力学学报

RESEARCH PROGRESS OF CONTACT FORCE MODELS IN THE COLLISION MECHANICS OF MULTIBODY SYSTEM

Wang Gengxiang, Ma Daolin, Liu Yang, Liu Caishan

Impact behavior is a ubiquitous phenomenon in multibody systems. The contact force model is a pivotal tool to predict the contact characteristics of multibody systems. At present, there are two kinds of contact models used for calculating impact behaviors: the static elastoplastic contact force model and the continuous contact force models with energy dissipation. There are many similarities and discrepancies among them in the impact dynamics of multibody systems. This review starts with the introduction of development history of these two kinds of contact models followed by their development progress and background illustrated in detail. Firstly, whether the initial impact velocity is contained in the denominator of damping term severs as a criterion to classify the continuous contact force model as two types of models that are the contact force model with hysteresis damping factor and the other one with viscous damping factor. The relationship between the power exponent and Hertz contact stiffness is analyzed. The problems in calculating the elastic-plastic contact collision behavior by using the existing continuous contact force models are discussed. Secondly, the static elastoplastic contact force models with the continuous transition between the pure elastic and full plastic are introduced, and its characteristic is illustrated when calculating the elastoplastic collision events. The coefficient of restitution acts as the bridge to connect the static elastoplastic contact model and dynamic dashpot model as a complete system. In order to sidestep the error from the Hertz contact stiffness in calculating the elastoplastic impact behavior, a new viscous damping factor is derived by means of the linear elastoplastic contact stiffness based on energy conservation. The intrinsic connection between the static elastoplastic model and the dashpot model is explored, which proves that the artificial damping describing energy dissipation is equivalent to the one generated by the discrepancy between the loading and unloading paths. In order to avoid the numerical singularity caused by the initial impact velocity in the denominator of damping when calculating the dynamic performance of granular matter, a continuous contact force model with viscous damping is obtained by solving a linear single degree of freedom underdamped vibration system. One-dimension chain is taken as the numerical example to validate that the new dashpot model is more accurate than the one used in the EDEM software. Finally, the current research status of impact dynamics of multibody systems is reviewed, and the development trend and future challenges of contact force models are briefly summarized.

2022, 54(12): 3239-3266. doi: 10.6052/0459-1879-22-266

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Editorial: Special topic on Meshless and Particle Methods

Zhang Xiong

2022, 54(12): 3267-3268. doi: 10.6052/0459-1879-22-557

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AN IMMERSED MULTI-MATERIAL FINITE VOLUME-MATERIAL POINT METOHD FOR STRUCTURAL DAMAGE UNDER BLAST LOADING

Ni Ruichen, Sun Zixian, Li Jiasheng, Zhang Xiong

Structural damage under blast loading always involves strong nonlinear shock-wave, extreme deformation, damage and breakage of solid structures, and strong fluid-solid interaction, which bring great difficulties to numerical simulation. In this paper, a novel immersed multi-material finite volume-material point method (iMMFV-MPM) is proposed to model the structural damage under blast loading. The multi-material finite volume method (MMFVM) is used to simulate the flow of explosives and surrounding air and specifically a TVD Riemann solver is adopted for shock simulation, while the material point method (MPM) is employed as solid solver for simulation of extreme deformation problem. The continuous-forcing immersed boundary method based on Lagrangian multiplier (lg-CFIBM) is extended to multi-material fluid to impose boundary conditions at the FSI interfaces. The lg-CFIBM can guarantee the boundary velocity conditions strictly at each time step and has no need to reconstruct FSI interfaces explicitly, which can effectively simulate the interaction between the explosion products and the building structure, the evolution of the shock wave around solid structure, and the dynamic fracture and topological change of the structure. Several numerical examples, including the damage pattern of a square reinforced concrete slab under close-in explosion, the structural damage of buildings under blast loading and the multi-chamber implosion tests, are simulated to verify and validate the proposed FSI algorithm, and numerical results are in good agreement with experiments.

2022, 54(12): 3269-3282. doi: 10.6052/0459-1879-22-446

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TOTAL LAGRANGIAN MATERIAL POINT METHOD FOR THE DYNAMIC ANALYSIS OF NEARLY INCOMPRESSIBLE SOFT MATERIALS

Zhang Zijian, Liu Zhenhai, Zhang Hongwu, Zheng Yonggang

The material point method (MPM) shows good performance in modeling nonlinear dynamic problems and has been widely used to simulate various types of large deformation dynamic problems. However, the classical MPM may suffer from the volumetric locking when modeling the dynamic responses of the incompressible or nearly incompressible materials, which reduces the computational accuracy and affects the convergence behavior greatly. In this work, a displacement-pressure mixed total Lagrangian material point method (TLMPM) with explicit time integration is proposed for the large deformation dynamic behavior of nearly incompressible soft materials. In this method, an equation about the hydrostatic pressure is introduced based on the volumetric part of the strain energy density of nearly incompressible soft materials. Then the introduced equation as well as the momentum equation is discretized within the framework of the explicit MPM and the total Lagrangian formulation is implemented to overcome the cell-crossing noise, which increases the computational accuracy for the problems involving large deformation. Furthermore, the B-spline interpolation functions with different orders are applied for displacement and pressure fields respectively and the mixed TLMPM is improved to increase the accuracy by introducing a remapping technique for the volumetric deformation. In addition, the staggered solving scheme is adopted and the displacement and the pressure are required to be solved sequentially in a single time step. Finally, several typical numerical examples are simulated by the mixed TLMPM and the convergence and accuracy are analyzed. The results demonstrate that the proposed mixed TLMPM is able to deal with the volumetric locking effectively and simulate the dynamic behavior of nearly incompressible soft materials involving large deformation accurately.

