Micro/macro-crack growth due to creep–fatigue dependency on time–temperature material behavior

The implicit character of micro-structural degradation is determined by specifying the time history of crack growth caused by creep–fatigue interaction at high temperature. A dual scale micro/macro-equivalent crack growth model is used to illustrate the underlying principle of multiscaling which can be applied equally well to nano/micro. A series of dual scale models can be connected to formulate triple or quadruple scale models. Temperature and time-dependent thermo-mechanical material properties are developed to dictate the design time history of creep–fatigue cracking that can serve as the master curve for health monitoring.In contrast to the conventional procedure of problem/solution approach by specifying the time- and temperature-dependent material properties as a priori, the desired solution is then defined for a class of anticipated loadings. A scheme for matching the loading history with the damage evolution is then obtained. The results depend on the initial crack size and the extent of creep in proportion to fatigue damage. The path dependent nature of damage is demonstrated by showing the range of the pertinent parameters that control the final destruction of the material. A possible scenario of 20 yr of life span for the 38Cr2Mo2VA ultra-high strength steel is used to develop the evolution of the micro-structural degradation. Three micro/macro-parameters μ^*, d^* and σ^* are used to exhibit the time-dependent variation of the material, geometry and load effects. They are necessary to reflect the scale transitory behavior of creep–fatigue damage. Once the algorithm is developed, the material can be tailor made to match the behavior. That is a different life span of the same material would alter the time behavior of μ^*, d^* and σ^* and hence the micro-structural degradation history. The one-to-one correspondence of the material micro-structure degradation history with that of damage by cracking is the essence of path dependency. Numerical results and graphs are obtained to demonstrate how the inherently implicit material micro-structure parameters can be evaluated from the uniaxial bulk material properties at the macroscopic scale.The combined behavior of creep and fatigue can be exhibited by specifying the parameter ξ with reference to the initial defect size a₀. Large ξ (0.90 and 0.85) gives critical crack size a_cr = 11–14 mm (at t < 20 yr) for a₀ about 1.3 mm. For small ξ (0.05 and 0.15), there results critical a_cr = 6–7 mm (at t < 20 yr) for a₀ about 0.7–0.8 mm. The initial crack is estimated to increase its length by an order of magnitude before triggering global to the instability. This also applies ξ ≈ 0.5 where creep interacts severely with fatigue. Fine tuning of a_cr and a₀ can be made to meet the condition oft = 20 yr.Trade off among load, material and geometric parameters are quantified such that the optimum conditions can be determined for the desired life qualified by the initial–final defect sizes. The scenario assumed in this work is indicative of the capability of the methodology. The initial–final defect sizes can be varied by re-designing the time–temperature material specifications. To reiterate, the uniqueness of solution requires the end result to match with the initial conditions for a given problem. This basic requirement has been accomplished by the dual scale micro/macro-crack growth model for creep and fatigue.     

Edge dislocations generated from a microcrack under initial residual stress of non-uniform distribution

Crystal nucleation gives rise to inhomogeneity in the crystal lattice. The prevailing stresses and strains caused by non-uniform cooling can create microcracks with residual stresses locked-in at the end segments. These stresses can have a non-uniform distribution where the amplitude can increase or decrease from the microcrack tip which is highly strained to generate edge dislocations under in-plane shear. A dual scale microdislocation crack model is considered by focusing attention near the microcrack tip singularity such that more than 10 orders of magnitude in lineal dimension can be covered from the atomic to the microscopic scale. The concept of a scale multiplier is employed to connect the microscopic and atomic scale results. Discontinuity at the cross-scaling location is necessitated by dividing the full range of the non-equilibrium process into two regions within which equilibrium mechanics can be used. When needed, additional mesoregions can be added to reduce the transient discontinuities.Solved in closed form is the solution for the generation of edge dislocations due to non-uniform residual stress distributions at the end segment of the microcrack tip which will henceforth be referred to simply as the “tip”. Three different Cases I, II and III will be considered where the residual stress will possess a peak at the different locations. Case I for the furthest away from the tip, Case II for the peak nearest to the tip and Case III for the peak in the middle of the residual stress segment. Compared are the scale multiplier α whose maximum value being one corresponding to no discontinuity at cross-scaling. Hence, small α corresponds to large discontinuity. For Cases I, II and III, αs are found, respectively as 0.17, 0.43 and 0.28. The largest discontinuity occurred at α = 0.17 when the peak of the residual stress is farthest away from the microcrack tip. The largest number of edge dislocations or imperfections are also generated for Case I. The precise location of the residual stress peak is related to the magnitude and the segment length of the residual stress. These findings are manifestation of the variety of non-homogeneities that can arise in a metal alloy during crystal formation, not to mention the prevailing conditions at the grain boundaries. The idea is not to account for the details per se but to test the sensitivities of the microscopic and atomic parameters involved. To this end, the energy density function for the dual scale model will be determined and discussed in connection with what has been emphasized.     

