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 → ∞.     

Asymptotic fields around an interfacial crack with a cohesive zone ahead of the crack tip

This paper considers an interfacial crack with a cohesive zone ahead of the crack tip in a linearly elastic isotropic bi-material and derives the mixed-mode asymptotic stress and displacement fields around the crack and cohesive zone under plane deformation conditions (plane stress or plane strain). The field solution is obtained using elliptic coordinates and complex functions and can be represented in terms of a complete set of complex eigenfunction terms. The imaginary portion of the eigenvalues is characterized by a bi-material mismatch parameter ε = arctanh(β)/π, where β is a Dundurs parameter, and the resulting fields do not contain stress singularity. The behaviors of “Mode I” type and “Mode II” type fields based on dominant eigenfunction terms are discussed in detail. For completeness, the counterpart for the Mode III solution is included in an appendix.     

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.     

Experimental investigation on strength and failure behavior of pre-cracked marble under conventional triaxial compression

Fractures in natural rocks have an important effect on the strength and failure behavior of rock mass, which are often evaluated in rock engineering practice. The theoretical evaluation of mechanical behavior of fractured rock mass has no satisfactory answer due to the role of confining pressure and crack geometry. Therefore, in this paper, conventional triaxial compression experiments were carried out to study the strength and failure behavior of marble samples with two pre-existing closed cracks in non-overlapping geometry. Based on the experimental results of a number of triaxial compression tests, the effect of crack coalescence on the axial supporting capacity and deformation property were investigated with different confining pressures. The results show that intact samples and flawed samples (marble with pre-existing cracks) have different deformation properties after peak stress, which change from brittleness to plasticity and ductility with the increase of confining pressure. The peak strength and failure mode are found depending not only on the geometry of flaw, but also on the confining pressure. The strength of flawed samples shows distinct non-linear behavior, which is in a better agreement with non-linear Hoek–Brown criterion than linear Mohr–Coulomb criterion. For a kind of rock that has been evaluated as a Hoek–Brown material, a new evaluation criterion is put forward by adopting optimal approximation polynomial theory, which can be used to confirm more precisely the strength parameters (cohesion and internal friction angle) of flawed samples. For intact samples, the marble leads to typical shear failure mode with a single fracture surface under different confining pressures, while for flawed samples, under uniaxial compression and a lower confining pressure (σ₃ = 10 MPa), tests for coarse and medium marble (the coarse and medium refer to the grain size) exhibit three basic failure modes, i.e., tensile mode, shear mode, and mixed mode (tensile and shear). Shear mode is associated with lower strength behavior. However, under higher confining pressures (σ₃ = 30 MPa), for coarse marble, the axial supporting capacity is not related to the geometry of flaw. The friction among crystal grains determines the strength behavior of coarse marble. For medium marble, the failure mode and deformation behavior are dependent on the crack coalescence in the sample. The present research provides increased understanding of the fundamental nature of rock failure under conventional triaxial compression.     

Pulsating flow in a planar diffuser upstream of automotive catalyst monoliths

The flow distribution across automotive exhaust catalysts has a significant effect on their conversion efficiency. The exhaust gas is pulsating and flow distribution is a function of engine operating condition, namely speed (frequency) and load (flow rate). This study reports on flow measurements made across catalyst monoliths placed downstream of a wide-angled planar diffuser presented with pulsating flow. Cycle-resolved particle image velocimetry (PIV) measurements were made in the diffuser and hot wire anemometry (HWA) downstream of the monoliths. The ratio of pulse period to residence time within the diffuser (defined as the J factor) characterises the flow distribution. During acceleration the flow remained attached to the diffuser walls for some distance before separating near the diffuser inlet later in the cycle. Two cases with J ∼ 3.5 resulted in very similar flow fields with the flow able to reattach downstream of the separation bubbles. With J = 6.8 separation occurred earlier with the flow field resembling, at the time of deceleration, the steady flow field. Increasing J from 3.5 to 6.8 resulted in greater flow maldistribution within the monoliths; steady flow producing the highest maldistribution in all cases for the same Re.     

