应用数学和力学

2008 Vol. 29, No. 7

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Investigation Into Field Structure at Mode Ⅲ Dynamically Propagating Crack Tip in Elastic-Visco-Plastic Materials

JIA Bin, WANG Zhen-qing, LI Yong-dong

2008, 29(7): 633-638.

Abstract(2684) PDF(487)

Abstract:
An elastic-viscoplastic mechanics model was adopted to investigate asymptotically the mode Ⅲ dynamically propagating crack tip field in elastic-viscoplastic materials.The stress and strain fields at crack tip possess the same powe-rlaw singularity under linear-hardening condition.And the singularity exponent is uniquely determined by the viscosity coefficient of the material.Numerical calculation results indicate that the motion parameter of crack propagating speed has little effect itself on zone structure at crack tip.The hardening coefficient dominates the structure of crack-tip field but the secondary plastic zone has little influence on the field.The viscosity of the material dominates the strength of stress and strain fields at crack tip while it does have certain influence on crack-tip field structure.The dynamic crack-tip field degenerates into the relevant quasistatic solution when the crack moving speed is zero.And the corresponding perfectly-plastic solution is recovered from the linear-hardening solution when the hardening coefficient becomes zero.

PSE as Applied to Problems of Transition in Compressible Boundary Layers

ZHANG Yong-ming, ZHOU Heng

2008, 29(7): 757-763.

Abstract(3117) PDF(615)

Abstract:
A new idea of using the parabolized stability equation(PSE) method to predict the laminarturbulent transition is proposed.It was tested in the prediction of the location of transition for compressible boundary layers on flat plates,and the results were compared with those obtained by direct numerical simulations(DNS).The agreement is satisfactory.The reason for the agreement was found to be that the PSE method does faithfully reproduce the mechanism leading to the breakdown process in laminar-turbulent transition,i.e.the modification of mean flow profile leads to a remarkable change of its stability characteristics.

Elasto-Plastic Postbuckling Analysis of Orthotropic Plates Including Damage Effects

TIAN Yan-ping, FU Yi-ming

2008, 29(7): 764-774.

Abstract(3064) PDF(616)

Abstract:
Based on the elasto-plastic mechanics and continuum damage theory,a yield criterion which is related to the spherical tensor of stress was proposed to describe the mixed hardening of damaged orthotropic materials,and the dimensionless form of which is isomorphic with the Mises criterion for isotropic materials.Furthermore,the incremental elasto-plastic damage constitutive equations and damage evolution equations were established.Based on the classical nonlinear plate theory, the incremental nonlinear equilibrium equations of orthotropic thin plates including damage effect were obtained,and the equations were solved by the finite difference method and iteration method.In the numerical examples,the effects of damage evolution and initial deflection on the elasto-plastic postbuckling of orthotropic plates are discussed in detail.

From the Second Gradient Operator and Second Category of Integral Theorems to Gauss or Spherical Mapping Invariants

YIN Ya-jun, WU Ji-ye, HUANG Ke-zhi, FAN Qin-shan

2008, 29(7): 775-782.

Abstract(2886) PDF(620)

Abstract:
Through the combination of the second gradient operator,the second category of integral theorems,the Gauss-curvature-based integral theorems and the Gauss(or spherical) mapping,a series of invariants or geometric conservation quantities under Gauss(or spherical) mapping were revealed.From these mapping invariants important transfor mations between original curved surface and the spherical surface were derived.The potential applications of these invariants and transformations to geometryare prospected.

Interaction Between Collinear Periodic Cracks in an Infinite Piezoelectric Body

CUI Zhi-jian, HU Hong-ping, YANG Feng

2008, 29(7): 783-789.

