Chinese Journal of Theoretical and Applied Mechanics was founded in 1957, is a comprehensive bimonthly academic periodical, administrated by the Chinese Academy of Sciences(CAS), sponsored by the Chinese Society of Theoretical and Applied Mechanics and undertaken by the Institute of Mechanics, CAS. The current Editor-in-Chief is academician Wei Yueguang. Previous editor-in-chiefs in chronological order: Prof. H. S. Tsien, Prof. Y. H. Kuo, Prof. Zheng Zhemin, Prof. Lin Tongji, Prof. Huang Kezhi, Prof. Wu Chengkang, Prof. Wang Ziqiang, Prof. Yang Wei, Prof. Zheng Quanshui... More>>
07 February 2021, Volume 53 Issue 2
Research Review
ON THE UNIFIED PHASE-FIELD THEORY FOR DAMAGE AND FAILURE IN SOLIDS AND STRUCTURES: THEORETICAL AND NUMERICAL ASPECTS1)
Wu Jianying
2021, 53(2):  301-329.  DOI: 10.6052/0459-1879-20-295
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Cracking-induced damage and fracture are the most commonly encountered failure modes of engineering materials and structures. In order to prevent such failure, it is a prerequisite in structural designs to understand how cracks nucleate, propagate, branch, coalesces and even fragmentation, etc., in solids, and more importantly, to quantify their adverse effects to the loss of integrity and even catastrophic collapse of structures. Aiming to provide a feasible approach in the modeling of damage and quasi-brittle failure in solids, this work presents systematically the theoretical and numerical aspects of the unified phase-field theory proposed recently by the author, with applications to a couple of representative benchmark problems. Being a variational approach for regularized cracks, this theory incorporates intrinsically the strength-based nucleation and energy-based propagation criteria, as well as the energy minimization-oriented path following criterion, in a standalone framework. Not only several popular phase-field models for brittle fracture can be recovered as particular examples, but also a novel model——the phase-field regularized cohesive zone model (or shortly, PF-CZM)——that applies to both brittle fracture and quasi-brittle failure, emerges naturally. This model can be numerically implemented in context of the coupled finite element method. In order to solve efficiently the discretized governing equations, several numerical algorithms are discussed, with the monolithic BFGS quasi-newton method being the most efficient one. Representative two- and three-dimensional numerical examples reveal that the PF-CZM is capable of reproducing complex fracture configurations in both brittle and quasi-brittle solids under quasi-static, dynamic and multi-physical environments. Remarkably, in all cases objective numerical predictions are achieved independent of the incorporated length scale and mesh discretization. Therefore, the PF-CZM can be used as a numerically predictive approach for the modeling of damage and failure in engineering structures. Finally, some research topics deserving further studies are suggested.

Fluid Mechanics
PREDICTION OF WAVE TRANSMISSION OF CULVERT BREAKWATER BASED ON CNN1)
Zhao Xizeng, Xu Tianyu, Xie Yulin, Lü Chaofan, Yao Yanming, Xie Jing, Chang Jiang
2021, 53(2):  330-338.  DOI: 10.6052/0459-1879-20-235
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Culvert breakwater is a common coastal engineering structure. At the same time, the wave energy can also be transmitted into the harbor through the culvert, which will affect the hydrodynamic characteristics and mooring stability of the harbor. The study of its wave transmission characteristics is closely related to the safety of relevant production equipment and the corresponding engineering economic cost. However, many scholars mainly focus on theoretical analysis, experimental simulation and numerical calculation for wave transmission of culvert type vertical breakwater. With the development of machine learning technology, the traditional hydrodynamic problems ushered in a new solution concept which has attracted many attentions in the field of physics and engineering. At present, many scholars have applied machine learning algorithm to wave related problems. The machine learning algorithm can autonomously learn the corresponding laws according to the training data set, and establish the prediction model of hydrodynamic characteristics by data mapping. In practical application, it does not need to solve the fluid motion control equation, and has high computational efficiency. In this paper, based on the convolutional neural network (CNN), the wave transmission characteristics of the culvert breakwater under different incident and different opening conditions are predicted. The corresponding training data set is generated by a CFD model for convolution neural network training. The CFD results are compared with physical results for validation. After the data mapping relationship between different working conditions and the corresponding wave transmission results are established, the wave transmission coefficient and wave characteristics of transmission wave under the new working conditions can be predicted rapidly. The results show that the trained convolutional neural network can calculate the corresponding results within 10 milliseconds with a relatively high accuracy. This study can provide a new idea for solving the problem of interaction between waves and coastal structures, and is of importance in engineering application.