Effect of side ratio on fluid flow and heat transfer from rectangular cylinders using the PANS method

A computational study of heat transfer from rectangular cylinders is carried out. Rectangular cylinders are distinguished based on the ratio of the length of streamwise face to the height of the cross-stream face (side ratio, R). The simulations were performed to understand the heat transfer in a flow field comprising separation, reattachment, vortex shedding and stagnation. The Partially-Averaged Navier–Stokes (PANS) modeling approach is used to solve the turbulent flow physics associated and the wall resolve approach is used for the near wall treatment because of the flow separation involved. The simulations were performed using a finite volume based opensource software, OpenFOAM, at Reynolds number (Re) = 22,000 for rectangular cylinder at constant temperature kept in an air stream. Two critical side ratios were obtained, R = 0.62 and 3.0. At R = 0.62, the maximum value of the drag coefficient (C_d) = 2.681 was observed which gradually reduced by 54% at R = 4.0. The base pressure coefficient and global Nusselt number also attained the maximum value at R = 0.62 and from R = 2.5 to 3.0 a sharp discontinuous increase by 140% in the Strouhal number was observed. At R = 0.62, it was observed that the separated flow reattaches at the trailing edge after rolling over the side face and therefore increases the overall Nusselt number. The phase averaging was also performed to analyze the unsteady behavior of heat transfer.     

Some exact solutions for rotating flows of a generalized Burgers’ fluid in cylindrical domains

The velocity field and the adequate shear stress corresponding to the flow of a generalized Burgers’ fluid model, between two infinite co-axial cylinders, are determined by means of Laplace and finite Hankel transforms. The motion is due to the inner cylinder that applies a time dependent torsional shear to the fluid. The solutions that have been obtained, presented in series form in terms of usual Bessel functions J₁( ? ), J₂( ? ), Y₁( ? ) and Y₂( ? ), satisfy all imposed initial and boundary conditions. Moreover, the corresponding solutions for Burgers’, Oldroyd-B, Maxwell, second grade, Newtonian fluids and large-time transient solutions for generalized Burgers’ fluid are also obtained as special cases of the present general solutions. The effect of various parameters on large-time and transient solutions of generalized Burgers’ fluid is also discussed. Furthermore, for small values of the material parameters, λ₂ and λ₄ or λ₁, λ₂, λ₃ and λ₄, the general solutions corresponding to generalized Burgers’ fluids are going to those for Oldroyd-B and Newtonian fluids, respectively. Finally, the influence of the pertinent parameters on the fluid motion, as well as a comparison between models, is shown by graphical illustrations.     

Flow instability in baffled channel flow

Flow instability in baffled channel flow, where thin baffles are mounted on both channel walls periodically in the direction of the main flow, has been numerically investigated. The geometry considered here can be regarded as a simple model for finned heat exchangers. The aim of this investigation is to understand how baffle interval (L) and Reynolds number (Re) influence the flow instability. With a fixed baffle length of one quarter of channel height (H), ratios of baffle interval to channel height (R_B = L/H) between 1 and 4 are considered. The critical Reynolds number of the primary instability, a Hopf bifurcation from steady flow to time-periodic flow, turned out to be minimum when R_B = 3.08. The friction factor (f) is strongly correlated with the critical Reynolds number for R_B ⩽ 2.5. For the particular cases of R_B = 1.456 and R_B = 1.0, we performed Floquet stability analysis in order to study the secondary instability through which time-periodic two-dimensional flow bifurcates into three-dimensional flow. The results obtained in this investigation are in good agreement with those computed from full simulations, and shed light on understanding and controlling flow characteristics in a finned heat exchanger, quite beneficial to its design.     

