应用数学和力学

2018 Vol. 39, No. 11

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A Study on the Constitutive Relation and the Flow of Spatial Fractional NonNewtonian Fluid in Circular Pipes

YANG Xu, LIANG Yingjie, SUN Hongguang, CHEN Wen

2018, 39(11): 1213-1226. doi: 10.21656/1000-0887.390153

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Abstract:
A thorough understanding of the flow behavior of non-Newtonian fluid is the first step for analyzing, predicting and controlling of pipe flow. Experiments indicate that non-Newtonian fluid is historically dependent on the procedure of shear flow. The constitutive model for fractional non-Newtonian fluid was established via the spatial fractional calculus approach. The velocity profile, the flux, the mean velocity, the pressure drop and the mean Reynolds number of the proposed model were also derived. In addition, a novel criterion for the flow state of fractional non-Newtonian fluid was proposed. The results show that, the shear stress of the non-Newtonian fluid can be described by the axial velocity distribution. For the fractional non-Newtonian fluid without yield shear stress, the larger the fractional order is, the more uniform the velocity distribution will be and the stronger the memory of the fluid will be. The magnitude of the fractional order reflects the memory of the fluid with respect to the global space. For the fractional non-Newtonian fluid with yield shear stress, the larger the fractional order is, the more uniform the velocity distribution in the velocity gradient region will be, and the smaller of the velocity in the core region will be. In this case, the magnitude of the fractional order reflects the memory of the fluid with respect to the local region. This study offers a new method for the modeling of memory characteristics of non-Newtonian fluid.

Parametric Vibration Responses of Supercritical Fluid-Conveying Pipes in 3∶1 Internal Resonance

ZHANG Kaikai, TAN Xia, DING Hu, CHEN Liqun

2018, 39(11): 1227-1235. doi: 10.21656/1000-0887.390121

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Abstract:
The parametric vibration responses were studied in the supercritical fluid-conveying pipes in the 3∶1 internal resonance condition. In the control equation, the non-normal static configuration of the pipe at the supercritical velocity was introduced, and the partial differential-integral control equation was obtained. The approximate analytic solution was deduced with the direct multiscale method, and the reliability of the approximate analytic results was verified with the Galerkin truncation method. Numerical examples show that, there exists energy transfer between different modes of the pipeline system in internal resonance. The influence of the parameter amplitude on the amplitude-frequency response was predicted based on approximate analytical results.

Numerical Simulation of Seawater Droplets Collision in Gaseous Environment

LIANG Wei, QI Xiaoni, YIN Qiang, ZHAO Qiang

2018, 39(11): 1236-1245. doi: 10.21656/1000-0887.390053

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Abstract:
Numerical simulation of binary collision between equal-sized seawater droplets was conducted with the VOF method, to reveal the mechanism and the influential factors for colliding seawater droplets in the seawater shower cooling tower. Veracity of the numerical models was firstly validated with the experimental results of Qian. The binary collision of equal-sized droplets were simulated with various Weber numbers and impact parameters under room temperature and pressure conditions. The Weber number ranged from 0.5 to 200 and the impact parameter ranged from 0 to 1. The simulation gave 3 different types of outcomes: coalescence, reflexive separation and stretching separation. The results show that the critical Weber number of head-on collision between coalescence and reflexive separation is 22. The We-x diagrams of various collision regimes of seawater droplets were also obtained.

A Modeling Method for Soil-FoundationWind Turbine Seismic Analysis Based on Foundation Impedance

WANG Jue, ZHOU Ding

2018, 39(11): 1246-1257. doi: 10.21656/1000-0887.390114

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Abstract:
A Chebyshev nested lumped-parameter model (LPM) was proposed to incorporate the frequency-dependent impedances into the time history analysis of soil-foundation-wind turbine systems. The complex Chebyshev polynomials were introduced to approximate the foundation impedances with the least squares curve-fitting technique. The computational program for seismic analysis of the system was developed based on the direct integration of the governing equation in the time domain. The validity and efficiency of the computational program were verified through the comparison examples. The time history analysis of a wind turbine supported by a pile group foundation under seismic excitation shows the universality and stability of the computational program. The advantage of the Chebyshev nested LPM is that it can express the frequency-dependent impedance accurately in the time domain and avoid the high-order numerical oscillation. The nested form of the model enables wider application to different kinds of foundations. In addition, as the present model adopts no mass unit, it avoids the problem of modifying the seismic loading at the node of the foundation and makes the analysis of practical engineering more convenient.

Application of a Coupled Interpolating Element-Free Galerkin Scaled Boundary Method and Finite Element Method in Fracture Analysis of Piezoelectric Materials

CHEN Shenshen, WANG Juan

2018, 39(11): 1258-1267. doi: 10.21656/1000-0887.390129

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Abstract:
The interpolating element-free Galerkin scaled boundary method (IEFG-SBM) is a semi-analytical method which only requires discretizing the boundary with the interpolating element-free Galerkin (EFG) method without fundamental solution. This method is very powerful to deal with fracture problems of piezoelectric materials. In order to further improve the applicability of the IEFG-SBM, a coupled IEFG-SBM and finite element method (FEM) for fracture analysis of piezoelectric materials was developed. The IEFG-SBM was utilized to model the domain close to the crack tip and the FEM was employed in the remaining domain. Based on continuity conditions at the interface between the IEFG-SBM sub-domain and the FEM sub-domain, the coupled formula of the proposed method can be conveniently derived. Finally, 2 numerical examples were presented to demonstrate the validity of the proposed method.

