力学学报

STATE OF THE ART OF CONSTITUTIVE RELATIONS OF HYPERELASTIC MATERIALS

Peng Xiangfeng, Li Luxian

Hyperelastic materials are commonly used in practical engineering with the prominent feature that a very large deformation may be produced under a force but the initial state can be completely recovered when the force is removed. Hyperelastic materials are typically nonlinear elastic ones, whose behaviors are in general characterized by their strain energy functions. For several decades, a lot of mathematical models and physical models have been proposed to study their constitutive relations through constructing the form of energy functions. However, a complete constitutive relation suitable for varied deformation modes and the entire deformation range is still the significant issue to expect in this field. This paper summarizes and analyzes the latest research status of constitutive relations of hyperelastic materials from three perspectives: (1) volume change modes including incompressible and compressible ones; (2) deformation modes such as uniaxial tension, shearing, biaxial tension and combined stretch and shear; (3) the entire range of deformation including small deformation, moderate deformation and large deformation. The latest progresses indicate that, in order to comprehensively describe experimental data of a given hyperelastic material and to apply it in practical problems, it is necessary to establish a complete constitutive relationship of compressible hyperelastic materials, which is suitable for varied deformation modes and the entire range of deformation. The authors suggest an implementation procedure for establishing the complete constitutive relationship of an actual hyperelastic material and an approach to construct the strain energy function of a compressible material.

2020, 52(5): 1221-1234. doi: 10.6052/0459-1879-20-189

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2020, 52(5): 1233-1234. doi: 10.6052/0459-1879-20-314

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THREE-DIMENSIONAL ANALYSIS OF A MAGNETOELECTROELASTIC HALF-SPACE UNDER AXISYMMETRIC TEMPERATURE LOADING

Hu Keqiang, Gao Cunfa, Zhong Zheng, Chen Zengtao

Considering the coupling effects between mechanical, electrical, magnetic and thermal fields, we have presented an analytical solution for the thermo-magneto-electro-elastic problem of a magnetoelectroelastic half-space under axisymmetric thermal loading based on the linear theory. Integral transform method and integral equation technique are applied to analytically solve the heat conduction equation, the governing equations of the magnetoelectroelastic material, and the mixed boundary value problem on the boundary of the magnetoelectroelastic half-space. A general closed-form solution is presented for the complementary and particular parts of the components of the displacement, electric potential and magnetic potential. Traction-free and open circuit electro-magnetic conditions are applied on the boundary surface and an integral form solution for the displacement, electric and magnetic potentials in the magnetoelectroelastic half-space has been successfully obtained. Temperature field in the half-space has been obtained analytically and the expression of the stresses, electric displacements and magnetic induction due to the temperature change applied on the surface are derived and given in an explicit closed form. Numerical results show that the temperature loading has much effect on the field distribution of the mechanical, electric and magnetic fields in the magnetoelectroelastic half-space. As the radius of the constant temperature loading increases, the distance from the region of the maximum normal stress to the free boundary will become larger, and the normal stress becomes much smaller in the regions outside of the circular region. The maximum shear stress appears just below the boundary surface at the edge of the circular region. The electric field is found to be intensified near to the boundary surface within the circular region, and similarly, intensities of the positive and negative magnetic fields are observed at different locations in the half-space under the temperature loading applied on the boundary. The results of this study are helpful for the design and manufacturing of smart materials/structures under thermal loading.

2020, 52(5): 1235-1244. doi: 10.6052/0459-1879-20-127

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FRICTIONAL TEMPERATURE ANALYSIS OF TWO-DIMENSIONAL ELASTO-PLASTIC WHEEL-RAIL SLIDING CONTACT WITH TEMPERATURE-DEPENDENT MATERIAL PROPERTIES

Fu Peilin, Ding Li, Zhao Jizhong, Zhang Xu, Kan Qianhua, Wang Ping

The accurate prediction of frictional temperature during wheel-rail sliding contact is of great significance to the studies for wear and fatigue performance of wheel-rail system. Current analytical or semi-analytical models of wheel-rail frictional temperature usually employ the elliptical distribution of contact pressure in the Hertz elastic contact theory and a single temperature-dependent material property, which differs from the actual heat-transfer state in many wheel-rail sliding contact conditions. Therefore, introducing the plastic correction of contact pressure and the temperature dependence of various properties in the calculation model of wheel-rail sliding temperature rise simultaneously could be great helpful to improve the accuracy of prediction result. Based on the elasto-plastic contact theory, considering the temperature dependence of thermal conductivity, specific heat capacity and friction coefficient simultaneously, the integration of thermal conductivity with respect to temperature is set as the quantity that needs to be solved by applying the Kirchhoff transformation method, and the nonlinear Fourier heat conduction equation is transformed into a corresponding simple partial differential equation with single variable coefficient, a unified implicit difference scheme with arbitrary form of temperature dependence is derived, and the influences of convection coefficient, vertical load, creepage and train speed on the temperature rise over rail surface are discussed, respectively. Results show that the convection coefficient has little effect on the temperature rise in the high-speed condition; the increase of creepage and train speed can enhance the friction power, and thus causes the increase of frictional temperature; and the maximum temperature rise also increases approximately linearly with the increasing vertical load. In addition, considering the temperature dependence of various thermophysical properties in the calculation model of temperature rise induced by the wheel-rail sliding contact can effectively avoid the overestimation of temperature rise, and the temperature-dependent friction coefficient has a more remarkable effect on the prediction result than thermal conductivity and specific heat capacity.

2020, 52(5): 1245-1254. doi: 10.6052/0459-1879-20-122

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A GENERALIZED THERMOELASTIC DIFFUSION PROBLEM OF THIN PLATE HEATED BY THE ULTRASHORT LASER PULSES

Li Yan, He Tianhu, Tian Xiaogeng

In recent years, ultrashort laser pulses are widely used in the fields of ultra-precision machining, optical storage and microelectronic manufacture due to the advantages of high power density, short duration and high machining accuracy. In the manuscript, the memory-dependent nonlocal generalized thermoelastic diffusion theory is established based on the L-S generalized thermoelastic diffusion theory as well as considering the memory-dependent effect and spatial nonlocal effect. The theory can accurately predict the thermoelastic diffusion responses of structures whose geometry size is equivalent to its internal characteristic scale. The control equations of the theory are derived, and the solution of the control equations are obtained based on the Laplace integral transformation. As a numerical example, the transient thermoelastic diffusion responses of a semi-infinite thin plate subjected to a non-Gaussian laser pulse and a chemical shock are studied. The variation of the temperature, chemical potential, displacement, stresses and concentration with different nonlocal parameters, thermal time delay factors and diffusion time delay factors are obtained. The results show that heat conduction has significant effect on mass transfer, while mass transfer has little effect on heat conduction; nonlocal parameter has significant influence on displacement and stress, but little effect on temperature, chemical potential and concentration. The establishment of this theory and the solution method are aimed at accurately predicting the transient responses of the heat and mass under the impact of mechanical loading, heat and chemical potential.

