Variation of crack-opening stresses in three-dimensions: finite thickness plate

Crack growth and closure behavior of a center cracked finite thickness plate subjected to constant amplitude cyclic load is investigated by means of a three-dimensional elastic-plastic finite-element analysis. Results are obtained for initial half crack length c_i to half plate thickness t ratios of c_i/t = 3.891 and 1.465 which shall be referred to, respectively, as thin and thick plate. A constant amplitude load with R = S_min/S_max = 0.1 and S_max/σ₀ = 0.25 is applied, where S stands for the stress amplitude and σ₀ the effective yield stress. Crack closure for the thinner plate is found to be largest at and near the free plate surface and to decrease toward the interior during the unloading portion of cyclic loading. The closure pattern stabilizes at the interior and exterior regions, respectively, for c_i/t = 3.981 at 0.34S_max and 0.56S_max and for c_i/t = 1.465 at 0.26S_max and 0.46S_max.A load-reduced displacement technique was used to determine crack-opening stresses at specified locations in the plate from the displacements calculated after 7th cycle (using unloading and reloading portions of cyclic loading). All locations were on the plate exterior surface and were located behind the crack tip and at the centerline of the crack. The opening stresses at the specified points as certain percentage of the maximum stress amplitude were obtained.     

Impact energy absorption by a rate-dependent locking target

The impact by an elastic cylindrical piston on a thin plate-like target resting on a rigid foundation is considered. The relationship between force F acting on the target and displacement x is given by F=kx+q dx/dt provided dx/dt≥0 and 0≤x<d (k, q and d≥0). When x=d locking occurs, and F can assume any value ≥kd without increase in x. The displacement is assumed to be completely irreversible. The motion of the impactor is assumed to be governed by the elementary wave equation and, since the target is thin, wave motion in the target is neglected. The energy W=εFdx and its components W _k=kεx dx (the energy absorbed in a corresponding quasistatic process) and W _q=qε(dx/dt)² dt (the excessive energy because of the rate-dependence) are determined in terms of the impact energy as functions of non-dimensional parameters representing k, q and d. With the aid of diagrams, it is shown under what circumstances locking occurs, and under what circumstances W _k or W _q, or both, are large.     

An experimental study of circular sand–water wall jets

Laboratory experiments were carried out to study the effects of sand particles on circular sand–water wall jets. Mean and turbulence characteristics of sand particles in the sand–water wall jets were measured for different sand concentrations c_o ranging from 0.5% to 2.5%. Effects of sand particle size on the centerline sand velocity of the jets were evaluated for sand size ranging from 0.21 mm to 0.54 mm. Interesting results with the range of measurements are presented in this paper. It was found that the centerline sand velocity of the wall jets with larger particle size were 15% higher than the jets with smaller particle size. Concentration profiles in the vertical direction showed a peak value at x/d = 5 (where x is the longitudinal distance from the nozzle and d is the nozzle diameter) and the sand concentration decreased linearly for x/d > 5. Experimental results showed that the turbulence level enhanced from the nozzle to x/d = 10. For sand–water wall jets with a higher concentration (c_o = 1.5–2.5%), the turbulence intensity became smaller than the corresponding single-phase wall jets by 34% due to turbulent modulation. A modified logarithmic formulation was introduced to model the longitudinal turbulent intensity at the centerline and along the axis of the jet.     

Experimental investigation of a confined flow downstream of a circular cylinder centred between two parallel walls

In this work, we present an experimental study of the wall confinement effect on the wake formation behind a circular cylinder of diameter d_c=10 mm and of length L_c=30d_c. The experiments were performed in a water tunnel with the dimensions (length=300d_c, height=3d_c, span L_c=30d_c). The confinement rate was r=1/3 and the Reynolds number was in the range of 30–277. The experiments were done using 2-D PIV measurements. The first instability was delayed by the confinement and the von Kármán vortices characteristics are different from the unconfined case. Proper orthogonal decomposition (POD) of the flow was used for a filtering purpose and to extract the energetic contribution of the different modes. A low-order representation of the flow, constructed from the first pair of modes in the well-defined region of the flow, shows that von Kármán vortices are equivalent to vanishing progressive waves. Measurements done above the cylinder show the presence of 3-D span instabilities showing great similarities with “Mode A” and “Mode B” found in the unconfined case.     

