应用数学和力学

2013 Vol. 34, No. 7

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Impact Behaviors and Damage Modes of Composites Under Low-Velocity Impact With Different Layup Thicknesses

HAO Kou-an, WANG Zhen-qing, ZHOU Li-min

2013, 34(7): 661-671. doi: 10.3879/j.issn.1000-0887.2013.07.001

Abstract(1137) PDF(1753)

Abstract:
Characteristics such as light weight and high strength, have made composite one of the most promising engineering materials, and its damage performance under impact has attracted many researchers’ attention. The low-velocity impact test based on the free-ball impact device was employed to study the impact properties of composites with different thickness. The features of damage modes were analyzed, and were summarized in aspects of relationships between deflection, velocity, contact force, energy and time. It was found that the energy absorbed by composites increased with the thickness. The energy balanced model was introduced based on the principle of conservation of total energy, and the resulting equation was solved with Newton-Raphson numerical technique. The energy balanced model results were compared with the experimental results.

Exact Analytic Solution for Laminated Plates With Free-Edges and Arbitrary Thickness

WANG De-cai, GUAN Qun, FAN Jia-rang

2013, 34(7): 672-686. doi: 10.3879/j.issn.1000-0887.2013.07.002

Abstract(1689) PDF(1061)

Abstract:
The problem of free-edges in three-dimensional elasticity is always a difficult one. The conditions that both normal stress and shear stress on the free edges equal zero are satisfied very difficultly. Based on the three-dimensional fundamental equations of elasticity and the state space method, the state equation for orthotropic plates was established through introduction boundary displacement function and consideration of all elastic constants of the orthotropic materials. Series expansion was carried out on the variables of the state equation. An exact solution was presented for laminated plates with arbitrary thickness by satisfaction of boundary conditions which can also satisfy the continuous conditions of stresses and displacements between plies of the laminates. The results of two examples show that the calculation process is simplified and the convergent solution can be achieved with less terms of series when the displacement function of free boundary is adopted. The numerical results with high accuracy can be obtained through comparison with the results of finite element. The results can be used as reference to numerical methods and semi-analytical methods.

Refined Model for Composite Sandwich Laminates of Moderate Thickness Based on the Variational Asymptotic Method

ZHONG Yi-feng, LIU Senlin, CHEN Yue-se, HUANG Bo-jie, ZHOU Xiao-ping

2013, 34(7): 687-696. doi: 10.3879/j.issn.1000-0887.2013.07.003

Abstract(1127) PDF(965)

Abstract:
In order to accurately predict the stress/strain distribution along the thickness direction, which is very important to the interface cracking of the composite sandwich laminate of moderate thickness, the small inherent parameter was used to rigidly decouple the original 3D plate into 1D analysis along the thickness direction and 2D nonlinear plate analysis. The 3D energy was approximately extended into a series of 2D energy functionals, in which the leading items were asymptotically corrected to match the original 3D energy as close as possible. Then, a refined model was built up without any field variable assumptions, and converted to the form of Reissner model for engineering applications. The cylindrical bending example of a sandwich plate with four layers shows that the 3D field reconstituted by this theory agrees better with the exact results than those by the first-order shear deformation theory and classical laminated theory; it’s amount of computation can be reduced up to 2~3 orders than 3D finite element method because the variational asymptotic model is an equivalent single-layer plate model, indicating a good tradeoff between the accuracy and efficiency.

Discontinuous Galerkin Finite Element Method Based on Rosenbrock-Type Exponential Integrator

CHEN Ye-fei, LI Wen-cheng, DENG Zi-chen>

2013, 34(7): 697-703. doi: 10.3879/j.issn.1000-0887.2013.07.004

Abstract(1627) PDF(1425)

Abstract:
A new Rosenbrocktype exponential discontinuous Galerkin method was developed. The method used discontinuous Galerkin method to discretize in space, while Rosenbrocktype exponential integrator method in time. Therefore, it not only has high accuracy to discretize in space, but also has excellent ability of explicit time marching with large time step. The numerical tests demonstrate that the method is effective for problems of 1D hyperbolic conservation laws.

A Four-Step Fractional Finite Element Method for Fluid-Structure Interaction

WANG Hua-kun, HONG Guo-jun, YANG Wen-yu, YU Guo-liang

2013, 34(7): 704-713. doi: 10.3879/j.issn.1000-0887.2013.07.005

Abstract(1425) PDF(1345)

Abstract:
A loosely-coupled algorithm for fluid-structure interaction based on arbitrary Lagrangian Eulerian(ALE) finite element method was proposed. The semi-implicit four-step fractional finite element method was extended to solve Navier-Stokes equations of ALE description, where the streamline upwind/Petrov-Galerkin (SUPG) stabilization term was added to the momentum equation to eliminate numerical oscillations of the velocity field. The temporal integration of the equation of motion for the structure was done with a Newmark-βalgorithm while the mesh updating was performed based on the modified Laplace equation solved by a standard Galerkin FEM. The entire deformation was imposed at each time step in order to avoid deterioration in mesh quality with long-term and large amplitude oscillations or deformations. The proposed method was applied to the numerical simulations on flow-induced vibrations of an elastically mounted circular cylinder with one and two degrees of freedom. Numerical results show good agreement with the existing solutions and suggest that the present method is competitive in terms of accuracy and efficiency.