A moving dislocation in a magneto–electro-elastic solid

This work considers the generalized plane problem of a moving dislocation in an anisotropic elastic medium with piezoelectric, piezomagnetic and magnetoelectric effects. The closed-form expressions for the elastic, electric and magnetic fields are obtained using the extended Stroh formalism for steady-state motion. The radial components, E_rand H_r, of the electric and magnetic fields as well as the hoop components, D_θ and B_θ, of electric displacement and magnetic flux density are found to be independent of θ in a polar coordinate system. This interesting phenomenon is proven to be is a consequence of the electric and magnetic fields, electric displacement and magnetic flux density that exhibit the singularity r⁻¹ near the dislocation core. As an illustrative example, the more explicit results for a moving dislocation in a transversely isotropic magneto–electro-elastic medium are provided and the behavior of the coupled fields is analyzed in detail.     

Kinetics of microcrack blunting ahead of macrocrack approaching shear wave speed

The fracturing of glass and tearing of rubber both involve the separation of material but their crack growth behavior can be quite different, particularly with reference to the distance of separation of the adjacent planes of material and the speed at which they separate. Relatively speaking, the former and the latter are recognized, respectively, to be fast and slow under normal conditions. Moreover, the crack tip radius of curvature in glass can be very sharp while that in the rubber can be very blunt. These changes in the geometric features of the crack or defect, however, have not been incorporated into the modeling of running cracks because the mathematical treatment makes use of the Galilean transformation where the crack opening distance or the change in the radius of curvature of the crack does not enter into the solution. Change in crack speed is accounted for only via the modulus of elasticity and mass density. For this simple reason, many of the dynamic features of the running crack have remained unexplained although speculations are not lacking. To begin with, the process of energy dissipation due to separation is affected by the microstructure of the material that distinguishes polycrystalline from amorphous form. Energy extracted from macroscopic reaches of a solid will travel to the atomic or smaller regions at different speeds at a given instance. It is not clear how many of the succeeding size scales should be included within a given time interval for an accurate prediction of the macroscopic dynamic crack characteristics. The minimum requirement would therefore necessitate the simultaneous treatment of two scales at the same time. This means that the analysis should capture the change in the macroscopic and microscopic features of a defect as it propagates. The discussion for a dual scale model has been invoked only very recently for a stationary crack. The objective of this work is to extend this effort to a crack running at constant speed beyond that of Rayleigh wave. Developed is a dual scale moving crack model containing microscopic damage ahead of a macroscopic crack with a gradual transition. This transitory region is referred to as the mesoscopic zone where the tractions prevail on the damaged portion of the material ahead of the original crack known as the restraining stresses, the magnitude of which depends on the geometry, material and loading. This damaged or restraining zone is not assumed arbitrarily nor assumed to be intrinsically a constant in the cohesive stress approach; it is determined for each step of crack advancement. For the range of micronotch bluntness with 0 < β < 30° and 0.2 σ_∞/σ₀ 0.5, there prevails a nearly constant restraining zone size as the crack approaches the shear wave speed. Note that β is the half micronotch angle and the applied stress ratio is σ_∞/σ₀ with σ₀ being the maximum of the restraining stress. For σ_∞/σ₀ equal to or less than 0.5, the macrocrack opening displacement COD is nearly constant and starts to decrease more quickly as the crack approaches the shear wave speed. For the present dual scale model where the normalized crack speed v/c_s increases with decreasing with the one-half microcrack tip angle β. There prevails a limit of crack tip bluntness that corresponds to β 36° and v/c_s 0.15. That is a crack cannot be maintained at a constant speed if the bluntness is increased beyond this limiting value. Such a feature is manifestation of the dependency of the restraining stress on crack velocity and the applied stress or the energy pumped into the system to maintain the crack at a constant velocity. More specifically, the transitory character from macro to micro is being determined as part of the unknown solution. Using the energy density function dW/dV as the indicator, plots are made in terms of the macrodistance ahead of the original crack while the microdefect bluntness can vary depending on the tip geometry. Such a generality has not been considered previously. The macro-dW/dV behavior with distance remains as the inverse r relation yielding a perfect hyperbola for the homogeneous material. This behavior is the same as the stationary crack. The micro-dW/dV relations are expressed in terms of a single undetermined parameter. Its evaluation is beyond the scope of this investigation although the qualitative behavior is expected to be similar to that for the stationary crack. To reiterate, what has been achieved as an objective is a model that accounts for the thickness of a running crack since the surface of separation representing damage at the macroscopic and microscopic scale is different. The transitory behavior from micro to macro is described by the state of affairs in the mesoscopic zone.     