2020, 52(5): 1255-1266. doi: 10.6052/0459-1879-20-118

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A GENERALIZED PIEZOELECTRIC-THERMOELASTIC THEORY WITH STRAIN RATE AND ITS APPLICATION

Li Jiwei, He Tianhu

The deformation of a large number of materials in engineering is between elasticity and viscosity, which exhibits both features of elastic solid and viscous fluid, that is, viscoelasticity. Viscoelasticity causes many mechanical relaxation phenomena, such as strain relaxation and hysteresis loss. In the investigation of transient response of multi-field problems subjected to thermal loading, it is especially important to take the phenomena such as thermal relaxation and the strain relaxation into consideration to accurately describe their transient response. For the transient response of the generalized piezoelectric-thermoelastic problems, although the generalized piezoelectric-thermoelastic model was established by taking into account the thermal relaxation, no strain relaxation is included so far. In present paper, by considering the strain relaxation in the process of deformation, a generalized piezoelectric-thermoelastic theory is theoretically established by extending Chandrasekharaiah's theory through taking strain rate into consideration. By means of the thermodynamic laws, the theory is formulated and the corresponding state equations and governing equations are obtained. In constitutive equation, a term of the product of a strain relaxation time and the strain rate is introduced, meanwhile, thermal relaxation time factors are included in constitutive equation and energy equation respectively. Subsequently, this theory is applied to investigating the dynamic response of a one-dimensional piezoelectric-thermoelastic problem subjected to a moving heat source. The Laplace transform and its numerical inverse transform are used to solve the problem, and the transient response under different strain relaxation time and heat source moving speed is obtained, that is, the distribution law of dimensionless temperature, displacement, stress and electric potential. The effect of strain rate on each physical quantity was investigated, and the results were presented in graphical form. The results show that the strain rate has a significant effect on the distributions of temperature, displacement, stress and electric potential.

2020, 52(5): 1267-1276. doi: 10.6052/0459-1879-20-120

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THERMOELASTIC WAVE PROPAGATION IN PLATES: AN IMPROVED LEGENDRE POLYNOMIAL APPROACH

Wang Xianhui, Li Fanglin, Liu Yujian, Chen Huitao, Yu Jiangong

In recent years, the research of thermoelastic coupled wave has greatly promoted the development of high temperature online detection and laser ultrasonic technology. For its small attenuation, long propagation distance and wide signal coverage, ultrasonic guided wave has become one of the rapid development directions in the field of nondestructive testing. However, the development of guided wave high temperature on-line detection and laser ultrasonic guided wave technology is slow. The key lies in the difficulty in solving the coupled thermoelastic wave equation and the difficulty in studying the propagation and attenuation characteristics. As an effective method, Legendre polynomial approach has been widely used to solve the problem of guided wave propagation since 1999. But there are two shortcomings in this method, which limit its further development and application. Two defects are: (1) Due to the Legendre polynomial and its derivative in integral kernel function, the integrals in the solution process leads to low calculation efficiency; (2) Only the thermoelastic guided wave propagation with isothermal boundary conditions can be treated. In order to solve these two defects, an improved Legendre polynomial method is proposed to solve the fractional thermoelastic guided waves in plates. The analytical integral instead of numerical integration in the available conventional Legendre polynomial approach, which greatly improves the calculation efficiency. A new treatment of the adiabatic boundary condition for the Legendre polynomial is developed by introducing the temperature gradient expansion based on the rectangular window function. Compared with the available data shows the validity of the improved method. Comparison with the CPU time between two approaches indicates the higher efficiency of the presented approach. Finally, the phase velocity dispersion curves, attenuation curves, the stress, displacement and temperature distributions for a plate with different fractional orders are analysed. The fractional order has weak influence on the elastic mode velocity, but it has considerable influence on the elastic mode attenuation.

2020, 52(5): 1277-1285. doi: 10.6052/0459-1879-20-124

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EXACT THERMO-ELECTRO-ELASTIC SOLUTION OF FUNCTIONALLY GRADED MULTILAYERED ONE-DIMENSIONAL QUASICRYSTAL CYLINDRICAL SHELLS

Li Yang, Qin Qinghua, Zhang Liangliang, Gao Yang

Layered structures made of two and more materials with different properties can meet the needs of industrial development. However, the abrupt change of material properties at the interface of the laminated structures can easily cause some interface problems, such as stress concentration, interface cracks, and interface delamination phenomena. Functionally graded materials refer to utilize a continuously changing component gradient instead of the original sudden change interface, which can eliminate or weaken the abrupt change of the physical properties and then increase the bonding strength of the layered structures. In this paper, the research object is the functionally graded multilayered one-dimensional quasicrystal cylindrical shells. By virtue of the pseudo-Stroh formalism and the propagator matrix method, we establish the layered one-dimensional quasicrystal cylindrical shells model with the material parameters following the power-law type distribution along its radius direction, and obtain the exact thermo-electro-elastic solution of the functionally graded layered one-dimensional quasicrystal cylindrical shells with simply supported boundary condition. Numerical examples are carried out to investigate the influences of the exponential factor on temperature, electric, phason and phonon fields of the functionally graded layered one-dimensional quasicrystal cylindrical shells subjected to both inner and outer surfaces temperature variations, especially the effects on physical quantities at the inner and outer surfaces of the layered one-dimensional quasicrystal cylindrical shells. The obtained results indicate that: the exponential factor can change the distribution characteristic of material parameters, which can cause a significant influence on the physical quantities in the temperature, electric, phason, and phonon fields; by increasing the exponential factor, the deformation at the internal surface induced by temperature stimuli is reduced and the strength of the layered one-dimensional quasicrystal cylindrical shells is improved. The results obtained in this paper can provide a reliable theoretical basis for the design and manufacture of functionally graded layered one-dimensional quasicrystal cylindrical shells.

