应用数学和力学

2015 Vol. 36, No. 4

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A Thermo-Mechanical Coupling Atom-Continuum Coupled Model and Its Algorithm

LI Hui, CUI Jun-zhi, LI Bo-wen

2015, 36(4): 343-351. doi: 10.3879/j.issn.1000-0887.2015.04.001

Abstract(1268) PDF(958)

Abstract:
A thermo-mechanical coupling atom-continuum coupled model and its efficient algorithm were studied systematically, and the formulae for calculation of thermodynamic and mechanical parameters of metallic materials in micro/nano-scale were obtained. The heterogeneous structure and deformation of polycrystalline clusters in micro- and nano-scale were studied by means of the‘structural deformation’part of the atomic motion. Then the lattice structural transformation of atomic clusters was related with the continuum deformation, and the expressions of free energy density, entropy density and internal energy density, which were dependent on the micro-structural deformation and thermal vibration, were derived under the quasi-harmonic approximation assumptions, with some transient thermodynamic and mechanical parameters given. The numerical simulation of a tensioning test on a Cu nanowire proves correctness of the proposed model and validity of the algorithm.

Elastic Analysis of Arbitrary Thin-Walled Structural Members Based on the Unified Analytical Beam Model With the Finite Nodal Line Method

GONG Yao-qing, SUN Bo, CHEN Xiao-dong

2015, 36(4): 352-361. doi: 10.3879/j.issn.1000-0887.2015.04.002

Abstract(734) PDF(686)

Abstract:
The traditional thin-walled beam theory is strongly dependent on the slenderness ratio, the shape of its cross section and the load form of a beam. In order to investigate the elastic behaviors of arbitrary thin-walled beams under arbitrary loads, a unified analytical model for all thin-walled structural members, named ‘unified analytical beam’(UAB), and a novel computational methodology titled ‘finite nodal line method’(FNLM) were presented. By means of UAB and FNLM, the elastic analysis of arbitrary thin-walled structural members can be conveniently carried out. Furthermore, when the properties of a beam problem are consistent with the application conditions for the traditional beam theory, an equal-precision solution to the problem will be obtained with both the FNLM and the traditional one. The computational comparison through several beam examples verifies the rationality and correctness of the proposed UAB and FNLM.

Inter-Belt Analysis of the Integral-Form Nonlocal Constitutive Relation

YAO Zheng, ZHENG Chang-liang

2015, 36(4): 362-370. doi: 10.3879/j.issn.1000-0887.2015.04.003

Abstract(1118) PDF(842)

Abstract:
Based on the Hamilton theory, the constitutive relation was investigated for the nonlocal linear elasticity originally proposed by Eringen. Eringen’s nonlocal equations can be written in the integral form and the differential form. The differential form with the relatively simple mathematical formulation, had been widely used in recent years. For the integral form of the nonlocal elastic theory, solving the integro-differential equations was challenging for numerical process. In the analytical structural mechanics, the integro-differential equations in time-delay problems had been solved with the inter-belt theory. According to the simulative relations between the analytical dynamics and the analytical structural mechanics, the inter-belt theory was introduced into the integral-form constitutive equations of the nonlocal theory, and hence the integro-differential equations were numerically solved with high precision. Then the fundamental theory and computational algorithm were applied to dynamic problems of nonlocal rod vibration. The numerical experiments demonstrate validity of the present method and potential of the symplectic system algorithm in solving nonlocal mechanics problems.

A Nonlinear Mechanical Model for Extensible Slender Rods

MA Gang, SUN Li-ping, AI Shang-mao, WANG Hong-wei

2015, 36(4): 371-377. doi: 10.3879/j.issn.1000-0887.2015.04.004

Abstract(1158) PDF(1525)

Abstract:
The polyester mooring line is a flexible rod liable to axial extension and bending deformation. Due to its large axial deformation in tension, the polyester line’s deformed configurations shall be distinguished from the undeformed ones, and the slender rod model based on the small extension hypothesis shall be improved in view of its nonnegligible length change. The global coordinates and slopes were used to replace the rotation angles in the Euler-Bernoulli beam to deal with the geometrically nonlinear problem of large spatial rotation based on Garrett’s slender rod theory. Then the point-to-point mapping method was proposed to correlate the 2 configurations before and after axial extension in the slender rod theory, with 2 element nodes, 1 intermediate point and 3 quadratic polynomial combined to derive the differential equations of motion for the slender rod element in tension. Through comparison between the numerical results and the analytical solution of an extensible cantilever beam example, convergence and accuracy of the proposed model are proved.

