Equivalent linearizations for practical hysteretic systems

Experimental testing of a friction damped base isolation system has indicated a need for a new model of friction damping and for an appropriate equivalent linearization technique. The model for the damping adopted is a combination of viscous damping, constant Coulomb friction and linear Coulomb friction.This model is incorporated into the equation of motion for a single-degree-of-freedom system and the exact solutions are given for free vibrations and for steady-state vibrations excited by a harmonic force. The exact solution is taken as a basis for an equivalent linearization technique that can be used in conjunction with conventional design spectra for a practical design of such a system.     

On unsteady unidirectional flows of a second grade fluid

Some properties of unsteady unidirectional flows of a fluid of second grade are considered for flows produced by the sudden application of a constant pressure gradient or by the impulsive motion of one or two boundaries. Exact analytical solutions for these flows are obtained and the results are compared with those of a Newtonian fluid. It is found that the stress at the initial time on the stationary boundary for flows generated by the impulsive motion of a boundary is infinite for a Newtonian fluid and is finite for a second grade fluid. Furthermore, it is shown that initially the stress on the stationary boundary, for flows started from rest by sudden application of a constant pressure gradient is zero for a Newtonian fluid and is not zero for a fluid of second grade. The required time to attain the asymptotic value of a second grade fluid is longer than that for a Newtonian fluid. It should be mentioned that the expressions for the flow properties, such as velocity, obtained by the Laplace transform method are exactly the same as the ones obtained for the Couette and Poiseuille flows and those which are constructed by the Fourier method. The solution of the governing equation for flows such as the flow over a plane wall and the Couette flow is in a series form which is slowly convergent for small values of time. To overcome the difficulty in the calculation of the value of the velocity for small values of time, a practical method is given. The other property of unsteady flows of a second grade fluid is that the no-slip boundary condition is sufficient for unsteady flows, but it is not sufficient for steady flows so that an additional condition is needed. In order to discuss the properties of unsteady unidirectional flows of a second grade fluid, some illustrative examples are given.     

Rotating beams and non-rotating beams with shared eigenpair for pinned-free boundary condition

In this paper we look for a rotating beam, with pinned-free boundary conditions, whose eigenpair (frequency and mode-shape) is same as that of a uniform non-rotating beam for a particular mode. It is seen that for any given mode, there exists a flexural stiffness function (FSF) for which the ith mode eigenpair of a rotating beam with uniform mass distribution, is identical to that of a corresponding non-rotating beam with same length and mass distribution. Inserting these derived FSF’s in a finite element code for a rotating pinned-free beam, the frequencies and mode shapes of a non-rotating pinned-free beam are obtained. For the first mode, a physically realistic equivalent rotating beam is possible, but for higher modes, the FSF has internal singularities. Strategies for addressing these singularities in the FSF for finite element analysis are provided. The proposed functions can be used as test functions for rotating beam codes and also for targeted destiffening of rotating beams.     

Singularities in 2D anisotropic potential problems in multi-material corners: Real variable approach

An analysis of singular solutions at corners consisting of several different homogeneous wedges is presented for anisotropic potential theory in plane. The concept of transfer matrix is applied for a singularity analysis first of single wedge problems and then of multi-material corner problems. Explicit forms of eigenequations for evaluation of singularity exponent in the case of multi-material corners are derived both for all combinations of homogeneous Neumann and Dirichlet boundary conditions at faces of open corners and for multi-material planes with singular interior points. Perfect transmission conditions at wedge interfaces are considered in both cases. It is proved that singularity exponents are real for open anisotropic multi-material corners, and a sufficient condition for the singularity exponents to be real for anisotropic multi-material planes is deduced. A case of a complex singularity exponent for an anisotropic multi-material plane is reported, apparently for the first time in potential theory. Simple expressions of eigenequations are presented first for open bi-material corners and bi-material planes and second for a crack terminating at a bi-material interface, as examples of application of the theory developed here. Analytical solutions of these eigenequations are presented for interface cracks with any combination of homogeneous boundary conditions along the interface crack faces, and also for a special case of a crack perpendicular to a bi-material interface. A numerical study of variation of the singularity exponent as a function of inclination of a crack terminating at a bi-material interface is presented.     

