应用数学和力学

2015 Vol. 36, No. 2

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Surface Effect on Propagation of Surface Waves in a Dielectric Elastomer Half Space Subject to Biasing Fields

ZHOU Wei-jian, CHEN Wei-qiu

2015, 36(2): 119-127. doi: 10.3879/j.issn.1000-0887.2015.02.001

Abstract(1334) PDF(1008)

Abstract:
The thin surface layer model was used to establish the 1st-order equivalent boundary conditions to account for the surface effect of dielectric elastomers. Based on the linear incremental theory of infinitesimal motions superimposed on the finite deformation of an electroelastic body, the equivalent boundary conditions were rigorously derived with the Stroh formula and the Ting method. The Stroh formalism was further used to deduce the governing equations for surface waves involving the surface effect. For compressible Neo-Hookean dielectric elastomers, the dispersion equations for the Love waves and generalized Rayleigh waves were derived and investigated numerically. It is found that the two types of waves are decoupled from each other, as in the classical cases. Different from the classical Rayleigh waves, the generalized Rayleigh waves with surface effect are size-dependent and uniqueness of the Rayleigh wave no longer exists. Numerical results indicate that it is possible to regulate and optimize the surface acoustic wave devices through application of appropriate biasing fields.

Fuzzy Elasto-Visco-Plasticity Analysis of Finite Deformation Based on the L-D Plastic Flow Rule

WANG Xi-gang, FU Ming-fu

2015, 36(2): 128-139. doi: 10.3879/j.issn.1000-0887.2015.02.002

Abstract(1273) PDF(922)

Abstract:
To dynamically analyze the finite deformation of rock and soil materials, the Green strain and the 2nd Kirchoff stress were used to describe their geometric nonlinearity. Through introduction of a membership function into the yield function, a fuzzy elasto-visco-plastic constitutive model based on the L-D plastic flow rule was built. According to the principle of nonlinear finite element, the numerical finite-deformation results corresponding to the dynamic triaxial soil tests were obtained, and compared with those traditional small-deformation results. The comparison shows that the finite-deformation results were closer to the dynamic triaxial tested results. Moreover, the fuzzy elasto-visco-plastic model well reflects the soil and rock dynamic properties under cyclic loading, and thus makes an effective way for soil and rock dynamic analysis, given its relatively simpler mathematical formulation and more convenience for finite element programming.

A Trefftz Finite Element Method for Solving Axisymmetric Poisson’s Equations

LIU Bo, WANG Ke-yong, WANG Ming-hong

2015, 36(2): 140-148. doi: 10.3879/j.issn.1000-0887.2015.02.003

Abstract(1618) PDF(1229)

Abstract:
A Trefftz finite element formulation was proposed for solving a kind of axisymmetric Poisson’s equations by means of the radial basis functions (RBFs). The non-zero right-hand side term brought the particular solution into the Trefftz intra-element field, which gave rise to domain integration related to the resultant element stiffness equation. The involved domain integration was eliminated through approximation of the particular solution with the RBFs. Furthermore, the‘boundary integration only’advantages were preserved for the Trefftz finite element method (TFEM). To obtain the particular solution, all elemental nodes and centroids in the whole solution domain were chosen as the fundamental interpolation points. In the meantime, a virtual boundary was constructed outside the solution domain, and a number of virtual points were selected as the additional interpolation points on the virtual boundary. Numerical examples demonstrate that the proposed method is valid and applicable.

High-Order Boundary Element Analysis of Temperature Fields in Thin-Walled Structures

HU Zong-jun, NIU Zhong-rong, CHENG Chang-zheng, ZHOU Huan-lin

2015, 36(2): 149-158. doi: 10.3879/j.issn.1000-0887.2015.02.004

Abstract(1041) PDF(885)

