2021 Vol. 42, No. 12

Display Method:
Solid Mechanics
Buckling Behaviors of Elastomers With Periodic Elliptical Holes Under Negative Pressure Activation
LIANG Guanpo, FU Yuxin, LOU Benliang, XIE Yuxin
2021, 42(12): 1221-1228. doi: 10.21656/1000-0887.420100
Abstract(231) HTML (118) PDF(62)

The buckling behaviors of 2D structures with periodic elliptical holes were studied through numerical simulations and theoretical analysis. A theoretical model was established for the modal analysis corresponding to different buckling modes. The analysis results indicate that, there is a transformation between the buckling modes of the 2D structure with elliptical holes, with the change of the geometrical parameters of the holes. The theoretical analysis and the numerical results match up. Furthermore, in the numerical simulation, a modified force boundary condition for a unit cell under negative pressure activation, being different from the displacement loading, should be considered to ensure the completeness conditions are fulfilled. The confusion in the application of appropriate boundary conditions for the unit cell will result in errors. The choice of the unit cell and the derivation of correct boundary conditions, combined with finite structures, were discussed.

Buckling Analysis of Stepped Columns Based on the Improved Fourier Series Method
LU Jianwei, BAO Siyuan, SHEN Feng
2021, 42(12): 1229-1237. doi: 10.21656/1000-0887.410373
Abstract(325) HTML (142) PDF(32)

The elastic buckling of stepped columns with variable cross sections was studied. Firstly, based on the improved Fourier series method, the displacement function of the column was established in the local coordinate system, then the linear equations for buckling loads were obtained with the constrained variational principle of potential energy. The problem was transformed into a matrix eigenvalue problem and the buckling load was obtained from solution of the matrix eigenvalues. Finally, the parameter values in the method were discussed through numerical examples, and the obtained results were compared with the finite element results and previous literature results so as to verify the accuracy of the method. In the presented model the translational and rotational springs were arranged at the 2 ends and the setback cross sections. The method can determine the buckling loads of stepped columns with various elastic boundary conditions accurately in engineering design.

A Symplectic Approach for Buckling Analysis of Natural Fiber Reinforced Composite Shells Under Hygrothermal Aging
ZHANG Junlin, NI Yiwen, LI Qingdong, TONG Zhenzhen, ZHOU Zhenhuan, XU Xinsheng
2021, 42(12): 1238-1247. doi: 10.21656/1000-0887.420018
Abstract(353) HTML (137) PDF(26)

The buckling analysis of natural fiber reinforced (NFRC) cylindrical shells was performed based on Reissner’s shell theory and the symplectic approach. The governing buckling equations for axially-compressed NFRC cylindrical shells were established in the Hamiltonian system. Therefore, the original problem was reduced to a symplectic eigen-problem in the symplectic space. Accurate critical loads and analytical buckling mode shapes were directly obtained from the symplectic eigenvalues and eigensolutions. With numerical examples, the effects of hygroscopic aging on the expressions of eigensolutions were investigated. In addition, the influences of the aging time, the fiber content and the geometric parameters on the buckling behavior of NFRC cylindrical shells were discussed in detail.

Effects of Structure Parameters on Dynamic Performances of Electrostatic Drive Micro-Machined Gyroscopes
LI Xinye, WANG Yaxue, ZHANG Huabiao, ZHANG Lijuan, YU Tao
2021, 42(12): 1248-1257. doi: 10.21656/1000-0887.410316
Abstract(141) HTML (72) PDF(11)

