On formulas for the Rayleigh wave velocity in pre-stressed compressible solids

In this paper, formulas for the velocity of Rayleigh waves in compressible isotropic solids subject to uniform initial deformations are derived using the theory of cubic equation. They are explicit, have simple algebraic forms, and hold for a general strain energy function. Unlike the previous investigations where the derived formulas for Rayleigh wave velocity are approximate and valid for only small enough values of pre-strains, this paper establishes exact formulas for Rayleigh wave velocity being valid for any range of pre-strains. When the prestresses are absent, the obtained formulas recover the Rayleigh wave velocity formula for compressible elastic solids. Since obtained formulas are explicit, exact and hold for any range of pre-strains, they are good tools for evaluating nondestructively prestresses of structures.     

A nonlinear dynamical systems analysis of child emotion displays in relation to family context and child adjustment: a cox hazard approach

This report examines how the relative attractor strengths of children's display of three emotion states, anger, sadness/fear, and neutral-engaged, are associated with exposure to maternal negative affect and care giving disruptions, and to child antisocial behavior and depression. Exposure to negative maternal affect was associated with a weaker attractor state for sadness or fear displays relative to those for anger and neutral-engaged displays. Exposure to care giving disruptions was associated with stronger attractor strength for anger and sadness/fear relative to that for neutral-engaged. Overt and covert antisocial behaviors were associated with weaker attractor states for sadness/fear displays relative that for the neutral-engaged displays. Overt antisocial behavior was associated with a stronger attractor state for anger displays relative to that for neutral-engaged displays, and covert antisocial behavior with a weaker attractor state for fear/sadness displays relative to that for neutral-engaged displays. Child depressive symptoms were marginally associated with a stronger attractor state for fear/sadness displays relative to neutral-engaged. The data suggest the attractor strengths for emotion display states are affected by social experience and that between-individual risk for various forms of psychopathology is related to the relative intra-individual attractor strength of various emotion displays in a multi-state emotion display system.     

Computer extension of perturbation series for mixed convection on a vertical plate: favourable and adverse flows

The mixed convection boundary layer on a vertical flat plate for uniform wall temperature/uniform heat flux is investigated using computer extension of perturbation series. The first thirteen terms for the uniform wall temperature case and the first ten terms for the uniform heat flux case are computed. The expansion, when transformed by Euler transform and other techniques, predicts the exact results to three-digit accuracy for all values of the streamwise coordinate, ξ, along the plate. For uniform wall temperature, the maximum error is 0.0087% for skin-friction and 0.00065% for heat-transfer. For uniform heat flux, the maximum error is 1.027% for skin-friction and 0.288% for wall temperature.     

Hypoplasticity theory for granular materials—II: Three-dimensional axially symmetric exact solutions

In part I of this paper, we consider the governing equations of hypoplasticity theory for two-dimensional steady quasi-static plane strain compressible gravity flow and determine some exact analytical solutions applying for certain special cases. Similarly, for the three-dimensional situation considered here in part II, we undertake a similar mathematical investigation to determine some simple solutions of the governing equations for three-dimensional steady quasi-static axially symmetric compressible gravity flow for hypoplastic granular materials. We again find that for certain special cases, we are able to determine some exact solutions for the stress and velocity profiles. We comment that hypoplasticity theory generally gives rise to complicated systems of coupled non-linear differential equations, for which the determination of any analytical solutions is not a trivial matter, and that the solutions determined here might be exploited as benchmarks for full numerical schemes.     

Renormalization Group Model of Large-Scale Turbulence in Porous Media

Renormalization group methods are used to develop a macroscopic (large-scale) turbulence model for incompressible flow in porous media. The model accounts for the large-distance and large-time behavior of velocity correlations generated by the momentum equation for a randomly stirred, incompressible flow. Utilizing the renormalization procedure, the transport equations for the large-scale modes and expressions for effective transport coefficients are obtained. Expressions for renormalized turbulent viscosity, which accounts for the ultraviolet subrange of the turbulent kinetic energy spectrum, are also obtained.     

