应用数学和力学

Buckling Critical Load Calculation and Analysis of Axially Impacted Cylindrical Shells Embedded in Elastic Media

GUI Yifei, MA Jianmin

2020, 41(4): 353-366. doi: 10.21656/1000-0887.400137

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Abstract:
The radial governing equations for cylindrical shells embedded in elastic media were established and solved. The bifurcation conditions for dynamic buckling before and after stress wave reflection were obtained, in view of the boundary conditions and the consistency conditions. The relationships among the critical buckling load, the position of the wave front reaching the cylindrical shell, the elastic medium stiffness and the ratio of the nonembedded depth to the shell length were acquired through numerical calculation. The numerical results show that, there exists the same trend in axisymmetric and nonaxisymmetric buckling modes of cylindrical shells embedded in elastic media. The larger the embedding depth and the elastic medium stiffness are, the harder the buckling will be. The critical buckling load experiences three stages of extremely slow growth, rapid growth and slow growth with stiffening of the elastic medium. Before stress wave reflection, buckling occurs only in the region of stress wave propagation. Before the reflected wave front goes through the interface, buckling occurs near the reflection end for a small critical load, and buckling covers the entire cylindrical shell as the axial mode order increases. After the reflected wave front goes through the interface, buckling always covers the entire cylindrical shell.

On the Programmed Kinematics Computation of Crank Rocker Mechanism Based on the Kinematics Bifurcation Position Analysis

YIN Xiaoli, LI Chunming

2020, 41(4): 367-375. doi: 10.21656/1000-0887.400173

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Abstract:
The crankrocker mechanism is a typical planar quabody mechanism. When the crank is a driven part, there is a kinematics bifurcation position (stuck position or dead point) with the collinear crank and rod. So it is difficult to achieve accurate programmed research. With the crank as the driving part, the same higher derivatives of the driven part position angles were obtained through differentiation of the vector projection equations and the angular velocity formula several times. With the rocker as the driving part, based on the L’Hopital’s rule, the derivative multiplication law, the chain derivative law and the vector equation graphic method, the kinematics parameters of 0/0 indefinites at the stuck position were obtained. The calculation formula was derived. So the foundation for accurate kinematics programmed simulation was made. The example shows that, 1) the order of derivation and solution of the vector projection equation system does not influence the final result; 2) the driven crank keeps the continuity of motion without impact at the stuck position.

Fracture Mechanics Analysis of Mode-Ⅲ Radial Multi Cracks on the Edge of a Hole With Surface Effects

FENG Guoyi, XIAO Junhua, SU Mengyu

2020, 41(4): 376-385. doi: 10.21656/1000-0887.400177

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Abstract:
The mode-Ⅲ fracture properties of the radial multi cracks on the edge of a nanoscale circular hole were theoretically investigated. Based on the Gurtin-Murdoch surface elasticity theory and the conformal mapping technique, the analytical solutions of the stress fields around the hole and cracks were given, and the closed form solution of the stress intensity factor at the crack tip was obtained. The size effect of the stress intensity factor was analyzed based on the solution. The effects of the crack number, the ratio of crack/hole radius and the surface defects on the stress intensity factor were discussed. The results indicate that, the dimensionless stress intensity factor has remarkable size-dependent effects when the sizes of the cracked hole are at the nanoscale. The variation of the stress intensity factor with the number of cracks is influenced by the crack/hole size ratio. The effects of the crack/hole size ratio on the stress intensity factors are restricted by the surface defects, and the effects of the surface properties on the stress intensity factor are also limited by the crack/hole size ratio. The surface effects on the stress intensity factors are independent of the number of cracks.

Analysis on Restrained Torsional Shear Stresses of Box Girders With Corrugated Steel Webs

WEI Na, ZHANG Yuanhai, YAO Xiaodong

2020, 41(4): 386-395. doi: 10.21656/1000-0887.390329

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Abstract:
Based on the theory of circumferential non-deformation, and combined with the calculation and analysis on constrained torsion of closed thin-wall members, the distribution and calculation of torsional shear stresses were addressed for composite box girders with corrugated steel webs. The distributions of free torsional shear stresses and warping torsional shear stresses were further demonstrated through deduction of the formula for calculating the shear flow of cantilever plates in restrained torsion. The relevant shortcomings in calculating shear stresses by existing references were pointed out. A simply supported composite box girder with corrugated steel webs was taken as an example to compare the analytical results with those from the ANSYS finite element method. The results show that, in the sections of box girders with corrugated steel webs, the torsional shear stresses are mainly undertaken by corrugated steel webs, then by concrete bottom plates. The maximum shear stress in the bottom plate occurs at its center and is almost half that in the webs. The torsional shear stresses in the top concrete plate and the cantilever plate are very small. The calculated results of torsional shear stresses are consistent with those with the finite element method on the whole. The shear stress is zero at the free tip of the cantilever plate and gradually increases and reaches the peak value with the distance away from the tip end.

