本研究以雙層法及微觀力學模型 Mori-Tanaka 模式，預測三相壓電壓磁纖維 複合材料之磁電耦合效應，並且搭配 COMSOL Multiphysics 有限元素法分析軟體 的數值驗證，證實理論結果之正確性。分析結果及相關發展在本章節依序列出。
5-1 結論
1. 建立三相含殼層纖維狀壓電壓磁複合材料之理論模型
由 COMSOL Multuphsics 證實，雙層法不僅可用於彈性行為分析，亦適 用於壓電壓磁複合材料等效性質預測。且對於含殼層內含物的分析上，雙層 法預測結果比直接 Mori-Tanaka 模式來的準確。
2. 尋找現有壓電壓磁材料之磁電耦合效應最佳化配置
本文探討的纖維狀(核心/殼層/母材)配置下，由最佳化分析結果得知：
若殼層與母材為同性質且經過適當配置之不同材料，可以得到E,11之最佳 化結果如：LNO/CFO/TD(壓電/壓磁/壓磁)及 TD/LNO/PVDF(壓磁/壓電/壓 電)。反之，若配置為不同性質，則最佳化結果由雙相材料所控制；對於E,33 而言，最佳化結果皆由雙相材料所控制。
3. 材料性質對磁電效應的影響
由第三章之最佳化分析結果前後，可以看出材料配置對磁電效應有重大 之影響，而藉由材料係數變動中可以得知：壓磁材料在變動介電性質方面皆 遠小於壓電材料介電性質之變動後所產生的E,11及E,33。另外，不論壓電 或壓磁材料在變動磁導率時，對E,33的影響皆微小至可忽略之程度，因此，
若先排除以上兩點將有助於簡化選擇材料之程序。
4. 磁電耦合效應在最佳化前後的差異性
最佳化前後得到的結果：在壓電/壓磁/壓磁配置下 LNO/CFO/TD 之E,11 值為-10.72(V/cmOe)比未最佳化前之 BTO/TD/CFO 配置之-0.5(V/cmOe)高出 約 21 倍、比雙相材料 LNO/TD 之-3.06(V/cmOe)高出約 3.5 倍；E,33最佳化 結果由 CFO/PZT-5J 雙相材料所控制，比 CFO/BTO 的 1.5(V/cmOe)高出約 4.1 倍。
5. 以界面材料模擬非完美交界面之可行性
由觀察結果得到，以適當的界面相材料做倍數乘積提升，致使磁電效應 逐漸低於完美假設下之值，反映著內含物與母材間之廣義應力、位移、電勢 能、磁勢能越不連續的現象。
6. 體積代表元素分析結果
以 COMSOL Multiphysics 建立正方形及正六邊形體積代表元素，模擬 纖維狀複合材料之結果顯示：以正六邊形體積代表元素預測的結果較為接近 理論分析之等效性質。但是，相對於正方型體積代表元素，其在處理資源上 以及求解時間過程中皆耗費甚鉅，因此，若分析僅需要取至體積百分比 0.5 以下時，可以直接採用正方形體積代表元素來進行分析，可減少建模以及處 理上的時間。
5-2 未來展望
1. 材料選擇多樣化
選擇多樣化的壓電及壓磁材料配置於材料中得到磁電效應之最佳化。另 外，也可於三相材料中配置一種非壓電或非壓磁材料來觀察對磁電效應之影 響。
2. 建立其它等效性質模型
雙層法的概念可藉由微觀力學模型來達到等效性質的預測，因此，除了 本文使用的 Mori-Tanaka 模式外，亦可由不同微觀力學模型來預測壓電壓磁 複合材料的等效性質。
3. 探討不同形式內含物對磁電耦合效應影響
本文所探討的是含殼層結構之複合材料分析，因此，也可以對散布於母 材內之兩種壓電或壓磁材料形式作深入的探討。
4. 建立非完美交界分析的正確模型
在本文中觀察到，壓電壓磁複合材料非完美交界面的問題可以界面材料 的方式分析，而後續的發展可以朝向界面材料的選擇與母材、內含物間之關 係以及界面材料倍數乘積對磁電效應折減(相對於完美界面假設下)的影響 等方面。
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附錄 A 磁電效應與材料係數間之關係
本附錄之目的在討論不同材料性質對磁電效應之影響，並以等高線的型式表 示。其中壓電相與壓磁相分別以 BaTiO3(BTO)與 CoFe2O4 (CFO)之材料係數作為 基礎，並在後續之不同組成下的分析中，逐步變動 C 、 κ 、 μ 、 q 、 e 觀察對磁 電電壓係數之影響。
在纖維狀複合材料分析中，由於E,11 與E,33 之間的差異性不容忽視，對於單 方面比較而言並不能代表整體磁電電壓係數之趨勢，因此，分析之步驟採兩部分 比較，先行分析E,11 ，其次再進行E,33 ，並在各組成之分析完成後，將最佳化之 結果做一彙整，並驗證在不同體積比 f 下，磁電電壓係數皆可比未變動前有更好 的表現。並由最佳化結果，選擇有助於提升磁電電壓係數之配置。
圖中設定部分：x 軸與 y 軸之數值變化，分別代表殼層(Shell)與核心(Core) 係數之改變，並以相對倍數表示；左上角代表母材之相對係數變化、右上角表示 變動係數後，磁電電壓係數與未變動前(E,110 、E,330 )之相對倍數關係，由圖中之 色階表示。部分圖式之座標軸改用對數型式表示，以利於分析後之觀察。