A MULTI-SCALE NETWORK FOR THE PREDICTION OF HYDRODYNAMICS IN UNDERWATER LAUNCH1)
Yue Jieshun, Quan Xiaobo, Ye Shuran, Wang Jingzhu, Wang Yiwei
2021, 53(2):  339-351.  DOI: 10.6052/0459-1879-20-186
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The prediction of cavitation and the hydrodynamic characteristics plays a significant role in the design of the underwater launched vehicle. In recent years, the artificial intelligence technology has become an important prediction method for these parameters. In order to quickly predict the dramatic changes of the bottom pressure in the underwater launching process, a multi-scale deep learning network is developed. This neural network model is based on a one-dimensional convolutional network (1DCNN) and established with an encoding-decoding network structure. The input data set is decomposed into a smooth part and fluctuating part through different sampling frequencies. A large-scale low-fidelity network and a small-scale high-fidelity network are trained separately to achieve the response and capture of different physical processes. Firstly, the bottom pressure under different launch conditions are obtained through numerical simulation, and the mechanism of bubble dynamics is constructed as a physical input data. Secondly, the data set is decomposed into two parts to train deep learning networks with two different scales respectively. Finally, two sets of output data based on two networks are integrated to establish a full prediction model. Testing and verification indicate that this newly developed multi-scale network can realize the fast and accurate prediction of the hydrodynamics of the underwater vehicle under various usual launch condition. The predicted bottom pressure curve during any stage, including the smooth stage, the transitional stage, as well as the frequency and magnitude of oscillation are consistent with the numerical simulation results. As a result, this method can provide a basis for the prediction of motion and trajectory of the underwater vehicle.

PARAMETRIC INSTABILITY OF LIQUID SHEETS SUBJECTED TO A TRANSVERSE AC ELECTRIC FIELD1)
Wang Tiehan, Fu Qingfei, Yang Lijun
2021, 53(2):  352-361.  DOI: 10.6052/0459-1879-20-300
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Parametric resonance occurs at the gas-liquid interface when a liquid sheet moving in a transverse AC electrical field. In order to obtain the dispersion relation of liquid sheet under AC electric field and to provide the theoretical basis for the analysis of the breaking behavior of liquid sheet, in this paper, the temporal parametric instability under DC and AC electric field were both analyzed in the leaky dielectric model. The leaky dielectric model is used to characterize the electrical properties of liquid. Since the mean flow is time-dependent function, the Floquet theory is used to solve the stability problem. In this paper, the electric field is defined as a mixed electric field coupled by part of an ac electric field and part of a DC electric field. The dimensionless dispersion relation between wave number and temporal growth rate can be derived as a matrix. According to this relationship, the influence of various liquid properties on the parametric instability were discussed. The effects of the ratio of gas-to-liquid density ($\rho )$, the Weber number (We), the Reynolds number (Re), the electrical Euler number ($Eu$), the relative relaxation time ($\tau )$ and the characteristic of the proportion of AC electric field ($Pr$) and the frequency of the electric field ($\varOmega )$ was concluded in this paper. As a conclusion, the electrical Euler number ($Eu$) influence the instability of both capillary unstable and parametric unstable region, the proportion of electric field ($Pr$) effects as a constant electric field force, the frequency of the AC electric field ($\varOmega )$ mainly influence the parametric instability region. In the experiment, in order to obtain parameter oscillation phenomenon more easily, increasing electrical Euler number ($Eu$) and reducing the frequency of AC electric field ($\varOmega )$ are founded as effective methods.

INFLUENCE OF WIND SHEAR ON AERODYNAMIC CHARACTERISTICS AND WAKE SHAPE OF WIND TURBINE BLADES1)
Xu Bofeng, Zhu Zixuan, Dai Chengjun, Cai Xin, Wang Tongguang, Zhao Zhenzhou
2021, 53(2):  362-372.  DOI: 10.6052/0459-1879-20-289
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A wind turbine usually operates in unsteady conditions, and the aerodynamic performance and wake of the rotor will change with the change of working conditions. Wind shear is the most common and long-term environment of wind turbine. It often affects the aerodynamic load, the shape of the wake and the overall performance of the wind turbine. So it is important to analyze the aerodynamic performance of blade of the wind turbine under the wind shear condition. In this paper, a time-marching free vortex wake method, which coupled with the wind shear model, is used to calculate the aerodynamic coefficient, thrust and wake shape change under different wind shear factors. At the same time, the influence of the wake shape variation on the induced velocity of the rotor rotating plane and the aerodynamic performance of the wind turbine blade are studied. The results show that: under the condition of wind shear inflow, with the increase of the wind shear factor, the fluctuation amplitude of aerodynamic coefficient of the wind turbine, which fluctuates periodically with time is increased, the average thrust decreases gradually, the tilted degree of wake increases and the tilted degree of wake under the center of hub is more obvious. The distortion of wake shape makes the distribution of the axial induced velocity factor in the rotational plane uneven, and makes the overall performance of wind turbine decrease and deviate greatly. There is a distinct difference of the induced effect on the rotational plane between the tilted wake and the symmetrical wake. The fluctuation amplitude of the aerodynamic coefficient induced by the tilted wake is larger than that induced by the symmetrical wake, and the deviation of the wave trough is more obvious than the wave crest. The more tilted the wake is, the more obvious the load asymmetry in the rotating plane is.

