A model for crack formation during active solid pyrolysis of a char-forming solid: crack patterns; surface area generation; volatile mass efflux

A model is developed for the formation and propagation of cracks in a material sample that is heated at its top surface, pyrolyses, and then thermally degrades to form char. In this work the sample is heated uniformly over its entire top surface by a hypothetical flame (a heat source). The pyrolysis mechanism is described by a one-step overall reaction that is dependent nonlinearly on the temperature (Arrhenius form). Stresses develop in response to the thermal degradation of the material by means of a shrinkage strain caused by local mass loss during pyrolysis. When the principal stress exceeds a prescribed threshold value, the material forms a local crack. Cracks are found to generally originate at the surface in response to heating, but occasionally they form in the bulk, away from ever-changing material boundaries. The resulting cracks evolve and form patterns whose characteristics are described. Quantities examined in detail are: the crack spacing in the pyrolysis zone; the crack length evolution; the formation and nature of crack loops which are defined as individual cracks that have joined to form loops that are disconnected from the remaining material; the formation of enhanced pyrolysis area; and the impact of all of the former quantities on mass flux. It is determined that the mass flux from the sample can be greatly enhanced over its nominal (non-cracking) counterpart. The mass efflux profile qualitatively resembles those observed in Cone Calorimeter tests.     

Fracture features of granite under various deformation conditions

Fracture mechanisms of dry and water-saturated granite samples and slip (displacement) along a ready fault have been studied by measuring acoustic emission signals. It has been found that disperse defect formation is observed in dry samples under mechanical load, then localization occurs, and a fracture source is formed, whose development results in macrofault formation. In water saturated samples, chaotic defect formation occurs in the entire volume, which leads to a high degree of material damage. At the closing deformation stage, several fault source zones are formed in which main cracks develop. In the case of slip along a ready fault, stoppers at crack edges are broken.     

Non-local theory solution of two collinear mode-I permeable cracks in a magnetoelectroelastic composite material plane

The non-local theory solution of two collinear mode-I permeable cracks in a magnetoelectroelastic composite material plane was investigated using the generalized Almansi's theorem and the Schmidt method. The problem was formulated through Fourier transform into two pairs of dual integral equations, in which the unknown variables are the jumps in displacements across the crack surfaces. To solve the dual integral equations, the displacement jumps across the crack surfaces were directly expanded as a series of Jacobi polynomials. Numerical examples were provided to show the effects of crack length, the distance between two collinear cracks and the lattice parameter on the stress field, the electric displacement field and the magnetic flux field near the crack tips. Unlike the classical elasticity solutions, it is found that no stress, electric displacement or magnetic flux singularities are present at the crack tips in a magnetoelectroelastic composite material plane. The non-local elastic solutions yield a finite hoop stress at the crack tip, thus allowing us to use the maximum stress as a fracture criterion.     

Stress-induced ultrasonic wave velocity anisotropy in fractured rock

The closure of cracks in rock under an applied compressive stress can significantly affect the permeability of the rock. Crack closure may be monitored using ultrasonic wave velocities, since these are significantly reduced in the presence of open cracks. When a non-hydrostatic compressive stress is applied to a rock, an initially isotropic distribution of cracks will become anisotropic and the rock will display an elastic anisotropy determined by the orientation distribution of those cracks remaining open. The crack orientation distribution function gives the probability of a crack having a given orientation with respect to a set of axes fixed in the rock. The coefficients W_Imn of a series expansion of this function in generalized Legendre functions can be obtained to order I = 4 from the angular variation of the elastic wave velocity. This allows construction of microfracture pole figures, which specify the orientation distribution of open cracks. The theory is applied to the measurements of Nur and Simmons, who applied a uniaxial compressive stress to a sample of Barre granite. Cracks with normals aligned along the stress direction are closed preferentially in agreement with the theory of Walsh. However, for crack normals perpendicular to the applied stress there is some evidence of crack opening that is not predicted by the theory. This is also observed in the electron microscope study of Batzle et al. and a possible mechanism is discussed.     

