Patent ID: 6205366
Filing Date: 2001-03-20
Classification: B21D

Abstract:
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, .DELTA..epsilon., 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.tbd.r.sub.x.sup.2 +r.sub.y.sup.2 -.alpha..sub.1 r.sub.x r.sub.y +.alpha..sub.2 r.sub.xy.sup.2 -k.sup.2.tbd.(1-.alpha..sub.1 /2)(r.sub.x +r.sub.y).sup.2 /2+(1+.alpha..sub.1 /2) (r.sub.x -r.sub.y).sup.2 /2+.alpha..sub.2 r.sub.xy.sup.2 -k.sup.2 =0where .alpha..sub.1 =2R/(1+R) and .alpha..sub.2 =2(1+2R)/(1+R),r.sub.i =.sigma..sub.i -a.sub.i, i=x, y, xy.DELTA.a.sub.i =A.DELTA.kb.sub.i.DELTA..epsilon..sub.i =.DELTA..epsilon..sub.i.sup.e +.DELTA..epsilon..sub.i.sup.p.DELTA..sigma.=H.DELTA..epsilon..sup.e.DELTA..epsilon..sub.i.sup.p =.DELTA..LAMBDA..differential.f/.differential..sigma..sub.iwhere .DELTA..LAMBDA.=.DELTA..epsilon..sup.p /2k.DELTA..sigma.=H(.DELTA..epsilon.-.DELTA..LAMBDA..differential.f/ .differential..sigma.).sigma.=.sigma..sup.E -.DELTA..LAMBDA.H.differential.f/.differential..sigma.where .sigma..sup.E is the elastic trial stress vector and .sigma..sup.E =.sigma..sub.0 +H.DELTA..epsilon.r=r.sup.E -.DELTA..LAMBDA.H.differential.f/.differential..sigma., where r.sup.E =.sigma..sup.E -aapplying the radial return method to the yield surface equation according to the following equations:r.sub.x +r.sub.y =(r.sub.x.sup.E +r.sub.y.sup.E)/[1+E.DELTA..LAMBDA.(2-.alpha..sub.1)/(1-.nu.)] (9a)r.sub.x -r.sub.y =(r.sub.x.sup.E -r.sub.y.sup.E)/[1+E.DELTA..LAMBDA.(2+.alpha..sub.1)/(1+.nu.)] (9b)r.sub.xy =r.sub.xy.sup.E /[1+E.DELTA..LAMBDA..alpha..sub.2 /(1+.nu.)] (9c)where E=Young's mnodulus.nu.=Poisson's ratio.DELTA..LAMBDA.=.DELTA..epsilon..sup.p /2Ysolving the yield surface equation by using Newton's method of iteration;obtaining a strain increment, .DELTA..epsilon., 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.tbd.r.sub.x.sup.2 +r.sub.y.sup.2 -.alpha..sub.1 r.sub.x r.sub.y +.alpha..sub.2 r.sub.xy.sup.2 -k.sup.2.tbd.(1-.alpha..sub.1 /2)(r.sub.x +r.sub.y).sup.2 /2+(1+.alpha..sub.1 /2)(r.sub.x -r.sub.y).sup.2 /2+.alpha..sub.2 r.sub.xy.sup.2 -k.sup.2 =0where .alpha..sub.1 =2R/(1+R) and .alpha..sub.2 =2(1+2R)/(1+R),r.sub.i =.sigma..sub.i -a.sub.i, i=x, y, xy.DELTA.a.sub.i =A.DELTA.kb.sub.i.DELTA..epsilon..sub.i =.DELTA..epsilon..sub.i.sup.e +.DELTA..epsilon..sub.i.sup.p.DELTA..sigma.=H.DELTA..epsilon..sup.eapplying the radial return method to the yield surface equation according to the following equations:r.sub.x.sup.E +r.sub.y.sup.E =[(.sigma..sub.0x -a.sub.x)+(.sigma..sub.0y -a.sub.y)]+.beta.(.DELTA..sigma..sub.x.sup.E +.DELTA..sigma..sub.y.sup.E)r.sub.x.sup.E -r.sub.y.sup.E =[(.sigma..sub.0x -a.sub.x)-(.sigma..sub.0y -a.sub.y)]+.beta.(.DELTA..sigma..sub.x.sup.E -.DELTA..sigma..sub.y.sup.E)r.sub.xy.sup.E =(.sigma..sub.0xy-a.sub.xy)+.beta..DELTA..sigma..sub.xy.sup.E ;solving the yield surface equation for .beta., whereby for .beta.<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 .beta..gtoreq.1; andfor .beta..gtoreq.1, the following equations are applied to the yield surface equation and the yield surface equation is solved using Newton's method of iteration:r.sub.x +r.sub.y =(r.sub.x.sup.E +r.sub.y.sup.E)/[1+E.DELTA..LAMBDA.(2-.alpha..sub.1)/(1-.nu.)]r.sub.x -r.sub.y =(r.sub.x.sup.E -r.sub.y.sup.E)/[1+E.DELTA..LAMBDA.(2+.alpha..sub.1)/(1+.nu.)]r.sub.xy =r.sub.xy.sup.E /[1+E.DELTA..LAMBDA..alpha..sub.2 /(1+.nu.)](1-.alpha..sub.1 /2)(r.sub.x +r.sub.y).sup.2 /2+(1+.alpha..sub.1 /2)(r.sub.x -r.sub.y).sup.2 /2+.alpha..sub.2 r.sub.xy.sup.2 -k.sup.2 =0.