Equilibrium of a pressurized plastic fluid in a wellbore crack

This paper investigates equilibrium of a pressurized plastic fluid invading a tensile wellbore crack in a linear elastic, permeable rock. The crack is initially filled by pore fluid at ambient pressure, that is immiscibly displaced by the plastic fluid invading from the wellbore. The plastic fluid comes to rest to form a “plug” within the elastically deformed crack when the limit equilibrium between the shear stresses generated at the fracture walls and the pressure drop between the wellbore wall and the crack tip is reached. The model assumes that the leak-off of the plastic fluid into the rock is negligible, while the displaced pore fluid in the crack tip region is freely exchanged with the surrounding permeable rock to maintain the ambient pressure level. When the crack length ? is small or large compared to the wellbore radius R, the problem reduces to that of a pressurized edge or Griffith’s crack, respectively, subjected to a uniform far-field confining stress. In these two end-member cases, the normalized solution for the net pressure distribution, the plug length, and the stress intensity factor at the crack tip is obtained as a function of two numbers – the normalized net fluid pressure at the crack inlet and at the crack tip (partial plugs only) – that embody the solution’s dependence on the wellbore and the far field loading, the fluid yield strength, and the rock modulus. In the general case of an intermediate crack length (? ～ R), the normalized solution is a function of two additional parameters, the length-to-radius ratio and a normalized measure of the far field stress anisotropy, respectively, which accurate approximation is devised from an end-member solution using a rescaling argument. The equilibrium plug solutions are used to evaluate the breakdown pressure, the critical wellbore pressure at which the crack propagation condition is first met, and to analyze the stability of the ensuing crack propagation.     

Experimental quantification of the plastic blunting process for stage II fatigue crack growth in one-phase metallic materials

The plastic blunting process during stage II fatigue crack growth was studied in pure polycrystalline Ni to investigate effects of strain localization and inelastic behavior on the kinematics of crack advance. Correlations were obtained between strain fields ahead of a fatigue crack, crack advance per cycle and crack growth kinetics. Strain fields were quantified using a combination of in situ loading experiments, scanning electron microscopy and digital image correlation for 8 < ΔK < 20 MPa m^1/2 and a fixed load ratio of 0.1. Results indicate that strain localized along a dominant deformation band, which was usually crystallographic and carried mostly pure shear for large loads and was of mixed character for lower loads. Instances of double deformation bands were observed, with bands acting either in a simultaneous or alternating fashion. It was found that the area integral of the opening strain for values larger than a given threshold, an “integrated” strain, had a power-law relationship with ΔK, with the exponent approximately equal to the Paris exponent (m). Therefore, the crack growth rate was proportional to the integrated strain. An analysis based on this correlation and the presence of dominant shear bands indicated that the integrated strain is related to the accumulated displacement in the band. This, in turn, is proportional to the product of the cyclic plastic zone radius and the average shear strain ahead of the tip, which represents a basic length scale for plastic blunting. Assumptions on the load dependence of these quantities, based on their observed spatial variation, allowed estimating $m=2\left(1+\frac{1}{1+n^{\prime }}\right)$, where n′ is the cyclic hardening exponent (0 < n′ < 1). This gives 3 < m < 4, which accounts for about 50% of the observed values of m between 1.5 and 6 for a wide variety of metallic materials.     

Tip gap flow characteristics in a turbine cascade equipped with pressure-side partial squealer rims

Tip gap flow characteristics and aerodynamic loss generations in a turbine cascade equipped with pressure-side partial squealer rims have been investigated with the variation of its rim height-to-span ratio (h_p/s) for a tip gap height of h/s = 1.36%. The results show that the tip gap flow is characterized not only by the incoming leakage flow over the pressure-side squealer rim but also by the upstream flow intrusion behind the rim. The incoming leakage flow tends to decelerate through the divergent tip gap flow channel and can hardly reach the blade suction side upstream of the mid-chord, due to the interaction with the upstream flow intrusion as well as due to the flow deceleration. A tip gap flow model has been proposed for h_p/s = 3.75%, and the effect of h_p/s on the tip surface flow is discussed in detail. With increasing h_p/s, the total-pressure loss coefficient mass-averaged all over the present measurement plane decreases steeply, has a minimum value for h_p/s = 1.88%, and then increases gradually. Its maximum reduction with respect to the plane tip result is evaluated to be 11.6%, which is found not better than that in the cavity squealer tip case.     

