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研究生： 沈孟韋
Shen, Meng-Wei
論文名稱： 負勁度複合材料之相場模擬研究
PHASE FIELD MODELING OF NEGATIVE STIFFNESS COMPOSITE MATERIALS
指導教授： 王雲哲
Wang, Yun-Che
學位類別： 碩士
Master
系所名稱： 工學院 - 土木工程學系
Department of Civil Engineering
論文出版年： 2017
畢業學年度： 105
語文別： 英文
論文頁數： 114
中文關鍵詞： 鐵彈性材料 、鐵彈疇 、有限元素法 、負普松 、負勁度 、相場模擬
外文關鍵詞： ferroelastic materials, ferroelastic domain, finite element method, auxetics, negative stiffness, phase field modeling
相關次數： 點閱：110  下載：6 
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    • 本論文藉由相場模擬方法探討二維鐵彈材料在鐵彈性相變情況時，其整體材料的力學性質變化。在本研究中並無直接使用負的彈性模數，主要探討因固-固相變所產生之負勁度效應。當模型同時受到外加應變與溫度變化的刺激下，可觀察到在相變附近時，整體材料因負勁度效應而使等效黏彈模數與阻尼有所提升，而這結果與實驗結果和古典複合材料理論對有負的模數情況下的預測一致。在整體模型所受溫度固定為0.5，也就是在系統具有兩個能量低點的情況下，不論外加力如何，可觀察到鐵彈疇的形成。若溫度固定為1.7，則不會有疇形成。而在只有溫度循環變化而無外力加載下，疇不會隨溫度變化而有消失與再形成現象。在同時有溫度循環與變形加載情況下，因在相變化附近所產生之負勁度效應，可發現在快速溫度變化下的單軸正弦加載可使等效勁度提高約15倍。而在其他像是雙軸剪力加載與二維體變形加載情況下，也可觀察到等效勁度的提升，但其提升幅度較單軸加載情況下低。此外，含裂紋的模型在相同條件下所觀察到的負勁度誘發之整體勁度提升幅度較正常相場模型來的低，且鐵彈疇相變會從裂紋尖端附近開始發生。而在泡沫鐵彈材料模型中，可觀察到無論是正普松還是負普松材料模型都與連續鐵彈體有類似的行為，但在泡沫模型中因相變所測得之等效應力變化會有因泡沫結構中枝幹影響而產生一些振盪情況。在一維鐵彈複合模型中，可觀察到在相變過程中，加載位移與鐵彈內含物的應力之間的關係相反。在相變發生時，可看到鐵彈內含物的應力會減少，而在彈性基材中在靠近加載點的應力卻有明顯增加。

    This research studies the effects of ferroelastic phase transition on the overall mechanical properties of the 2D materials by the phase field simulation (PFS) method. Without directly using negative values in elastic moduli, PFS enables the study of negative-stiffness effects due to the phase transformation. Under simultaneous straining and temperature cycling, enhancements on effective viscoelastic modulus and damping in the vicinity of phase transition are observed, in consistence with classical composite theory with negative modulus and experiment. Regardless of externally applied stress, ferroelastic domains may be formed when constant temperature is set to T=0.5 since the system possessing two energy minimal points. If the constant temperature is set to be T=1.7, no domains will be formed. Under no external stresses, domains will not re-form under temperature cycling. The uniaxial deformation under the fast temperature rate may enhance effective stiffness due to the negative stiffness effects generated from phase transition by 15 folds. Enhancements on effective stiffness are also observed in biaxial shear and 2D bulk deformation, but their magnitudes are less than those in the uniaxial mode. The cracked ferroelastic materials show less NS-induced enhancement than those intact ones, and the domain evolution would start from crack tips. Foamed ferroelastic materials, auxetic or non-auxetic, show similar behavior as continuum ferroelastic media, but some oscillatory responses were observed in stress response of the foams, which may be due to the responses of each rib in the foam.

    CHINESE ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Goals and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Negative stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.2 Landau’s theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.3 Phase field modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Theoretical considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1 Phase field modeling of one-dimensional systems . . . . . . . . . . . . . . . . 6 2.2 Phase field modeling of two-dimensional systems . . . . . . . . . . . . . . . . 6 2.2.1 Effects of thermal expansion . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.2 Weak form for ferroelastic and ferroelectric . . . . . . . . . . . . . . . 12 2.3 Phase field modeling of three-dimensional systems . . . . . . . . . . . . . . . 14 3 Numerical considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.1 Finite element solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2.1 Phase field model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2.2 Electric field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2.3 Thermal expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2.4 Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.1 Static energy profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.2 Initial chess-board pattern polarization under constant temperature . . . . . . . 30 4.2.1 No mechanical loading . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2.2 With uniaxial sinusoidal straining . . . . . . . . . . . . . . . . . . . . 32 4.3 Initial chess-board pattern polarization with temperature cycling . . . . . . . . 45 4.4 Monotonic straining under temperature cycling for directly detecting negative stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.5 Initial chess-board pattern polarization with triangular-wave temperature cycling and sinusoidal straining . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.5.1 Convergence study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.5.2 Uniaxial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.5.3 Biaxial shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.5.4 Two-dimensional bulk deformation . . . . . . . . . . . . . . . . . . . 67 4.6 Ferroelastic material with cracks . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.6.1 Notch-shape crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.6.2 Penny-shape crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.7 Ferroelastic materials with foamed microstructure . . . . . . . . . . . . . . . . 85 4.8 One-dimensional ferroelectric phase field model . . . . . . . . . . . . . . . . . 93 4.9 One-dimensional ferroelastic phase field model with thermal expansion . . . . 94 4.10 One-dimensional ferroelastic composite phase field modeling . . . . . . . . . . 96 5 Conclusions and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 APPENDICES Appendix A: Cases studied in this thesis . . . . . . . . . . . . . . . . . . . . . . . 107 Appendix B: Derivation of the normalized quantities . . . . . . . . . . . . . . . . 108 Appendix C: Phase-field modeling of crack propagation . . . . . . . . . . . . . . 110 VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 vi

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