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Generalized covariant differentiation and axiom-based tensor analysis

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Applied Mathematics and Mechanics(English Edition) 2016年第3期37卷 379-394页

作者： yajun yin Department of Engineering Mechanics School of Aerospace EngineeringTsinghua University

This paper reports the new progresses in the axiomatization of tensor anal- ysis, including the thought of axiomatization, the concept of generalized components, the axiom of covariant form invariability, the axiomatized definition, the algebraic structure, the transformation group, and the simple calculation of generalized covariant differentia- tions. These progresses strengthen the tendency of the axiomatization of tensor analysis.

关键词： tensor analysis axiom of covariant form invariability generalized compo-nent generalized covariant differentiation covariant differential transformation group

Generalized Covariant Derivative with Respect to Time in Flat Space(Ⅰ):Eulerian Description

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Acta Mechanica Solida Sinica 2016年第4期29卷 345-358页

作者： yajun yin Department of Engineering Mechanics School of Aerospace EngineeringTsinghua University

This paper reports a new derivative in the Eulerian description in flat space-the generalized covariant derivative with respect to time. The following contents are included:(a) the restricted covariant derivative with respect to time for Eulerian component is defined;(b) the postulate of the covariant form invariability in time field is set up;(c) the generalized covariant derivative with respect to time for generalized Eulerian component is defined;(d) the algebraic structure of the generalized covariant derivative with respect to time is made clear;(e) the covariant differential transformation group in time filed is derived. These progresses reveal the covariant form invariability of Eulerian space and time.

关键词： Eulerian description covariant form invariability generalized Eulerian component generalized covariant derivative with respect to time covariant differential transformation group

Generalized Covariant Derivative with Respect to Time in Flat Space(Ⅱ):Lagrangian Description

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Acta Mechanica Solida Sinica 2016年第4期29卷 359-370页

作者： yajun yin Department of Engineering Mechanics School of Aerospace Engineering Tsinghua University

The previous paper reported a new derivative in the Eulerian description in flat space—the generalized covariant derivative of generalized Eulerian component with respect to time. This paper extends the thought from the Eulerian description to the Lagrangian description:on the basis of the postulate of covariant form invariability in time field, we define a new derivative in the Lagrangian description in flat space—the generalized covariant derivative of generalized Lagrangian component with respect to time. Besides, the covariant differential transformation group is set up. The covariant form invariability of Lagrangian space-time is ascertained.

关键词： Lagrangian description the postulate of covariant form invariability generalized Lagrangian component generalized covariant derivative with respect to time covariant differential transformation group

The Particle Time Derivative of the Characteristic Geometric Quantities on Soft Curved Surface in Lagrangian Description

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Acta Mechanica Solida Sinica 2018年第6期31卷 683-697页

作者： yajun yin Department of Engineering Mechanics School of Aerospace EngineeringTsinghua UniversityBeijing 100084China

This paper studies the particle time derivatives of the characteristic geometric quantities on soft curved surfaces in Lagrangian description.On the basis of differential geometry, the calculation formulas for the particle time derivatives of the base vectors,metric tensor, Christoffel symbol,unit normal vector,curvature tensor and scalar curvatures on soft curved surface are derived.The limitations of particle time derivatives,e.***.the non-covariance,are pointed out.This research paves the way for studying particle time derivative of any tensor field on soft curved surface.

关键词： Soft curved surfaces Lagrangian description Characteristic geometric quantities Particle time derivative

ARNOLD CIRCULATION AND MULTI-OPTIMAL DYNAMIC CONTROLLING MECHANISMS IN DRAGONFLY WINGS

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Acta Mechanica Solida Sinica 2013年第3期26卷 237-244页

作者： Hongxiao Zhao yajun yin Zheng Zhong School of Aerospace Engineering and Applied Mechanics Tongji University School of Aerospace Department of Engineering MechanicsTsinghua University

This paper aims to reveal the multi-optimal mechanisms for dynamic control in drag- onfly wings. By combining the Arnold circulation with such micro/nano structures as the hollow inside constructions of the pterostigma, veins and spikes, dragonfly wings can create variable mass, variable rotating inertia and variable natural frequency. This marvelous ability enables dragonflies to overcome the contradictory requirements of both light-weight-wing and heavy-weight-wing, and displays the multi-optimal mechanisms for the excellent flying ability and dynamic control capac- ity of dragonflies. These results provide new perspectives for understanding the wings＇ functions and new inspirations for bionic manufactures.

