Slip motion and stability of a single degree of freedom elastic system with rate and state dependent friction

We consider quasistatic motion and stability of a single degree of freedom elastic system undergoing frictional slip. The system is represented by a block (slider) slipping at speed V and connected by a spring of stiffness k to a point at which motion is enforced at speed V₀ We adopt rate and state dependent frictional constitutive relations for the slider which describe approximately experimental results of Dieterich and Ruina over a range of slip speeds V. In the simplest relation the friction stress depends additively on a term A In V and a state variable θ; the state variable θ evolves, with a characteristic slip distance, to the value ? B In V, where the constants A, B are assumed to satisfy B > A > 0. Limited results are presented based on a similar friction law using two state variables.Linearized stability analysis predicts constant slip rate motion at V₀ to change from stable to unstable with a decrease in the spring stiffness k below a critical value k_cr. At neutral stability oscillations in slip rate are predicted. A nonlinear analysis of slip motions given here uses the Hopf bifurcation technique, direct determination of phase plane trajectories, Liapunov methods and numerical integration of the equations of motion. Small but finite amplitude limit cycles exist for one value of k, if one state variable is used. With two state variables oscillations exist for a small range of k which undergo period doubling and then lead to apparently chaotic motions as k is decreased.Perturbations from steady sliding are imposed by step changes in the imposed load point motion. Three cases are considered: (1) the load point speed V₀ is suddenly increased; (2) the load point is stopped for some time and then moved again at a constant rate; and (3) the load point displacement suddenly jumps and then stops. In all cases, for all values of k:, sufficiently large perturbations lead to instability. Primary conclusions are: (1) ‘stick-slip’ instability is possible in systems for which steady sliding is stable, and (2) physical manifestation of quasistatic oscillations is sensitive to material properties, stiffness, and the nature and magnitude of load perturbations.     

Heat transfer in a second-order fluid for flow around a circular cylinder

The temperature distribution within the thermal boundary layer region due to the flow of an incompressible second-order fluid around a heated circular cylinder, maintained at a constant temperature higher than that of the fluid at infinity, has been obtained near the forward stagnation point by series expansion. The graph of the Nusselt number Nu for the Prandtl number P = 25 and the Eckert number E = 0.1 indicates that the non-Newtonian effect is to increase the heat flux from the cylinder to the liquid in the region 0° ? θ ? 15.7° and to decrease it in the region 15.7° < θ ? 0₀ where θ is the angular distance on the cylinder measured from the forward stagnation point. The critical point θ₀ at which Nu = 0, that is, where the effect of the frictional heating balances the effect of the temperature difference and there is no heat flux either from the cylinder or from the liquid, shifts towards the forward stagnation point with the increase of non-Newtonian effects.     

State of stress at the vertex of a quarter-infinite crack in a half-space

Spherical coordinates are r, θ, φ. The half-space extends in θ < π/2. The crack occurs along φ = 0. The region to be investigated is the solid space-triangle (or cone) between the three planes θ = π/2, φ = +0 and φ = 2π ? 0, which planes are to be taken stress-free.In this space-angle a state of stress is considered in terms of the cartesian stress components σ_xx = r^λ?_xx(λ, θ, φ); σ_xy = r^λ(λ, θ, φ); etc. Possible values λ are determined from a characteristic (or eigenvalue) equation, expressing the condition that a determinant of infinite order is equal to zero. The root of λ which gives the most serious state of stress in the vertex region (the region r → 0) is the root closest to the limiting value Re λ > ?3/2. Knowledge of this state of stress, or at least of this value of λ is essential in the determination of the three-dimensional state of stress around a crack in a plate for distances of order of the plate thickenss.Along the front of the quarter-infinite crack (z-axis) the so called stress-intensity factor behaves like z^λ+½ (z → 0) and thus tends to zero, respectively to infinity, accordingly to Re λ being >?½ or <?½. But in the region z → 0 the notion stress-intensity factor loses its meaning. The required state of stress passes into the well-known state of plane strain around a crack tip if Poisson's ratio (v) tends to zero. The computed state of stress for the incompressible medium (v = ½) is confirmed by experiment.     