Abstract(2929) PDF(537)

Abstract:
The problem of collinear periodic cracks in an infinite piezoelectric body is studied.Effect of saturation strips at the crack-tips was taken into account.By means of Stroh formalism and the conformal mapping technique,the general periodic solutions for collinear cracks were obtained.The stress intensity factors and the size of saturation strips were derived analytically and their dependencies on the ratio of the periodicity to half-length of the crack h/l were analyzed in detail.Numerical results show that:1) when h/l is higher than 4.0,the stress intensity factors become almost identical to the ones for a single crack in an infinite piezoelectric body.It indicates that the interaction between cracks can be ignored in establishing the criterion for crack initiation in the case that h is larger than 4.10;2) the speed of saturation strip size of periodic cracks approaching one of a single crack depends on the electric load applied at infinity.In general,a larger electric load applied at infinity is with a slower approaching speed.

Study of Numerical Errors in DNS/LES

YANG Xiao-long, FU Song

2008, 29(7): 790-798.

Abstract(2965) PDF(700)

Abstract:
By comparing the energy spectrum and total kinetic energy,the effects of numerical errors (which arise from aliasing and discretization errors),subgrid-scale(SGS) models and their interactions on direct numerical simulation(DNS) and large eddy simulation(LES) were investigated in detail.The decaying isotropic turbulence was chosen as the test case.In order to simulate complex geometries,both spectral method and Pad?compact difference schemes were studied.The truncated Navier-Stokes(TNS) equation model with Pad discrete filter was adopted as SGS model.It is shown that for DNS,discretization error plays a key role in the simulation.Low order difference scheme may be not suitable.While for LES,it is found that SGS model can not only represent the effect of small scales to large scales,but also dump the numerical errors.Thus low order discretization scheme can also obtain reasonable results.

A Stencillike Volume-of-Fluid (VOF) Method for Tracking Free Interface

LI Xiao-wei, FAN Jun-fei

2008, 29(7): 799-805.

Abstract(3004) PDF(656)

Abstract:
A stencillike volume-of-fluid (VOF) method is proposed for tracking free interface.A stencil on a grid cell is worked out according to the normal direction of the interface,in which,only three interface positions are possible in 2D cases,and the interface can be reconstructed by only requiring the known local volume fraction information.On the other hand,the fluid-occupying-length was defined on each side of the stencil,through which a unified fluid occupying volume model and the unified algorithm can be resulted for solving the interface advection equation.The method is suitable for arbitrary geometry of the grid cell,and is extendible to 3D cases.The typical numerical examples show that the current method can give "sharp" result for tracking free interface.

Large Deflection of Circular Membrane Under Concentrated Force

JIN Cong-rui

2008, 29(7): 806-812.

Abstract(3368) PDF(673)

Abstract:
The analytical solution of Fêppl-Hencky membrane with rigidly clamped boundary condition under concentrated force was provided.The stability of nonlinear circular membrane has been investigated.

A Boussinesq Model With Improved Nonlinearity and Dispersion

ZHANG Dian-xin, TAO Jian-hua

2008, 29(7): 813-824.

Abstract(2868) PDF(612)

Abstract:
A new form of Boussinesq model over uneven bottom is derived.In the new model,the nonlinearity is improved without increasing the orders of the highest derivative of the differential equations.The dispersion relationship of the model was improved to the order of Pad (2,2)by adjusting a parameter in the model based on the long wave approximation.The analysis of the linear dispersion, linear shoaling and nonlinearity of the present model shows that the nonlinearity,dispersion and shoaling of this model are improved.The numerical results obtained for the present model were compared with the experimental data,and it is found that the predicted results agree with the experimental data.

Three Kinds of Nonlinear-Dispersive Waves in Finite Deformation Elastic Rods

ZHANG Shan-yuan, LIU Zhi-fang

2008, 29(7): 825-832.

