Determining the stress intensity factor by using the principle of minimum potential energy

Expressing the total potential energy of the system of a cracked body П by Williams’ infinite series solution of stress and displacement components containing coefficients A_n(n = 1,2,...), we obtain a set of simultaneous linear equations of unknown coefficients A_n by using the principle of minimum potential energy. When the set of equations is solved, the stress intensity factor K₁ can be easily determined. It is equal to √2πaA₁ Take a sample plate as an example. A single-edgc-cracked plate under tension, with the ratio of crack length to the width of the plate being 0.5 and the ratio of half plate height to the width of the plate being 2.0 and 2. 5, has been calculated. Only 20 - 30 coefficients are taken, and the errors in stress intensity factors are within 5%.     

Two-Parameter Model of a Mode I Crack in an Elastoplastic Body under Plane-Strain conditions

The Dugdale crack model is generalized to the case of plane strain. The governing equations are set up to determine the stresses in the plastic zone. Numerical results from specific problems are analyzed and compared with those for plane stress state and other cases. A relationship between the crack model and K _I-T theory is established in the case of small-scale yielding at the crack tip __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 6, pp. 44–55, June 2005.     

Contribution of point defects to the temperature anomaly of the yield stress in single crystals of the Ni<Subscript>3</Subscript>Ge alloy

The temperature dependence of the yield stress τ_* Ni ₃ Ge single crystals is studied. The temperature dependence τ_*(T) in the high-temperature region (above 420 K) is found to be conditioned by thermally activated accumulation of the density of non-screw components of superdislocations. Interaction of point defects with edge dislocations and its effect on the temperature anomaly of the yield stress in Ni ₃ Ge single crystals are analyzed. The calculated results are found to agree with experimental data. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 154–161, July–August, 2007.     

Thermoelastic investigations for fatigue life assessment

An investigation is presented on the suitability and accuracy of a thermoelastic technique for the analysis of fatigue cracks. The stress intensity factor ranges ΔK _I and ΔK _II are determined from thermoelastic data recorded from around the tip of a sharp slot in a steel specimen under biaxial load, in order to assess the accuracy of the technique. ΔK _I and ΔK _II are determined to within 4% and 9% of a theoretical prediction, respectively. The results from a similar test on a fatigue crack under biaxial load are also presented. These show that thermoelastic stress analysis is a rapid and accurate way of analyzing mixed-mode fatigue cracks. A discussion is given on the potential of thermoelastic stress analysis of propagating cracks.     

Edge fracture in cone-plate and parallel plate flows

Edge fracture is an instability of cone-plate and parallel plate flows of viscoelastic liquids and suspensions, characterised by the formation of a `crack' or indentation at a critical shear rate on the free surface of the liquid. A study is undertaken of the theoretical, experimental and computational aspects of edge fracture. The Tanner-Keentok theory of edge fracture in second-order liquids is re-examined and is approximately extended to cover the Criminale-Ericksen-Filbey (CEF) model. The second-order theory shows that the stress distribution on the semi-circular crack is not constant, requiring an average to be taken of the stress; this affects the proportionality constant, K in the edge fracture equation −N _2c = KΓ/a, where N _2c is the critical second normal stress difference, Γ is the surface tension coefficient and a is the fracture diameter. When the minimum stress is used, K = 2/3 as found by Tanner and Keentok (1983). Consideration is given to the sources of experimental error, including secondary flow and slip (wall effect). The effect of inertia on edge fracture is derived. A video camera was used to record the inception and development of edge fracture in four viscoelastic liquids and two suspensions. The recorded image was then measured to obtain the fracture diameter. The edge fracture phenomenon was examined to find its dependence on the physical dimensions of the flow (i.e. parallel plate gap or cone angle), on the surface tension coefficient, on the critical shear rate and on the critical second normal stress difference. The critical second normal stress difference was found to depend on the surface tension coefficient and the fracture diameter, as shown by the theory of Tanner and Keentok (1983); however, the experimental data were best fitted by the equation −N _2c = 1.095Γ/a. It was found that edge fracture in viscoelastic liquids depends on the Reynolds number, which is in good agreement with the inertial theory of edge fracture. Edge fracture in lubricating grease and toothpaste is broadly consistent with the CEF model of edge fracture. A finite volume method program was used to simulate the flow of a viscoelastic liquid, obeying the modified Phan-Thien-Tanner model, to obtain the velocity and stress distribution in parallel plate flow in three dimensions. Stress concentrations of the second normal stress difference (N ₂) were found in the plane of the crack; the velocity distribution shows a secondary flow tending to aid crack formation if N ₂ is negative, and a secondary flow tending to suppress crack formation if N ₂ is positive. Received: 4 January 1999 Accepted: 19 May 1999     

Analysis of dynamic mixed-mode isochromatics

The mixed-mode, elastodynamic state of stress in the neighborhood of a constant-velocity crack tip is used to generate numerically unsymmetric isochromatics. Unsymmetry associated with the third-order terms of a mixed-mode stress field, with and without the Mode II singular stress term, is also investigated. In extractingK _I from an unsymmetric isochromatic pattern, errors in the Mode I fracture parameters due to the assumed presence ofK _II in aK _I stress field were found to be significant when data are taken more than 4 mm from the crack tip. Paper was presented at V International Congress on Experimental Mechanics held in Montreal, Quebec, Canada on June 10–15, 1984.     

