EFFECT OF ANISOTROPIC YIELD CRITERION ON THE SPRINGBACK IN PLANE STRAIN PURE BENDING

Abstract

Plastic anisotropy is one of the multiple material representations significantly affecting the springback simulations. The present paper studies the influence of the plastic anisotropy over the springback in plane strain pure bending by means of the exact semi-analytical solution. The yield criterion and the constitutive equations for the orthotropic material with consideration of the crystal lattice constants and parameters of the crystallographic texture are used taking into account the anisotropy.

Keywords

• anisotropy
• crystallographic orientation
• yield criterion
• springback
• bending
• plane strain
• elastic/plastic solution

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As it is evident from the production analysis of the Russian plants the metal loss in sheet metal forming processes only amounts up to 45-50%: metal utilization factor in aircraft industry is about 40%, in engine industry - 30% and in automotive industry - 50%. Presumably, waste is much higher due to the collateral loss (about 10-15%). In a greater degree, these losses arise from the undesirable anisotropy of the blank properties. For most materials the anisotropy of physical and mechanical properties is a rule rather than an exception [1]. Generally, anisotropy, which stems from the crystal structure of materials, determines processing and service characteristics of the products. As to metal forming processes, the undesirable anisotropy not only distorts the shapes and sizes of the parts, but also limits the formability, causing the excessive thinning of the material [2]. Therefore, in order to compensate thinning, it is necessary to increase the initial sizes of the blanks, which in turn results in the metal loss, the weight growth of the construction, the technological cycle extension, and, consequently, in the numerous additional expenses (equipment, areas, power resources, workers, etc). However, as it follows from the numerous researches [3-6], the influence of anisotropy has more advantages rather than disadvantages. For example, the considerable achievements in material science of the electric steels (e.g. increasing magnetic induction and capacitance, cutting electric power losses) became possible mainly due to developing efficient anisotropy of physical properties [7-8].

From the above it appears, firstly, the instant necessity of the anisotropy impact assessing in processing and service characteristics of products; secondly, the necessity of developing special methods for obtaining and efficient application of anisotropy. Unfortunately, the imperfection of the process design techniques does not allow considering anisotropy of the blank properties [9-10] and, as a result, makes the excessive metal consumption inevitable at the production stage.

A comprehensive overview on springback that follows a sheet forming process when the forming loads are removed from the workpiece has been provided in [11]. It is emphasized that plastic anisotropy is one of the various material representations that significantly affects the springback simulations. In the present paper, the effect of the springback plastic anisotropy in the plane strain pure bending is studied by means of the exact semi-analytical solution. 2

Theory of bending at large strain

The theory of the elastic/plastic plane strain bending of an incompressible anisotropic sheet obeying the quadratic Hill yield criterion [12] has been presented in the article [13]. Ultimately, the main results of this work are represented in this section. The solution is based on the following mapping between the Eulerian Cartesian coordinate system (x, y) and the Lagrangian coordinate system ( , ) Hx  a  as2 cos 2a   as , Hy  a  as2 sin 2a  , ( 1 ) where H is the initial thickness of the sheet, s is an arbitrary function of a , which is a function of the time, t , and a  0 at t  0 . It follows from Eq. ( 1 ) that x   H and y  H at a  0 if s  1 4 at a  0.

It can be verified by the inspection through the application of l’Hospital’s rule to Eq. ( 1 ), with the use of Eq. ( 2 ), as a  0 . Eq. ( 1 ) and ( 2 ) a transformation description of the rectangle defined at the initial instant a  0 , by the equations x  H , x  0 and y  L (or, in the Lagrangian coordinates, by the equations   1,   0 and    L H ) into the shape determined by the two circular arcs, AD and CB, and the two straight lines, AD and CB (Fig. 1).

It is convenient to introduce a moving cylindrical coordinate system ( r, ) by the equations of transformation ( 2 ) Hx  as  r cHos and Hy  r sHin . T   ln  saas  , T  ln  saas   2 in the region 1     2 a and ( 3 ) ( 6 ) where X , Y and Z are the tensile yield stresses in the  - ,  - and z - directions,   1 ln2 4  a  s  

T p p  ln  4s  , 4   1 ln2 4  a  s  

T p in the region  2 a     1 a . Here p  ln 4s   4 4 p

ln 4  a  s  s  T  2a  1 4a2 4 ,  1 a   exp  p 2  4s 4a

,  2 a 

X 2Y 2Z 2 X 2Y 2Z 2 Y 2  Z 2   Y 4Z 4  X 4 Y 2  Z 2 2 exp  p 2  4s

, , p  4a 2T G , respectively, and G is the shear modulus of elasticity.

The bending moment per unit length is defined by

rAB M    rdr.

rCD Using Eq. ( 1 ) and ( 3 ), this equation can be transformed to

  R02  R12  m  T2HM2  1a 01T d , where m is the dimensionless bending moment.

