In-plane perturbation of the tunnel-crack under shear loading II: Determination of the fundamental kernel

For any plane crack in an infinite isotropic elastic body subjected to some constant loading, Bueckner–Rice's weight function theory gives the variation of the stress intensity factors due to a small coplanar perturbation of the crack front. This variation involves the initial SIF, some geometry independent quantities and an integral extended over the front, the “fundamental kernel” of which is linked to the weight functions and thus depends on the geometry considered. The aim of this paper is to determine this fundamental kernel for the tunnel-crack. The component of this kernel linked to purely tensile loadings has been obtained by Leblond et al. [Int. J. Solids Struct. 33 (1996) 1995]; hence only shear loadings are considered here. The method consists in applying Bueckner–Rice's formula to some point-force loadings and special perturbations of the crack front which preserve the crack shape while modifying its size and orientation. This procedure yields integrodifferential equations on the components of the fundamental kernel. A Fourier transform in the direction of the crack front then yields ordinary differential equations, that are solved numerically prior to final Fourier inversion.     

Statistics of the deformation of the front of a tunnel-crack propagating in some inhomogeneous medium

We present a statistical analysis of some geometrical features of the front of a tensile tunnel-crack propagating quasistatically, according to some Paris-type law, in some elastic solid with spatially varying Paris constant. The work is based on an earlier formula of the authors, which provides the first-order change of the distribution of the mode I stress intensity factor along the front of a tunnel-crack, arising from some small but otherwise arbitrary in-plane perturbation of this front. The quantities studied include the power spectrum and the autocorrelation function of the deviation of the two parts of the front from reference straight lines, the autocorrelation function of the derivative of this deviation in the direction of the crack front, the mean squared fluctuation of the deviation, and its correlation distance. The various measures of the magnitude of the deviation of the front from straightness are all found to increase in time at a considerable rate, which means in some sense that the “wavyness” of the front continuously grows. However, the correlation distance of the deviation also increases, which mitigates the preceding conclusion, since it means in another sense that the crack front tends to “straighten back” in time. Also, comparisons are made with the cases of a semi-infinite crack propagating quasistatically or dynamically, using some results of Rice and coworkers for the latter case. The rate of growth of the various measures of the magnitude of the deviation from straightness is much larger for the tunnel-crack than for the semi-infinite one. This is because the finite width of the tunnel-crack induces a “destabilizing” effect of the straight configuration of the front for sinusoidal perturbations with large wavelengths, which is typical of such finite crack geometries.     

Wave front analysis of a plane compressional pulse scattered by a cylindrical elastic inclusion

The plane-strain problem of a stress pulse striking an elastic circular cylindrical inclusion embedded in an infinite elastic medium is treated. The method used determines dominant stress singularities that arise at wave fronts from the focusing of waves refracted into the interior. It is found that a necessary and sufficient condition for the existence of a propagating stress singularity is that the incident pulse has a step discontinuity at its front. The asymptotic wave front behavior of the first few P and SV waves to focus are determined explicitly and it is shown that the contribution from other waves are less important. In the exterior, it is found that in most composite materials the reflected waves have a singularity at their wave front which depends on the angle of reflection. Also the wave front behavior of the first few singular transmitted waves is given explicitly.The analysis is based on the use of a Watson-type lemma, developed here, and Friedlander's method [5]. The lemma relates the asymptotic behavior of the solution at the wave front to the asymptotic behavior of its Fourier transform on time for large values of the transform parameter. Friedlander's method is used to represent the solution in terms of angularly propagating wave forms. This method employs integral transforms on both time and θ, the circumferential coordinate. The θ inversion integral is asymptotically evaluated for large values of the time transform parameter by use of appropriate asymptotics for Bessel and Hankel functions and the method of stationary phase. The Watson-type lemma is then used to determine the behavior of the solution at singular wave fronts.The Watson-type lemma is generally applicable to problems which involve singular loadings or focusing in which wave front behavior is important. It yields the behavior of singular wave fronts whether or not the singular wave is the first to arrive. This application extends Friedlander's method to an interior region and physically interprets the resulting representation in terms of ray theory.     