COMPUTATIONAL INVESTIGATIONS FOR AERODYNAMIC PERFORMANCE OF BIO-INSPIRED DELTA-WING BASED ON SWIFT-WING1)
Zhang Qing, Ye Zhengyin
2021, 53(2):  373-385.  DOI: 10.6052/0459-1879-20-265
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Aiming at aerodynamic configuration for micro aerial vehicle at the low-Reynolds number flow regime, a group of bio-inspired non-slender delta wings ($\varLambda =50^{\circ}$) similar to swift wings with different leading edge bluntness was designed. To quantitatively investigate the aerodynamic effect caused by the trailing edge tapering of the delta wing, a set of generic delta wings with the same sweep angle was designed for comparisons. In order to deeply investigate the evolution characteristics of the leading edge vortex and the overall aerodynamic characteristics of the bio-inspired delta wing, the numerical simulation method was used to explore the leading edge vortex structure and the overall aerodynamic characteristics at different angles of attack in detail under low Reynolds number flow $(Re=1.58\times 10^{4})$. Computational results show that, the leading edge bluntness and trailing edge tapering have significant effect on the vortex intensity and vortex breakdown position of the leading edge vortex of the generic delta wing and swift delta wing. Compared to the blunt leading edge, the sharp leading edge increased the pressure difference between the upper and lower surfaces, so it has more pronounced vortex intensity and more significant lift enhancement for models with sharp leading edge. Comparing to generic delta wing configuration, the bevel angle of the leading edge of the bio-inspired delta wing results in higher drag, and the trailing edge tapering makes the vortex breakdown position fixed at the trailing edge, so the entire upper wing surface remains at low pressure, resulting in greater overall lift. Since the lift increases more obviously at low angles of attack, the aerodynamic efficiency of the bionic delta wing is significantly greater than that of generic delta wing at low angles of attack. These conclusions are of great values in revealing the flight mechanism of birds and the design of bionic micro aerial vehicles in the near future.

ANALYSIS OF THE PROMOTING EFFECT OF ANNULAR GROOVE ON THE COALESCENCE OF VENTILATED CAVITY1)
Sun Longquan, Yan Hao, Ma Guihui, Zhao Jipeng
2021, 53(2):  386-394.  DOI: 10.6052/0459-1879-20-271
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The load distribution and the water-existing attitude of the vehicle will be affected by the natural cavitation on the shoulder of vehicles during the water-exit process. In engineering, active ventilation by opening vent at shoulder is often employed to improve the mechanical environment of vehicle surface so as to solve such problems. This article is aimed at solving the problem that the ventilated cavity circumferential coalescence of underwater vehicles is unsatisfactory. Using the VOF (volume of fluid) multi-phase flow model and dynamic grid technique of dynamic layering based on finite volume method, the mechanism of adding a small-scale annular groove at the downstream of the vent to promote the cavity coalescence was numerically investigated. And the promotion effect of the annular groove on cavity coalescence at different development stages and different working conditions was also studied. The results show that the cavity coalescence is greatly improved by the annular groove. Flow separation in boundary layer occurs when the incoming flow passes through the annular groove in the moving coordinate system because of flow expansion. The induced entrainment of annular groove, on the one hand, retards the axial development of cavity, and promotes the generation of a larger circumferential shear vortex along with expanding in the circumferential direction. On the other hand, part of the ventilation gas is sucked into the annular groove. The gas sucked into the groove is squeezed and broken contributing to the circumferential coalescence. The coalescence cavity in the groove overflows and leaks to promote the upward movement of the cavity coalescence boundary. In addition, under different working conditions, the shape and internal pressure of the cavity become more stable for the groove improving the internal flow state of the cavity.

VORTEX-INDUCED VIBRATION AND WAKE TRACING MECHANISM OF HARBOR SEAL WHISKER: A DIRECT NUMERICAL SIMULATION1)
Song Liqun, Ji Chunning, Zhang Xiaona
2021, 53(2):  395-412.  DOI: 10.6052/0459-1879-20-268
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Even without eyesight and hearing, harbor seal can identify and track the wake of swimming fish in the water by its whiskers with a special shape. From the biomimetic point of view, study on the vibration responses and the tracking mechanisms of harbor seal whisker in wake flow contributes to the development of a new-type underwater sensor. In this paper, direct numerical simulation of the vortex-induced vibration of the harbor seal whisker in uniform and wake flow with a Reynolds number of $Re=300$ and a reduced velocity of $U_{\rm r}=6.0$ was performed by applying the iterative immersed boundary method. The vibration characteristics and the wake structures of the whisker models are investigated and compared with those of a circular cylinder and an elliptical cylinder with the same equivalent diameter. The effects of different structural shapes on the vibration characteristics and wake structures are analyzed, and the sensing ability and tracking mechanisms of the harbor seal whisker are discussed. The simulation results show that the whisker model can significantly reduce the drag force and suppress the vibration responses in uniform flow. It undergoes a chaos motion with a very low amplitude, which provides a pure signal background for harbor seal's whisker for sensoring. However, in wake flow, the vibration response of the whisker model increases significantly, being stable and periodic. As a result, the whisker model has a higher signal-to-noise ratio and sensitivity than other cylinders. This reveals the mechanism of harbor seal using its whiskers to identify and track the wake of swimming fish in the water, which is of great significance for the development of a new-type underwater detector.

