The paper investigates continuously changing wrinkle patterns of thin films bonded to a gradient substrate. Three types of gradient substrates including exponential, power-law, and symmetry models are considered. The Galerkin method is used to discretize the governing equation of film bonded to gradient substrates. The wavelength and the normalized amplitude of the wrinkles for substrates of various material gradients are obtained. The numerical simulation based on the finite element method (FEM) is used to evolve the wrinkle patterns. The result agrees well with that of the analytical model. It is concluded that localization of wrinkle patterns strongly depends on the material gradient. The critical membrane force depends on both the minimum value of wrinkle stiffness and the gradient of wrinkle stiffness when the wrinkle stiffness is at its minimum. This work provides a better understanding for local wrinkle formation caused by gradient substrates.
The vibration of a longitudinally moving rectangular plate submersed in an infinite liquid domain is studied analytically with the Rayleigh-Ritz method. The liquid is assumed to be incompressible, inviscid, and irrotational, and the velocity potential is used to describe the fluid velocity in the whole liquid field. The classical thin plate theory is used to derive mechanical energies of the traveling plate. As derivative of transverse displacement with respect to time in the compatibility condition equation exists, an exponential function is introduced to depict the dynamic deformation of the moving plate. It is shown that this exponential function works well with the RayleighRitz method. A convergence study shows a quick convergence speed for the immersed moving plate. Furthermore, the parametric study is carried out to demonstrate the effect of system parameters including the moving speed, the plate location, the liquid depth, the plate-liquid ratio, and the boundary condition. Results show that the above system parameters have significant influence on the vibration characteristics of the immersed moving plate. To extend the study, the method of added virtual mass incremental (AVMI) factor is used. The results show good agreement with those from the Rayleigh-Ritz method.
In this paper, Donnell's shell theory and smeared stiffeners technique are improved to analyze the postbuckling and buckling behaviors of circular cylindrical shells of stiffened thin functionally graded material (FGM) sandwich under an axial loading on elastic foundations, and the shells are considered in a thermal environment. The shells are stiffened by FGM rings and stringers. A general sigmoid law and a general power law are proposed. Thermal elements of the shells and reinforcement stiffeners are considered. Explicit expressions to find critical loads and postbuckling load-deflection curves are obtained by applying the Galerkin method and choosing the three-term approximate solution of deflection. Numerical results show various effects of temperature, elastic foundation, stiffeners, material and geometrical properties, and the ratio between face sheet thickness and total thickness on the nonlinear behavior of shells.
A fast algorithm is proposed to predict penetration trajectory in simulation of normal and oblique penetration of a rigid steel projectile into a limestone target. The algorithm is designed based on the idea of isolation between the projectile and the target. Corresponding factors of influence are considered, including analytical load model, cratering effect, free surface effect, and separation-reattachment phenomenon. Besides, a method of cavity ring is used to study the process of cavity expansion. Further, description of the projectile's three-dimensional gesture is presented. As a result, the algorithm is coded for fast calculation, named PENE3D. A series of cases with selected normal and oblique penetrations are simulated by the algorithm. The predictions agree with the results of tests, showing that the proposed algorithm is fast and effective in simulation of the penetration process and prediction of the penetration trajectory.
The double Hopf bifurcation of a composite laminated piezoelectric plate with combined external and internal excitations is studied. Using a multiple scale method, the average equations are obtained in two coordinates. The bifurcation response equations of the composite laminated piezoelectric plate with the primary parameter resonance, i.e., 1:3 internal resonance, are achieved. Then, the bifurcation feature of bifurcation equations is considered using the singularity theory. A bifurcation diagram is obtained on the parameter plane. Different steady state solutions of the average equations are analyzed. By numerical simulation, periodic vibration and quasi-periodic vibration responses of the composite laminated piezoelectric plate are obtained.
Large eddy simulation (LES) using the Smagorinsky eddy viscosity model is added to the two-dimensional nine velocity components (D2Q9) lattice Boltzmann equation (LBE) with multi-relaxation-time (MRT) to simulate incompressible turbulent cavity flows with the Reynolds numbers up to 1×107. To improve the computation efficiency of LBM on the numerical simulations of turbulent flows, the massively parallel computing power from a graphic processing unit (GPU) with a computing unified device architecture (CUDA) is introduced into the MRT-LBE-LES model. The model performs well, compared with the results from others, with an increase of 76 times in computation efficiency. It appears that the higher the Reynolds numbers is, the smaller the Smagorinsky constant should be, if the lattice number is fixed. Also, for a selected high Reynolds number and a selected proper Smagorinsky constant, there is a minimum requirement for the lattice number so that the Smagorinsky eddy viscosity will not be excessively large.
The reduced-order model (ROM) for the two-dimensional supersonic cavity flow based on proper orthogonal decomposition (POD) and Galerkin projection is investigated. Presently, popular ROMs in cavity flows are based on an isentropic assumption, valid only for flows at low or moderate Mach numbers. A new ROM is constructed involving primitive variables of the fully compressible Navier-Stokes (N-S) equations, which is suitable for flows at high Mach numbers. Compared with the direct numerical simulation (DNS) results, the proposed model predicts flow dynamics (e.g., dominant frequency and amplitude) accurately for supersonic cavity flows, and is robust. The comparison between the present transient flow fields and those of the DNS shows that the proposed ROM can capture self-sustained oscillations of a shear layer. In addition, the present model reduction method can be easily extended to other supersonic flows.
In this paper, the effect of fluid in a tunnel of Corti (TC) on organ of Corti (OC) is studied. A three-dimensional OC model including basilar membrane (BM), tectorial membrane (TM), inner and outer hair cells (OHCs), and reticular lamina (RL) is established by COMSOL. An initial pressure is applied to the fluid in the TC. The frequency response of the structure is analyzed, and the displacement of the BM is achieved. The results are in good agreement with the experimental data, confirming validity of the finite element model. Based on the model, the effect of fluid in the TC on the OC is studied. The results show that, when the pressure gradient is absent in the fluid, with the increase of the initial fluid pressure, the displacement of the BM increases. However, when the initial fluid pressure increases to a certain value, the increase rate of the displacement of the BM becomes very slow. The movement of the fluid amplifies the BM movement. Furthermore, the movement of the fluid can strengthen the movement of the OHCs and the shear movement of the stereocilia, especially in the vicinity of the characteristic frequency at which the amplification effect reaches a peak. Nevertheless, a pressure gradient in the fluid affects the BM movement.
In this paper, two new existence theorems of solutions to inverse variational and quasi-variational inequality problems are proved using the Fan-Knaster-KuratowskiMazurkiewicz (KKM) theorem and the Kakutani-Fan-Glicksberg fixed point theorem. Upper semicontinuity and lower semicontinuity of the solution mapping and the approximate solution mapping to the parametric inverse variational inequality problem are also discussed under some suitable conditions. An application to a road pricing problem is given.