Mixed Mode Dynamic Fracture in Particulate Reinforced Functionally Graded Materials

A detailed analytical and experimental investigation is presented to understand the dynamic fracture behavior of functionally graded materials (FGMs) under mode I and mixed mode loading conditions. Crack-tip stress, strain and displacement fields for a mixed mode crack propagating at an angle from the direction of property gradation were obtained through an asymptotic analysis coupled with a displacement potential approach. This was followed by a comprehensive series of experiments to gain further insight into the behavior of propagating cracks in FGMs. Dynamic photoelasticity coupled with high-speed photography was used to obtain crack tip velocities and dynamic stress fields around the propagating cracks. Birefringent coatings were used to conduct the photoelastic study due to the opaqueness of the FGMs. Dynamic fracture experiments were performed using different specimen geometries to develop a dynamic constitutive fracture relationship between the mode I dynamic stress intensity factor (K _ID ) and crack-tip velocity ( ) for FGMs with the crack moving in the direction of increasing fracture toughness. A similar -K _ID relation was also obtained for matrix material (polyester) for comparison purposes. The results obtained show that crack propagation velocities in FGMs were about 80% higher than the polyester matrix. Crack arrest toughness was found to be about 10% lower than the value of local fracture toughness in FGMs.     

Oscillatory flow in microchannels

Commonly used, lumped-parameter expressions for the impedance of an incompressible viscous fluid subjected to harmonic oscillations in a channel were compared with exact expressions, based on solutions of the Navier-Stokes equations for slots and channels of circular and rectangular cross-section, and were found to differ by as much as 30% in amplitude. These differences resulted in predicted discrepancies by as much as 400% in frequency response amplitude for simple second-order systems based on size scales and frequencies encountered in microfluidic devices. These predictions were verified experimentally for rectangular microchannels and indicate that underdamped fluidic systems operating near the corner frequency of any included flow channel should be modeled with exact expressions for impedance to avoid potentially large errors in predicted behavior.List of symbols A Channel cross-sectional area (m²) - A_c Membrane area (m²) - a Rectangular duct and slot half-width or radius (m) - b Rectangular duct half-depth and slot depth (m) - C Capacitance (m³/Pa) - C - D_h Channel hydraulic diameter (m) - E Voltage (V) - f Darcy friction factor - F Force (N) - I Channel inertance (Pa s²/m³) - i - Imaginary part of a complex number - J_k Bessel function of the first kind of order k - System transfer function - K Sum of minor loss factors - k Membrane stiffness (N/m) - L Channel length (m) - n Outward unit normal vector - P Fluid pressure (Pa) - p_n - Q Volumetric flow rate (m³/s) - R Channel resistance (Pa s/m³) - Real part of a complex number - Re Reynolds number, - V Velocity (m/s) - _V Volume (m³) - w Axial component of velocity (m/s) - Harmonic amplitude of membrane centerline displacement - Fluid impedance (kg/m⁴ s) - Duct aspect ratio, b/a - ² Nondimensional frequency parameter, - Nondimensional corner frequency, - Membrane shape factor - C/C - µ Fluid dynamic viscosity (Pa s) - Fluid kinematic viscosity (m²/s) - Mass density (kg/m³) - Radian frequency - _c R_s/I_s cutoff or corner frequency - _n Undamped natural frequency - Channel shape parameter in Eqs. 29 and 30 - Damping ratio - ( )_e Exact property - ( )_s Simplified property - (^–) Spatial average - Complex quantity     

Resolvent Estimates and Maximal Regularity in Weighted L q -spaces of the Stokes Operator in an Infinite Cylinder

Let be an infinite cylinder of , n ≥ 3, with a bounded cross-section of C ^1,1-class. We study resolvent estimates and maximal regularity of the Stokes operator in for 1 < q, r < ∞ and for arbitrary Muckenhoupt weights ω ∈ A _r with respect to x′ ∈ Σ. The proofs use an operator-valued Fourier multiplier theorem and techniques of unconditional Schauder decompositions based on the -boundedness of the family of solution operators for a system in Σ parametrized by the phase variable of the one-dimensional partial Fourier transform. Supported by the Gottlieb Daimler- und Karl Benz-Stiftung, grant no. S025/02-10/03.     

