Discussion on the Leibniz rule and Laplace transform of fractional derivatives using series representation

ABSTRACT

Taylor series is a useful mathematical tool when describing and constructing a function. With the series representation, some properties of fractional calculus can be revealed clearly. On this basis, the Lebiniz rule and Laplace transform of fractional calculus is investigated. It is analytically shown that the commonly used Leibniz rule cannot be applied for Caputo derivative. Similarly, the well-known Laplace transform of Riemann–Liouville derivative is doubtful for n-th continuously differentiable function. After pointing out such problems, the exact formula of Caputo Leibniz rule and the explanation of Riemann–Liouville Laplace transform are presented. Finally, three illustrative examples are revisited to confirm the obtained results.     

自适应混合遗传算法优化团簇

提出一种用于原子团簇结构优化的方法,该方法把具有全局寻优能力的自适应遗传算法与基于牛顿法思想提出的局部优化方法相结合.碳团簇的结构优化用于验证新方法的合理性,计算结果与自适应遗传算法的结果相比较,证明所提出的局部优化方法能够有效地搜索到局部极值,计算结果和混合遗传算法的结果进行对比,证明提出的自适应混合遗传算法能有效地解决"早熟"现象,并且通过对C₁₂的四次计算,表明该算法具有一定的稳定性.     

Stability for nonlinear fractional order systems: an indirect approach

In this paper, we first verify that fractional order systems using Caputo’s or Riemann–Liouville’s derivative can be represented by the continuous frequency distributed model with initial value carefully allocated. Then, the relation of the stability between the fractional order system and its corresponding integer order system is discussed and it is proven that stability of integer order system implies the stability of its corresponding fractional order system under some mild conditions. Moreover, we extend the stability theorems to the finite-dimensional case since fractional order systems are always implemented by approximation. Some illustrative examples are finally provided to show the usage and effectiveness of the proposed stability theorems.     

CONSERVATION LAWS IN FINITE MICROCRACKING BRITTLE SOLIDS

This paper addresses the conservation laws in finite brittle solids with microcracks. The discussion is limited to the 2-D cases. First, after considering the combination of the Pseudo-Traction Method and the indirect Boundary Element Method, a versatile method for solving multi-crack interacting problems in finite plane solids is proposed, by which the fracture parameters （SIF and path-independent integrals） can be calculated with a desirable accuracy. Second, with the aid of the method proposed, the roles the conservation laws play in the fracture analysis for finite microcracking solids are studied. It is concluded that the conservation laws do play important roles in not only the fracture analysis but also the analysis of damage and stability for the finite microcracking system. Finally, the physical interpretation of the M-integral is discussed further. An explicit relation between the M-integral and the crack face area, i.e., M = GS, has been discovered using the analytical method, which can shed some light on the Damage Mechanics issues from a different perspective.     

谷胱甘肽在汞表面吸附与自组装行为的原子力显微镜研究

用原子力显微镜研究谷胱甘肽（GSH）在汞表面的吸附与自组装状态。谷胱甘肽在汞表面首先形成非均匀地堡状吸附体,大部分单个吸附体高约10-13mn,半高直径约20nm。约每6个单吸附体以近似等边三角形形态自组装成一簇。随着吸附时间的延长,GSH可在汞表面自组装成约15nm的多分子层紧密层。与此同时,在紧密吸附层上可自组装成高约90—170nm、半高直径约130nm的特大吸附簇。     

界面微裂纹屏蔽J_k矢量投影守恒

用理论推导和电算实践证明，尽管两相材料界面裂纹Ｊ积分的显函数表达式与均质材料中不同，尽管界面裂纹尖端固有的 １／２＋ｉε振荡奇异性和张开滑移型固有的耦合造成近尖区应力场分布的复杂性，但作者在均质材料微裂纹屏蔽问题中发现的Ｊ积分再分配关系 Ｊｋ矢量投影守恒定理在两相材料界面微裂纹屏蔽问题中仍然成立．这再次说明，远场Ｊ积分向界面裂纹尖端传递过程中跨越微裂纹群是有损失的．这个损失可用Ｊｋ矢量在界面裂纹延线坐标轴上的投影来定量地评估     