On corner flows of Oldroyd-B fluids

Exact series solutions for planar creeping flows of Oldroyd-B fluids in the neighbourhood of sharp corners are presented and discussed. Both reentrant and non-reentrant sectors are considered. For reentrant sectors it is shown that more than one type of series solution can exist formally, one type exhibiting Newtonian-like asymptotic behaviour at the corner, away from walls, and another type exhibiting the same kind of asymptotics as an Upper Convected Maxwell (UCM) fluid. The solutions which are Newtonian-like away from walls are shown to develop non-integrable stress singularities at the walls when the no-slip velocity boundary condition is imposed. These mathematical solutions are therefore inadmissible from the physical viewpoint under no-slip conditions. An inadmissible solution, with stress singularities which are not everywhere integrable, is identified among the solutions of UCM-type. For a 270° reentrant sector the radial behaviour of the normal stress is everywhere r^−0.613. In the viscometric region near a wall, the radial normal stress σ^rr behaves like (rε)^−0.613, where ε is the angle made with the wall. In addition σ^rθ is infinite (not integrable) at the wall even when r is non-zero. Another UCM-type solution has a normal stress behaviour away from walls which is r^−0.985 for 270° sector. Again, this solution has a non-integrable stress singularity and is therefore inadmissible. Finally, for non-reentrant sectors it is shown that the flow is always Newtonian-like away from walls.     

Crack repair using an elastic filler

The effect of repairing a crack in an elastic body using an elastic filler is examined in terms of the stress intensity levels generated at the crack tip. The effect of the filler is to change the stress field singularity from order 1/r^1/2 to 1/r^(1-λ) where r is the distance from the crack tip, and λ is the solution to a simple transcendental equation. The singularity power (1-λ) varies from (the unfilled crack limit) to 1 (the fully repaired crack), depending primarily on the scaled shear modulus ratio γ_r defined by G₂/G₁=γ_rε, where 2πε is the (small) crack angle, and the indices (1, 2) refer to base and filler material properties, respectively. The fully repaired limit is effectively reached for γ_r≈10, so that fillers with surprisingly small shear modulus ratios can be effectively used to repair cracks. This fits in with observations in the mining industry, where materials with G₂/G₁ of the order of 10^-3 have been found to be effective for stabilizing the walls of tunnels. The results are also relevant for the repair of cracks in thin elastic sheets.     

