塑性力學三次降伏面之幾何分析

Geometric Analysis of Cubic Yield Surfaces in Plasticity

黃仲均(Chung-Chun Huang)

指導教授 ： 洪宏基

國立臺灣大學/工學院/土木工程學研究所/碩士(2019年)

https://doi.org/10.6342/NTU201902330

摘要

本論文延續國立臺灣大學力學聲響振動實驗室所進行之降伏面實驗與理論研究。本實驗室致力於觀察材料鋁合金6061在軸扭環三向應力空間中初始降伏面與接續降伏面之演化過程，並發展出降伏面擬合方法，探討其降伏面演化行為。本實驗室使用經退火狀態之同一試棒給予不同的預應變路徑在不同環向應力下之進行軸扭平面刺探，並建立一套自動化實驗流程以排除人為判斷，探尋在不同的預應變路徑下三向應力空間之降伏點，並利用三次降伏函數擬合出三向應力空間中之降伏面。本論文亦重新檢驗[L.-W. Liu and H.-K. Hong: A description of three-dimensional yield surfaces by cubic polynomials, Journal of Engineering Mechanics, Vol. 143, no. 11, 2017.]所提出之三次多項式擬合方法，並給予其新的數學證明，使得該降伏函數不論在力學上抑或是幾何上具有更堅強的理論架構。藉由修改後的降伏函數，降伏面之扭曲現象更可以被清楚觀察。本研究中亦推導出該三次降伏函數的正則降伏函數，此函數可用於推導三刺降伏面的關聯性塑流，亦可探討其與應力不變量之關係。另外，本研究也提出了新的彈塑性模型，並可運用龐加萊群求出材料在不同時間的應力應變狀態，此模型亦可用於解釋大部分降伏面的演化模式包括平移、旋轉、扭曲與等向伸縮。本研究亦研究出該彈塑性模型所有塑性參數的求得方式，並與實驗資料做比對與討論。

關鍵字

軸扭環應力空間 ； 三向降伏面 ； 降伏面擬合 ； 降伏面演化 ； 龐加萊群

並列摘要

This thesis carries on the researches on elastoplastic theory in yield surface evolution experiments conducted in NTU MSV lab. Our lab devotes to observing the evolution processes of yield surfaces in three dimensional axial-torsional-hoop stress space for alloy Al 6061. Also, we develop the fitting methods of the yield surfaces and study the behavior of the evolution of the yield surfaces. Thin-walled tubes at annealed state were subjected to multiple pre-strain paths in order to probe the yield points of the yield surfaces in two dimensional axial-torsional stress space in our experiments. In addition, our lab developed a fitting method that can fit the yield surfaces in three-dimensional stress space well. This fitting method proposed by [L.-W. Liu and H.-K. Hong: A description of three-dimensional yield surfaces by cubic polynomials, Journal of Engineering Mechanics, Vol. 143, no. 11, 2017.] is polished in this thesis by a new mathematical proof, which is helpful for us to observe the distorted cubic yield surfaces. Moreover, this thesis also proposes two elastoplastic models which can describe four different modes of yield surface evolution. The exact solutions of the models can be calculated by Poincaré groups. Also, procedures to determine the material constants of the models are developed, and the experimental data analyses of the two models are also conducted and discussed.

並列關鍵字

Axial-torsional-hoop stress space ； Three-dimensional yield surfaces ； Fitting of yield surfaces ； Evolution of yield surfaces ； Poincaré group

參考文獻

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[2] R.-C. Chang, Experiment and analysis on evolution of yield surface in three-dimensional stress space for aluminum alloy. Master Thesis, Department of Civil Engineering, National Taiwan University, 2012.

[3] C.-Y. Cheng, On the evolution of yield surfaces in three-dimensional stress space and some results of aluminum alloy. Master Thesis, Department of Civil Engineering, National Taiwan University, 2011.

[4] M.-K. Chung, Fitting Method and Geometric Analysis of Evolution of Yield Surfaces in Axial-Torsional-Hoop Stress Space. Master Thesis, Department of Civil Engineering, National Taiwan University, 2018.

[5] H. P. Feigenbaum and Y. F. Dafalias, “Directional distortional hardening in metal plasticity within thermodynamics,” International Journal of Solids and Structures, vol. 44, no. 22, pp. 7526–7542, 2007.