2020, 52(5): 1286-1294. doi: 10.6052/0459-1879-20-126

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ANALYSIS OF THE PIEZOTRONIC EFFECT OF A PIEZOELETRIC SEMICONDUCTOR FIBER UNDER MUTIPLE LOCAL TEMPERATURE LOADINGS

Cheng Ruoran, Zhang Chunli

The polarization potential resulted from a temperature change through a series coupling effects can significantly change physical fields in piezoelectric semiconductor (PS) structures, which has been found important engineering applications in wearable electronics and temperature-related semiconductor electron devices. Using the one-dimensional multi-field coupling model for thermo-piezoelectric semiconductors, we study the effect of multiple local temperature changes on the behaviors of PS fibers. Based on the linearized current constitutive relations, we obtain the analytical solutions. For instance, a PS fiber under two local temperature changes is numerically studied. The effect of local temperature changes on the distributions of the displacement, potential, electric displacement, polarization and carrier are examined. For large temperature changes, we conduct a nonlinear analysis by COMSOL using the nonlinear current model. The numerical results show that the potential barrier and well produced by the temperature changes through a series of coupling effects depend on the magnitude of the temperature change and the thermal load position. This provides a useful theoretical guidance in the designing of devices.

2020, 52(5): 1295-1303. doi: 10.6052/0459-1879-20-128

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DEEP-LEARNING-BASED INTELLIGENT FORCE MEASUREMENT SYSTEM USING IN A SHOCK TUNNEL

Wang Yunpeng, Yang Ruixin, Nie Shaojun, Jiang Zonglin

Aerodynamic force measurement in high-enthalpy flow is very important for the design and optimization of hypersonic vehicles. Currently, impulse facilities are used for generating high-temperature and high-pressure driving gas to simulate the high-enthalpy flow with hypersonic flight-conditions, such as a shock tunnel. However, when force tests are conducted in an impulse facility, the inertial force has a large influence on the measuring results, which creates low-frequency vibrations of the test model and its motion cannot be addressed through digital filtering. In the case of a few milliseconds of test time, the structural design of the six-component balance is greatly challenged. Therefore, dynamic calibration becomes very important for improving the precision and accuracy of force measurement during short-duration. A new method, deep-learning-based single-vector dynamic self-calibration of the force measurement system, and intelligent force measurement system are proposed for obtaining high-accurate aerodynamic force in impulse facilities. One of the main features of this dynamic calibration method is the calibration of the overall force measurement system, not just the balance. Applying this method, the calibrated force measurement system is the wind tunnel test object, which ensures the consistency of calibration and application. In the evaluation, the test verification has achieved relatively ideal results, the large-scale low-frequency vibration interference has been basically eliminated, and the accuracy and reliability of the force measurement in impulse facility have been greatly improved.

2020, 52(5): 1304-1313. doi: 10.6052/0459-1879-20-190

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A RESIDUAL-BASED UNRESOLVED-SCALE FINITE ELEMENT MODELLING FOR IMPLICT LARGE EDDY SIMULATION

Chen Linfeng

In consistence with large and small scales in turbulent flows, shape function space can be divided into resolved and unresolved scale spaces in a frame of finite element method. Introducing the same decomposition of the weighting function space, the variational formulations of Navier-Stokes equations can be divided into two systems of equations: resolved- and unresolved-scale equations. Generally, only the resolved-scale equation is computed, and the unresolved scales are modeled. Based on the unresolved-scale equations, an approximate residual-based unresolved-scale modeling is proposed in the present study. The large-scale equations are then computed by substituting the unresolved-scale modeling. The method is called residual-based large eddy simulation, in which unlike in the classical LES a filtering for Navier-Stokes equations is needed, multiscale decomposition is instead used. Numerical simulations of a turbulent channel flow are implemented with in-house codes of the residual-based large eddy simulation. The results show that, with a low number of elements, the mean streamwise velocity obtained using the present method is in agreement with the DNS data in the inner layer, and it is slightly overpredicted in the outer layer. Underprediction of the Reynolds stress by the present method causes a reduction of turbulence intensity transportation from the streamwise direction to the normal direction. Isosurfaces of the streamwise velocity reveals its capability of capturing the large-eddy structures. Meanwhile, low-speed streaks can be clearly observed in the sublayer near the wall.

2020, 52(5): 1314-1322. doi: 10.6052/0459-1879-20-055

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MODELLING AND CALCULATION OF THE TURBULENT BOUNDARY LAYER ON A ROTATING CYLINDER SURFACE WITH STRONG SUCTION

Ievgen Mochalin, Lin Jingwen, Cai Jiancheng, Volodymyr Brazhenko, E Shiju

A simple, fast and adequate approach to the turbulent boundary layer calculation on the surface of a rotating permeable cylinder has been elaborated for the case of a strong suction through the cylinder surface. Firstly, the rotational gap flow between two concentric cylinders was analyzed theoretically; the outer cylinder is stationary, and the inner porous cylinder is rotating with suction condition. Based on the fact that the stationary outer cylinder does not influence the flow near the rotating inner one, it can be treated as a boundary layer on the surface of rotating permeable cylinder, and an analytical expression for the circumferential velocity distribution is obtained. Secondly, the Cebeci-Smith two-layer algebraic turbulence model has been adjusted to account for centrifugal force field (streamlines curvature), wall suction, and low-Reynolds-number effects. Analytical corrections and empirical coefficients are used to tune the model for the specific conditions of coupled influence of the factors mentioned above. The calibration database was used which has been obtained by detailed numerical simulation based on the Reynolds stress turbulent model. The numerical simulation approach has been comprehensively verified in the known study for the specific flow conditions under consideration. Finally, the solution algorithm based on generalized Cebeci-Smith two-layer algebraic turbulence model was offered to solve the boundary layer flow over the rotating porous cylinder surface. The algorithm is suited for the situation of flow uniformity in the azimuthal and axial directions that required a special iterative procedure to be elaborated. The results of the algebraic turbulence model with different combinations of the rotational speed and the suction velocity agree well with the simulation results of the Reynolds stress turbulent model. It is demonstrated that the method developed reproduces also the laminar boundary layer at the same initial conditions when the detailed numerical simulation predicts the stable laminar flow in the inner cylinder boundary layer.

