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2022年 第43卷 第4期    刊出日期：2022-04-01

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论文

Thickness dependence of viscoelastic stress relaxation of laminated microbeams due to mismatch strain

Xiaosheng CAI, Nenghui ZHANG, Hanlin LIU

2022, 43(4):  467-478.  doi:10.1007/s10483-022-2841-5

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Time-dependent behaviors due to various mismatch strains are very important to the reliability of micro-/nano-devices. This paper aims at presenting an analytical model to study the viscoelastic stress relaxation of the laminated microbeam caused by mismatch strain. Firstly, Zhang’s two-variable method is used to establish a mechanical model for predicting the quasi-static stress relaxation of the laminated microbeam. Secondly, the related analytical solutions are obtained by combining the differential method and the eigenvalue method in the temporal domain. Finally, the influence of the substrateto-film thickness/modulus ratio on the relaxation responses of the laminated microbeam subject to a step load of the mismatch strain is studied. The results show that the present predictions are consistent with the previous theoretical studies. Furthermore, the thickness dependence of stress relaxation time of the laminated microbeam is jointly determined by the intrinsic structural evolution factors and tension-bending coupling state; the stress relaxation time can be controlled by adjusting the substrate-to-film thickness/modulus ratio.



Natural dynamic characteristics of a circular cylindrical Timoshenko tube made of three-directional functionally graded material

Ye TANG, Jiye XU, Tianzhi YANG

2022, 43(4):  479-496.  doi:10.1007/s10483-022-2839-6

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The natural dynamic characteristics of a circular cylindrical tube made of three-directional (3D) functional graded material (FGM) based on the Timoshenko beam theory are investigated. Hamilton’s principle is utilized to derive the novel motion equations of the tube, considering the interactions among the longitudinal, transverse, and rotation deformations. By dint of the differential quadrature method (DQM), the governing equations are discretized to conduct the analysis of natural dynamic characteristics. The Ritz method, in conjunction with the finite element method (FEM), is introduced to verify the present results. It is found that the asymmetric modes in the tube are controlled by the 3D FGM, which exhibit more complicated shapes compared with the unidirectional (1D) and bi-directional (2D) FGM cases. Numerical examples illustrate the effects of the axial, radial, and circumferential FGM indexes as well as the supported edges on the natural dynamic characteristics in detail. It is notable that the obtained results are beneficial for accurate design of smart structures composed from multi-directional FGM.

Many-body dissipative particle dynamics with energy conservation: temperature-dependent long-term attractive interaction

Jie LI, Kaixuan ZHANG, Chensen LIN, Lanlan XIAO, Yang LIU, Shuo CHEN

2022, 43(4):  497-506.  doi:10.1007/s10483-022-2840-7

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Heat and mass transfer during the process of liquid droplet dynamic behaviors has attracted much attention in decades. At mesoscopic scale, numerical simulations of liquid droplets motion, such as impacting, sliding, and coalescence, have been widely studied by using the particle-based method named many-body dissipative particle dynamics (MDPD). However, the detailed information on heat transfer needs further description. This paper develops a modified MDPD with energy conservation (MDPDE) by introducing a temperature-dependent long-term attractive interaction. By fitting or deriving the expressions of the strength of the attractive force, the exponent of the weight function in the dissipative force, and the mesoscopic heat friction coe–cient about temperature, we calculate the viscosity, self-diffusivity, thermal conductivity, and surface tension, and obtain the Schmidt number Sc, the Prandtl number Pr, and the Ohnesorge number Oh for 273 K to 373 K. The simulation data of MDPDE coincide well with the experimental data of water, indicating that our model can be used to simulate the dynamic behaviors of liquid water. Furthermore, we compare the equilibrium contact angle of droplets wetting on solid surfaces with that calculated from three interfacial tensions by MDPDE simulations. The coincident results not only stand for the validation of Young’s equation at mesoscale, but manifest the reliability of our MDPDE model and applicability to the cases with free surfaces. Our model can be extended to study the multiphase flow with complex heat and mass transfer.

