摘 要 本文的目的在於應用變分方法探討中立半徑模式對於上界限解的影響，且視中立半徑於鍛粗過程中為一函數型態，以取代一般將其視為常數的假設。本文首先根據流函數（Stream Function）所定義呈函數型態的動可容速度場及變形場，建立鍛粗加工所需的上界限能量消耗的總功率，其次以變分法（Variational Method）最小化總功率的泛函數以獲得靜可容邊界條件，並應用於求取上界限解。本文所採用之數值解析，係利用IMSL［25］最佳化程式"NCONF"。 為驗證理論分析的妥適性，本文採用圓環及圓盤鍛粗加工的實驗結果［5,6,9,18］和理論計值互做比較及討論。此外，本文亦探討各種加工因素，如摩擦條件（Friction Condition）、高度縮減率（Reduction in Height）對上界限解的影響。本文結果顯示，在考慮中立半徑為一函數型態且滿足靜可容邊界條件下，對於校正曲線、圓環及圓盤桶脹輪廓的預測有較佳的結果。 Abstract The objective of this thesis is to investigate the effect of neutral radius on the upper-bound solution of the upset forging of rings using a variational approach. To this end, the neutral radius of the upset ring is considered an explicit function of polynomials during deformation process, and the upper-bound energy equation is formulate in terms of the explicit function as well as the velocity field derived from the theory of stream function. Since the velocity field was considered unknown functions before the upper-bound energy equation is extremized, a variational approach is required. As a result, a set of boundary conditions can be derived. To determine the upper-bound solution, the set of boundary conditions, which include the so-called natural boundary conditions, was imposed in the optimization procedure. In numerical analysis, an optimization program "NCONF" of the IMSL［25］is utilized. Some experimental results of the upset disks［6,18］and ring［5,9］were employed for discussion and comparison among the theories. It also discussion the effect of working factors, such as the friction condition and the reduction in height on upper-bound solution. From the result we can clearly indicate that the solution in predicting the calibration curve as well as the bulged profiles of the upset disks and ring would be better when it is based on the model of neutral radius considered, and satisfies the natural boundary condition.