Interaction of thermal contact resistance and frictional heating in thermoelastic instability

Thermoelastic contact problems can posess non-unique and/or unstable steady-state solutions if there is frictional heating or if there is a pressure-dependent thermal contact resistance at the interface. These two effects have been extensively studied in isolation, but their possible interaction has never been investigated. In this paper, we consider an idealized problem in which a thermoelastic rod slides against a rigid plane with both frictional heating and a contact resistance. For sufficiently low sliding speeds, the results are qualitatively similar to those with no sliding. In particular, there is always an odd number of steady-state solutions; if the steady-state is unique it is stable and if it is non-unique, stable and unstable solutions alternate, with the outlying solutions being stable. However, we identify a sliding speed V₀ above which the number of steady states is always even (including zero, implying possible non-existence of a steady-state) and again stable and unstable states alternate. A parallel numerical study shows that for V>V₀ there are some initial conditions from which the contact pressure grows without limit in time, whereas for V<V₀ the system will always tend to one of the stable steady states.     

Numerical study on the oscillation of a transient bubble near a confined free surface for droplet generation

In the present work, the oscillation of a spark-created bubble near a confined water–air interface and the ensuing droplet generation and ejection are studied numerically using the boundary element method. The interface is accorded by the top opening of either one of the following symmetrical configurations, which are distinguished by the value of angle between their vertical symmetry axis and lateral wall (i.e., θ): (i) a centrally perforated horizontal flat plate (θ = 90°) and (ii) vertically placed cylinder (θ = 0°), nozzle (θ > 0°) and diffuser (θ < 0°). Furthermore, the influences of the effective parameters such as the strength parameter (i.e., the intensity of local energy input), the bubble-free surface distance (standoff distance) and the nozzle size on the bubble dynamics and droplet formation and ejection processes are investigated. It was found that the moment at which the bubble attained its maximum volume was advanced as θ increased. In addition, by decreasing θ the attraction of the bubble toward the free surface during its expansion phase and its migration from the free surface during its contraction phase became stronger. Furthermore, for the nozzle case, by increasing θ, the volume of the droplet was increased. It was also found that by increasing the strength parameter, the volume of the droplet increased and its pinch-off happened earlier. Finally, as the standoff distance was increased, the volume of the droplet increased and its pinch-off was delayed.     

Numerical investigation of the influence of incidence angle on asymmetrical turbine blade model showerhead film cooling effectiveness

The influence of various incidence angles on film cooling effectiveness of an axial turbine blade cascade with leading edge ejection from two rows of cooling holes is numerically investigated. The rows are located in the vicinity of the stagnation line. One row is located on the suction side and the other one is on the pressure side. The predicted pressure field for various blowing ratios (M = 0.7, 1.1 and 1.5) is compared with available experimental results at the design condition. Moreover, the effect of various incidence angles (?10°, ?5°, 0°, 5° and 10°) at three blowing rates is investigated by analyzing the results of both laterally averaged and area averaged values of adiabatic film cooling effectiveness. Numerical results indicate that the incidence angle can strongly affect the thermal protection of the blade at low blowing ratio but becomes less dominant at high blowing ratio. In fact, for the low blowing ratio, a small change in the incidence angle that relates to the design condition can deeply affect the thermal protection of the blade, which is evident from the laterally and area averaged film cooling effectiveness distributions.     

