应用数学和力学

2013 Vol. 34, No. 6

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Symplectic Group of the Transfer Matrix Converges to the Symplectic Lie Group

ZHONG Wan-xie, GAO Qiang

2013, 34(6): 547-551. doi: 10.3879/j.issn.1000-0887.2013.06.001

Abstract(1829) PDF(1275)

Abstract:
By using action variational principle, the transfer symplectic matrix for the discrete integral of the Hamiltonian canonical equation was given. Then the Lie algebra corresponding to the Hamiltonian canonical equation was given. When the time step tends to zero, that the symplectic group of the transfer matrix for discrete integrator converges to the symplectic Lie group of the continuoustime differential equation of the Hamiltonian system was proved.

Study on Ovalization of Submarine Pipe During Deep Water Reel-Lay

ZHANG Jiu-ju, DUAN Meng-lan, MA Jian-min, HU Xian-wei

2013, 34(6): 552-563. doi: 10.3879/j.issn.1000-0887.2013.06.002

Abstract(1974) PDF(1611)

Abstract:
During the reel-lay installation in deep water, large deformation occurs on the pipeline, leading to ovalization, instability or even local buckling. Ovality theory was established based on strain-energy method and Ritz method. Also, finite element analysis was employed by software, ABAQUS. The results obtained from the former procedures were compared with those from modified Brazier and modified von Kármán, serving as a good verification of the former solution. In addition, intact and defective pipelines are separately simulated by ABAQUS, suggesting that ovality is subject to pipelinediameter, wallthickness, initial bending curvature and bending curvature, et al. The law between ovality and geometrical parameters is further obtained. The above research of ovality is of certain interest to reel-lay installation.

Nonlinear Stability of a Double-Deck Reticulated Truncated Circular Shallow Spherical Shell

LU Ying-hua, LIU Ren-huai, WANG Fan

2013, 34(6): 564-575. doi: 10.3879/j.issn.1000-0887.2013.06.003

Abstract(1713) PDF(1198)

Abstract:
By means of the modified iteration method,the nonlinear stability problem of a double-deck reticulated truncated circular shallow spherical shell under uniform pressure was investigated. According to the fundamental equations of double-deck reticulated circular shallow spherical shells,the critical buckling load for the shell with two types of boundary conditions was obtained and the effect of geometric parameters of the shell on the critical buckling load was discussed.

An Explicit Sub-Stepping Stress Integration Method and Its Applications in Numerical Simulations of SMA

CHEN Xi, ZHANG Jian, LIU Jian-kun

2013, 34(6): 576-585. doi: 10.3879/j.issn.1000-0887.2013.06.004

Abstract(1769) PDF(1414)

Abstract:
Shape memory alloy (SMA) has complex thermomechanical constitutive relation, thus its numerical simulations demand reliable and efficient stress integration algorithms. The implicit returnmapping stress point algorithms, which have been successfully applied to such materials, may encounter convergence difficulties when loading conditions are complicated or load steps are large. Hence, an explicit substepping stress integration method with automatic local error control was proposed for the simulation of the thermomechanical constitutive relation of shape memory alloys. By investigating several numerical examples, the efficiency of the proposed method and the implicit returnmapping stress point algorithm were evaluated and compared. Numerical results indicate that the number of global sub-steps dominates the entire analyzing time for large-scale computations.The proposed modified Euler automatic sub-stepping scheme leads to less global sub-steps so that the computing time is significantly reduced. Therefore, the explicit sub-stepping stress integration method has the potential for large-scale SMA simulations and computations.

Evaluation on Stress Intensity Factors at the Crack Tip Under Dynamic Loads Using Extended Finite Element Methods

JIANG Shou-yan, DU Cheng-bin

2013, 34(6): 586-597. doi: 10.3879/j.issn.1000-0887.2013.06.005

Abstract(2132) PDF(1634)

Abstract:
In the framework of the extended finite element methods (XFEM), the extraction of dynamic stress intensity factors (DSIFs) for stationary cracks being subjected to dynamic loads was detaily studied. Having constructed the approximation of dynamic XFEM, the derivation of governing equation for dynamic XFEM was presented. The Newmark implicit algorithm was used for time integration. Meanwhile, a mass lumping strategy for XFEM implicit dynamics was proposed. In addition, the interaction integral method was given for evaluating DSIFs. Compared with the interaction integral method for evaluating stress intensity factors (SIFs) of cracks under static conditions, the contribution of inertial effects was added to the interaction integral method for evaluating DSIFs. The numerical illustrations show that the XFEM can evaluate accurately DSIFs and the proposed mass lumping strategy is also quite effective. To obtain DSIFs correctly, the inertial effects on interaction integral cannot be ignored.

Fracture Process Zone Size Based on Secondary Elastic Crack Tip Stress Solution

DUAN Shu-jin, ZHANG Yan-long, AN Rui-mei

2013, 34(6): 598-605. doi: 10.3879/j.issn.1000-0887.2013.06.006

Abstract(1866) PDF(1060)

Abstract:
The contour and characteristic sizes of a microcrack zone ahead of a fracture process zone (PFZ) were derived by the local solution based on Westergaard stress function with the secondary elastic crack tip stress. The critical sizes of FPZ were yielded out by the use of a power exponent tensile strain softening model under the maximum tensile stress criterion and the maximum tensile strain criterion. Based on the first elastic crack tip stress expression and the secondary elastic crack tip stress expression by Westergaard stress function, Muskhelishvili stress function and DuanNakagawa model, the critical sizes of FPZ were compared. The discussions show that the size of a microcrack zone and the critical size of FPZ increase with the decreasing Poisson ratio, and approach that of the maximum stress criterion. The contour and characteristic size of a microcrack zone and the critical sizes of FPZ based on the secondary elastic crack tip stress solution are bigger than the one based on the first elastic crack tip stress solution. The critical size of FPZ increases with the increasing tensile strain softening index. The accuracy of critical size of FPZ based on the secondary elastic crack tip stress solution is much higher than the one based on the first elastic crack tip stress solution.

