A catastrophic model for vibrational buckling of elastic arches

This paper represents an attempt at the application of catastrophe theory to thedynamic stability of engineering structures.The authors not only obtain a catastrophicmodel of vibrational buckling of elastic arches,but also give the critical condition of losingstability.     

Lattice-Boltzmann model for compressible perfect gases

We present an adaptive lattice Boltzmann model to simulate supersonic flows. The particle velocities are determined by the mean velocity and internal energy. The adaptive nature of particle velocities permits the mean flow to have high Mach number. A particle potential energy is introduced so that the model is suitable for the perfect gas with arbitrary specific heat ratio. The Navier-Stokes equations are derived by the Chapman-Enskog method from the BGK Boltzmann equation. As preliminary tests, two kinds of simulations have been performed on hexagonal lattices. One is the one-dimensional simulation for sinusoidal velocity distributions. The velocity distributions are compared with the analytical solution and the measured viscosity is compared with the theoretical values. The agreements are basically good. However, the discretion error may cause some non-isotropic effects. The other simulation is the 29 degree shock reflection. The project supported by the National Natural Science Foundation of China (Grant Nos. 19672030 and 19972037) and by the Scientific Research Foundation for Returned Overseas Chinese Scholars, State Education Ministry     

Resonant phenomena in a cylindrical shell containing a spherical inclusion and immersed in an elastic medium

A method is proposed to investigate the behavior of an axisymmetric system consisting of an infinite thin elastic cylindrical shell immersed in an infinite elastic medium, filled with a perfect compressible fluid, and containing an oscillating spherical inclusion. The system is subjected to periodic excitation. The task is to detect so-called resonant phenomena, to establish conditions that cause them, and to examine the possibilities of using the characteristic parameters of such a hydroelastic system to influence these conditions. The method allows transforming the general solutions of mathematical physics equations from one coordinate system to another to obtain exact analytic solutions (in the form of Fourier series) to interaction problems for systems of rigid and elastic bodies __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 7, pp. 82–97, July 2006.     

Computational properties of vertical grids for a nonhydrostatic anelastic model

Computational dispersion properties of all vertically staggered grids, which are presently available, are analysed in terms of frequency and group velocity components using the second-order centre difference scheme for a nonhydrostatic anelastic approximation system with a general method. The inertial-gravitational waves with a horizontal scale of a hundred-, ten- and one-kilometres are considered. The comparison analysis shows that the Charny-Phillips (CP) and Lorenz grids are suitable for waves at all abovementioned horizontal scales, while the Lorenz time staggered and Charny-Phillips time staggered grids are applicable only to waves with a horizontal scale less than 10 km. The unstaggered (N) grid is not suitable for simulating waves at any horizontal scale. In an idealised flow numerical test, the result on the CP grid has much less error than that on the N grid.     

Some multiplicity results for an elastic beam equation at resonance

This paper deals with multiplicity results for nonlinear elastic equations of the type-d~4u/dx~4+π~4u+g(x,u)=e(x)(0相似文献   

A theoretical model for the prediction of thermal expansion behaviour of particulate composites

The thermal expansion coefficient of particle-reinforced polymers was evaluated using a theoretical model which takes into account the adhesion efficiency between the inclusions and the matrix — an important factor affecting the thermomechanical properties of a composite. To measure the adhesion efficiency a boundary interphase, i.e. a layer between the matrix and the fillers having a structure and properties different from those of the constituent phases, was considered. This layer is assumed to have varying properties.To obtain information concerning the properties and extent of the interphase, an experimental study of the thermal behaviour of aluminium-epoxy composites was undertaken. Differential Scanning Calorimetry (DSC) measurements were performed to evaluate heat capacity with respect to temperature. In addition, the effects of different factors, such as heating rate and filler concentration on the glass transition temperature of the composite, were examined. The sudden changes in heat capacity values in the glass transition region were used to estimate the extent of the boundary interphase according to an existing theory.Finally, the values of the thermal expansion coefficient, predicted by this model, were compared with theoretical results obtained by other authors and with experimental results.     