A modified fractional-order thermo-viscoelastic model and its application to a polymer micro-rod heated by a moving heat source

Wei PENG, Like CHEN, Tianhu HE

2022, 43(4):  507-522.  doi:10.1007/s10483-022-2835-9

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Classical thermo-viscoelastic models may be challenged to predict the precise thermo-mechanical behavior of viscoelastic materials without considering the memory-dependent effect. Meanwhile, with the miniaturization of devices, the size-dependent effect on elastic deformation is becoming more and more important. To capture the memory-dependent effect and the size-dependent effect, the present study aims at developing a modified fractional-order thermo-viscoelastic coupling model at the microscale to account for two fundamentally distinct fractional-order models which govern the memory-dependent features of thermal conduction and stress-strain relation, respectively. Then, the modified theory is used to study the dynamic response of a polymer micro-rod heated by a moving heat source. The governing equations are obtained and solved by the Laplace transform method. In calculation, the effects of the fractional-order parameter, the fractional-order strain parameter, the mechanical relaxation parameter, and the nonlocal parameter on the variations of the considered variables are analyzed and discussed in detail.

Static and free vibration analysis of four-parameter continuous grading elliptical sandwich plates

M. H. YAS, F. AKHLAGHI, S. KAMARIAN, A. H. YAS

2022, 43(4):  523-536.  doi:10.1007/s10483-022-2834-7

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In the present study, the static and dynamic analyses of elliptical functionally graded sandwich (FGS) plates are investigated. The constituent materials of the sandwich plates are ceramic and metal so that the core is made of pure metal, while the face sheets consist of a combination of metal and ceramic according to a four-parameter power-law distribution. Different material profiles such as classic, symmetric, and asymmetric can be obtained using the applied generalized power-law distribution relation. The analysis is performed based on the classical laminated plate theory (CLPT) and the Ritz method. The effects of four parameters in the material distribution relation as well as different geometric parameters on the deflection and natural frequencies of elliptical FGS plates are studied. The results of this study show that with a proper distribution of materials, the optimal static and dynamic behavior can be achieved. The results also indicate that the generalized power-law distribution has significant effects on the natural frequencies of elliptical FGS plates. For example, although the frequency parameter of a plate with ceramic face sheets is more than the one with metal face sheets, the use of larger amounts of ceramic does not necessarily increase the natural frequency of the structure.

On Chien’s question to the Hu-Washizu three-field functional and variational principle

Bohua SUN

2022, 43(4):  537-546.  doi:10.1007/s10483-022-2838-5

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There is an open question, namely Chien’s question, in construction of a generalized functional in elasticity, i.e., why the stress-strain relation can still be derived from the Hu-Washizu generalized variational principle while the Lagrangian multiplier method is applied in vain? This study shows that the generalized variational principle can only be understood and implemented correctly within the framework of thermodynamics. This investigation finds that as long as the functional has one of the combinations (A(εij)-σ_ijε_ij) or (B(σ_ij)-σ_ijε_ij), its corresponding variational principle will produce the stress-strain relation without the need to introduce extra constraints by the Lagrangian multiplier method. This research proves that the Hu-Washizu functional Π_HW(u_ij, ε_ij; σ_ij) is real three-field functional, and resolves the historic academic controversy on the issue of constructing a three-field functional.

Slip effects on unsteady mixed convection of hybrid nanofluid flow near the stagnation point

N. A. ZAINAL, R. NAZAR, K. NAGANTHRAN, I. POP

2022, 43(4):  547-556.  doi:10.1007/s10483-022-2823-6

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The unsteady mixed convection of the Al₂O₃-Cu/H₂O hybrid nanofluid flow near the stagnation point past a vertical plate is analyzed. The bvp4c technique is used to solve the resulting ordinary differential equations. The combined effects of the velocity and thermal slip are addressed. The effects of different relevant physical parameters are studied numerically. The results show that the heat transfer rate is reduced when the volume fraction of the nanoparticles increases, while the unsteadiness parameter has an opposite effect in the opposing flow. The presence of the slip parameter is proven to increase the skin friction coe–cient while reduce the local Nusselt number in the buoyancy opposing flow. A contradictory result is observed in the buoyancy assisting flow. Meanwhile, the heat transfer rate is reduced in the buoyancy of the assisting and opposing flows when the thermal slip effect is considered.

