Method of applying the radial return method to the anisotropic hardening rule of plasticity to sheet metal forming processes

A method for predicting deformation of a sheet of metal during a draw forming process designed to form the sheet metal into a part, said method for use with a computer having a memory and sheet forming tools, said method comprising the steps of:

• obtaining a strain increment, Δε, for a load step associated with initial loading, unloading and reloading without a break in the yield surface of the sheet metal in the sheet forming tools;
• calculating the total stress for the strain increment by Mroz's hardening rule according to the following yield surface equation;
• f≡rx2+ry2−α1rxry+α2rxy2−k2≡(1−α1/2)(rx+ry)2/2+(1+α1/2) (rx−ry)2/2+α2rxy2−k2=0
• where α1=2R/(1+R) and α2=2(1+2R)/(1+R),
• ri=σi−ai, i=x, y, xy
• Δai=AΔkbi
• Δεi=Δεie+Δεip
• Δσ=HΔεe
• Δεip=ΔΛ∂f/∂σi
• where ΔΛ=Δεp/2k
• Δσ=H(Δε−ΔΛ∂f/∂σ)
• σ=σE−ΔΛH∂f/∂σ
• where σE is the elastic trial stress vector and σE=σ0+HΔε
• r=rE−ΔΛH∂f/∂σ, where rE=σE−a
• applying the radial return method to the yield surface equation according to the following equations:
• rx+ry=(rxE+ryE)/[1+EΔΛ(2−α1)/(1−ν)] (9a)
• rx−ry=(rxE−ryE)/[1+EΔΛ(2+α1)/(1+ν)] (9b)
• rxy=rxyE/[1+EΔΛα2/(1+ν)] (9c)
• where E=Young's mnodulus
• ν=Poisson's ratio
• ΔΛ=Δεp/2Y
• solving the yield surface equation by using Newton's method of iteration;
• obtaining a strain increment, Δε, for a load step associated with reloading with a possible break in the yield surface of the sheet metal in the sheet forming tools;
• calculating the total stress for the strain increment by Mroz's hardening rule according to the following yield surface equation:
• f≡rx2+ry2−α1rxry+α2rxy2−k2≡(1−α1/2)(rx+ry)2/2+(1+α1/2)(rx−ry)2/2+α2rxy2−k2=0
• where α1=2R/(1+R) and α2=2(1+2R)/(1+R),
• ri=σi−ai, i=x, y, xy
• Δai=AΔkbi
• Δεi=Δεie+Δεip
• Δσ=HΔεe
• applying the radial return method to the yield surface equation according to the following equations:
• rxE+ryE=[(σ0x−ax)+(σ0y−ay)]+β(ΔσxE+ΔσyE)
• rxE−ryE=[(σ0x−ax)−(σ0y−ay)]+β(ΔσxE−ΔσyE)
• rxyE=(σ0xy−axy)+βΔσxyE;
• solving the yield surface equation for β, whereby for β<1, existing data is used to determine the total stress and reset the strain increment;
• repeat the step of calculating the total stress for the strain increment using Mroz's hardening rule until β≧1;
• and
• for β≧1, the following equations are applied to the yield surface equation and the yield surface equation is solved using Newton's method of iteration:
• rx+ry=(rxE+ryE)/[1+EΔΛ(2−α1)/(1−ν)]
• rx−ry=(rxE−ryE)/[1+EΔΛ(2+α1)/(1+ν)]
• rxy=rxyE/[1+EΔΛα2/(1+ν)]
• (1−α1/2)(rx+ry)2/2+(1+α1/2)(rx−ry)2/2+α2rxy2−k2=0.