The effect of cross flow on one-dimensional spectra measured using hot wires

Expressions were developed to estimate the cross-flow error that occurs in the one-dimensional velocity spectra determined by applying Taylors frozen field hypothesis to measurements with single- and cross-wire probes. The cross-flow error and the error caused by the unsteady convection of the small-scale motions were evaluated for typical measurements. It was found that the cross-flow error could be significant in inertial range of the measured one-dimensional spectra, and was much larger than the error caused by the unsteady convection of the small-scale motions in the one-dimensional spectra of the cross-stream velocity components, and . The results indicate that the one-dimensional spectra of the streamwise velocity component measured with a single-wire probe should be significantly more accurate than the spectra measured with a cross-wire probe. The cross-flow error in the one-dimensional spectra also becomes much less important in the dissipation range of the measured spectra.

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Effect of surface layers on the constriction resistance of an isothermal spot. Part II: Analytical results for thick layers

Solution of a non-homogeneous Fredholm integral equation of the second kind [1], which forms the basis for the evaluation of the constriction resistance of an isothermal circular spot on a half-space covered with a surface layer of different material, is considered for the case when the ratio, , of layer thickness to spot radius is larger than unity. The kernel of the integral equation is expanded into an infinite series in ascending odd-powers of (1/) and an approximate kernel accurate to (^–(2M+1)) is derived therefrom by terminating the series after an arbitrary but finite number of terms, M. The approximate kernel is rearranged into a degenerate form and the integral equation with this approximate kernel is reduced to a system of M linear equations. An explicit analytical solution is obtained for a four-term approximation of the kernel and the resulting constriction resistance is shown to be accurate to (^–9). Solutions of lower orders of accuracy with respect to (1/) are deduced from the four-term solution. The analytical approximations are compared with very accurate numerical solutions and it is shown that the (^–9)-approximation predicts the constriction resistance exceedingly well for any 1 over a four orders of magnitude variation of layer-to-substrate conductivity ratio for both conducting and insulating layers. It is further shown that, for all practical purposes, an (^–3)-approximation gives results of adequate accuracy for > 2.     

Pulsed-wire measurements in the near-wall layer in a reattaching separated flow

Pulsed-wire velocity measurements have been made in the near-wall layer, including the viscous sublayer, beneath a separated flow. A method for correcting the error caused by fluctuations in velocity gradient is given, extending the work of Schober et al. (1998). The measurements show that the r.m.s. of the streamwise velocity fluctuations scale closely in accordance with an inner-layer scaling, where the velocity scale, , is based on the r.m.s. of the wall shear stress fluctuations (measured by means of a pulsed-wire shear stress probe), rather than the mean wall shear stress. The effects of velocity gradient are only significant beneath of 10 or less.List of symbols C Calibration constant - f Function representing mean velocity - h_f Height of fence above splitter plate surface - L Length scale of outer-layer structures - s Distance between pulsed and sensor wires - u r.m.s. of U - Velocity scale based on r.m.s of wall shear stress fluctuation - U Instantaneous velocity in x-direction - U_m Instantaneous measured velocity in x-direction - U_r Free-stream reference velocity - x Streamwise direction from separation point - y Distance from splitter plate surface, in normal direction - X_r Length of separation bubble - ₀ Thickness scale in oscillating layer - Blasius laminar boundary layer parameter - Density - Wall shear stress - r.m.s. of wall shear stress fluctuation - Frequency of oscillating layer - Kinematic viscosity - Overbar denotes time average     

Methods of Small and Large δ in the Nonlinear Dynamics – A Comparative Analysis

New asymptotic approaches for dynamical systems containing a power nonlinear term x ⁿ are proposed and analyzed. Two natural limiting cases are studied: n 1 + , 1 and n . In the firstcase, the 'small method' (SM)is used and its applicability for dynamical problems with the nonlinearterm sin as well as the usefulness of the SMfor the problem with small denominators are outlined. For n , a new asymptotic approach is proposed(conditionally we call it the 'large method' –LM). Error estimations lead to the followingconclusion: the LM may be used, even for smalln, whereas the SM has a narrow application area. Both of the discussed approaches overlap all values ofthe parameter n.     

