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研究生：	楊培崙 Yang, Pei-lun
論文名稱：	微極彈性材料之壓印問題 Indentation problems of elastic micropolar 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：	2009
畢業學年度：	97
語文別：	英文
論文頁數：	68
中文關鍵詞：	負勁度複合材、微極性固體
外文關鍵詞：	micropolar, viscoelastic, Cosserats
相關次數：	點閱：56 下載：1
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發展高勁度及高阻尼材料是目前研究尖端材料的方向之一。複合材料理論預
測，含負勁度的二元複合材料，具有異常的高勁度及高阻尼。但近年的研究顯
示，此類複合材料的穩定性仍有待深入了解。本研究探討一特殊的微極性固體
材料，其楊氏勁度為負值，並將此負勁度當成內含物，埋入線彈性基材中，探
討此複合系統的整體勁度。本文先討論一個二維離散系統，並分析其穩定性，
結果顯示高勁度及高阻尼確實存在，但穩定性比一維系統弱。然後、使用
COMSOL軟體，分析負勁度複合材料的有效勁度，所分析的三個例子如下，(1)
平面應變方板含負勁度圓形內含物、(2) 二維Bousinesq問題含負勁度層、
(3)Hertz接觸問題含負勁度層。三個例子均顯示，有效勁度可藉由負勁度內含物
而大幅提升。應力分析顯示，負勁度與正勁度交界面上，承受相當大的應力，
此現象可解釋為何前人的實驗重複性不佳。連續體的穩定性分析，不在本論文
的討論範圍。此外、微極性固體材料所具備的尺寸效應，亦不在本論文的討論
範圍。

In this thesis, a special Cosserats medium (a kind of micropolar medium),
exhibiting negative stiffness, is considered for studying its modulus and viscoelastic
damping properties in the form of a composite material. First, a two dimensional
discrete model, hexagonal lattice structure, containing pre-chosen negative stiffness
components was studied. The stability of the structure is analyzed in accordance with
the Lyapunov indirect theorem. Strong motivations for this study come from the
needs of fundamental understanding of negative stiffness effects on structures
consisting of many degrees of freedom, as well as the symmetry of the structures. We
use the finite element method, combined with the technique of the state space
representation, to analyze the two-dimensional, nested hexagons composed of twoforce
components that obey the constitutive relation of the standard linear solid.
Effective stiffness and damping anomalies in terms of peaks and anti-peaks are
observed in the meta-stable regime when the amount of negative stiffness is
selectively tuned. According to the Lyapunov indirect stability theorem, the system as
a whole is stable when the tuning stiffness is greater than a critical negative value, but
meta-stable otherwise. The degree of meta-stability depends on viscosity in the
components. Furthermore, stiffness anomalies may be easier to be experimentally
observed from compression tests than hydrostatic or shear tests. Beyond the discrete
model, three continuum models were studied. Namely, the circular inclusion in a
square plate and two dimensional Boussinesq problem with sandwiched substrate.
Both of the models show anomalies in effective stiffness when negative-stiffness
inclusion is embedded.

摘 要............................................................................................................................................................... I
ABSTRACT.................................................................................................................................................II
ACKNOWLEDGEMENTS..................................................................................................................... III
TABLE OF CONTENTS......................................................................................................................... IV
LIST OF TABLES ......................................................................................................................................V
LIST OF FIGURES.................................................................................................................................. VI
CHAPTER 1 INTRODUCTION............................................................................................................1
1.1 MOTIVATIONS AND GOALS...............................................................................................................1
1.2 LITERATURE SURVEY .......................................................................................................................2
1.3 OUTLINE OF THIS THESIS ..................................................................................................................3
CHAPTER 2 THEORY ...........................................................................................................................4
2.1 DISCRETE SYSTEMS ..........................................................................................................................4
2.1.1 Equations of motion and loading conditions .........................................................................5
2.1.2 Constitutive relations...............................................................................................................6
2.1.3 State-space representation ......................................................................................................7
2.1.4 Linear viscoelastic damping and stability.............................................................................7
2.2 INDENTATION THEORY (CONTACT THEORY)....................................................................................8
2.3 CONTINUUM COSSERATS MECHANICS ..........................................................................................10
CHAPTER 3 COMPUTATION – COMSOL MULTIPHYSICS ..................................................14
3.1 GENERAL NUMERICAL CONSIDERATIONS IN COMSOL................................................................14
3.2 CONTACT MODELING IN COMSOL ...............................................................................................14
CHAPTER 4 RESULTS AND DISCUSSION...................................................................................17
4.1 A TWO-DIMENSIONAL DISCRETE MODEL .......................................................................................17
4.1.1 Effective stiffness and damping anomalies ..........................................................................17
4.1.2. Results of stability analysis..................................................................................................22
4.2 A CIRCULAR INCLUSION IN A SQUARE PLATE (PLANE ELASTICITY)..............................................25
4.3 DEAD LOAD ON SANDWICHED SUBSTRATE (2D BOUSSINESQ PROBLEM)....................................28
4.4 TWO-DIMENSIONAL CONTACT PROBLEM......................................................................................31
CHAPTER 5 CONCLUSIONS AND FUTURE WORKS................................................................35
5.1 CONCLUSIONS................................................................................................................................35
5.2 FUTURE WORKS ..............................................................................................................................35
APPENDIX A: CIRCULAR INCLUSION IN THE SQUARE PLATE............................................37
APPENDIX B: SANDWICHED SUBSTRATE.....................................................................................42
APPENDIX C: TWO-DIMENSIONAL CONTACT PROBLEM......................................................63
LIST OF REFERENCES ..........................................................................................................................67
                                    

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校內：2011-02-12公開
校外：2019-02-12公開

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