2022, 54(12): 3344-3351. doi: 10.6052/0459-1879-22-471

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RESEARCH ADVANCES ON THE COLLOCATION METHODS BASED ON THE PHYSICAL-INFORMED KERNEL FUNCTIONS

Fu Zhuojia, Li Mingjuan, Xi Qiang, Xu Wenzhi, Liu Qingguo

In the past few decades, although traditional computational methods such as finite element have been successfully used in many scientific and engineering fields, they still face several challenging problems such as expensive computational cost, low computational efficiency, and difficulty in mesh generation in the numerical simulation of wave propagation under infinite domain, large-scale-ratio structures, engineering inverse problems and moving boundary problems. This paper introduces a class of collocation discretization techniques based on physical-informed kernel function to efficiently solve the above-mentioned problems. The key issue in the physical-informed kernel function collocation methods is to construct the related basis functions, which includes the physical information of the considered differential governing equation. Based on these physical-informed kernel functions, these methods do not need/only need a few collocation nodes to discretize the considered differential governing equations, which may effectively improve the computational efficiency. In this paper, several typical physical-informed kernel functions that satisfy common-used homogeneous differential equations, such as the fundamental solutions, the harmonic functions, the radial Trefftz functions and the T-complete functions and so on, are firstly introduced. After that, the ways to construct the physical-informed kernel functions for nonhomogeneous differential equations, inhomogeneous differential equations, unsteady-state differential equations and implicit differential equations are introduced in turn. Then according to the characteristics of the considered problems, the global collocation scheme or the localized collocation scheme is selected to establish the corresponding physical-informed kernel function collocation method. Finally, four typical examples are given to verify the effectiveness of the physical-informed kernel function collocation methods proposed in this paper.

2022, 54(12): 3352-3365. doi: 10.6052/0459-1879-22-485

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THE OPTIMIZATION OF RIBS POSITION BASED ON STIFFENED PLATES MESHLESS MODEL WITH NONLINEARITY

Peng Linxin, Li Zhixian, Xiang Jiacheng, Qin Xia

In the stiffened plate's meshless model, the ribs' position is critical to the mechanical performance of the stiffened plate under various working conditions. Based on the first-order shear deformation theory and the moving-least square approximation, a meshless model of the stiffened non-rectangular plate considering nonlinearity is proposed and the position of the ribs is optimized based on the genetic algorithm. Firstly, the meshless model of the stiffened plate is obtained by discretizing the plate and ribs with discrete nodes. Secondly, the bending governing equation for the geometrically nonlinear problem of the stiffened non-rectangular plate is derived from the Von Karman large deflection theory. Then, the governing equation for the free vibration problem of the stiffened non-rectangular plate is derived from the Hamilton principle. Finally, the genetic algorithm is introduced with the position of the ribs as the design variable and the minimal deflection or the maximal natural frequency of the center point of the non-rectangular stiffened plate as the objective function to optimize the position of ribs. In the process of ribs' position optimization considering the influence of geometric nonlinearity, only the displacement transformation matrix needs to be recalculated when the ribs' position changed, and the mesh reconstruction is totally avoided.In this paper, first taking the single-rib rhombus plate under global load as an example, the comparison with the theoretical results is carried out and the validity of the method is verified. Then, taking the minimum center point deflection and the maximum natural frequency of the stiffened plate as the optimization objective, the stiffened plates with different shapes and different rib' arrangements under local load were optimized, and then the convergence and stability of the proposed method were studied.

2022, 54(12): 3366-3382. doi: 10.6052/0459-1879-22-433

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A CONSISTENT AND EFFICIENT METHOD FOR IMPOSING MESHFREE ESSENTIAL BOUNDARY CONDITIONS VIA HELLINGER-REISSNER VARIATIONAL PRINCIPLE

Wu Junchao, Wu Xinyu, Zhao Yaobing, Wang Dongdong

Galerkin meshfree methods with arbitrary order smooth shape functions exhibit superior accuracy advantages in structural analysis. However, the smooth meshfree shape functions generally do not have the interpolatory property and thus the enforcement of essential boundary conditions in Galerkin meshfree methods is not trivial. The variationally consistent Nitsche’s method shows very good performance regarding convergence and stability and is widely used to impose essential boundary conditions. In this work, a consistent and efficient method is proposed to impose meshfree essential boundary conditions. The proposed method is based upon the Hellinger-Reissner (HR) principle, where the displacements are represented by the conventional meshfree shape functions and the stresses are approximated by reproducing kernel smoothed gradients defined in each background integration cells. The resulting meshfree discrete equations share almost identical forms with those derived from Nitsche’s method. It is shown that the stabilized term in Nitsche’s method is a natural outcome from the HR variational principle, but there is absolutely no need to use any artificial parameter to maintain the coercivity of stiffness matrix. Moreover, under the reproducing kernel gradient smoothing framework, the costly derivatives of conventional meshfree shape functions are completely avoided and the integration constraint is automatically fulfilled. Numerical results demonstrate that the proposed approach and Nitsche’s method yield comparable solution accuracy, nonetheless, much higher efficiency is observed for the proposed methodology that imposes the essential boundary conditions for Galerkin meshfree formulation via the HR variational principle.

