Discrete fractal fracture mechanics

A modification of the classical theory of brittle fracture of solids is offered by relating discrete nature of crack propagation to the fractal geometry of the crack. The new model incorporates all previously considered theories of fracture processes, in particular the Griffith [Griffith AA. The phenomenon of rupture and flow in solids. Philos Trans Roy Soc Lond 1921;A221:163-398] theory, its contemporary extension known as LEFM and the most recently developed Quantized Fracture Mechanics (QFM) by Pugno and Ruoff [Pugno N, Ruoff RS. Quantized fracture mechanics. Philos Mag 2004;84(27):2829-45]. Using an equivalent smooth blunt crack for a given fractal crack, we find that assuming that radius of curvature of the blunt crack is a material property, the crack roughens while propagating. In other words, fractal dimension at the crack tip is a monotonically increasing function of the nominal crack length, i.e., the presence of the Mirror-Mist-Hackle phenomenon is analytically demonstrated.     

Stochastic multiscale fracture analysis of three-dimensional functionally graded composites

A new moment-modified polynomial dimensional decomposition (PDD) method is presented for stochastic multiscale fracture analysis of three-dimensional, particle-matrix, functionally graded materials (FGMs) subject to arbitrary boundary conditions. The method involves Fourier-polynomial expansions of component functions by orthonormal polynomial bases, an additive control variate in conjunction with Monte Carlo simulation for calculating the expansion coefficients, and a moment-modified random output to account for the effects of particle locations and geometry. A numerical verification conducted on a two-dimensional FGM reveals that the new method, notably the univariate PDD method, produces the same crude Monte Carlo results with a five-fold reduction in the computational effort. The numerical results from a three-dimensional, edge-cracked, FGM specimen under a mixed-mode deformation demonstrate that the statistical moments or probability distributions of crack-driving forces and the conditional probability of fracture initiation can be efficiently generated by the univariate PDD method. There exist significant variations in the probabilistic characteristics of the stress-intensity factors and fracture-initiation probability along the crack front. Furthermore, the results are insensitive to the subdomain size from concurrent multiscale analysis, which, if selected judiciously, leads to computationally efficient estimates of the probabilistic solutions.     

Stochastic analysis of groundwater flow in a two‐dimensional generalized fractal field

Abstract

Stochastic analysis of groundwater flow in a generalized fractal field is performed in this study. The random field is described by fractional Levy motion (fLm), which is a generalized version of traditional fractional Brownian motion (fBm) and is superior to describe a field with a high degree of variability. A truncated power variogram of the fLm is derived using the weighted superposition of mutually uncorrelated exponential variograms. When the Levy index of fLm α equals 2, the fBm is recovered. When the upper and lower cutoffs of the truncated power variogram are close, the stationary exponential model can be well approximated. First‐order perturbation analyses of flow in a two‐dimensional fLm field are performed and results are compared to those in the stationary exponential and fractal fBm fields. Since the proposed general fractal model has broader applications than the stationary and fBm models, it is versatile enough to simulate flow in different scenarios and provide more accurate modeling results.     

The fourth mode of fracture in fractal fracture mechanics

This paper offers a systematic approach for obtaining the order of stress singularity for different self-similar and self-affine fractal cracks. Mode II and Mode III fractal cracks are studied and are shown to introduce the same order of stress singularity as Mode I fractal cracks do. In addition to these three classical modes, a Mode IV is discovered, which is a consequence of the fractal fracture. It is shown that, for this mode, stress has a weaker singularity than it does in the classical modes of fracture when self-affine fractal cracks are considered, and stress has the same order of singularity when self-similar cracks are considered. Considering this new mode of fracture, some single-mode problems of classical fracture mechanics could be mixed-mode problems in fractal fracture mechanics. By imposing a continuous transition from fractal to classical stress and displacement fields, the complete forms of the stress and displacement fields around the tip of a fractal crack are found. Then a universal relationship between fractal and classical stress intensity factors is derived. It is demonstrated that for a Mode IV fractal crack, only one of the stress components is singular; the other stress components are identically zero. Finally, stress singularity for three-dimensional bodies with self-affine fractal cracks is studied. As in the two-dimensional case, the fourth mode of fracture introduces a weaker stress singularity for self-affine fractal cracks than classical modes of fracture do.     