Multi-scale and multi-order singularity approach to non-equilibrium mechanics: Coupling of atomic-micro-macro damage

Offered in this work is the development of a macro/meso/micro model that covers the lineal scale of 10⁻¹¹ to 10⁰ by application of the volume energy density function. Boundary constraints and defect geometries are shown to play a role at the smaller scale in the same way as those at the macroscopic scale. Different orders of stress (or energy density) singularities are used to describe the defect geometry and prevailing constraint via the boundary conditions in a way similar to singularity adopted in classical fracture mechanics. Two classes of singularities have been identified in addition to classical one without violating the finiteness conditions of the local displacement and energy density. Still the connection of results from the different scales is no small task and is made possible by application of a scale multiplier. It is determined by considering the interactive effects of the parameters at the different scales from the atomic to the macroscopic. Unlike the classical boundary value problem approach, application of the scale multiplier has led to closed-form asymptotic multiscale solutions that otherwise would not have been made possible. The procedure is demonstrated for the anti-plane shear of a macro-micro-atomic model that accounts for imperfection at the different scales Published in Prikladnaya Mekhanika, Vol. 42, No. 1, pp. 3–22, January 2006.     

Boundary-layer flow of Reiner–Philippoff fluids past a stretching wedge

This paper presents a numerical analysis of the steady boundary-layer flow of a Reiner–Philippoff fluid induced by a 90° stretching wedge in a variable free stream. The governing partial differential equations are converted into a set of two ordinary differential equations by the use of a similarity transformation. The flow is therefore governed by a stretching velocity parameter λ and two non-Newtonian fluid parameters γ and μ₀. The variation of the skin friction, as well as other flow characteristics, as a function of the governing parameters is presented graphically and tabulated. A stability analysis has also been performed for this self-similar flow based on linear disturbances to the steady similarity solutions. The results presented in this paper reveal that there are no multiple (dual) solutions for the present problem and the unique solution is stable.     

A Numerical Study of Micro-Heterogeneity Effects on Upscaled Properties of Two-Phase Flow in Porous Media

Commonly, capillary pressure–saturation–relative permeability (P ^c–S–K _r) relationships are obtained by means of laboratory experiments carried out on soil samples that are up to 10–12 cm long. In obtaining these relationships, it is implicitly assumed that the soil sample is homogeneous. However, it is well known that even at such scales, some micro-heterogeneities may exist. These heterogeneous regions will have distinct multiphase flow properties and will affect saturation and distribution of wetting and non-wetting phases within the soil sample. This, in turn, may affect the measured two-phase flow relationships. In the present work, numerical simulations have been carried out to investigate how the variations in nature, amount, and distribution of sub-sample scale heterogeneities affect P ^c–S–K _r relationships for dense non-aqueous phase liquid (DNAPL) and water flow. Fourteen combinations of sand types and heterogeneous patterns have been defined. These include binary combinations of coarse sand imbedded in fine sand and vice versa. The domains size is chosen so that it represents typical laboratory samples used in the measurements of P ^c–S–K _r curves. Upscaled drainage and imbibition P ^c–S–K _r relationships for various heterogeneity patterns have been obtained and compared in order to determine the relative significance of the heterogeneity patterns. Our results show that for micro-heterogeneities of the type shown here, the upscaled P ^c–S curve mainly follows the corresponding curve for the background sand. Only irreducible water saturation (in drainage) and residual DNAPL saturation (in imbibition) are affected by the presence and intensity of heterogeneities.     

Liquid–crystalline nematic polymers revisited

A new model for nematic polymers is proposed, based on the probability ψ(u,u,t) for a macromolecule to be oriented along direction u while embedded in a u environment created by its neighbours. The potential of the internal forces is written Φ(u,u) accordingly. The free energy contains a contribution ν Φ + k_BT ln ψ where the brackets mean an average over the probability distribution, while ν is the (uniform) polymer number density. An equation is derived for the time-evolution of the order parameter S = uu − I/3, together with an expression for the stress tensor. These two results offer a generalization of the Doi Model in so far as they include a distortional energy, analogue to the Frank elastic energy for low molecular mass nematics. Extending the Maier–Saupe variational procedure, we specify the way that the internal potential Φ(u,u) must be written for it to favour non-zero values of the order parameter, while giving a penalty to situations with gradients of the order parameter. The result is quite different from the potential proposed a decade ago by Marrucci and Greco (their Φ depends on u only), while it has a clear connection with the so-called Landau-de Gennes (LdG) tensor models, which are based on a free-energy depending on the order parameter and its gradients.     