2022, 54(12): 3283-3296. doi: 10.6052/0459-1879-22-151

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IMPROVEMENT OF THE TOTAL LAGRANGIAN SPH AND ITS APPLICATION IN IMPACT PROBLEMS

Wang Lu, Xu Fei, Yang Yang

SPH (smoothed particle hydrodynamics) has its natural advantages in dealing with the large deformation of the material, fracture and crack propagation due to the absence of mesh distortion. However, the tensile instability which is an inherent defect encountered in the conventional SPH, is an obstacle for further applying SPH in computational solid mechanics. TL-SPH (total Lagrangian-SPH) is an effective measure to improve the tensile instability, but it still faces some defects. For example, the accuracy may be not enough at the boundary region for the truncated supported domain of the particle. The interface conditions are difficult to be implemented strictly, and the crack propagation cannot be presented under the Total Lagrangian frame. So, first of all, TL-SPH is coupled with the high-order SPH method, which can achieve second-order accuracy. Moreover, the high-order method is simplified by reducing the number of neighbor particles to save the calculational cost, and TL-SFPM (TL-simplified finite particle method) method is proposed with a reasonable neighbor particles selection mode. Secondly, TL-SPH method is combined with the DFPM (discontinuous finite particle method), which can improve the accuracy of the interface. A contact algorithm based on the Riemann solution is proposed by establishing the Riemann model between two particles with different materials. Then the fluid-solid contact algorithm and the solid-solid contact algorithm are introduced, respectively. Moreover, to capture the damage form of the solid under external load, a particle damage model based on the total Lagrangian frame is proposed. Finally, the rationality and accuracy of the proposed TL-SFPM method, the contact algorithm and the damage model are verified by cases of the fluid-solid impact and solid-solid impact, which further extends the application of TL-SPH method in the calculation of solid impact problems. The results of the dam break with an elastic baffle and the bullet impacting target plate also demonstrate the algorithms proposed in this paper has a wide application prospect for simulation of fluid-solid interaction and solid impact problems.

2022, 54(12): 3297-3309. doi: 10.6052/0459-1879-22-214

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PERIDYNAMIC THERMOMECHANICAL COUPLING MODEL WITH PHASE CHANGE AND SIMULATION OF FREEZING FAILURE OF POROUS MEDIA

Li Xing, Gu Xin, Xia Xiaozhou, Chen Aijiu, Zhang Qing

A wide range of natural and industrial processes involve the phenomenon of heat and mass transport in porous media. At low temperatures, the transported substance in porous media may undergo a phase change, which may induce material damage and even lead to structural failure. The prediction of this kind of failure phenomenon needs refined modeling to reflect the phase change process and the failure characteristics of materials. In the framework of peridynamic, the classical heat conduction equation is rewritten by using the enthalpy method, a thermal-mechanical coupling model considering the phase transition of substances is established, and the numerical calculation method of staggered solution is developed. The following problems are simulated with the established model, including the angular freezing of square plates, the thermally induced deformation of square plates, and the freezing failure of porous media. The phase transformation characteristics, temperature, deformation distribution of square plate freezing, and the freezing failure process of porous media are obtained by simulation, which are in good agreement with the results of experiments and other numerical methods. The research shows that the peridynamic thermomechanical coupling model established in this paper can reflect the nonlocal effect of materials and the influence of the latent heat of material phase change, accurately capture the evolution characteristics of the liquid-solid interface during the phase change process, and reproduce the process of material phase change, thermal deformation of matrix and freezing failure in porous media. This method breaks through the bottleneck of the traditional continuity model in solving this kind of failure problem and provides an effective way for in-depth research on the freezing and thawing failure process and failure mechanism of porous media.

2022, 54(12): 3310-3318. doi: 10.6052/0459-1879-22-521

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SIMULATION OF THE MOTION OF AN ELASTIC HULL IN REGULAR WAVES BASED ON MPS-FEM METHOD

Huang Congyi, Zhao Weiwen, Wan Decheng

A ship always encounters waves and may move with six degrees of freedom in the naval architecture and ocean engineering. The ship can be regarded as a rigid body simply when the motion amplitude is small. However, when the wave gets severe, the ship's motion amplitude get large and the ship hull may deforms a lot. In this situation, ship's elasticity may effects the pressure on the hull and the ship response motion, which cannot be ignored. Therefore, it is of great significance to simulate the motion of an elastic ship in waves and to study the influence of the hull elasticity, which can improve the ship performance and the navigation safety. Moving particle semi-implicit (MPS) method is a mesh free particle method based on Lagrangian representation. This method has its unique advantages in simulating problems with large deformation characteristics of free surfaces. As a traditional structural solution method, finite element method (FEM) has been widely used and has been proved with good stability, accuracy and robustness. In this paper, the advantages of MPS method and FEM method are combined and the in-house fluid-structure interaction solver MPSFEM-SJTU is used to simulate the motions of rigid and elastic hulls in regular waves. The impact of hull elasticity on the hull motion response and the pressure on the hull is analyzed. Firstly, the effect of regular wave length on the motion response of hull is studied by simulating the motion of a rigid hull in regular waves with different wavelengths. Then the motions of rigid and elastic hull in regular waves are simulated respectively. The results show that the motion amplitude of rigid hull, both pitch and heave, are greater than those of the elastic hull. and the pressure near the midship of elastic hull is greater than that of rigid hull. For the pressure distribution on elastic and hull surface, the pressure at the bottom near the midship is greater than that on the rigid hull due to the bending of the elastic ship.