Abstract(3163) PDF(594)

Abstract:
On the basis of classical linear theory on longitudinal,torsional and flexural waves in thin elastic rods,taking finite deformation and dispersive effects into consideration,three kinds of nonlinear evolution equations were derived.Qualitative analyses of three kinds of nonlinear equation were completed.It is shown that these equations have homoclinic or heteroclinic orbits on the phase plane, which correspond to solitary wave or shock wave solution respectively.Based on the principle of homogeneous balance,these equations were resolved by Jacobi elliptic function expansion method.The results show that the existence of solitary wave solution and shock wave solution are possible under certain conditions.These conclusions are consistent with that of the qualitative analysis.

Multi-Symplectic Method for Generalized Boussinesq Equation

HU Wei-peng, DENG Zi-chen

2008, 29(7): 839-845.

Abstract(3141) PDF(570)

Abstract:
Generalized Boussinesq equation,representing a group of important nonlinear equations, possesses many interesting properties.The multi-symplectic formulations of which in Hamilton space were introduced.Then an implicit multi-symplectic scheme equivalent to the multi-symplectic Box scheme was constructed to solve the partial differential equations(PDEs) that were derived from the generalized Boussinesq equation.The numerical experiments on the soliton solutions of the generalized Boussinesq equation were also reported.Finally,the results of which show that the multi-symplectic method is an efficient algorithm with excellent long-time numerical behaviors for nonlinear partial differential equation.

Exact Solution of Spatial Warping Curved Beams in Natural Coordinates

ZHU Li-li, ZHAO Ying-hua

2008, 29(7): 846-854.

Abstract(3037) PDF(895)

Abstract:
The purpose is to present an exact analytical solution of the spatial curved beam under multiple loads based on the existed theory.The transverse shear deformation and torsion-related warping effects are taken into account.By sing this solution,a plane curved beam subjected to uniform vertical loads and torsions was analyzed.The accuracy and the efficiency of present theory were demonstrated by comparing its numerical results with Heins.solution.Besides,the effects of the transverse shear deformation and torsion-related warping on the deformations of the beam were discussed.

Viscous Flow With Free Surface Motion by Least Square Finite Element Method

TANG Bo, LI Jun-feng, WANG Tian-shu

2008, 29(7): 855-863.

Abstract(2531) PDF(529)

Abstract:
A method for simulation of free surface problems is presented.Based on the viscous incompressible Navier-Stokes equations,the space discretization of the flow was obtained by least square finite element method and the time evolution was obtained by finite difference method.Lagrangian description was used to track the free surface.The results here were compared with experimental dam break results,including water collapse in 2D rectangular section and in 3D cylinder section.Good agreement was achieved for the distance of surge front as well as the height of residual column.

Note on “Block H-Matrices and Spectrum of Block Matrices”

LIU Jian-zhou, HUANG Ze-jun

2008, 29(7): 864-870.

Abstract(2683) PDF(682)

Abstract:
Some results of "Block H-matrices and spectrum of block matrices" are consummated.Furthermore,a new bound for eigenvalues of block matrices was given and some examples are given to show the advantages of this new result.

Analysis on Temperature Field at the Time of Pulse Current Discharge in Metal Structure With Elliptical Embedding Crack

FU Yu-ming, TIAN Zhen-guo, ZHENG Li-juan, LI Wei

2008, 29(7): 871-876.

Abstract(2406) PDF(673)

Abstract:
Temperature field at the time of pulse current discharge in metal structure with elliptical embedding crack was theoretically analyzed.In the progress of getting the temperature field,the current flows past an elliptical embedding crack is similar to the fluid flows past a barrier according to similarity principle.The boundary condition deriving from this theory was introduced so that the distribution of the current density and the temperature field expressions are gotten.The basic theoretical study for the actual application of spatial crack arrest is carried on by electromagnetic heating.

Several Properties of New Ellipsoids

SHEN Ya-jun, YUAN Jun

2008, 29(7): 877-882.

Abstract(2577) PDF(588)

Abstract:
The polytope whose new ellipsoid is a ball was first characterized.Furthermore,some properties for operator Gamma (-2) were proved and some inequalities were obtained.