Solid Mechanics
STUDY ON FRACTURE TOUGHNESS OF SEMICONDUCTOR MATERIAL USING VICKERS AND BERKOVICH INDENTERS1)
Liu Ming, Hou Dongyang, Gao Chenghui
2021, 53(2):  413-423.  DOI: 10.6052/0459-1879-20-349
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The indentation method is one of the commonly used methods to determine fracture toughness ($K_{\rm IC})$ of brittle materials. One of the challenges is to obtain a suitable equation of the materials from various equations according to different materials and indenters. Therefore, fracture toughness tests with pyramid indenters (Vickers indenter and Berkovich indenter) were conducted on Si (111) and 4H-SiC (0001) under various loads. The crack length $c$ generated in the Vickers indentation experiments were statistically analyzed, and thirteen equations were selected to calculate the fracture toughness of semiconductor materials at room temperature. The applicability of the indentation test was evaluated, based on a comparative analysis with the results of the scratch test. The results show that to eliminate the inherent discreteness of crack length $c$ generated in the Vickers indentation experiment, multiple indentation tests (at least thirty tests) need to be conducted. The ratio of crack length $c$ over the indentation diagonal length $a$ increases with an increase in the applied load $P$. The crack types of the materials depend on $P$: Palmqvist crack system appears for low loads and Median crack system appears for high loads. Compared with the average fracture toughness (0.96~MPa,$\cdot$,$\sqrt{\rm m}$ and 2.89~MPa,$\cdot$,$\sqrt{\rm m}$, respectively) of Si (111) and 4H-SiC (0001) obtained by micro scratch test, based on linear elastic fracture mechanics (LEFM), the appropriate equations was obtained for both Vickers and Berkovich indenters for the same as material, but which can not be obtained for both Si (111) and 4H-SiC (0001) under the same as indenter from thirteen equations. The fracture toughness of semiconductor materials are best calculated by an expression develope from the Median crack system, and the relationship between fracture toughness being obtained with Vickers indenter and that of with Berkovich indenter is not theoretically 1.073 times, which should be 1.13$\pm$0.01.

DYNAMIC DUCTILE-BRITTLE TRANSITION AND FRACTURE TOUGHNESS MEASUREMENT OF METAL UNDER INTERMEDIATE-LOW LOADING VELOCITIES1)
Li Yilei, Yao Di, Qiao Hongwei, Li Xihua, Zhang Kun, Sun Lei, Yan Xiao, Li Pengzhou
2021, 53(2):  424-436.  DOI: 10.6052/0459-1879-20-304
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The phenomenon of ductile-brittle transition and the measurement of dynamic fracture toughness of metallic materials under impact loading are important parts of the research on dynamic mechanical properties of metal materials. In view of the lack of understanding of ductile-brittle transition of metallic materials under impact loading and the difficulty in measuring the dynamic $J$-resistance curve of ductile materials at relative low loading rate, a method is proposed to measure the ductile-brittle transition process of 15MnTi and 11MnNiMo steels at different loading rates, and the effect of crack tip constraint on the rate change of dynamic ductile-brittle transition of the two materials, by using high-speed material testing machine and its corresponding special fixtures. The dynamic fracture toughness of 15MnTi steel under low loading rate was measured by adjusting the length of compression bar and changing the crack propagation by means of the brake of upper roller. The experimental results indicate that the CT specimen of 15MnTi steel is characterized as ductile fracture when loading velocity is lower than 0.025~m/s, and the fracture character of CT specimen of 15MnTi steel is ductile-brittle combination when loading velocity is between 0.1~m/s and 0.5~m/s, and brittle fracture of the CT specimen of 15MnTi steel starts from 0.5~m/s. The phenomenon of brittle fracture followed by ductile fracture for the CT specimen of 11MnNiMo steel occurs when the loading rate is greater than 1.5~m/s. The dynamic brittle fracture rate of 15MnTi and 11MnNiMo steels is significantly affected by crack tip constraint, and the dynamic brittle fracture rate of the material decreases obviously with the increase of in-plane constraint and out of plane constraint. It is also found that in the three-point bending tests, the fracture toughness of 15MnTi steel decreases slowly with the increase of loading rate, when the loading rate is lower than 8788~MPa$\cdot$mm/s.

