应用数学和力学

2021 Vol. 42, No. 1

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Dynamic Failure Simulation of Metal Materials and Structures Under Blast and Impact Loading

LIU Zhanli, CHU Dongyang, WANG Tao, WANG Yigang

2021, 42(1): 1-14. doi: 10.21656/1000-0887.410262

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Abstract:
Studying the dynamic failure laws of metal materials and structures under blast and impact loading through numerical simulation is of great significance for characterizing the damage effects of explosive shock and designing novel impact-resistant structures. The metals’ failure under strong dynamic loading involves multiple complex physical processes, such as the large deformation, the thermo-mechanical coupling and the material state changes. These complex physical processes bring great challenges to numerical simulation, including the geometric description of complex dynamic failure modes such as cracks and shear bands, the determination of failure criteria and the description of plasticity-damage coupled evolution, etc. In response to these challenging issues, a theoretical and computational thermal elasto-plastic phase field model was established based on the energy variational principle to describe metals' dynamic failure. For the model, a unified description of the crack and shear band was realized, and an efficient solving strategy of explicit finite elements was proposed. The model was further applied to 3 typical metals’ dynamic failure issues under blast and impact loading: the transition between brittle and ductile failure modes of metals, the self-organization of adiabatic shear bands (ASBs) and the transition between failure modes of thin-walled disks under shock waves. The results verify the accuracy of the theoretical model and the robustness of the computational model. This work lays a foundation for the subsequent development of damage assessment and protective structure design against blast and impact loading based on simulation.

Bending and Buckling of Functionally Graded Nanoplates Based on the Nonlocal Strain Gradient Theory

WANG Pingyuan, LI Cheng, YAO Linquan

2021, 42(1): 15-26. doi: 10.21656/1000-0887.410188

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Abstract:
The bending and buckling of functionally graded nanoplates in intelligent devices (e.g., nanorobots) were studied based on the nonlocal strain gradient theory. The motion equations in general cases were derived, and then reduced to bending and buckling in special cases. The effects of the nonlocal scale parameter, the material characteristic scale parameter, the gradient index and the geometric size on the bending deflection and the critical buckling load were acquired and analyzed in detail. The results show that, the maximum bending deflections under different higherorder continuum mechanics theories increase with the gradient index. The deflection goes lower for the square nanoplate. The thicker the nanoplate is, the smaller the bending deflection will be. The maximum deflection increases with the nonlocal scale parameter but decreases with the material characteristic scale parameter. The critical buckling load decreases with the gradient index, and increases with the thickness and the aspect ratio. When the nonlocal scale parameter increases, the critical buckling load will decrease, but will increase with the material characteristic scale parameter. The softening and hardening mechanisms exist in higherorder bending and buckling of the functionally graded nanoplates, and the coupling effect between 2 internal characteristic parameters also occurs. When the nonlocal scale is greater than the material characteristic scale, the nonlocal effect will dominate in the mechanical properties of functionally graded nanoplates, otherwise the strain gradient effect will play a leading role. The analytical solutions also show that, when the nonlocal scale is equal to the material characteristic scale, the results based on the nonlocal strain gradient theory will degenerate into the corresponding classical ones.

Application of the Wavelet Galerkin Method to Solution of Nonlinear Bifurcation Problems

ZHANG Lei, TANG Conggang, WANG Dequan, LIU Bing

2021, 42(1): 27-35. doi: 10.21656/1000-0887.410085

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Abstract:
Application of the wavelet Galerkin method to solution of nonlinear bifurcation problems was studied through a typical Bratu problem. Firstly, 1D and 2D Bratu equations were discretized with the Coiflet based wavelet Galerkin method, then both the pseudo arclength scheme for tracing solution curves and the extended equations for calculating limit bifurcation points were derived in the case of 1parameter Bratu problems, similarly both the pseudo arclength scheme for tracing solution surfaces and the extended equations for solving cusp bifurcation points were also derived in the case of 2parameter Bratu problems. Numerical results show that, the wavelet Galerkin method not only has higher accuracy during bifurcation point calculation, but also is capable of capturing fold lines and cusp catastrophe quantitatively in the case of 2parameter bifurcation problems. This example exhibits the specific procedure of numerical bifurcation analysis based on the wavelet Galerkin method and demonstrates its potential for capturing complex bifurcation behaviors of multiparameter problems.