Spatial and Temporal Characteristics of OH in Turbulent Opposed-Jet Double Flames

Simultaneous high repetition-rate, two-point hydroxyl (OH) time-series measurements with associated PLIF/PIV measurements are employed to investigate spatio–temporal scales and flame-velocity interactions in turbulent opposed jets sustaining methane-air double flames. For a fuel-side equivalence ratio, ϕ _B  = 1.2, a rich premixed flame exists on the fuel side while a diffusion flame exists on the air side of the stagnation plane. The bulk Reynolds number (Re) and strain rate (SR) can be adjusted to generate flames at ϕ _B  = 1.2 with both well separated and completely merged flame fronts. Simultaneous PLIF/PIV measurements highlight distinct spatial OH structures of the premixed and diffusive fronts corresponding to variations in the flow field. The self-propagating tendency of the rich premixed front causes large-scale wrinkling, thereby enhancing the OH contour length by 15% as compared to the diffusive front. Two-point OH time-series measurements are implemented to quantify both spatial and temporal fluctuations via study of radial length and time scales. In general, these integral length and time scales follow similar trends and reach a minimum at the axial location of peak [OH]. In comparison to merged double flames having higher Re and SR, greater OH fluctuations are observed in the rich-premixed front as compared to the diffusive front for a well separated double flame. Because of the developing turbulence, the OH length scales exhibit reduced axial gradients across the reaction zone for higher Re in comparison to lower Re. A stochastic time-series simulation, using a state relationship based on a joint mixture fraction and progress variable, is utilized to extract estimated scalar time scales from those of measured OH. The simulations indicate that the hydroxyl fluctuations in double flames are only twice those of the underlying conserved scalar. “Turbulent Opposed-Jet Double Flames” is submitted for consideration as a full length article to Flow Turbulence and Combustion.     

The Structure of Precipitation Fronts for Finite Relaxation Time

When convection is parameterized in an atmospheric circulation model, what types of waves are supported by the parameterization? Several studies have addressed this question by finding the linear waves of simplified tropical climate models with convective parameterizations. In this paper’s simplified tropical climate model, convection is parameterized by a nonlinear precipitation term, and the nonlinearity gives rise to precipitation front solutions. Precipitation fronts are solutions where the spatial domain is divided into two regions, and the precipitation (and other model variables) changes abruptly at the boundary of the two regions. In one region the water vapor is below saturation and there is no precipitation, and in the other region the water vapor is above saturation level and precipitation is nonzero. The boundary between the two regions is a free boundary that moves at a constant speed. It is shown that only certain front speeds are allowed. The three types of fronts that exist for this model are drying fronts, slow moistening fronts, and fast moistening fronts. Both types of moistening fronts violate Lax’s stability criterion, but they are robustly realizable in numerical experiments that use finite relaxation times. Remarkably, here it is shown that all three types of fronts are robustly realizable analytically for finite relaxation time. All three types of fronts may be physically unreasonable if the front spans an unrealistically large physical distance; this depends on various model parameters, which are investigated below. From the viewpoint of applied mathematics, these model equations exhibit novel phenomena as well as features in common with the established applied mathematical theories of relaxation limits for conservation laws and waves in reacting gas flows.     

Temperature analysis of one-dimensional NiTi shape memory alloys under different loading rates and boundary conditions

Superelastic polycrystalline NiTi shape memory alloys under tensile loading accompany the strain localization and propagation phenomena. Experiments showed that the number of moving phase fronts and the mechanical behavior are very sensitive to the loading rate due to the release/absorption of latent heat and the material’s inherent temperature sensitivity of the transformation stress. In this paper, the moving heat source method based on the heat diffusion equation is used to study the temperature evolution of one-dimensional superelastic NiTi specimen under different loading rates and boundary conditions with moving heat sources or a uniform heat source. Comparisons of temperature variations with different boundary conditions show that the heat exchange at the boundaries plays a major role in the nonuniform temperature profile that directly relates to the localized deformation. Analytical relation between the front temperature of a single phase front, the inherent Clausius–Clapeyron relation (sensitivity of the material’s transformation stress with temperature), heat transfer boundary conditions and the loading rate is established to analyze the nucleation of new phase fronts. Finally, the rate-dependent stress hysteresis is also simply discussed by using the results of temperature analyses.     