Entropy generation due to laser step input pulse heating

Laser heating of surfaces results in thermal expansion of the substrate material in the region irradiated by a laser beam. In this case, the thermodynamic irreversibility associated with the thermal process is involved with temperature and thermal stress fields. In the present study, entropy analysis is carried out to quantify the thermodynamic irreversibility pertinent to laser pulse heating process. The formulation of entropy generation due to temperature and stress fields is presented and entropy generation is simulated for steel substrate. It is found that the rapid rise of surface displacement in the early heating period results in high rate of entropy generation due to stress field in the surface region while entropy generation due to temperature field increases steadily with increasing depth from the surface. c ₁ Wave speed in the solid (m/s) - c ₁* Dimensionless wave speed - c ₂ Constant - C _p Specific heat (J/kg.K) - E Elastic modules (Pa) - I Power intensity (W/m²) - I ₁ Power intensity after surface reflection (W/m²) - I _o Laser peak power intensity (W/m²) - k Thermal conductivity (W/m.K) - r _f Reflection coefficient - s Laplace variable - S Entropy generation rate (W/m³K) - S* Dimensionless entropy generation rate - T(x, t) Temperature (K) - T*(x*, t*) Dimensionless temperature - Temperature in Laplace domain (K) - Dimensionless reference temperature - t Time (s) - t* Dimensionless time - U Displacement (m) - U* Dimensionless displacement (U) - W* _lost Dimensionless lost work - x Spatial coordinate (m) - x* Dimensionless distance (x) - Thermal diffusivity (m²/s) - _T Thermal expansion coefficient (1/K) - Poissons ratio - Absorption coefficient (1/m) - Density (kg/m³) - _x Thermal stress (Pa) - _x * Dimensionless thermal stress      

Optimal cupolas of uniform strength: Spherical M-shells and axisymmetric T-shells

Summary The first part of this paper is concerned with the optimal design of spherical cupolas obeying the von Mises yield condition. Five different load combinations, which all include selfweight, are investigated. The second part of the paper deals with the optimal quadratic meridional shape of cupolas obeying the Tresca yield condition, considering selfweight plus the weight of a non-carrying uniform cover. It is established that at long spans some non-spherical Tresca cupolas are much more economical than spherical ones.

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Optimale Kuppeln gleicher Festigkeit: Kugelschalen und axialsymmetrische Schalen

Übersicht Im ersten Teil dieser Arbeit wird der optimale Entwurf sphärischer Kuppeln behandelt, wobei die von Misessche Fließbewegung zugrunde gelegt wird. Fünf verschiedene Lastkombinationen werden untersucht. Der zweite Teil befaßt sich mit der optimalen quadratischen Form des Meridians von Kuppeln, die der Fließbedingung von Tresca folgen.

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List of Symbols a_k, b_k, c_k, A_k, B_k, C_k coefficients used in series solutions - A, B constants in the nondimensional equation of the meridional curve - normal component of the load per unit area of the middle surface - meridional and circumferential forces per unit width - radial pressure per unit area of the middle surface, - skin weight per unit area of the middle surface, - vertical external load per unit horizontal area, - base radius, - R radius of convergence - s - cupola thickness, - u, w subsidiary functions for quadratic cupolas - vertical component of the load per unit area of middle surface - resultant vertical force on a cupola segment - structural weight of cupola, - combined weight of cupola and skin, - distance from the axis of rotation, - vertical distance from the shell apex, - z auxiliary variable in series solutions - specific weight of structural material of cupola - radius of the middle surface, - uniaxial yield stress - meridional stress, - circumferential stress, - _a, _b, _c, _d, _e subsidiary variables used in evaluating the meridional stress - auxiliary function used in series solutions This paper constitutes the third part of a study of shell optimization which was initiated and planned by the late Prof. W. Prager     