An Improved Precise Runge-Kutta Method for Structural Dynamic Equations

ZHANG Ji-feng, DENG Zi-chen, ZHANG Kai

2015, 36(4): 378-385. doi: 10.3879/j.issn.1000-0887.2015.04.005

Abstract(1118) PDF(879)

Abstract:
Based on the precise Runge-Kutta method, in view of the characteristics of the non-homogeneous terms of the state space equations and the particular distribution of the loads, a new improved precise Runge-Kutta method was presented for solving the structural dynamic equations. Through partitioning of the related state space matrices, the improved method not only inherited the advantage of high precision of the precise Runge-Kutta method, but also greatly promoted the computational efficiency, making it suitable for solving large-scale structural dynamic problems and conducting long-time simulations. The results of numerical examples show the correctness and validity of the proposed simplified method.

Application of the PseudoArclength Continuation Algorithm to Investigate the SizeDependent Pull-in Instability of the Electrostatically Actuated MEMS

LIANG Bin-bin, ZHANG Long, WANG Bing-lei, ZHOU Shen-jie

2015, 36(4): 386-392. doi: 10.3879/j.issn.1000-0887.2015.04.006

Abstract(1234) PDF(923)

Abstract:
In the study of the electrostatically actuated MEMS (microelectromechanical systems), based on the strain gradient elasticity theory, the governing equations for the microbeam are nonlinear differential equations that are difficult to solve. The mathematical model for this problem is of essential bifurcation, and the solution branches of the equations have inflection points. The iteration process can’t go through the inflection points with the local continuation method. Therefore, the generalized differential quadrature method was applied to discretize and reduce the order of the governing equations, and the pseudoarclength continuation algorithm was used to enable the iteration process to go smoothly through the inflection points, with the complete solution curve calculated. The numerical results show that the pseudoarclength continuation algorithm makes an effective way precisely solving the nonlinear highorder differential equations with bifurcation phenomenon embedded, and helps to accurately predict the pullin voltage of the electrostatically actuated MEMS.

Effects of Boundary Damping on the Elastic Rod’s Response to Longitudinal Impact

JIAO Xiao-juan, MA Jian-min

2015, 36(4): 393-403. doi: 10.3879/j.issn.1000-0887.2015.04.007

Abstract(773) PDF(1037)

Abstract:
The longitudinal impact between a rigid body and an elastic rod with boundary stiffness and damping conditions was discussed, the analytical velocity and stress distributions during the 1st impact wave period were derived for the rod, and the condition that the impact time duration equals 2 times of the value of the rod length divided by the wave velocity, was also investigated. Then several examples were calculated to discuss the effects of the boundary damping on the velocity and stress distributions in the rod in the cases of different mass ratios between the rigid body and the rod, and different connection stiffnesses. It is shown that the boundary damping influences the values of reflection wave velocity and stress wave front, and the velocity and stress distributions in the rod become gentler with a higher boundary damping. Given a larger boundary stiffness, the effects of the boundary damping on the velocity and stress in the rod will be more significant. It is also indicated that the impact time duration is not related to the initial impact velocity, but to the mass ratio, the boundary stiffness, the boundary damping and the rod length. Especially, for a larger mass ratio, the boundary damping has greater influence on the stress of the rod’s impact end.

Growth Trend of Thermal Grown Oxide in TBCs Under Chemical Oxidation Effect

CHAI Yi-jun, LIN Chen, LI Yue-ming

2015, 36(4): 404-413. doi: 10.3879/j.issn.1000-0887.2015.04.008

Abstract(1266) PDF(952)

Abstract:
A user element subroutine of ABAQUS was developed, based on the Fick’s law involving the oxidation effect and the constitutive relationship of the 2-phase material in the oxidation zones according to the Voigt model. With a 2D element model which was constructed to reflect the real interface morphology of the thermal barrier coatings, numerical analysis of the oxidation effect on the growth trend of thermal grown oxide (TGO) and the stresses along the top coat (TC)-TGO and the bond coat (BC)-TGO interfaces was made. The numerical results show that, without the oxidation effect only uniform growth of TGO is predicted, while with the oxidation effect non-uniform growth of TGO is achieved. The stresses along the TC-TGO and BC-TGO interfaces with the oxidation effect are at higher levels compared to those without the oxidation effect. Moreover, the greater the oxidation effect is, the more irregular the growth of TGO becomes.

