应用数学和力学

2024 Vol. 45, No. 2

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Spectical Topic of Southern Conference on Computational Mechanics
Dynamics and Control
Solid Mechanics

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2024, 45(2)

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Spectical Topic of Southern Conference on Computational Mechanics

Smoothed Finite Element Analysis of Contact and Large Deformation Problems

FAN Yajie, LI Yan, LI Zhongpan, CHEN Huijian, FENG Zhiqiang

2024, 45(2): 127-143. doi: 10.21656/1000-0887.440251

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Abstract:
Rubber material is widely used in practical engineering due to its good seismic and energy absorption effect. However, the collision of hyperelastic materials is a strong nonlinear problem. It is of great significance to analyze the contact collision and large deformation of hyperelastic materials to improve the buffering performance of the device. The smoothed finite element method (S-FEM) is a weak form of numerical calculation method. Compared with the traditional finite element method, the smoothed finite element method has low requirements on the mesh quality, allows the element to undergo large deformation during the calculation process, where the construction of the smooth domain is more flexible. The S-FEM has high accuracy without additional degrees of freedom. Based on the S-FEM, the double potential method was applied to contact calculation, with both advantages of the S-FEM in calculating large deformation problems and advantages of the double potential method in solving contact force fully used. In comparison with the numerical results of finite element software MSC. Marc, the results of the proposed algorithm were verified with high accuracy and good energy conservation, and the effects of the friction coefficient on the collision body were analyzed.

Symplectic Isogeometric Analysis Coupling Method for Interfacial Fracture of Piezoelectric Quasicrystal Composites With Notches

YANG Zhenting, WANG Yajing, NIE Xueyang, XU Xinsheng, ZHOU Zhenhuan

2024, 45(2): 144-154. doi: 10.21656/1000-0887.440247

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A high-precision semi numerical and semi analytical method for interfacial fracture problem of piezoelectric quasicrystals (PQCs)/piezoelectric crystals (PZCs)/elastic material composites with notches was developed. Firstly, the Hamiltonian system was introduced and the Hamiltonian dual equations for the 3-material composite were formulated. The higher order partial differential governing equations were transformed into a set of ordinary differential equations. Secondly, the symplectic eigenvalues and eigensolutions were obtained through separation of variables. The physical quantities were expressed with the expansion of symplectic series. Finally, a symplectic isogeometric analysis (IGA) coupling equation was derived through combination of the symplectic series and the IGA. The analytical expressions of the physical quantities near the notch tip and the intensity factors were derived.

Study on Constitutive Relations and Boundary Value Problems of Granular Materials Based on Artificial Neural Networks

ZHANG Guangjiang, YANG Deze, CHU Xihua

2024, 45(2): 155-166. doi: 10.21656/1000-0887.440248

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Granular materials are widely used in engineering practice, where the numerical simulation of boundary value problems related to granular materials is of great significance. By the artificial neural network algorithm, the discrete element method based on discrete particle models and the finite element method based on continuous models were organically combined to solve the boundary value problems of granular materials. A new and complete model and the solution were formed, namely, the micro-macroscopic 2-scale model and its solution system for offline calculation of the meso model. Specifically, the principal stress, the principal strain, and the corresponding stress-strain matrix of a granular material were first obtained based on the discrete element method. Then an artificial neural network model was built in the main space to describe the constitutive relationship of the granular material by an artificial neural network algorithm. Finally, the artificial neural network model was imported into ABAQUS to solve the boundary value problem of the granular material with the user-defined material subroutine UMAT. Through the numerical tests of plate compression and slope stability, and the comparison with the solution results of the classical elastoplastic model, it is seen that the trained artificial neural network model can effectively reflect the constitutive relationship of granular materials, and can be used in practice to solve boundary value problems. The results show the feasibility of the solution scheme.

