Modeling the brittle–ductile transition in ferritic steels. Part II: analysis of scatter in fracture toughness

A dislocation simulation model has been proposed to predict the brittle–ductile transition in ferritic steels in Part I. Here we extend the model to address the problem of inherent scatter in fracture toughness measurements. We carried out a series of Monte Carlo simulations using distributions of microcracks situated on the plane of a main macrocrack. Detailed statistical analysis of the simulation results showed the following: (a) fracture is initiated at one of the microcracks whose size is at the tail of the size distribution function, and (b) the inherent scatter arises from the distribution in the size of the critical microcrack that initiates the fracture and not from the variation of the location of the critical microcrack. Utilizing the weakest-link theory, Weibull analysis shows good agreement with the Weibull modulus values obtained from fracture toughness measurements.     

HIGH-CYCLE FATIGUE BEHAVIOUR OF SPHEROIDAL GRAPHITE CAST IRON

An expression for the cumulative failure probability of a structure is proposed for cyclic loading conditions. This expression is dependent on an initial flaw distribution and a microcrack propagation law. Two sets of experiments were carried out on specimens made of spheroidal graphite cast iron. These specimens are tested under cyclic tension with two different load ratios. The initial flaw distribution is experimentally identified from microscopic observations. The crack propagation law parameters are identified from experimental results obtained with a load ratio R = 0.1. The expression for the failure probability is then used to predict experimental data obtained with a load ratio R = −1.     

Evaluation of the basic formulations for the cumulative probability of brittle fracture with two different spatial distributions of microcracks

The random distribution of microcracks in terms of their size, shape, orientation and spatial location has direct impact on the cumulative probability of brittle fracture induced failure, with the effect of spatial distribution being rarely explored. Recently, two weakest link theory‐based formulations for the cumulative probability of brittle fracture induced failure have been proposed for the spatial distribution of microcracks obeying the Poisson postulates and the uniform distribution, respectively. This work compares these two new formulations with the currently commonly adopted one built on the Poisson postulates under both the uniform and the non‐uniform uniaxial loading conditions. It is concluded that under general loading conditions involving non‐uniform stress states, the existing formulation is equivalent to or closely approximate to neither of the two new formulations thus should be discarded, because of its inaccurate derivation. The new formulations are featured with unique symmetry or self‐similarity in their expressions. Their capability in revealing the size effect or the scaling law of failure is highlighted and validated by a set of published uniaxial and biaxial flexural strength data of brittle material.     

Critical linkages of coalescing microcracks under stress loading

ABSTRACT The fracture process of brittle materials with randomly orientated microcracks critically depends on strong interactions among microcracks and the coalescence path that leads to a fatal crack. In this paper, a model based on the coalescence process for planar orientated microcracks is presented. An energy ratio is defined as the competition between the potential energy release and the new crack surface energy in each coalescence step, which is a token of the excessive driving force for microcrack propagation. A critical linkage dictates the coalescence of microcracks under stress loading. Probabilities of microcrack coalescence dominated by the first linkage and subsequent linkages are analysed for collinear and wavy microcrack arrays in detail.     

Mode II delamination failure mechanisms of polymer matrix composites

The failure process of mode II delamination fracture is studied on the basis of the microscopic matrix failure modes (microcracks and hackles) as well as fracture mechanics principles. The crack tip matrix stresses leading to delamination is analysed by examining an adhesive bond with a crack analogous to a delamination crack in the resin layer of a composite. Such crack tip stresses induce matrix microcracks involving two major events: (a) single microcrack initiation and (b) development of multiple microcracks with regular spacing. The microcrack initiation shear stress τ* is found by the use of fracture mechanics to be related to certain resin properties (shear modulus G and mode I fracture toughness GIC) and microcrack length of the order of the resin layer thickness t (related to resin content). The more or less regular microcrack spacing S deduced from shear lag considerations can be related to resin properties GIC, G, τy (resin yield strength) and t. The multiple microcracks reduce the effective resin modulus and strongly affect the subsequent microcrack coalescence process. As a result of the detailed analysis of the failure process, mode II laminate fracture toughness GIIC can be quantitatively expressed as a function of resin GIC and (τ2y/G). The failure process modelled is used to interpret the mode II delamination behaviour of several carbon/epoxy systems studied here and that reported in the literature. This study reveals the critical importance of resin fracture (GIC related) and deformation (yielding) mechanisms in controlling mode II delamination resistance of laminated composites. This revised version was published online in November 2006 with corrections to the Cover Date.     