A Boltzmann based model for open channel flows

A finite volume, Boltzmann Bhatnagar–Gross–Krook (BGK) numerical model for one‐ and two‐dimensional unsteady open channel flows is formulated and applied. The BGK scheme satisfies the entropy condition and thus prevents unphysical shocks. In addition, the van Leer limiter and the collision term ensure that the BGK scheme admits oscillation‐free solutions only. The accuracy and efficiency of the BGK scheme are demonstrated through the following examples: (i) strong shock waves, (ii) extreme expansion waves, (iii) a combination of strong shock waves and extreme expansion waves, and (iv) one‐ and two‐dimensional dam break problems. These test cases are performed for a variety of Courant numbers (C_r), with the only condition being C_r≤1. All the computational results are free of spurious oscillations and unphysical shocks (i.e., expansion shocks). In addition, comparisons of numerical tests with measured data from dam break laboratory experiments show good agreement for C_r≤0.6. This reduction in the stability domain is due to the explicit integration of the friction term. Furthermore, BGK schemes are easily extended to multidimensional problems and do not require characteristic decomposition. The proposed scheme is second‐order in both space and time when the external forces are zero and second‐order in space but first‐order in time when the external forces are non‐zero. However, since all the test cases presented are either for zero or small values of external forces, the results tend to maintain second‐order accuracy. In problems where the external forces become significant, it is possible to improve the order of accuracy of the scheme in time by, for example, applying the Runge–Kutta method in the integration of the external forces. Copyright © 2001 John Wiley & Sons, Ltd.     

A stabilized mixed formulation for unsteady Brinkman equation based on the method of horizontal lines

In this paper, we present a stabilized mixed formulation for unsteady Brinkman equation. The formulation is systematically derived based on the variational multiscale formalism and the method of horizontal lines. The derivation does not need the assumption that the fine‐scale variables do not depend on the time, which is the case with the conventional derivation of multiscale stabilized formulations for transient mixed problems. An expression for the stabilization parameter is obtained in terms of a bubble function, and appropriate bubble functions for various finite elements are also presented. Under the proposed formulation, equal‐order interpolation for the velocity and pressure (which is computationally the most convenient) is stable. Representative numerical results are presented to illustrate the performance of the proposed formulation. Spatial and temporal convergence studies are also performed, and the proposed formulation performed well. Copyright © 2011 John Wiley & Sons, Ltd.     

Bi-variable damage model for fatigue life prediction of metal components

Based on the theory of continuum damage mechanics,a bi-variable damage mechanics model is developed,which,according to thermodynamics,is accessible to derivation of damage driving force,damage evolution equation and damage evolution criteria. Furthermore,damage evolution equations of time rate are established by the generalized Drucker’s postulate. The damage evolution equation of cycle rate is obtained by integrating the time damage evolution equations,and the fatigue life prediction method for smooth specimens under repeated loading with constant strain amplitude is constructed. Likewise,for notched specimens under the repeated loading with constant strain amplitude,the fatigue life prediction method is obtained on the ground of the theory of conservative integral in damage mechanics. Thus,the material parameters in the damage evolution equation can be obtained by reference to the fatigue test results of standard specimens with stress concentration factor equal to 1,2 and 3.     

UNCONDITIONAL STABLE SOLUTIONS OF THE EULER EQUATIONS FOR TWO－AND THREE－D WINGS IN ARBITRARY MOTION

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Existence and Integral Representations of Weak Solutions for Elastic Plates with Cracks

The existence and continuous dependence on the data are investigated in Sobolev spaces for the problem of bending of a Reissner-Mindlin-type plate weakened by a crack when the displacements or the moments and force are prescribed along the two sides of the crack. The cases of both an infinite and a finite plate are considered, and representations are sought for the solutions in terms of single layer and double layer potentials with distributional densities. This revised version was published online in August 2006 with corrections to the Cover Date.     

Interaction of differently shaped bodies in a potential flow of perfect compressible fluid: Axisymmetric internal problem

An approach is proposed to investigate an axisymmetric system consisting of an infinite thin elastic cylindrical shell that contains a potential flow of perfect compressible fluid and a periodically vibrating spherical inclusion. The approach emerged as part of a project devoted to developing methods to bring plugged oil wells back into production by the Vibration Theory Department of the S. P. Timoshenko Institute of Mechanics. This mathematical approach allows transforming the general solutions to equations of mathematical physics from one coordinate system to another to obtain an exact analytical solution (in the form of Fourier series) to interaction problems for systems of rigid and elastic bodies __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 9, pp. 16–31, September 2006.     

Dynamical equations of various types for finite deformable polar elastic media

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Evolutionary model of finite-strain thermoelasticity

The equation of state of finite-strain thermoelasticity is obtained using a formalized approach to constructing constitutive relations for complex media under the assumption of closeness of intermediate and current configurations. A variational formulation of the coupled thermoelastic problem is proposed. The constitutive equation, the heat-conduction equation, the relations for internal energy, free energy, and entropy, and the variational formulation of the coupled problem of finite-strain thermoelasticity are tested on the problem of uniaxial extension of a bar. The model adequately describes experimental data for elastomers, such as entropic elasticity, temperature inversion, and temperature variation during an adiabatic process. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 3, pp. 184–196, May–June, 2008.     