變動幅度部分：彈性係數變動範圍取 0.025 至 1 倍之間、介電常數κ採 0.1 至 10 倍之間、磁導率 μ 採 0.001 至 10 倍之間、壓電係數 e 與壓磁係數 q 採 1 至 8 倍之變動幅度。
A-1 壓電/壓磁/壓磁
Cr,Core = CPE/CBTOCr,Matrix = CPM/CCFO=0.025 *E,11/0E,11 Cr,Core = CPE/CBTO
Cr,Matrix = CPM/CCFO=0.125 *E,11/0E,11 Cr,Core = CPE/CBTO
Cr,Matrix = CPM/CCFO=0.375 *E,11/0E,11 Cr,Core = CPE/CBTO
Cr,Matrix = CPM/CCFO=0.5 *E,11/0E,11 Cr,Core = CPE/CBTO
Cr,Matrix = CPM/CCFO=0.75 *E,11/0E,11 Cr,Core = CPE/CBTO
Cr,Matrix = CPM/CCFO=1 *E,11/0E,11
r,Shell =
PM/
CFO
r,Core =PE/BTO
r,Matrix = PM/CFO= 0.1 *E,11/0E,11
r,Core =PE/BTO
r,Matrix = PM/CFO= 1 *E,11/0E,11
r,Core =PE/BTO
r,Matrix = PM/CFO= 10 *E,11/0E,11
qr,Shell = qPM/qCFO er,Core = ePE/eBTO
qr,Matrix = qPM/qCFO =1 *E,11/0E,11 er,Core = ePE/eBTO
qr,Matrix = qPM/qCFO =3 *E,11/0E,11 er,Core = ePE/eBTO
qr,Matrix = qPM/qCFO =5 *E,11/0E,11 er,Core = ePE/eBTO
qr,Matrix = qPM/qCFO =8 *E,11/0E,11 圖 A-4(a)說明：圖中 BTO/CFO/CFO 代表未變動前之分析結果；Cr,core=0.025、
Cr,shell=0.025 以及 Cr,matrix=0.025 分別代表相對於未變動前，核心、殼層以及母材
之彈性常數 C 乘上 0.025 之分析結果。其餘各圖所表示之涵義相同。
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.09
-0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0
Volume Fraction of Inclusion
* E,11 (V/cmOe)
PE/PM/PM, = 0.8
BTO/CFO/CFO Cr,Core =0.025 Cr,Shell =0.025 Cr,Matrix=0.025
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.18 -0.16 -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0
Volume Fraction of Inclusion
* E,11 (V/cmOe)
PE/PM/PM, = 0.8
BTO/CFO/CFO
r,Core =0.1
r,Shell =1
r,Matrix=0.1
(a)變動彈性係數 (b)變動介電常數
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-2.5 -2 -1.5 -1 -0.5 0
Volume Fraction of Inclusion
* E,11 (V/cmOe)
PE/PM/PM, = 0.8
BTO/CFO/CFO
r,Core =10
r,Shell =10
r,Matrix=0.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-2.5 -2 -1.5 -1 -0.5 0
Volume Fraction of Inclusion
* E,11 (V/cmOe)
PE/PM/PM, = 0.8
BTO/CFO/CFO er,Core =3.5 qr,Shell =1 qr,Matrix=8
(c)變動磁導率 (d)變動壓電、壓磁係數
圖 A-4 E,11最佳化參數與內含物體積比 f 間之關係(壓電/壓磁/壓磁)
圖 A-4 E,11最佳化參數與內含物體積比 f 間之關係(壓電/壓磁/壓磁)