应用数学和力学

2011 Vol. 32, No. 11

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On a Fractional-Order Maxwell Model of Seabed Mud and Its Effect to Surface-Wave Damping

XIA Yue-zhang, ZHU Ke-qin

2011, 32(11): 1265-1273. doi: 10.3879/j.issn.1000-0887.2011.11.001

Abstract(1616) PDF(984)

Abstract:
A fractional-order Maxwell model was used to describe viscoelastic seabed mud.The experimental data of real mud were fitted well by the fractional-order Maxwell model,which had fewer parameters than the traditional one.This model was then applied to investigate the effect of mud on surface-wave damping.The damping rate of a linear monochromatic wave was obtained.The elastic resonance of the mud layer was observed,which led to the peaks of the damping rate.The damping rate was a sum of modal damping rates,which indicated the wave damping induced by mud motions of particular modes.Analysis shows that near resonance,the total damping rate is dominated by the damping rate of the corresponding mode.

MHD Effects on Free Convective Flow Over Moving Semi-Infinite Vertical Cylinder With Temperature Oscillation

P. Loganthan, M. Kannan, P. Ganesan

2011, 32(11): 1274-1282. doi: 10.3879/j.issn.1000-0887.2011.11.002

Abstract(1866) PDF(839)

Abstract:
Numerical solutions of MHD effects on free convective flow an incompressible viscous fluid past a moving semi-infinite vertical cylinder with temperature oscillation was presented.The dimensionless, unsteady,non-linear and coupled governing partial differential equations were solved using an implicit finite difference method of Crank-Nicolson type.The velocity,temperature and concentration profiles were studied for various parameters.The local as well as average skin-friction,Nusselt number and Sherwood number were also analyzed and presented graphically.The present results are compared with available results in literature and are found to be in good agreement.

Effects of Residual Interface Stress on Effective Thermal Expansion Coefficient of Particle-Filled Composite

HUANG Ru-chao, CHEN Yong-qiang

2011, 32(11): 1283-1293. doi: 10.3879/j.issn.1000-0887.2011.11.003

Abstract(1461) PDF(844)

Abstract:
The "three configurations" based surface/interface energy theory proposed by Huang et al was used to study the effective properties of thermal elastic nanocomposites.Particular emphasis was placed on the discussions of the influence of the residual interface stress on the thermal expansion coefficient of the said composites.First,the thermo-elastic interface constitutive relations expressed in terms of the first kind Piola-Kirchhoff interface stress and the Lagrangian description of the generalized Young-Laplace equation were presented.Second,the Hashin's composite sphere assemblage(CSA)was taken as the representative volume element(RVE),and the elastic deformations from the stress-free configuration to the reference configuration and from the reference configuration to the current configuration were calculated.Based on the above calculations,an analytical expression of the effective thermal expansion coefficient of thermo-elastic composite was derived.It is shown that the "residual" interface stress has a significant effect on the thermal expansion properties of the thermo-elastic nanocomposites.

Nonstationary Probability Densities of System Response of Strongly Nonlinear Single-Degree-of-Freedom System Subject to Modulated White Noise Excitation

JIN Xiao-ling, HUANG Zhi-long, LEUNG Andrew Y T

2011, 32(11): 1294-1305. doi: 10.3879/j.issn.1000-0887.2011.11.004

Abstract(1560) PDF(1136)

Abstract:
The nonstationary probability densities of system response of a single-degree-of-freedom system with lightly nonlinear damping and strongly nonlinear stiffness subject to dulated white noise excitation were studied.Using the stochastic averaging method based on the generalized harmonic functions,the averaged Fokker-Planck-Kolmogorov equation governing the nonstationary probability density of the amplitude was derived.The solution of the equation was approximated by a series expansion in terms of a set of properly selected basis functions with time-dependent coefficients.According to the Galerkin method,the time-dependent coefficients can be solved from a set of first-order linear differential equations.Then the semi-analytical formulae of the nonstationary probability density of the amplitude response as well as the nonstationary probability density of the state response and the statistic moments of the amplitude response can be obtained.A van der Pol-Duffing oscillator subject to modulated white noise was given as an example to illustrate the proposed procedures.The effects of the system parameters,such as linear damping coefficient and nonlinear stiffness coefficient,on the system response were discussed.

Exact Solutions of Two Semi-Infinite Collinear Cracks in a Piezoelectric Strip

LU Zi-xing, LIU Ping, GUO Jun-hong

2011, 32(11): 1306-1313. doi: 10.3879/j.issn.1000-0887.2011.11.005

Abstract(1502) PDF(727)

Abstract:
Using the complex variable function method and the technique of the conformal mapping,the fracture problem of two semi-infinite collinear cracks in a piezoelectric strip was studied under the antiplane shear stress and in-plane electric load on the partial crack surface.The analytic solutions of the field intensity factors and the mechanical strain energy release rate were derived under the assumption that the surface of crack was electrically impermeable.The present results can be reduced to the well-known solutions for a purely elastic material in the absence of electric load.Moreover,when the distance between the two crack tips tends to infinity,the analytic solutions of a semi-infinite crack in a piezoelectric strip were obtained.Finally,numerical examples were given to show the influences of the loaded crack length,the height of strip,the distance between the two crack tips,and the applied mechanical/electric loads on the mechanical strain energy release rate.It shows that the material is easier to fail when the distance between the two crack tips becomes shorter.Moreover,the mechanical/electric loading has greater influence on the propagation of the left crack than that of the right one.