A dilatancy model of tensile macrocracks in compressed rock

A model is developed for tensile fracture under compression for a brittle material with microcracks. The final stage of failure with the formation of macroscopic-splitting cracks is considered. Pre-existing microcracks act as a converter of compression into tension in one direction. This results in the nucleation of other tensile microcracks. Rupture of spacings between the microcracks generates a mode I macrocrack parallel to the direction of maximum compression. Crack propagation is due to sliding along planes that are inclined to the compression microcrack surfaces and is stimulated by the forces distributed along the interacting macrocrack surfaces. Equilibrium, stability and growth of cracks are studied on the basis of the theory of fracture mechanics under the assumption of the plane strain state. The behaviour of both short and long macrocracks are analysed. Parameters of the model are evaluated with the help of data from fracture experiments on some rocks.     

Time-variant reliability global sensitivity analysis with single-loop Kriging model combined with importance sampling

This paper develops a novel failure probability-based global sensitivity index by introducing the Bayes formula into the moment-independent global sensitivity index to approximate the effect of input random variables or stochastic processes on the time-variant reliability. The proposed global sensitivity index can estimate the effect of uncertain inputs on the time-variant reliability by comparing the difference between the unconditional probability density function of input variables and the conditional probability density function in failure state of input variables. Furthermore, a single-loop active learning Kriging method combined with metamodel-based importance sampling is employed to improve the computational efficiency. The accuracy of the results obtained by Kriging model is verified by the reference results provided by the Monte Carlo simulation. Four examples are investigated to demonstrate the significance of the proposed failure probability-based global sensitivity index and the effectiveness of the computational method.     

A stochastic damage model for the rupture prediction of a multi-phase solid

The stochastic damage model, which has been proposed in Part I of this paper, is utilized to analyze quantitatively the effects of uncertainties in locations, orientations and numbers of microcracks at the macro-tip. This is accomplished by introducing a computer simulation model which incorporates a statistical characterization of geometrical parameters of a random microcrack array. The counteracting effects of microcracking on the fracture toughness, namely, toughness enhancement and toughness degradation, are explored statistically through the change of the location and size of the near tip damage zone. The effects of changes in the geometric configuration and density of microcracks at the macro-tip are also examined through the present statistical approach. The validity of the present model is verified by comparing the obtained statistical distribution with the analytic model based on the Neville distribution function. A very good fit achieved by the use of the Neville function demonstrates the potential of the present damage model, in predicting the inherent statistical distribution of the fracture toughness from the intrinsic random microdefects.     

A three-dimensional micromechanics model for the damage of brittle materials based on the growth and unilateral effect of elliptic microcracks

Based on the analysis of a representative elliptic microcrack embedded in a RVE, the additional compliance tensor induced by an embedded opening/closed microcrack is derived, and that corresponding to the kinked growth of a closed elliptic microcrack is also derived by making use of its approximately equivalent simplification. The effect of the microcracks is analyzed with the Taylor’s scheme by introducing an appropriate probability density function. A three-dimensional micromechanics damage model is obtained for brittle materials, assuming numerous randomly distributed elliptic microcracks and taking into account their deformation, frictional sliding, growth and kinked growth.     

An approximate scheme for considering effects of microcrack interaction on the overall constitutive relation of brittle solids under complex loading

Summary Presented in this paper is a three-dimensional, micromechanical evolutionary damage model enabling the calculation of an overall constitutive relation of microcrack-weakened brittle materials under complex loading. An approximate scheme is proposed to determine the effects of microcrack interaction on the overall constitutive relation under complex loading. All microcracks are assumed to be embedded in an approximate effective medium that is weakened by uniformly distributed microcracks of the same radius depending upon the actual damage state. This elastic moduli of this approximate effective medium can be calculated by the well-established Taylor's model, self-consistent method, differential method, or other effective medium methods. The effective compliance tensor uncluding the influences of microcrack interaction is formulated for brittle solids under arbitrary tensile loading. This approximate method improves the accuracy of the Taylor model by implementing the effects of microcrack interaction in the overall constitutive relation and avoids the cumbersome computation of the self-consistent method in general loading cases.     