2022, 54(12): 3319-3332. doi: 10.6052/0459-1879-22-468

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A MULTI-RESOLUTION PD-SPH COUPLING APPROACH FOR STRUCTURAL FAILURE UNDER FLUID-STRUCTURE INTERACTION

Yao Xuehao, Chen Ding, Wu Liwei, Huang Dan

Structural failure under fluid-structure interaction (FSI) is a type of strong nonlinear problem, which involves structural motion, deformation and failure as well as complex free-surface flows. Considering the respective advantages of peridynamics (PD) and smoothed particle hydrodynamics (SPH) as well as their computational efficiency, a multi-resolution PD-SPH coupling approach suitable for solving complicated FSI-concerned structural failure problems was proposed. The fluid and solid are discretized and solved by using SPH and PD approaches with different spatial and temporal resolutions, respectively. To achieve the precise satisfaction of interface boundary conditions, the fluid-structure interface is treated by using virtual particle technology, in which the same smoothing length of virtual particles as fluid particles is adopted. The modeling and analysis for two benchmark tests: large deformation of an elastic plate with hydrostatic pressure, and dam-break flow through an elastic gate, show that the presented multi-resolution PD-SPH coupling strategy and approach is suitable for simulating fluid-structure-interaction problems with satisfactory accuracy and efficiency. Further, the process of hydraulic fracture in Koyna gravity dam with an initial crack is simulated, and the cracking path in the simulation agrees well with available literature results, which indicates that the proposed coupling approach is appropriate for solving FSI-concerned structural failure problems. Finally, the proposed coupling strategy and numerical approach is employed to investigate the collapse process of a concrete slab due to fluid flow impacting, and the whole process of the concrete slab fracture as well as the motion of the fluid are captured with high accuracy. The results show that the proposed multi-resolution PD-SPH coupling approach may provide a potential alternative to simulate the process of structural failure under fluid-structure interaction.

2022, 54(12): 3333-3343. doi: 10.6052/0459-1879-22-268

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DYNAMIC FRACTURE ANALYSIS WITH THE FRAGILE POINTS METHOD

Shen Baoying, Wang Song, Li Mingjing, Dong Leiting

Impact resistant structures in engineering are likely to undergo dynamic fracture when they are subjected to impact or explosion. Restricting the dynamic fracture has been a key method to reinforce structures’ impact resistance. Thus, an accurate prediction on structures’ fracture behavior under dynamic loads is needed. Numerical simulation has been an important tool for the prediction of dynamic fracture. But, finite element method, commonly used in engineering practices, has some difficulties in fracture simulations, such as mesh distortion and inserting crack explicitly. The recently proposed fragile points method (FPM) is a discontinuous Galerkin meshless method which is suitable for fracture simulations. This paper aims on extending the FPM to analyze dynamic fracture problems. On the one hand, taken the weak form meshless methods as references, the FPM uses points and subdomains to discretize the problem domains. The shape function of an FPM subdomain is determined based on the point cloud in its supporting domain, and thus the FPM is not sensitive to mesh distortion. On the other hand, taken the discontinuous Galerkin finite element method as a reference, piece-wise continuous trial functions are used in the FPM, and the interior interface numerical flux correction is introduced in the weak formulations to guarantee the consistency and stabilization of the FPM. Thus, explicit cracks can be easily introduced in the FPM models. This paper starts with the introduction of the core idea and discretization method of FPM. Then the derivation of the equation of motion in weak form for the dynamic FPM is presented. After that the explicit dynamic solution scheme of the FPM is established. Finally some examples are employed to verify the dynamic FPM’s capability regarding the prediction of stress wave propagation and dynamic fracture.

2022, 54(12): 3383-3397. doi: 10.6052/0459-1879-22-498

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STABILITY ANALYSIS OF THERMOCAPILLARY LIQUID LAYERS WITH TWO FREE SURFACES FOR A BINGHAM FLUID

Wang Sheng, Hu Kaixin

Thermocapillary convection refers to the fluid motion driven by the temperature-induced surface tension gradient. It mainly exists in the microgravity environment such as space or small-scale flow dominated by surface tension. In many industrial fields, such as crystal growth, polymer processing, inkjet printing, and microfluidic, product quality is closely related to thermocapillary convection. 3D printing is an important technology in space manufacturing, which can support the long-term manned operation and maintenance of the space station in orbit and realize on-demand manufacturing. This paper takes the spatial 3D printing of polymer fluids as the application background, the stability of thermocapillary liquid layers with two free surfaces for a Bingham fluid is studied by using the linear stability analysis. The function relation between the critical Marangoni number (Ma_c) and Prandtl number (Pr) at different Bingham number (B) is obtained. The flow field and energy mechanism of the critical mode are analyzed. It is found that the critical modes include the streamwise wave and the oblique wave, which are related to B, Bi and the vertical temperature difference (Q) between two interfaces. The increase of B and Bi will enhance the stability. When Q = 0, there are two kinds temperature distribution, which are symmetric and antisymmetric. When Q > 0, the increase of Pr will destabilize the flow. The perturbation temperature is distributed in the whole flow field at small Pr, and the perturbation temperature is zero in the plug region at large Pr. The energy analysis shows that the main energy source of perturbation energy is the work done by surface tension,but for small Pr, the basic flow also makes some contributions.