Parametric stress-concentration factors in greenstick bone fracture

The reported work is a part of an ongoing research program concerned with structural analysis of fractured long bone and methods of internal fixation. The stress-concentration factors for equine metacarpus bones containing greenstick fractures and “through” fractures (surgically repaired) were determined for the compression, flexural and torsional modes of loading based on whole bone (unfractured) strengths. The greenstick type of fracture was simulated with saw cuts at the mid-span of the bone, and the parameters varied were depth of fracture and orientation of fracture. All specimens consisted of fresh dead bone which had been placed in a freezer within 4 hr after expiration. The maximum stress-concentration factors for the simulated greenstick fractures studied were about 3.4 for compression, 4.3 for torsion and 16 for flexure. The stressconcentration factors for fractured bones surgically repaired with commercial plates were about 3.0 for compression, 2.7 for torsion and 6.1 for flexure.     

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.     

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.     

Green's function for SH-waves in a cylindrically monoclinic material

Green's function for SH-waves in a cylindrically monoclinic material is considered for impulsive and time-harmonic sources. Closed form expressions for the Green's function are derived for a few limited values of anisotropic parameters. A very interesting time development of the wave front shape is illustrated and the wave front singularity is discussed for the transient SH-wave. Contours of the displacement amplitude for the time-harmonic wave are also shown.     

Axisymmetric compressive buckling of multi-walled carbon nanotubes under different boundary conditions

The paper studies the axisymmetric compressive buckling behavior of multi-walled carbon nanotubes (MWNTs) under different boundary conditions based on continuum mechanics model. A buckling condition is derived for determining the critical buckling load and associated buckling mode of MWNTs, and numerical results are worked out for MWNTs with different aspect ratios under fixed and simply supported boundary conditions. It is shown that the critical buckling load of MWNTs is insensitive to boundary conditions, except for nanotubes with smaller radii and very small aspect ratio. The associated buckling modes for different layers of MWNTs are in-phase, and the buckling displacement ratios for different layers are independent of the boundary conditions and the length of MWNTs. Moreover, for simply supported boundary conditions, the critical buckling load is compared with the corresponding one for axial compressive buckling, which indicates that the critical buckling load for axial compressive buckling can be well approximated by the corresponding one for axisymmetric compressive buckling. In particular, for axial compressive buckling of double-walled carbon nanotubes, an analytical expression is given for approximating the critical buckling load. The present investigation may be of some help in further understanding the mechanical properties of MWNTs.     

Minimum entropy generation for laminar flow at constant wall temperature in a circular duct for optimum design

Entropy generation for hydrodynamically and thermally developing laminar flow in a circular duct at constant wall temperature is investigated. Optimum temperature difference between fluid inlet temperature and wall temperature is determined for minimum entropy generation rate. An equation for optimum temperature difference at minimum entropy generation rate is given. It is seen that the optimum dimensionless temperature difference is a finite value for a duct length approaching zero for low temperature differences. It is possible to calculate the optimum pressure loss for given wall and inlet fluid temperature differences and corresponding optimum dimensions of a circular duct can be determined with these results. An example for the dimensioning of an air-conditioning heat exchanger tube for given wall and air inlet temperatures and heat transferred is presented.     

The re-examination of the weakly nonlinear theory of hydrodynamic stability

The weakly nonlinear theory has been widely applied in the problem of hydrodynamicstability and also in other fields.However,although its application has been successful forsome problems,yet,for other problems,the results obtained are not satisfactory,especiallyfor problems like transition or the evolution of the vortex in the free shear flow,for whichthe goal of the theoretical investigation is not seeking for a steady state,but predicting anevolutional process.In this paper,we shall examine the reason for the unsuccessfulness andsuggest ways for its amendment.     

Mechanical energy balances for a capillary viscometer

The entrance and exit flow processes for a cylindrical geometry are analyzed by writing macroscopic mechanical energy balances for a capillary viscometer. These equations can be used to compute the entrance and exit excess dissipation integrals from measured pressure differences if viscometric normal stress data are available for the material of interest. Upper and lower bounds are derived for these integrals for cases when high shear rate normal stress data are not available. The utilization of macroscopic mechanical energy balances in the interpretation of capillary viscometer results is illustrated using numerical solutions for a Maxwell fluid and experimental pressure drop data for high density polythylene.     