Contact Force Distribution and Anisotropic Analysis in Dense Granular Flow Between the Shearing Parallel Plates

MENG Fan-jing, LIU kun, WANG wei

2013, 34(7): 714-723. doi: 10.3879/j.issn.1000-0887.2013.07.006

Abstract(1433) PDF(1203)

Abstract:
The probability distribution of contact force, anisotropy of contact force network, the friction influence on macro rheological and micro force chain distribution in dense granular flow between shearing parallel plates were discussed. A discrete element numerical analysis model was established for study. The numerical analysis results show that the probability distribution of contact force confoms to power law; the contact angle comply with exponent law and the average normal contact force oscillates up and down randomly with the average contact angle chage; the magnitude of wave velocity is a key evaluation index for macro rheology smoothness, and in micro force chain aspect, super force chain number increases notably when the granular flow is not smooth between the shearing parallel plates.

Global Uniform Asymptotic Stability of Memristor-Based Recurrent Neural Networks With Time Delays

HU Jin, SONG Qian-kun

2013, 34(7): 724-735. doi: 10.3879/j.issn.1000-0887.2013.07.007

Abstract(1401) PDF(1113)

Abstract:
Memristor is a newly prototyped nonlinear circuit device. Its value is not unique and changes according to the value of the magnitude and polarity of the voltage applied to it. Due to this feature, the memristor has the function of memory and broad potential applications of memristors have been reported in various fields. A simplified mathematical model was proposed to characterize the pinched hysteresis feature of the memristor and a memristor-based recurrent neural network model was given. With the theory of differential inclusion, Lyapunov approach and homeomorphism theory, the existence and uniqueness of the equilibrium point of the model were obtained, and the global uniform asymptotic stability of memristor-based recurrent neural networks was also obtained. Finally, the simulation result shows the efficiency of the theorem.

Precise Iterative Refinement of Solution for Ill-Conditioned Systems of Linear Algebraic Equations

ZHANG Wen-zhi, HUANG Pei-yan

2013, 34(7): 736-741. doi: 10.3879/j.issn.1000-0887.2013.07.008

Abstract(1412) PDF(1357)

Abstract:
A precise iterative refinement of solution for ill-conditioned systems of linear algebraic equations was proposed. First, the ill-conditioned matrix was improved through introduction of a small parametrr, and then via the precise integration method, a highly precise method was provided for the inversion of the improved matrix. Both the theoretical convergence analysis and numerical examples show the efficiency and accuracy of the method.

Numerical Analysis on Thermal Stress of Multilayer Materials Combined Structures for a Lightweight Thermal Protection System

ZHANG Lang, LI Xue-wu, XIA Jian-zhong

2013, 34(7): 742-749. doi: 10.3879/j.issn.1000-0887.2013.07.009

Abstract(1739) PDF(1942)

Abstract:
A design concept of using multilayer structure for the top face-sheet of a lightweight thermal protection system was proposed. To obtain the different structural performances, three schemes were considered according to the different stacking orders of two kinds of insulation materials. With the mechanical and thermal loads during aircraft reentry considered, internal thermal stress of the multilayer structure was simulated. Details of shear stresses, temperature of the bottom surface and y-direction displacement were obtained. The results show that the shear stresses occur at the edge area, and the distributions are anti-symmetrical along the midline. The usage of the materials with high thermal conductivity and heat capacity can reduce the temperature gradient and then the thermal stresses and deformation. With the uniform temperature field, the difference of the thermal expansion coefficients of two materials is positively related with the shear stress between the two materials. It is suggested that, there is a great potential of optimization design for the multilayer structure through different combinations of materials in different stacking orders.

Quadratically Consistent Meshfree Methods for Heat Conduction in Steady State

WANG Bing-bing, GAO Xin, DUAN Qing-lin

2013, 34(7): 750-755. doi: 10.3879/j.issn.1000-0887.2013.07.010

Abstract(1240) PDF(1130)

Abstract:
The quadratically consistent 3-point integration method (QC3) was extended to the meshfree analysis of heat conduction problem in steady state. Numerical results show that, in comparison with the standard triangle integration method and the existing 1-point integration method which only satisfies the linear consistency, the proposed QC3 method not only passes the quadratic patch test exactly, but also exhibits significant superiority in terms of accuracy, convergence and efficiency.

A New Method of Obtaining Timoshenko’s Shear Coefficients

WANG Le, WANG Liang

2013, 34(7): 756-763. doi: 10.3879/j.issn.1000-0887.2013.07.011

Abstract(2208) PDF(2456)

Abstract:
With the effects of shear deformation of slender beams considered in Timoshenko beam theory, a new method of obtaining Timoshenko’s shear coefficients was derived. First, the exact solutions of the cross section’s shear stress distribution of the cantilever beam under the action of pure bending were used, and the new expressions of various cross sections were obtained based on energy principle. Then, the exact solutions of the cross section’s shear stress distribution of the cantilever beam under action of bending and torsion were derived and the coefficients obtained. The results show that the coefficients decrease when the terminal force departs from the bending center. The results are smaller than those given by Cowper because his theory doesn’t include the influence of shear stress perpendicular to the terminal force, and solution of the new method proved better.

Weak Resonant Double Hopf Bifurcation of n van der Pol Oscillators With Delay Coupling

WANG Wan-yong, CHEN Li-juan

2013, 34(7): 764-770. doi: 10.3879/j.issn.1000-0887.2013.07.012

Abstract(1491) PDF(1053)

Abstract:
Weak resonant double Hopf bifurcation of nvan der Pol oscillators with delay coupling was investigated. With an extended method of multiple scales, the complex amplitude equations were obtained. With the complex amplitudes expressed in a polar form, the complex amplitude equations were reduced to a two dimensional real amplitude system. The equilibria and their stability of the real amplitud equations were studied, and the dynamics around 2∶5 resonant point unfolded and classified. Some interesting phenomena are found, such as amplitude death, periodic solution and bistability, etc. Validity of the analytical results is proved by their consistency with numerical simulations.