2022, 54(12): 3398-3407. doi: 10.6052/0459-1879-22-364

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INVESTIGATION ON FLUID DYNAMICS IN A CAPILLARY TUBE UNDER MICROGRAVITY BASED ON THE MAGNETIC COMPENSATION EXPERIMENT

Jin Yupeng, Xiao Mingkun, Qiu Yi’nan, Wang Tianxiang, Yang Guang, Huang Yonghua, Wu Jingyi

Due to the dominance of capillary force, the flow characteristics of fluid in microgravity environment are essentially different from those in normal gravity environment. Based on the principle of magnetic compensation, an experiment platform simulating the flow under microgravity with high tunability is established on the ground. The accuracy of the experimental system is verified by comparing the experimental data with the theoretical models, and the dynamic flow behavior of water-based magnetic fluid in vertical capillary tube under different equivalent gravity levels is studied. By comparing the experimental data with the different theoretical model solutions, the feasibility of using the magnetic compensation method to carry out the investigation on microgravity flow is verified. The average deviation between the experimental results obtained by the magnetic compensation method and the two theoretical model solutions using different dynamic contact angle models is 7.1% and 13.7% respectively. Furthermore, the influence of factors such as pipe diameter, equivalent gravity level and dynamic advancing contact angle on the dynamic flow characteristics in the capillary tube has been quantitatively studied. In a near zero-gravity environment, the flow development process can be divided into three stages where the liquid level h has a linear relationship with $ {t}^{2} $, t, $\sqrt t $successively. The pipe diameter has a complicated affect on the capillary climbing process. The influence of pipe diameter on flow does not change linearly with pipe diameter and its influence on the flow velocity is different among flow stages. For the capillary flow in the vertical direction, the greater the equivalent gravitational acceleration, the worse the capillary climbing ability of the magnetic fluid in the tube, and the more difficult it is to observe the existence of the first capillary climbing stage. Under the same conditions, the larger the dynamic advancing contact angle of the fluid, the smaller the capillary climbing velocity.

2022, 54(12): 3408-3417. doi: 10.6052/0459-1879-22-346

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INVESTIGATION ON MULTIGRID FEATURES OF THE FINITE VOLUME METHOD WITH WALSH BASIS FUNCTION

Wang Gang, Gan Yuan, Ren Jiong

The finite volume method with Walsh basis functions (FVM-WBF) is a novel numerical method with the ability to capture discontinuity inside grid. The numerical resolution of the FVM-WBF method can be effectively improved by increasing the number of basis functions, but the explosive growth of computation and the decrease of convergence speed will also appear simultaneously. To relieve the computational costs due to the increasing of the number of basis functions, the scales of the piecewise continuous mean value subdomains inside the grid cell, which are dominated by different levels of Walsh basis functions and their coefficients, have been analyzed. It is found that FVM-WBF method implicitly has scale characteristics similar to multigrid. Based on this finding, an FVM-WBF method combined with multigrid strategy is presented. In time integration stage of steady flow simulation, this newly developed FVM-WBF method defines the maximal time step for each level of Walsh basis function according to their influence scales and the corresponding numerical scheme stability constraint. As a result, the numerical error of different wavelengths in the process of time advancing is quickly eliminated and the convergence can be accelerated. Several test cases are selected to evaluate the multigrid features of the presented FVM-WBF method, including the low speed inviscid flow over two-dimensional cylinder and a set of inviscid steady flow with different Mach number around NACA0012 airfoil. The numerical results confirm that the newly developed FVM-WBF method has the key characteristics of multigrid, and convergence rate can be greatly enhanced only by adjusting the time step without any additional processing and computational costs.

2022, 54(12): 3418-3429. doi: 10.6052/0459-1879-22-281

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FRACTURE BEHAVIOR OF PERIODIC POROUS STRUCTURES BY PHASE FIELD METHOD

Ying Yuxuan, Huang Wei, Ma Yu-E, Peng Fan

Periodic porous structures have excellent characteristics such as low mass, low density, high specific strength, sound insulation, and they are also well satisfy the needs for structural-functional integration, which have a wide range of applications in many fields. At present, the mechanical response and fracture behavior of periodic porous structures under complex loads have been poorly investigated. In this paper, we use a combination of micro-mechanics method and phase field method to investigate the crack initiation location, crack propagation path, fracture mode and the ultimate strength of periodic porous structures under combined multiaxial loading based on a two-dimensional representative volume element (RVE) model with the periodic boundary condition (PBC) that can implement multiaxial proportional loading. Numerical simulation results in this paper show that all cracks in the periodic porous structure established in this paper initiate from the edge of the holes and propagate consequently along the horizontal direction under the uniaxial tensile loading in the vertical direction. Secondly, under the biaxial loadings in both vertical and horizontal directions, the ultimate strength of the periodic porous structure gradually increases with the increase of the horizontal tensile loading. When the horizontal load is equal to the vertical load, the fracture pattern exhibits as orthogonal cross-type cracking and the ultimate strength reaches the maximum value. Thirdly, the in-plane shear stress simultaneously acted on the RVE model of the periodic porous structure results in a significant decrease of the ultimate strength and the variations of initiation location and propagation trajectory of the hole-edge cracks. Hence, the fracture pattern of periodic porous structure subjected to combined multiaxial loadings changes from the single S-type cracking to the double arc-type cracking, and cracks extend toward adjacent holes in horizontal direction. Finally, with the increase of the horizontal tensile loading, cracks initiate diagonally at the edge of the holes and propagate along the 45-degree direction which lead to the oblique cracking of periodic porous structure.

