicting the electroelastic moduli of composites with spherical and cylindrical inclusions, numerical evaluation of integrals over the composite microstructure is required in order to obtain the corresponding expressions for a general ellipsoidal particle in a piezoelectric matrix. The electroelastic moduli of piezoelectric composites predicted by the analytical model developed in the present study demonstrate excellent agreement with results obtained from three-dimensional finite-element models for several piezoelectric systems that exhibit varying degrees of elastic anisotropy.     

On the implementation of anisotropic yield functions into finite strain problems of sheet metal forming

In this article the implementation of anisotropic yield functions into finite element investigations of orthotropic sheets with planar anisotropy is discussed within a plane-stress context. Special attention is focused on the proper treatment of the orientation of the anisotropic axes during deformation into the finite-strain range. As an example problem the hydrostatic bulging of a membrane is considered in conjunction with a recently proposed anisotropic yield function. It is shown that the aspects of the plane-stress assumption, which do not come into consideration in isotropic analyses, can play an important role on the accuracy of the solution when the rotation of the orthotropic axes enters the computation directly due to the presence of material anisotropy. When the material anisotropy is considered and when the deformation of the workpiece is not limited to the plane of the undeformed sheet (such as cup drawing, hydrostatic bulging, etc.), the numerical experiments indicate that the only correct formulation is the one based on numerically imposing the requirement that for the plane-stress application, the in-plane material axes have to remain in the plane of the sheet during the deformation.     

Flange earing and its control on deepdrawing of anisotropy circular sheets

The Hill's quadric anisotropy yield function and the Barlat-Lian anisotropy yield function describing well anisotropy sheet metal with stronger texture are introduced into a quadric-flow corner constitutive theory of elastic-plastic finite deformation suitable for deformation localization analysis. And then, the elastic-plastic large deformation finite element formulation based on the virtual power principle and the discrete Kirchhoff shell element model including the yield functions and the constitutive theory are established. The focus of the present research is on the numerical simulation of the flange earing of the deep-drawing of anisotropy circular sheets, based on the investigated results, the schemes for controlling the flange earing are proposed. Supported by NSFC(No. 19832020) and National Automobile Dynamic Simulation Laboratory of China.