关键词： dragonfly wings veins and pterostigma Arnold circulation multi-optimal control-ling mechanism

NONLINEAR VIBRATION OF THE COUPLED STRUCTURE OF SUSPENDED-CABLE-STAYED BEAM-1:2 INTERNAL RESONANCE

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Acta Mechanica Solida Sinica 2014年第5期27卷 467-476页

作者： Kun Huang Qi Feng yajun yin School of Aerospace Engineering and Applied Mechanics Tongji University Department of Engineering Mechanics School of Aerospace Tsinghua University

Through the Galerkin method the nonlinear ordinary differential equations （ODEs） in time are obtained from the nonlinear partial differential equations （PDEs） to describe the mo- tion of the coupled structure of a suspended-cable-stayed beam. In the PDEs, the curvature of main cables and the deformation of cable stays are taken into account. The dynamics of the struc- ture is investigated based on the ODEs when the structure is subjected to a harmonic excitation in the presence of both high-frequency principle resonance and 1：2 internal resonance. It is found that there are typical jumps and saturation phenomena of the vibration amplitude in the struc- ture. And the structure may present quasi-periodic vibration or chaos, if the stiffness of the cable stays membrane and frequency of external excitation are disturbed.

关键词： the coupled structure of suspended-cable-stayed beam the multiscale method 1：2internal resonance Hopf bifurcation chaos

Curvature-Driven Forces Based on Natural Exponential Pair Potential at Micro/Nanoscales

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Acta Mechanica Solida Sinica 2019年第2期32卷 133-147页

作者： Dan Wang yajun yin Zheng Zhong Zhili Hu State Key Laboratory of Mechanics and Control of Mechanical Structures College of Aerospace Engineering Nanjing University of Aeronautics and Astronautics Nanjing 211100 China Department of Engineering Mechanics School of Aerospace Tsinghua University Beijing 100084 China School of Aerospace Engineering and Applied Mechanics Tongji University Shanghai 200092 China

The natural exponential potential (Ce^R/λ0) widely exists at micro/nanoscales;this paper studies the interaction potential between a curved-surface body and an outside particle base on the natural exponential potential. Mat hematical derivation proves t hat the int er act ion potential can be expressed as a function of curvatures. Then, idealized numerical experiments are designed to verify the accuracy of the curvature-based potential. The driving forces exerted on the particle are discussed and confirmed to be a function of curvatures and the gradient of curvatures, which may explain some abnormal movements at micro/nanoscales.

关键词： Micro/nanoscales Curvature-driven forces Natural exponential pair potential Curvature-based potential

Relations between cubic equation, stress tensor decomposition, and von Mises yield criterion

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Applied Mathematics and Mechanics(English Edition) 2015年第10期36卷 1359-1370页

作者： Haoyuan GUO Liyuan ZHANG yajun yin Yongxin GAO Department of Engineering Mechanics Tsinghua University School of Mechanical Engineering University of Science and Technology Beijing Department of Mathematics Tianjin University

Inspired by Cardano＇s method for solving cubic scalar equations, the addi- tive decomposition of spherical/deviatoric tensor （DSDT） is revisited from a new view- point. This decomposition simplifies the cubic tensor equation, decouples the spher- ical/deviatoric strain energy density, and lays the foundation for the von Mises yield criterion. Besides, it is verified that under the precondition of energy decoupling and the simplest form, the DSDT is the only possible form of the additive decomposition with physical meanings.

关键词： Cardano＇s method Caylay-Hamilton theorem cubic tensor equation decomposition of spherical/deviatoric tensor （DSDT） von Mises yield criterion

Duality in interaction potentials for curved surface bodies and inside particles

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Applied Mathematics and Mechanics(English Edition) 2017年第8期38卷 1071-1090页

作者： Dan WANG yajun yin Jiye WU Zheng ZHONG Department of Engineering Mechanics Tsinghua University Department of Civil Engineering Nanjing Tech University Department of Engineering Mechanics Tongji University

Based on the viewpoint of duality, this paper studies the interaction between a curved surface body and an inside particle. By convex/concave bodies with geometric duality, interaction potentials of particles located outside and inside the curved surface bodies are shown to have duality. With duality, the curvature-based potential between a curved surface body and an inside particle is derived. Furthermore, the normal and tangential driving forces exerted on the particle are studied and expressed as a function of curvatures and curvature gradients. Numerical experiments are designed to test accuracy of the curvature-based potential.

关键词： curved duality curvature concave tangential viewpoint convex intersection body radius