Continuum shape sensitivity analysis of mixed-mode fracture using fractal finite element method

This paper presents a new fractal finite element based method for continuum-based shape sensitivity analysis for a crack in a homogeneous, isotropic, and two-dimensional linear-elastic body subject to mixed-mode (modes I and II) loading conditions. The method is based on the material derivative concept of continuum mechanics, and direct differentiation. Unlike virtual crack extension techniques, no mesh perturbation is needed in the proposed method to calculate the sensitivity of stress-intensity factors. Since the governing variational equation is differentiated prior to the process of discretization, the resulting sensitivity equations predicts the first-order sensitivity of J-integral or mode-I and mode-II stress-intensity factors, K_I and K_II, more efficiently and accurately than the finite-difference methods. Unlike the integral based methods such as J-integral or M-integral no special finite elements and post-processing are needed to determine the first-order sensitivity of J-integral or K_I and K_II. Also a parametric study is carried out to examine the effects of the similarity ratio, the number of transformation terms, and the integration order on the quality of the numerical solutions. Four numerical examples which include both mode-I and mixed-mode problems, are presented to calculate the first-order derivative of the J-integral or stress-intensity factors. The results show that first-order sensitivities of J-integral or stress-intensity factors obtained using the proposed method are in excellent agreement with the reference solutions obtained using the finite-difference method for the structural and crack geometries considered in this study.     

Stochastic penetration of smooth and fractal basin boundaries under noise excitation

Recent work on the escape of a sinusoidally driven oscillator from a universal cubic potential well has elucidated the complex patterns of attractor and basin bifurcations that govern the escape process. Optimal escape, under a minimum forcing magnitude, occurs at a forcing frequency of about 80 per cent of the small-amplitude linear natural frequency, and at this forcing frequency we have identified a significant and dramatic erosion of the safe basin of attraction, triggered by a homoclinic tangency, that would seriously impair the engineering integrity of a practical system long before the final chaotic instability of the constrained attractor. Introducing a superimposed noise excitation, we here quantify this in terms of a stochastic integrity measure, and correlate this with the geometric changes experienced by the deterministic basin of attraction     

Stochastic fracture analysis of laminated composite plate with arbitrary cracks using X-FEM

In this paper, second order statistics of mixed mode stress intensity factors (MSIFs) and crack propagation analysis of the symmetric angle ply laminated composite plate with through thickness arbitrary curve cracks subjected to tensile and shear stress is presented. The fracture behaviour is analysed using extended finite element method (X-FEM). The cracks like line, semi elliptical, semi circular and arbitrary curves are considered for the detailed numerical study. The material properties, lamination angle, loading, crack width and crack depth are modelled as independent, combine uncorrelated and correlated input random Gaussian variables. The interaction integral (M-integral) is adopted for calculating the MSIFs. The second order perturbation technique and Monte Carlo simulations are proposed to obtain the mean and coefficient of variance of MSIFs by random change in input system parameters. This work signifies the accurate and realistic evaluation of fracture response by handling the various levels of uncertainties. The effect of crack propagation on MSIFs using tensile and shear stresses using global tracking algorithm is also highlighted.     

Application of fractal fracture mechanics to polymers and polymeric composites

It is shown that the fundamental concepts of fractal fracture mechanics can be applied both to polymers and polymeric composites and to metals and ceramics. The critical crack opening displacement can be chosen as a scale of fracture of polymeric materials. The results obtained for polymers and polymeric composites are described by the same sigmoidal dependence, which means that the regularities of fracture processes in these materials are common.Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 40, No. 4, pp. 53–57, July–August, 2004.     

Stochastic fracture mechanics using polynomial chaos

Crack propagation in metals has long been recognized as a stochastic process. As a consequence, crack propagation rates have been modeled as random variables or as random processes of the continuous. On the other hand, polynomial chaos is a known powerful tool to represent general second order random variables or processes. Hence, it is natural to use polynomial chaos to represent random crack propagation data: nevertheless, no such application has been found in the published literature. In the present article, the large replicate experimental results of Virkler et al. and Ghonem and Dore are used to illustrate how polynomial chaos can be used to obtain accurate representations of random crack propagation data. Hermite polynomials indexed in stationary Gaussian stochastic processes are used to represent the logarithm of crack propagation rates as a function of the logarithm of stress intensity factor ranges. As a result, crack propagation rates become log-normally distributed, as observed from experimental data. The Karhunen–Loève expansion is used to represent the Gaussian process in the polynomial chaos basis. The analytical polynomial chaos representations derived herein are shown to be very accurate, and can be employed in predicting the reliability of structural components subject to fatigue.     