A Nonlinear Viscoelastic Model for Silicon Rubber Foam Cushion Considering Time-Varying Evolution Characteristics

FAN Zhigeng, WAN Qiang, NIU Hongpan, JIN Fan

2024, 45(2): 167-174. doi: 10.21656/1000-0887.440249

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Abstract:
Based on the common theory of viscoelasticity, the mechanical model for aging silicon rubber foam cushion and the continuous loading prediction model for long-term stress relaxation silicon foam cushion, were obtained, in view of the 2 aging mechanisms, the nonlinear material characteristics, the loading process, the stress relaxation history, the aging effect and the degradation difference of each motion element of viscoelastic model. The mechanism of the model is clear, and it can reflect the information of the material service history and the corresponding effects on mechanics of the silicon rubber foam cushion.

Homogenization Modeling of Single-Phase Gas Local Flow in Porous Media

LI Shuguang, QU Kai

2024, 45(2): 175-183. doi: 10.21656/1000-0887.440246

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Abstract:
The application of asymptotic homogenization method was investigated based on the filtration theory for single-phase gas, and the mathematical model and numerical method for the gas flow at the pore scale were developed. With the asymptotic homogenization method, a local problem of periodic cells was established to describe the local flow process of a single-phase gas at the pore scale of the periodic porous medium. The special mathematical properties and physical significance of the local problem were discussed. With a simplified approach based on symmetric and antisymmetric extensions, a least squares finite element method for the local problem was proposed, to overcome the numerical difficulties due to averaging operators and periodic boundary conditions. The solution of the local problem was obtained with accurate local velocity and pressure distributions in a single pore, and with gas permeability evaluation of porous media only in knowledge of the pore geometry. Beyond the local problem, the analytical solution of the Poiseuille flow in microtubes was obtained through theoretical analysis, to verify the proposed mathematical model and the numerical algorithm. Finally, a 3D periodic porous structure was considered, and numerical results of local flow in a single pore and permeability coefficients in porous media were obtained.

Dynamics and Control

Synergistic Effects of Impact and Attack Angles on Anti-Penetration Performances of Thin Aramid Laminates

JI Haibo, WANG Xin, SU Jinbo, LI Zhen, WANG Pengfei, JU Yuanyuan, LU Tianjian

2024, 45(2): 184-196. doi: 10.21656/1000-0887.440084

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Abstract:
A 3D finite element (FE) simulation model was built to investigate the synergistic effects of impact and attack angles on the penetration resistance of relatively thin aramid laminates to flat-noded bullets. Two scenarios of the impact responses of a 4-mm-thick aramid laminate were calculated, i.e., considering the impact angle alone and considering the impact and attack angles together. The penetration resistance of the aramid laminate was reflected by the residual velocity of the bullet, the ballistic limit velocity and the perforation energy threshold of the target plate. Deformation and damage mechanisms of the laminate under different impact conditions were also analyzed. The main findings are: (ⅰ) the ballistic limit velocity decreases and then increases with the initial impact angle; (ⅱ) with the increasing impact velocity and the decreasing initial impact angle, the changes of the impact angle and the attack angle both tend to decrease; (ⅲ) for a fixed impact angle, a negative attack angle is not conducive to penetration, but a positive attack angle is conducive to penetration.

Wave Propagation in Functionally Graded Piezoelectric Nanoshells

WANG Xinte, LIU Juan, HU Biao, ZHANG Bo, SHEN Huoming

2024, 45(2): 197-207. doi: 10.21656/1000-0887.440057

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The waves propagation characteristics in porous functionally graded piezoelectric nanoshells were investigated based on the nonlocal strain gradient theory. The governing equations were developed under Hamilton's principle and the 1st-order shear theory. The scale-dependent characteristic equations were obtained through combination of the nonlocal strain gradient theory and the harmonic solutions. The effects of the scale parameter, the wave number, the gradient index, the thickness, the porosity and the voltage on the wave propagation characteristics were discussed numerically. The results show that, the influences of the nonlocal parameter and the strain gradient parameter on the wave propagation frequency are closely related to the wave number, and the larger the wave number is in a certain range, the greater the influence of scale parameters on the frequency will be. In addition, the porosity and the gradient index have a coupling effect on the frequency.