Shear Effect Analysis on Plates Based on the 4-Variable Refined Plate Theory

ZHOU Tao, SONG Yanqi

2018, 39(11): 1268-1281. doi: 10.21656/1000-0887.390066

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Abstract:
For moderately thick plates or composite plates, due to the low ratio of the transverse shear modulus to the in-plane extension modulus, the shear effects signify much to the mechanical behavior of structures. Based on the 2-variable refined plate theory (RPT), the discussion about the shear effects on the bending of plates was carried out. The RPT was introduced briefly at first, then numerical examples of simply supported rectangular plates subjected to uniformly distributed load were given, with a focus on the geometry and property effects on the shear effects. The shear effects increase with the plate thickness, especially dramatically when the width-thickness ratio is less than 10; the shear effects increase with the ratio of the transverse shear modulus to the in-plane extension modulus; under the same condition, the proportion of the “shear” deflection in the orthotropic plate is always larger than that in the isotropic plate, and the difference of shear effects between these 2 kinds of plates becomes more significant with the increase of the thickness or the aspect ratio; the shear effects in the isotropic plates are more sensitive to the aspect ratios, and the proportion of the “shear” deflection in the isotropic plate decreases with the aspect ratio, but decreases first and increases later in the orthotropic plate.

A Separate Solution Method for Shear Lag Effects in Box Girders and Parameter Analysis

ZHANG Yuyuan, ZHANG Yuanhai, ZHANG Hui

2018, 39(11): 1282-1291. doi: 10.21656/1000-0887.380073

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Abstract:
With the additional deflection caused by the shear lag effect as a generalized displacement, the shear lag deformation of the box girder was separated from the elementary beam deflection as an independent state for analysis. The governing differential equations and the corresponding boundary conditions were established with the energy variational method. According to the simply supported boundary conditions, the analytical solutions of the additional deflection and the longitudinal stress were deduced under the concentrated load and the uniform load. The longitudinal stress analysis shows that the stresses calculated with the present method were in good agreement with those with the spline function method, validating the rationality of the present method. The deflection study shows that the additional deflection of shear lag diminishes from the span center section to the 2 ends. The shear lag additional deflection at the span center section reaches respectively 2.57% and 3.03% of the elementary beam deflection of the simply supported box girder under the uniform load and the concentrated load. With the increases of the height-span and width-span ratios, the additional deflection at the span center section of the box girder gradually decreases. In particular, the influence of the width-span ratio on the additional deflection is far greater than that of the height-span ratio.

Study on Symmetries and Conserved Quantities of Mechanical Multibody System Collision Dynamics

ZHENG Mingliang, FENG Xian, LI Wenxia, CAO Yalin

2018, 39(11): 1292-1299. doi: 10.21656/1000-0887.380291

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Abstract:
To provide a powerful new tool for quantitative and qualitative analysis of collision dynamics in complex mechanical multibody systems, the symmetry theory in modern analytical mechanics was introduced into the study of mechanical multibody external collision dynamics. Firstly, the Euler-Lagrange equation of collision dynamics was derived based on the momentum method; secondly, the group theory was introduced, then, according to the invariance principle, the condition equations for the Noether symmetry and the Lie symmetry were obtained and the corresponding conserved quantity form was got, which made possible an effective approach to the analytic integral theory for dynamic equations. Finally, the collision dynamics of a planar open-loop 2-connecting-rod mechanism was taken as an example for application and analysis. Research shows that deeper mechanical laws and motion characteristics of mechanical multibody system collision dynamics can be obtained by means of symmetries and conserved quantities, and the results also lay a theoretical foundation for more precise dynamic optimal design and advanced control.

Exponential Stability of Traveling Wavefronts for ReactionDiffusion Equations With Delayed Nonlocal Responses

LI Panxiao

2018, 39(11): 1300-1312. doi: 10.21656/1000-0887.380336

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Abstract:
A spatially nonlocal diffusion model with a class of delayed nonlocal responses was considered. The asymptotic stability and the convergence rate of the traveling wavefronts were mainly studied. Through construction of weighted functions and establishment of a comparison principle for the related linear equations, the conclusion that if the initial function is within a bounded distance from a certain traveling wavefront with respect to a weighted maximum norm, the solution satisfying the initial value will converge to the traveling wavefront exponentially in time, was proved, and the exponential convergence rate was also obtained.

Exact Traveling Wave Solutions and Bifurcations of (2+1)-Dimensional Space-Time Fractional-Order Nizhnik-Novikov-Veslov Equations

JIANG Lin, SUN Yuhuai, ZHANG Xue, HONG Yun

2018, 39(11): 1313-1322. doi: 10.21656/1000-0887.380299

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Abstract:
The (2+1)-dimensional space-time fractional-order Nizhnik-Novikov-Veslov equations were transformed into ordinary differential equations through the fractional complex transform, then the Hamiltonian and the bifurcation phase portraits for the corresponding plane system to the equations were got with the bifurcation method for dynamical systems. According to the tracks in the phase portraits, solitary wave solutions, blow-up wave solutions, periodic wave solutions and periodic blow-up wave solutions to the equations were obtained. Relations between the traveling wave solutions were also discussed.