微裂纹屏蔽问题中守恒积分投影关系

对连续体损伤力学在微裂纹屏蔽问题中应用的Ｊ积分守情理假设提出质疑，用理论分析和电算实践证明了远场Ｊ积分在微裂纹损伤区中的再分配关系。即ＪＫ矢量的投影守恒关系。在这一关系中，被Ｈｅｒｒｍａｎｎ所轻视的Ｊ２分量起着十分重要的作用，本文的研究表明，Ｏｒｔｉｚ理论应考虑到远场Ｊ积分在损伤区中的损失，并通过计及这一损失做出必要的修正。     

Investigation of the crack problem in non-local piezoelectric materials under combined electromechanical loadings

The present investigation of the crack problem in piezoelectric materials is performed based on the non-local theory. After some manipulations, the impermeable crack, the permeable crack （the crack gap is full of NaCI solution）, and the semi-permeable crack （the crack gap is full of air or silicon oil） are reduced to a uniform formulation by assuming the normal electric displacement on the crack surfaces to be an unknown variable. Thus, a triple integral equation with the unknown normal electric displacement is established. By using the Newton iterative method and solving the triple integral equation, it is found that the normal electric displacement on the crack surfaces is no longer a constant as determined by previous studies, rather, it depends upon the remote combined electromechanical loadings. Numerical results of the stresses and electric displacement fields show that there are no singularities at the crack tips so that the stresses remain finite. It is of great significance that the concrete electric boundary condition on the crack surfaces exerts significant influence on the near-tip fields and in this way plays an important role in evaluating the crack stability in the non-local piezoelectric materials. More specifically, the impermeable crack model always overestimates the finite stresses at the crack tips, whereas the permeable crack model always underestimates them.     

Further investigation of interaction between interface macrocrack and parallel microcracks in bimaterial anisotropic solids

In this paper, with the aid of superimposing technique and the Pseudo Traction Method (PTM), the interaction problem between an interface macrocrack and parallel microcracks in the process zone in bimaterial anisotropic solids is reduced to a system of integral equations. After the integral equations are solved numerically, a conservation law among three kinds ofJ-integrals is obtained which are induced from the interface macrocrack tip, the microcrack and the remote field, respectively. This conservation law reveals that the microcrack shielding effect in such materials could be considered as the redistribution of the remoteJ-integral. The project supported by the National Natural Science Foundation of China, and the Doctorate Foundation of Xi'an Jiaotong University     

非平衡等离子体对甲烷——氧扩散火焰影响的实验研究

利用自主设计的等离子喷注器采用介质阻挡放电方式产生非平衡等离子体,首先利用纹影技术、热电偶、单点红外测温等多种诊断方法实验研究了纯氧放电等离子体的电学特性、热效应及气动效应,然后通过可见光和化学自发辐射成像技术获得了火焰形态及特征参数,详细分析了等离子体对甲烷--纯氧扩散火焰形态和释热的影响,并计算了放电功率及费效比. 结果表明, 燃烧导致放电电流显著增大,其中电压幅值与氧气流速对放电电流大小的影响规律正好相反;与空气等离子体相比, 相同流量与电压条件下氧等离子体放电功率较高,但其发光强度明显较弱; 氧等离子体热效应微弱, 对燃烧的影响可以忽略,放电反应中释热过程主要由含氧组分决定;放电产生了具有3个速度分量的诱导射流, 增大了氧射流角,且电压越大越显著.等离子体主要通过气动效应改变了燃料与氧化剂的掺混,使得一定条件下火焰变得更稳定、释热更强.在所研究的范围内等离子体作用的费效比最低仅为2.2％,大流量、小混合比更有利.