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(4)e1=FLD0+e2(0.784854−0.008565 e2)
and in the right hand side of the FLD where e2<0, the values of e1 and e2 are related to each other as in below equation
(5)e1=FLD0+e2(0.027254 e2−1.1965)
The FLD0 is the engineering failure strain in plain strain condition where e2=0. The statistical value of FLD0 can be calculated using below equation
(6)FLD0=n0.2116(23.25+356.1C1)
The value of C1 for thicknesses (t0) less than 0.29972 mm is equal to t0/25.4. However for larger thickness values, C1 is considered to be equal to 0.0118.
Calculating the values of e1 and e2, the true major and minor strains of the FLD (ɛ1 and ɛ2) are calculated using below equation
(7)ɛ1=ln(1+e1100)ɛ2=ln(1+e2100)
Using the statistical FLD as the ductile fracture criterion, for each element of the blank in FE model, the minor strain of this element is used and the major failure strain is calculated using the present method. If the true strain value of this element is less than the calculated failure strain, the element is considered to be formed without any fracture and vise versa.
M-K Method Using Hill's Nonquadratic Yielding Criterion.
Dariani and Azodi, based on the M-K method, calculated the FLD using the Hill's nonquadratic yielding criterion [35]. According to the Hill's nonquadratic yielding criterion, in the plane stress state and for in-plane isotropic materials, the sheet metal yields when the following Eq. (8) is satisfied
(8)|σ1+σ2|M+(2r+1)|σ1−σ2|M=2(r+1)σMe
where M is the Hill index, r is the normal anisotropy ratio, σ1 and σ2 are the principal stresses. They suggested that the Hill index (M) of 2.5–4 has good agreement with the experimental FLD of different materials.
轴对称拉深部件的自动化工艺顺序设计和有限元模拟有限元仿真模块：
在这个模块中，通过自动模拟Theabaqus/EXPLICIT有限元软件获得产品的工艺步骤。该模块创建ABAQUS输入并将数据提交给Theabaqus/EXPLICIT软件执行。毛坯和模具将分别采用壳体和刚性元素。将毛坯建模在第一建模阶段，但在下一阶段的节点以及前一个阶段毛坯信心将被导入为初始状态。系统在自动执行ABAQUS软件后，能获得相应的信息间隔。运用成形极限图（FLD）或者韧性断裂准则系统检查该过程是否成功。如果不成功，系统将自动更改合适的工艺参数，即冲孔圆弧半径，模具圆弧半径，拉深比，摩擦系数和压边力。论文网
最初运用有限元模拟计算出的压边力，是利用公式（2）（参考文献36）：
压边力BHF=Pbh×Abh=10−3c[(R0rp−1)3+0.01R0t0]Su×π(R20−(rp+rd)2) 公式（2）
其中Pbh为坯料夹持压力，Abh为毛坯与坯料夹持间的接触面积，R0为初始毛坯半径，rp是冲孔圆弧半径，t0为初始毛坯厚度，rd为模具圆弧半径，Su为板材的拉深强度。因素c为常数，其大小可由使用者决定（通常取2到3之间）。默认c值为2。有限元法通过计算出的压边力进行模拟。该系统确定防皱杯状物在垂直方向进行拉深操作时，所有杯状物的节点距离。如果有一个节点距离大于预定值（默认为金属板厚度的10%），那么该杯状物起皱。如果在有限元模拟中观察到起皱现象，系统会增加影响压边力的c值，随后会重新计算压边力，重新进行有限元模拟。
如果在有限元模拟中预测出会出现断裂现象，那么系统将检测凹模圆弧半径和冲孔圆弧半径是否在安全区域中。如果它们并不处于安全区域中，所选择的参数将转移至安全区域，并更改其工艺步骤。如果它们处于安全区域中，系统将降低拉深比，并会将其对下一拉深阶段的影响应用于设计工艺步骤中。下一拉深阶段也是模拟该过程，该过程将一直进行到所有拉深阶段都成功为止。