基于幅度调制的非线性超声相控阵成像方法*

超声相控阵是超声检测领域常用的材料缺陷定位和成像技术,可便捷快速地对裂纹、孔洞等缺陷进行成像。但是,传统超声相控阵方法对较小缺陷如闭合裂纹不太敏感。非线性超声信号因对材料性能退化以及微小缺陷敏感而广受关注。本文针对疲劳闭合裂纹检测,提出一种基于幅度调制的非线性超声相控阵成像方法,通过测量物理聚焦和虚拟聚焦两种聚焦模式下超声扩散场中的声能差,并将其作为非线性参量,实现疲劳裂纹闭合部分的定位成像和定量表征。将该法应用于7075铝合金试样疲劳裂纹的实验测量,并研究了扩散场信号延迟时间对非线性超声相控阵成像结果的影响。结果表明：相较于传统超声相控阵全聚焦法,基于幅度调制的非线性超声相控阵成像方法能够更准确地定位和成像疲劳裂纹闭合部分;延迟时间的选择对疲劳裂纹长度的表征精度影响较大,本文研究了该延迟时间的选择方法并实现了检测结果的优化。     

In situ estimation of crack concentration in hard rock

Results of a 2004 field experiment aimed at determining the quadratic nonlinearity parameter in granite that forms the shore of the Ladoga lake are presented. The measurements were based on the observation of the nonlinear interaction between monochromatic waves excited by two hydroacoustic radiators of 1 kW each positioned near the shore. The initial level of nonlinear distortions was much lower than the level of the received difference-frequency signal. The quadratic nonlinearity parameter proved to be higher than that reported in the majority of publications. An assumption was put forward that the high nonlinearity of granite is caused by the high concentration of cracks in it. Pieces of rock were taken from the measurement site, and rectangular samples were prepared from them. The samples were studied by the acoustic spectroscopy method in laboratory conditions. As a result, estimates of crack concentration were obtained, which proved to be consistent with the field measurements of the quadratic nonlinearity parameter. Thus, the possibility of estimating the crack concentration in situ from the measurements of the quadratic nonlinearity parameter was demonstrated.     

Modeling of thermomechanical fracture of functionally graded materials with respect to multiple crack interaction

Different aspects of thermomechanical fracture of functionally graded materials (FGMs) are considered. Among them are the crack interaction problems in a functionally graded coating on a homogeneous substrate (FGM/H). The interaction between systems of edge cracks is investigated, as well as, how this mutual interaction influences the fracture process and the formation of crack patterns. The problem is formulated with respect to singular integral equations which are referred to the boundary equation methods. The FGM properties are modeled by exponential functions. The main fracture characteristics are calculated, namely, the stress intensity factors, the angles of deviation of the cracks from their initial propagation direction and the critical stresses when the crack starts to propagate. The last two characteristics are calculated using an appropriate fracture criterion. The problem contains different parameters, such as the geometry (location and orientation of cracks, their lengths, and the width of the FGM layer) and material parameters, i.e. the inhomogeneity parameters of elastic and thermal coefficients of the functionally graded material. The influence of these parameters on the thermo-mechanical fracture of FGM/H is investigated. As examples the following real material combinations are discussed: TiC/SiC, Al₂O₃/MoSi₂, MoSi₂/SiC, ZrO₂/nickel and ZrO₂/steel.     

激光激发瑞利波检测表面倾斜缺陷的研究

研究激光激发瑞利波检测样品的表面倾斜缺陷。基于频域热弹耦合方程,采用有限元方法建立激光激发瑞利波检测倾斜缺陷的数值模型,研究倾斜缺陷的检测机理。数值计算含不同的长度及倾斜角度缺陷的样品中瞬态位移波形,分析瑞利波在倾斜缺陷处模式转换的过程,研究各种瑞利波的传播路径。在此基础上,研究缺陷宽度与材料黏性对瑞利波传播及缺陷检测的影响。结果表明:瑞利波在缺陷处产生的反射及透射瑞利波的到达观测点的时间可以检测缺陷位置和长度,瑞利波在缺陷的底部发生模式转换产生的切变波可以检测缺陷倾斜角度。数值结果和已有的实验结果一致,从而为表面倾斜缺陷的检测提供有效的理论依据。      

Detection and evaluation of the depth of surface cracks in conductive materials by using a loop antenna