Boundary element simulations of spheres settling in circular,square and triangular conduits

Numerical simulations of a spherical particle sedimenting in circular, triangular and square conduits containing a viscous, inertialess, Newtonian fluid were investigated using the Boundary Element Method (BEM). Settling velocities and pressure drops for spheres falling along the centre-lines of the conduits were computed for a definitive range of sphere sizes. The numerical simulations for the settling velocities showed good agreement with existing experimental data. The most accurate analytic solution for a sphere settling along the axis of a circular conduit produced results which were almost indistinguishable from the present BEM calculations. For a sphere falling along the centre-line of a square conduit, the BEM calculations for small spheres agreed well with analytic results. No analytic results for a sphere falling along the axis of a triangular conduit were available for comparison. Extrapolation of the BEM predictions for the pressure drops, to infinitely small spheres, showed remarkable agreement with analytic results. For the circular conduit, the sphere's settling velocity and angular velocity were computed as a function of drop position for small, medium and large spheres. Excellent agreement with a reflection solution was achieved for the small sphere. In addition, end effects were investigated for centre-line drops and compared where possible with available experimental data and analytic results.Los Alamos National Laboratory, Los Alamos, New Mexico, USA.     

An implicit characteristic-flux-averaging method for the euler equations for real gases

A formulation of an implicit characteristic-flux-averaging method for the unsteady Euler equations with real gas effects is presented. Incorporation of a real gas into a general equation of state is achieved by considering the pressure as a function of density and specific internal energy. The Ricmann solver as well as the flux-split algorithm are modified by introducing the pressure derivatives with respect to density and internal energy. Expressions for calculating the values of the flow variables for a real gas at the cell faces are derived. The Jacobian matrices and the eigenvectors are defined for a general equation of state. The solution of the system of equations is obtained by using a mesh-sequencing method for acceleration of the convergence. Finally, a test case for a simple form of equation of state displays the differences from the corresponding solution for an ideal gas.     

Development of co-current air–water flow in a vertical pipe

Measurements of the cross-sectional distribution of the gas fraction and bubble size distributions were conducted in a vertical pipe with an inner diameter of 51.2 mm and a length of about 3 m for air/water bubbly and slug flow regimes. The use of a wire-mesh sensor obtained a high resolution of the gas fraction data in space as well as in time. From this data, time averaged values for the two-dimensional gas fraction profiles were decomposed into a large number of bubble size classes. This allowed the extraction of the radial gas fraction profiles for a given range of bubble sizes as well as data for local bubble size distributions. The structure of the flow can be characterized by such data. The measurements were performed for up to 10 different inlet lengths and for about 100 combinations of gas and liquid volume flow rates. The data is very useful for the development and validation of meso-scale models to account for the forces acting on a bubble in a shear liquid flow and models for bubble coalescence and break-up. Such models are necessary for the validation of CFD codes for the simulation of bubbly flows.     

An Improved IMPES Method for Two-Phase Flow in Porous Media

In this paper we develop and numerically study an improved IMPES method for solving a partial differential coupled system for two-phase flow in a three-dimensional porous medium. This improved method utilizes an adaptive control strategy on the choice of a time step for saturation and takes a much larger time step for pressure than for the saturation. Through a stability analysis and a comparison with a simultaneous solution method, we show that this improved IMPES method is effective and efficient for the numerical simulation of two-phase flow and it is capable of solving two-phase coning problems.     