Convergence and Exact Solutions of the Spline Finite Strip Method Using a Unitary Transformation Approach

Jackson KONG, Dick THUNG

2011, 32(11): 1314-1328. doi: 10.3879/j.issn.1000-0887.2011.11.006

Abstract(1646) PDF(733)

Abstract:
The spline finite strip method was one of the most popular numerical methods for analyzing prismatic structures.Efficacy and convergence of the method had been demonstrated in previous studies by comparing only numerical results with analytical results of some benchmark problems.To date,no mathematical exact solutions of the method or its explicit forms of error terms had been derived to demonstrate analytically its convergence.As such,mathematical exact solutions of spline finite strips in plate analysis were derived using a unitary transformation approach(abbreviated as the U-transformation method herein).These exact solutions were presented for the first time in open literature.Unlike the conventional spline finite strip method which involves assembly of the global system of matrix equation and its numerical solution,the U-transformation method decoupled the global matrix equation into one involving only two unknowns,thus rendering exact solutions of the spline finite strip to be derived explicitly.By taking Taylor's series expansion of the exact solution,error terms and convergence rates were also derived explicitly and compared directly with other numerical methods.In this regard,it was found that the spline finite strip method converged at the same rate as a non-conforming finite element,yet involving smaller number of unknowns compared to the latter.The convergence rate was also found superior to the conventional finite difference method.

Propagation of Harmonic Waves Through a Micro Gap With Consideration of Frictional Contact

CHEN Xiao-yun, YU Gui-lan

2011, 32(11): 1329-1341. doi: 10.3879/j.issn.1000-0887.2011.11.007

Abstract(1707) PDF(712)

Abstract:
Propagation of elastic waves through a micro gap between two solids with consideration of frictional contact was investigated.By using the Fourier analysis technique and corrective solution method, the nonlinear boundary problem was reduced to a set of algebraic equations.Numerical results exhibit the locations and extents of separation,slip and stick zones,the interface tractions and the energy partition. The effects of gap width,frictional coefficients and the incident angle on the wave transmission were discussed in detail.The results show that higher harmonics are generated due to the local contact/slip at the interface.The dependence of reflection and refraction coefficients for higher harmonics on the initial gap width may provide us a theoretical foundation for non-destructive testing of a pre-existing micro gap.

Separation Work Analysis of Cohesive Law and Consistently Coupled Cohesive Law

HE Ming-hua, XIN Ke-gui

2011, 32(11): 1342-1351. doi: 10.3879/j.issn.1000-0887.2011.11.008

Abstract(1152) PDF(713)

Abstract:
An appropriate coupled cohesive law for predicting mixed mode failure was established by combining normal separation and tangential separation of surface in cohesive zone model and cohesive element method.Xu-Needleman exponential cohesive law with fully shearing failure mechanism was one of the most popular models in literature.Based on the proposed consistently coupled rule/principle,Xu-Needle-man law with fully shearing failure mechanism was proved to be a non-consistently coupled cohesive law by analyzing surface separation work.It is shown that Xu-Needleman is only valid in mixed mode fracture when the normal separation work equals to the tangential separation one.Based on the consistently coupled principle and the modification of Xu-Needleman law,a consistently coupled cohesive(CCC law)was given.It is shown that the proposed CCC law has already overcome the non-consistency defect of Xu-Needleman Law with great promise in mixed mode analysis.

John Haussermann, K. Vajravelu, Robert A. Van Gorder

2011, 32(11): 1352-1360. doi: 10.3879/j.issn.1000-0887.2011.11.009

Abstract(1647) PDF(738)

Abstract:
The Lin-Reissner-Tsien equation describes unsteady transonic flows under the transonic approximation.The equation was reduced to an ordinary differential equation via a similarity transformation. The resulting equation was then solved analytically,and in some cases even exactly.Numerical simulations were provided for the cases in which there were no exact solutions.Traveling wave solutions were also obtained.

Bilevel Generalized Mixed Equilibrium Problems Involving Generalized Mixed Variational-Like Inequality Problems in Reflexive Banach Spaces

DING Xie-ping

2011, 32(11): 1361-1377. doi: 10.3879/j.issn.1000-0887.2011.11.010

Abstract(1611) PDF(785)

Abstract:
A new class of bilevel generalized mixed equilibrium problems(BGMEP)involving generalized mixed variational-like inequality problems was introduced and studied in reflexive Banach spaces. First,an auxiliary generalized mixed equilibrium problem(AGMEP)to compute the approximate solutions of the bilevel generalized mixed equilibrium problems involving generalized mixed variational-like inequality problems was introduced.By using a minimax inequality,the existence and uniqueness of solutions of the AGMEP was proved under quite mild conditions without any coercive assumptions.By using auxiliary principle technique,new iterative algorithm to compute the approximate solutions of the BGMEP were suggested and analyzed.The strong convergence of the iterative sequences generated by the algorithms was proved under quite mild conditions without any coercive assumptions.These results are new and generalize some recent results in this field.

Sloshing Simulation of Standing Wave With a Time-Independent Finite Difference Method for Euler Equations

LUO Zhi-qiang, CHEN Zhi-min

2011, 32(11): 1378-1390. doi: 10.3879/j.issn.1000-0887.2011.11.011

Abstract(1870) PDF(850)

Abstract:
The numerical solutions of standing wave for Euler equations with nonlinear free surface boundary condition in a two dimensional tank were solved.The irregular tank was mapped onto a fixed square domain through proper mapping functions and a staggered mesh system was employed in a two dimensional tank in order to calculate the elevation of the transient fluid.A time-independent finite difference method, which was developed by Bang-fuh Chen,was applied and was used to solve Euler equations for incompressible and inviscid fluid.The numerical solutions agree well with analytic solutions and previously published results.The nonlinear and beating phenomena are very clear and the sloshing of surge and heave motions with initial standing wave are presented.