2020, 52(5): 1323-1333. doi: 10.6052/0459-1879-20-032

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THE INFLUENCE OF GLOBAL-DIRECTION STENCIL ON GRADIENT AND HIGH-ORDER DERIVATIVES RECONSTRUCTION OF UNSTRUCTURED FINITE VOLUME METHODS

Kong Lingfa, Dong Yidao, Liu Wei

The accuracy of unstructured finite volume methods is greatly influenced by different stencils. In previous work, based on the existing problems of local-direction stencil, we explored a more concise global-direction stencil selection method for the second-order unstructured finite volume solver, and stencil cells selected by this novel stencil selection method are always along the boundary normal and circumferential directions even on grids with high aspect ratio. As a result, the variation of flowfield is effectively captured, and flow anisotropy are well reflected. In addition, the novel method is topology-independent, since global directions are determined by the flowfield, while the local directions are strongly coupled with the grid. Therefore, the complex process of advancing front as well as local directions estimation are completely avoided in the novel stencil selection method, and the phenomenon that stencil cells deviate from the boundary normal vector is effectively eliminated on high-aspect-ratio triangular grids. What's more, a better computational accuracy and lower truncation errors on the second-order accurate finite volume solver are obtained by the employment of global-direction stencil. In order to further test the effectiveness of global-direction stencil on high-order unstructured finite volume methods, we will preliminarily utilize this stencil to test the effect of gradient and high-order derivatives reconstruction. After verification, computational errors of global-direction stencil are lower than that of local-direction stencil, and also lower than that of commonly used vertex-neighbor stencil on different grid types. Besides, errors of variable and derivatives at the Gauss point obtained by global-direction stencil are also the lowest among three methods we tested. Therefore, the global-direction stencil is well performed on gradient as well as high-order derivatives reconstruction, and it is feasible to extend this novel stencil selection method to high-order unstructured finite volume methods.

2020, 52(5): 1334-1349. doi: 10.6052/0459-1879-20-093

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PIV MEASUREMENT OF CLOSE IMPINGING JET ON FLAT PLATE

Hu Jianjun, Zhu Qing, Wang Meid, Jin Yaolan, Wang Simin, Kong Xiangdong

The time-resolved particle image velocimetry (TR-PIV) method was used to measure the flow field of a jet impinging on a flat plate in two orthogonal plane at very close distance. The effects of impinging distance and Reynolds number on the flow characteristics and the vortex topology in the clearance were analyzed. The results show that there are three kinds of typical vortex structure in the clearance, namely, double vortex ring mode, single vortex ring mode and complete entrainment mode. However, under the condition of turbulent state with large flow rate, the jet may break through the vortex ring and form random high-speed outflow. The occurrence of each flow pattern is mainly related to the strength of wall constraints. The energy transportation and flow loss of three typical flow patterns are investigated by vorticity analysis. The results show that the energy of the jet is transmitted outward through the vortex-ring mode at close impingement. In the double vortex ring mode, the two vortex rings have opposite rotational direction. Due to the constraint of the end face, both vortex rings are strictly constrained within the end face of the jet nozzle. The strength of the primary vortex ring is significantly greater than that of the secondary vortex ring. Finally, the proper orthogonal decomposition (POD) method is used to analyze the flow mode and energy distribution of the impinging jet. The first ten modal analysis of single and double vortex mode show that the energy fluctuation occurs in a paired pattern at a lower order, which indicates that both the primary and secondary vortex rings have good symmetry. Meanwhile, in the double vortex ring mode, the primary vortex ring is the dominant large-scale flow structure. The first three modes of the complete entrainment mode show that the energy of the jet is concentrated in the upstream of the jet and decreases sharply with the turbulent diffusion.

2020, 52(5): 1350-1361. doi: 10.6052/0459-1879-20-142

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ARCJET ABLATION EXPERIMENT TO SIMULATE THE CHELYABINSK ASTEROID ENTRY

Luo Yue, Wang Lei, Dang Leining, Liu Jinbo, Zhang Jun, Liu Sen

Ablation is one of the most important phenomenon when an asteroid enters the earth atmosphere at hypervelocity, which largely determines the mass loss, flight trajectory, and even radiation characteristics of the asteroid. To research the typical ablation process of asteroids when entering the earth atmosphere, experiments were conducted in an arc heater to simulate the typical conditions (velocity: 6 km/s, height: 17 km, diameter: 1 m) of Chelyabinsk asteroid event. The blunt-shaped test samples with the head radius of 20mm were made by carbon steel and basalt. In this work, the ablation process of test samples were clearly recorded, in which the melt flow of two different materials and the spallation of fragments as well as vaporization of basalt were observed. The evolutions of emission spectroscopy, recession profile and surface temperature profile during the whole process were acquired. The results indicate that the ablation phenomenon and the mechanism of mass loss of two materials are obviously different: Under the impact of the high-temperature flow, the carbon steel was sputtering into mass of tiny droplets which were washed away by the flow rapidly, while the mass loss of basalt were the shear flow of molten matter with small amount of massive spalling and evaporation spraying. All samples were exposed for four seconds in the plasma stream, mass loss and recession of the basalt and carbon steel were [37.9 g, 7.3 mm] and [72.7 g, 13.1 mm] respectively. The estimated effective enthalpy of ablation was 2.6 MJ/kg, the component measured by emission spectroscopy conforms to the electron microscopy (EDS) scanning.

2020, 52(5): 1362-1370. doi: 10.6052/0459-1879-20-081

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RESEARCH ON IMPROVEMENTS OF LRN TURBULENCE MODEL BASED ON FLOW AROUND AUTOMOBILE BODY

Qiang Guanglin, Yang Yi, Chen Zhen, Gu Zhengqi, Zhang Yong

The flow around automobiles was modularized into typical local flows in this paper. Through analyzing the characteristics of typical local flows, it is verified that the capture ability of turbulence model to transition is the key to accurately simulate the flow around automobiles. The paper optimized the solutions of steady-state and transient-state problems by analyzing the separation and transition mechanism of the flow, promoted the prediction ability of turbulence model for transition and improved the accuracy of turbulence model for automobile flow field simulation. For the steady-state solution of the flow around automobiles, by introducing the streamline curvature factor and the response threshold into the low Reynolds number (LRN) $k$-$\varepsilon $ model proposed by Jones and Lauder, a modified low Reynolds number turbulence model (modified LRN $k$-$\varepsilon $) which can predict transition more accurately was obtained. This model alleviated the problems of the original model's over-dependence on the turbulent dissipation rate and the insufficient prediction of the total stress development. For the transient-state solution, by analyzing the characteristics of the RNAS(Reynolds-averaged Navier-Stokes equations)/LES(large eddy simulation) mixed turbulence model, introducing the constrained large eddy simulation (CLES) method and the modified LRN $k$-$\varepsilon $ turbulence model proposed in this paper, a transition LRN CLES model that can accurately predict the transition was proposed. These improved models were applied to the simulation of the external flow field and buffeting noise of a real automobile model respectively. Computations were carried out using the ANSYS Fluent solver. The calculation results were compared with the simulation results of the commonly turbulence models, HD-2 wind tunnel test results and real vehicle road test results, it show that the improved turbulence models can more accurately simulate the steady-state and transient-state characteristics of the complex real automobiles, which provides a reliable theoretical basis and effective numerical solution method for the study of automotive aerodynamic.