HOMOGENEOUS AND INHOMOGENEOUS SWELLING AND PARAMETRIC STUDY OF HYDROGELS CONSIDERING THE MICROSTRUCTURAL DEFORMATION1)
Liu Yan, Wang Huiming
2021, 53(2):  437-447.  DOI: 10.6052/0459-1879-20-368
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An analytical model is proposed to analyze the swelling-induced deformation of the hydrogels considering the microstructural deformation of the polymer chains. It is assumed that the chain is constrained to a tube-like space due to the action of the surrounding chains, and the chains experience the non-affine deformation. Based on the presented model, we studied the free swelling deformation, the swelling along one direction with the pre-stretches in other two directions and the constraint swelling of a spherical hydrogel with a rigid core. For the free swelling case, the deformation is homogeneous and the deformation of the network is equal to that of the chains. The stretches of the network and the chains increase with the increase of the number of polymer segments and decrease with the increase of the density of the chains. The stretches also increase with the increase of the effective tube geometry parameter (ETGP). The effect of the ETGP on the swelling-induced stretch gradually disappears when the external solvent pressure exceeds a certain value. For the swelling along one direction with pre-stretches in other two directions, the equilibrium state with which the stretch of micro-chain equals to that of macro-network can be reached. At this state, the stresses in the hydrogel become zero. For the constraint swelling of a spherical hydrogel with a rigid core, the hydrogel deforms inhomogeneously and the swelling-induced radial stretch is different from the tangential stretch. At the region near the rigid core, the stretch of micro-chain and the stretch of macro-network in the radial direction are larger than that of the free swelling state. At the region far from the rigid core, the stretch of the micro-chain and the radial and tangential stretches of the macro-network approach to that of the free swelling state. The osmotic pressure decreases with the increase of the ETGP, while the volume fraction of the solvent molecules increases with the increase of the ETGP. The presented model can be used to predict the swelling-induced deformation of micro-chain.

BUCKLING LANDSCAPE OF CAN UNDER THE COMBINDE ACTION OF AXIAL COMPRESSION-TORSION-LATERAL POKING-INTERNAL PRESSURE
Song Guangkai, Sun Bohua
2021, 53(2):  448-466.  DOI: 10.6052/0459-1879-20-315
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The cylindrical shell structure has been widely used in the various fields. However, the cylindrical shells are liable to catastrophic buckling, because of the notorious imperfection. The aim of this work is to investigate the buckling of the cylindrical shells based on the non-linear finite element analysis program ABAQUS and applied to the buckling analysis of cans. Firstly, the numerical simulation method was used to verify the buckling test results of the can by Virot et al. In order to obtain some qualitative results of buckling,the buckling behaviour of the cylindrical shells will be investigated. In this article, we focus on the effect of different load combinations and different geometric parameters of the cylindrical shells. The straightforward, simple analysis of the buckling of the cylindrical shells under the axially compressed-lateral perturbation load is presented. We show that the three-dimensional curves of external force-buckling load-displacement called landscape. The numerical results indicate that: the phenomenon of "cliff" appears in the force-displacement curves of specimens under the action of lateral pressure-axially compressed-torsional load; It will be appreciated that the torsional is not conducive to the stability of the specimen and makes the specimen sensitive to the initial imperfections; For specimen under axially compressed-torsional load, in this paper, the plane with zero bearing capacity is defined as "sea level" to distinguish the failure modes of specimens; The results of specimens with different boundary conditions shows that the bearing capacity of the cylindrical shells can be improved with fixed boundaries. The internal pressure can greatly improve the bearing capacity and stability of the structure and reduce the imperfection-sensitivity.

A FINITE ELEMENT COLLOCATION METHOD WITH SMOOTHED NODAL GRADIENTS1)
Fan Liheng, Wang Dongdong, Liu Yuxiang, Du Honghui
2021, 53(2):  467-481.  DOI: 10.6052/0459-1879-20-361
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The collocation formulation has the salient advantages of simplicity and efficiency, but it requires the employment of high order gradients of shape functions associated with certain discretized strategies. The conventional finite element shape functions are usually C$^{0}$ continuous and thus cannot be directly adopted for the collocation analysis. This work presents a finite element collocation method through introducing a set of smoothed gradients of finite element shape functions. In the proposed formulation, the first order nodal smoothed gradients of finite element shape functions are defined with the aid of the general gradient smoothing methodology. Subsequently, the first order smoothed gradients of finite element shape functions are realized by selecting the finite element shape functions as the kernel functions for gradient smoothing. A further differential operation on the first order smoothed gradients then leads to the desired second order smoothed gradients of finite element shape functions, where it is noted that the conventional first order gradients are replaced by the first order smoothed gradients of finite element shape functions. It is theoretically proven that the proposed smoothed gradients of linear finite element shape functions not only meet the first order gradient reproducing conditions that are also satisfied by the conventional gradients of finite element shape functions, but also meet the second order gradient reproducing conditions for uniform meshes that cannot be fulfilled by the conventional finite element formulation. The proposed smoothed gradients of finite element shape functions enable a second order accurate finite element collocation formalism regarding both $L_{2}$ and $H_{1}$ errors, which is one order higher than the conventional linear finite element method in term of $H_{1}$ error, i.e., a superconvergence is achieved by the proposed finite element collocation method with smoothed nodal gradients. Numerical results well demonstrate the convergence and accuracy of the proposed finite element collocation method with smoothed nodal gradients, particularly the superior convergence and accuracy over the conventional finite element method according to the $H_{1}$ or energy errors.