Phenomenological study of parabolic and spherical indentation of elastic-ideally plastic material

A phenomenological study of parabolic and spherical indentation of elastic ideally plastic materials was carried out by using precise results of finite elements calculations. The study shows that no “pseudo-Hertzian” regime occurs during spherical indentation. As soon as the yield stress of the indented material is exceeded, a deviation from the, purely elastic Hertzian contact behaviour is found. Two elastic–plastic regimes and two plastic regimes are observed for materials of very large Young modulus to Yield stress ratio, E/σ_y. The first elastic–plastic regime corresponds to a strong evolution of the indented plastic zone. The first plastic regime corresponds to the commonly called “fully plastic regime”, in which the average indentation pressure is constant and equal to about three times the yield stress of the indented material. In this regime, the contact depth to penetration depth ratio tends toward a constant value, i.e. h_c/h = 1.47. h_c/h is only constant for very low values of yield strain (σ_y/E lower than 5 × 10^?6) when aE¹/Rσ_y is higher than 10,000. The second plastic regime corresponds to a decrease in the average indentation pressure and to a steeper increase in the pile-up. For materials with very large E/σ_y ratio, the second plastic regime appears when the value of the non-dimensional contact radius a/R is lower than 0.01. In the case of spherical and parabolic indentation, results show that the first plastic regime exists only for elastic-ideally plastic materials having an E/σ_y ratio higher than approximately 2.000.     

Revisiting the perforation of ductile plates by sharp-nosed rigid projectiles

The process of ductile plate perforation by sharp-nosed rigid projectiles is further examined in this work through 2D numerical simulations. We highlight various features concerning the effective resisting stress (σ_r) which a finite thickness plate, with a flow stress of Y_t, exerts on the projectile during perforation. In particular, we show that the normalized resisting stress (σ_r/Y_t) can be represented as a unique function of the normalized thickness of the plate (H/D, where H is plate thickness and D is projectile diameter), for a large range of normalized thicknesses. Our simulations for very thin target plates show that the penetration process is achieved through the well-known dishing mechanism, where the target material is pushed forward by the projectile’s nose. An important observation, which emerges from our simulations, is that the transition between the dishing and the hole enlargement mechanisms takes place at a normalized thickness of about H/D = 1/3. We also find that the normalized resistive stress for intermediate plate thicknesses, 1/3 < H/D < 1.0, is relatively constant at a value of σ_r/Y_t = 2.0. This range of thicknesses conforms to a state of quasi plane stress in the plates. For thicker plates (H/D > 1) the σ_r/Y_t ratio increases monotonically to values which represent the resistance to penetration of semi-infinite targets, where the stress state is characterized by plane strain conditions. Using a simple model, which is based on energy conservation, we can predict the values of the ballistic limit velocities for many projectile/target combinations, provided the perforation is done through the ductile hole enlargement mechanism. Good agreement is demonstrated between predictions from our model and experimental data from different sources, strongly enhancing the confidence in both the validity and usefulness of our model.     

Experimental investigation of a developing two-phase bubbly flow in horizontal pipe

Experimental results for various water and air superficial velocities in developing adiabatic horizontal two-phase pipe flow are presented. Flow pattern maps derived from videos exhibit a new boundary line in intermittent regime. This transition from water dominant to water–gas coordinated regimes corresponds to a new transition criterion C_T = 2, derived from a generalized representation with the dimensionless coordinates of Taitel and Dukler.Velocity, turbulent kinetic energy and dissipation rate, void fraction and bubble size radial profiles measured at 40 pipe diameters for J_L = 4.42 m/s by hot film velocimetry and optical probes confirm this transition: the gas influence is not continuous but strongly increases beyond J_G = 0.06 m/s. The maximum dissipation rate, derived from spectra, is increased in two-phase flow by a factor 5 with respect to the single phase case.The axial evolution of the bubble intercept length histograms also reveal the flow organization in horizontal layers, driven by buoyancy effects. Bubble coalescence is attested by a maximum bubble intercept evolving from 2.5 to 4.5 mm along the pipe. Turbulence generated by the bubbles is also manifest by the 4-fold increase of the maximum turbulent dissipation rate along the pipe.     