2020, 52(5): 1371-1382. doi: 10.6052/0459-1879-20-095

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ANLYTICAL SOLUTION OF THERMOELASTIC DAMPING IN RECTANGULAR MINDLIN MICRO PLATES

Ma Hangkong, Zhou Chenyang, Li Shirong

Analytical solution of thermoelastic damping for rectangular Mindlin micro plate with the four edges simply supported is presented for the first time. Based on the Mindlin plate theory and the one-way coupled heat conduct theory, governing differential equations for thermo-elastically coupled free vibration of the micro plate are formulated. Ignoring the in-plane variation of the temperature gradient, analytical solution of temperature field in terms of the kinematic parameters is obtained under the adiabatic boundary conditions at the top and bottom surfaces. Furthermore, the equations of structural vibration including the thermal bending moment are transformed into a fourth-order partial differential equation only in terms of the deflection. By using the mathematical similarity between the eigenvalue problems under the simply supported boundary conditions, analytical solution of the complex natural frequency of the Mindlin plate is expressed in terms of the frequency of isothermal Kirchhoff plate. Then the inverse quality factor which represents the level of the thermoelastic damping is obtained. Finally, the effects of the shear deformation, the material and the geometry parameters on the thermoelastic damping are examined in detail by the numerical results. The numerical results show that thermoelastic damping estimated by Mindlin plate theory is less than that by Kirchhoff plate theory. The difference between the values evaluated by the two plate theories becomes very significant near the critical thickness. Moreover, along with the increase of the side-to-thickness aspect ratio,the maximum of the thermoelastic damping in micro Mindlin plate increase monotonically, however, that of micro Kirchhoff plate keeps constant.

2020, 52(5): 1383-1393. doi: 10.6052/0459-1879-20-175

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ANALYSIS OF 3-D NOTCHED/CRACKED STRUCTURES BY USING EXTENDED BOUNDARY ELEMENT METHOD

Li Cong, Niu Zhongrong, Hu Zongjun, Hu Bin

According to the theory of linear elasticity, the conventional numerical methods are difficult to calculate the singular stress fields of three dimensional V-notched/cracked structures because of the stress singularity in the V-notch/crack tip region. In this paper, the extended boundary element method (XBEM) is first proposed to calculate the whole displacement and stress fields of three dimensional V-notch/crack structures. Firstly, the three dimensional V-notched/cracked structure is divided into two parts, which are a small sectoral column around the notch/crack tip and the outer region without the tip sectorial column. The displacement and stress components in the small sector column are expressed as the asymptotic series expansions with respect to the radial coordinate from the tip. The stress singular orders and the associated displacement and stress eigen-functions in the tip region are determined by the interpolating matrix method. The amplitude coefficients in the asymptotic series expansions are taken as the basic unknowns. Secondly, the boundary element method is used to analyze the three dimensional V-notched/cracked structure removed the small sector column. Hence, the whole displacement and stress fields of both the tip region and outer region are obtained by combining the boundary element analysis and the asymptotic series expansions of the displacement and stress fields in the notch/crack tip region, where the XBEM has the characteristics of the semi-analytic approach. The XBEM is suitable for the displacement and stress analysis of the three dimensional V-notched/cracked structures, and its solution can accurately describe the displacement and stress fields from the notch/crack tip to the whole region of the V-notched/cracked structures. Finally, two typical examples are given to demonstrate the effectiveness and accuracy of the extended boundary element method.

2020, 52(5): 1394-1408. doi: 10.6052/0459-1879-20-129

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A MULTIAXIAL HIGH CYCLE FATIGUE LIFE PREDICTION MODEL CONSIDERING THE EFFECT OF MEAN STRESS FOR TENSION-TORSION LOADINGS WITH SAME FREQUENCY

Li Jing, Tong Xiaolong, Yang Shuo, Qiu Yuanying

The influence of normal mean stress on fatigue life prediction has been well reflected for most of the critical plane-based high cycle fatigue life prediction models, whereas the effect of shear mean stress isn't well considered in these models. It is found that the fatigue life of 7075-T651 aluminum alloy is substantially reduced due to the existing of the shear mean stress by analyzing the experimental data of this aluminum alloy, which is similar to the effect of tensile mean stress. Therefore, nonconservative predictions maybe obtained under the loading paths with substantial shear mean stresses for these life prediction models ignoring the effect of mean shear stress. In order to estimate the fatigue life better, a new critical plane-based multiaxial high cycle fatigue life prediction model is proposed to take into account the effects of both normal and shear mean stresses. In the proposed model, the strain-based Fatemi-Socie criterion is first extended to the high cycle fatigue life prediction. And then a stress-based Fatemi-Socie criterion is developed. The shear and normal Walker factors are introduced in the developed criterion to consider the effects of shear and normal mean stresses, respectively. Both the shear and normal Walker factors vary from 0 to 1, which reflects the sensitivity of the material to shear and normal mean stresses. Procedures to determine the damage parameters acting on the critical plane and to calculate the constants contained in the proposed model are all presented. The proposed model is valid for the metallic materials with the ratio ${0.5<\tau _{-1} } / {\sigma _{-1} }<0.8$. Comparisons between test results of 5 kinds of metallic materials and model predictions under 12 types of loading paths with different mean stress levels showed that the proposed model presents relatively accurate predictions. Most of the predictions are fell within a life factor of $\pm $3.

