兩相複合黏彈性材料的等效彈性常數及阻尼，可因負勁度內含物而大幅增加或減少。本研究利用有限元素法模擬方形板中含有圓形內含物的力學性質，並假設內含物的楊氏模數為負值。方形板被施以單軸均勻拉力。在改變內含物幾何尺寸與α值 (內含物之楊氏模數與方形板楊氏模數之比值) 中發現，此負勁度複合材料在適當選取的幾何尺寸及α值下，具有異常的整體等效楊氏模數(以受拉力邊上的平均應力除以受拉力邊上的平均應變而得)，應變能及內含物-基材界面應力對負勁度亦有劇烈的反應。

對於負勁度複合材料之穩定性，本文運用滿足Lyapunov穩定理論的彈性應變能論述，藉由對應理論延伸純彈性理論結果至黏彈性低頻域範圍。研究發現三維黏彈性複合材料的勁度及阻尼峰值落於不穩定區域，但二維平面應變複合材料之勁度峰值落於穩定區域，原因為z方向提供額外的位移束制。

為運用有限元素法研究負勁度內含物對黏彈性材料的影響，本研究使用 DMA（動態機械分析儀）和HTVS(高溫黏彈性頻譜儀)先對 PMMA（聚甲基丙烯酸甲酯）進行實驗量測，結果發現其最大的消散模數(loss tangent)是約為0.07，而且此Debye alpha peak 發生在1Hz附近。其次，將實驗測得的消散模數帶入有限元素計算中，驗證此數值方法可正確的算出受扭力下的消散模數。最後，進行二維平面應變黏彈性複合材料含顆粒內含物的 Boussinesq 有限元分析，結果顯示如負勁度顆粒內含物接近表面，則整體等效勁度為負值。負勁度對整體消散模數的影響仍待進一步探討。

Recent development in composites containing phase-transforming particles, such as vanadium dioxide or barium titanate, reveals that the overall stiffness and viscoelastic damping of the composites may be unbounded. Negative stiffness can be induced from phase transformation, as predicted by the Landau phase transformation theory. Although this unbounded phenomenon is theoretically supported with the composite homogenization theory, detailed stress analyses of the composites are still lacking. In this work, we analyze the stress distribution of the Hashin-Shtrikman composite and its two-dimensional variant, namely a circular inclusion in a square plate, under the assumption that the Young’s modulus of the inclusion is negative. Assumption of negative stiffness is a priori in the present analysis. For stress analysis, a closed form solution for the HS model and finite element solutions for the 2D composite are presented. A static loading condition is adopted to estimate the effective modulus of the composites by the ratio of stress to average strain on the loading edges. It is found that the interfacial stresses between the circular inclusion and matrix increase dramatically when the negative stiffness is so tuned that overall stiffness is unbounded. Furthermore, it is found that stress distributions in the inclusion are not uniform, contrary to Eshelby’s uniformity theorem, which states, for two-phase, infinite composites, the inclusion’s stress distribution is uniform when the shape of the inclusion has higher symmetry than an ellipse. The stability of the composites is discussed from the viewpoint of deterioration of perfect interface conditions due to excessive interfacial stresses.
Stability sufficient conditions, based on the energy argument, are consistent with the Lyapunov stability theory. It is found that the extreme properties of the 3D viscoelastic composites require negative stiffness in the unstable regime. However, the plane-strain 2D composite shows anomalous mechanical properties in the stable regime. The rationale may be due to additional displacment constraints from the out-of-plane direction. These stability results are valid in the low frequency limit when the materials are viscoelastic in accordance with the correspondence principle.
In order to extend our calculations to viscoelastic materials, finite element analysis on the standard linear solid under torsion was conducted. The calculated results were compared with data from DMA (Dynamic Mechanical Analyzer) and HTVS (high temperature viscoelastic spectroscopy) experiments. The storage and loss modulus of PMMA (polymethyl methacrylate) under tension, bending and torsion at various driving frequencies and temperatures were experimentally measured. The experimental measured loss tangent of the materials was fed in the finite element analysis to study the viscoelastic behavior of a cantilever beam under dynamic torsion. It was found that the largest loss tangent measured from HTVS experiments was 0.07, occurred at its Debye alpha peak around 1 Hz.
Our viscoelastic finite element analysis showed that numerical and experimental results were consistent. As for plane viscoelastic composites containing a negative stiffness inclusion under Boussinesq loading, it was found that the location of the inclusion can drastically affect the overall properties of the composite. When the inclusion was near the loading surface, the effective stiffness was negative. On the other hand, when the inclusion in far away from the loading, the overall stiffness remains positive. These results shed lights on future experimental investigations. The effects of negative stiffness on overall damping of the plane viscoelastic composite requires further theoretical and numerical studies.

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