3D Analytical Solutions of Stress Fields at Griffith Crack Tips in Functionally Graded Plates

SUN Yeli, SHEN Lulu, YANG Bo

2021, 42(1): 36-48. doi: 10.21656/1000-0887.410143

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Abstract:
Based on the generalized England-Spencer plate theory, the 3D stress field at the Griffith crack tip in a transversely isotropic functionally graded plate was investigated. With the material parameters assumed to vary continuously and arbitrarily along the thickness direction, by means of the complex function theory and the conformal mapping technology, the analytical results of the 3D stress field at the Griffith crack tip under loading at infinity and uniform internal pressure were obtained respectively. With the material reduced to an isotropic homogeneous material, the validity of the solution was verified through comparison with the classical 2D solution. Numerical examples were given to demonstrate the effects of material gradient factors and load conditions on the stress field.

Numerical Simulation of Topology Optimization Technique for Tank Sloshing Suppression

WEI Zhijun, SHEN Limin, GUAN Hui, SUN Mingjing, WU Chuijie

2021, 42(1): 49-57. doi: 10.21656/1000-0887.410206

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Abstract:
The problem of liquid sloshing exists widely in the fields of ships and ocean engineering. When the external excitation frequency is close to the natural frequency of the fluid in the tank, it is easy to produce violent sloshing and great forces, thus causing structural damage. Therefore, it is of great significance to study the effective method to control the impact of liquid sloshing. A numerical program was developed to study the sloshing problem in rectangular tanks with the topology optimization technique. In the numerical program the finite difference method was used to solve the homogeneous and incompressible 3D unsteady Navier-Stokes equations. The free surface was captured with the VOF/PLIC method and a topology optimization program based on the optimal control theory was applied to optimize the shape of baffles in the tank. The sloshing problem with given-shape double baffles and topologically optimized double baffles in the tank was calculated respectively and the kinematics and dynamic characteristics of the flow field with baffles were analyzed. The results show that, the topology optimization of the baffle shape brings better effects of sloshing suppression, which provides a new research idea for the sloshing problems in the fields of ships, ocean engineering and aerospace.

Effects of Proplet on the Deformation of Elastic Gradient Thin Substrate

YANG Yonglin, WANG Xu, LI Xing

2021, 42(1): 58-70. doi: 10.21656/1000-0887.410175

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Abstract:
The phenomenon of droplet wetting has potential significance in cell deformation research and design as well as fabrication of soft devices. In view of the linear tension at the 3-phase contact lines, the gradient thin substrate deformation caused by liquid droplets was studied. Firstly, the constitutive equations of the substrate deformation were solved with the integral transformation method, and the normal displacement expression of the deformation was given. Secondly, the substrate deformation was discussed with different types of elastic moduli of no gradient, the exponential gradient and the power gradient. Finally, the variations of the substrate displacement with the droplet size, the elastic modulus, the linear tension and the gradient index were given. The numerical results show that, with the increases of the elastic modulus and the gradient index, the wetting ridge will go higher and the deformation larger. The smaller the linear tension and the characteristic depth are, the higher the peak displacement value and the larger the deformation will be. When the droplet radius is smaller, the symmetry of the wetting ridge will be better.