From Griffith's theory to the fractal fracture mechanics

Conclusion The above analysis shows the necessity for further development of fractal fracture mechanics at micro-, meso-, and macrolevels, using fractal theory and the general principle of synergetics.A. A. Baikov Institute of Metallurgy, Russian Academy of Sciences, Moscow. Published in Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 29, No. 3, pp. 101–106, May–June, 1993.     

Stochastic process of fracture in granular materials

By regarding the coefficient of particle friction of granular materials as a random variable distributed on the particle surface, the mechanism of particle sliding is interpreted as a stochastic process. Axial, shearing and volumetric strains are defined with regard to the deformation of a microscopic regular assembly of uniform spheres. For illustration, these strains are calculated for a uniform distribution of the coefficient of particle friction and applied to some triaxial test results reported elsewhere.     

Stochastic analysis of the fracture toughness of polymeric nanoparticle composites using polynomial chaos expansions

The fracture energy is a substantial material property that measures the ability of materials to resist crack growth. The reinforcement of the epoxy polymers by nanosize fillers improves significantly their toughness. The fracture mechanism of the produced polymeric nanocomposites is influenced by different parameters. This paper presents a methodology for stochastic modelling of the fracture in polymer/particle nanocomposites. For this purpose, we generated a 2D finite element model containing an epoxy matrix and rigid nanoparticles surrounded by an interphase zone. The crack propagation was modelled by the phantom node method. The stochastic model is based on six uncertain parameters: the volume fraction and the diameter of the nanoparticles, Young’s modulus and the maximum allowable principal stress of the epoxy matrix, the interphase zone thickness and its Young’s modulus. Considering the uncertainties in input parameters, a polynomial chaos expansion surrogate model is constructed followed by a sensitivity analysis. The variance in the fracture energy was mostly influenced by the maximum allowable principal stress and Young’s modulus of the epoxy matrix.     

On the fractal dimension of fracture surfaces of concrete elements

The problem of the relation between the fractal dimension of a fractured surface and the fracture toughness expressed by the stress intensity factor is investigated. The theoretical conditions for such assumptions are discussed. Collected experimental results and new tests performed onconcrete specimens subjected to Mode II fracture seem to confirm that relation within the scope of materials tested and with certain necessary restrictions.     

基于分形理论的岩石冲击破坏分形分析

利用大直径分离式霍普金森压杆试验系统,对绢云母石英片岩和砂岩进行冲击压缩试验, 采用不同等级标准筛对岩石冲击破碎后试块进行筛分统计,运用分形几何理论,计算出冲击荷载作用下两种岩石破碎块度分布的分形维数,研究冲击速度对块度分维影响,分析两种岩石动态抗压强度随块度分维的变化关系。试验结果表明,绢云母石英片岩块度分维大部分集中在1.9~2.4之间,砂岩集中在2.5~3.0之间;两种岩石分形维数随冲击速度的升高呈上升趋势,近似线性正比关系;绢云母石英片岩动态抗压强度与块度分维无明显函数关系,砂岩动态抗压强度随块度分维的增大呈增加趋势。采用分形维数对岩石试件在冲击破碎过程中动态抗压强度的变化进行定量描述,为探索岩石动态破碎分形特征与冲击力学性能之间的内在规律,开辟新的研究途径。     

Application of fractal geometry to damage development and brittle fracture in materials

This article represents a first attempt to apply the concepts of fractal geometry to damage development in materials. Two extremes of material behavior were considered. In the case of unstable brittle propagation of microcracks, application of fractal geometry and weakest link statistics lead to the well-known Weibull distribution for fracture stress. The Weibull slope (shape parameter) is equal to twice the fractal dimension of the flaw distribution. For the case of stable microcrack growth, a number of simplifying assumptions led to a closed-form expression for the change in effective modulus with an increment of damage. Although an oversimplification, this latter analysis illustrates the potential value of fractal geometry in modeling damage development. Future work will be directed towards developing these initial results further.     

The fractal geometry of Si3N4 wear and fracture surfaces

Good mechanical properties and chemical stability at high temperatures make silicon nitride a good candidate as an advanced engine material. Much research has been done to characterize the mechanical strength and resistance of crack propagation in this material. In this paper, we use fractal analysis to study the geometry of Si3N4 fracture and wear surfaces. We found that the geometries of the failure surfaces as characterized by the fractal dimensional increment, D*, under different failure stress states are similar for the same brittle material, but different for different brittle materials. The similar D* in an identical brittle material implies that the failure process in the material is the same regardless of loading mode, i.e., mode I or mixed-mode stress. The fractal technique is shown to be useful for correlating the fractal dimension to the material properties and fracture-surface topography. This revised version was published online in November 2006 with corrections to the Cover Date.