应用数学和力学

The Bi-Potential Theory Applied to Non-Associated Constitutive Laws

ZHOU Yang-jing, FENG Zhi-qiang, NING Po

2015, 36(8): 787-804. doi: 10.3879/j.issn.1000-0887.2015.08.001

Abstract(1452) PDF(849)

Abstract:
The explicit integration algorithm based on the traditional plastic mechanics framework and the implicit integration algorithm proposed by Simo-Taylor were 2 classic constitutive integration algorithms widely used in solid mechanics. These 2 algorithms were reviewed respectively with the 2 corresponding classic non-associated constitutive models: the Drucker-Prager model and the Armstrong-Frederick model as the examples. Then, according to the bi-potential theory and with the bi-potential concept applied to the material free energy, solid materials were divided into explicit standard materials and implicit standard ones. It was verified that the 2 classic integration algorithms both can effectively deal with explicit standard materials. However, in dealing with implicit standard materials, the orthogonality cannot be guaranteed in a unified form with the classic methods. The bi-potential algorithm has its own advantage in dealing with both explicit and implicit standard materials. The solution existence of the bi-potential integration algorithm was derived based on the variational principle. Furthermore, the results of the bi-potential algorithm and the classic algorithms were compared through calculation of the Drucker-Prager and Armstrong-Frederick models, and the accuracy and stability of the bi-potential algorithm were proved.

Multi-Scale Analysis of Piezoelectric Energy Harvesters With Magnetic Oscillators

ZHAO Jian, ZHANG Guo-ce, CHEN Li-qun

2015, 36(8): 805-813. doi: 10.3879/j.issn.1000-0887.2015.08.002

Abstract(1287) PDF(968)

Abstract:
A piezoelectric energy harvester with a magnetic oscillator was studied. The dynamic equation was derived via introduction of coordinate transform based on the equilibrium configuration. The Taylor series expansion method was employed to deal with the nonlinear function of the magnetic force. The multi-scale method was applied to obtain the steady-state periodic solutions of the system. The solvability condition and the amplitude-frequency relationship were derived through elimination of the secular terms. Then the Runge-Kutta method was used to numerically calculate the system’s forced vibration time history and give the amplitude-frequency response characteristics and instability boundary of the 1st 2 primary resonance cases. The results show that the multi-scale analysis yields uniformly valid solutions of high accuracy, and provides a theoretic base for the optimal design of piezoelectric energy harvesters with magnetic oscillators.

Bandgap Properties of Periodic 4-Point Star-Shaped Honeycomb Materials With Negative Poisson’s Ratios

YUN Hao, DENG Zi-chen, ZHU Zhi-wei

2015, 36(8): 814-820. doi: 10.3879/j.issn.1000-0887.2015.08.003

Abstract(1404) PDF(1048)

Abstract:
The bandgap properties of the periodic 4-point star-shaped honeycomb materials with negative Poisson’s ratios were investigated. The in-plane wave propagation in the honeycomb material was analyzed with the finite element method and according to the Bloch theorem. Attention was devoted to determining the influence of the unit cell geometry on the bandgaps. The results show that the 4-point star-shaped honeycomb material has wide bandgaps with relatively stable locations and widths, and the local rotation resonance of the star cells makes the main cause for the formation of the lowest-order bandgaps of the materials. The above bandgap properties of the 4-point star-shaped honeycomb material endow itself with potential application values in the fields of vibration attenuation and noise reduction.

Elastoplastic Damage Analysis of Powder Metallurgy Superalloy Based on the Cosserat Continuum Theory

ZHANG Cheng-cheng, YANG Dong-sheng, REN Yuan, ZHANG Sheng

2015, 36(8): 821-832. doi: 10.3879/j.issn.1000-0887.2015.08.004

Abstract(1337) PDF(711)

Abstract:
The development of the powder metallurgy (PM) superalloy applied in turbine discs of high thrust-weight ratio aeroengines was reviewed. Based on the Cosserat continuum theory, the elastoplastic damage model for the PM superalloy was developed in which the microscopic properties were modeled with the characteristic lengths of the material. Furthermore, the mesh dependency of the localized bands in the analysis of softening problems was eliminated in the present model. For the softening problems, massive iterations were usually needed with the classical elastoplastic methods and may not converge in some cases. Based on the parametric variational principle, the original nonlinear problem was transferred to a complementary problem, then the efficiency and convergence of the solution were greatly improved. Finally, several numerical examples demonstrate the validity of the proposed method.

Lie Group and Lie Algebra Modeling for Numerical Calculation of Rigid Body Dynamics

BAI Long, DONG Zhi-feng, GE Xin-sheng

2015, 36(8): 833-843. doi: 10.3879/j.issn.1000-0887.2015.08.005

Abstract(1720) PDF(1225)

Abstract:
The Lie group dynamics equation for rigid bodies was derived based on the exponent mapping equivalence relationship between the Lie group and Lie algebra. The discrete Lie group variational integrator was derived according to the discrete variation theory. The momentum conservation of the 2 Lie group equations was demonstrated. The Lie group dynamics equation was processed so that every part has the same dimension and the equation can be solved with the RungeKutta method directly. The RungeKutta method to directly solve the Lie group dynamics equation with different dimensions was also built. The Lie group variational integrator was solved with the Lie algebraic transform, the Cayley transform and Newton iteration, respectively. The computation results of the 3 algorithms are highly identical to each other, the structure conservation and momentum conservation both have high precisions.

