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Understanding of the basic properties of the positive semi-definite tensor is a prerequisite for its extensive applications in theoretical and practical fields, especially for its square-root. Uniqueness of the square-root of a positive semi-definite tensor is proven in this paper without resorting to the notion of eigenvalues, eigenvectors and the spectral decomposition of the second-order symmetric tensor. 相似文献
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Recent experiments and molecule dynamics simulations have shown that adhesion droplets on conical surfaces may move spontaneously
and directionally. Besides, this spontaneous and directional motion is independent of the hydrophilicity and hydrophobicity
of the conical surfaces. Aimed at this important phenomenon, a general theoretical explanation is provided from the viewpoint
of the geometrization of micro/nano mechanics on curved surfaces. In the extrinsic mechanics on micro/nano soft curved surfaces,
we disclose that the curvatures and their extrinsic gradients form the driving forces on the curved spaces. This paper focuses
on the intrinsic mechanics on micro/nano hard curved surfaces and the experiment on the spontaneous and directional motion.
Based on the pair potentials of particles, the interactions between an isolated particle and a micro/nano hard curved surface
are studied, and the geometric foundation for the interactions between the particle and the hard curved surface is analyzed.
The following results are derived: (a) Whatever the exponents in the pair potentials may be, the potential of the particle/hard
curved surface is always of the unified curvature form, i.e., the potential is always a unified function of the mean curvature
and the Gaussian curvature of the curved surface. (b) On the basis of the curvature-based potential, the geometrization of
the micro/nano mechanics on hard curved surfaces may be realized. (c) Similar to the extrinsic mechanics on micro/nano soft
curved surfaces, in the intrinsic mechanics on micro/nano hard curved surfaces, the curvatures and their intrinsic gradients
form the driving forces on the curved spaces. In other words, either on soft curved surfaces or hard curved surfaces and either
in the extrinsic mechanics or the intrinsic mechanics, the curvatures and their gradients are all essential factors for the
driving forces on the curved spaces. (d) The direction of the driving force induced by the hard curved surface is independent
of the hydrophilicity and hydrophobicity of the curved surface, explaining the experimental phenomenon of the spontaneous
and directional motion. 相似文献
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研究半正定张量半正定方根的唯一性问题.避开了二阶张量特征值的概念和对称二阶张量谱分解定理,运用简单的预备知识,直接证明了二阶半正定张量半正定方根的唯一性. 相似文献
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近期的实验和分子动力学模拟均表明:圆锥面上粘附液滴能自发地定向运动,且自发定向运动的方向与粘附面的亲水、疏水性质无关.针对这一重要现象,拟从曲面微纳米力学几何化的角度,提供一般性的理论解释.借助于粒子对势,研究了孤立粒子与微纳米硬曲面之间的相互作用,分析了粒子/硬曲面相互作用的几何学基础.可以证实:(a) 粒子/硬曲面的作用势均具有统一的曲率化形式,均可以统一地表达成曲面平均曲率和Gauss曲率的函数;(b) 基于曲率化的作用势,能够实现曲面微纳米力学的几何化;(c) 曲率与曲率的内蕴梯度构成卷曲空间上的驱动力;(d) 驱动力方向与曲面的亲水、疏水性质无关,解释了自发定向运动实验. 相似文献
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Three-phase line tensions may become crucial in the adhesion of miero-nano or small droplets on solid planes. In this paper we study for the first time the nonlinear effects in adhesion spanning the full range of physically possible parameters of surface tension, line tension, and droplet size. It is shown that the nonlinear adhesion solution spaces can be characterized into four regions. Within each region the adhesion behaves essentially the same. Especially, inside the characteristic regions with violent nonlinearities, the co-existence of multiple adhesion states for given materials is disclosed. Besides, two common fixed points in the solution space are revealed. These new results are consistent with numerical analysis and experimental observations reported in the literatures. 相似文献
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