A new method is developed for detection and evaluation of the depth of surface cracks in conductive materials. The main components of the measurement equipment are a conductive wire and a loop antenna, both of which are located as close as possible to the surface of the investigated material, while the antenna loop is perpendicular to the surface and parallel to the wire. Supplying of electrical potential with microwave frequency f and amplitude |_s| to the wire results in spreading of induced current close beneath the surface of the material. The induced current redistributes in the vicinity of surface cracks, and some of the current paths flow along the crack walls. This leads to generation of a time-variable magnetic field mostly within the volume of the crack, which is detected by the loop antenna. A theoretical analysis is presented of the contributions of the different types of generated magnetic fields to the potential drop with amplitude |U_s| measured by the antenna. The distribution of the ratio R(dB)=|U|/|U_s| is measured for specimens of paramagnetic and ferromagnetic steel which contain three artificial cracks with depths of 0.5 mm, 1 mm, and 2 mm, and frequency f=300 MHz of the supplied electrical potential. It turns out that artificial cracks with a depth of d 0.5 mm, located at distances from x=3 mm to x=15 mm from the wire, can be detected. The largest sensitivity of crack detection is achieved for x4 mm when the long axis of a crack is perpendicular to the wire. The crack response of the measured ratio R is proportional to the crack depth when the loop of the antenna is located above or slightly outside an artificial crack, and at least 4 mm from the wire. This relation could be used for evaluation of the depth of real surface cracks. The notable decrement of the crack response outside the crack can indicate the location of the crack tips on the surface of the specimen. It is shown that a loop antenna with a diameter of 7 mm provides the largest crack response, as well as that the developed method can be applied to both paramagnetic and ferromagnetic materials. An example of the detection of a real crack by an antenna is reported for the first time. PACS 81.70.Ex; 84.40.Ba; 84.32.Ff     

A numerical analytic method for electromagnetic radiation accompanying with fracture of rocks

This paper studies Rabinovitch’s compression experiments on granite and chalk and proposes an oscillating dipole model to analyse and simulate the electromagnetic radiation phenomenon caused by fracture of rocks. Our model assumes that the electromagnetic radiation pulses are initiated by vibrations of the charged rock grains on the tips of the crack. The vibrations of the rock grains are stimulated by the pulses of the cracks. Our simulations show comparable results with Rabinovitch’s compression experiments. From the simulation results, it verifies an assumption that the crack width is inversely proportional to the circular frequency electromagnetic radiation, which is presented by Rabinovitch et al. The simulation results also imply that, by using our oscillating dipole model together with Rabinovitch’s two equations about the crack length and crack width, we can quantitatively analyse and simulate the electromagnetic radiation phenomenon, which is induced from the fracture of the rocks.     

Strength evaluation of brittle ceramics with surface defects subjected to thermal shock

The thermal shock strength of a ceramic material plate with a semi-elliptical surface crack is evaluated in this paper. The thermal stress distribution at the plate without crack is expressed by a polynomial form. The calculated thermal stresses are used to obtain the thermal stress intensity factors of semi-elliptical surface cracks using the geometric functions. Variations of thermal stress intensity factor with crack depth and time are obtained. For the strength evaluation of the material, crack growth analysis is conducted. In addition, the thermal shock resistance of the ceramic material plate is evaluated by the stress-based failure criterion and the toughness-based failure criterion. The critical temperature at which crack propagation starts is obtained. Applicability of the two failure criteria in thermal shock resistance evaluation of ceramic is identified and the importance of the semi-elliptical surface defects on the thermal shock resistance behaviour of ceramic materials is justified.     

Nonextensive statistical analysis of the data on the high-speed impact fracture of solids

The high-speed detection of impact-induced cracks in samples of materials that differ strongly in their degree of homogeneity (fused quartz and granite) has been performed by the acoustic emission method. The experimental energy distributions of the acoustic emission signals that correspond to the energy distribution in the events of the nucleation, growth, and coalescence of the microcracks have been interpreted in terms of the Tsallis statistics, which has been developed to generalize classical thermodynamics over the case of nonequilibrium systems. This allowed us to estimate the degree of correlation in the process of crack formation and to compare the energy release densities in various materials and at various stages of the impact fracture.     