A numerical analysis of flat internal cracks under mixed mode loading

A numerical/analytical approach is proposed to determine the stress intensity factors K_I, K_II, and K_III of a 3D internal crack. The main point of this approach is the meshing technique that can model very sharp crack fronts. The meshing technique is based on an elliptical coordinate transformation that starts from a circular crack. It allows the obtainment of a curved crack front with elements normal to the crack front. Remarkable accuracy can be obtained for elliptical crack fronts with axes ratio smaller that 0.01. Accuracy demonstration is provided for cylindrical element with an inclined internal crack subjected to uni-axial tension. This case corresponds to crack propagation for all three modes of loading, the solution of which can checked with references’ results.     

Plane-strain state of an elastoplastic body with a crack under mixed-mode loading

A mixed-mode (I + II) crack model with a plastic strip on its continuation under plane strain is proposed. The stress components within the strip are determined from the yield conditions, stress limitation, and relationship between the normal stress components defined via the principal stress state. The crack parameters are analyzed for the Mises yield condition. In the quasibrittle case, the governing system of equations includes stress intensity factors K _I, K _II, and T-stresses     

Dynamic fracture of a kane-mindlin plate

We study dynamic crack problems for an elastic plate by using Kane-Mindlin's kinematic assumptions. The general solutions of the Laplace transformed displacements and stresses are first derived. Path independent integrals for stationary cracks subjected to transient loads and steadily growing cracks are deduced. For a stationary crack in a very thin plate subjected to impact loads, the crack tip dynamic stress intensity factor (DSIF), K₁(t), is related to the far field plane stress one, K₁⁰(t), by where ν is Poisson's ratio. For a crack steadily growing with speed V, the crack tip DSIF, K₁(V), is given by where K₁⁰(V) is the plane stress DSIF and A(V) and B(V) are known functions of V. These results are applied to compute the DSIF for a semi-infinite stationary crack in an unbounded plate subjected to impact pressure on the crack faces. The results of DSIF for a finite crack in an infinite plate under uniform impact pressure on the crack surfaces show that for each plate thickness, the maximum DSIF is higher than that for the plane stress case.     

Oscillatory Couette flow in the presence of an inclined magnetic field

Oscillatory MHD Couette flow of electrically conducting fluid between two parallel plates in a rotating system in the presence of an inclined magnetic field is considered when the upper plate is held at rest and the lower plate oscillates non-torsionally . An exact solution of the governing equations has been obtained by using Laplace transform technique. Asymptotic behavior of the solution is analyzed for M ² ≪1, K ² ≪1 and ω ≪1 and for large M ², K ² and ω. Numerical results of velocities are depicted graphically and the frictional shearing stresses are presented in tables. It is found that a thin boundary layer is formed near the lower plate, for large values of rotation parameter K ², Hartman number M ² and frequency parameter ω. The thickness of this boundary layer increases with increase in inclination of the magnetic field with the axis of rotation.     

A comparison of three different methods for measuring both normal stress differences of viscoelastic liquids in torsional rheometers

A novel pressure sensor plate (normal stress sensor (NSS) from RheoSense, Inc.) was adapted to an Advanced Rheometrics Expansion System rheometer in order to measure the radial pressure profile for a standard viscoelastic fluid, a poly(isobutylene) solution, during cone–plate and parallel-plate shearing flows at room temperature. We observed in our previous experimental work that use of the NSS in cone-and-plate shearing flow is suitable for determining the first and second normal stress differences N ₁ and N ₂ of various complex fluids. This is true, in part, because the uniformity of the shear rate at small cone angles ensures the existence of a simple linear relationship between the pressure [i.e., the vertical diagonal component of the total stress tensor (Π₂₂)] and the logarithm of the radial position r (Christiansen and coworkers, Magda et al.). However, both normal stress differences can also be calculated from the radial pressure distribution measured in parallel-plate torsional flows. This approach has rarely been attempted, perhaps because of the additional complication that the shear rate value increases linearly with radial position. In this work, three different methods are used to investigate N ₁ and N ₂ as a function of shear rate in steady shear flow. These methods are: (1) pressure distribution cone–plate (PDCP) method, (2) pressure distribution parallel-plate (PDPP) method, and (3) total force cone–plate parallel-plate (TFCPPP) method. Good agreement was obtained between N ₁ and N ₂ values obtained from the PDCP and PDPP methods. However, the measured N ₁ values were 10–15% below the certified values for the standard poly(isobutylene) solution at higher shear rates. The TFCPPP method yielded N ₁ values that were in better agreement with the certified values but gave positive N ₂ values at most shear rates, in striking disagreement with published results for the standard poly(isobutylene) solution.