Calculation of the shock-wave adiabatics for some heterogeneous mixtures

A method has been proposed [1] for calculating shock-wave adiabatics for two-phase mixtures which does not use the additivity principle, and this is here extended to a medium consisting of n phases; the shock-wave adiabatic is derived and compared with experiment for a three-phase mixture; an alloy consisting of copper, zinc, and lead (brass). It is shown for a paraffin-tungsten mixture that the adiabatics for the individual phases deviate from ones for the corresponding solid materials. The known equations of state for aluminum and epoxide resin have been used to determine the shock-wave adiabatic for such a mixture. A comparison is made with the analogous calculation made from the additivity principle.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 130–137, July–August, 1973.I am indebted to V. N. Nikolaevskii for suggesting the topics and for useful discussion.     

A sharp condition for existence of an inertial manifold

It is shown that a perturbation argument that guarantees persistence of inertial (invariant and exponentially attracting) manifolds for linear perturbations of linear evolution equations applies also when the perturbation is nonlinear. This gives a simple but sharp condition for existence of inertial manifolds for semi-linear parabolic as well as for some nonlinear hyperbolic equations. Fourier transform of the explicitly given equation for the tracking solution together with the Plancherel's theorem for Banach valued functions are used.     

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.     

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.     

Convective heat transfer from two rotating circular cylinders in tandem arrangement by using lattice Boltzmann method

The numerical investigation of the two-dimensional laminar flow past two ro- tating circular cylinders in the tandem arrangement is conducted by the lattice Boltzmann method. The numerical strategy is used for dealing with curved and moving boundaries of the second-order accuracy for velocity and temperature fields. The effects of various rotational speed ratios and gap spacing are studied with the Reynolds number of 100 and the Prandtl number of 0.71. A varied range of rotational speed ratios are investigated for four different gap spacing, i.e., 3.0, 1.5, 0.7, and 0.2. The results show that, for the first cylinder, the lift and drag coefficients for large gap spacing are similar to those for a single cylinder; for the second cylinder, the lift coefficient descends with the increase in the angular velocity for all gap spacing, while the drag coefficient ascends except for the gap spacing of 3.0. The results of the averaged periodic Nusselt number on the surface of the cylinders show that, for small distances between the cylinders and low angular velocities, conduction is a dominant mechanism of heat transfer, but for large distances and high angular velocities, convection is the main mechanism of heat transfer.     

An integral representation of the surface-impedance tensor for incompressible elastic materials

A fundamental result in anisotropic elasticity and surface-wave theory is the integral representation for the surface-impedance tensor first derived by Barnett and Lothe in 1973. However, this representation is only valid for compressible materials but not valid for incompressible materials. In this paper the corresponding integral representation for the surface-impedance tensor valid for incompressible materials is derived and is used to establish the uniqueness of surface-wave speed and to obtain an expression for the tensor Green's function for the infinite space. Mathematics subject classifications (2000) 74B05, 74B15, 74B20, 74J15     

Indentation by nominally flat or conical indenters with rounded corners

Axisymmetric indentation of a flat surface is considered: specifically, the case of flat-ended indenter with rounded edges, and the case of a shallow cone with a rounded tip. Analytical solutions are obtained for the normal and sequential tangential loading, in both full or partial slip conditions (with the Cattaneofn9Mindlin approximation) , and for the complete interior stress field in all these conditions.Implications for strength of the contact are discussed with reference to metallic or brittle materials, with the intention to shed more light in particular to the understanding of common fretting fatigue or indentation testings with nominally flat or conical indenters. It is found that the strength of the contact, which is nominally zero for perfectly sharp flat or conical indenters, is well defined even for a small radius of curvature. This is particularly true for the flat indenter, for which the strength is even significantly higher than for the classical Hertzian indenter for a wide range of geometrical and loading conditions, rendering it very attractive for design purposes.