Generalized covariant differentiation and axiom-based tensor analysis

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.

Abstraction and operator characterization of fractal ladder viscoelastic hyper-cell for ligaments and tendons

This study investigates the viscoelastic behavior of soft bio-fibres in association with their fractal structures. A spring-dashpot fractal network with the self-similar topology, named the -type fractal ladder hyper-cell (FLHC), is abstracted from the micro/nano-structure of ligaments and tendons (LTs). Its constitutive operator is derived by the Heaviside operational calculus, which is of intrinsic fractional order. In terms of this operator, the long-term viscoelastic relaxation of bio-fibres arising from the fractal ladder topology is expounded. In addition, the fractional-order viscoelastic constitutive equation is obtained based on the FLHC of LTs, and its results are consistent with those of available human knee and spinal LT relaxation experiments. Results on the constitutive equation of FLHCs are formulated into two propositions. The multidisciplinary invariance and implications from the fractal ladder pattern of bio-fibres are also discussed.

Surface evolution caused by curvature driven forces based on natural exponential pair potential

Based on the natural exponential pair potential, the interaction potential between curved surface body and on surface particle is studied. Firstly, the interaction potential is written as a function of curvatures through the differential geometry. Secondly, idealized numerical experiments are designed to test the accuracy of curvature-based potential. Then, the driving forces induced by curvatures are analyzed, which confirms that micro/nano curved surface body can induce driving forces, curvatures and the gradient of curvatures are the essential elements forming the driving forces. Finally, by combing with the curvature based potential and driving forces, the movements of on surface particles and the evolution of surface morphology of curved surface body are predicted.

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

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.

Effects of heat treatments on microstructures of TiAl alloys

This study aims to investigate the effects of heat treatments on the microstructure ofγ-TiAl alloys.Two Ti-47Al-2Cr-2Nb alloy ingots were manufactured by casting method and then heat-treated in two types of heat treatments.Their microstructures were studied by both optical and scanning electron microscopies.The chemical compositions of two ingots were determined as well.The ingot with lower Al content only obtains lamellar structures while the one higher in Al content obtains nearly lamellar and duplex structures after heat treatment within1270 to 1185℃.A small amount of B2 phase is found to be precipitated in both as-cast and heat-treated microstructures.They are distributed at grain boundaries when holding at a higher temperature,such as 1260℃.However,B2 phase is precipitated at grain boundaries and in colony interiors simultaneously after heat treatments happened at 1185℃.Furthermore,the effects of heat treatments on grain refinement and other microstructural parameters are discussed.

Fundamental kinematics laws of interstitial fluid flows on vascular walls

In the previous studies,the phenomenon that the interstitial fluid(ISF)can flow along tunica adventitia of the arteries and veins in both human and animal bodies was reported.On the basis of these studies,this paper aims to:(i)summarize the basic properties of the ISF flows in the walls of arteries and veins,(ii)combine the basic properties with axiomaticism and abstract the axiom for ISF flows,and(iii)propose three fundamental laws of the ISF flow,(i.e.,the existence law,the homotropic law and the reverse law).The three laws provide solid theoretical basement for exploring the kinematic patterns of interstitial fluid flow in the cardiovascular system.

Duality in interaction potentials for curved surface bodies and inside particles

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.

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

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.

The Role of Soft Vein Joints in Dragonfly Flight

Dragonflies are excellent flyers among insects and their flight ability is closely related to the architecture and material properties of their wings. The veins are main structure components of a dragonfly wing, which are found to be connected by resilin with high elasticity at some joints. A three-dimensional （3D） finite element model of dragonfly wing considering the soft vein joints is developed, with some simplifications. Passive deformation under aerodynamic loads and active flapping motion of the wing are both studied. The functions of soft vein joints in dragonfly flight are concluded. In passive deformation, the chordwise flexibility is improved by soft vein joints and the wing is cambered under loads, increasing the action area with air. In active flapping, the wing rigidity in spanwise direction is maintained to achieve the required amplitude. As a result, both the passive deformation and the active control of flapping work well in dragonfly flight. The present study may also inspire the design of biomimetic Flapping Micro Air Vehicles （FMAVs）.