Constant moving crack in a magnetoelectroelastic material under anti-plane shear loading

Analytical solutions for an anti-plane Griffith moving crack inside an infinite magnetoelectroelastic medium under the conditions of permeable crack faces are formulated using integral transform method. The far-field anti-plane mechanical shear and in-plane electrical and magnetic loadings are applied to the magnetoelectroelastic material. Expressions for stresses, electric displacements and magnetic inductions in the vicinity of the crack tip are derived. Field intensity factors for magnetoelectroelastic material are obtained. The stresses, electric displacements and magnetic inductions at the crack tip show inverse square root singularities. The moving speed of the crack have influence on the dynamic electric displacement intensity factor (DEDIF) and the dynamic magnetic induction intensity factor (DMIIF), while the dynamic stress intensity factor (DSIF) does not depend on the velocity of the moving crack. When the crack is moving at very lower or very higher speeds, the crack will propagate along its original plane; while in the range of M_c1 < M < M_c2, the propagation of the crack possibly brings about the branch phenomena in magnetoelectroelastic media.     

Relationships between the J-integral and the crack opening displacement for stationary and extending cracks

Relationships between the J-integral and the crack opening displacement δ_t are obtained by exploiting the dominance of the Hutchinson-Rice-Rosengren singularity in the crack-tip region. The coefficient d_n that relates J to δ_t, is dependent on the material deformation properties and is independent of crack configuration under small-scale yielding conditions. For low hardening materials, d_n appears to be configuration dependent in the fully yielded state. Similarly, the slope of the J-resistance curve is relatable to an operationally defined crack opening angle if J-controlled crack growth conditions are met. These relationships are corroborated by finite element results for the complete regime of elastic-plastic deformation and experimental data for A533B steels, HY-80 steels and several other ductile metals.     

Unsteady gravity-driven slender rivulets of a power-law fluid

Unsteady gravity-driven flow of a thin slender rivulet of a non-Newtonian power-law fluid on a plane inclined at an angle α to the horizontal is considered. Unsteady similarity solutions are obtained for both converging sessile rivulets (when 0 < α < π/2) in the case x < 0 with t < 0, and diverging pendent rivulets (when π/2 < α < π) in the case x > 0 with t > 0, where x denotes a coordinate measured down the plane and t denotes time. Numerical and asymptotic methods are used to show that for each value of the power-law index N there are two physically realisable solutions, with cross-sectional profiles that are ‘single-humped’ and ‘double-humped’, respectively. Each solution predicts that at any time t the rivulet widens or narrows according to |x | ^{(2N+1)/2(N+1)} and thickens or thins according to |x | ^N/(N+1) as it flows down the plane; moreover, at any station x, it widens or narrows according to |t | ^?N/2(N+1) and thickens or thins according to |t | ^?N/(N+1). The length of a truncated rivulet of fixed volume is found to behave according to |t | ^N/(2N+1).     

Analysis of deformation-induced crack tip toughening in ductile single crystals by nano-indentation

This paper reports on the experimental examination of the deformation characteristics near a crack tip in a cyclically work-hardened copper single crystal using a 2D surface scans with nano-indentation. The experimental methodology enables the characterization of the primary deformation field near a crack tip via the modulation of the imposed secondary deformation field by a nano-indentation. In a heavily deformed 4-point bend specimen, the measurements showed an existence of an asymptotic field around the crack tip at a distance of R ⩾ 2.5J/σ₀. The measurements also showed the qualitative details of toughness evolution within the high-gradient deformation field around the crack tip. The nature of dislocation distribution (i.e. statistically distributed vs. distributions necessitated by geometry) around the crack tip is quantified. The measurements indicate the dominance of the geometrically necessary dislocation within the finite deformation zone ahead of the tip up to a distance of R ≈ 3J/σ₀. Thereafter, it is confined in radial rays coinciding with the sector boundaries around the crack tip. These measurements elucidate the origin of the inhomogeneous hardening and the size dependent macroscopic response close to crack tip.     