Shear Strength Formula for Reinforced Concrete Beams Without Web Reinforcements gainst Size Effect

LUO Lin, WANG Qi-zhi

2013, 34(6): 606-619. doi: 10.3879/j.issn.1000-0887.2013.06.007

Abstract(1844) PDF(1043)

Abstract:
Based on the limit equilibrium method, equilibrium equations were established by studying the free body composed of the critical inclined crack and the top crosssection from reinforced concrete beams without web reinforcements. Analyzing the relationships between principal stresses of the critical shear compression zone and damage forms, stress analysis of the critical shear compression zone was done. Then by applying Bazant’s size effect law, a shear strength formula of reinforced concrete beams without stirrups for both diagonal tension failure and shear compression failure was obtained, where the critical shear compression zone height was the unknown parameter. And then by LevenbergMarguardt nonlinear leastsquare curve fitting on Bazant’s test database, exponential relationship between the height of the critical shear compression zone and parameters was gotten. Finally, comparison with Collins’ test database shows that the obtained shear strength formula can be demonstrated to agree more favorably with the test database than those from ACI 318-08 and GB 50010-2010, and close with Bazant’s shear strength formula.

Large Deformation Beam Element Based on Curvature Interpolation

ZHANG Zhi-gang, QI Zhao-hui, WU Zhi-gang

2013, 34(6): 620-629. doi: 10.3879/j.issn.1000-0887.2013.06.008

Abstract(1999) PDF(1382)

Abstract:
The shape functions of linear beam element, which will cause the false strain when large rotation occurs, does not apply to geometric nonlinear analysis. Because of the coherence of the interpolation of the displacement and angle, traditional geometric nonlinear beam element is often caused by problems such as shear locking. A plane large deformation beam element was proposed, by use of the interpolation of curvature and the functional relationship between the curvature and nodal displacements . The element node forces and displacements were expressed as a function of the curvature. Essentially the interpolation of the beam curvature is strain interpolation，which ensures that the element rigid body motion does not produce false node force; the shear locking in traditional element is avoided because the beam centroid displacement is expressed as a function of curvature. Thus this method is especially suitable for geometry nonlinear analysis of the beam. The numerical examples show the truth and validity of the proposed method.

A Novel Method to Obtain the Elasticity Solutions of Anisotropic Plane Beam Subjected to Arbitrary Loads

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

2013, 34(6): 630-642. doi: 10.3879/j.issn.1000-0887.2013.06.009

Abstract(1644) PDF(972)

Abstract:
A novel method was given to obtain the elasticity solutions for linear anisotropic plane beam subjected to arbitrary loads with various ends conditions by solving functional equations. Comparing this general method with traditional trial-and-error method, the most advantage was that there was no need to guess the form of stress function and obtain the solutions directly. The united equations for solving boundary value problem of anisotropic plane beam were found and several examples showed the correctness of this general method. A new way was also provided to derive the elasticity solutions of plane beam subjected to arbitrary loads with various ends conditions.

Stability of the Sets of Efficient Points of Vector-Valued Optimization Problems

ZHAO Yong, PENG Zai-yun, ZHANG Shi-sheng

2013, 34(6): 643-650. doi: 10.3879/j.issn.1000-0887.2013.06.010

Abstract(2226) PDF(1128)

Abstract:
By using quasi C-convex function and recession cone property, the stability of efficient points sets to vector optimization problems without the assumption of compactness was established.The lower part of the Painlevé-Kuratowski convergence of the sets for efficient points of perturbed problems to the corresponding efficient sets for the vector optimization problems was obtained, where the perturbation was performed on both the objective function and the feasible set. These results extend and improve the corresponding ones in the literature (Attouch H, Riahi H. Stability results for Ekeland’s ε-variational principle and cone extremal solution; Huang X X. Stability in vector-valued and set-valued optimization), then examples are given to illustrate our main results.

Exact Travelling Wave Solutions for the (2+1)-Dimensional ZK-MEW Equation by Using an Improved Algebra Method

HAN Zhong, ZHANG Yu-feng, ZHAO Zhong-long

2013, 34(6): 651-660. doi: 10.3879/j.issn.1000-0887.2013.06.011

Abstract(1709) PDF(1036)

Abstract:
Based upon an improved unified algebra method and implement in the symbolic computation system Mathematica, the (2+1)-dimensional ZakharovKuznetsov modified equal width equation was considered. This method converted the work of constructing exact travelling wave solutions for an equation into solving a system of nonlinear algebra equations(NLAEs). After solving the system of nonlinear algebra equations, abundant general form solutions are obtained, which including rational function solutions, trigonometric function solutions, hyperbolic function solutions, Jacobi elliptic function solutions, Weierstrass elliptic function solutions. The profiles of some obtained solutions are also given out.