Elastic-Plastic Flange Wrinkling of Circular Plates in Deep Drawing Process Farzad Moayyedian, Mehran Kadkhodayan Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran fa_mo577@stu-mail.um.ac.ir, kadkhoda@um.ac.ir Keywords: Elastic/plastic wrinkling, energy method, bifurcation functional, circular plate, deep drawing process. Abstract. This paper deals with two-dimensional plane stress wrinkling model of elastic/plastic annular plate. Based on energy method and nonlinearity of strain-displacement law, a bifurcation functional in polar coordinate is derived analytically. This technique leads to the critical conditions for the onset of the elastic/plastic wrinkling of the flange during the deep-drawing process. Tresca yield criterion along with deformation theory of plasticity are utilized and the material of the plate is assumed to behave perfectly plastic. Moreover, the influence of the blankholder upon wrinkling and on the number of the generated waves is quantitatively predicted by the suggested scheme. The main advantage of the proposed solution is its better agreement with the experimental and analytical results found by the other resarchers. Introduction Wrinkling is one of the major modes of the failure in automotive sheet pressing operations. This behavior is caused by excessive compressive stresses during the forming. In a deep-drawing operation an initially flat round blank is drawn over a die by a cylindrical punch, Fig. (1). The annular parts of the blanks are subjected to a radial tensile stress, while in the circumferential direction compressive stress is generated during drawing, Fig. (2). For particular drawing-tool dimensions and blank thicknesses, there is a critical blank diameter/thickness ratio. The critical stress causes the plastic buckling of the annular part of the blanks so that an undesirable mode of deformation ensues with the generated waves in the flange [1, 2], Fig. (3). A bifurcation functional was proposed by Hutchinson based on Hill general theory of uniqueness and also bifurcation in elastic-plastic solids. This functional is given as; 1 F ( M ij ij N ij ij0 t ij w,i w, j ) ds , (1) 2 where S denotes the region of the shell middle surface over which the wrinkles appear, w the buckling displacement, t the thickness of the plate, N ij the force resultants, M ij the couple resultants (per unit width), ij the curvature tensor and ij0 the stretch strain tensor. This bifurcation functional contains the total energy for wrinkling occurrence. On the other words, for some non-zero displacement fields, the state of F 0 corresponds to the critical conditions for wrinkles to occur [3]. In this paper, the complete form of this bifurcation functional is found by using energy method. In plastic analysis, to have a closed-form analytical solution, Tresca yield criterion and also deformation theory of plasticity are used. Therefore the complete form of the functional (1) is suggested for elastic-plastic materials. The results show that the proposed solutions for the critical conditions of the onset of the wrinkling have better agreement with the other avilable experemental findings. Elastic-plastic flange wrinkling 0 The nonlinear Lagrangian strain tensor is used in this paper is ij ij z ij , where; 0 u 0 1 w 2 rr r 2 ( r ) 0 u 0 1 v 0 1 1 w 2 ( ) r r 2 r 0 1 1 u 0 v0 v0 w 1 w r 2 [ r r r ( r ) ( r ) ] 2w , 11 r2 1 w 1 2w , 22 r r r 2 2 1 2w 1 w 2 . 12 r r r (2) u 0 and v0 are the displacements in the middle plane of the plate, in r and directions, respectively. The buckling displacement normal to the plane of the plate in the z direction is shown by w . The strain energy can be condensed in the following form for plane stress problem [4]; h b 2 2 h b 2 2 0 0 1 U 2a 1 h ij ij dz r dr d 2 a 2 ( rr rr 2 r r ) dz r dr d . (3) h 2 The force ( N ij ) and the moment ( M ij ) resultants are found in terms of the displacement fields as; h h 2 2 0 0 N dz ij ij h h Cijkl ( ij z kl ) dz h Cijkl ij , 2 2 h h 2 2 h3 0 M z dz C ( z ) z dz C ijkl kl . ij ij ijkl ij kl h h 12 2 2 (4) which C ijkl can be elastic or