Microcracks size growth prediction based on microdefects nucleation number

The mechanical reason for rock and concrete failure is trans-scale fracture, which can be divided into three phases: (1) microcrack evolution, (2) macrocrack nucleation, (3) macrocrack growth and run-through. Using the idea that a microcrack could be regarded as a well-organized aggregation of nucleated microdefects, the size growth model of the largest microcrack based on the accumulated number of microdefect nucleation is established. In order to test the validity of the model, trans-scale fracture of a plate made of heterogeneous material is numerically simulated to display the microcrack’s evolution. Statistical analysis of the number and sizes of the microcracks indicates that the predicted size of the largest microcrack according to the model is in close agreement with the measured crack size prior to peak stress, but not at all close to the measured values after the peak. At the end of the paper, some remaining problems are proposed for the further work.     

A statistical approach for the assessment of reliability in ceramic materials from ultrasonic velocity measurement: Cumulative Flaw Length Theory

A primary objective of statistical fracture approach is to predict the probability of failure of a component for an arbitrary stress state when the failure statistics are known. This study introduces the fundamentals and application of a new approach to characterize the mechanical behaviour of high temperature ceramic materials, including refractory materials, by coupling non-destructive methods, in particular ultrasonic velocity measurement, and the Batdorf statistical fracture theory. A new approach, termed Cumulative Flaw Length Theory (CFLT), has been developed for the case of macroscopically homogeneous isotropic materials containing randomly oriented microcracks uniformly distributed in a location subjected to non-uniform multiaxial stresses. A function representing the number of cracks per unit volume is estimated based on the histograms of ultrasonic velocity measurements. This function is used without additional assumptions to determine the probability of fracture under an arbitrary stress condition. Two different cordierite-mullite high temperature ceramic materials were characterized under the assumptions of this theory to provide experimental evidence to support the model.     

A SIMPLE RELIABILITY MODEL FOR THE FATIGUE FAILURE OF REPAIRABLE OFFSHORE STRUCTURES

A simple reliability model for fatigue failure of tubular welded joints used in the construction of offshore oil and gas platforms is proposed. The stress-life data obtained from large-scale fatigue tests conducted on tubular joints is used as the starting point in the analysis and is combined with a fracture mechanics model to estimate the distribution of initial defect sizes. This initial defect distribution is hypothetical but it agrees well with other initial defect distributions quoted in the literature and when used with the fracture mechanics model results in failure probabilities identical to those obtained from the stress-life data. This calculated distribution of initial defect sizes is modified as a result of crack growth under cyclic loading and the probability of failure as a function of fatigue cycles is calculated. The failure probability is modified by inspection, and repair and the results illustrate the trade-off between inspection sensitivity and inspection interval for any desired reliability.     

Microcrack nucleation and growth in elastic lamellar solids

The nucleation and growth of microcracks in elastic lamellar microstructures is studied numerically. The analyses are carried out within a framework where the continuum is characterized by two constitutive relations: one relating the stress and strain in the bulk material and the other relating the traction and separation across a specified set of cohesive surfaces. In such a framework, fracture initiation and crack growth, including micro-crack nucleation ahead of the main crack, arise naturally as a consequence of the imposed loading, without any additional assumptions concerning criteria for crack growth, crack path selection or micro-crack nucleation. Full transient analyses are carried out and plane strain conditions are assumed. The specific problem analyzed is a compact tension specimen with two regions of differing lamellar orientation separated by a fracture resistant layer of finite width d, which is small compared to the physical dimensions of the specimen. An initial crack, normal to the applied loading, is assumed to exist in the first region whose lamellar orientation is fixed. The lamellar orientation of the second region, , is varied, as is the thickness of the fracture resistant layer. It is found that microcrack nucleation in the second region is highly sensitive to the lamellar orientation in that region for small values of d. However, microcrack nucleation becomes rather insensitive to with increasing d. It is also shown that a linear elastic fracture mechanics model with one adjustable parameter gives good agreement with the numerical results for fracture initiation.     

