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.     

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.     

Stress field interaction and strain energy distribution between a stationary main crack and its process zone

The damage process zone developed by brittle materials in front of a macrocrack is simulated by means of a distribution of microcracks. Crack mutual interactions are taken into account by means of a numerical technique, based on a displacement discontinuity boundary element method that is able of considering both the macrocrack–microcrack and microcrack–microcrack interactions inside the process zone. In the frame of linear elastic fracture mechanics the stress field at each crack tip and the related elastic strain energy are calculated. The main features of the interaction phenomena turn out to be almost independent of the microcrack density. Some considerations both on the shielding and amplification effects on the main crack and on the strain energy distribution between cracks give explanation to experimental evidence and prove that crack interaction is not such a short-range effect as sometimes expected.     

Elastic interaction of a crack with a random array of microcracks

Two issues are addressed in the paper. The first deals with a new characterization of a random array of microcracks in terms of spatial distributions of the sizes, orientations and density of microcracks. The second is a formulation of crack-microcrack array interaction in terms of the distributions. The interaction problem is posed as a system of two coupled singular integral equations with respect to a vector field of an average microcrack opening and the main crack opening displacement. The approach is illustrated by a numerical solution of the interaction problem for two particular configurations of a random array of microcracks. It is shown that the morphology of the microcrack array may strongly alter elastic fields in the vicinity of the main crack. Specifically, it is found that a variation in the microcrack length distribution (for fixed distributions of microcrack orientation and density) has a profound effect on the solution. The main crack opening displacement is evaluated to compare the theory with the experimental observations.     

Analysis on interaction of numerous microcracks

A novel numerical method is proposed in this paper to consider the interaction of microcracks when their number is large. To determine the stress intensity factors of a microcrack surrounded by numerous or even countless microcracks, the solid is divided into two regions. The interaction of the microcracks in an elliptical or circular subregion around the considered microcrack is calculated directly by using a micromechanics method, whereas the influence of all other microcracks is incorporated by appropriately modifying the far-field stress. This simplified scheme yields a satisfactorily accurate estimate of stress intensity factors, and then provides an efficient tool for analyzing some deformation and failure phenomena associated with microcracking damage. As an example of its various potential applications, this method is used to determine the effective elastic moduli of a solid containing either uniformly or non-uniformly microcracks.     

THE DAMAGE PROCESS IN A FINITE-SIZED BRITTLE SPECIMEN WITH INTERACTING MICROCRACKS

Abstract— The whole damage process in a finite sized specimen with interacting microcracks is simulated by a method combining the closed form crack solutions with boundary elements. Interactions among microcracks and boundary elements are taken into account with an explicit interaction matrix. A coalescence criterion is assumed to rule the intersection behaviour and propagation arrest. The fatal coalescence cluster resulting in the failure of the specimen, out of many intersections of propagating microcracks, is identified with a particular coalescence matrix. The numerical model proposed in this paper can be used to simulate the damage process in a brittle specimen of any shape, under arbitrary plane stress conditions.     

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.     

The effect of fatigue microcracks on rapid catastrophic failure in Al---SiC composites

The fatigue resistance of discontinuous reinforced aluminum metal matrix composites is greatly influenced by the rapid nucleation and growth of a very large number of microcracks. These microcracks, which result from an abundance of potential crack initiation sites, are often retarded in growth by the presence of reinforcement through crack trapping. With continued fatigue, microcracking becomes so extensive that it induces widespread coalescence leading to increasingly larger microcracks. Inevitably, some of these large microcracks link together to form the fatal crack and instability takes place very shortly afterwards at unusually small critical crack sizes. The present study examines the form of catastrophic failure in smooth specimens of a 2124 aluminum alloy reinforced with silicon carbide whiskers. Experimental observations of microcrack initiation and stage-by-stage growth through to final failure are reported. Effort is directed at characterizing the distribution and orientation of microcracks present, particularly when linkage(s) result in the formation of the fatal crack, and developing a geometric probability method for predicting coalescence. Results show that microcracks initiate and grow preferentially in arrays which maximize the energy release rate. Near the end of life, the interaction of some microcracks brings about large increases in their stress intensity factor leading to coalescence and fatal crack formation.     

Numerical investigation of dynamic crack branching under biaxial loading

Dynamic crack growth and branching of a running crack under various biaxial loading conditions in homogeneous and heterogeneous brittle or quasi-brittle materials is investigated numerically using RFPA2D (two-dimensional rock failure process analysis)-Dynamic program which is fully parallelized with OpenMP directives on Windows. Six 2D models were set up to examine the effect of biaxial dynamic loading and heterogeneity on crack growth. The numerical simulation vividly depicts the whole evolution of crack and captured the crack path and the angles between branches. The path of crack propagation for homogenous materials is straight trajectory while for heterogeneous materials is curved. Increasing the ratio of the loading stress in x-direction to the stress in y-direction, the macroscopic angles between branches become larger. Some parasitic small cracks are also observed in simulation. For heterogeneous brittle and quasi-brittle materials coalescence of the microcracks is the mechanism of dynamic crack growth and branching. The crack tip propagation velocity is determined by material properties and independent of loading conditions.     