Study on the Transient Temperature Field Based on the Fractional Heat Conduction Equation for Laser Heating

XU Guang-ying, WANG Jin-bao, HAN Zhi

2015, 36(8): 844-854. doi: 10.3879/j.issn.1000-0887.2015.08.006

Abstract(1576) PDF(1487)

Abstract:
Based on the fractional Taylor series expansion principle, the 1st-order fractional approximate heat conduction constitutive equation was formulated through expansion of the single-phase lag model. Combined with the energy equation, the fractional heat conduction equations were built for short pulse laser heating, and the Laplace transform was applied to solve the equations and obtain the analytical solution of the volumetric heat source temperature field of the non-Gauss time type. The properties of the temperature wave influenced by the fractional order were investigated based on specific examples. The thermal wave velocity decreases and its amplitude increases with the fractional order. The fractional heat conduction equation is applicable for depicting the intermediate heat conduction process between that of the Fourier diffusion equation and that of the thermal wave equation. The correlation between the heat conduction mechanism and the fractional derivative terms in the fractional heat conduction equation was also fully discussed.

Creep Analysis of Simply Supported-to-Continuous Box Girders Under Shear Lag Effect

SUN Yong-xin, LIN Peng-zhen

2015, 36(8): 855-864. doi: 10.3879/j.issn.1000-0887.2015.08.007

Abstract(1394) PDF(947)

Abstract:
To analyze how creep interacts with shear lag effect in concrete box girders, the calculation formulae for the creep secondary internal force and stress were derived with the energy variation method. With a simply supported-to-continuous box girder for example, the creep bending moment and stress were calculated under the shear lag effect, and the shear lag coefficients were obtained under the influence of creep. The results show that the shear lag effect increases the creep stress significantly. Under the shear lag effect, the creep variation at the intermediate bearing section is the maximum, where the secondary bending moment increases by 42.56% and the flange normal stress above the web plate decreases by 8.5%.

A Real-Time Identification Method for the Bottomhole Gas Invasion State in Managed Pressure Drilling

HE Miao, LIU Gong-hui, LI Jun, ZHANG Tao, LI Meng-bo, GUO Qing-feng

2015, 36(8): 865-874. doi: 10.3879/j.issn.1000-0887.2015.08.008

Abstract(1086) PDF(852)

Abstract:
Real-time identification of the bottomhole gas invasion state is the key to well control during managed pressure drilling (MPD), which is directly related to the selection of follow-up operations involving the gas circulation exhaust based on standpipe pressure control or the conventional shut-in procedure. In view of the factors of gas migration expansion and dissolution, it is pointed out here that the equilibrium between outlet flow and inlet flow does not directly mean the stop of bottomhole gas invasion, and there is a precedence relationship between them. Based on the rapidly increasing wellhead back pressure control method, an MPD wellbore-formation coupling model for multiphase variable-mass flow was established, and the finite difference method was used to iteratively solve this model. The calculated results agreed well with the experimental measured ones. The simulation results show that the obvious inflexion point of the outlet flow is a sign for distinguishing the precedence relationship between the outlet-inlet flow equilibrium and the bottomhole gas invasion stop. Once the bottomhole gas invasion stops, the 2nd derivative of the standpipe pressure will quickly drop to near zero and keep stable thereafter. According to the above parameter variation characteristics, a real-time identification method for the bottomhole gas invasion state based on the outlet and inlet flow as well as the real-time monitored standpipe pressure was proposed. This study has guiding significance for the improvement of the MPD well control theory.

A Reduced-Order Extrapolating FDM for Conserved High-Order Anisotropic Traffic Flow Models

LUO Zhen-dong, XU Yuan

2015, 36(8): 875-886. doi: 10.3879/j.issn.1000-0887.2015.08.009

Abstract(1601) PDF(986)

Abstract:
A reduced-order extrapolating finite difference method (FDM) with sufficiently high accuracy and very few degrees of freedom for conserved high-order anisotropic traffic flow models was established by means of the Godunov fluid method and the POD technique. The error estimate of the reduced-order approximate solutions and the algorithm implementation of the reduced-order extrapolating difference scheme were presented. Finally, a numerical example was used to illustrate that the results of the proposed method were consistent with those of the classic difference scheme. Moreover, the high efficiency and sufficient accuracy of the reduced-order extrapolating simulation method are shown.

A Numerical Method for the Solutions to Nonlinear Dynamic Systems Based on Cubic Spline Interpolation Functions

LI Peng-zhu, LI Feng-jun, LI Xing, ZHOU Yue-ting

2015, 36(8): 887-896. doi: 10.3879/j.issn.1000-0887.2015.08.010

Abstract(1924) PDF(1278)

Abstract:
The cubic spline interpolation function has good convergence, stability and 2nd-order smoothness. A numerical method for the solutions to nonlinear dynamic systems was constructed with the cubic spline interpolation functions. Advantages and disadvantages were compared between this method and the previous numerical methods for nonlinear dynamic systems, with the error estimation conducted in the 2 numerical examples. The results show that the numerical method derived out of the cubic spline interpolation functions has faster convergence rate and higher accuracy than the existing methods, and has good approximation to the analytical solutions to nonlinear dynamic systems.

2015 Vol. 36, No. 8