The interaction and the surface crack of single-crystal silicon induced by a millisecond laser

When the silicon material is irradiated by laser, it absorbs the laser energy leading to the temperature rise and the thermal stress. The damage effect includes melting, vaporation and thermal stress damage. Once the thermal stress exceeds the stress strength the crack will initiate. The silicon surface cracks induced by a millisecond laser are investigated. The experimental results show that three types of cracks are generated including cleavage crack, radial crack and circumferential crack. The cleavage crack is located within the laser spot. The radial crack and circumferential crack are located outside the laser spot. A two-dimensional spatial axisymmetric model of silicon irradiated by a 1064 nm millisecond laser is established. To assess what stresses generate and explain the generation mechanism of the different cracks, the thermal stress fields during laser irradiation and the cooling process are obtained using finite element method. The radial stress and hoop stress within the laser spot are tensile stress after the laser irradiation. The temperature in the center is the highest but the thermal stress in the center is not always highest during the laser irradiation. The cleavage cracks are induced by the tensile stress after the laser irradiation. The radial crack and the circumferential crack are generated during the laser irradiation.     

Solution of collinear cracks in an electrostrictive solid

A general treatment is presented of the two-dimensional problem of N collinear cracks in an infinite electrostrictive material subjected to remote electric loads based on the complex variable method combined with analytical extension of the complex variable functions. First, for the case of permeable cracks, general solutions for the electric potentials, Maxwell stresses, electroelastic stresses and stress intensity factors are derived. As specific examples, explicit and concise results are obtained for the cases of one crack and two collinear cracks. Then, these results are extended to the cases of impermeable and conducting collinear cracks, respectively. It is found that, in general, the total stresses always have the classical singularity of the r ^{- 1/2} type at the crack tips, whereas the Maxwell stresses have an r ^{- 1} singularity for the above three crack models. Finally, it is concluded that the applied electric field may either enhance or retard crack growth depending on the electric boundary conditions adopted on the crack faces, and the Maxwell stresses on the crack faces and at infinity.     

Estimating the sizes of surface cracks based on Hall element measurements of the leakage magnetic field and a dipole model of a crack

A method is proposed for estimating the sizes of surface cracks in magnetic materials. The method is based on applying a magnetic field, then determining the leakage magnetic field in the vicinity of a crack by moving a Hall element on the surface of the material along one or two scanning lines crossing the crack, and measuring the corresponding Hall voltage distribution. A dipole model of a crack is utilized, in which a surface crack is considered as being full of magnetic dipoles aligned parallel to the applied field, and whose density varies linearly along the depth of the crack. Analytical expressions are derived for the z-component of the intensity of the leakage magnetic field, and for the measured Hall voltage in the vicinity of a crack with an arbitrary cross-section along its long axis when it is perpendicular to the applied field. The crack sizes and the parameters of the distribution of magnetic dipoles along the crack depth are computed by crack inversion, which represents a regression for the Hall voltage distribution. A variable theoretical Hall voltage distribution is fitted to the measured Hall voltage distribution by minimizing the corresponding RMS error, which gives the unknown parameters at the end of the minimization. Hall voltage distributions are measured on ferromagnetic steel samples containing one artificial surface crack. Some crack inversions are performed for estimating the maximum crack depth and the crack width of cracks with rectangular and isosceles triangular cross-sections along the long crack axis. The accuracy of these crack inversions increases by utilizing either Hall voltage distributions measured along only one of the scanning lines, instead of along both scanning lines, or by using more precisely measured Hall voltage distributions. The fast and accurate estimation of the maximum crack depth and the crack width by such crack inversions could be important for pipeline inspection. Other crack inversions are performed for determining the cross-section along the long axis of the investigated cracks with satisfactory results. Received: 2 March 2001 / Accepted: 10 April 2001 / Published online: 25 July 2001     

Formation and propagation of cracks during cyclic deformation

Summary Starting from observation of the geometric features of formation and propagation of cracks in cycled single crystals of copper a model was developed for fatigue cracking. The only and experimentally well established assumption of the model is that the slip during cyclic deformation tends to be coarse. By the cooperation of stress concentrations at the slip steps and the hardening of slip planes activated locally two slip systems (with different slip planes and Burgers vectors) are activated alternatively so that a crack develops from the slip step. It propagates without monotonically increasing the hardening at the crack tip. The coarse slip produces sharp slip steps at the surface for crack formation and prohibits crack blunting during propagation. In contrast to other models the one described can show how the irreversible process of crack formation and propagation can take place despite completely symmetrical push-pull stresses. The whole crack is formed merely by the motion of dislocations present in the material so that but comparatively small stresses are needed. As, moreover, no thermally activated processes are necessary, fatigue at 4·2°K can be explained too. The strong dependence of fatigue on the state of the surface can also be accounted for since the cracks form at the surface steps. Materials which tend to coarse slip even in unidirectional tests are expected to fatigue easily. This is corroborated experimentally. Finally, many details of crack geometry can be explained in terms of the model.Published in Z. f. Metallkunde58 (1967), 780.     