Numerical study on local entropy generation in a burner fueled with various fuels

This study considers numerical simulations of the combustions of hydrogen and various hydrocarbons with air, including 21% oxygen and 79% nitrogen, in a burner and the numerical solution of the local entropy generation rate due to the high temperature and velocity gradients in the combustion chamber. The combustion is simulated for the fuel mass flow rates providing the same heat transfer rate to the combustion chamber in the each fuel case. The effects of (only in the case of H₂ fuel) and equivalence ratio () on the combustion and entropy generation rate are investigated for the different (from 5,000 to 10,000 W) and s (from 0.5 to 1.0). The numerical calculation of combustion is performed individually for all cases with the help of the Fluent CFD code. Furthermore, a computer program has been developed to numerically calculate the volumetric entropy generation rate distributions and the other thermodynamic parameters by using the results of the calculations performed with the FLUENT code. The calculations bring out that the maximum reaction rates decrease with the increase of (or the decrease of ). The large positive and negative temperature gradients occur in the axial direction, nonetheless, the increase of significantly reduces them. The calculations bring out also that with the increase of from 0.5 to 1.0, the volumetric local entropy generation rates decrease about 4% and that the merit numbers increase about 16%.     

The Boundary Riemann Solver Coming from the Real Vanishing Viscosity Approximation

We study the limit of the hyperbolic–parabolic approximation

Control of unstable oscillations in a confined and bounded injecting wall channel

The analysis of a confined flow field generated through two separated injecting walls was carried out by studying the effect of a mass flow rate difference between the two walls. Such a parameter has been found to play a major role in unstable flow field behaviour. Concretely speaking, we have identified two vortex shedding phenomena, i.e. the main flow and the wall vortex shedding phenomena. Results clearly show that when the mass flow rate is increased at the second injecting wall, wall vortex coherence is enhanced and impinging of such structures forces a coupling phenomenon to develop between flow field dynamics and acoustics. On the other hand, only a 15% mass flow rate difference of the first injecting block is sufficient to prevent such coupling between acoustics and vortex shedding phenomenon. Consequently, the resonance phenomenon is pronouncedly weakened and significant oscillation reduction is achieved.Nomenclature sound velocity (m/s) - nth longitudinal acoustic mode (Hz) - h_t height of the nozzle throat (m) - h_c channel height (m) - l length between the edge of the second injecting block and the nozzle location (m) - L channel length (m) - P mean pressure at the front-head (Pa) - P fluctuating pressure (Pa) - q_m, q₁, q₂ total, first and second injecting block mass flow rate (kg/s) - S_x normalised power spectral density of the x fluctuations (Hz^-1) - s=w h_c characteristic surface area (m²) - T temperature of the flow (K) - u, v longitudinal and lateral velocity component (m/s) - u_X longitudinal mean velocity at X location (m/s) - u_m maximum longitudinal velocity (m/s) - u, v longitudinal and lateral fluctuating velocity (m/s) - characteristic acoustic velocity (m/s) - v_w wall injection velocity (m/s) - w channel width (m) - X, Y, Z non-dimensional axis normalised respectively by l, h_c and w - dynamic viscosity (kg/ms) - density (kg/m³) - time delay (s)Dimensionless parameters turbulence intensity - Mach number - Re_c= M a h_c/ Reynolds number - Re_w= v_w h_c/ wall injection Reynolds number - correlation coefficient of pressure and velocity fluctuations - normalised longitudinal velocity component - parameter of unbalanced mass flow rate between the two injecting blocks - specific heat ratio - coherence function     

