摘 要 本文之目的在於驗證變分上界限法應用於圓環鍛粗加工問題的妥適性，並討論自然邊界條件及中立面幾何形狀對上界限解的影響。 對於圓環鍛粗問題的上界限理論解析，以往的研究重點，在於建立滿足動可容邊界條件(kinematically admissible boundary conditions)的速度場，並藉以求得最小值的上界限成形功率為目標。惟其解析結果，雖可提供實務加工上對成形所需功率計值的參考，但對於被加工件的變形行為分析與實驗結果卻有相當的差距。為此，本文利用了變分上界限法(Variational Upper-Bound method, VUB method)，以期在成形功率與變形行為分析雙方面均能提供更為精確的預測。變分上界限法因應用變分原理(variational principle)以最小化上界限能量方程式，故可同時獲得動可容邊界條件以及一組統御質點塑流變形的自然邊界條件(natural boundary conditions)。前述的自然邊界條件係以變分方法推導而得，且被應用於最佳化求解過程中。 本文的另一個研究目的，則在探討中立面幾何形狀的效應，因為經由實驗及有限元素的結果發現圓環鍛粗之中立面，未如以往上界限解所假設的圓柱形狀。為達此目的，因此本文將中立面視為一較高階的函數而求解，以期待所獲得之上界限解會有所改進。 為驗證本文所採方法的妥適性及應用性，其結果除和MARC有限元素解互做比較討論外，亦利用實驗所得之圓盤及圓環鍛粗加工之桶脹輪廓以及校正曲線(即在不同摩擦條件下，圓環最小內徑變化率對縮減量之關係曲線)的數據直接驗證。這結果證明了，在考慮中立面為一函數型態且滿足自然邊界條件下，不僅能影響環鍛時的校正曲線及桶脹輪廓，亦可改善成形能量。 ABSTRACT The objective of this thesis is to verify the validity of the variational upper-bound(VUB) method in analysis of upset forging of rings. The effect of natural boundary conditions and geometrical shape of the neutral surface on the upper-bound solution would be investigated in this project as well. As for the upper-bound method on the theoretical analysis of upset forging of rings, the most research interests concentrate on the kinematically admissible velocity field for a lower upper-bound solution of energy dissipation. The analytical result could provide some important reference for the metal forming process. However, a large gap exists between the theoretical prediction on deformation behavior and the experimental results. Therefore, in this research work, a variational upper-bound (VUB) method is used. It is the method that determines an upper-bound solution using variational calculus. Consequently, in addition to the kinematically boundary condition, a set of natural boundary conditions (NBCs) can be derived theoretically and can be applied to approximate the solution. These NBCs were found to affect the upper-bound solution of energy dissipation as well as the pattern of metal deformation significantly. The other objective of this thesis is to investigate the effect of the geometrical shape of the neutral surface, since the results of experiment and FEM has shown that the neutral surface of the upset forging of ring is not a cylindrical shape as was assumed in the upper-bound solution. For this purpose, the radius of the neutral surface, which was assumed as a constant, will be assumed as a function of higher order in analysis. It is expected that the solution determined in this manner should be improved. In order to verify the validity and applicability of the present method, the result obtained in this research proposal will be compared and discussed with FEM solution, and also experimentally verified by the bulged profiles of upset disks and rings, and the calibration curves which indicate the relation between the decrease in minimum internal diameter of upset rings and the reduction under different interfacial friction conditions. The result proves that while considering the neutral surface as a function form and satisfying the natural boundary conditions, not only the calibration curves and bulged profiles in upset ring will be influenced, but also the forming energy will be improved.