Shock wave diffraction on multi-facetted and curved walls

The two-dimensional diffraction of a shock wave over a wall made up of a series of plane and/or curved sections is considered. The analysis is based on the theory presented by, for the interaction of an originally plane shock wave with a corner. A method is presented by which the shock profile may be determined for a wall of any shape and for any incident Mach number, in regions where the characteristics form a simple wave. Comparisons are made between experimental measurements and theoretical predictions for convex walls consisting of a number of facets, and for circular arcs, for a range of incident shock wave Mach numbers. It is shown that the theory gives a satisfactory prediction of the wave shape, which improves as the Mach number increases. Modifications in the flow field behind the shock, compared to that for a simple corner made up of two plane walls is discussed, particularly relating to flow separation. For circular arc concave walls a inverse Mach reflection results experimentally, leading to regular reflection, for which the theory is of no use. PACS 47.40.Nm     

Contact problems for a two-layer elastic wedge

We obtain integral equations for plane contact problems for a two-layer wedge (composite) under three types of boundary conditions on one of its sides (absence of stresses, sliding, or rigid fixation). The composite consists of two wedges completely linked with each other, which have different opening angles and elasticity parameters. Using the symbols (Mellin transforms) of the kernels of integral equations for the two-layer wedge, one can derive the symbols of the kernels of integral equations for symmetric problems about a crack in a three-layer wedge or a three-layer strip and for contact problems for a two-layer strip (by passing to the limit in a special way). The complex zeros of the Mellin transform determine the asymptotics of the normal contact pressure at the corner point of the composite as the contact region approaches this point. It is important that this asymptotics is also preserved in three-dimensional contact problems as the die enters the edge of a two-layer wedge (outside the corner points of the die itself). Taking into account this asymptotics, we obtain solutions of the contact problems as the die enters the vertex of the composite. We show that by appropriately choosing the materials and the internal angle of the two-layer wedge one can avoid contact pressure oscillations at the vertex, which occur in the case of a homogeneous wedge and result in loss of contact. The contact pressure at the wedge vertex can be made zero for a composite, while for a homogeneous wedge with the same opening angle it increases unboundedly. We construct asymptotic solutions of the contact problems for a plane die located relatively close or to the vertex of a two-layer wedge or relatively far from the vertex. The asymptotic and other methods were earlier used to solve similar plane contact problems for a homogeneous wedge [1, 2]. In the case of sliding fixation of one of the sides of a plane homogeneous wedge, the closed solution of the contact problem is known for a die entering the corner point [3, p. 131]. Two-dimensional contact problems were studied for a truncated wedge [4] and for a wedge supported by a rod of equal resistance [5]. The out-of-plane shear vibrations were studied for wedge-shaped composites [6, 7]. The spatial contact problems were considered for a homogeneous wedge [8]. The plane contact problem was analyzed for a continuously inhomogeneous wedge one of whose sides was rigidly fixed (the shear modulus continuously depends on the angular coordinate and the Poisson coefficient is constant). For a two-layer composite, which is studied in the present paper, the kernel symbol has different asymptotic properties, which are used in asymptotic methods for solving the problem. A similar distinction of the symbol properties takes place in contact problems for a continuously inhomogeneous layer and a layered packet.     

Surface instability of an elastic half space with material properties varying with depth

If a body with a stiffer surface layer is loaded in compression, a surface wrinkling instability may be developed. A bifurcation analysis is presented for determining the critical load for the onset of wrinkling and the associated wavelength for materials in which the elastic modulus is an arbitrary function of depth. The analysis leads to an eigenvalue problem involving a pair of linear ordinary differential equations with variable coefficients which are discretized and solved using the finite element method.The method is validated by comparison with classical results for a uniform layer on a dissimilar substrate. Results are then given for materials with exponential and error-function gradation of elastic modulus and for a homogeneous body with thermoelastically induced compressive stresses.     

