Effect of natural boundary condition and the neutral surface of nonlinear type on the upper-bound solution to upset forging of rings using a variational approach

A general upper-bound formulation of stream function of upset forging of ring using a variational approach

“…Although the problem being considered has been discussed elsewhere [15][16][17] using a variational approach also, the major difference between the present and previous works is that the function used to establish energy dissipation equation is different physically. Specifically, in the previous works [15][16][17] the function, assumed implicitly in prior, was interpreted as velocity field; however, in the present work it is introduced to represent a stream function also assumed implicitly in prior, and thus any equation derived would be pertinent to the stream function itself rather than the velocity field. As the stream function may be known to be a potential of the velocity and/or strain-rate fields, it is expected that the formulation to be derived for the stream function would become more widespread in applications for the problem considered.…”

“…Meanwhile, the constraint on surface traction or force condition due to friction existing on interfacial frictional interface, which, we believe, would also affect metal flow in upset forging of ring or disk as experimentally observed in [1,2,[11][12][13][14], has never been appropriately taken into consideration unless a variational approach is involved to derive it as reported in Refs. [15][16][17].…”

“…It was initially proposed by Yeh and Yang [18] and has been successfully applied to the problem of cup ironing [18], plane strain extrusion [19,20] and upset forging of ring or disk [15][16][17]. In the forgoing cited articles where the VUB method was applied, an upper-bound energy dissipation equation has been established as a functional of the velocity field, assumed as an implicit function in prior, and then extremized using the technique of variational calculus in an attempt to derive an equilibrium equation along with a set of associated boundary condition of the velocity field.…”

“…The derived boundary condition, which was termed as statically or naturally boundary condition and had never been accounted for in traditional upper-bound analyses such as in [1][2][3][4][5][6][7][8][9][10][21][22][23][24][25], was finally imposed to aim to determine upper-bound solution. The upper-bound solution determined in this way has been shortly termed the VUB solution, and has been shown to be better than the corresponding upper-bound solution that ignores the statically boundary condition in predicting the pattern of metal flow as well as the load required for a variety of metal forming problems [15][16][17]19,20]. Although the problem being considered has been discussed elsewhere [15][16][17] using a variational approach also, the major difference between the present and previous works is that the function used to establish energy dissipation equation is different physically.…”

“…The upper-bound solution determined in this way has been shortly termed the VUB solution, and has been shown to be better than the corresponding upper-bound solution that ignores the statically boundary condition in predicting the pattern of metal flow as well as the load required for a variety of metal forming problems [15][16][17]19,20]. Although the problem being considered has been discussed elsewhere [15][16][17] using a variational approach also, the major difference between the present and previous works is that the function used to establish energy dissipation equation is different physically. Specifically, in the previous works [15][16][17] the function, assumed implicitly in prior, was interpreted as velocity field; however, in the present work it is introduced to represent a stream function also assumed implicitly in prior, and thus any equation derived would be pertinent to the stream function itself rather than the velocity field.…”