Permanent Fronts in Two-Phase Flows in a Porous Medium

A family of exact solutions for a model of a one-dimensional horizontal flow of two immiscible, incompressible fluids in a porous medium, including the effects of capillary pressure, is obtained analytically by solving the governing singular parabolic nonlinear diffusion equation. Each solution has the form of a permanent front propagating with a constant velocity. It is shown that, for every propagation velocity, there exists a set of permanent fronts all of which are moving with this velocity in an inflowing wetting–outflowing non-wetting flow configuration. Global bifurcations of this set, with the front velocity as a bifurcation parameter, are investigated analytically and numerically in detail in the case when the permeabilities and the capillary pressure are linear functions of the wetting phase saturation. Main results for the nonlinear Brooks–Corey model are also presented. In both models three global bifurcations occur. By using a geometric dynamical system approach, the nonlinear stability of the permanent fronts is established analytically. Based on the permanent front solutions, an interpretation of the dynamics of an arbitrary front of finite extent in the model is given as follows. The instantaneous upstream (downstream) velocity of an arbitrary non-quasistationary front is equal to the velocity of a permanent front whose shape coincides up to two leading orders with the instantaneous shape of the non-quasistationary front at the upstream (respectively, downstream) location. The upstream and downstream locations of the front undergo instantaneous translations governed by modified nonsingular hyperbolic equations. The portion of the front in between these locations undergoes a diffusive redistribution governed by a nonsingular nonlinear parabolic diffusion equation. We have proposed a numerical approach based on a parabolic–hyperbolic domain decomposition for computing non-quasistationary fronts.     

层结流体中湍流锋面特征的实验研究

在线性分层的水体中；由拖动坚条形栅格产生湍流锋面，通过阴影可视化技术考察了锋面的流态和运动行为，将锋面迹线经过图像处理，获得了锋面运动特征的有关信息：运动速度随时间的增长关系；锋面坍塌的平均无因次特征时间为NTC＝2．9；它随湍流Froude数Fri的增大而减小；锋面的特征尺度Hf与栅格运动的参数有关；当湍流演化成内波时，对锋面与内波运动间的关系进行了初步探讨和分析．     

Growth of discontinuities in fluids with internal relaxation

The growth equation for weak discontinuities headed by wave fronts of arbitrary shape in a relaxing gas flow is derived along the orthogonal trajectories of the wave fronts. An explicit criteria for the growth and decay of weak discontinuities along their orthogonal trajectories is given. It is concluded that the internal relaxation processes in the flow as well as the wave front curvature both have a stabilizing effect on the tendency of the wave surface to grow into a shock in the sense that they cause the shock formation time to increase.     

The stability of growth of a through-wall circumferential crack in a pipe subject to combined bending and tensile loadings

The stability of growth of a through-wall circumferential crack in a pipe is analysed for the case where the material has a high crack growth resistance, the analysis being based on the tearing modulus procedure. Rotations and lateral displacements are applied at the ends of the pipe, and this allows the combined effects of bending and tensile loadings on the stability of crack growth to be assessed. The general conclusion is that tensile loadings can have an adverse effect on crack stability, in accord with the conclusion reached in the author's earlier studies of plane strain crack growth in a beam. The stability results are compared with those obtained by Tada, Paris and Gamble, who allowed the tensile loadings to affect the position of the neutral axis, but did not consider instability in terms of the deformations produced by these loadings.     

Dynamics of propagating phase boundaries in NiTi

Propagating boundaries of phase transformation have been generated in polycrystalline NiTi specimens under a tensile impact loading condition. Multiple strain gages were used to monitor the time evolution of the strain at different spatial locations in the specimen. Nucleation and propagation of multiple phase fronts were detected in these experiments; the phase front speed was found to be in the range between 37 and 370 m/s. The strain measurements were interpreted through the one-dimensional analysis of Abeyaratne and Knowles [1997. On the kinetics of an austenite→martensite phase transformation induced by impact in Cu-Al-Ni shape-memory alloy. Acta Mater. 45, 1671-1683] and a model of partial phase transformation in the polycrystalline specimen. The driving force for the motion of the phase front was evaluated from the measurements in order to establish the kinetic relation.     