In view of the cubic nonlinear stiffness and the nonlinear electrostatic force in fraction form, a 2DOF model was analyzed with the harmonic balance method and the residue theorem, and the effects of structure parameters on dynamic performances of micro-machined gyroscopes were studied. The variations of the capacitance with the driving force frequency and the carrier angular velocity were obtained for different thicknesses and gaps of driving electrode comb teeth, different electrode plate areas and different detecting electrode gaps. In addition, the variations of sensitivity and nonlinearity with these structure parameters were also presented. It is found that, the variation curves of the detection capacitance with the driving force frequency show obvious nonlinear characteristics. In other words, the 2nd peak leans rightward, which results in jumping. The effects of thicknesses and gaps of driving electrode comb teeth, and gaps between detecting electrode plates on the variation curves of the capacitance with the carrier angular velocity are much greater than those of detecting electrode plate areas. The variations of sensitivity and nonlinearity with thicknesses and gaps of driving electrode comb teeth and detecting electrode plate areas, are approximately linear, however, those with gaps between detecting electrode plates are nonlinear.

Asymptotic Solutions of Plastic Stress and Displacement at V-Notch Tips Under Anti-Plane Shear
LI Cong, HU Bin, NIU Zhongrong
2021, 42(12): 1258-1275. doi: 10.21656/1000-0887.420045
Abstract(318) HTML (130) PDF(28)

An efficient method was developed to determine the first- and high-order terms of asymptotic solutions of plastic stress and displacement near V-notch tips under anti-plane shear in power-law hardening materials. Through introduction of the asymptotic series expansions of stress and displacement fields around the V-notch tip into the fundamental equations of the elastoplastic theory, the governing ordinary differential equations (ODEs) with the stress and displacement eigen-functions were established. Then the interpolating matrix method was employed to solve the resulting nonlinear and linear ODEs. Consequently, the high-order stress exponents and the associated eigen-solutions were obtained. The presented method, being capable of dealing with the V-notches with arbitrary opening angles and strain hardening indexes under anti-plane shear, has the advantages of great versatility and high accuracy. Typical examples were given to demonstrate the accuracy and effectiveness of this method.

A Stress Analysis Model for CFRP Repaired Defective Steel Plates
ZHOU Daocheng, CHANG Lichen
2021, 42(12): 1276-1286. doi: 10.21656/1000-0887.420006
Abstract(235) HTML (209) PDF(15)

In the repairing method with CFRP for steel structures with corrosion defects, the stresses of CFRP and adhesive layers are key to determine the bearing capacity of CFRP repaired structures. Based on the assumption of plane sections, the distributions of stresses and strains under bending moments were obtained; based on the adhesive shear model, the relationship between the adhesive shear stress and the displacements of the CFRP as well as the steel plate was obtained; based on the force balance, the stress relationship of the CFRP and the steel plate was obtained. Combined with the relationships between various materials, the analytical stress distribution solutions of the CFRP and the adhesive layer under the combined action of the axial force and the bending moment were derived. Numerical analysis was also conducted to calculate the defective steel plate repaired with CFRP bonded on both sides, and the results are consistent with the analytical ones. The stress distribution characteristics of the defective steel plate with CFRP bonded on both sides and the possible failure position of the component were obtained, which provides a basis for calculation of the ultimate bearing capacity of the component.

Upper Bound Analysis on the Stability of Cracked Slopes at Tunnel Entrance Subjected to Pore Water Pressure
ZHANG Jingwu, ZHANG Haiyang, LI Mingdong, GUAN Zhongmin
2021, 42(12): 1287-1295. doi: 10.21656/1000-0887.420054
Abstract(372) HTML (147) PDF(28)

The stability of the cracked slope at tunnel entrance subjected to pore water pressure was studied. The upper-bound limit analysis method and the logarithmic spiral rotation failure mechanism were adopted. A formula for calculating the stability coefficient reflecting the critical slope height was derived. The rationality of the proposed method was verified through comparison of the calculated results with those without effects of the pore water pressure. The distribution of the most critical crack position at the top of slope and the safety factor of the slope were studied by an example analysis. The results show that, the greater the crack depth is, the larger the internal friction angle and the slope angle will be, and the shallower the water level is, the closer the crack position will be to the edge of the slope. The greater the pore water pressure coefficient and the cracking depth are, the smaller the stability coefficient of the slope top will be. The deeper the crack is, the higher the pore water pressure coefficient and the steeper the slope will be, and the more unstable the slope will be. However, the lower the water level is, the more stable the slope will be.