Influence of temperature on a carbon–fibre epoxy composite subjected to static and fatigue loading under mode-I delamination

In this paper, interlaminar crack initiation and propagation under mode-I with static and fatigue loading of a composite material are experimentally assessed for different test temperatures. The material under study is made of a 3501-6 epoxy matrix reinforced with AS4 unidirectional carbon fibres, with a symmetric laminate configuration [0°]_16/S. In the experimental programme, DCB specimens were tested under static and fatigue loading. Based on the results obtained from static tests, fatigue tests were programmed to analyse the mode-I fatigue behaviour, so the necessary number of cycles was calculated for initiation and propagation of the crack at the different temperatures. G–N curves were determined under fatigue loading, N being the number of cycles at which delamination begins for a given energy release rate. G_ICmax–a, a–N and da/dN–a curves were also determined for different G_cr rates (90%, 85%, 75%, etc.) and different test temperatures: 90 °C, 50 °C, 20 °C, 0 °C, ?30 °C and ?60 °C.     

An experimental investigation on the tensile moduli and strengths of graphite/epoxy laminates

The results of a series of tensile tests on graphite/epoxy [0°]_8S and [0°/±30°/0°]_2S laminates at rates varying from 0.002 in/min to 2 in./min are reported. The loads are applied at various angles to the fiber directions in each case. The rate-dependent behavior of the stress-strain response is assessed. Evidence is presented to indicate that failure first occurs on inner plies. Also, evidence is presented to indicate that, in some cases, moduli increase with increased stress or strain level. Lamination theory is used to predict moduli and comparisons with experiment are given. This theory is also used in conjunction with three failure theories to predict ultimate strengths with varying degrees of success. Further, two approaches to ply unloading after first-ply failure are used and discussed. One is a standard method found in the literature while the other is a proposed ‘strength-of-materials’ type of technique which is computationally much simpler.     

A transition phenomenon of ice formation in water flow between two horizontal parallel plates

Experiments have been performed to investigate the icetransition profiles and heat-transfer characteristics for water flows between two horizontal parallel plates. The experiments are carried out under the condition that upper plate is cooled at uniform temperature kept less than freezing temperature of water, while the lower plate is heated at uniform temperature kept higher than the temperature of water flow. The temperatures of the upper and lower plates range from ?8 to ?14°C and from 10 to 60 °C, respectively, with inlet-water temperature varied from 1.5 to 4.5 °C. The cooling and heating temperature ratios, θ_c and θ_h, are ranging from 1.78 to 9.33 and from 1.22 to 39, respectively. By using three kinds of heightH of 16, 30 and 40 mm between the horizontal parallel plates, the Reynolds and Grashof numbers are varied from 3.2 × 10² to 1.5 × 10⁴ and from 3.4 × 10³ to 8.97 × 10⁶, respectively. As a result of this investigation two ice-transition modes are observed. The first ice-transition mode is due to an interruption of upper and lower thermal boundary layers, while the second mode is due to an instability of laminar boundary layer formed on water-ice interface. In order to determine the kind of ice-transition mode, criterion correlation formulas including the Reynolds numberRe _H , Grashof numberGr _H , and heating temperature ratio θ_h are determined and may be written as follows: For thermal icetransition mode (th.I.T.M.)Re _H /(Gr _H ·θ _h )^0.23<1.6×10^?3 and for hydrodynamical ice-transition mode (hy.I.T.M.)Re _H /(Gr _H ·θ _h )^0.23>2.3×10^?3 By introducing the freezing parameterB _f , correlation equations for local and mean Nusselt numbers along the water-ice interface at steady-state condition are determined. From the current experimental results it is found that the local Nusselt number may be described as the following equation:Nu _x =0.835 Re _H ^0.278 · B _f ^0.834 ·x/H)^?0.139     

Wavefronts for a cooperative tridiagonal system of differential equations

Consider the infinite system of nonlinear differential equations \(\dot u\) _n =f(_n?1, u_n, u_n+1),nε?, wherefεC ¹,D ₁ f > 0,D ₃ f>0, andf(0, 0, 0) = 0 =f(1, 1, 1). Existence of wavefronts—i.e., solutions of the formu _n (t) = U(n + ct), wherecε?,U(? ∞) = 0,U(+∞) = 1, andU is strictly increasing—is shown for functionsf which satisfy the condition: there existsa, 0<a<1, such thatf(x, x,x)<0 for 0<x<a andf(x, x, x) > 0 fora < x < 1.     