2020, 52(5): 1409-1421. doi: 10.6052/0459-1879-20-066

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FUNDAMENTAL FREQUENCY MAXIMIZATION DESIGN FOR CONTINUOUS FIBER-REINFORCED COMPOSITE STRUCTURES

Cheng Changzheng, Bian Guangyao, Wang Xuan, Long Kai, Li Jingchuang, Wu Qiaoguo

Compared with traditional metal-materials, fiber-reinforced composite materials have better performance in many aspects such as strength, stiffness, and fracture resistance. At present, fiber-reinforced composite materials have been widely used in automotive, aerospace, and other industrial fields. This paper proposes a topology optimization method for solving the fundamental frequency maximization of undamped free vibration of continuous fiber-reinforced composite structures. To achieve the simultaneous optimization of the structural topological configuration and the fiber angle. A dynamic topological optimization model is established with the permitted material usage as the constraint and the structure's first-order eigenvalue as the objective function. The model includes density design variables that characterizes the topological configuration of the structure and angular design variables that characterizes the fiber orientation. The analytical sensitivity formulas of the objective function of eigenvalue with respect to density design variables and angle design variables are derived in detail, and the method of moving asymptotes (MMA) is used to solve the optimization problem. Finally, three numerical examples are performed to verify the effectiveness of the proposed method, which includes a static optimization example with the stiffness maximization as the goal and two dynamic optimization examples with the first-order eigenvalue maximization as the goal. The results show that the proposed method can achieve a stable iterative history and fast convergence, and can effectively improve the structural frequency while achieving the integrated optimization of the structural topological configuration and the fiber angle.

2020, 52(5): 1422-1430. doi: 10.6052/0459-1879-20-083

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AN INTERVAL FINITE ELEMENT METHOD BASED ON THE NEUMANN SERIES EXPANSION

Wu Pengge, Ni Bingyu, Jiang Chao

Uncertainty is common in the practical engineering. The interval finite element method is an interval method which introduces the numerical computational method of finite element to structural uncertainty analysis. The aim of the interval finite element analysis is to obtain the upper and lower response bounds of the structure with interval uncertain parameters, where solving the interval finite element equilibrium equations is the key issue. But the solution of interval linear equations belongs to a class of NP-hard problems which are often difficult to solve. This paper classifies and defines a type of linearly decomposable interval finite element problems, which exist commonly in practical engineering. To solve this type of problems, an interval finite element method based on Neumann series is proposed. It is named as the linearly decomposable interval finite element problem if the stiffness matrix in the interval finite element analysis formulation can be expressed as a linear superposition of a set of independent interval variables when the interval uncertain parameter is expressed as a linear superposition form of the independent interval variables. For this kind of problems, the inverse of the stiffness matrix can be represented by its Neumann series expansion. Thus the explicit expressions of structural responses with interval variables can be then obtained, with which the upper and lower bounds of the structural response can be solved efficiently. Finally, two numerical examples show the effectiveness and accuracy of the proposed method.

2020, 52(5): 1431-1442. doi: 10.6052/0459-1879-20-152

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DYNAMIC MECHANICAL BEHAVIOR AND PENETRATION PERFORMANCE OF WFeNiMo HIGH-ENTROPY ALLOY

Chen Haihua, Zhang Xianfeng, Xiong Wei, Liu Chuang, Wei Haiyang, Wang Haiying, Dai Lanhong

In order to investigate the deformation behavior and penetration performance of WFeNiMo high-entropy alloy under different strain rates, the static mechanical properties of the high-entropy alloy was tested by universal material testing machine and the dynamic mechanical properties of the high-entropy alloy was tested by the SHPB (split Hopkinson pressure bar). The micro mechanism of deformation characteristics of the alloy under different strain rates was also discussed. Based on the ballistic gun test platform, the fragments penetration performance of the high-entropy alloy and the typical tungsten alloy (93W-4.9Ni-2.1Fe, wt%) to the finite thickness steel target was studied. The relationship between the penetration process of the two kinds of alloy fragments and the target damage characteristics, the energy consumption of penetration and the impact velocity was analyzed. The results show that the yield strength and strain rate of the high-entropy alloy and tungsten alloy present a positively correlation. The yield strength of the high-entropy alloy is higher than the tungsten alloy under the same strain rate. With the increase of strain rate of deformation, the high-entropy alloy develops from the brittle fracture, quasi-cleavage with the mixing of tough and brittle characters to the fracture deformation mode with adhesive characteristics. The high-entropy alloy has a strong local adiabatic deformation ability and high shear sensitivity when the fragments penetrate into the thin steel targets. The energy consumption of the high-entropy alloy fragments penetrating into the target is lower than the tungsten alloy fragments under the same impact velocity. The high--entropy alloy has excellent mechanical properties and superior performance in the penetration ability. In addition to the traditional shear plug effect, there is a certain energy release characteristic when the thin target is impacted at high speed by the high-entropy alloy fragments and it has a good application prospect in the field of the preformed fragments.

2020, 52(5): 1443-1453. doi: 10.6052/0459-1879-20-166

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EQUIVALENT DYNAMICS MODELING AND ANALYSIS OF LIQUID-FILLED SPACECRAFT WITH FUEL CONSUMPTION

Liu Feng, Yue Baozeng, Ma Bole, Shen Yunfeng

The liquid in the tank of in-orbit spacecraft may experiences different types of motion, mainly including the overall rigid motion of liquid respect to the tank, the lateral sloshing of the liquid free surface, and the liquid rotation starting which will maybe eventually graduate into the rotary sloshing of the liquid free surface and/or the spinning motion of liquid, etc. The composite 3DOF-rigid-pendulum sloshing model is able to describe all of these main motion types of liquid, and it has been validated to be effective for the analysis of the liquid sloshing dynamics during the rotation-starting period. In this paper, the composite rigid-pendulum model is developed to investigate the large motion coupling dynamics of a variable-mass liquid-filled spacecraft with nonlinear liquid slosh and fuel consumption, by taking account of the variation of equivalent model parameters over liquid-fill ratio. Based on the Lagrangian equations, the orbital-attitude-slosh coupling dynamics model of a liquid-filled spacecraft is equivalently established by using the composite rigid-pendulum model to replace the liquid in a spherical tank. Then, simulations of a three-axis stabilized large angle attitude maneuver and a zero-impulse orbital maneuver of spacecraft are given for the coupling dynamics response analysis of the liquid-filled spacecraft system. Simulation results show that the liquid motion with respect to the tank will cause the position offset of spacecraft, and the orbital velocity of spacecraft will not converge to zero when the spacecraft executes zero-impulse maneuvers with fuel consumption; and that the liquid is likely to experience violent and complicated sloshing motion, which may cause the instability in the rigid-body motion of spacecraft, during the orbital maneuvering when the tank is eccentrically installed in the rigid hub.