Damping Ratio Analysis of Rectangular TLD Tanks With Hydrodynamic Interaction Effects Between Baffles

ZHONG Wenkun, WU Jiurong, SUN Lianyang

2021, 42(1): 71-81. doi: 10.21656/1000-0887.410154

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Abstract:
The estimation formulas for linear damping ratios of rectangular TLD tanks equipped with bottom-mounted vertical baffles or symmetrically wall-mounted horizontal baffles were derived based on the energy dissipation principle under sinusoidal excitation. The formula for the damping ratio of the tank was revised through introduction of the velocity potential function correction factor to consider the hydrodynamic interaction effect between the baffles. Furthermore, several shaking table tests were conducted on a scaled-down rectangular TLD water tank to validate the proposed analytical model for damping ratio estimation. Comparisons between predicted and measured damping ratios show that, the hydrodynamic interaction effects between the baffles with small spacings could reduce the damping ratio of the rectangular water tank. The estimated damping ratio of the rectangular water tank has higher accuracy with the proposed method in view of hydrodynamic interaction effects between baffles.

A Hybrid Neural Network Data Fusion Algorithm Based on Time Series

ZHANG Qiaoling, GAO Shuping, HE Di, CHENG Mengfei

2021, 42(1): 82-91. doi: 10.21656/1000-0887.410056

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Abstract:
For traditional data fusion algorithms, the fusion performance of high-noise, large-scale and complex-structure time series data is poor. A hybrid neural network data fusion algorithm (i.e. the SCLG algorithm) was proposed to solve this problem. Firstly, the time series data were decomposed and reconstructed with the singular spectrum analysis algorithm to eliminate noise. Secondly, the spatial and short-term characteristics of the data were extracted by means of the deep convolutional neural network. Thirdly, the long short-term memory neural network and the gated recurrent unit neural network were introduced to extract data features in the time dimension. Finally, the fully connected layer was applied to integrate the main information and output the final decision. The experimental results from the SP&500 and AQI data sets show that, the proposed algorithm is superior to DCNN, CNN-LSTM and FDL in terms of fusion performance and stability.

Existence and Blow-Up Phenomena of Solutions to Heat Equations With Variable Coefficients Under Nonlinear Boundary Conditions

LI Yuanfei, XIAO Shengzhong, CHEN Xuejiao

2021, 42(1): 92-101. doi: 10.21656/1000-0887.410091

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Abstract:
Under nonlinear boundary conditions, the heat equations with variable coefficients defined on Ω were considered, with Ω∝R^N(N≥2) as a bounded convex region. By means of the technique of differential inequalities, the conditions under which the blowup will definitely occur were derived and the upper bound of the blowup time was determined. Meanwhile, with certain restrictions on the nonlinear terms, the global existence of the solution was obtained. At the blowup moment, the lower bound of the blowup time was also got.

Attractors of Coupled Beam Equations With Structural Damping Under Nonlinear Boundary Conditions

ZHANG Ting, ZHANG Jianwen

2021, 42(1): 102-112. doi: 10.21656/1000-0887.410089

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Abstract:
Attractors of the system of coupled beam equations with rotational inertia and structural damping under nonlinear boundary conditions were studied. Firstly, the existence and uniqueness of the global solution were proved by means of the Faedo-Galerkin method. Secondly, the existence of the bounded absorbing set in the system and the asymptotic smoothness of the related solution semigroup were also proved. Finally, the existence of the global attractor was given.

Lower Bounds of the Blow-up Time for a Class of Reaction Diffusion Equations

XU Ran, TIAN Ya, QIN Yao

2021, 42(1): 113-122. doi: 10.21656/1000-0887.410160

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Abstract:
The blow-up behaviors in a bounded domain were considered for a class of reaction diffusion equations with nonlinear nonlocal heat sources and time-dependent-coefficient heat sink, under the Dirichlet, the Neumann and the Robin boundary conditions respectively.Through construction of auxiliary functions and appropriate conditions for the time-dependent coefficients, with the Sobolev inequality, the H?lder inequality and the Payne and Schaefer integral inequality etc., the lower bounds of the blow-up time of solutions were given for the blow-up occurring in a finite time.