Abstract:
The geometric features of 3-node elements in the 2D BEM were analyzed, and the relative distance (namely the approach degree) from a source point to a high-order element was defined. Based on the geometric features, the approximate kernel functions were constructed with the same II-type singularity as the nearly singular kernel functions. For the nearly singular integrals, the dominant singular parts were separated from the original kernel functions through subtraction. After subtraction of the approximate kernel functions, the original kernel functions were rid of near singularity and turned into the sum of two integrals, of which one was a regular integral to be evaluated accurately with the conventional Gaussian quadrature, the other was a singular integral to be calculated with a series of analytical formulae derived herein. Then a new semi-analytical algorithm was established to compute the nearly singular integrals for the high-order elements effectively. In verification, the new method was applied to calculate several temperature field examples of thin-body structures for 2D potential boundary element analysis. The results indicate that the presented high-order-element semi-analytic algorithm takes full advantage of the BEM and has highly improved calculation accuracy and efficiency.

Stress Analyses of Double Lap Bonding Joint Using Viscoelastic Constitutive Model

ZHANG Pan, XU Ying-jie, WANG Hai-bin, GU Jing-wei

2015, 36(2): 159-166. doi: 10.3879/j.issn.1000-0887.2015.02.005

Abstract(1345) PDF(1131)

Abstract:
The finite element model of steel-rubber double lap bonding joint was created. The generalized Maxwell viscoelastic constitutive model was employed to model the time-dependent mechanical properties of adhesive. The Yeoh constitutive model was applied to describe the super elasticity of rubber. The influence of loading time on the shear stress of adhesive layer was analyzed. Computed results clearly show that the absolute value of shear stress decreases with the increase of loading time. In addition, the influence of adhesive thickness on the shear stress of adhesive layer is analyzed as well. With the increase of adhesive thickness, the absolute value of shear stress firstly increases obviously.

Formulation and Programming of a Dynamic Constitutive Model at High Strain Rates in ABAQUS for the 304NG Stainless Steel

LAN Bin, YE Xian-hui, SONG Shun-cheng, WU Wan-jun, WANG Bi-hao

2015, 36(2): 167-177. doi: 10.3879/j.issn.1000-0887.2015.02.006

Abstract(1496) PDF(1907)

Abstract:
The 304NG stainless steel is commonly used in reactor internal structural members. High strain rate dynamic characteristics of this material have important influences on the structural responses under impact loads. However, there is no suitable constitutive model for the high strain rate dynamic behaviors of this material in existing FEM programs as yet. Based on the dynamic tensile tests of the 304NG stainless steel, a new dynamic constitutive model for it was proposed. With the radial return algorithm and the stable dichotomy iteration method, a UMAT subroutine for the ratedependent model was written into ABAQUS, with the implicit stress update algorithm achieved. Then the dynamic FEM analyses of several examples were performed to verify the UMAT subroutine. The results indicate that the proposed dynamic constitutive model is in good agreement with the test data. The UMAT subroutine is helpful to be applied to response analysis of similar structures under impact loads.

A Uniformly Valid Series Solution to Sakiadis Flow

XU Ding

2015, 36(2): 178-189. doi: 10.3879/j.issn.1000-0887.2015.02.007

Abstract(1102) PDF(921)

Abstract:
In order to overcome the major mathematical difficulties in Sakiadis flow due to the semi-infinite flow domain and the asymptotic far field boundary condition, transformations were introduced for both the related independent variables and functions simultaneously, to convert the semi-infinite domain to a finite one and the asymptotic boundary condition to a convenient form. Then, based on the fixed point theory in functional analysis, the deduced nonlinear differential equation was solved, and an approximate semi-analytical series solution to Sakiadis flow was obtained. The calculation results show that the solution is uniformly valid in the semi-infinite domain, and the fixed point method makes an effective way to achieve approximate analytical solutions to differential equations.