High-order LES of turbulent heat transfer in a rotor–stator cavity

The present work examines the turbulent flow in an enclosed rotor–stator system subjected to heat transfer effects. Besides their fundamental importance as three-dimensional prototype flows, such flows arise in many industrial applications but also in many geophysical and astrophysical settings. Large eddy simulations (LES) are here performed using a spectral vanishing viscosity technique. The LES results have already been favorably compared to velocity measurements in the isothermal case (Séverac, E., Poncet, S., Serre, E., Chauve, M.P., 2007. Large eddy simulation and measurements of turbulent enclosed rotor–stator flows. Phys. Fluids, 19, 085113) for a large range of Reynolds numbers 10⁵Re=Ωb²/ν10⁶, in an annular cavity of large aspect ratio G=(b-a)/H=5 and weak curvature parameter R_m=(b-a)/(b+a)=1.8 (a,b the inner and outer radii of the rotor and H the interdisk spacing). The purpose of this paper is to extend these previous results in the non-isothermal case using the Boussinesq approximation to take into account the buoyancy effects. Thus, the effects of thermal convection have been examined for a turbulent flow Re=10⁶ of air in the same rotor–stator system for Rayleigh numbers up to Ra=10⁸. These LES results provide accurate, instantaneous quantities which are of interest in understanding the physics of turbulent flows and heat transfers in an interdisk cavity. Even at high Rayleigh numbers, the structure of the iso-values of the instantaneous normal temperature gradient at the disk surfaces resembles the one of the iso-values of the tangential velocity with large spiral arms along the rotor and more thin structures along the stator. The averaged results show small effects of density variation on the mean and turbulent fields. The turbulent Prandtl number is a decreasing function of the distance to the wall with 1.4 close to the disks and about 0.3 in the outer layers. The local Nusselt number is found to be proportional to the local Reynolds number to the power 0.7. The evolution of the averaged Bolgiano length scale L_B with the Rayleigh number indicates that temperature fluctuations may have a large influence on the dynamics only at the largest scales of the system for Ra10⁷, since L_B remains lower than the thermal boundary layer thicknesses.     

Logarithmic singularity of an elastic composite wedge subjected to out-of-the-plane extensional strain

When an elastic composite wedge is not under a plane strain deformation, an out-of-the-plane extensional strain exists. The singularity analysis for the stresses at the apex of the composite wedge reduces to a system of non-homogeneous linear equations. When the composite wedge consists of two anisotropic elastic materials, it is shown that the stresses have the (ln r) term for all combinations of wedge angles with few exceptions. The same is true when the materials are isotropic except that the (ln r) term may appear in the form of r(ln r) in the displacements only. For these isotropic composite wedges therefore the stresses are bounded, though not continuous, at the apex. However, there are isotropic composite wedges for which the stress singularity is logarithmic. Conditions are given for isotropic composite wedges for which the stresses are (a) uniform, (b) non-uniform but bounded and (c) logarithmic. Unlike the r^−λ singularity, the existence of the (ln r) term does not depend on the complete boundary conditions.     

Parallelization of pseudo-particle modeling and its application in simulating gas–solid fluidization

Pseudo-Particle Modeling (PPM) is a particle method proposed by Ge and Li in 1996 [Ge, W., & Li, J. (1996). Pseudo-particle approach to hydrodynamics of particle–fluid systems. In M. Kwauk & J. Li (Eds.), Proceedings of the 5th international conference on circulating fluidized bed (pp. 260–265). Beijing: Science Press] and has been used to explore the microscopic mechanism in complex particle–fluid systems. But as a particle method, high computational cost remains a main obstacle for its large-scale application; therefore, parallel implementation of this method is highly desirable. Parallelization of two-dimensional PPM was carried out by spatial decomposition in this paper. The time costs of the major functions in the program were analyzed and the program was then optimized for higher efficiency by dynamic load balancing and resetting of particle arrays. Finally, simulation on a gas–solid fluidized bed with 102,400 solid particles and 1.8 × 10⁷ pseudo-particles was performed successfully with this code, indicating its scalability in future applications.     

