Applied Mathematics and Mechanics (English Edition)

Flowing simulation of injection molded parts with micro-channel

崔志香司军辉刘春太申长雨

2014, 35(3): 269-276. doi:10.1007/s10483-014-1789-7

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In the micro-molding of component with a micro-sized channel, the ability for polymer melt to flowing into the micro-channel in a macro-sized part is a big challenge. The multidimensional flow behaviors are included in the injection molding the macrocomponent with a micro-channel. In this case, a simplified model is used to analyze the flow behaviors of the macro-sized part within a micro-channel. The flow behaviors in the macro-cavity are estimated by using the finite element and finite difference methods. The influence of the injection rate, micro-channel size, heat transfer coefficient, and mold temperature on the flowing distance is investigated based on the non-isothermal analytic method. The results show that an increase in the radius of the micro-channel and mold temperature can improve effectively the flowing distance in the micro-channel.

Region dependent fracture resistance behavior of human dentin based on numerical simulation

徐远志安兵兵张东升汪饶饶

2014, 35(3): 277-284. doi:10.1007/s10483-014-1790-8

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Dentin has a hierarchical structure and is composed of numerous tubules whose diameters and densities vary with the distances to the dentin-enamel junction. The unique structure determines the mechanical performance of dentin. In this study, a multiscale model, which is based on the combination of the virtual multidimensional internal bond (VMIB) theory and the Monte Carlo method, is used to simulate the fracture behavior of human dentin. Numerical simulations reveal that human dentin exhibits a graded resistance curve (R-curve). Among the three regions of dentin, superficial dentin shows the strongest resistance to crack propagation, and deep dentin has the weakest resistance. In addition, the predictions of fracture toughness of middle dentin agree well with the experimentally reported values, suggesting that the proposed model can be used to characterize the fracture behavior of human dentin comprehensively and properly.

Combined stress waves with phase transition in thin-walled tubes

宋卿争唐志平

2014, 35(3): 285-296. doi:10.1007/s10483-014-1791-7

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The incremental constitutive relation and governing equations with combined stresses for phase transition wave propagation in a thin-walled tube are established based on the phase transition criterion considering both the hydrostatic pressure and the deviatoric stress. It is found that the centers of the initial and subsequent phase transition ellipses are shifted along the σ-axis in the στ-plane due to the tension-compression asymmetry induced by the hydrostatic pressure. The wave solution offers the “fast” and “slow” phase transition waves under combined longitudinal and torsional stresses in the phase transition region. The results show some new stress paths and wave structures in a thin-walled tube with phase transition, differing from those of conventional elastic-plastic materials.

Bending of simply-supported circular timber beam strengthened with fiber reinforced polymer

杨骁杨峥文群

2014, 35(3): 297-310. doi:10.1007/s10483-014-1792-x

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The bending behavior of the circular cross-section timber beam strengthened with a fiber reinforced polymer (FRP) sheet is investigated. The tight bonding is on the interface between the surface of a timber beam and the reinforcement layer of the FRP sheet. An analytical expression for the bending moment-curvature relation is presented, and its failure modes are analyzed. The governing equation for nonlinear small deflection of the FRP-strengthened circular timber beam is established, and the corresponding numerical method is given. The bending deformation of the simply-supported circular timber beam strengthened with the carbon fiber reinforced polymer (CFRP) sheet subject to a uniform load is studied numerically. The influence of the angle and thickness of the CFRP layer as well as the timber strength on the bending deflection of the FRPstrengthened circular timber beam is examined. It is revealed that, with the increases of the thickness and angle, the deflection of the CFRP-strengthened circular timber beam is decreased, and its carrying capacity and ductility are increased. However, when the angle of the layer reaches a certain value, the deflection will no longer decrease with the increase of the angle. At the same time, the nonlinear bending moment-curvature relation of the CFRP-strengthened circular timber beam is simplified as an approximate bilinear constitutive model. The approximate deflections of the simply-supported circular timber beam strengthened with the CFRP sheet are obtained. The results are compared with the linearly elastic bending and nonlinear bending models, showing that the mid-span deflections of a CFRP-strengthened circular timber beam with the approximate bilinear constitutive model are greater than those with the nonlinear constitutive model. The results of its stiffness analysis are on the safe side.

Numerical methods for solving singular integral equations obtained by fracture mechanical analysis of cracked wedge

M. GHADIRI H. GHASEMI

2014, 35(3): 311-324. doi:10.1007/s10483-014-1793-6

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The numerical solutions to the singular integral equations obtained by the fracture mechanical analyses of a cracked wedge under three different conditions are considered. The three considered conditions are: (i) a radial crack on a wedge with a nonfinite radius under the traction-traction boundary condition, (ii) a radial crack on a wedge with a finite radius under the traction-traction boundary condition, and (iii) a radial crack on a finite radius wedge under the traction-displacement boundary condition. According to the boundary conditions, the extracted singular integral equations have different forms. Numerical methods are used to solve the obtained coupled singular integral equations, where the Gauss-Legendre and the Gauss-Chebyshev polynomials are used to approximate the responses of the singular integral equations. The results are presented in figures and compared with those obtained by the analytical response. The results show that the obtained Gauss-Chebyshev polynomial response is closer to the analytical response.

