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hardening (kinematic hardening, isotropic hardening
and directional hardening), Zhang and Hu (1998)
developed a mathematical model for predicting sheet
springback of bending and calculated residual stress distribution through thickness after springback. Ac-
cording to exponential hardening law and Von Mises
yield criterion, Buranathiti and Cao (2004) proposed
an analytical model for predicting springback after
sheet straight flanging. However, since these re-
searches ignored the effects of neutral surface shift
and contact pressure between the sheet and die, they
were limited to the springback problem of bending
ratio Ri/t0≥5. For the bending ratio Ri/t0≥5, the effects
of contact pressure, transverse stress, neutral surface
shifting on sheet springback of bending are signifi-
cant, with the analysis models which ignore them
being inaccurate as the bending ratio decreases. Hill
(1950) proposed a bending theory in which the effect
of transverse stress was considered, but applied the
ideal elastic-plastic material model, which cannot
reflect the real material property. Chen (1962) ex-
tended Hill’s work to exponential hardening model,
but he ignored the effects of bending arm and contact
pressure. Robinson (2000) estimated the errors in-
troduced by the ignorance of transverse stress and
concluded that the transverse stress was an important
factor in the bending analysis. And the contact pres-
sure is another factor ignored by previous analysis
(Tekaslan et al., 2006; Tekiner, 2004). By using a
simplified method in which the transverse stress of
shell element is considered, Cho et al.(2002) con-
cluded that transverse stress induced by contact
pressure has much influence on the sheet metal
forming and springback analysis.
An analytical model is proposed in this paper to
predict the sheet springback of V-bending. This
model is based on Hill’s yielding criterion and plane
strain condition, and takes into account contact
pressure, transverse stress, neutral surface shifting
and sheet thickness thinning. The effects of contact
pressure, the length of bending arm between the
punch and die, neutral surface shifting and sheet
thickness thinning on sheet springback were studied.
The predicted results by the analytical model were
compared with FEM simulating results. And the cal-
culation of bending ability—the limit bending ratio
(Ri/t0) for a given sheet metal thickness and material
properties is also presented in this paper.
considered as bending under the actions of contact
pressure q and bending moment M as shown in Fig.1.
The following assumptions are applied: (1) The sheet
is wide enough relative to its thickness. Therefore the
strain in the width direction is zero; (2) Straight lines
perpendicular to the neutral surface remain straight
during the V-bending process; (3) Volume conserva-
tion is kept during V-bending process; (4) Bausch-
inger effect is neglected and only elastic deformation
occurs during the unloading process.
Sheet thinning and neutral surface shifting after
V-bending
According to Assumptions (1) and (3), the area
of sheet cross-section remains constant during
V-bending process, that is:
where L0, t0 are initial sheet length and thickness,
respectively; θ is the bending angle; Ri, Ro are the
radii of concave and convex surface.
Since the neutral layer length remains constant
during V-bending, the radius of neutral surface Rn can
be defined as: 摘要：模具分离后，薄片材料应力分布的重新分配会引起回弹。精确预测回弹片模具的设计是非常重要的。根据希尔的屈服准则和平面应变条件，分析模型的提出在本纸，这需要考虑到的长度弯曲手臂冲床和模具之间，横向应力，中性表面转移和板材厚度变薄上上的接触压力的影响V形弯曲的片材的回弹。这个分析模型所预测的结果表明，接触的压力和横向应力有很大的影响回弹时的弯曲比（冲头半径为板材厚度的比值）是不到5。接触压力下降时，弯曲手臂的长度上升，这意着更短的长度弯曲手臂，将导致更大的回弹。中性面移回弹的效果小于接触压力随弯曲比。然而，这项研究表明，厚度变薄对于回弹的影响可以忽略不计。采用有限元法（FEM）模拟结果比较显示，符合与有限元仿真结果的分析模型所预测的结果。除此之外，该弯曲的能力对于一个给定的片材的厚度和材料特性的极限弯曲比率也被确定。