Dynamic interaction of various beams with the underlying soil – finite and infinite, half-space and Winkler models

Various beams lying on the elastic half-space and subjected to a harmonic load are analyzed by a double numerical integration in wavenumber domain. The compliances of the beam–soil systems are presented for a wide frequency range and for a number of realistic parameter sets. Generally, the soil stiffness G has a strong influence on the low-frequency beam compliance whereas the beam parameters EI and m^′ are more important for the high-frequency compliance. An important parameter is the elastic length l=(EI/G)^1/4 of the beam–soil system. Around the corresponding frequency ω_l=v_S/l, the wave velocity of the combined beam–soil system changes from the Rayleigh wave v_R≈v_S to the bending wave velocity v_B and the combined beam–soil wave has typically a strong damping. The interaction frequency ω_l is found not far from the characteristic frequency ω₀=(G/m^′)^1/2 where an amplification compared to the static compliance is observed for special parameter constellations. In contrast, real foundation beams show no resonance effects as they are highly damped by the radiation into the soil. At medium and high frequencies, asymptotes for the compliance of the beam–soil system are found, u/P(ρv_Paiω)^−3/4 in case of the dominating damping and u/P(−m^′ω²)^−3/4 for high frequencies. The low-frequency compliance of the coupled beam–soil system can be approximated by u/P1/Gl, but it also depends weakly on the width a of the foundation. All numerical results of different beam–soil systems are evaluated to yield a unique relation u/P₀=f(a/l). The integral transform method is also applied to ballasted and slab tracks of railway lines, showing the influence of train speed on the deformation of the track beam. The presented results of infinite beams on half-space are compared with results of finite beams and with infinite beams on a Winkler support. Approximating Winkler parameters are given for realistic foundation-soil systems which are useful when vehicle-track interaction is analyzed for the prediction of railway induced vibration.     

Effect of inertia on thermoelastic flow instability

Thermal effects induced by viscous heating cause thermoelastic flow instabilities in curvilinear shear flows of viscoelastic polymer solutions. These instabilities could be tracked experimentally by changing the fluid temperature T₀ to span the parameter space. In this work, the influence of T₀ on the stability boundary of the Taylor–Couette flow of an Oldroyd-B fluid is studied. The upper bound of the stability boundary in the Weissenberg number (We)–Nahme number (Na) space is given by the critical conditions corresponding to the extension of the time-dependent isothermal eigensolution. Initially, as T₀ is increased, the critical Weissenberg number, We_c, associated with this upper branch increases. Increasing T₀ beyond a certain value T^* causes the thermoelastic mode of instability to manifest. This occurs in the limit as We/Pe → 0, where Pe denotes the Péclet number. In this limit, the fluid relaxation time is much smaller than the time scale of thermal diffusion. T₀ = T^* represents a turning point in the We_c–Na_c curve. Consequently, the stability boundary is multi-valued for a wide range of Na values. Since the relaxation time and viscosity of the fluid decrease with increasing T₀, the elasticity number, defined as the ratio of the fluid relaxation time to the time scale of viscous diffusion, also decreases. Hence, O(10) values of the Reynolds number could be realized at the onset of instability if T₀ is sufficiently large. This sets limits for the temperature range that can be used in experiments if inertial effects are to be minimized.     