elastic-plastic stiffness matrix and h is the thickness of the plate. The external force ( N̂ ij ) is acting in the middle plane of the plate (i.e. z 0 ). Considering this force, and also using the first term of Eq. (2), it is easy to show that the external work can be written in the following form [2-4]; b 2 b 2 b 2 (5) 0 0 WE Nˆ ij ij rdrd Nˆ ij ij0 rdrd [ Nˆ 1111 Nˆ 22 022 2 Nˆ 1212 ]rdrd . a 0 a 0 a 0 Having the strain energy and the external work in hand and considering Nˆ ij h ij , the total potential energy function can be formed; U WE b 2 1 h3 2w 1 w 1 2 w 2 {C1111( 2 ) 2 C 2222[ ( )] 2 a 0 12 r r r 2 r 2 w 1 w 1 2 w 1 2 w 1 w 2 )[ ( )] 4C1212[ ( )] rdrd } 2 2 r r r r r r r b 2 u u 1 1 w 1 v0 1 1 w 2 2 h{C1111[ 0 ( ) 2 ]2 C 2222[ 0 ( ) ] 2a 0 r 2 r r r 2 r 2C1122 ( u 0 1 w 2 u 0 1 v0 1 1 w 2 ( ) ][ ( ) ] r 2 r r r 2 r v v 1 1 u0 w 1 w 2 4C1212 ( [ 0 0 ( )( )]) }rdrd 2 r r r r r b 2 u u 1 w 1 v0 1 1 w 2 [h{ r [ 0 ( ) 2 ] [ 0 ( ) ]}rdrd r 2 r r r 2 r a 0 2C1122[ (6) e Now the critical condition for elastic wrinkling can be obtained by substituting Cijkl instead of C ijkl for plane stress problem, in Eq. (6). To obtain the critical load and critical wave number the prebuckled elastic isotropic stress distribution in an annular plate subjected to a radial stress p along its inner edge can be investigated [1] as; p a2 b2 ( 1) , r b2 a2 r 2 2 2 p a (1 b ) . b2 a2 r2 (7) In this study, it is assumed that the displacement fields of the flange for a deep drawn cup have the following form [10, 11]; w(r, ) c (r a) (1 cos n) , (8) u 0 (r, ) d r cos n , v (r, ) e r sin n , 0 where c , d and e are the constants and n is the number of generated waves in the flange [3]. It is obvious that any admissible bifurcation mode in Eq. (8), satisfies the geometrical boundary conditions u, v 0 and w 0 at the inner edge ( r a ) and also the kinematical constraints contain w(r , ) 0 , u0 (r , ) 0 and v0 (r , ) 0 for a r b . By substituting the elastic stress distribution from Eq. (7) and the displacements u 0 , v0 and w from Eq. (8) into the functional (6), the following relationship may be found; c2D 2 E (m, n, ) 1 b 2 hE [ F (m, n, ) c 4 G (m, n, ) c 2 e 2 64 (1 ) H (m, n, )c d I (m, n, )d J (m, n)e K (m, n, )de] 2 2 2 c 2 b 2 tp (9) L(m, n). 2 The suggested functional may be shown in the following matrix form; F c c2 M 11 0 e 0 0 d 0 M 22 M 32 M 42 0 M 23 M 33 M 43 0 c M 24 c 2 . M 34 d M 44 e (10) Now, the critical conditions for the onset of elastic wrinkling can be written as [2, 3]; Det ( M ij ) 0, [ Det ( M ij )] 0. n (11) The validity of the deformation theory in plasticity is limited to the monotonically increasing loading in which:(1) the stress components are increased nearly proportionally in a loading process, known as proportional loading; and (2) no unloading occurs. Hence, because these conditions are acceptable in this case, it is preferred to use the deformation theory rather than the incremental ep theory of plasticity. For the perfectly plastic materials the Cijkl is determined by the following relationship [5]; f f C epqkl mn pq , f e f Crstu rs tu e Cijmn ep e Cijkl Cijkl (12) e where f is a proper yield criterion suited for this problem and Cijkl is the elastic coefficient of the ep hook’s law. To obtain Cijkl , the Tresca yield criterion can be used as f r Y 0. In this function, Y is the yield stress. For the plane stress problem (i.e. 33 13 23 0 ) a simple plane stress relation is found. It should be noted that the plastic stress distribution in the flange before the wrinkling is similar to the axisymmetric problem as [1]; b r Y ln( r ) 0, Y [ln( b ) 1] 0. r (13) ep Utilizing this plastic stress distribution, displacement fields of Eq. (8) and obtained Cijkl , we have; t 3 c 2 E ep 1 tE b 2 E ( m , n , ) [ F ep (m, n, ) c 4 G ep (m, n) d 2 H ep (m, n)e 2 128 (1 2 ) 96(1 2 ) F (14) t b 2 c 2 Y ep I (m, n)de J (m, n)c e K (m, n)c d ] L (m, n). 