Influence of single rectangular groove on the flow past a circular cylinder

In the present study, flow control mechanism of single groove on a circular cylinder surface is presented experimentally using Particle image velocimetry (PIV). A square shaped groove is patterned longitudinally on the surface of the cylinder with a diameter of 50 mm. The flow characteristics are studied as a function of angular position of the groove from the forward stagnation point of the cylinder within 0° ≤ θ ≤ 150°. In the current work, instantaneous and time-averaged flow data such as vorticity, ω streamline, Ψ streamwise, u/U_o and transverse, v/U_o velocity components, turbulent kinetic energy, TKE and RMS of streamwise, u_rms and transverse, v_rms velocity components are utilized in order to present the results of quantitative analyses. Furthermore, Strouhal numbers are calculated using Karman vortex shedding frequency, f_k obtained from single point spectral analysis. It is concluded that a critical angular position of the groove, θ = 80° is observed. The flow separation is controlled within 0° ≤ θ < 80°. At θ = 80°, the flow separation starts to occur in the upstream direction. The instability within the shear layer is also induced on grooved side of the cylinder with frequencies different than Karman vortex shedding frequency, f_k.     

Numerical study of solidification of a drop with a growth angle difference

This study presents the direct numerical results of a drop solidifying on a plate, in which the difference between the growth angles is considered. The drop is two-dimensional with the presence of the left and right triple points, and the method used is a front-tracking technique. The growth angles at the right (ϕ_gr1) and left (ϕ_gr2) triple points are not equal, i.e. Δϕ_gr = ϕ_gr1–ϕ_gr2 ≠ 0°. Unlike the identical growth angles, the growth angle difference results in an asymmetric drop after complete solidification. In the presence of the solid-to-liquid density ratio ρ_sl < 1.0 (i.e. volume expansion), the tip of the solidified drop shifts more to the right as Δϕ_gr increases in the range of 0°–12°. In addition, the angle at the solidified drop top (i.e. tip angle) increases with Δϕ_gr. We also pay attention to the effects of some other parameters (such as the wetting angle ϕ₀, the growth angle ϕ_gr1 and ρ_sl) on the solidification process with the growth angle difference. The results reveal that the growth angle varied in the range of 6°–24° has a minor effect on the movement of the tip to the right while the tip shift increases with an increase in ϕ₀ in the range of 60°–130° or with a decrease in ρ_sl in the range of 0.8–1.1. The tip angle increases with an increase in ρ_sl or with a decrease in ϕ_gr1 or ϕ₀. We also investigate the solidification process under the influence of the Bond number.     

Large Strain Shear Compression Test of Sheet Metal Specimens

A hybrid experimental-computational procedure to establish accurate true stress-plastic strain curve of sheet metal specimen covering the large plastic strain region using shear compression test data is described. A new shear compression jig assembly with a machined gage slot inclined at 35° to the horizontal plane of the assembly is designed and fabricated. The novel design of the shear compression jig assembly fulfills the requirement to maintain a uniform volume of yielded material with characteristic maximum plastic strain level across the gage region of the Shear Compression Metal Sheet (SCMS) specimen. The approach relies on a one-to-one correlation between measured global load–displacement response of the shear compression jig assembly with SCMS specimen to the local stress-plastic strain behavior of the material. Such correlations have been demonstrated using finite element (FE) simulation of the shear compression test. Coefficients of the proposed correlations and their dependency on relative plastic modulus were determined. The procedure has been established for materials with relative plastic modulus in the range 5?×?10^?4?<?(E _p /E)?<?0.01. It can be readily extended to materials with relative plastic modulus values beyond the range considered in this study. Nonlinear characteristic hardening of the material could be established through piecewise linear consideration of the measured load–displacement curve. Validity of the procedure is established by close comparison of measured and FE-predicted load–displacement curve when the provisional hardening curve is employed as input material data in the simulation. The procedure has successfully been demonstrated in establishing the true stress-plastic strain curve of a demonstrator 0.0627C steel SCMS specimen to a plastic strain level of 49.2 pct.     