Adaptive methods of lines for one-dimensional reaction-diffusion equations

Adaptive and non-adaptive finite difference methods are used to study one-dimensional reaction-diffusion equations whose solutions are characterized by the presence of steep, fast-moving flame fronts. Three non-adaptive techniques based on the methods of lines are described. The first technique uses a finite volume method and yields a system of non-linear, first-order, ordinary differential equations in time. The second technique uses time linearization, discretizes the time derivatives and yields a linear, second-order, ordinary differential equation in space, which is solved by means of three-point, fourth-order accurate, compact differences. The third technique takes advantage of the disparity in the time scales of the reaction and diffusion processes, splits the reaction--diffusion operator into a sequence of reaction and diffusion operators and solves the diffusion operator by means of either a finite volume method or a three-point, fourth-order accurate compact difference expression. The non-adaptive methods of lines presented in this paper may use equaliy or non-equally spaced fixed grids and require a large number of grid points to solve accurately one-dimensional problems characterized by the presence of steep, fast-moving fronts. Three adaptive methods for the solution of reaction-diffusion equations are considered. The first adaptive technique is static and uses a subequidistribution principle to determine the grid points, avoid mesh tangling and node overtaking and obtain smooth grids. The second adaptive technique is dynamic, uses an equidistribution principle with spatial and temporal smoothing and yields a system of first-order, non-linear, ordinary differential equations for the grid point motion. The third adaptive technique is hybrid, combines some features of static and dynamic methods, and uses a predictor-corrector strategy to predict the grid and solve for the dependent variables, respectively. The three adaptive techniques presented in this paper use physical co-ordinates and may employ finite volume or three-point, compact methods. The adaptive and non-adaptive finite difference methods presented in the paper are used to study a decomposition chemical reaction characterized by a scalar, one-dimensional reaction-diffusion equation, the propagation of a one-dimensional, confined, laminar flame in Cartesian co-ordinates and the Dwyer-Sanders model of one-dimensional flame propagation. It is shown that the adaptive moving method presented in this paper requires a smaller number of grid points than adaptive static, adaptive hybrid and non-adaptive methods. The adaptive hybrid method requires a smaller time step than adaptive static techniques, due to the lag between the grid prediction and the solution of the dependent variables. Non-adaptive methods of lines may yield temperature oscillations in front of and behind the flame front if Crank-Nicolson techniques are used to evaluate the time derivatives. Fourth-order accurate methods of lines in space yield larger temperature oscillations than second-order accurate methods of lines, and the magnitude of these oscillations decreases as the time step is decreased. It is also shown that three-point, fourth-order accurate discretizations of the spatial derivatives require the same number of grid points as second-order accurate, finite volume methods, in order to resolve accurately the structure of steep, fast-moving flame fronts.     

Lüders front propagation in low carbon steels

T characteristics of Lüders front propagation in low carbon steel wire and thin sheet specimens have been investigated by tensile straining at constant extension rates at room temperature. The specimen variables of grain size and matrix structure were both shown to be important in determining the Lüders front velocity. The testing variables which affected the velocity were the specimen extension rate and the number of fronts but not the specimen strain rate. The unique relationship obtained between the Lüders stress and Lüders strain for a particular steel is a consequence of the strain hardening behaviour of the steel in that, for each Lüders stress, the associated Lüders strain will be dictated by a homogeneous stress-strain relationship.     

In-plane perturbation of the tunnel-crack under shear loading I: bifurcation and stability of the straight configuration of the front

One considers a planar tunnel-crack embedded in an infinite isotropic brittle solid and loaded in mode 2+3 through some uniform shear remote loading. The crack front is slightly perturbed within the crack plane, from its rectilinear configuration. Part I of this work investigates the two following questions: Is there a wavy “bifurcated” configuration of the front for which the energy release rate is uniform along it? Will any given perturbation decay or grow during propagation? To address these problems, the distribution of the stress intensity factors (SIF) and the energy release rate along the perturbed front is derived using Bueckner–Rice's weight function theory. A “critical” sinusoidal bifurcated configuration of the front is found; both its wavelength and the “phase difference” between the fore and rear parts of the front depend upon the ratio of the initial (prior to perturbation of the front) mode 2 and 3 SIF. Also, it is shown that the straight configuration of the front is stable versus perturbations with wavelength smaller than the critical one but unstable versus perturbations with wavelength larger than it. This conclusion is similar to those derived by Gao and Rice and the authors for analogous problems.     

