具梯度效應多鐵纖維複合材料之廣義反平面問題

Multiferroic Fibrous Composites with Gradient Effects under Generalized Anti-Plane Deformation

李庚恩(Geng-En Li)

指導教授 ： 郭心怡

國立交通大學/工學院/土木工程系所/碩士(2018年)

摘要

本研究探討具梯度效應多鐵纖維複合材料之勢能及場域。由過往實驗結果得知，材料在縮小到微米、奈米尺寸後，傳統線性理論不再適用於描述梯度效應，本研究將Aifantis(1992)及Yue等人(2014)提出的應變梯度理論推廣至有力學場、電學場及磁學場多重耦合的多鐵纖維複合材料，透過特徵值方法解耦此多重物理場耦合問題，並以應變梯度理論分析各物理場之特徵長度對廣義勢能(位移、電勢能及磁勢能)及廣義應變場(應變、電場及磁場)的分布影響。
數值分析部分先以具纖維狀微型孔洞之壓電材料 進行分析，結果與Yue等人相比有良好的一致性，之後再分析PZT-4 – CoFe2O4之纖維複合材料其特徵長度對廣義勢能及廣義應變場的影響。結果顯示廣義勢能極值不再發生於材料交界面上，而是會隨特徵長度漸增向母材或內含物方向偏移。另外，母材廣義應變場會隨母材或內含物特徵長度對半徑比例減少而趨近無梯度效應情形，但內含物廣義應變場主要受交界面上的應變連續性影響而非受特徵長度影響。

關鍵字

壓電壓磁纖維複合材料 ； 梯度效應 ； 應變梯度 ； 電場梯度 ； 磁場梯度 ； 廣義反平面問題

並列摘要

We study the potentials and fields of multiferroic fibrous composites with gradient effects. Earlier experiment shows that when the material sizes come to micrometers or nanometers, the classic elasticity theory is no longer valid due to the gradient effect. In this research, we extend the strain gradient theory proposed by Aifantis (1992) and Yue et. al. (2014) to the multiferroic fibrous composites, which contain the coupling among mechanics, electricity, and magnetism. We use the eigenvalue method to decouple the multi-physics problem and use strain gradient theory to investigate the influences of each characteristic length to the generalized potentials and generalized strains.
Numerical calculations are first demonstrated for piezoelectric material PZT-4 with micro-void and are shown in good agreement with earlier study. Following, we study the behavior of PZT-4 – CoFe2O4 composites and investigate the influences of the characteristic length to the generalized potentials and generalized strains. The results show that the extremum of the generalized potential no longer appears at the interface but offsets to the matrix or inclusion direction when the characteristic length increases. Further, the generalized strains of matrix approach to the classical result of the case with no gradient effects when the characteristic length decreases. In contrast to the generalized strains of matrix, the generalized strains of inclusion are mainly
affected by the strain continuous interface condition instead of the characteristic length.

並列關鍵字

Multiferroic fibrous composites ； Gradient effects ； Strain gradient ； Electric field gradient ； Magnetic field gradient ； Generalized anti-plane deformation

參考文獻

Aifantis, E. C., 1992. “On the role of gradients in the localization of deformation and fracture,” International Journal of Engineering and Science, 30, 1279.

Altan, B. S., Aifantis, E. C., 1997. “On Some Aspects in the Special Theory of Gradient Elasticity,” Journal of the Mechanical Behavior of Materials, 8, 231.

Arvanitakis, A. I., Kalpakides, V. K., Hadjigeorgiou, E. P., 2011. “Electric field gradients and spontaneous quadrupoles in elastic ferroelectrics,” Acta Mechanica, 218, 269.

Astrov, D. N., 1960. “The magnetoelectric effect in aniferromagnetics,” Soviet Physics-JETP, 11, 708.

Benveniste, Y., 1995. “Magnetoelectric effect in fibrous composite with piezoelectric and piezomagnetic phases,” Physical review B, 51, 16424.