表面垂直裂痕诱发瑞利波散射的数值分析

激光激发的声表面波为材料表面缺陷的检测提供了有力的工具.针对含缺陷材料在模型边界上的复杂性,建立了基于平面应变的有限元模型并选取了相同厚度但含有不同深度的表面裂痕的单层铝板进行了对比计算,得到了声表面波经过不同深度的表面裂痕时产生的反射及透射信号波形的时域特征.进而引入了基于Wigner-Ville分布理论的时-频分析方法计算裂痕前、后散射的瞬态表面波的能量在时间-频率平面内分布的情形.结果显示：声表面波接近中心频率的某一频率成分在经过深度小于其中心波长的表面缺陷时,随着裂痕深度的增加,对应于该频率的反射系数呈现单调递增的趋势;而透射系数呈现递减的特征,这一结果可以为激光超声检测表面缺陷提供一种定量的表征手段.     

Multi-cracks detection of a beam-like structure based on the on-vehicle vibration signal and wavelet analysis

In this paper, a new method for detecting a multi-cracked beam-like structure subjected to a moving vehicle is presented. The crack model is adopted from fracture mechanics. The dynamic response of the bridge-vehicle system is measured directly from the moving vehicle. When moving along the structure, the moving vehicle causes small distortions in the dynamic response of the bridge-vehicle system at the crack locations. In general, these small distortions are difficult to detect visually. However, wavelet transform has recently emerged to be an effective method of detecting such small distortions. Large values (peaks) in the wavelet transform indicate the existence of the cracks. The locations of the cracks are pinpointed by positions of peaks of the wavelet transform and the velocity of the moving vehicle. Numerical results show that the method can detect cracks as small as 10% of the beam height. The proposed method is applicable for low velocity-movements while high velocity-movements are not recommended. The method presents an idea for measuring the vibration directly from the vehicle for crack detection problem in practice.     

SH-wave propagation and scattering in periodically layered composites with a damaged layer

Plane SH-wave propagation in periodically layered elastic composites with a damaged layer is investigated. Two different models are developed to approximate the damaged layer, namely, a periodic array of cracks and continuously distributed springs in the layer. In the first model, the total wave field in the elastic stack of layers with cracks is described as a sum of incident wave field modeled by the transfer matrix method and the scattered wave field governed by an integral representation in terms of the crack-opening-displacements on the crack-faces. The integral equation derived from the boundary conditions on the crack-faces is solved numerically by a Galerkin method. By using Bloch–Floquet theorem the crack-opening-displacements for a periodic array of cracks are expressed by the crack-opening-displacement on a reference crack. In the spring model, the spring constant is estimated by the material properties and the crack density and the modified transfer matrix method is used to compute the wave reflection and transmission coefficients. Numerical results obtained by both models are presented and discussed. Special attention of the analysis is devoted to wave transmissions and reflections, band gaps, wave localization and resonance phenomena due to damages. The influences of the damage types (periodic cracks and stochastic cracks approximated by distributed springs) on the wave field pattern and the band gaps are analyzed.     

Resonance blocking of traveling waves by a system of cracks in an elastic layer

Wave processes that occur in an elastic layer when waves traveling in it are diffracted by a system of horizontal cracks are investigated. Integral representations of wave fields are constructed in terms of the convolution of Green’s matrices and unknown jumps of displacements at the cracks. The displacement jumps are determined from the boundary integral equations, which are obtained from the initial boundary-value problem with the boundary conditions at crack faces being satisfied. The spectrum of the integral operator is studied for different variants of mutual crack arrangement and is compared with the spectrum of the corresponding operators for individual cracks; the relationship between the spectrum and the blocking effects is analyzed. The possibility of obtaining an extended frequency band of waveguide blocking in the case of groups of cracks is demonstrated.