2022, 54(12): 3430-3443. doi: 10.6052/0459-1879-22-411

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COMPLETE CONSTITUTIVE RELATION OF HYPERELASTIC MATERIALS FOR TRELOAR’S EXPERIMENTAL DATA

Han Lei, Wang Xintong, Li Luxian

Hyperelastic material is a typical one widely used in many fields such as aerospace engineering and civil industrials. However, due to the property of nonlinear large deformation, the constitutive behavior of hyperelastic materials is extremely complex and the models are quite different in form. Starting from the strain energy function, a complete constitutive relation of hyperelastic materials is studied within the theoretical framework of continuum mechanics in this paper. Firstly, the feature is analyzed for the experimental curves under three essential deformation modes like uniaxial tension, equibiaxial tension and pure shear, which are conducted by Treloar for a vulcanized rubber hyperelastic material. Next, the same stress conditions of the three deformation modes are summarized in detail, based on which the constitutive relationship is derived in a same manner in terms of the stress and the principal stretch ratio in the loading direction for the three modes according to the hyperelastic constitutive theory. The constitutive behaviors of two typical power-law strain energy functions, namely $I_1^m$ and $I_2^m$, are accordingly studied for the three essential modes. The experimental curves are divided into the initial regime and the remaining regime, and then the neo-Hookean model is adopted for the initial regime while the power-law functions with variable exponents are used for the remaining regime. The complete constitutive model is eventually established after the model parameters are identified by minimizing the overall error functional of the three modes. The responses are re-predicted for the three essential deformation modes, and the results agree better with the experimental than other models available in published literature. The present work indicates that a complete constitutive relation can be obtained for a hyperelastic material in light of the experimental curves with whole deformation range under multiple deformation modes, which is therefore instructive and meaningful to theoretical research and engineering application of complex practical problems such as fracture of hyperelastic materials.

2022, 54(12): 3444-3455. doi: 10.6052/0459-1879-22-317

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A NEW STOCHASTIC DAMAGE CONSTITUTIVE MODEL OF CONCRETE CONSIDERING STRAIN RATE EFFECT

Guo Chenggong, Li Jie

Due to the complex and randomly distributed components of concrete materials, the mechanical behavior of concrete materials inevitably exhibits nonlinearity and randomness. In addition, the mechanical properties of concrete materials are sensitive to strain rates. This work established the nano-micro-meso stochastic damage model to comprehensively reflect the three basic properties of nonlinearity, randomness, and strain rate sensitivity in the mechanical behavior of concrete. By introducing the rate process theory to describe the growth rate of nano-cracks, the related energy dissipation process can be obtained. The nanoscale analysis is upscaled to the micro scale by a crack hierarchy model, and the expression for micro energy dissipation is derived. The nano-micro-meso stochastic damage constitutive model for concrete was established by combining the micro energy dissipation expression with the micro-meso stochastic fracture model. In the meantime, a time-dependent energy barrier is used for analyzing the bond surviving probability under different loading rates. Assuming the evolution of reaction coordinate is governed by the Langevin equation, the surviving probability can be obtained by solving the corresponding Fokker-Planck-Kolmogorov equation. The results reveal that the increase of dynamic strength resulted from the competitive mechanism of loading rate and bond breaking rate. Since the first eigenvalue of the Fokker-Planck-Kolmogorov equation corresponds to the rate process theory, it is not applicable when the loading rate is too high. According to the analysis aforementioned, it is assumed that the interaction between micro-cracks is linearly related to the corresponding logarithmic strain rate, so the energy dissipation rate of the micro-spring is related to the strain rate, extending from the static constitutive model to the dynamic constitutive model. Numerical examples show that the proposed model can simultaneously reflect the nonlinearity, randomness, and strain rate sensitivity in the mechanical behavior of concrete materials. The correctness of the proposed model is verified by comparison with the relevant experimental results.