Numerical Simulations of 2D Free Decaying Flow in an Unbounded Domain

YIN Zhao-hua, David C Montgomery

2015, 36(2): 190-197. doi: 10.3879/j.issn.1000-0887.2015.02.008

Abstract(980) PDF(933)

Abstract:
The fluid motion in an unbounded domain is an appealing and difficult problem in fluid mechanics. The 2D unbounded free decaying flow was studied and simulated with the traditional extended domain Fourier spectral scheme and the newly developed Hermite spectral algorithm, respectively. The results show that, in the case of only samesigned vortices existing in the domain at the beginning of simulations, both methods give correct results; on the other hand, in the case of positive and negative vortices coexisting initially, the new Hermite spectral method still gives satisfactory results for the problem efficiently even after longtime simulation, but the traditional Fourier method hardly yields correct results even in a greatly extended computing domain. Moreover, the numerical simulations of the examples with the Hermite spectral method prove the existence of the theoretically predicted Oseen vortices.

3D Simulation of Explosive Detonation With the Material Point Method

WANG Yu-xin, LI Xiao-jie, WANG Xiao-hong, YAN Hong-hao, SUN Ming

2015, 36(2): 198-206. doi: 10.3879/j.issn.1000-0887.2015.02.009

Abstract(1260) PDF(877)

Abstract:
It is difficult to simulate the explosive detonation of multi-lighter points and describe the assembling energy of detonation waves with the existent finite element methods. Particularly, computational efficiency and precision of the finite element methods are limited due to mesh distortion. Therefore, the explosive detonation of 2 lighters and even more lighters was numerically simulated with the material point method based on the explicit integration algorithm, to give results fitting the theoretical solution well. The proposed method not only avoids the re-meshing procedure for distorted elements, but also makes an effective way to the numerical simulation of the explosive detonation.

Numerical Analysis of Grease Lubrication for Radial Spherical Plain Bearings in Rotation and Swing Motion

TANG Zhan-qi, LIU Kun, WANG Wei, LIU Xiao-jun

2015, 36(2): 207-219. doi: 10.3879/j.issn.1000-0887.2015.02.010

Abstract(1331) PDF(919)

Abstract:
Formulation of the 3D lubrication model for spherical bearings was proposed. The rotation of the inner ring, the inner ring tilt caused by the axle journal misalignment and the swing angular velocity of the inner ring were taken into account in this model, and the modified Reynolds equation in the spherical coordinate system was obtained to depict non-Newtonian fluid lubrication. Moreover, with the lubrication model aforementioned, the lubrication problems of radial spherical plain bearings were analysed numerically based on the Ostwald rheological model for greases. The pressure distribution, the maximum pressure, the load capacity and the flow rate of the grease lubricant film were studied with variations of the grease power-law index, the inner ring inclination angle and the swing angular velocity, respectively. It is found that the grease lubricant film generates obvious hydrodynamic effects. With the other parameters fixed, the power-law index has significant influences on the maximum pressure and the load capacity of the grease film. Compared with Newtonian fluid, the shear-thickening fluid is observed to increase the values of film maximum pressure, load capacity and axial flow rate, but the shear-thinning fluid is observed to act contrarily. The inner ring inclination angle has little effect on the maximum pressure and the load capacity, while the swing angular velocity has much.

Numerical Simulation of the Geometric Models for Particle-Reinforced Composite Materials

SONG Lai-zhong, LI Yao-hui

2015, 36(2): 220-232. doi: 10.3879/j.issn.1000-0887.2015.02.011

Abstract(1117) PDF(1114)

Abstract:
To simulate the meso-level structures of multiphase particle-reinforced composite materials, such as concrete and soil-rock mixture, the corresponding reinforcing particles were assumed as convex polyhedrons. Firstly, the method was developed for obtaining an arbitrary polyhedron as well as its parametric equation through transformation of a randomly chosen octahedron. Secondly, the conditions for determining whether a spatial point is in the interior or exterior of a given convex polyhedron were presented. Moreover, the procedure for calculating the distance between a spatial point and a convex polyhedron or the distance between 2 polyhedrons was established, based on which the algorithm to generate a region with abundant randomly distributed convex polyhedrons was proposed. In order to increase the content of the polyhedrons in the region to be simulated, the descent algorithm was presented. The numerical results show that, for the 2-graded distribution, the content of the generated polyhedrons in the simulated region reaches 35% (volume ratio), with higher content realizable through combination of this method and those previously published. The work provides an effective way to build realistic meso-level geometric models for particle-reinforced composite materials.