Non-axisymmetrical vibration of elastic circular plate on layered transversely isotropic saturated ground

The non-axisymmetrical vibration of elastic circular plate resting on a layered transversely isotropic saturated ground was studied.First,the 3-d dynamic equations in cylindrical coordinate for transversely isotropic saturated soils were transformed into a group of governing differential equations with 1-order by the technique of Fourier ex- panding with respect to azimuth,and the state equation is established by Hankel integral transform method,furthermore the transfer matrixes within layered media are derived based on the solutions of the state equation.Secondly,by the transfer matrixes,the general solutions of dynamic response for layered transversely isotropic saturated ground excited by an arbitrary harmonic force were established under the boundary conditions, drainage conditions on the surface of.ground as well as the contact conditions.Thirdly, the problem was led to a pair of dual integral equations describing the mixed boundary- value problem which can be reduced to the Fredholm integral equations of the second kind solved by numerical procedure easily.At the end of this paper,a numerical result concerning vertical and radical displacements both the surface of saturated ground and plate is evaluated.     

On radii of absorbing sets for Kuramoto-Sivashinsky equation

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Rotary oscillations of axi‐symmetric bodies in an axi‐symmetric viscous flow with slip: Numerical solutions for sphere and spheroids

Rotary oscillations of several axi‐symmetric bodies in axi‐symmetric viscous flows with slip are investigated. A numerical method based on the Green's function technique is used wherein the relevant Helmholtz equation, as obtained from the unsteady Stokes equation, is converted into a surface integral equation. The technique is benchmarked against a known analytical solution, and accurate numerical results for local stress and torque on spheres and spheroids as function of the frequency parameter and the slip coefficients are obtained. It is found that in all cases, slip reduces stress and torque, and increasingly so with the increasing frequency parameter. The method discussed here can be potentially extended to the realistic case of an oscillating disk viscometer. Copyright © 2003 John Wiley & Sons, Ltd.     

An operator splitting scheme for the stream‐function formulation of unsteady Navier–Stokes equations

A fictitious time is introduced into the unsteady equation of the stream function rendering it into a higher‐order ultra‐parabolic equation. The convergence with respect to the fictitious time (we call the latter ‘internal iterations’) allows one to obtain fully implicit nonlinear scheme in full time steps for the physical‐time variable. For particular choice of the artificial time increment, the scheme in full time steps is of second‐order of approximation. For the solution of the internal iteration, a fractional‐step scheme is proposed based on the splitting of the combination of the Laplace, bi‐harmonic and advection operators. A judicious choice for the time staggering of the different parts of the nonlinear advective terms allows us to prove that the internal iterations are unconditionally stable and convergent. We assess the number of operations needed per time step and show computational effectiveness of the proposed scheme. We prove that when the internal iterations converge, the scheme is second‐order in physical time and space, nonlinear, implicit and absolutely stable. The performance of the scheme is demonstrated for the flow created by oscillatory motion of the lid of a square cavity. All theoretical findings are demonstrated practically. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical solution of the unsteady Euler equations for airfoils using approximate boundary conditions

This paper presents an efficient numerical method for solving the unsteady Euler equations on stationary rectilinear grids. Boundary conditions on the surface of an airfoil are implemented by using their first-order expansions on the mean chord line. The method is not restricted to flows with small disturbances since there are no restrictions on the mean angle of attack of the airfoil. The mathematical formulation and the numerical implementation of the wall boundary conditions in a fully implicit time-accurate finite-volume Euler scheme are described. Unsteady transonic flows about an oscillating NACA 0012 airfoil are calculated. Computational results compare well with Euler solutions by the full boundary conditions on a body-fitted curvilinear grid and published experimental data. This study establishes the feasibility for computing unsteady fluid-structure interaction problems, where the use of a stationary rectilinear grid offers substantial advantages in saving computer time and program design since it does not require the generation and implementation of time-dependent body-fitted grids.     

PERTURBATION ANALYSIS FOR WAVE EQUATION OF NONLINEAR ELASTIC ROD

The longitudinal oscillation of a nonlinear elastic rod with lateral inertia was studied. Based on the far field and simple wave theory, a nonlinear Schrodinger (NLS) equation was established under the assumption of small amplitude and long wavelength. It is found that there are NLS envelop solitons in this system. Finally the soliton solution of the NLS equation was presented.