Magnetohydrodynamic Electroosmotic Flow in Zeta Potential Patterned Micro-Parallel Channels

WANG Shuang, JIAN Yongjun

2020, 41(4): 396-405. doi: 10.21656/1000-0887.400151

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Abstract:
Two-dimensional magnetohydrodynamic (MHD) electroosmotic flow (EOF) in zeta potential patterned micro-parallel channels was studied. The flow was driven by the combination of the Lorentz force and the electric field force produced due to an externally imposed vertical magnetic field and two horizontal electric fields. The analytical solutions of stream function and velocity distribution were obtained under the condition of hydrodynamic slippage. The variations of velocities with related non-dimensional parameters, such as Hartmann number Ha,slip length B and electrokinetic width K were addressed in detail. Results show that, the patterned charged surfaces induce a vertical velocity component leading to the formation of the vortexes. Also, the magnitudes of velocities increase with slip length B and electrokinetic width K.Moreover, it is interesting to note that the magnitudes of velocities become small with the increasing value of Ha, unlike the situation where there exists a critical value of Ha in one-dimensional flow. The present theoretical results can be utilized to design efficient microfluidic devices.

Phragmén-Lindelöf Type Results for Non-Standard Stokes Flow Equations Around Semi-Infinite Cylinder

LI Yuanfei

2020, 41(4): 406-419. doi: 10.21656/1000-0887.400272

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Abstract:
The initial-boundary value problem of nonstandard Stokes fluid equations defined around semi-infinite cylinder was considered, in which the nonlinear boundary condition was applied to the finite end of the cylinder and the zero boundary condition was satisfied on the side face of the cylinder. In the appropriate range of initial conditions, the differential inequality technique was used to obtain the Phragmén-Lindelöf results of Stokes fluid equations. In the case of decay, it is proved that ‘total energy’ can be controlled by known data items.

Limits of Riemann Solutions for Generalized Chaplygin Gas Magnetohydrodynamic Euler Equations With Source Terms

YU Kangning, GUO Lihui

2020, 41(4): 420-437. doi: 10.21656/1000-0887.400122

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Abstract:
The asymptotic behaviors of Riemann solutions for generalized Chaplygin gas magnetohydrodynamic Euler equations with source terms were considered. The self-similarity of the solutions is no longer true due to the inhomogeneous term. They will converge to Riemann solutions for zero-pressure flow transport equations when pressure and magnetic induction disappear at the same time, and δ-shock wave and vacuum will appear in the solutions. The solutions will converge to Riemann solutions for generalized Chaplygin gas Euler equations with inhomogeneous terms in the case of vanishment of magnetic induction, additionally only δ-shock wave appears in the solutions.

Dynamic Behaviors of Filippov Ecosystems Induced by Allee Effects

QIN Wenjie, GUAN Haiyan, WANG Peipei, TANG Guangyao

2020, 41(4): 438-447. doi: 10.21656/1000-0887.400169

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Abstract:
A non-smooth Filippov predatorprey system induced by Allee effects was proposed. The sliding domain, the sliding mode dynamics and the existence of several equilibria were discussed by means of the qualitative theory and method related to the Filippov system. Furthermore, the sliding mode bifurcation, the boundary focus bifurcation and the global dynamic behaviors were given through numerical simulations. The results indicate that, the intensity of Allee effects could make the dynamics of population become unstable, and may be unfavorable for the protection of endangered species.

An Evaluation Method for Impacts of a New Satellite Hall on Boarding Gates

YANG Pengfei, LI Ting, RU Hongwu, CHEN Haibin

2020, 41(4): 448-457. doi: 10.21656/1000-0887.400189

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Abstract:
In order to solve the passenger flow saturation in the airport terminal, the addition of a satellite hall was used to realize passenger diversion. Based on the method of single-objective integer linear programming and multi-objective optimization, the boarding gate assignment optimization network model and the multi-objective optimization model were constructed respectively. The boarding gate assignment optimization network algorithm was used to screen the common boarding gate, establish the objective function and list the constraint conditions, and the objective and constraint method were used to solve the established model. On this basis, according to the idea of objective modeling, a flight gate assignment model with the shortest overall process time for transit passengers and the smallest number of gates was established. According to MATLAB calculations, 303 flights can be accommodated normally by 42 gates.

Vector Variational-Like Inequalities and Vector Optimization Problems Involving ρ-(η,d)-B Invexity on Riemannian Manifolds

LIU Shuang, MO Dingyong, ZHOU Zhiang

2020, 41(4): 458-466. doi: 10.21656/1000-0887.400227

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Abstract:
A class of optimality problems involving the generalized directional derivatives were studied on Riemannian manifolds. Firstly, by means of the generalized directional derivative, three concepts of the ρ-(η,d)-B invex function, the pseudo ρ-(η,d)-B invex function and the quasi ρ-(η,d)-B invex function on Riemannian manifolds were introduced. Secondly, the relations between the solution to variational inequalities and the solution to the optimization problem on Riemannian manifolds were discussed. Finally, the Kuhn-Tucker sufficient condition for the optimality problem was established.

2020 Vol. 41, No. 4