2020, 52(5): 1454-1464. doi: 10.6052/0459-1879-20-027

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DYNAMICS MODELING AND EXPERIMENT OF A FLEXIBLE CAPTURING MECHANISM IN A SPACE MANIPULATOR

Li Haiquan, Liang Jianxun, Wu Shuang, Liu Qian, Zhang Wenming

An end-effector with flexible capturing mechanisms, which can accomplish on-orbit capturing operations with large tolerance, is a vital component of a large-scale space manipulator. Dynamics modeling and theoretical analysis of the flexible capturing mechanisms are very important for on-orbit servicing task simulation and prediction. In this paper, a dynamics model of a flexible capturing mechanism with three cables in a space end-effector is developed firstly. The three-dimensional absolute nodal coordinate formulation (ANCF) is used to create nonlinear finite elements of flexible cables. Both bending and longitudinal deformation of the cables are considered, furthermore, contact between the flexible cables and the rigid target is analyzed by introducing an intermediate cylinder reference coordinate system. Then, an experiment with passive spring suspension is built to validate the proposed model and signals of both motions and forces of the target are collected and compared with the simulation results. The comparison shows that the values of the simulation match well with the experimental measuring ones. The presented model could be used as supplements for the two-dimension planar air-bearing experiment and could be used for capturing task simulations of large-scale space manipulators. At last, capturing simulations of two representative on-orbit operations are conducted by co-simulation with the proposed model and a dynamics model of a large-scale space manipulator. One of the operations is the soft capturing process of an inspection task on the spacecraft surface, the other is the soft capturing of a floating target. The main difference between the two simulations is that the target in the first simulation is fixed on the base of the manipulator. Results of these simulations show that the soft capturing process can be accomplished on the prescribed condition.

2020, 52(5): 1465-1474. doi: 10.6052/0459-1879-20-106

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FIXED POINT CHAOS AND FOLD/FOLD BURSTING OF A CLASS OF DUFFING SYSTEMS AND THE MECHANISM ANALYSIS

Chen Yani, Meng Wenjing, Qian Youhua

In this paper, a class of bistable Duffing type system with two slow variables under new materials is explored. The system is simulated by time history diagram, phase diagram and bifurcation diagram, then the dynamic mechanism of the system under different parameters is analyzed theoretically. Firstly, this manuscript describes that when the amplitude parameter value is greater than 1, the system may exhibit fixed point chaos and explains the reason of fixed point chaos. Secondly, this manuscript introduces the phenomenon of Fold/Fold bursting in parameter space which is caused by the movement of the system from one side of the saddle-node surface to the other side. We also call it saddle-node bursting. In fact, when the system passes through the saddle-node surface, the number of equilibrium points changes. Then this manuscript uses the path of longitudinal parabolic to explain the mechanism of Fold/Fold bursting. And it is found that regardless of the value of constant coefficient term and amplitude, as long as a certain relationship is satisfied, there will always be Fold/Fold bursting. Next this manuscript uses the linear path to discuss the influence of newly added constant coefficient term. It is found that the position where the path intersects the saddle-node surface will affect the symmetry of the bursting, and the span of the path will affect the magnitude of the bursting oscillation. Finally, this manuscript uses the multiple inflection curve path to discuss the phenomenon when two incentive terms have specific relation. When $n=3$, the change of the constant coefficient term will make the system show Fold/Fold bursting with different times, and the maximum can reach triple bursting. Moreover, it is found that if you can find a path that can be divided into $n$ segments, and each segment will have an intersection with the saddle surface, then $n$ times Fold/Fold bursting will occur.

2020, 52(5): 1475-1484. doi: 10.6052/0459-1879-20-098

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ANALYSIS OF FRICTION INDUCED VIBRATION UNDER THE ACTIVE DISTURBANCE REJECTION CONTROL FRAMEWORK

Piao Minnan, Wang Ying, Zhou Yajing, Sun Mingwei, Zhang Xinhua, Chen Zengqiang

Active disturbance rejection control(ADRC) is a practical control method with a two-degree-of-freedom structure. Due to its capability of handling multifarious disturbances in a straightforward and effective manner, ADRC has been successfully applied to many mechanical systems. However, the limit cycle vibration may be induced when employing the ADRC for mechanical systems with friction. At present, there is no precise analysis work about the friction induced vibration under the ADRC framework. Therefore, this paper investigates this problem by using the analysis tools of nonlinear dynamic systems. First, two representative friction models, static switch model and dynamic LuGre model, respectively, are considered, and active disturbance rejection controllers of different orders are designed for a class of second-order motion systems. Equivalent forms of the controllers are obtained and their relationships with the proportional-integral-derivative(PID) controller are revealed. Then, the limit cycle is calculated by using the shooting method combined with the pseudo arc-length continuation approach. Based on the Floquet theory, the stability, occurrence and type of bifurcation of the limit cycle can be determined. In addition, the local stability of the equilibrium points is analyzed based on the Jacobian matrix and approximate numerical method. Finally, the effects of the model and parameter of friction, the order and parameters of the ADRC on the limit cycle are investigated by numerical calculations. As shown by the calculation results, the parameter $\beta$, which determines the negative slope of the Stribeck effect, has a significant effect. When $\beta>1$, closed-loop systems with these two friction models have the same characteristics. Cyclic fold bifurcation(CFB) of the limit cycle occurs and the set of equilibrium points is locally stable. However, characteristics of these two closed-loop systems are totally different when $\beta<1$. As for the ADRC order, it is found that the order does not affect the conclusions in terms of the existence and stability of the limit cycle, and the stability of the set of equilibrium points. Moreover, low-order ADRC has superior performance in tackling the conflict between the friction compensation and stability robustness. These results can provide some guidelines on the understanding of practical phenomena, selection of the ADRC order, and parameter tuning.