Layers of interstitial fluid flow along a“slit-shaped”vascular adventitia

Interstitial fluid(ISF)flow through vascular adventitia has been discovered recently.However,its kinetic pattern was unclear.We used histological and topographical identification to observe ISF flow along venous vessels in rabbits.By magnetic resonance imaging(MRI)in live subjects,the inherent pathways of ISF flow from the ankle dermis through the legs,abdomen,and thorax were enhanced by paramagnetic contrast.By fluorescence stereomicroscopy and layer-by-layer dissection after the rabbits were sacrificed,the perivascular and adventitial connective tissues(PACTs)along the saphenous veins and inferior vena cava were found to be stained by sodium fluorescein from the ankle dermis,which coincided with the findings by MRI.The direction of ISF transport in a venous PACT pathway was the same as that of venous blood flow.By confocal microscopy and histological analysis,the stained PACT pathways were verified to be the fibrous connective tissues,consisting of longitudinally assembled fibers.Real-time observations by fluorescence stereomicroscopy revealed at least two types of spaces for ISF flow:one along adventitial fibers and another one between the vascular adventitia and its covering fascia.Using nanoparticles and surfactants,a PACT pathway was found to be accessible by a nanoparticle of<100 nm and contained two parts:a transport channel and an absorptive part.The calculated velocity of continuous ISF flow along fibers of the PACT pathway was 3.6-15.6 mm/s.These data revealed that a PACT pathway was a"slit-shaped"porous biomaterial,comprising a longitudinal transport channel and an absorptive part for imbibition.The use of surfactants suggested that interfacial tension might play an essential role in layers of continuous ISF flow along vascular vessels.A hypothetical"gel pump"is proposed based on interfacial tension and interactions to regulate ISF flow.These experimental findings may inspire future studies to explore the physiological and pathophysiological functions of vascular ISF or interfacial fluid flow among interstitial connective tissues throughout the body.

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

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.

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

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.

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

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.

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

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.g.the non-covariance,are pointed out.This research paves the way for studying particle time derivative of any tensor field on soft curved surface.

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

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.

Algebraic equations and non-integer orders of fractal operators abstracted from biomechanics

The previous study implies that the fractional operator may stem from the motion in fractal space or the motion of fractal structure abstracted from biological systems.This study is devoted to answering another essential question,i.e.,what determines the order of the fractional operator in fractal structure?This paper generalizes the concept of the fractal cell defined in the previous paper,explores the tree-like and net-like fractal structures with higher-order topology,abstracts two classes of higherorder fractal operators,and derives the algebraic equations satisfied by the fractal operators to answer this question.It is proved that the solutions of the algebraic equations for fractal operators are deterministically related to the fractional-time operators that are usually of fractional orders.By the Vieta theorem,the relation between the solutions of algebraic equations for fractal operators and the physical-component operators is clarified,and the duality constraints between them are revealed.The solutions of the fractal operators show that the topological invariants of the fractal cells are one of the essential factors in determining the fractional orders.A conjecture on the specific order of the fractional-time operator in fractal structure is proposed.

CURVATURE-BASED INTERACTION POTENTIAL BETWEEN MICRO/NANO PLANAR CURVE AND PARTICLE LOCATED OUTSIDE THE CURVE

Based on the negative exponential pair-potential （I/R）n, the interaction potential between the micro/nano planar curve and the particle located outside the curve is studied. We verified that, whatever the value of exponent n may be the potential of particle/plane-curve is always of unified curvature form. Furthermore, we proved that the driving forces acted on the particle may be induced by the highly curved micro/nano curve, and the curvature and gradient of curvature are confirmed to be the essential factors forming the driving force. Through the idealized numerical experiments, the accuracy and reliability of the curvature-based potential are examined.