因材料固有的尺度效應，在微觀尺度下，其微結構效應的影響將不可忽略，傳統古典彈性力學無法適切描述材料的變形行為，利用Cosserat彈性理論可以做更精確地分析。本文將比較Cosserat彈性理論與傳統古典彈性力學的不同，其主要不同點在於Cosserat彈性理論多出了三個獨立微旋轉參數，所以在組成律、平衡方程式和幾何方程式均含有旋轉量的影響，並於此理論基礎下，推導材料為中心對稱及非中心對稱聖維南柱體純拉伸作用下的變形場，檢視推導出的變形場與在傳統古典彈性力學下的異同。另本文也探討均質、等向且中心對稱之Cosserat彈性材料圓柱受到扭轉作用下中性內含物的問題，利用推導出均質、等向且中心對稱之Cosserat彈性材料圓柱受到扭轉作用下的變形場解，建立fiber和coating的關係式，並解得滿足平衡方程式的通解，再利用邊界條件關係式，建立求解待定係數及內含物等效材料係數的方程組，最後推導內含物等效材料係數的關係式，使得內含物置入基質後，內含物外基質部分的場量函數並不會因此而改變。

Under certain length scale, the framework of classical elasticity will not be capable to suitably characterize the deformation behavior of materials, due to its intrinsic length scales in the microstructures. This work is motivated by a recent interest of designing metamaterials with negative mass density and negative elastic moduli, in which a number of known structures possess a macroscopic behavior with Cosserat elasticity. We first review the progress of research works in Cosserat elasticity. A preliminary mathematical framework of Cosserat elasticity, in which, in addition to three displacements, three microrotations are incorporated in the formulation to describe the deformation behavior. Based on the mathematical framework, we derived the deformation field of a Saint-Venant cylinder under simple tension with both centrosymmetric and non-centrosymmetric constitutive relations. We also examine the possibility of neutral inclusion under torsion in a homogeneous, isotropic and centrosymmetric Cosserat elastic cylinder. The known solution of a homogeneous, isotropic and centrosymmetric Cosserat elastic cylinder under torsion is utilized to derive the possible combination of material parameters for a coated cylinder so that, under the twisting deformation mode, the fields in the matrix remain unchanged. This was generally referred to as neutral inclusion in torsion. We derive the general equilibrium equations for the fiber and for the coating. We utilize boundary conditions to construct system of equations for solving undetermined coefficients and effective modulus. This work is a preliminary study to look into the possibility of neutral composite cylinders.

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