“Thermal layer” effect in MHD: Interaction of a parallel shock with a layer of decreased density

Interaction of a parallel fast MHD shock with a layer of decreased density is discussed using ideal MHD approach. This is an extrapolation of gas dynamic thermal layer effect on ideal MHD. Computer simulations show that a magnetic field of a moderate intensity ( 1) may change the character of the flow for intermediate Mach numbers (M 5) and a new raking regime may occur which is not observed in the absence of a magnetic field. Self similar precursor analogous to that in gas dynamics may develop in the case of highM and low density in the layer but magnetic forces essentially decrease its growth rate. This problem appears in connection with cosmical shock propagation where planetary magnetic tails play the role of the thermal layer, and it may also be observed in the laboratory when the shock is strong enough to heat the walls ahead of it.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

RADAR ECHO DELAY IN THE(Ω,Aμν)-FIELD THEORY

The present work aims to consider the.fourth test of general relativity theory by Shapiro.using radar echo delay in Yu’s(Ω,A_μν)-field theory.     

Analytical solutions of one-dimensional discrete dynamical systems with chaotic behaviour

This paper studies exact analytical solutions in closed form of the difference equation

Investigation of the secondary corner vortex in a benchmark turbulent backward-facing step using cross-correlation particle imaging velocimetry

An experimental study of a turbulent backward-facing step (BFS) was undertaken to investigate the vortex structures behind the step. Attention was given to the secondary vortex because of its poor representation in literature and its potential for evaluating computational turbulence models. A 2D, cross-correlation particle image velocimeter (PIV) was developed, which allowed measurement of the highly turbulent, reversing step flow. Global, high resolution data was obtained for the cross-sectional plane of the BFS and for several other planes parallel to it. Measurement planes across the step revealed the 3D nature of the secondary vortex and an unexpected flow structure was identified. The secondary vortex was found to traverse across the flow, from the cross-sectional plane towards the step edge–sidewall corner.List of symbols AR aspect ratio - d particle displacement (m) - d error in particle displacement (m) - D expansion channel height (mm) - D₀ inlet channel height (mm) - ER expansion ratio - H step height (mm) - N number of samples - Re_H Reynolds number based on step height - Sp(x,y) centre coordinates of primary vortex (mm) - Ss(x,y) centre coordinates of secondary vortex (mm) - t laser pulse separation time (s) - t error in pulse separation time (s) - U horizontal velocity (m/s) - ̄ mean horizontal velocity (m/s) - horizontal velocity variance (m²/s²) - inlet centreline mean velocity (m/s) - inlet centreline velocity variance (m²/s²) - V vertical velocity (m/s) - VM velocity magnitude (m/s) - VM error in velocity magnitude (m/s) - W step width (mm) - x length dimension (mm) - y height dimension (mm) - z width dimension (mm) - Xr shear layer reattachment point (mm) - Xr reattachment point for infinite step width (mm) - Xs secondary vortex separation point (mm) - Ys secondary vortex reattachment point (mm) - U velocity error (m/s) - mean velocity error estimate (m/s) - velocity variance error estimate (m²/s²) - _bot bottom inlet boundary layer thickness (mm) - _top top inlet boundary layer thickness (mm) - ₉₉ 0.99 boundary layer thickness (mm)     

Thin-Walled Beams: the Case of the Rectangular Cross-Section

In this paper we present an asymptotic analysis of the three-dimensional problem for a thin linearly elastic cantilever =×(0,l) with rectangular cross-section of sides and ², as goes to zero. Under suitable assumptions on the given loads, we show that the three-dimensional problem converges in a variational sense to the classical one-dimensional model for extension, flexure and torsion of thin-walled beams. Mathematics Subject Classifications (2000) 474K20, 74B10, 49J45.     