Fatigue Failure Analysis and Structural Improvement of PCB Solder Joints Under Thermal Cycles

SU Pei-lin, LI Tao, PENG Xiong-qi

2015, 36(4): 414-422. doi: 10.3879/j.issn.1000-0887.2015.04.009

Abstract(1217) PDF(926)

Abstract:
To investigate the fatigue failure of PCB solder joints under thermal impact cycles, a 3D finite element model of PCB with the traditional structure was built. For comparison, an improved structural model with solder pads added at the QFN corners was also built. Based on the measurement of Young’s moduli and thermal expansion coefficients of FR4 by experiment, the thermomechanical FE analysis was conducted on the 2 models in ANSYS. The thermal fatigue life of the PCB solder joints was evaluated by means of the modified CoffinManson equation according to the FEA results. It is found that the peak equivalent plastic strain at the PCB solder joints is significantly reduced after addition of the solder pads at the QFN corners, and the thermal fatigue life of the PCB solder joints was thus greatly improved.

A B-Spline Curve Fitting Algorithm Based on Contour Key Points

HAN Jiang, JIANG Ben-chi, XIA Lian, LI Da-zhu

2015, 36(4): 423-431. doi: 10.3879/j.issn.1000-0887.2015.04.010

Abstract(2240) PDF(2179)

Abstract:
Aimed at the sliced contour data points of point cloud in the reverse engineering, a B-spline curve fitting method based on contour key points was presented. Under the premise of keeping the shape fidelity, first, the scanned strip point set was resampled with an equidistance method and the contour key points were selected, in turn an initial interpolation curve was built. Next, the curve deviation values were calculated with a neighborhood point comparison method, and a new key point was added where the curve deviation value exceeded the specified tolerance, then a new interpolation curve was gained. The above procedure was repeated until the fitting curve reached expected accuracy. The numerical experiments show that, for the B-spline fitting of dense sectional scanned points, the proposed algorithm effectively compresses the number of key points and bears high computational efficiency. At the same time, since the distribution of key points accurately reflects the fitting curve’s curvature changes, this method also makes one promising iteration step in the curve approximation under deviation constraints.

Analysis of Unsteady Blood Flow in the Human Aortic Bifurcation

LIU Ying, ZHANG De-fa, BI Yong-qiang, WANG Meng-hong

2015, 36(4): 432-439. doi: 10.3879/j.issn.1000-0887.2015.04.011

Abstract(1065) PDF(965)

Abstract:
Based on the 3D unsteady Navier-Stokes equations as the governing equations, the blood flow in the human aortic bifurcation was simulated by means of the non-Newtonian blood model with the computational fluid dynamics method. The influence of blood flow characteristic parameter distributions on the formation of atherosclerotic plaques at the feature points within a cardiac cycle was investigated, and a comparison of the wall pressure and wall shear stress parameters was made between the Newtonian blood model and the non-Newtonian blood model. The results show that, compared to the Newtonian blood model, the non-Newtonian blood model is more suitable for the description of real blood flow characteristics. Larger blood stagnation areas, more complex wall pressure and wall shear stress distributions exist around the lateral walls of the bifurcation blood vessels during the systolic cycle, more probably causing deposition of fat particles, platelets and fibrin in the blood vessels which are liable to wall injury and reconstruction, in turn formation of atherosclerotic plaques. However, smaller blood stagnation areas, less changes of wall pressure and wall shear stress distributions happen around the lateral walls of the bifurcation blood vessels during the diastolic cycle, with less influence on the formation of atherosclerotic plaques.

A New Method for Accurate Calculation of the Bingham Fluid Axial Laminar Flow Pressure Drop and Velocity Distribution in Circular Pipes

ZHAO Shan-shan, WU Xiao-ming, LIU Yao-dong

2015, 36(4): 440-448. doi: 10.3879/j.issn.1000-0887.2015.04.012

Abstract(1623) PDF(1983)

Abstract:
Based on the Bingham model, the analytical calculations of axial laminar flow pressure drop in circular pipes often involve the general formula for viscous fluid, with the Bingham fluid constitutive equations introduced into the calculation to get only the analytical solution of pressure drop. A new method combining the Bingham fluid constitutive equation and the motion equation to establish the mechanical equilibrium equations was proposed, to solve the circular pipe laminar flow nonlinear equations according to the algebraic equation radical solution theory, and directly get the analytical solution of the axial laminar flow pressure drop, the velocity flow core zone radius and the flow velocity. The results show that, the direct influential factors on the Bingham fluid pipe laminar flow velocity are flow rate, plastic viscosity and yield value, and the flow core zone width is proportional to the yield value and inversely proportional to the flow velocity or the plastic viscosity. Besides, the wider the velocity flow core zone is, the lower the flow velocity in the core zone is.