Nonlocal Vibration, Buckling and Bending of 1D Layered Quasicrystal Nanobeams

YUAN Qingdan, GUO Junhong

2024, 45(2): 208-219. doi: 10.21656/1000-0887.440260

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Abstract:
Based on the nonlocal theory, a 1D layered nano-quasicrystal (QC) simply supported beam model was established to investigate the free vibration, buckling behavior, and bending deformation of nano-QC beams. The pseudo-Stroh formula was used to derive the governing equations for the nanobeam. Using the transfer matrix method, exact solutions of the natural frequency, the critical buckling load, the generalized displacement and the generalized stress for bending problems of layered nano-QC beams was obtained under simply supported boundary conditions. The effects of the height-span ratio, the layer thickness ratio, the stacking sequence, and the nonlocal effect on the natural frequency, the critical buckling load and the bending deformation of layered nano-QC simply supported beams were analyzed. The results show that, the natural frequency and the critical buckling load decrease with increasing nonlocal parameter. The bigger the outer-layer quasicrystal elastic constant is, the higher the natural frequency and the buckling critical load will be. The stacking sequence has a significant effect on the mechanical behavior of nano-QC beams. The obtained exact solution provides a reference for various numerical methods and experimental results of nanoscale beam structures.

Solid Mechanics

Nearly Incompressible Elasto-Plastic Analysis of Extra-DOF-Free Generalized Finite Elements

MA Jinwei, DUAN Qinglin

2024, 45(2): 220-226. doi: 10.21656/1000-0887.440067

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The volumetric locking of the conventional finite element method (FEM) was investigated through the nearly incompressible elasto-plastic analysis, and the extra-DOF-free reinforcement function was introduced in the framework of the generalized finite element method (GFEM) for improvement of this problem. Through the introduction of the reinforcement function, a richer approximation space was obtained for the interpolation function, of which the ability of correctly reflecting structural deformations was promoted under the approximate volumetric invariance constraint. Furthermore, the construction of the reinforcement function is independent of the extra degrees of freedom, which eliminates the linear dependency in the traditional GFEM. Different triggering conditions and manifestations of volumetric locking of the conventional FEM were found in linear elastic, hyperelastic, and plastic analyses. Three classical numerical examples show that, the extra-DOF-free GFEM can effectively alleviate the volumetric locking in all analyses, and obtain accurate and reasonable results.

The 2D Adhesive Contact of the Functionally Graded Piezoelectric Coating Under a Conducting Indenter

HAN Lifu, LIU Tiejun

2024, 45(2): 227-244. doi: 10.21656/1000-0887.440238

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Nano-indentation experiments are an important means of studying the mechanical properties and surface morphology of materials. With the decrease of the contact area, the adhesion between the indenter and the contact surface of the specimen cannot be ignored. Therefore, the adhesion effect plays an important role in the contact problem under the action of the indenter. The functional graded piezoelectric material (FGPM) has the advantages of both graded and piezoelectric materials, and can effectively avoid contact damage and failure of coatings. The adhesive contact problem of FGPMs under conducting indenters was studied. With exponentially changing material parameters of the FGPM coating, based on the Maugis adhesive model, the control singular integral equation for the 2D frictionless adhesive contact problem of the FGPM coating under the conducting indenter, was obtained through the Fourier integral transform, and the Erdogan-Gupta numerical method was used to solve the equation. The effects of the adhesive stress, the graded parameter and the charge of the indenter on the electro-mechanical coupling response were obtained. The results provide a theoretical basis for improving the contact behavior of material surfaces with FGPM coatings, and help design piezoelectric structures and devices.

Numerical Conformal Mappings From Multiply Connected Regions Onto Annular Domains With Slits

ZHAO Xin, LÜ Yibin

2024, 45(2): 245-252. doi: 10.21656/1000-0887.440134

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A numerical method was proposed based on the charge simulation method for calculating conformal mappings from the bounded high connectivity regions onto unit annular domains with logarithmic spiral slits. The bi-conjugate residual (BiCR) method was used to solve the constraint equation system acquired with the Dirichlet boundary conditions, obtain the simulated charge, and further construct a high-precision approximate conformal mapping function. Numerical examples show the effectiveness of the proposed method applied to simulate flow over the spiral point vortex in the bounded high connectivity domains.