8 ep ep 2 2 The above functional can be written in matrix form similar to Eq. (10), and therefore the critical conditions for the onset of wrinkling can be as Eq. (11). According to Fig. (1), when a spring-type blankholder is used, assuming the spring coefficient of the blankholder as ( K ), its strain energy can be written as 1 2 K (u 02max v02max wmax ) . To obtain the effect of blankholder upon onset of wrinkling 2 this energy should be added to Eq. (14). Results and discussions In this part, the critical conditions for the onset of elastic and plastic wrinkling in a deep drawn cup are investigated for steel ( E 200 Gpa , 0.3 ) and Aluminum ( E 70 Gpa , 0.3 ). The second condition of Eq. (11) yields to M 11 0 . From this equation ncr can be found and its variation with n a ) is shown in Fig. (4). After finding ncr , pcr is also calculated. A comparison between the b current obtained results and Yu’s results [1] has been shown in Fig. (5). It may be observed from Fig. (6) that the Aluminum wrinkles under the lower loads than the steel for the same value of a ( 1 ). The similar procedure can be used for plastic wrinkling. Figures (7) and (8) illustrate the b (1 variation of n and E t with ( a ) and the comparison of current results with some available 1 b Y b experimental data [1]. It can be observed that there is a good agreement between the suggested nonlinear solution and the experimental one. Figures (7) and (8) demonstrate the onset of wrinkling for the flange and its comparison with the Geckeler and Senior results. It can be realized from Fig. (8) that the Geckeler's result can give a good approximation for the small values of ( 1 a ), (i.e. narrow b flanges), but its error rises as the width of the flange increases [1]. It may be observed that for the plastic wrinkling also Aluminum wrinkles under the lower loads than steel for the same value of ( 1 a ), Fig. (9). Moreover, increasing BHF (blankholder force) increases the number of waves for b the same value of ( 1 a ) and also postpone the onset of wrinkling, Figs. (10) and (11). b Conclusions Using energy method, the elastic and plastic wrinkling of the flange in deep drawing process have been studied analytically. The proposed bifurcation functional is more general than the ones of previously proposed methods. A closed-form analytical solution is formed based on Tresca yield criterion and deformation theory of plasticity and perfectly plastic material. The suggested technique leads to the critical loads and the numbers of the generated waves. The influence of the blankholder upon wrinkling and also on the number of the generated waves was quantitatively predicted by the suggested scheme. The obtained analytical results have a good agreement with the experimental data. References [1] T. X. Yu and W. Johnson: International Journal of Mechanical Sciences, Vol. 24 (1982), p. 175-88. [2] E. Chu and Y. Xu: Journal of Mechanics and Physics of Solids, Vol. 43 (2001), p. 1421-1440. [3] J. P. Corria and G., Ferron: Journal of Materials and Processing Technology, Vol. 155-156 (2004), p. 1604-1610. [4] J. N. Reddy: Energy Principles and Variation Methods in Applied Mechanics (John Wiley & Sons, 2002). [5] A. Khan and S. Hung: Continuum theory of plasticity (John Wiley & sons, 1995). Figure 1. Deep drawing process Figure 2. The flange is modeled with cylindrical punch. as an annular plate with radial stress distribution in their inner edges. Figure 3. The waves produce in the flange. 12 10 n 8 6 4 2 0 0 0.1 0.2 0.3 1-a/b 0.4 0.5 0.6 Figure. 4. The number of waves for the elastic wrinkling of the flang. Figure 5. The critical load for the elastic wrinkling of the flange. 18 16 14 12 n 10 8 6 Present Result Yu's Result Experimental Results 4 2 0 Figure 6. Comparing the onset of elastic wrinkling for steel and Aluminium. Figure 8. The limitation of plastic wrinkling for the flange. Figure 10. Number of the generated waves in the flange with Blankholder. 0 0.05 0.1 0.15 0.2 1-a/b 0.25 0.3 0.35 0.4 Figure 7. The number of generated waves in the plastic flange wrinkling of the annular plate. Figure 9. Comparing the onset of plastic wrinkling for steel and Aluminium. Figure 11. The onset of the plastic wrinkling for different blankholder force(BHF).