In the case of the purely elastic unloading, the stress increments are 2    4 ln  r   R0 

T p C1  R12 ln  R1 R0   R0    r  1 , T  4p C1  R12 Rln02R1R12R0  ln  Rr0  1  Rr202 1, 1 C1  m f p  H 2   R12 R02 12  4 ln  R1  .

2  R0    R12 R02  ln  R1 R0   R0  Here m f , R0 and R1 are the values of m , rCD and rAB at the end of loading. Also, r  H  a f

 as 2ff , where a f and s f are the values of a and s at the end of loading. Using Eq. ( 7 ), in which a and s should be replaced with a f and s f from Eq. ( 11 ), the variation of ( 7 ) ( 8 ) ( 9 ) ( 10 ) ( 11 )  and  with r H at the end of loading is determined in parametric form with  being the parameter. Then, the distribution of the residual stresses are found from res  f   , res  f   .

The radius rCD after unloading, Ru , is determined as R

u  1 C1  R0

C1  R02  R12  2R12 ln  R1 R0 

. 3

Material model

where hi , ki , li are Miller’s indices, which determine the i-th direction in the crystal with respect to the principal axes of anisotropy.

Considering the tension along the principal axes, let us determine the tensile yield stresses from Eq. ( 13 ): Let us consider the bending of orthotropic rolled sheet. The principal axes of orthotropy coincide with the coordinate lines of the Cartesian coordinate system: x, y and z are thickness, transverse and rolling directions, respectively, i.e. in the case under consideration the bending line coincides with rolling direction. Therefore, the yield function proposed in [14] is: 23  xx  yy 2 12  yy  zz 2 31  zz  xx 2  4  23  x2y   12  2yz   31  z2x   2 e2q .

 5  5  5  2   2   2   Here  eq is the equivalent stress,  ij are the components of the stress tensor in the Cartesian system of coordinates. The generalized anisotropy factors ij are defined as 15 A 1  3  2 A  i   j  k 

1  ij  1   , ( 14 ) 5  where A is the anisotropy factor of a crystal lattice, i are the parameters of the crystallographic texture. The anisotropy factor is defined through the compliance constants of crystal lattice S1111 , S1122 and S 2323 For a certain crystallographic orientation hkl uvw the parameters of the crystalloA  S1111  S1122 .

2S 2323 graphic texture are defined as  i  hi2ki2  ki2li2  li2hi2 ,  hi2  ki2  li2 2 ( 12 ) ( 13 ) ( 15 ) ( 16 ) 12 23 Y 

, where Z is the tensile yield stress in the rolling direction.

Thus, the material model considers the crystal lattice constants ( A ), i.e. chemical composition of alloy, and the crystallographic texture ( i ), i.e. its thermo-mechanical treatment. Using this model, it is possible to analyze the influence of the ideal crystallographic orientations over the springback in bending. 4

Numerical results

Let us consider the copper sheet for which components of compliance tensor are S1111  15.0 TPa-1; S1122  6.30 TPa-1 and S 2323  3.33 TPa-1 [15], i.e. A  3.203 (Eq. ( 15 )); the tensile yield stress in the rolling direction Z is 340 MPa; the shear modulus of elasticity G is 44 GPa. As an example, let us take the following ideal crystallographic orientations: copper {112}<111>, brass {110}<112> (rolling components) and Goss {110}<001> (recrystallization component). Also, as a comparison let us consider the isotropic case, for which A  1 or i  1 5 [16], i.e. ij  1 . The parameters of crystallographic texture i , generalized anisotropy factors factors ij and parameter p calculated using Eq. ( 16 ), ( 14 ) and (18) for the stated components are listed in Table 1. The dependence of the bending moment on the radius of the concave surface at the beginning of the process is shown in Fig. 2. It is seen from this diagram that the effect of anisotropy is revealed at the very beginning of the process only. Note that the required bending moment for some ideal crystallographic orientations can be bigger ({112}<111>) or smaller ({110}<112>, {110}<001>) in comparison with the isotropic material. The dependence of the springback Ru R0 on value of radius rCD is depicted in Fig. 3. The figure shows that the springback decreases as the deformation at the end of loading increases. All other things being equal the orientations {110}<112> and {110}<001> cause bigger springback than the isotropic material and orientation {112}<111>, for which the springback is minimal. The influence of the ideal crystallographic orientations upon the thickness distribution of the residual stresses is illustrated in Fig. 4. In particular, the radial stress is shown H  60 . The figures speciin Fig. 4a and the circumferential stress in Fig. 4b at rCD fied reveal that the effect of anisotropy is most significant at the central part of the specimen. The distribution of the residual stresses outside the central part of the specimen is almost independent from the crystallographic orientation except for {110}<001>.

a) b) The plane strain pure bending of materials textured has been investigated. It has been stated that depending on the ideal crystallographic orientations the springback and the distribution of the residual stresses change grossly. Note that the springback of the textured metal sheet can be bigger ({110}<112>, {110}<001>) or smaller ({112}<111>) in comparison with the isotropic material.

Acknowledgements

The reported study was funded by RFBR according to the research project №16-3800495.

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