Simulations of microcracking in the process region of ceramics with a cell model

Microcracking around a macrocrack and the consequential toughening in polycrystalline ceramics are simulated. The objective is to check the hypothesis that the suppression mechanism for opening of off-side microcracks does not work in certain ceramic materials because cracks open up, assisted by residual stresses, at a much reduced load in some grains, while, still encountering a high resistance to crack growth at the grain boundaries. A two-dimensional cell model of a polycrystalline material is investigated. Each cell represents one grain. The load-deformation law for the cell is assumed to contain two load peaks. The first peak is associated with microcrack nucleation in the grain, while the second peak is related to the resistance that the microcrack meets at the grain boundary. The cells are included in a finite element model. Grain to grain variations, for instance due to residual stresses, are taken into account by a Weibull distribution of the first load peak. Results from the simulations show that variations of the propensity for microcrack nucleation between different grains constitute a major factor responsible for the generation of microcrack clouds. Such cloud formation would otherwise be impeded by unloading effects from central microcracks. In addition, and in accordance with observations, the simulations also show high fracture energies (compared to what would be expected for a typically brittle material), as well as a period of stable crack propagation.     

Representation and problem solving with Distribution Envelope Determination (DEnv)

Distribution Envelope Determination (DEnv) is a method for computing the CDFs of random variables whose samples are a function of samples of other random variable(s), termed inputs. DEnv computes envelopes around these CDFs when there is uncertainty about the precise form of the probability distribution describing any input. For example, inputs whose distribution functions have means and variances known only to within intervals can be handled. More generally, inputs can be handled if the set of all plausible cumulative distributions describing each input can be enclosed between left and right envelopes. Results will typically be in the form of envelopes when inputs are envelopes, when the dependency relationship of the inputs is unspecified, or both. For example in the case of specific input distribution functions with unspecified dependency relationships, each of the infinite number of possible dependency relationships would imply some specific output distribution, and the set of all such output distributions can be bounded with envelopes. The DEnv algorithm is a way to obtain the bounding envelopes. DEnv is implemented in a tool which is used to solve problems from a benchmark set.     

Insights into the effects of tensile and compressive loadings on microstructure dependent fracture of trabecular bone

The micro-scale finite element models used in the past to understand yielding failure of trabecular bone have not addressed the microcrack formation and its effect on microstructure dependent fracture. An understanding of microcrack based failure mechanisms can be important to develop insights into response of trabecular bone to external loading before final failure. With this goal, we analyze tensile and compressive fracture failure at two different ages in two trabecular bone micrographs obtained from an ovine femur using a recently developed cohesive finite element method (CFEM) framework. The results and analyses indicate that examined trabecular microstructures are optimally designed for resisting compressive loading. Under tensile loading, initial damage in a microstructure is localized in a single random trabecula. Final microstructure failure occurs immediately after the failure of the trabecula. However, under compressive loading, failure of the first trabecula does not precede immediate complete failure of microstructure. Under compression the propagation fracture toughness (characterized by change in energy release rate as a function of crack density) increases with increase in crack density. However, under tension the propagation fracture toughness decreases with increasing crack density. The fracture mechanism remains unaffected by age variation. Effect of tissue property random variation on the variation in fracture strength diminishes under tension and increases under compression with increase in the age. Overall, results indicate that structural arrangement of the trabecular bone (besides the hierarchical chemical composition) can be an important contributor to its unique fracture resistance properties.     

岩石爆破损伤模型及评述

岩石爆破损伤模型是将岩石的动态断裂作为一个连续的损伤累积过程来处理,其基本点是建立损伤变量与岩石内微裂纹密度的关系,并预测在爆炸载荷作用下岩石的损伤和破坏过程。在分析研究现有岩石爆破损伤模型的基础上,对模型中损伤变量的定义及有效体积模量进行了评述,并指出了现有岩石爆破损伤模型存在的不足及发展方向     

Quantitative Relation Between Acoustic Emission Signal Amplitude and Arrested Cleavage Microcrack Size

The possibility of obtaining a quantitative relation between acoustic emission (AE) signal amplitudes and arrested cleavage microcrack sizes in the partially transformed coarse grained heat affected zone of a structural steels is explored. Interrupted fracture mechanics tests are performed, and the size of measured arrested cleavage microcracks are compared with recoded AE signal amplitudes. It is shown that the experimentally measured relationship between the arrested microcrack size and AE amplitude closely follows a theoretical relation derived by Lysak (1996). The results may provide quantitative data as input to further development of micromechanically based cleavage fracture models for steels.