The stability of crack growth in a beam subject to combined bending and tensile loadings: I: Beam built-in at one end

The stability of crack growth in a beam subject to combined bending and tensile loadings is examined for the case where the material is very ductile and where the beam is built-in at one end. Application of a displacement at the free end, and variation of the angle which the displacement makes with the beam, allows the combined effects of bending and tensile loadings on the stability of crack growth to be assessed. The stability analysis is based on the tearing modulus procedure, and the general conclusion is that tensile loadings can promote crack instability.     

Generalized Traveling Waves in Disordered Media: Existence, Uniqueness, and Stability

We prove existence, uniqueness, and stability of transition fronts (generalized traveling waves) for reaction-diffusion equations in cylindrical domains with general inhomogeneous ignition reactions. We also show uniform convergence of solutions with exponentially decaying initial data to time translations of the front. In the case of stationary ergodic reactions, the fronts are proved to propagate with a deterministic positive speed. Our results extend to reaction-advection-diffusion equations with periodic advection and diffusion.     

The stability of crack growth in a beam subject to combined bending and tensile loadings. II: Beam subject to rotations and lateral displacements at the ends

The stability of crack growth in a beam subject to combined bending and tensile loadings is examined for the case where the material is very ductile and where the beam is built-in at one end. Application of a displacement at the free end, and variation of the angle which the displacement makes with the beam, allows the combined effects of bending and tensile loadings on the stability of crack growth to be assessed. The stability analysis is based on the tearing modulus procedure, and the general conclusion is that tensile loadings can promote crack instability.     

In-plane perturbation of a system of two coplanar slit-cracks – II: Case of close inner crack fronts or distant outer ones

The study of the in-plane perturbation of a system of two coplanar slit-cracks carried out in Part I is specialized to the case where the distance between the inner crack fronts is small, or equivalently that between the outer fronts large. The limit process involved is complex because of appearance of a “boundary layer” in the limiting case considered; this boundary layer occurs near the origin in the Fourier space used to determine the unknown components of the fundamental kernel looked for. A technique of matched asymptotic expansions is used to tackle this difficulty.The problem is thus reduced to determining two unknown functions only, which characterize the “interactions” between the two inner fronts. These functions obey a system of nonlinear differential equations in Fourier’s space, which are solved analytically near the origin and numerically in general. The results evidence a very slow decrease of long-range interactions between distinct points on the same front or distinct ones. This represents a striking difference with respect to the cases considered earlier of a single semi-infinite crack and a single slit-crack.     

Inelastic deformation and energy dissipation in ceramics: A mechanism-based constitutive model

A mechanism-based constitutive model is presented for the inelastic deformation and fracture of ceramics. The model comprises four essential features: (i) micro-crack extension rates based on stress-intensity calculations and a crack growth law, (ii) the effect of the crack density on the stiffness, inclusive of crack closure, (iii) plasticity at high confining pressures, and (iv) initial flaws that scale with the grain size. Predictions of stress/strain responses for a range of stress states demonstrate that the model captures the transition from deformation by micro-cracking at low triaxiality to plastic slip at high triaxialities. Moreover, natural outcomes of the model include dilation (or bulking) upon micro-cracking, as well as the increase in the shear strength of the damaged ceramic with increasing triaxiality. Cavity expansion calculations are used to extract some key physics relevant to penetration. Three domains have been identified: (i) quasi-static, where the ceramic fails due to the outward propagation of a compression damage front, (ii) intermediate velocity, where an outward propagating compression damage front is accompanied by an inward propagating tensile (or spallation) front caused by the reflection of the elastic wave from the outer surface and (iii) high velocity, wherein plastic deformation initiates at the inner surface of the shell followed by spalling within a tensile damage front when the elastic wave reflects from the outer surface. Consistent with experimental observations, the cavity pressure is sensitive to the grain size under quasi-static conditions but relatively insensitive under dynamic loadings.     

Energy domain integral applied to solve center and double-edge crack problems in three-dimensions

A three-dimensional Boundary Element Method (BEM) implementation of the energy domain integral for the numerical computation of the crack energy release rate is presented in this paper. The domain expression of the energy release rate is naturally compatible with the BEM, since stresses, strains and derivatives of displacements at internal points can be evaluated using the appropriate boundary integral equations. The pointwise crack energy release rate is evaluated along the three-dimensional crack front over a cylindrical domains that surround a segment of the crack front. The accuracy of the implementation is demonstrated by solving several problems, which include geometries containing straight as well as curved crack fronts.