第六章 結論與建議
6.2 建議
科技日新月異，材料亦推陳出新，本論文無法涵蓋至大部分之材料，僅提出一種較 為廣義的分析方法及模式作為後續研究之參考，因此建議可以對於較新穎且具有實際 用途之智能材料進行分析，例如功能梯度性材料 (functionally graded material, FGM)。
根據本研究所得電磁彈奇異性階數之結果，可對於更加複雜之幾何形狀尖端進行 應力、電位移與磁通量分析，嘗試推求其強度因子。因本研究中電磁彈性與壓電材料之 極化方向毋須平行迴轉體之旋轉軸或垂直於楔形體之中平面，則材料係數經座標轉換 後均為 θ 之函數，故求取強度因子之過程中仍可能有許多困難點需克服。
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表 3.1 材料性質
材料係數 PZT-4a PZT-5Ha PZT-6Bb PZT-6B(Im.)b BaTiO3c CoFe2O4c
BaTiO3-CoFe2O4d
(VI= 0.5)
彈性係數 (GPa)
¯
c11 139.0 126.0 168.0 168.0 166.0 286.0 226.0
¯
c12 77.8 55.0 60.0 60.0 77.0 173.0 125.0
¯
c13 74.3 53.0 60.0 60.0 78.0 170.5 124.0
¯
c33 115.0 117.0 163.0 163.0 162.0 269.5 216.0
¯
c44 25.6 35.3 27.1 27.1 43.0 45.3 44.0
壓電係數 (C/m2)
¯
e11 -5.2 17.0 4.6 43.0 11.6 - 5.8
¯
e31 15.1 -6.5 -0.9 -14.0 -4.4 - -2.2
¯
e33 12.7 23.3 7.1 36.0 18.6 - 9.3
介電係數
×10−10(F/m)
¯
η11 64.6 151.0 36.0 200.0 112.0 - 56.4
¯
η33 56.2 130.0 34.0 247.0 126.0 - 63.5
壓磁係數 (N/Am)
d¯11 - - - 550.0 275.0
d¯33 - - - 580.3 290.2
d¯33 - - - 699.7 350.0
磁導係數
×10−6(Ns2/C2)
¯
µ11 - - - - 5.0 590.0 297.0
¯
µ33 - - - - 10.0 157.0 83.5
電磁耦合係數
×10−10(Ns/VC)
¯
g11 - - - - 0.0 0.0 0.05367
¯
g33 - - - - 0.0 0.0 27.375
a
為參考 Li 等人 [52]
b
為參考 Xu 與 Mutoh[53]
c
為參考 Huang 與 Kuo[64]
d
為參考 Ramón 等人 [63]
表 3.2 壓電迴轉體之 Re[λ1] 收斂性分析
幾何形狀 材料 1/材料 2 子域數
級數解項數
文獻
5 6 7 9 11 13 15
β = 90◦ β1= 90◦
CdSe/
PZT-5H
2 0.9363 0.9348 0.9357 0.9377 0.9387 0.9383 0.9380
0.9381∗ 4 0.9379 0.9381 0.9382 0.9381 0.9381 0.9381 0.9381
6 0.9381 0.9381 0.9381 0.9381 0.9381 0.9381 0.9381 8 0.9381 0.9381 0.9381 0.9381 0.9381 0.9381 0.9381
CdSe/
PZT-6B
2 0.9268 0.9242 0.9308 0.9302 0.9280 0.9272 0.9278
0.9281∗ 4 0.9286 0.9289 0.9279 0.9281 0.9281 0.9281 0.9281
6 0.9281 0.9281 0.9281 0.9281 0.9281 0.9281 0.9281 8 0.9281 0.9281 0.9281 0.9281 0.9281 0.9281 0.9281 2
CdSe/
BaTiO3
2 0.8949 0.9588 0.9429 0.9172 0.9394 0.9256 0.9284
0.9429∗ 4 0.9436 0.9430 0.9430 0.9430 0.9429 0.9429 0.9429
6 0.9429 0.9428 0.9428 0.9429 0.9429 0.9429 0.9429 8 0.9429 0.9428 0.9429 0.9429 0.9429 0.9429 0.9429
PZT-6B/
PZT-6B(Im.)