Dynamics, Vibration and Control
LOCKING ALLEVIATION TECHNIQUES OF TWO TYPES OF BEAM ELEMENTS BASED ON THE LOCAL FRAME FORMULATION1)
Tang Huiying, Zhang Zhijuan, Liu Cheng, Liu Shaokui
2021, 53(2):  482-495.  DOI: 10.6052/0459-1879-20-274
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For rigid-flexible coupling dynamic problems with large rotation and large deformation, the modeling method based on the local frame formulation (LFF) of SE(3) group can avoid geometrically nonlinear problem caused by the rigid-body motion. In discretized flexible multibody systems, the generalized mass matrix and the tangent stiffness matrix are invariant under the arbitrary rigid-body motion, which can improve computational efficiency significantly. In the finite element method, locking is the main reason for low convergence rate of elements, such as shear and Poisson locking in beam elements. Mixed methods are effective strategies to alleviate locking in beam and plate/shell elements. In these methods, not only the displacement field but also the stress field and the strain field are discretized, which can increase the accuracy of stress and strain. Based on the local frame formulation, the paper studies locking alleviation techniques of several beam elements, including geometrically exact beam formulation (GEBF) and absolute nodal coordinate formulation (ANCF) beam elements. The Hu-Washizu variational principle is used to alleviate shear locking in the geometrically exact beam, while the strain split method is used to eliminate Poisson locking in the fully parameterized ANCF beam. Numerical examples show that the proposed beam elements based on the local frame formulation can eliminate geometrically nonlinearity caused by the rigid-body motion and can minimize the updating times of mass matrices and tangent stiffness matrices when modeling flexible multibody systems with high rotational speed or large deformation. After locking alleviation, the convergence rate of the above beam elements improves significantly.

NONLINEAR DYNAMIC ANALYSIS OF A VAN DER POL-MATHIEU EQUATION WITH EXTERNAL EXCITATION1)
Huang Jianliang, Wang Teng, Chen Shuhui
2021, 53(2):  496-510.  DOI: 10.6052/0459-1879-20-310
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The periodic responses and quasi-periodic motions of a van der Pol-Mathieu equation subjected to three excitations, i.e., self-excited, parametric excitation, and external excitation, are studied in this paper. A new characteristic is observed that the spectra of the quasi-periodic motions contain uniformly spaced sideband frequencies. Firstly, the traditional incremental harmonic balance (IHB) method is used to obtain periodic responses of the van der Pol-Mathieu equation and to trace their nonlinear frequency response curves automaically. Then the Floquet theory is used to analyze stability of the periodic responses and their bifurcations. Based on the characteristic that the spectra of quasi-periodic motions contain two incommensurate basic frequencies, i.e., the excitation frequency and a priori unknown frequency related to uniformly spaced sideband frequencies. Then the IHB method with two time-scales basing on the two basic frequencies is formulated to accurately calculate all frequency components and their corresponding amplitudes even at critical points. All the results obtained from the IHB method with two time-scales are in excellent agreement with those from numerical integration using the fourth-order Runge-Kutta method. Finally, this investigation reveals rich dynamic characteristics of the van der Pol-Mathieu equation in a range of excitation frequencies.

OUTPUT CHARACTERISTIC ANALYSIS OF SINGLE GIMBAL CONTROL MOMENT GYROSCOPE1)
Huang Zhilai, Li Xinyuan, Jin Dongping
2021, 53(2):  511-523.  DOI: 10.6052/0459-1879-20-306
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The single-gimbal control moment gyroscope (SCMG), which is widely used in aerospace field, has the advantage of torque amplification effect. It is based on the principle of torque amplification with some hypotheses. In this paper, the output characteristics of SCMG are analyzed without those hypotheses. By considering the motion of the mounting base, the output torque model of SCMG with a two-dimensional input and three-dimensional output is obtained, in which the adjustable and nonadjustable parts are identified. In order to analyze the output characteristics of SCMG, two parameters are defined. One is the ratio of the norms about the SCMG's output to input torque vectors. The other is the ratio of the norm about the SCMG's used and unused torque vector, which is to represent the utilization ratio of the SCMG's output torque. In all feasible regions, the results show that the characteristic parameters of torque output are is not always greater than 1, i.e., SCMG does not always has torque amplification effect and efficient torque utilization, which are closely related to the state of SCMG. Finally, for the spacecraft attitude maneuver task with two SCMGs, the simulation of non-diagonal singular robust control and optimal control is completed. It is found that the control effect is closely related to the output characteristic parameters which are determined by the system state. At the same time, the optimal control with a SCMG is used to realize the three-dimensional attitude maneuver of a spacecraft based on the three-dimensional output characteristics of SCMG. The simulation results show that the SCMG always has the torque amplification effect and the efficient torque utilization in the process of optimal control.