Applied Mathematics
The Wave Speed Signs for Bistable Traveling Wave Solutions in 3-Species Competition-Diffusion Systems
ZHENG Jingpan
2021, 42(12): 1296-1305. doi: 10.21656/1000-0887.420093
Abstract(210) PDF(71)
In the bistable competition-diffusion model, the wave speed signs for the traveling waves can predict which species are more dominant and will eventually occupy the whole habitat. Therefore, it is of great biological significance to study the speed signs for the traveling waves. Firstly, the Lotka-Volterra competition-diffusion system was transformed into a cooperative system. Under the comparison principle, the comparison theorem for the bistable wave speed and the specific upper-lower solution wave speeds of wave profile equations was obtained. Then, according to the comparison theorem and through construction of suitable upper-lower solutions, some sufficient conditions for determining the bistable traveling wave speed signs were obtained. The results help predict and control the competition results of biological populations.
Analysis and Application of a COVID-19 Transmission Model With Tracking and Isolation Measures
2021, 42(12): 1306-1316. doi: 10.21656/1000-0887.420107
Abstract(342) PDF(41)

The novel coronavirus epidemic, appearing at the end of December 2019, spread rapidly due to the large-scale population movement in the Spring Festival travel rush in 2020. Since January 23, 2020, China has taken various measures to effectively control the epidemic. For example, the closure of Wuhan, the tracking and isolation of close contacts of confirmed cases, and the home isolation of Hubei people, etc. Based on the actual transmission of novel coronavirus (COVID-19) in Shanxi province, a dynamic model was established for tracking and isolation of close contacts with imported and confirmed cases. Without regard to the imported cases, the dynamic behavior of the model was analyzed. By means of the case data of novel coronavirus in Shanxi province, the real-time reproduction number was calculated. It is found that the closure of villages and streets in Shanxi province on January 25, 2020 effectively controls the spread of COVID-19 epidemic, that is, the real-time reproduction number is less than 1, which verifies the effectiveness of prevention and control measures from a macro perspective. Further, through the numerical simulation of the model, it is concluded that the prevention and control strategy for early infected patients isolated for 14 days is reasonable and effective; the earlier the closure of Wuhan is, the smaller the scale of infected people will be; the larger number of tracked and isolated contacts of confirmed cases is, the smaller the size of the patients will be.

Existence of Critical Traveling Wave Solutions for a Class of Discrete Diffusion SIR Models With Nonlinear Incidence and Time Delay
ZHANG Xiaoyan
2021, 42(12): 1317-1326. doi: 10.21656/1000-0887.420111
Abstract(195) PDF(49)
The existence of critical traveling wave solutions for a class of discrete diffusion SIR models with nonlinear incidence and time delay were studied. Under the condition that the total population is not a constant, the upper and lower solutions method and the Schauder fixed point theorem were used to prove the existence of the solution on a finite interval. Furthermore, the existence of critical traveling wave solutions was proved on the real number field through limit arguments. Finally, with the fluctuation lemma and the proof by contradiction, the asymptotic boundary of the critical traveling wave was obtained.
Dynamics of a Class of Stochastic SIRS Infectious Disease Models With Both Logistic Birth and Markov Switching
HE Xueqing
2021, 42(12): 1327-1337. doi: 10.21656/1000-0887.420140
Abstract(171) PDF(43)
A class of stochastic SIRS infectious disease models with both logistic birth and Markov switching were investigated. The uniqueness of the existence of a globally positive solution to the stochastic infectious disease model was first analyzed through construction of suitable V functions and then by means of Itô’s formula. Afterwards, the results of the existence of an ergodic smooth distribution for the solution of the model and the sufficient conditions for the extinction of the disease were discussed. Finally, numerical examples were given to illustrate the conclusions.