2022, 54(12): 3456-3467. doi: 10.6052/0459-1879-22-306

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SEMI-ANALYTICAL TRANSIENT SOLUTIONS FOR STRONG NONLINEAR SYSTEMS EXCITED BY POISSON WHITE NOISE

Ye Wenwei, Chen Lincong, Sun Jian-Qiao

Random perturbations are common in nature and engineering, and most of them exhibit inherent non-Gaussian properties. Thus, it may lead to huge errors if Gaussian excitation is used for modeling. As a typical and important non-Gaussian excitation model, Poisson white noise has attracted extensive attention. At present, the dynamic characteristic analysis of the system subjected to Poisson white noise is mainly focused on the study of the stationary response, while the solution of the transient response is still difficult and needs further development. In this paper, an efficient semi-analytical method based on radial basis function neural networks (RBF-NN) are proposed for transient response prediction of single-degree-of-freedom strong nonlinear systems under Poisson white noise excitation. Firstly, the transient solution of the generalized Fokker-Plank-Kolmogorov (FPK) equation is expressed as a set of Gaussian RBF-NN with unknown time-varying weight coefficients. Then, the finite difference method is applied to discretize and approximate the time derivative term, and the loss function with time recurrence is constructed by the random sampling technique. Finally, the time-varying optimal weight coefficients can be determined by minimizing the loss function through the Lagrange multiplier method. As examples, two classical strong nonlinear systems are investigated, and the solutions are validated by the Monte Carlo simulation (MCS) method. The results show that the transient probability density functions (PDFs) obtained by the proposed scheme agree well with the MCS data, and the algorithm has high computational efficiency. In the whole evolution process of the system response, the proposed scheme can effectively capture the complex nonlinear characteristics of the system response at each moment. Furthermore, the high precision semi-analytical transient solution obtained by the proposed scheme can not only be used as a benchmark to test the accuracy of other nonlinear random vibration analysis methods, but also has great potential application value for the structural optimum design.

2022, 54(12): 3468-3476. doi: 10.6052/0459-1879-22-381

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STABILIZING UNSTABLE PERIODIC TRAJECTORIES OF CHAOTIC SYSTEMS WITH TIME-VARYING SWITCHING DELAYED FEEDBACK CONTROL

Zeng Jianjian, Zheng Yuanguang

In order to improve the effect of the classical delayed feedback control in stabilizing the unstable periodic trajectory and expand the stability region, the time-varying switching strategy is used to modify the classical delayed feedback control, which leads to the method of time-varying switching delayed feedback control. The control signal of the time-varying switching delayed feedback control only exists in specific time intervals, and there is no control signal in other time intervals, which is different from the fixed control signal in the classical delayed feedback control. Through case studies, the specific performance of time-varying switching delayed feedback control in stabilizing unstable periodic trajectory is investigated. The maximum conditional Lyapunov exponent of the controlled periodic trajectory is calculated as a function of the feedback strength. The relationship between the stability region of the controlled periodic trajectory and the switching frequency is obtained. The results show that with the increase of switching frequency, the stable region of the controlled periodic trajectory changes non smoothly. The stability region of the time-varying switching delayed feedback control is significantly larger than that of the classical delayed feedback control when the switching frequency is properly selected. In the engineering practice of chaos control, the control signal is often constrained. To achieve the stable control of the target periodic trajectory, the controlled periodic trajectory needs to have a large enough stable region. Therefore, compared with the classical time-delay feedback control, the time-varying switching time-delay feedback control proposed in this paper has a wider application prospect.

2022, 54(12): 3477-3485. doi: 10.6052/0459-1879-22-361

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RIGID-DISCRETE COUPLING DYNAMIC ANALYSIS OF ROBOT MONO-PEDAL SYSTEM JUMPING IN SAND

Sun Hao, Liu Zhuyong, Liu Jinyang

In the process of planetary exploration, it involves the landing and movement of the probe on the earth, as well as the collection, storage and return of some sample materials. Therefore, it is necessary to establish a dynamic model of the motion of the probe robot on the sand, so as to optimize the system configuration. In recent years, the studies on jumping detection machinery have received considerable attentions. In this paper, the discrete element method is used to simulate the particle field deformation. The multibody dynamics method is used to model the mechanical system. Then the coupling dynamics simulation and analysis are carried out for the jumping problem of the robot single foot system on the sand. Based on Prandtl-Reissne theory of classical soil mechanics, starting from the form of pressure stratification and momentum transfer of particle field, a modified Poncelet formula is proposed while the inertia force dynamic resistance term describing particle intrusion resistance is modified. The modified formula adds supplementary items related to rigid body acceleration and intrusion depth, and no new fitting coefficient is added compared with the original formula. By comparing with the results of discrete element simulation, it is shown that the proposed modified Poncelet formula can more accurately calculate the sand and soil invasion resistance of the mechanical foot than the original Poncelet formula. Especially, it shows better convergence when reaching a certain invasion depth. Finally, the influence of different size and shape of the mechanical leg's foot on the jumping effect in sand is analyzed, and the approximate calculation formula of the volume of the conical foot and the cylindrical foot on the jumping effect is presented. The simulation results show that the conical sole will replace the volume of the consolidation zone of the particle field. Furthermore, the influence of particles in the consolidation zone of robot foot on the invasion resistance is discussed. This study will expand the rigid-discrete coupling dynamics theory, and provide technical support for the system design of the new type probe moving on the planetary soil.

