Wrinkling initiation and growth in modified Yoshida buckling test: Finite element analysis and experimental comparison

Abstract

Wrinkling is one of the major defects in sheet metal products and may also play a significant role in the wear of the tool. The initiation and growth of wrinkles are influenced by many factors such as stress ratios, mechanical properties of the sheet material, geometry of the workpiece, contact condition, etc. It is difficult to analyze the wrinkling initiation and growth considering all the factors because the effects of the factors are very complex and the wrinkling behavior may show a wide scatter of data even for small deviations of factors. In this study, the 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 found 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 effect of magnitude of perturbation on the wrinkling behavior can be avoided by the Newton-type iteration method. 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. For the verification of the analysis, the postbuckling of columns and circular plates are analyzed by finite element analysis using the bifurcation algorithm introduced in the study, and the results are compared with the exact solutions. In order to investigate the effects of geometry and stress ratio on the wrinkling initiation and growth, a modified Yoshida buckling test is proposed as an improved effective buckling test. In the modified Yoshida buckling test, the dimensions of the sheet specimen are varied to change the stress ratio and the degree of constraint. The finite element analysis is carried out for the modified Yoshida buckling test and compared with the experimental results.

Introduction

Wrinkling is one of the major defects in sheet metal forming processes together with tearing, springback and other geometric and surface defects. Wrinkling may be a serious obstacle to implementing the forming process and assembling the parts, and may also play a significant role in the wear of the tool. In order to improve the productivity and quality of products, the wrinkling problem must be essentially resolved. Recently, sheet metal forming processes are widely used in various industrial sectors such as in automotive, electric home appliance, and aircraft industries, and also the needs for high precision and high value-added products are increasing. In order to reduce the process development time, the prediction of those defects and the modification of design in the design stage are needed.

The initiation and growth of wrinkles are influenced by many factors such as stress ratio, mechanical properties of the sheet material, geometry of the workpiece, and contact conditions. It is difficult to analyze the wrinkling initiation and growth considering all the factors because the effects of the factors are very complex and the wrinkling behavior may show a wide scatter of data for small deviation of factors as is common in instability phenomena. Due to these difficulties, the study on wrinkling has been carried out case by case for a given process and generalized wrinkling criterion that can be used effectively for various processes has not been proposed. Many analytic approaches [1], [2], [3], [4] about wrinkling have been carried out by using simple analytical bifurcation theory. They investigated the wrinkling phenomena of circular plates and studied the effect of blankholding force upon wrinkling initiation and number of wrinkling waves quantitatively. Simple analytical bifurcation analysis can give a useful estimate of elastic-plastic buckling when the plate has an elementary shape and is subject to boundary conditions that are easily prescribed. Wrinkling has also been studied by the finite element analysis together with the bifurcation algorithm widely used in buckling problem [5], [6]. There are two types of wrinkling analysis using the finite element method; the bifurcation analysis [6] of perfect structure and the nonbifurcation analysis [7], [8] employing initial imperfection. The nonbifurcation analysis employing initial imperfections, sometimes, gives more reasonable results than the bifurcation analysis because all real structures have inherent imperfections, such as material nonuniformity or geometric unevenness. However, the results obtained by nonbifurcation analysis are sensitive to initial imperfections. In the present study, therefore, a bifurcation algorithm of perfect structure is introduced into the finite element method in order to analyze wrinkling behavior of sheet metal more exactly and more rigorously.

In most of buckling analyses, the incremental solution procedures are carried out based on load increment. Then, the postbuckling behavior can be analyzed by using a proper singular point passing algorithm such as the arc length method. In the analysis of sheet metal forming processes, however, it is convenient to obtain incremental solution based on displacement increment rather than load increment. In the analysis of wrinkling in sheet metal forming processes, therefore, it is difficult to employ the singular point passing algorithm that can improve the convergence near a singular point. Only a branching scheme that branches the solution to secondary path can be employed. Near the bifurcation point, therefore, the convergence may become worse. Wang and Lee [6] adopted artificial scalar multiple of eigenvector as an assumed solution at the bifurcation point in order to overcome the convergence problem. The solutions at later steps, however, were affected by the size of artificial multiple. In the present study, a scalar multiple of eigenvector at the bifurcation point is used as initial guess for the Newton–Raphson iteration procedures. Then, the artificiality of the size of scalar multiple is eliminated. The scalar multiplied to the eigenvector is determined to minimize the deformation energy. For the verification of the analysis, the postbuckling of columns and circular plates are subjected to finite element analysis by using the bifurcation algorithm introduced in the study, and the results are compared with the analytic solutions.

Yoshida [9] proposed the buckling test (Yoshida buckling test) in order to assess the wrinkle formation tendencies of sheet metals. In the Yoshida buckling test, the square metal sheet is stretched in one diagonal direction as shown in Fig. 1. Tensile deformation in the stretching direction (y) causes the compressive stress in the transverse direction (x) in the central region of the specimen due to the geometric constraint of the outside rigid region, and thus, the induced compressive stress causes the wrinkle formation. Thereafter, some research works about the Yoshida buckling test have been reported. Szacinski and Thomson [10] investigated the effect of mechanical properties such as work hardening coefficient, anisotropic constant, and yield strength on the wrinkling growth in Yoshida buckling test by experiments. Tomita and Shindo [7] analyzed the Yoshida buckling test by the finite element method using elastic-plastic shell element and employing the initial imperfection for the analysis of wrinkling. Wang and Lee [6] implemented the bifurcation theory proposed by Hill [11] to the finite element method and analyzed the Yoshida buckling test. More recently, Di and Thomson [12] introduced the neural network theory in the analysis of the Yoshida buckling test and predicted the wrinkling limit.

Yoshida buckling test is effective for assessing the wrinkling tendencies of sheet metal. By the Yoshida buckling test, however, various deformation characteristics for various stress ratios cannot be modeled because the geometry of the specimen is fixed. Yoshida proposed the biaxial diagonal tension [9] in which, however, the wrinkling formation is over constrained by grips. In this study, a modified Yoshida buckling test is also proposed in order to investigate the effect of geometry and stress ratios on the wrinkling initiation and growth. The bifurcation algorithm proposed by Riks [13] is introduced for the finite element analysis of the modified Yoshida buckling test. The finite element formulation is based on the incremental deformation theory [14], [15] and elastic-plastic material modeling. The continuum-based resultant shell element is employed and the normal anisotropy of the sheet metal is considered. The incremental solution procedures are carried out based on displacement increment.