Thermo-elastic crack branching in general anisotropic media

Thermal fields may exist in addition to mechanical loading, for example, due to short term exposure to fire. In this paper, the branching of cracks in the presence of combined thermal and mechanical loads is investigated for general anisotropic media by employing the theory of Stroh’s dislocation formalism, extended to thermo-elasticity in matrix notation. A general solution to the thermo-elastic crack problem for an anisotropic material under arbitrary loading is obtained in a compact form. Green’s functions are also presented for a thermal dislocation (heat vortex) and a conventional dislocation (or, referred as mechanical dislocation), which are formulated considering the cuts located at an arbitrary angle with respect to the x₁ axis of the coordinate system (x₁, x₂, x₃). Using the derived compact expressions, the interaction between the crack and the dislocation is studied and a closed form solution for this interaction is obtained. The branching portion of the thermo-elastic crack is modelled as a continuous distribution of dislocations. This problem is then converted into a set of singular integral equations. Numerical results are presented to illustrate the possible effects of thermal loading on the propagation of the branched crack.     

Over-tip leakage flow and loss in a turbine cascade equipped with suction-side partial squealers

Over-tip leakage flow and loss in a turbine cascade equipped with suction-side partial squealer rims have been investigated for the squealer rim height-to-span ratios (h_st/s) of 0.94%, 1.88%. 3.75%, and 5.63% in the case of a tip gap height-to-span ratio of h/s = 1.36%. The casing wall and tip surface visualizations for h_st/s = 3.75% show that most of the incoming tip leakage flow tends to accelerate through a convergent (nozzle-like) tip gap flow channel and penetrates into the neighboring blade flow passage even upstream of the mid-chord in the form of a wall jet, whereas the rest of it is entrapped by the suction-side squealer rim, flows backward, and is separated from the tip surface along a backward flow separation line. Therefore, the tip surface can be divided into a separation bubble and a backward flow area by the backward flow separation line. A qualitative tip gap flow model for the suction-side squealer tip is suggested in this study. For the present suction-side squealer tip, the total pressure loss coefficient mass-averaged throughout the present measurement plane decreases consistently with increasing h_st/s and is higher than that for the cavity squealer tip or the pressure-side squealer tip regardless of h_st/s.     

Flow evolution of a turbulent submerged two-dimensional rectangular free jet of air. Average Particle Image Velocimetry (PIV) visualizations and measurements

The paper presents average flow visualizations and measurements, obtained with the Particle Image Velocimetry (PIV) technique, of a submerged rectangular free jet of air in the range of Reynolds numbers from Re = 35,300 to Re = 2200, where the Reynolds number is defined according to the hydraulic diameter of a rectangular slot of height H. According to the literature, just after the exit of the jet there is a zone of flow, called zone of flow establishment, containing the region of mixing fluid, at the border with the stagnant fluid, and the potential core, where velocity on the centerline maintains a value almost equal to the exit one. After this zone is present the zone of established flow or fully developed region. The goal of the paper is to show, with average PIV visualizations and measurements, that, before the zone of flow establishment is present a region of flow, never mentioned by the literature and called undisturbed region of flow, with a length, L_U, which decreases with the increase of the Reynolds number. The main characteristics of the undisturbed region is the fact that the velocity profile maintains almost equal to the exit one, and can also be identified by a constant height of the average PIV visualizations, with length, L_CH, or by a constant turbulence on the centerline, with length L_CT. The average PIV velocity and turbulence measurements are compared to those performed with the Hot Film Anemometry (HFA) technique. The average PIV visualizations show that the region of constant height has a length L_CH which increases from L_CH = H at Re = 35,300 to L_CH = 4–5H at Re = 2200. The PIV measurements on the centerline of the jet show that turbulence remains constant at the level of the exit for a length, L_CT, which increases from L_CT = H at Re = 35,300 to L_CT = 4–5H at Re = 2200. The PIV measurements show that velocity remains constant at the exit level for a length, L_U, which increases from L_U = H at Re = 35,300 to L_U = 6H at Re = 2200 and is called undisturbed region of flow. In turbulent flow the length L_U is almost equal to the lengths of the regions of constant height, L_CH, and constant turbulence, L_CT. In laminar flow, Re = 2200, the length of the undisturbed region of flow, L_U, is greater than the lengths of the regions of constant height and turbulence, L_CT = L_CH = 4–5H. The average PIV and HFA velocity measurements confirm that the length of potential core, L_P, increases from L_P = 4–5H at Re = 35,300 to L_P = 7–8H at Re = 2200, and are compared to the previous experimental and theoretical results of the literature in the zone of mixing fluid and in the fully developed region with a good agreement.