2022, 54(12): 3486-3495. doi: 10.6052/0459-1879-22-405

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HUMAN-MACHINE COUPLING DYNAMICS AND ASSISTANCE PERFORMANCE ANALYSIS OF AN ANKLE EXOSKELETON

Gao Yuqing, Jin Wei, Xu Jian, Fang Hongbin

The ankle joint provides the largest joint torque during human lower limb motions. Therefore, ankle exoskeletons have received major attention in the research of lower limb augmented exoskeletons. Walking of a human equipped with an exoskeleton is a typical dynamics problem, while the research on human-exoskeleton coupling dynamics is still at an early stage. Concentrated on the cable-driven ankle exoskeleton, this paper developed a human-machine coupled dynamic model considering foot-ground interaction forces, human joint torques, and exoskeleton torques, by integrating the robot forward kinematics method and the Lagrange's equation, where the foot-ground interaction force was described by the Kelvin-Voigt model together with the Coulomb’s dry friction model, the human joint torque was generated by the PD control with the particle swarm optimization, and the assistive exoskeleton torque was determined by an upper-level controller in accordance with the human gait cycle. Through model-based dynamic simulations, this paper systematically analyzed the effect of the ankle exoskeleton assistance on human walking from the perspectives of the angle, torque, power, and work of the human ankle. It was demonstrated that when walking at a speed between 2.0 km/h and 6.5 km/h, human wearing the exoskeleton can achieve at least a 24.84% reduction in average ankle torque and at least a 24.69% reduction in ankle work. Musculoskeletal modeling and predictive simulations based on the SCONE were also performed in this paper. The simulation results showed that at a speed of 3.6km/h, wearing the exoskeleton can effectively reduce the peak level of soleus activation and the RMS value of the EMG signal by 6.21%, thereby validating the effect of the ankle exoskeleton assistance from a physiological perspective. Based on the results of this paper, the dynamic modeling and analysis method of human-exoskeleton coupled systems is further improved. The assistance mechanism of the ankle exoskeleton for walking is confirmed and interpreted from the perspectives of dynamics and physiology. This research also provides a theoretical basis for future experimental studies of lower-limb exoskeletons.

2022, 54(12): 3496-3512. doi: 10.6052/0459-1879-22-472

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A FULLY IMPLICIT AND MONOLITHIC PARALLEL DECOMPOSITION METHOD FOR 3D FLUID-SOLID INTERACTION PROBLEMS

Deng Xiaomao, Liao Ziju

Numerical methods based on unstructured meshes for the three-dimensional fluid-solid interaction problems have many applications in science and engineering. Most of the existing algorithms are based on the partitioned approach that the equations for the fluid and solid are solved separately using existing solvers by enabling them to share interface data with one another. The convergence of the partitioned approach is sometimes difficult to achieve because the method is basically a Gauss-Seidel type process and it may encounter the instability problem of the so-called added mass effect. Moreover, the parallel scalability of the solution algorithm is also an important issue when solving the large-scale problem. In contrast, the monolithic approach shows a more robust convergence and also eliminates the added mass effect even for complicated problems. In this work, a fully implicit and monolithic scalable parallel algorithm based on domain decomposition method is developed for the three-dimensional unsteady fluid-solid interaction problem. The governing equations are established based on the arbitrary Lagrangian-Eulerian framework, and a stabilized unstructured finite element method is employed for the discretization in space and a second-order fully implicit backward differentiation formula in time. An inexact Newton-Krylov method together with a restricted additive Schwarz preconditioner is constructed to solve the large, sparse system of nonlinear algebraic equations resulted from the discretization. The accuracy of the numerical method is verified by a benchmark problem of flows around an elastic obstacle. The numerical performance tests show that the fully implicit and monolithic method has good stability with large time step sizes and good robustness under different physical parameters, and a parallel efficiency of 91% was achieved for 3072 processor cores on the “Tianhe 2” supercomputer. The experimental results show that the proposed numerical method is expected to be applied for the numerical simulation of large-scale fluid-structure interaction problems in complex regions.

2022, 54(12): 3513-3523. doi: 10.6052/0459-1879-22-398

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A HYBRID TOPOLOGY OPTIMIZATION METHOD OF SIMP AND MMC CONSIDERING PRECISE CONTROL OF MINIMUM SIZE

Lian Ruichao, Jing Shikai, Li Ying, Xiao Dengbao, Chen Yang

Topology optimization is an advanced design method, which has been successfully used to solve multidisciplinary optimization problems, but there are still many obstacles to the reliable use of topology optimization results in engineering manufacturing, such as tiny holes or boundary cracks and hinges in structural design. An effective means to solve the above problems is to consider the minimum size control of the structure in the topology optimization design stage. In the topology optimization method considering the minimum size control, the boundaries of the widely used solid isotropic material with penalization (SIMP) method optimization result are usually blurred and not smooth, and moving morphable component (MMC), which contains precise geometric information, has a strong dependence on the initial layout of components. This paper proposes a hybrid topology optimization method of the SIMP and MMC considering precise control of minimum size. The proposed method inherits the advantages of both and avoids their respective disadvantages. In this method, a mapping method from the SIMP optimization results to the initial layout of MMC components is firstly proposed, which uses the active contour without edges (ACWE) algorithm to obtain the topological boundary contour data of the SIMP and the geometric parameter matrix of the components. Secondly, the topological description function model of the multi-deformable component with the semicircular end is established by introducing three length variables of the component. Finally, a topology optimization model that considers the minimum size control of the structure is constructed with the component thickness variable as the constraint. The effectiveness of the proposed method is verified by the minimum compliance problem and the compliance mechanism problem. The numerical results show that, the proposed method can achieve precise control of the minimum size of the overall structure and obtain a globally smooth topological structure boundary only by setting the lower limit of the component thickness variable without additional constraints.

2022, 54(12): 3524-3537. doi: 10.6052/0459-1879-22-283

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