本文在於延伸變分上界限法(VUB)的應用性，進一步考慮同時滿足自然邊界條件（即摩擦應力條件）及材料加工硬化條件下，利用變分的方法最小化上界限能量方程式求解圓盤及圓環鍛粗加工問題。根據上述方法所獲得理論計值，除以圓環及圓盤鍛粗的實驗結果做驗證外，亦探討加工因素，如胚料與模具邊界的摩擦條件、半高度縮減率對圓盤、圓環鍛粗加工速度場、塑性變形場及等效應變場的影響。結果證實，同時考慮自然邊界條件及材料加工硬化條件下的UB法其預測鍛粗變形行為的能力的確是最好的。

This article is to extend the applicability of the variational upper-bound method for the problem of upset forging of ring and cylinder by satisfying the natural boundary conditions (ie, the friction stress conditions) and accounting for material work-hardening characteristics. variational upper-bound solution can be determined by minimizing the upper-bound energy equation. Based on the above theoretical prediction, it is not only to verify by the experimental results of the ring and cylinder upset forging, but also to discuss the processing factors, such as the boundary friction conditions between the material and the mold, the half-height reduction rate of the disk and cylinder upsetting to affect the velocity field, the plastic deformation field and the equivalent strain field. The results confirm that taking into account the natural boundary conditions and material hardening characteristics of upper-bound method forecast upset forging deformation behavior is indeed the best.

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