2020, 52(5): 1485-1497. doi: 10.6052/0459-1879-20-149

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VIBRO-IMPACT DYNAMICS OF PIPE CONVEYING FLUID SUBJECTED TO RIGID CLEARANCE CONSTRAINT

Wang Yikun, Wang Lin, Ni Qiao, Yang Mo, Liu Dezheng, Qin Tao

The vibro-impact dynamics due to loose constraints have become one of the key scientific problems in the dynamical system of pipes conveying fluid. The impact force modeled by smoothed nonlinear springs varies continuously with time and displacement of the vibrating pipe, which cannot exactly capture the non-smooth characteristics of the saltation of state vectors for the pipes before and after an impact. In this paper, a non-smooth mathematical model of simply supported pipes conveying pulsating fluid, subjected to a rigid constraint somewhere along the pipe length is established, with consideration of the effect of the values of clearance and coefficient of restitution of the constraint. Especially, the periodic and aperiodic oscillations are investigated under various pulsating frequencies of the internal fluid. The transition matrices of the displacement and velocity of all nodes along the pipe before and after impact were derived based on a Galerkin's approach. The nonlinear equations of motion are solved via a fourth-order Runge-Kutta method, by applying the impact boundary conditions. Results show that the pipe is capable of displaying interesting vibro-impact behaviors in the presence of the rigid clearance constraint with the variation of pulsating frequency of the flowing fluid. With a clearance close to the maximum displacement of the pipe without constraint, periodic vibro-impact behaviors are observed with multiple impacts. The vibration velocities before the pipe impacts on the edge of the rigid clearance constraint decrease to zero gradually with the displacement invariant, which is called a dynamical stick-slip motion, also known as a typical non-smooth phenomenon. By decreasing the value of coefficient of restitution, the responses of the pipe may change from periodic vibrations to chaotic ones. This work provides an attractive strategy for further understanding of the nonlinear impact dynamics of pipes conveying fluid subjected to rigid clearance constraint based on non-smooth theories.

2020, 52(5): 1498-1508. doi: 10.6052/0459-1879-20-137

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ANALYTICAL STUDY ON GROUND VIBRATION INDUCED BY MOVING VEHICLE

Wang Li'an, Zhao Jianchang, Wang Zuowei

Based on vehicle dynamics, the coupling vibration model between vehicle and ground is established, and the contact model of elastic roller is adopted to reflect the tyre tolerance. At the same time, the longitudinal and vertical forces between the wheel and the ground are considered, the system dynamic control equation is constructed, and the analytical solution of the surface vibration displacement is obtained by using Fourier and Laplace integral transformation. In the numerical example, the inverse discrete Fourier transform and Crump's method are used to do the numerical inversion, and the time domain solution of the surface vibration displacement is obtained, the influence of the parameters of the surface vibration displacement is analyzed. The results show that the surface irregularity has the most significant influence on the wheel-earth interaction, and the more uneven the ground, the greater the wheel-earth interaction and the greater the surface vibration displacement. The influence of vehicle speed on the wheel-ground force is limited, but it has a great influence on the excitation frequency of load. When vehicle speed increases, the excitation frequency increases, and the surface vibration displacement increases accordingly. At a low speed, tire inclusivity has a certain effect on the wheel-ground force and surface vibration. With the increase of tire inflation pressure, the wheel-ground force and surface vibration displacement increase, but with the increase of speed, this effect will gradually disappear.

2020, 52(5): 1509-1518. doi: 10.6052/0459-1879-20-033

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ANISOTROPIC STRENGTH CRITERION BASED ON T CRITERION AND THE TRANSFORMATION STRESS METHOD

Wan Zheng, Meng Da

Geotechnical materials are usually distributed horizontally in layers, which can be regarded as transverse isotropic materials. Transverse isotropy has a significant effect on the deformation and strength of geotechnical materials. Based on the proposed t-strength criterion, the t-strength criterion is proposed on the physical concept of the existence of an effective slip plane in the isotropic element body, and the failure condition is fulfilled when the ratio of the principal shear stress to the principal normal stress on the space effective slip plane reaches a certain threshold. Effective slip plane in space and physical deposition surface, based on the position relationship between the two surfaces in the space, the idea is proposed that two surfaces angle as a characterization of transverse isotropy can influence the shear strength parameters, and assume that when the angle of value, the greater the strength of the anisotropy of the contribution degree is, the greater the corresponding stress ratio strength value is, on the contrary, the corresponding smaller stress ratio strength value is. Based on the above ideas, the two surfaces angle is regarded as a parameter to construct anisotropic stress ratio strength formula, and use the stress strength formula for correction, a new anisotropic criterion has been put forward based on t strength criterion, finally transverse isotropic t criterion formula of strength criterion is proposed. On the basis of the above criteria, considering transverse isotropic stress space into isotropic stress space, based on the rule of anisotropic t, transformation equation is derived based on the transverse isotropic strength t criterion. It can be used to convert the traditional p and q as the variables of the isotropic constitutive model into the transverse isotropic three-dimensional constitutive model. By predicting the strength of rock and soil materials and the stress-strain relationship test data under true triaxial condition, the validity and applicability of the proposed transverse isotropic t criterion and its stress transformation formula are verified.

2020, 52(5): 1519-1537. doi: 10.6052/0459-1879-20-138

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STUDY ON DEBONDING FAILURE MODEL OF POLYUREA-BASED COATING WITH HIGH VELOCITY WATER FLOW BASED ON COHESIVE ZONE MODEL

Li Bingqi, Zhang Zhenyu, Li Bin, Liu Xiaonan, Yang Xuhui

The surface of the flood discharge building is usually sprayed with polyurea-based coating to improve the impact resistance. However, the research on the debonding failure mechanism of the anti-wear polyurea-based coating under the action of high-speed water flow velocity is still blank. Based on the flow pattern of high-velocity water flow, the mechanical model of high-velocity water flow to flood-discharge building was determined, and the load on flood discharge building caused by high-velocity water flow mainly includes drag force, impact force, fluctuating force and lifting force. The cohesive zone model was used to characterize the debonding failure process of the interface between polyurea-based coating and flood-reducing building, and the debonding failure model of polyurea-coating with high-speed water flow was established, and the finite element formal equations, constitutive relationship, damage initiation principle, evolution principle and contact and collision model of the model are given. The relationship between stress-displacement in the process of debonding failure was obtained by the debonding failure tests, and the variation law between peeling failure stress and inclination angle of interface was obtained. According to the peeling failure test, the parameters of debonding failure model were obtained, and the model was also verified. The test results were in good agreement with the model calculation results, which provides a theoretical basis for the anti-shock and wear-resistant design of flood discharge building.

2020, 52(5): 1538-1546. doi: 10.6052/0459-1879-20-064

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