2 0.98792 0.98475 0.98641 0.98793 0.98713 0.98828 0.98792
0.98724∗∗
4 0.98742 0.98732 0.98731 0.98720 0.98613 0.98725 0.98724 6 0.98802 0.98764 0.98764 0.98733 0.98724 0.98724 0.98724 8 0.98730 0.98724 0.98723 0.98724 0.98724 0.98724 0.98724
β = 90◦ β1= 180◦
PZT-6B/
PZT-6B(Im.)
3 0.54766 0.53669 0.52792 0.52053 0.52716 0.52670 0.53197
0.52819∗∗
6 0.52694 0.52758 0.52801 0.52836 0.52819 0.52818 0.52820 9 0.52803 0.52809 0.52823 0.52820 0.52820 0.52820 0.52820
註：本表中各分析案例之邊界條件均為 FOFO
∗
為 Li 等人 [52] 之結果
∗∗
為 Xu 與 Mutoh[53] 之結果
表 5.1 壓電楔形體之 Re[λ1] 收斂性分析
幾何形狀
材料一 / 材料二
邊界條件 子域數
級數解項數
文獻
6 7 8 9 10 12 14 15
β = 180◦ β1= 180◦
PZT-4
FOFO
3 0.4978 0.4916 0.4417 0.4980 0.4998 0.4750 0.5000 0.4999
0.5000† 4 0.4963 0.4993 0.4999 0.4999 0.4984 0.4999 0.4999 0.4999
6 0.4993 0.4999 0.4999 0.5000 0.5000 0.4999 0.5000 0.5000 8 0.4999 0.4999 0.4999 0.4999 0.5000 0.5000 0.4999 0.5000
FOCC
3 0.1969 0.2052 0.1602 0.1718 0.1965 0.1724 0.1954 0.1895
0.1869‡ 4 0.1895 0.1847 0.1877 0.1879 0.1857 0.1877 0.1865 0.1869
6 0.1869 0.1869 0.1869 0.1869 0.1869 0.1869 0.1869 0.1869 8 0.1869 0.1869 0.1869 0.1869 0.1869 0.1869 0.1869 0.1869
β = 90◦ β1= 90◦
PZT-4 FOCC
2 0.3751 0.3749 0.3740 0.3736 0.3735 0.3741 0.3737 0.3738
0.3739‡ 3 0.3737 0.3738 0.3739 0.3739 0.3739 0.3739 0.3739 0.3737
4 0.3739 0.3739 0.3739 0.3739 0.3739 0.3739 0.3739 0.3739 6 0.3739 0.3739 0.3739 0.3739 0.3739 0.3739 0.3739 0.3739
†
為 Sosa 與 Pak[39] 之結果
‡
為 Hwu 與 Ikeda[37] 之結果
圖 1.1 幾何或材料不連續示意圖
圖 1.2 材料可能缺陷示意圖
圖 2.1 含尖角之雙材料旋轉體
圖 2.2 迴轉體幾何形狀
圖 2.3 圓柱 (r, Z) 與尖角 (ρ, ξ)座標系統
圖 2.4 迴轉體子域 ξ∈[ξ0, ξn]
(a) (b)
圖 3.1 單 PZT-4 壓電迴轉體 λ1 隨 θ 之變化
(a) (b)
圖 3.2 單 PZT-6B 壓電迴轉體 λ1 隨 θ 之變化
(a) (b)
圖 3.3 不同邊界條件下單 PZT-4 迴轉體 λ1 隨 β 之變化
(a) (b)
圖 3.4 不同邊界條件下單 PZT-6B 迴轉體 λ1 隨 β 之變化
(a) (b)
(c) (d)
圖 3.5 雙壓電迴轉體 Re[λ1]隨 θ 之變化
(a) (b)
(c) (d)
圖 3.6 不同邊界條件下雙壓電迴轉體 Re[λ1] 之變化
(a) (b)
(c) (d)
圖 3.7 壓電材料/Si 迴轉體 Re[λ1] 隨 θ 之變化
(a) (b)
(c) (d)
圖 3.8 不同邊界條件下壓電材料/Si 迴轉體 Re[λ1] 隨β 之變化
(a) (b)
(c) (d)