On a Generalized Nonlinear K–ε Model and the Use of Extended Thermodynamics in Turbulence

A resent extension of the nonlinear K–ε model is critically discussed from a basic theoretical standpoint. While it was said in the paper that this model was formulated to incorporate relaxation effects, it will be shown that the model is incapable of describing one of the most basic such turbulent flows as is obvious but is described for clarity. It will be shown in detail that this generalized nonlinear K–ε model yields erroneous results for the Reynolds stress tensor when the mean strains are set to zero in a turbulent flow – the return-to-isotropy problem which is one of the most elementary relaxational turbulent flows. It is clear that K–ε type models cannot describe relaxation effects. While their general formalism can describe relaxation effects, the nonlinear K–ε model – which the paper is centered on – cannot. The deviatoric part of the Reynolds stress tensor is predicted to be zero when it actually only gradually relaxes to zero. Since this model was formulated by using the extended thermodynamics, it too will be critically assessed. It will be argued that there is an unsubstantial physical basis for the use of extended thermodynamics in turbulence. The role of Material Frame-Indifference and the implications for future research in turbulence modeling are also discussed. Received 19 February 1998 and accepted 23 October 1998     

Mixed Convection in Water Near the Density Extremum Along a Vertical Plate with Sinusoidal Surface Temperature Variation Embedded in a Porous Medium

A steady laminar boundary layer flowing along a vertical plate immersed in a Darcy–Brinkman porous medium saturated with water at 4°C is studied. The plate temperature varies sinusoidally along the plate between 0 and 8°C where the density of water varies parabolically and is almost symmetrical at about 4°C. Except for the existence of the buoyancy force, it is assumed that either the plate moves upwards or the ambient water moves upwards (moving stream). The results are obtained with the direct numerical solution of the boundary layer equations taking into account the temperature dependence of water thermophysical properties (ρ, μ and c _p). Results are presented for the wall temperature gradient and the wall shear stress along the plate for free convection and mixed convection. Temperature and velocity profiles are also presented.     

Flow of a yield stress fluid over a rotating surface

We study the flow of yield stress fluids over a rotating surface when both the viscoelastic solid behavior below a critical deformation (γ _c) and liquid properties beyond γ _c can play a significant role. We review the detailed characteristics of the flow in the solid regime in the specific case of a pure elongational strain (large height to radius ratio). We, in particular, show that there exists a critical rotation velocity (ω _c) associated with the transition from the solid to the liquid regime. We then consider the specific case of lubricational regime (small height to radius ratio) in the liquid regime. In that case we describe the different possible evolutions of the equilibrium shape of the material as a function of the rotation velocity (ω), from which we extrapolate the transient shape evolutions as ω increases. We show that for a sufficiently large rotation velocity the sample separates into two parts, one remaining at rest around the rotation axis, the other going on moving radially. These predictions are then compared with systematic spin-coating tests under increasing rotation velocity ramps followed by a plateau at ω _f with typical yield stress fluids. It appears that there exists a critical velocity below which the material undergoes a limited elongation and beyond which it starts to spread significantly over the solid surface. For a larger ω _f value the sample forms a thick peripheral roll, leaving behind it a thin layer of fluid at rest relatively to the disc. These characteristics are in qualitative agreement with the theoretical predictions. Beyond a sufficiently large ω _f value this roll eventually spreads radially in the form of thin fingers. Moreover, in agreement with the theory in the lubricational regime, the different curves of deformation vs ω fall along a master curve when the rotation velocity is scaled by ω _c for different accelerations, different sample radii, or different material yield stress. The final thickness of the deposit seems to be mainly governed by the displacement of the roll, the characteristics of which take their origin in the initial stage of the spreading, including the solid–liquid transition.     