An adaptive multilevel wavelet framework for scale‐selective WENO reconstruction schemes

We put forth a dynamic computing framework for scale‐selective adaptation of weighted essential nonoscillatory (WENO) schemes for the simulation of hyperbolic conservation laws exhibiting strong discontinuities. A multilevel wavelet‐based multiresolution procedure, embedded in a conservative finite volume formulation, is used for a twofold purpose. (i) a dynamic grid adaptation of the solution field for redistributing grid points optimally (in some sense) according to the underlying flow structures, and (ii) a dynamic minimization of the in built artificial dissipation of WENO schemes. Taking advantage of the structure detection properties of this multiresolution algorithm, the nonlinear weights of the conventional WENO implementation are selectively modified to ensure lower dissipation in smoother areas. This modification is implemented through a linear transition from the fifth‐order upwind stencil at the coarsest regions of the adaptive grid to a fully nonlinear fifth‐order WENO scheme at areas of high irregularity. Therefore, our computing algorithm consists of a dynamic grid adaptation strategy, a scale‐selective state reconstruction, a conservative flux calculation, and a total variation diminishing Runge‐Kutta scheme for time advancement. Results are presented for canonical examples drawn from the inviscid Burgers, shallow water, Euler, and magnetohydrodynamic equations. Our findings represent a novel direction for providing a scale‐selective dissipation process without a compromise on shock capturing behavior for conservation laws, which would be a strong contender for dynamic implicit large eddy simulation approaches.     

Simulation of glass cutting with an impinging hot air jet

Thermal cutting of glass sheet due to an impinging hot air jet is simulated and analyzed. Induced thermal stresses due to the moving heat source can be used to stably initiate and attract a crack toward the jet axis. Relative motion between the jet and glass sheet then can be used to cut the glass sheet. This paper presents a theoretical study of this process for straight cuts. Process simulation is accomplished by analyzing the coupled temperature and stress fields together with the fracture mechanics criteria for the crack growth.A finite element remeshing technique is employed for the analysis and singular elements are used around the crack tip for a more precise computing of the stress intensity factor. It is shown that a certain minimum air jet temperature for a given nozzle velocity and a certain maximum air jet velocity for a given temperature are required for continuous cutting. The results of the simulation show a good agreement with the published results in the literature. However a variating nature is detected for the distance between the crack tip and the air jet nozzle from a starting value to the steady-state one.     

Overall damage and elastoplastic deformation in fibrous metal matrix composites

A phenomenological constitutive model for fibrous composite materials with a ductile matrix is postulated incorporating damage mechanics with micromechanical behavior. The model is first formulated in an undamaged composite system and then transformed consistently achieved in terms of an overall damage tensor M for the whole composite. In the process of formulating this model, interesting results are obtained demonstrating the necessity of using a non-associated flow rule for plasticity in the damaged composite system together with a Hill's type yield criterion. It is also shown that using a Ziegler-Prager kinematic hardening rule for the ductile matrix leads to a general kinematic hardening rule for the composite that is a combination of a generalized Ziegler-Prager model and a Phillips-type model. Finally, an explicit expression for the elastoplastic stiffness tensor for the damaged composite is obtained.     

Griffith crack moving in a piezoelectric strip

Summary  The dynamic problem of an impermeable crack of constant length 2a propagating along a piezoelectric ceramic strip is considered under the action of uniform anti-plane shear stress and uniform electric field. The integral transform technique is employed to reduce the mixed-boundary-value problem to a singular integral equation. For the case of a crack moving in the mid-plane, explicit analytic expressions for the electroelastic field and the field intensity factors are obtained, while for an eccentric crack moving along a piezoelectric strip, numerical results are determined via the Lobatto–Chebyshev collocation method for solving a resulting singular integral equation. The results reveal that the electric-displacement intensity factor is independent of the crack velocity, while other field intensity factors depend on the crack velocity when referred to the moving coordinate system. If the crack velocity vanishes, the present results reduce to those for a stationary crack in a piezoelectric strip. In contrast to the results for a stationary crack, applied stress gives rise to a singular electric field and applied electric field results in a singular stress for a moving crack in a piezoelectric strip. Received 14 August 2001; accepted for publication 24 September 2002 The author is indebted to the AAM Reviewers for their helpful suggestions for improving this paper. The work was supported by the National Natural Science Foundation of China under Grant 70272043.     