Absorption spectroscopy of mid and neighboring Z plasmas: Iron, nickel,copper and germanium

Opacities of four medium Z element plasmas (iron, nickel, copper and germanium) have been measured at the LULI-2000 facility in similar conditions: temperatures between 15 and 25 eV and densities between 2 and 10 mg/cm³, in a wavelength range (8–18 Å) including the strong 2p–3d structures.Two laser beams from the LULI facility were used in the nanosecond-picosecond configuration. The NANO-2000 beam (at λ = 0.53 μm) heated a gold hohlraum with an energy between 30 and 150 J with a duration of 0.6 ns. Samples covering half a hohlraum hole were thus radiatively heated. The picosecond pulse PICO-2000 beam (at λ = 1.053 μm) has been used to produce a short (about 10 ps) X-ray backlighter in order to reduce time variations of temperatures and densities during the measurement. A crystal high-resolution spectrometer was used as the main diagnostic to record at the same time the non-absorbed and the absorbed backlighter spectra. Radiation temperatures were measured using a broadband spectrometer. 1D and 2D simulations have been performed in order to estimate hydrodynamic plasmas parameters.The measured spectra have been compared with theoretical ones obtained using either the superconfiguration code SCO or the detailed term accounting code HULLAC. These comparisons allow us to check the modeling of the statistical broadening and of the spin-orbit splitting of the 2p–3d transitions and related effects such as the interaction between relativistic subconfigurations belonging to the same non-relativistic configuration.     

MOTIONS, FORCES AND MODE TRANSITIONS IN VORTEX-INDUCED VIBRATIONS AT LOW MASS-DAMPING

These experiments, involving the transverse oscillations of an elastically mounted rigid cylinder at very low mass and damping, have shown that there exist two distinct types of response in such systems, depending on whether one has a low combined mass-damping parameter (low m^*ζ), or a high mass-damping (highm^*ζ ). For our low m^*ζ, we find three modes of response, which are denoted as an initial amplitude branch, an upper branch and a lower branch. For the classical Feng-type response, at highm^*ζ , there exist only two response branches, namely the initial and lower branches. The peak amplitude of these vibrating systems is principally dependent on the mass-damping (m^*ζ), whereas the regime of synchronization (measured by the range of velocity U^*) is dependent primarily on the mass ratio, m^*ζ. At low (m^*ζ), the transition between initial and upper response branches involves a hysteresis, which contrasts with the intermittent switching of modes found, using the Hilbert transform, for the transition between upper–lower branches. A 180° jump in phase angle φ is found only when the flow jumps between the upper–lower branches of response. The good collapse of peak-amplitude data, over a wide range of mass ratios (m^*=1–20), when plotted against (m^*+C_A) ζ in the “Griffin” plot, demonstrates that the use of a combined parameter is valid down to at least (m^*+C_A)ζ 0·006. This is two orders of magnitude below the “limit” that had previously been stipulated in the literature, (m^*+C_A) ζ>0·4. Using the actual oscillating frequency (f) rather than the still-water natural frequency (f_N), to form a normalized velocity (U^*/f^*), also called “true” reduced velocity in recent studies, we find an excellent collapse of data for a set of response amplitude plots, over a wide range of mass ratiosm^* . Such a collapse of response plots cannot be predicted a priori, and appears to be the first time such a collapse of data sets has been made in free vibration. The response branches match very well the Williamson–Roshko (Williamson & Roshko 1988) map of vortex wake patterns from forced vibration studies. Visualization of the modes indicates that the initial branch is associated with the 2S mode of vortex formation, while the Lower branch corresponds with the 2P mode. Simultaneous measurements of lift and drag have been made with the displacement, and show a large amplification of maximum, mean and fluctuating forces on the body, which is not unexpected. It is possible to simply estimate the lift force and phase using the displacement amplitude and frequency. This approach is reasonable only for very low m^*.     

Series solution of the temperature distribution in the Falkner–Skan wedge flow by the homotopy analysis method

A new analytical method, namely the homotopy analysis method (HAM), has been applied to investigate the temperature field associated with the Falkner–Skan boundary-layer problem, and a series solution is provided in this paper. The results of the present work show agreement with those of numerical solutions in a large range of Prandtl numbers (0 < Pr ≤ 100), which demonstrates the validity of the present analysis.