Low Reynolds number flow over a square cylinder with a detached flat plate

A numerical study of the alteration of a square cylinder wake using a detached downstream thin flat plate is presented. The wake is generated by a uniform flow of Reynolds number 150 based on the side length of the cylinder, D. The sensitivity of the near wake structure to the downstream position of the plate is investigated by varying the gap distance (G) along the wake centerline in the range 0 ⩽ G ⩽ 7D for a constant plate length of L = D. A critical gap distance is observed to occur at G_c ≈ 2.3D that indicates the existence of two flow regimes. Regime I is characterised by vortex formation occurring downstream of the gap while for regime II, formation occurs within the gap. By varying the plate length and gap distance, a condition is found where significant unsteady total lift reduction can occur. The root mean square lift reduction is limited by an unsteady stall process on the plate.     

Multiple scattering of flexural waves in a semi-infinite thin plate with a cutout

The multiple scattering of flexural waves and dynamic stress concentration in a semi-infinite thin plate with a cutout are investigated, and the expressions of this problem are obtained. The analytical solutions of wave fields are expressed by employing the wave function expansion method and the expanded mode coefficients are solved by satisfying the boundary condition of the cutout. The image method is used to satisfy the traction free boundary condition of the plate. As an example, the numerical results of dynamic stress concentration factors are graphically presented and discussed. Numerical results show that the analytical results of the scattered waves and dynamic stress in semi-infinite plates are significantly different from those in infinite plates when the ratio of distance b/a is relatively little. In the region of low frequency and long wavelength, the maximum dynamic stress concentration factors occur on the illuminated side of the scattering body with θ = π, but not at the edge of the cutout with θ = π/2. As the incidence frequency increases (the wavelength becomes short), the dynamic stress on the illuminated side of the cutout decreases, however, the dynamic stress on the shadow side increases.     

Asymptotic crack-tip fields for dynamic crack growth in compressive power-law hardening materials

The effect of material compressibility on the stress and strain fields for a mode-I crack propagating steadily in a power-law hardening material is investigated under plane strain conditions. The plastic deformation of materials is characterized by the J₂ flow theory within the framework of isotropic hardening and infinitesimal displacement gradient. The asymptotic solutions developed by the present authors [Zhu, X.K., Hwang K.C., 2002. Dynamic crack-tip field for tensile cracks propagating in power-law hardening materials. International Journal of Fracture 115, 323–342] for incompressible hardening materials are extended in this work to the compressible hardening materials. The results show that all stresses, strains, and particle velocities in the asymptotic fields are fully continuous and bounded without elastic unloading near the dynamic crack tip. The stress field contains two free parameters σ_eq0 and s₃₃₀ that cannot be determined in the asymptotic analysis, and can be determined from the full-field solutions. For the given values of σ_eq0 and s₃₃₀, all field quantities around the crack tip are determined through numerical integration, and then the effects of the hardening exponent n, the Poisson ratio ν, and the crack growth speed M on the asymptotic fields are studied. The approximate behaviors of the proposed solutions are discussed in the limit of ν → 0.5 or n → ∞.     

An improved atomistic simulation based mixed-mode cohesive zone law considering non-planar crack growth

A novel and improved atomistic simulation based cohesive zone law characterizing interfacial debonding is developed which explicitly accounts for the non-planarity of the crack propagation. Group of atoms in the simulation constituting cohesive zones which are used to obtain local stress and crack opening displacement data are determined dynamically during the non-planar crack growth as they cannot be determined apriori. The methodology is used to study the debonding of Σ5 (2 1 0)/[0 0 1] symmetric tilt grain boundary interface in a Cu bicrystal under several mixed mode loading conditions. Simulations show that such bicrystalline specimen exhibits three types of energy dissipative mechanisms – shear coupled GB migration (SCM) away from the crack-tips, change in spacial orientation of GB structural units rendering highly disordered grain boundary near the crack tips and brittle intergranular fracture. Which combination of these three deformation mechanism will be active influencing the degree of non-planarity of the crack propagation at various stages of loading depends on the loading mode-mixity. As the ratio of shear component of the loading parallel to the GB plane and normal to the tilt axis with respect to the normal loading increases (thereby increasing the mode-mixity), overall strain-to-failure also increases and SCM tends to become the dominant deformation mechanism. Through this framework, analytical functional forms and parameters describing cohesive laws for both normal and shear traction as a function of the mode-mixity of the loading and crack opening displacement are predicted.     