AN OPTIMAL GRASP PLANNER FOR SPACE ROBOTS USING CLAMPED B-SPLIE1)
Wang Mingming, Luo Jianjun, Yu Min
2021, 53(2):  524-538.  DOI: 10.6052/0459-1879-20-114
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Space manipulator is one of the key technologies to carry out on-orbit servicing and maintenance missions in the future. Until now, it is still a vast challenging mission to capture a non-cooperative target satellite by using a space robot, especially when the motion of the target satellite is tumbling. How to design a feasible and optimal grasping strategy is very important for the successfully capturing of non-cooperative target. Based on the concept of the Clamped B Spline, this paper investigates an optimal grasp planner for a kinematically redundant space manipulator to capture an arbitrarily rotating target, such as space debris, dysfunctional satellites, etc. The kinematics and dynamics of the space robotic system and non-cooperative target in pre- and post-capture phases are firstly introduced as the foundation for designing the grasp planner. With consideration of the kineto-statics duality of the non-cooperative target captured by a space robot, the concept of the force manipulability ellipsoid was derived and employed as an optimization index in the following grasp planning strategy design. Subsequently, the space robotic optimal grasping time and the target's terminal motion states are determined with consideration of the robotic capability map, the target motion prediction and the grasping direction of the space robotic end-effector. Furthermore, the joint trajectories are parameterized with time normalization using the clamped B-spline curves. The grasp planner of the space robot is then transformed as a multi-constraint, multi-objective nonlinear optimization issue with consideration of the space robotic joint angle, velocity, collision avoidance and end-effector's grasping cone limits, and solved by a constrained particle swarm optimization algorithm with adaptive inertia parameters. The designed grasp planning strategy is applied to a seven degree-of-freedom kinematically redundant manipulator mounted on a free-floating spacecraft base, and the successful capturing of a tumbling target satellite in space is realized. Simulation results are presented and demonstrated the feasibility and effectiveness of the proposed method.

Biomechanics, Engineering and Interdiscipliary Mechanics
BIOMECHANICS AND FUNCTIONAL REGULATIONS OF CD44-LIGAND INTERACTIONS1)
Li Linda, Ding Qihan, Chen Shenbao, Lü Shouqin, Long Mian, Guo Xingming
2021, 53(2):  539-553.  DOI: 10.6052/0459-1879-20-313
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As a widely expressed cellular adhesion molecule, type I transmembrane glycoprotein CD44 is crucial in cell proliferation, differentiation, migration, angiogenesis and other biological processes to induce intracellular signal transduction and regulate tissue homeostasis. Especially, cell adhesion dynamics mediated by CD44-selectin and CD44-hyaluronic acid (HA) interactions play key roles in classic inflammatory cascade, tumor metastasis, or tissue-specific liver immunity. This review discussed the progresses and remaining issues of CD44 selectin and CD44-HA interactions in various aspects of cellular adhesion dynamics, two- and three-dimensional molecular reaction kinetics, atomic microstructural features, and intracellular signal transduction pathways. Nowadays, the importance of mechanical and physical factors to biological activities has been gradually accepted by scientific community. New concepts such as mechanomedicine, mechanoimmunology and mechanomics have been put forward one after another. Under physiological or pathological conditions, cell adhesion mediated by CD44-ligand interactions are regulated by in vivo mechanical and physical cues such as blood shear or tissue stiffness, but their regulatory mechanisms are still unclear. From that on, future perspectives related to CD44-ligand interaction were also proposed in this review as follows: how mechanical and physical factors regulate cellular adhesion dynamics and intrinsic mechanism mediated by CD44-ligand interactions; what the mechanical regulation features of molecular reaction kinetics of CD44-ligand interactions and corresponding structural bases are; and how the atomic-level microstructures of CD44-ligand binding evolve dynamically under mechanical forces. This review provides clues for further understanding the biological functions and structure-function relationship of CD44-ligand interactions.

A TIME-DOMAIN ARTIFICIAL BOUNDARY CONDITION FOR VECTOR WAVE IN MULTILAYERED WAVEGUIDE1)
Wu Lihua, Zhao Mi, Du Xiuli
2021, 53(2):  554-567.  DOI: 10.6052/0459-1879-20-213
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A time-domain artificial boundary condition (ABC) is proposed to simulate the in-plane vector wave in a linear elastic multilayered waveguide with Rayleigh damping. The ABC is stable and can be seamlessly coupled with the finite element method. First, the vector wave equations of the multilayered waveguide are simplified to two scalar wave equations, which are decoupled in both $x$ and $y$ directions. Then, based on the scaled boundary finite element method, semi-discrete frequency-domain dynamic stiffness in the modal space is obtained. The dynamic stiffness can be approximately expressed as matrix continued fraction. Finally, the continued fraction is converted to the time-domain ABC by introducing the auxiliary variable technique. In this method, the parameters affecting the calculation accuracy and efficiency include the mode number $n$, the order $J$ of continued fraction, and the distance $L$ from the artificial boundary to the region of interest. Numerical examples show that only the mode numbers of the infinite domain excited by the load have to be used. $J$=3 can be taken generally. The value of $L$ is independent of the size of the underground structure. But it is proportional to the total height $H$ of the soil layer, and the relation coefficient is related to the material parameters of the soil layer.