Analytical solutions for long waves on a circular island with combined topographies

We derived analytical solutions for long waves on a circular island with combined topographies. The inner region is a circular island on a flat bottom and the outer region is a cylindrical island on a shoal. The solution for the circular island on a flat bottom was developed previously by Jung et al. (2010) [13]. The solution for the cylindrical island on a shoal was developed in the present study using the methods of separation of variables, Taylor series expansion, and Frobenius series. The present solution for the cylindrical island on a shoal was verified by comparing with previous solutions which was developed for limited cases. Also, some cases were investigated with different values of wave period, shoal vertex level, and power of the radius. The solutions for two different types of real islands were obtained using the solution for the combined topographies. The present solution for the combined topographies can be applied more accurately to the real island than the previous solution for the single bottom topography.     

Weakly-singular,weak-form integral equations for cracks in three-dimensional anisotropic media

Singularity-reduced integral relations are developed for displacement discontinuities in three-dimensional, anisotropic linearly elastic media. An isolated displacement discontinuity is considered first, and a systematic procedure is followed to develop relations for the displacement and stress fields induced by the discontinuity. The singularity-reduced relation for the stress is particularly important since it is in a form which allows a weakly-singular, weak-form traction integral equation to be readily established. The integral relations obtained for a general displacement discontinuity are then specialized to an isolated crack and to dislocations; the relations for dislocations are introduced to emphasize their direct connection to corresponding results for cracks and to allow earlier independent findings for these two types of discontinuities to be put into proper context. Next, the singularity-reduced integral equations obtained for an isolated crack are extended to allow treatment of cracks in a finite domain, and a pair of weakly-singular, weak-form displacement and traction integral equations is established. These integral equations can be combined to obtain a final formulation which is in a symmetric form, and in this way they serve as the basis for a weakly-singular, symmetric Galerkin boundary element method suitable for analysis of cracks in anisotropic media.     

可压缩多介质流动的间断有限元方法

采用间断有限元方法、LS方法和通量装配技术相结合,建立了一种计算可压缩多介质流动的有效 方法。计算中以光滑Heavside函数构造流体比热比和重新初始化方程中的符号距离函数,并采用通量装配 技术抑制界面附近的非物理振荡。为解决可压缩多介质流动提供一种新的手段。     

Three-dimensional isothermal lava flows over a non-axisymmetric conical surface

Within the thin-layer approximation for a highly-viscous heavy incompressible fluid, a hydrodynamicmodel of a 3D isothermal lava flow over a non-axisymmetric conical surface is constructed. Using analytical methods, a self-similar solution for the law of leading-edge propagation is obtained in the case of a flow from a non-axisymmetric source located at the apex of a conical surface with smoothly varying properties. In the case of a flow over a substantially non-axisymmetric surface, it is shown that there exists a self-similar solution for the free-surface shape and the law of leading-edge motion. This solution is studied numerically for particular examples of the substrate surface and the source. In the general case of a non-self-similar flow over a substantially non-axisymmetric conical surface, a local analytical solution is obtained for the free-surface shape and the velocity field near the leading flow front.     

Effective elastic properties of periodic composite medium

A new method is presented for calculating the bulk effective elastic stiffness tensor of a two-component composite with a periodic microstructure. The basic features of this method are similar to the one introduced by Bergman and Dunn (1992) for the dielectric problem. It is based on a Fourier representation of an integro-differential equation for the displacement field, which is used to produce a continued-fraction expansion for the elastic moduli. The method enabled us to include a much larger number of Fourier components than some previously proposed Fourier methods. Consequently our method provides the possibility of performing reliable calculations of the effective elastic tensor of periodic composites that are neither dilute nor low contrast, and are not restricted to arrays of nonoverlapping inclusions. We present results for a cubic array of nonoverlapping spheres, intended to serve as a test of quality, as well as results for a cubic array of overlapping spheres and a two dimensional hexagonal array of circles (a model for a fiber reinforced material) for comparison with previous work.