Heat transfer due to impinging co-axial jets and the jets’ fluid flow characteristics

This investigation had multiple goals. One goal was to obtain definitive information about the heat transfer characteristics of co-axial impinging jets, and this was achieved by measurements of the stagnation-point, surface-distribution and average heat transfer coefficients. These results are parameterized by the Reynolds number Re which ranged from 5000 to 25,000, the dimensionless separation distance between the jet exit and the impingement plate H/D (4–12), and the ratio of the inner diameters of the inner and outer pipes d/D (0–0.55). The d/D = 0 case corresponds to a single circular jet. The other major goal of this work was to quantify the velocity field of co-axial free jets (impingement plate removed). The velocity-field study included both measurements of the mean velocity and the turbulence intensity.It was found that the variation of the stagnation-point heat transfer coefficient with d/D attained a maximum at d/D = 0.55. Furthermore, the variation of the local heat transfer coefficient across the impingement surface was more peaked for d/D = 0 and became flatter with decreasing d/D. This suggests that for cooling a broad expanse of surface, co-axial jets of high d/D are preferable. On the other hand, for localized cooling, the single jet (d/D = 0) performed the best. In general, for a given Reynolds number, a co-axial jet yields higher heat transfer coefficients than a single jet. Off-axis velocity peaks were encountered for the jets with d/D = 0.105. The measurements of turbulence intensity yielded values as high as 18%.     

Flow-induced force fluctuations on a sphere at high Strouhal number

An empirical model is developed to estimate the broadband unsteady force spectrum induced on a rigid sphere in a nominally steady, uniform flow. The Reynolds number is sub-critical, and the frequency range considered is above the low-mode Strouhal shedding frequency of the sphere (0.5⩽fd/U₀⩽100, where f is the frequency, d is the diameter, and U₀ is the mean flow speed). The model uses the separation of variables assumption for the cross-power spectral densities of the surface pressure fluctuations. The assumption is shown to be a proper engineering approximation except in the lower part of the considered frequency range. In addition, the flow-induced unsteady lift and drag forces are measured independently of each other using towed spheres in a basin of water. Both estimations, from the empirical model and the data measured in the tow tank, show that the dimensionless power spectral densities of broadband unsteady lift and drag forces are constant for fd/U₀<1, and (fd/U₀)⁻³ dependent for 1⩽fd/U₀⩽100. The model predicts that the broadband spectral density of the unsteady lift force is about 5 dB higher than that of the unsteady drag force, while the measured data show the level difference between 3 and 7 dB. The empirical model presented here has application in predicting the flow-induced noise of underwater hydrophones that sense acoustic particle velocity or acceleration.     

Transient heating effects in high pressure Diesel injector nozzles

The tendency of today’s fuel injection systems to reach injection pressures up to 3000 bar in order to meet forthcoming emission regulations may significantly increase liquid temperatures due to friction heating; this paper identifies numerically the importance of fuel pressurization, phase-change due to cavitation, wall heat transfer and needle valve motion on the fluid heating induced in high pressure Diesel fuel injectors. These parameters affect the nozzle discharge coefficient (C_d), fuel exit temperature, cavitation volume fraction and temperature distribution within the nozzle. Variable fuel properties, being a function of the local pressure and temperature are found necessary in order to simulate accurately the effects of depressurization and heating induced by friction forces. Comparison of CFD predictions against a 0-D thermodynamic model, indicates that although the mean exit temperature increase relative to the initial fuel temperature is proportional to (1 − C_d²) at fixed needle positions, it can significantly deviate from this value when the motion of the needle valve, controlling the opening and closing of the injection process, is taken into consideration. Increasing the inlet pressure from 2000 bar, which is the pressure utilized in today’s fuel systems to 3000 bar, results to significantly increased fluid temperatures above the boiling point of the Diesel fuel components and therefore regions of potential heterogeneous fuel boiling are identified.     