Elasto-plastic buckling and post-buckling analysis of sandwich plates with functionally graded metal-metal face sheets and interfacial damage

傅衣铭邵雪飞陈央

2014, 35(3): 325-344. doi:10.1007/s10483-014-1794-7

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Based on the elasto-plastic theory, considering the effect of spherical stress tensor on the elasto-plastic deformation and using the slicing treatment to deal with the plasticity of functionally graded coatings, the elasto-plastic increment constitutive equations of the sandwich plates with functionally graded metal-metal face sheets can be derived. Applying the weak bonded theory to the interfacial constitutive relation and taking into account the geometric nonlinearity, the nonlinear increment differential equilibrium equations of the sandwich plates with functionally graded metal-metal face sheets are obtained by the minimum potential energy principle. The finite difference method and the iterative method are used to obtain the post-buckling path. When the effect of geometrical nonlinearity of the plate is ignored, the elasto-plastic critical buckling load of the sandwich plates with functionally graded metal-metal face sheets can be solved by the Galerkin method and the iterative method. In the numerical examples, the effects
of the interface damages, the induced load ratio, the functionally graded index, and the geometry parameters on the elasto-plastic post-buckling path and the elasto-plastic critical buckling load are investigated.

Algorithmic tangent modulus at finite strains based on multiplicative decomposition

李朝君冯吉利

2014, 35(3): 345-358. doi:10.1007/s10483-014-1795-6

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The algorithmic tangent modulus at finite strains in current configuration plays an important role in the nonlinear finite element method. In this work, the exact tensorial forms of the algorithmic tangent modulus at finite strains are derived in the principal space and their corresponding matrix expressions are also presented. The algorithmic tangent modulus consists of two terms. The first term depends on a specific yield surface, while the second term is independent of the specific yield surface. The elastoplastic matrix in the principal space associated with the specific yield surface is derived by the logarithmic strains in terms of the local multiplicative decomposition. The Drucker-Prager yield function of elastoplastic material is used as a numerical example to verify the present algorithmic tangent modulus at finite strains.

Nonlinear evolution of Klebanoff type second mode disturbances in supersonic flat-plate boundary layer

禹旻罗纪生

2014, 35(3): 359-368. doi:10.1007/s10483-014-1796-8

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Studying the evolution of 3D disturbances is of crucial theoretical importance for understanding the transition process. The present study concerns the nonlinear evolution of second mode unstable disturbances in a supersonic boundary layer by the numerical simulation, and discusses the selectivity of 3D disturbances and possibility to transition. The results indicate that a Klebanoff type nonlinear interaction between 2D and 3D disturbances with the same frequency may amplify a band of 3D disturbances centered at a finite spanwise wavenumber. That is, certain 3D disturbances can be selectively and rapidly amplified by the unstable 2D disturbances, and certain small-scale 3D structures will appear.

New explicit multi-symplectic scheme for nonlinear wave equation

李昊辰孙建强秦孟兆

2014, 35(3): 369-380. doi:10.1007/s10483-014-1797-6

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Based on splitting multi-symplectic structures, a new multi-symplectic scheme is proposed and applied to a nonlinear wave equation. The explicit multi-symplectic scheme of the nonlinear wave equation is obtained, and the corresponding multi-symplectic conservation property is proved. The backward error analysis shows that the explicit multi-symplectic scheme has good accuracy. The sine-Gordon equation and the Klein-Gordon equation are simulated by an explicit multi-symplectic scheme. The numerical results show that the new explicit multi-symplectic scheme can well simulate the solitary wave behaviors of the nonlinear wave equation and approximately preserve the relative energy error of the equation.

Asymptotic approximation method for elliptic variational inequality of first kind

李曦袁达明

2014, 35(3): 381-390. doi:10.1007/s10483-014-1798-x

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An asymptotic approximation method is proposed to solve a particular elliptic variational inequality of first kind associated with unilateral obstacle problems. In this method, the free boundary is first captured, and then the method of the fundamental solution (MFS) is used to find the solution of the Dirichlet problem for Laplace’s equation in the non-coincidence set. Numerical examples are given to show the efficiency of the method.

Effective numerical approach with complete damage transfer under multi-step loading

赵士洋薛璞彭雄奇王琰

2014, 35(3): 391-402. doi:10.1007/s10483-014-1799-8

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An effective numerical approach is proposed for structural damage analysis, especially for structures which may be damaged at multiple locations to different extents. The structural damage state is represented by defining a set of field-variable functions, and the degradation of material properties is described. Then, through customization of the commercial finite element software ABAQUS/Standard, the damage state is directly assigned to the integration points of elements, thereby avoiding modification of the finite element model. Afterwards, the damaged structures by ABAQUS/Standard is conducted numerically. Finally, this approach is verified by simulating the growth of delamination for a pre-delaminated composite plate and a composite plate under impact loading, respectively.

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