Wakes of elliptical cylinders at low Reynolds number

The wakes of elliptical cylinders are numerically investigated at a Reynolds number Re_D = 150. ANSYS-Fluent, based on the finite volume method, is used to simulate two-dimensional Newtonian fluid flow. The cylinder cross-sectional aspect ratio (AR) is varied from 0.25 to 1.0 (circular cylinder), and the angle of attack (α) of the cylinder is changed as α = 0° – 90°. With the changes in AR and α, three distinct wake patterns (patterns I, II, III) are observed, associated with different characteristics of fluid forces. Steady wake (pattern I) is characterised by two steady bubbles forming behind the cylinder, occurring at AR < 0.37 and α < 2.5°. Time-mean drag and fluctuating lift coefficients are small. Pattern II refers to Karman wake followed by steady wake (AR ≥ 0.37 – 0.67, depending on α) with the Karman street transitioning to two steady shear layers downstream. An inflection angle α_i is identified where the time-mean drag of the elliptical cylinder is identical to that of a circular cylinder. Pattern III is the Karman wake followed by secondary wake (AR ≤ 0.67, α > 52°), where the Karman street forming behind the cylinder is modified to a secondary vortex street with a low frequency. The Time-mean drag coefficient is maximum for this pattern.     

Enhanced nucleate boiling on copper micro-porous surfaces

Saturation boiling of PF-5060 dielectric liquid on Cu micro-porous surface layers (95, 139, 171, 197 and 220-μm thick) is investigated. These layers are deposited on 10 × 10 mm Cu substrates using two-stage electrochemical process. The basic micro-structure, obtained in the first stage using current density of 3 A/cm² for 15–44 s, depending on thickness, is strengthened by continuing electrochemical deposition using much lower current density for 10’s of minutes. For conditioned surface layers, after a few successive boiling tests, the pool boiling curves are reproducible and the temperature excursion prior to boiling incipience is either eliminated or reduced <7 K. Present nucleate boiling results are markedly better than those reported for dielectric liquids on micro- and macro-structured surfaces. Present values of CHF (22.7–27.8 W/cm²) and h_MNB (2.05–13.5 W/cm² K) are ∼40–70% higher than and >17 times those reported on plane surfaces (<16 W/cm² and ∼0.8 W/cm² K). Best results are those of the 171-μm thick layer: CHF of 27.8 W/cm² occurs at ΔT_sat of only 2.1 K and h_MNB of 13.5 W/cm² K occurs at ΔT_sat = 2.0 K.     

Terminal velocity of a Taylor drop in a vertical pipe

A scaling analysis based on the field equations for two phases and the jump conditions at the interface is carried out to deduce a balance of forces acting on a Taylor drop rising through stagnant liquid in a vertical pipe. The force balance is utilized to deduce a functional form of an empirical correlation of terminal velocity of the Taylor drop. Undetermined coefficients in the correlation are evaluated by making use of available correlations for two limiting cases, i.e. extremely high and low Reynolds number Taylor bubbles in large pipes. Terminal velocity data obtained by interface tracking simulations are also used to determine the coefficients. The proposed correlation expresses the Froude number Fr as a function of the drop Reynolds number Re_D, the Eötvös number Eo_D and the viscosity ratio μ^*. Comparisons between the correlation, simulations and experimental data confirm that the proposed correlation is applicable to Taylor drops under various conditions, i.e., 0.002 < Re_D < 4960, 4.8 < Eo_D < 228, 0 ? μ^* ? 70, 1 < N < 14700, −12 < log M < 4, and d/D < 1.6, where N is the inverse viscosity number, M the Morton number, d the sphere-volume equivalent drop diameter and D the pipe diameter.     