High-precision stress determination in photoelasticity

Zikang XU, Yongsheng HAN, Hongliang SHAO, Zhilong SU, Ge HE, Dongsheng ZHANG

2022, 43(4):  557-570.  doi:10.1007/s10483-022-2830-9

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Stress separation is usually achieved by solving differential equations of equilibrium after parameter determination from isochromatics and isoclinics. The numerical error resulting from the stress determination is a main concern as it is always a function of parameters in discretization. To improve the accuracy of stress calculation, a novel meshless barycentric rational interpolation collocation method (BRICM) is proposed. The derivatives of the shear stress on the calculation path are determined by using the differential matrix which converts the differential form of the equations of equilibrium into a series of algebraic equations. The advantage of the proposed method is that the auxiliary lines, grids, and error accumulation which are commonly used in traditional shear difference methods (SDMs) are not required. Simulation and experimental results indicate that the proposed meshless method is able to provide high computational accuracy in the full-field stress determination.

Research on data assimilation strategy of turbulent separated flow over airfoil

Ying ZHANG, Lin DU, Weiwei ZHANG, Zichen DENG

2022, 43(4):  571-586.  doi:10.1007/s10483-022-2827-7

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In order to increase the accuracy of turbulence field reconstruction, this paper combines experimental observation and numerical simulation to develop and establish a data assimilation framework, and apply it to the study of S809 low-speed and high-angle airfoil flow. The method is based on the ensemble transform Kalman filter (ETKF) algorithm, which improves the disturbance strategy of the ensemble members and enhances the richness of the initial members by screening high flow field sensitivity constants, increasing the constant disturbance dimensions and designing a fine disturbance interval. The results show that the pressure distribution on the airfoil surface after assimilation is closer to the experimental value than that of the standard Spalart-Allmaras (S-A) model. The separated vortex estimated by filtering is fuller, and the eddy viscosity field information is more abundant, which is physically consistent with the observation information. Therefore, the data assimilation method based on the improved ensemble strategy can more accurately and effectively describe complex turbulence phenomena.

Solutions for a class of Hamiltonian systems on time scales with non-local boundary conditions

Yongfang WEI, Suiming SHANG, Zhanbing BAI

2022, 43(4):  587-602.  doi:10.1007/s10483-022-2832-9

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In this work, the solvability of a class of second-order Hamiltonian systems on time scales is generalized to non-local boundary conditions. The measurements obtained by non-local conditions are more accurate than those given by local conditions in some problems. Compared with the known results, this work establishes the variational structure in an appropriate Sobolev’s space. Then, by applying the mountain pass theorem and symmetric mountain pass theorem, the existence and multiplicity of the solutions are obtained. Finally, some examples with numerical simulation results are given to illustrate the correctness of the results obtained.

Condensed Galerkin element of degree m for first-order initial-value problem with O(h^2m+2) super-convergent nodal solutions

Si YUAN, Quan YUAN

2022, 43(4):  603-614.  doi:10.1007/s10483-022-2831-6

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A new type of Galerkin finite element for first-order initial-value problems (IVPs) is proposed. Both the trial and test functions employ the same m-degreed polynomials. The adjoint equation is used to eliminate one degree of freedom (DOF) from the test function, and then the so-called condensed test function and its consequent condensed Galerkin element are constructed. It is mathematically proved and numerically verified that the condensed element produces the super-convergent nodal solutions of O(h2m+2), which is equivalent to the order of accuracy by the conventional element of degree m+1. Some related properties are addressed, and typical numerical examples of both linear and nonlinear IVPs of both a single equation and a system of equations are presented to show the validity and effectiveness of the proposed element.