PORE-SCALE LATTICE BOLTZMANN MODELING OF SOIL WATER DISTRIBUTION1)
Hu Wulong, Liu Guofeng, Yan Shilin, Fan Yanwei
2021, 53(2):  568-579.  DOI: 10.6052/0459-1879-20-198
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Water in soil controls almost all physical and biogeochemical processes in terrestrial ecosystems and correctly describing its distribution and flow is critical in human development and ecological environment protection. Water distribution at pore scale is modulated by a multitude of abiotic and biotic factors such as the exudates secreted by plant roots and microbes, which could alter soil wettability and water surface tension. The combined impact of all these factors can be described by a single parameter, the contact angle. Practical studies on soil water distribution normally focus on large scale using continuum approaches by volumetrically averaging the microscopic processes out, but it is the physical and biochemical processes occurring in the pores that underpin the emerging phenomena at large scales. Studying the microscopic mechanisms underlying the microscopic water distribution is hence essential to improving the understanding of macroscopic phenomena. Since it is difficult to observe the water distribution at pore-scale due to the complexity of pores structure and the opaque nature of the soils, pore-scale modelling in combination with tomography has been increasingly used to bridge this gap. In this paper, we numerically investigated how a change in the contact angle reshaped water distribution using the Lattice Boltzmann model and X-ray computed tomography. Two soils with contrasting structures were acquired using X-ray computed tomography and they were then segmented to binary images consisting of pore and solid voxels. Water distribution in pore spaces of the soils was assumed to be controlled by capillary force and was simulated using a modified two-phase lattice Boltzmann model. The results show that with the contact angle increasing, the impact of the pore diameter on water distribution in both soils waned, and that a change in the contact angle also led to a change in the channel diameter for fluid flow and interfacial areas between liquid, solid and gas. It was found that as the contact angle decreased, the channel diameter for the liquid decreased while that for the gas increased first followed by a decline. The density of the liquid water was independent of the contact angle, but the density of the vapor decreased significantly as the contact angle increased. The effects of saturation on density of the vapor also increased as the contact angle decreased.

THE LATERAL DISTRIBUTION OF DEPTH-AVERAGED VELOCITY IN CONSECUTIVE BENDS WITH POOL-POINT BAR1)
Liu Yujiao, Yu Minghui, Tian Haoyong
2021, 53(2):  580-588.  DOI: 10.6052/0459-1879-20-208
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This paper proposes an analytical approach to modeling the lateral distribution of depth-averaged streamwise velocity for flow in consecutive bends with pool-point bar based on the depth-integrated Navier-Stokes equations. The additional secondary flow and yet fully developed flow are assumed to be a linear function of the lateral distance. Then, the model for calculating the average vertical velocity distribution along the cross section of the pool region and the point bar regions is presented, and it is applied to consecutive bends with pool-point bar. By calibrating the calculated parameters from the measured data, the model can calculate the average longitudinal velocity distribution of vertical cross-section under different outlet water depth. The modeled results agree well with experimental data. The value rules of the parameters have analyzed for different water depth and along the cross-sections. Sensitivity analysis is performed on the parameters which showed that the region division of the coefficient of the first degree term in the linear hypothesis has great influence on the size and position of the peak flow velocity. The region division of the constant term value is according to the traverse gradient. The region edge between the flat bed and the sloping bed of the constant term has a significant influence on the results. According to the sensitivity of parameters, the mean value of parameters along the flume is presented as a reference value. The transverse distribution of the secondary flow term and the additional stress term in the depth-integrated Navier--Stokes equations along the experimental channel is discussed to further understand the applicability of the linear hypothesis. The results show that the linear hypothesis is suitable for the curve path in the flume. The research results are helpful to understand the longitudinal velocity distribution characteristics and the formation mechanism of the consecutive bends with pool-point bar.

NEW PROGRESS IN THE FIELD OF EXTRAORDINARY ENVIRONMENTAL MECHANICS——REVIEW OF THE CHINESE JOURNAL OF THEORETICAL AND APPLIED MECHANICS SYMPOSIUM ON EXTREME MECHANICS1)
Jiang Zonglin, Liu Junli, Yuan Chaokai, Chen Haixuan, Lu Xiyun
2021, 53(2):  589-599.  DOI: 10.6052/0459-1879-20-442
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This article introduces the background knowledge of extraordinary environmental mechanics and reviews the frontier progress in the field of extreme mechanics that was reported on the Chinese Journal of Theoretical and Applied Mechanics (CJTAM) symposium. This article focuses on research directions that are critical for the national security, such as the deep sea, deep space and high temperature and hypersonic flow, and introduces the significant achievements and the latest research progress in the field of extraordinary environmental mechanics. Through this conference, the editorial department of CJTAM is exploring a new communication mode, which can timely convey the cutting-edge and groundbreaking achievements to researchers, thus supporting the research and development in related engineering and technology fields. This article also summarizes the research fields involved in the symposium, hoping to promote research and communication within the field of extraordinary environmental mechanics.

Former Editors-in-Chief
12th' Editorial Committee
• Editor-in-Chief Wei Yueguang
• Editor-in-Chief
Wei Yueguang
Associate Editor-in-Chief
Chen Liqun　　Feng Xiating 　　　 Lu Xiyun
Wang Yuesheng　　　Wang Guangqian