The flow regime in the turbulent near wake of a hemisphere

The work presented is a wind tunnel study of the near wake region behind a hemisphere immersed in three different turbulent boundary layers. In particular, the effect of different boundary layer profiles on the generation and distribution of near wake vorticity and on the mean recirculation region is examined. Visualization of the flow around a hemisphere has been undertaken, using models in a water channel, in order to obtain qualitative information concerning the wake structure.List of symbols C _p pressure coefficient, - D diameter of hemisphere - n vortex shedding frequency - p pressure on model surface - p ₀ static pressure - Re Reynolds number, - St Strouhal number, - U, V, W local mean velocity components - mean freestream velocity inX direction - U ^* shear velocity, - u, v, w velocity fluctuations inX, Y andZ directions - X Cartesian coordinate in longitudinal direction - Y Cartesian coordinate in lateral direction - Z Cartesian coordinate in direction perpendicular to the wall - it* boundary layer displacement thickness, - diameter of model surface roughness - elevation angleI - O boundary layer momentum thickness, - _w wall shearing stress - dynamic viscosity of fluid - density of fluid - streamfunction - _x longitudinal component of vorticity, - _y lateral component of vorticity, - _z vertical component of vorticity, This paper was presented at the Ninth symposium on turbulence, University of Missouri-Rolla, October 1–3, 1984     

Unsteady flow in an annulus between two concentric rotating spheres

An analysis is presented for the unsteady laminar flow of an incompressible Newtonian fluid in an annulus between two concentric spheres rotating about a common axis of symmetry. A solution of the Navier-Stokes equations is obtained by employing an iterative technique. The solution is valid for small values of Reynolds numbers and acceleration parameters of the spheres. In applying the results of this analysis to a rotationally accelerating sphere, a virtual moment of intertia is introduced to account for the local inertia of the fluid.Nomenclature R _i radius of the inner sphere - R _o radius of the outer sphere - radial coordinate - r dimensionless radial coordinate, - meridional coordinate - azimuthal coordinate - time - t dimensionless time, - Re _i instantaneous Reynolds number of the inner sphere, _i R _k ² / - Re _o instantaneous Reynolds number of the outer sphere, _o R _o ² / - radial velocity component - V _r dimensionless radial velocity component, - meridional velocity component - V dimensionless meridional velocity component, - azimuthal velocity component - V dimensionless azimuthal velocity component, - viscous torque - T dimensionless viscous torque, - viscous torque at surface of inner sphere - T _i dimensionless viscous torque at surface of inner sphere, - viscous torque at surface of outer sphere - T _o dimensionless viscous torque at surface of outer sphere, - externally applied torque on inner sphere - T _p,i dimensionless applied torque on inner sphere, - moment of inertia of inner sphere - Z _i dimensionless moment of inertia of inner sphere, - virtual moment of inertia of inner sphere - Z _i,v dimensionless virtual moment of inertia of inner sphere, - virtual moment of inertia of outer sphere - _i instantaneous angular velocity of the inner sphere - _o instantaneous angular velocity of the outer sphere - density of fluid - viscosity of fluid - kinematic viscosity of fluid,/ - radius ratio,R _i/R _o - swirl function, - dimensionless swirl function, - stream function - dimensionless stream function, - _i acceleration parameter for the inner sphere, - _o acceleration parameter for the outer sphere, - shear stress - _r dimensionless shear stress,      

Full Measure Reducibility for Generic One-parameter Family of Quasi-periodic Linear Systems

Let be the set of m × m matrices A(λ) depending analytically on a parameter λ in a closed interval . Consider one-parameter families of quasi-periodic linear differential equations: , where is analytic and sufficiently small. We prove that there is an open and dense set in , such that for each the equation can be reduced to an equation with constant coefficients by a quasi-periodic linear transformation for almost all in Lebesgue measure sense provided that g is sufficiently small. The result gives an affirmative answer to a conjecture of Eliasson (In: Proceeding of Symposia in Pure Mathematics). Dedicated to Professor Zhifen Zhang on the occasion of her 80th birthday