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研究生:李奇富
研究生(外文):Chi-Fu Li
論文名稱:彈塑性模式混合控制的計算方法
論文名稱(外文):The numerical algorithms for elastoplastic models under mixed control
指導教授:劉進賢
指導教授(外文):Chein-Shan Liu
學位類別:碩士
校院名稱:國立臺灣海洋大學
系所名稱:機械與輪機工程學系
學門:工程學門
學類:機械工程學類
論文種類:學術論文
論文出版年:2004
畢業學年度:92
語文別:中文
論文頁數:110
中文關鍵詞:保群算法混合控制相對誤差等誤差圖
外文關鍵詞:Group preserving schemeMixed-controlledRelative errorsIsoerror mapsArmstrong-FrederickPrager
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本文中我們嘗試在5維的應力空間中使用保群算法尋找有背應力的Armstrong-Frederic走動硬化模式的正確解,及利用混合控制的想法去重建Prager走動硬化模式,希望能達到傳統控制所不能作的.更進一步的,我們添加材料函數的觀念使這個模式可以描述更複雜的應變硬化現象,並且簡短的討論混合控制失效的情況,並與本模式比較.
首先我們使非線性動態系統,轉變成為一個局部李形式增廣動態系統,在此同時使用Cayley 轉換和指數轉換運用在增廣空間中發展群的保持算法.這個方法具有在每一個時間步長增加之後自動更新在錐形上的增廣狀態點的能力.靠著投射,我們可以得到狀態空間 的數值方法,其形式類似Euler法,但是具有可調整的權重因子.經由與傳統的徑向回復法比較,可以確定保群理論的正確性與效率性.
一個好的數值方法應該有精準的應力大小性與應力方向性.相對誤差用來確定應力大小性.等誤差圖用來確定應力方向性.由這一系列的討論與圖表,我們可以確定此數值方法的優異表現.
In this paper we attempt to use group preserving scheme (GPS) in five-dimensional space to search numerical solution of the Armstrong-Frederick kinematic hardening model with back stress, and employ the mixed-controlled idea to re-constitute the Prager kinematic hardening model by hoping to reach the result that the traditional strain or stress control can not be applied. Furthermore, we make the model more complicated with material function being employed and discuss briefly the mixed-control model failure, and compare it to the original model to get more useful information.
In this thesis we first convert the nonlinear dynamical system,into an augmented dynamical system of Lie type ,locally. In doing so, the inherent symmetry group and the (null) cone structure of nonlinear dynamical system are brought out; then the Cayley transformation and the exponential transformation are utilized to develop group preserving schemes in the augmented space. The schemes are capable of updating the augmented state point to locate automatically on the cone at the end of each time increment. By projection we thus obtain the numerical schemes on state space , which have the form similar to the Euler scheme but with the weighting factor being adaptive. The classic radial return method (RRM) is employed to compare with group preserving scheme, which ascertains the accuracy and efficiency of the latter scheme.
A good scheme should be accurate both in stress magnitude and in stress orientation. The relative errors in strain and stress are employed to assess the stress magnitude, and the isoerror maps are employed to check the stress orientation. A series of compassions and Figures are utilized to assess the accuracy and efficiency of our algorithms.
誌謝 I
摘要 III
Abstract IV
Contents VI
Figure Contents IX
Chapter 1 Introduction 1
1.1 Overview 1
1.2 Literature review 2
1.3 Organization 5
Chapter 2 Geometrical numerical algorithms for plasticity model with Armstrong-Frederick kinematic hardening rule under strain and stress controls 6
2.1 Introduction 6
2.2 Armstrong-Frederick model 7
Part one: Strain-controlled case 10
2.3 Integral representations 10
2.4 Internal symmetry under stain control 13
2.5 Geometrical integrator under strain control 16
2.6 Radial return method under strain control 19
Part two: Stress-controlled case 22
2.7 Internal symmetry under stress control 22
2.8 Geometrical integrator under stress control 24
2.9 Radial return method under stress control 25
Part Three: Numerical comparisons and conclusions 26
2.10 Numerical results and comparisons 26
2.10.1 Stress relative error 26
2.10.2 Isoerror maps 28
2.11 Conclusions 29
Chapter 3 Mixed control of elastoplastic model with Prager hardening rule and numerical computation 35
3.1 Introduction 35
3.2 Exact linearization of the flow model under strain control 38
3.2.1 Operators of generalized strain 38
3.2.2 Switch of plastic irreversibility under strain control 39
3.2.3 Linear system under strain control 40
3.3 Exact linearization of the flow model under stress control 41
3.3.1 Operators of generalized stress 41
3.3.2 Switch of plastic irreversibility under stress control 41
3.3.3 Linear system under stress control 42
3.4 Mixed stress-strain control of the flow model 43
3.5 Quasilinear mixed-control equations 44
3.6 Twin-cone structure and internal symmetry groups 46
3.7 Group-preserving scheme and numerical examples 50
3.8 Realization of mixed-control equation in a pseudo-Riemann manifold 52
3.9 Radial return method 57
3.10 Numerical examples and conclusions 58
3.10.1 Numerical examples 58
3.10.2 Conclusions 63
Chapter 4 Mixed hardening of material model under mixed stress-strain control and numerical implementation 64
4.1 Introduction 64
4.2 Quasi linearization of the flow model under strain control 67
4.2.1 Operators of generalized strain 67
4.2.2 Switch of plastic irreversibility under strain control 68
4.2.3 Quasilinear system under strain control 69
4.3 Quasi-linearization of the flow model under stress control 70
4.3.1 Operators of generalized stress 70
4.3.2 Switch of plastic irreversibility under stress control 71
4.3.3 Linear system under stress control 71
4.4 Mixed control of the flow model 72
4.5 Mixed-mode failure and mixed-mode loading condition 74
4.5.1 Mixed-mode failure 74
4.5.2 Loading condition, straining condition and mixed-mode loading condition 75
4.6 Quasilinear mixed-control equations 76
4.7 Twin-cone structure and internal symmetry groups 79
4.8 Group-preserving scheme and numerical examples 82
4.9 Realization of mixed-control equation in a pseudo-Riemann manifold 85
4.10 Numerical examples and conclusions 89
4.10.1 Numerical examples 89
4.10.2 Conclusions 95
Chapter 5 Conclusions 96
5.1 Conclusions 96
5.2 Future advance 97
References 98
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