Numerical simulation of stress intensity factors via the thermoelastic technique

The purpose of this study is to investigate the accuracy of the least squares method for finding the in-plane stress intensity factorsK _I andK _II using thermoelastic data from isotropic materials. To fully understand the idealized condition ofK _I andK _II calculated from thermoelastic experiments, the total stress field calculated from finite element analysis is used to take the place of data obtained from real thermoelastic experiments. In the finite element analysis, theJ-integral is also calculated to compare with (K _I ² +K _II ² )/E evaluated by the least squares method. The stress fields near the crack tip are dominated by the two stress intensity factors; however, the edge effect will cause inaccuracy of the thermoelastic data near the crack tip. Furthermore, the scan area of thermoelastic experiments cannot be too small. Therefore, we suggest that three or four terms of stress function be included in the least squares method for evaluating stress intensity factors via the thermoelastic technique. In the idealized condition, the error can be smaller than 3 percent from our numerical simulations. If only ther ^–1/2 term (K _I andK _II ) is included in the least squares method, even in the idealized case the error can be up to 20 percent.     

Computational flow and heat transfer of a row of circular jets impinging on a concave surface

A computational investigation is carried out to study the flow and heat transfer from a row of circular jets impinging on a concave surface. The computational domain simulates the impingement cooling zone of a gas turbine nozzle guide vane. The parameters, which are varied in the study include jet Reynolds number (Re _d = 5000–67800), inter-jet distance to jet diameter ratio (c/d = 3.33 and 4.67) and target plate distance to jet diameter ratio (H/d = 1, 3 and 4). The flow field, predicted with K-ω turbulence model and using Fluent 6.2.16, is characterized with the presence of a pair of counter rotating vortices, an upwash fountain flow and entrainment. The local pressure coefficient and Nusselt number variations along the concave plate are presented and these values are found to under predict the available experimental data by about 12%.     

Embedded fine-grid method applied to three-dimensional stress-intensity-factor analysis

The purpose of this paper is to extend the embedded fine-grid method to three-dimensional stress-intensity-factor analysis. The embedded crossed fine grids give two components of displacement and are capable of calculatingK _I without the assumption of the plane-strain condition along a crack or a notch front. It is not valid to assume the plane-strain condition to calculateK _I in the vicinity of a free surface, whereK _I is influenced by a free surface (plane-stress condition). In this paper, a dyeing and bleaching process is considered to reproduce the crossed fine grids on an epoxy plate. By using these grids as embedded grids, the distributions ofK _I along notch fronts in SEN specimens with various different thicknesses and side grooves are studied. The influences of a free surface and the side-groove effects on the distribution ofK _I are discussed. Paper was presented at 1982 SESA/JSME Spring Meeting held in Oahu and Maui, HI on May 24–29.     

MHD mixed convective heat transfer flow about an inclined plate

Mixed convection heat transfer about a semi-infinite inclined plate in the presence of magneto and thermal radiation effects is studied. The fluid is assumed to be incompressible and dense. The nonlinear coupled parabolic partial differential equations governing the flow are transformed into the non-similar boundary layer equations, which are then solved numerically using the Keller box method. The effects of the mixed convection parameter R _i, the angle of inclination α, the magnetic parameter M and the radiation–conduction parameter R _d on the velocity and temperature profiles as well as on the local skin friction and local heat transfer parameters. For some specific values of the governing parameters, the results are compared with those available in the literature and a fairly good agreement is obtained.     

Topological Semi-Conjugacies Between Adding Machines

Let K = (k ₁, k ₂, . . .) be a sequence of positive integers, and be a topological group with a metric and an additive operation given. The adding machine f _K is the addition-by-one map on Σ _K . Buescu and Stewart (Ergodic Theory Dynam Syst 15:271–290, 1995), Block and Keesling (Topology Appl 140:151–161, 2004) and Banks (Ergodic Theory Dynam Syst 17:505–529, 1997) obtained several equivalent conditions for which two adding machines are topologically conjugate, respectively. In this paper, we have a further discussion about semi-conjugate relationship between adding machines, and give some necessary and sufficient conditions for an adding machine to be semi-conjugate to another one. Moveover, we also prove that a transitive translation on a compact subgroup of Σ _K is isometrically conjugate to an adding machine. Dedicated to Professor Zhifen Zhang on the occasion of her 80th birthday     

The interaction between crack and electric dipole of piezoelectricity

Discrete dipoles located near the crack tip play an important role in nonlinear electric field induced fracture of piezoelectric ceramics. A physico-mathematical model of dipole is constructed of two generalized concentrated piezoelectric forces with equal density and opposite sign. The interaction between crack and electric dipole in piezoelectricity is analyzed. The closed form solutions, including those for stress and electric displacement, crack opening displacement and electric potential, are obtained. The function of piezoelectric anisotropic direction,p _a (θ)=cosθ+p _a sinθ, can be used to express the influence of a dipole's direction. In the case that a dipole locates near crack tip, the piezoelectric stress intensity factor is a power function with −3/2 index of the distance between dipole and crack tip. Supported by National Natural Science Foundation of China(No. 10072033)