A new 1-D composite lumped model facilitates the algebraic calculation of local temperatures, mean temperatures, and total heat transfer in simple bodies

Estimates of spatio-temporal temperatures and total heat transfer in simple bodies (large plate, long cylinder and sphere) have been done in the past with the help of charts and recently by way of algebraic correlation equations; both avenues are valid for large times mostly, i.e., dimensionless times or Fourier numbers τ?≥?0.2. The imposing time restriction comes from the utilization of the truncated ‘one-term series’ because the Fourier infinite series diverge for very short times approaching zero. The central goal of this technical paper is to predict the mean, surface and center temperatures, as well as the total heat transfer in those simple bodies by implementing a 1-D composite lumped analysis united to a sound physics-based computational procedure. The effortless combined methodology is new. It brings forth a handful of compact algebraic equations covering the full gamma of Biot numbers (0?<?Bi?<?100) and all dimensionless times, 0?<?τ?<?∞.     

Large bending behavior of creased paperboard. I. Experimental investigations

The large bending behavior of a creased paperboard is studied in the range of rotation θ ? [0°, 180°] – new results, apparently not reported previously in literature – with the aim to point out some crucial aspect involved in an adaptive robotic manipulation of the industrial cartons.The loading tests show a great variability of the mechanical behavior, depending dramatically on the crease indentation depth (also for the specimens obtained from the same carton): (a) when the damage induced during the crease formation is relatively small, the bending response is unusually complex: the moment constitutive function, m_L(θ), presents (up to) two peaks followed by unstable branches; (b) for greater indentation, the m_L(θ) is monotone.In the unloading case the response m_U(θ) is always monotone and is practically independent of the formation conditions of the crease. These behaviors can be easily described analytically using (piecewise) third degree splines.In a companion paper, the erection of a typical carton corner with unstable constitutive behavior is fully analyzed to detect the possible criticalities.     

Heat and mass transfer characteristics of the self-similar boundary-layer flows induced by continuous surfaces stretched with rapidly decreasing velocities

The mechanical and thermal characteristics of the self-similar boundary-layer flows induced by continuous surfaces stretched with rapidly decreasing power-law velocities U _w∝x ^m , m<?1 are considered. Comparing to the well studied cases of the increasing stretching velocities (m>0) several new features of basic significance have been found. Thus: (i) for m<?1 the boundary layer equations admit self-similar solutions only if a lateral suction is applied; (ii) the dimensionless suction velocity f _w<0 must be strong enough, i.e. f _w<f _w,max(m) where f _w,max(m) depends on m so that its absolute maximum max (f _w,max(m))=?2.279 is reached for m→?∞, while for m→?1, f _w,max(m)→?∞; (iii) the case {m→?∞, f _w,max(m)=?2.279} of the flow boundary value problem is isomorphic to the stretching problems with exponentially decreasing velocities U _w∝e ^ax with arbitrary a<0; (iv) for any fixed m<?1 and f _w<f _w,max(m) the flow problem admits a non-denumerable infinity of multiple solutions corresponding to the values of the dimensionless skin friction f ^″(0)≡s belonging to a finite interval s∈ [s _min(f _w,m), s _max(f _w,m)]; (v) the solution is only unique for f _w=f _w,max(m) where s=s _min(f _w,m)= s _max(f _w,m) holds; (vi) to every one of the multiple solutions of the flow problem there corresponds a unique solution of the heat transfer problem with a wall temperature distribution T _w?T _∞∝x ⁿ and a well defined and distinct value of the dimensionless wall temperature gradient ?^′(0), except for the cases n=(|m|?1)/2 where ?^′(0) has the same value ?^′(0)=Pr·f _w for the whole class of flow solutions with s∈[s _min(f _w,m), s _max(f _w,m)]; (vii) for f _w→?∞ one obtains the `asymptotic suction profiles' corresponding to s=s _min(f _w,m)?f _w and ?^′(0)?Pr·f _w in an explicit analytic form. The paper includes several examples which illustrate the dependence of the heat and fluid flows induced by surfaces stretching with rapidly decreasing velocities on the physical parameters f _w, m, n and Pr.