圖 3.9 單 BaTiO3-CoFe2O4 迴轉體λ1 隨 θ 之變化
(a) (b)
(c)
圖 3.10 不同邊界條件下單 BaTiO3-CoFe2O4 迴轉體 λ1 隨 β 之變化
(a) (b)
(c) (d)
圖 3.11 雙 BaTiO3-CoFe2O4 迴轉體 λ1 隨θ 之變化
(a) (b)
(c) (d)
圖 3.12 不同邊界條件下雙 BaTiO3-CoFe2O4 迴轉體 Re[λ1]隨 β 之變化
(a) (b)
(c) (d)
圖 3.13 BaTiO3-CoFe2O4/Si 迴轉體 λ1 隨 θ 之變化
(a) (b)
(c) (d)
圖 3.14 不同邊界條件下 BaTiO3-CoFe2O4/Si 迴轉體 Re[λ1]隨 β 之變化
(a) (b)
(c) (d)
圖 3.15 BaTiO3-CoFe2O4/PZT-4 迴轉體 Re[λ1]隨 θ 之變化
(a) (b)
(c) (d)
圖 3.16 BaTiO3-CoFe2O4/PZT-6B 迴轉體 Re[λ1]隨 θ 之變化
(a) (b)
(c) (d)
圖 3.17 不同邊界條件下 BaTiO3-CoFe2O4/PZT-4 迴轉體 Re[λ1] 隨β 之變化
(a) (b)
(c) (d)
圖 3.18 不同邊界條件下 BaTiO3-CoFe2O4/PZT-6B 迴轉體 Re[λ1] 隨 β 之變化
圖 4.1 楔形體示意圖
圖 4.2 楔形體 (r, θ) 座標系統
圖 4.3 楔形體子域 θ∈[ θ0, θn]
圖 4.4 楔形體幾何示意圖
(a)
(b)
圖 5.1 不同邊界條件下單壓電材料楔形體λ1 隨 β 之變化
(a) (b)
(c) (d)
圖 5.2 不同邊界條件下單壓電材料楔形體 Re[λ1]隨 γ 之變化
(a)
(b)
圖 5.3 不同邊界條件下單壓電材料楔形體 Re[λ1]隨 θ 之變化
(a)
(b)
圖 5.4 不同邊界條件下雙壓電材料楔形體 Re[λ1]隨 β 之變化
(a)
(b)
(c)
110
(a)
(b)
圖 5.6 不同邊界條件下雙壓電材料楔形體 Re[λ1]隨 θ 之變化
(a)
(b)
圖 5.7 不同邊界條件下壓電材料/Si 楔形體 Re[λ1] 隨β 之變化
(a) (b)
(c) (d)
圖 5.8 不同邊界條件下壓電材料/Si 楔形體 Re[λ1]隨 γ 之變化
(a)
(b)
圖 5.9 不同邊界條件下壓電材料/Si 楔形體 Re[λ1] 隨θ 之變化
(a)
(b)
圖 5.10 不同邊界條件下單 BaTiO3-CoFe2O4 楔形體 λ1 隨 β 之變化
(a)
(b)
圖 5.11 不同邊界條件下單 BaTiO3-CoFe2O4 楔形體 Re[λ1] 隨γ 之變化
(c)
(d)
圖 5.11 不同邊界條件下單 BaTiO3-CoFe2O4 楔形體 Re[λ1] 隨γ 之變化
(a)
(b)
圖 5.12 不同邊界條件下單 BaTiO3-CoFe2O4 楔形體 Re[λ1]隨 θ 之變化
(a) (b)
(c) (d)
圖 5.13 不同邊界條件下雙 BaTiO3-CoFe2O4 楔形體 Re[λ1]隨 β 之變化
(a) (b)
(c)
圖 5.14 不同邊界條件下雙 BaTiO3-CoFe2O4 楔形體 Re[λ1] 隨γ 之變化
(a)
(b)
圖 5.15 不同邊界條件下雙 BaTiO3-CoFe2O4 楔形體 Re[λ1]隨 θ 之變化
(a) (b)
(c)
圖 5.16 不同邊界條件下 BaTiO3-CoFe2O4/Si 楔形體 Re[λ1]隨 β 之變化
(a) (b)
(c) (d)
圖 5.17 不同邊界條件下 BaTiO3-CoFe2O4/Si 楔形體 Re[λ1]隨 γ 之變化
(a)
(b)
圖 5.18 不同邊界條件下 BaTiO3-CoFe2O4/Si 楔形體 Re[λ1]隨 θ 之變化
(a) (b)
(c) (d)
圖 5.19 不同邊界條件下 BaTiO3-CoFe2O4/PZT-4 楔形體 Re[λ1] 隨β 之變化
(a) (b)
(c) (d)
圖 5.20 不同邊界條件下 BaTiO3-CoFe2O4/PZT-6B 楔形體 Re[λ1] 隨 β 之變化
(a) (b)
(c) (d)
圖 5.21 不同邊界條件下 BaTiO3-CoFe2O4/PZT-4 楔形體 Re[λ1]隨 γ 之變化
(a) (b)
(c) (d)
圖 5.22 不同邊界條件下 BaTiO3-CoFe2O4/PZT-6B 楔形體 Re[λ1]隨 γ 之變化
(a) (b)