Turbulence modulation in a particle-laden flow over a hemisphere-roughened wall

Turbulence modulation by the inertia particles in a spatially developing turbulent boundary layer flow over a hemisphere-roughened wall was investigated using the direct numerical simulation method. The Eulerian and Lagrangian approaches were used for the gas- and particle-phases, respectively. An immersed boundary method was employed to resolve the hemispherical roughness element. The hemispheres were staggered in the downstream direction and arranged periodically in the streamwise and spanwise directions with spacing of p_x/d= 4 and p_z/d= 2 (where p_x and p_z are the streamwise and spanwise spacing of the hemispheres, and d is the diameter). The effects of particles on the turbulent coherent structures, turbulent statistics and quadrant events were analyzed. The results show that the addition of particles significantly damps the vortices structures and increases the length scales of streak structures. Compared with the particle-laden flow over the smooth wall, the existence of the wall roughness decreases the mean streamwise velocity in the near wall region, and makes the peaks of Reynolds stresses profiles shift up. In addition, the existence of particles also increases the percentage contributions to Reynolds shear stress from the Q4 events, however, decreases the percentage contributions from other quadrant events.     

Experimental study of single bubble motion in a liquid metal column exposed to a DC magnetic field

The motion of single Argon bubbles rising in the eutectic alloy GaInSn under the influence of a DC longitudinal magnetic field (parallel to the direction of bubble motion) was examined. The magnetic field strength was varied up to 0.3 T corresponding to a magnetic interaction parameter N (which measures the ratio of electromagnetic forces to inertial forces) slightly greater than 1. The liquid metal was at rest in a cylindrical container. Bubble and liquid velocities were measured using ultrasound Doppler velocimetry (UDV). The measured bubble terminal velocity showed oscillations indicating a zigzag movement of ellipsoidal bubbles. For small bubbles (d_e ⩽ 4.6 mm) an increase of the drag coefficient with increasing magnetic interaction parameter N was observed, whereas for larger bubbles (d_e ⩾ 5.4 mm) the application of the magnetic field reduces the drag coefficient. The measurements revealed a distinct electromagnetic damping of the bubble induced liquid velocity leading to more rectilinear bubble trajectories when the magnetic field is applied. Moreover, significant modifications of the bubble wake structure were observed. Raising of the magnetic field strength caused an enlargement of the eddies in the wake. The Strouhal number decreases with increasing magnetic interaction parameter N.     

An alternative method to study the cooling of glasses from melt to solid phase

The aim of this preliminary work was to present a novel method, suitable to investigate the glass cooling, from melt to solid state, based on a fast, non-usual and easy microwave method. The following glass system xBaO · (100?x)B₂O₃ (x = 0% and 40%) was selected as an example for this study. The melt was poured inside a piece of waveguide and then, its cooling was monitored by the microwave signal as a function of time. The variations in the signal can provide valuable informations about some structural changes that take place during the cooling stages, such as relaxation processes. This method can be useful to investigate the cooling and heating of other materials, opening new possibilities for investigation of dielectric behavior of materials under high temperatures.     

Theoretical stress analysis of intersecting cylindrical shells subjected to external loads transmitted through branch pipes

A thin shell theoretical solution of two normally intersecting cylindrical shells subjected to thrust-out force and three kinds of moments transmitted through branch pipes is presented in this paper. The solutions of modified Morley equation, which can be applicable up to ρ₀ = d/D ⩽ 0.8 and λ = d/(DT)^1/2 ⩽ 8 and the order of accuracy is raised to O(T/D), for the four loading cases are given. The accurate continuity conditions of generalized forces and displacements at the intersecting curve of two cylindrical shells for the four loading cases and the condition of the uniqueness of displacements are derived in this paper. The presented results are verified by experimental and numerical results successfully. They are in agreement with WRC Bulletin 297 when d/D is small.