圖 5.23 不同邊界條件下 BaTiO3-CoFe2O4/PZT-4 楔形體 Re[λ1] 隨θ 之變化
(a) (b)
圖 5.24 不同邊界條件下 BaTiO3-CoFe2O4/PZT-6B 楔形體 Re[λ1] 隨 θ 之變化
附錄 A：座標系統轉換
[c] = [T]σ[K] [¯c] [K]T[T]−1ε , (A.1)
[e] = [T]D[L] [ˆe] [K]T[T]−1ε , (A.2)
[d] = [T]B[L][d¯]
[K]T[T]−1ε , (A.3)
[η] = [T]D[L] [¯η] [L]T[T]−1E , (A.4)
[µ] = [T]B[L] [¯µ] [L]T[T]−1H (A.5)
其中
[T]σ =
cos2θ sin2θ 0 2 cos θ sin θ 0 0 sin2θ cos2θ 0 −2 cos θ sin θ 0 0
0 0 1 0 0 0
− cos θ sin θ cos θ sin θ 0 cos2θ− sin2θ 0 0
0 0 0 0 cos θ sin θ
0 0 0 0 − sin θ cos θ
[T]ε =
cos2θ sin2θ 0 cos θ sin θ 0 0 sin2θ cos2θ 0 − cos θ sin θ 0 0
0 0 1 0 0 0
−2 cos θ sin θ 2 cos θ sin θ 0 cos2θ− sin2θ 0 0
0 0 0 0 cos θ sin θ
0 0 0 0 − sin θ cos θ
[T]E = [T]D = [T]B = [T]H =
附錄 B：以位移、電勢與磁勢表示應力、電位移
+ c66
附錄 C：式 (4.3) 之變係數
r6(θ) = c24
個人簡歷
胡政甯
Cheng-Ning Hu 出生日期：1984.05.24電子郵件：[email protected]
學歷
台北縣私立南山高級中學 普通科 1999.09 ∼2002.06 國立交通大學 土木工程學系 大學部 2002.09 ∼2005.06 國立交通大學 土木工程學系 結構工程組 碩士班 2005.09 ∼2006.06 國立交通大學 土木工程學系 結構工程組 博士班 2006.09 ∼2013.06
經歷
國立交通大學 材料科學與工程學系 兼任講師 2013.03 ∼2013.07
證照
民國 101 年 公務高考 結構工程高考 及格
民國 101 年 專技高考 結構工程技師 錄取
獲獎紀錄
92 學年度 第一學期 土木工程學系 書卷獎
92 學年度 第一學期 土木工程學系 書卷獎
93 學年度 財團法人榮民榮眷基金會 績優學生獎學金 94 學年度 第一學期 土木工程研究所 書卷獎
99 學年度 第二學期 財團法人中興工程顧問社 優秀學生獎學金
101 學年度 財團法人中華工程顧問司 第九屆工程科技獎學金
102 學年度 中華民國斐陶斐榮譽學會 榮譽會員
發表期刊
1 C.S. Huang, C.N. Hu, Geometrically induced stress singularities in a piezoelectric body of revolution, Computers and Structures, pp1681-1696, 2011.
2 C.S. Huang, C.N. Hu, Three-Dimensional analyses of stress singularities at the vertex of a piezoelectric wedge, Applied Mathematical Modelling, pp 4517-4537, 2013.
3 C.S. Huang, C.N. Hu, Singularity Analysis for a Magneto-electro-elastic Body of Revolution, Composite Structures, pp 55-70, 2013
研討會論文
1 C.C. Chen, W.M. Yu, C.N. Hu, Self-Centering steel moment resisting frames with supplemental column, 2006
2 王榮進、陳誠直、翁正強、蘇晴茂、胡政甯、蔡煒銘，Ductile behavior of steel beam to encased steel reinforced concrete column connections, 中華民國第九屆結 構工程研討會，2008