Geometry effects of plane microdamages on the material deformation behavior

Using the energy-based method accounting for interaction of crack faces we have derived equations for the effective elastic characteristics of materials weakened by a system of elliptic cracks randomly distributed over the bulk of material. The effect of microcrack geometry on the effective elastic moduli of the damaged material is investigated.     

Simulation of crack propagation resistance in brittle media of high porosity

Summary The crack propagation resistance through a porous or microstructurally heterogeneous brittle solid with local variability in strength and stiffness has been simulated. Specifically, the simulation probes the behavior of porous brittle materials in the range of porosity less than those of cellular materials and greater than those of microstructures that are in the category of dilute porosity. The simulation plane consists of a triangular network of points interacting with each other through both linear central force springs and bond angle springs, incorporating an appropriate element of a noncentral force contribution. Explicit microstructural details were incorporated into the model and the simulation was first carried out under conditions of uniaxial tensile strain in order to investigate the mechanisms of subcritical damage evolution, leading to quasi-homogeneous fracture. In order to investigate material strength and stiffness variability on the scale of a representative volume element for coherent fracture events in a crack tip stress gradient, the explicit microstructural results were incorporated into a simulation with boundary conditions characteristic of the displacement field of an infinite Mode I crack. To impart some 3D realism to the primarily 2D simulations a special 2D super-element was devised, which incorporated variability information as might be sampled by a crack front in three dimensions. For a given porosity, in general, only small differences were found between nominally diverse microstructures in terms of their tensile toughness, maximum strength and elastic moduli. The strongest dependence of the overall fracture toughness was found to come from the average porosity. The variability in local element strength and stiffness on the scale of the porosity produced highly tortuous crack paths, roughly on the scale of the chosen representative volume element. The tortuosity of the crack was largest where local variability of strength and stiffness was uncorrelated. Examples of microcrack toughening and crack bridging were observed.     

Experimental study on mechanical behavior of brittle marble samples containing different flaws under uniaxial compression

Uniaxial compression experiments were carried out for the marble samples (located in the eastern ground of China) with different pre-existing flaws in non-overlapping geometry by the rock mechanics servo-controlled testing system. Based on the experimental results of complete axial stress-axial strain curves, the effect of flaw geometry on the strength and deformation behavior of marble samples is made a detailed analysis. Compared with the intact marble sample, the marble samples with different pre-existing flaws show the localization deformation failure. The uniaxial compressive strength (UCS), elastic modulus and peak axial strain of marble samples with pre-existing flaws are all lower than that of intact marble sample, and the reduction extent is closely related to the geometry of pre-existing flaws. The crack coalescence were observed and characterized from internal tips of different pre-existing flaws in brittle marble sample. Eight different crack types were identified based on their geometry and crack propagation mechanism (tensile, shear and compressive) for two pre-existing flaws, which can be used to analyze the failure mode and cracking process of marble sample containing different flaws in uniaxial compression. In the end, the influence of the crack coalescence on the strength and deformation failure behavior of brittle marble sample is analyzed under uniaxial compression. The present research provides increased understanding of the fundamental nature of rock failure under uniaxial compression.     

Analysis of stable crack growth in brittle materials Part I: A process zone model

The phenomenon of stable crack growth in brittle materials is considered where stable crack growth is modeled as the formation of an elongated process zone of microcracks and isolated intact ligaments ahead of a stationary main crack. The role of residual stress in protecting the intact ligaments is explored via the hypothesis that compressive zones sorrounding isolated ligaments under residual tension protect these ligaments and result in stable crack growth and toughening of the microcracked material. The dependence of process zone behavior on either a plane strain or an axisymmetric representation of microcrack distribution is considered. Numerical results based on the above hypothesis indicate that interaction between residual stress and microcracking can lead to stable crack growth with attendant toughness enhancement.     

An Altgernating Method Based on the VNA Solution for Analysis of Damaged Solid Containing Arbitrarily Distributed Elliptical Microcracks

This paper presents the development of an alternating method for the interaction analysis of arbitrary distributed numerous elliptical microcracks. The complete analytical solutions (VNA solutions) for a single elliptical crack in an infinite solid, subject to arbitrary crack-face tractions, are implemented in the present alternating method, together with the coordinate transformations for stress tensors. First, the present method is verified by solving the problems of two interacting cracks for which accurate numerical solutions have been obtained previously. Next, the present method demonstrates obtaining efficient and accurate solutions for the problems of many interacting elliptical cracks, which cannot be solved in a practical sense by the ordinary numerical methods such as the finite element method. Furthermore, damaged solids containing periodically distributed elliptical microcracks are analyzed by the present alternating method. The effective elastic moduli are evaluated for varying microcrack density. Detailed structures of the interactions in the damaged solids are visualized and clarified. This revised version was published online in August 2006 with corrections to the Cover Date.     

An alternating method based on the VNA solution for analysis of damaged solid containing arbitrarily distributed elliptical microcracks

This paper presents the development of an alternating method for the interaction analysis of arbitrary distributed numerous elliptical microcracks. The complete analytical solutions (VNA solutions) for a single elliptical crack in an infinite solid, subject to arbitrary crack-face tractions, are implemented in the present alternating method, together with the coordinate transformations for stress tensors. First, the present method is verified by solving the problems of two interacting cracks for which accurate numerical solutions have been obtained previously. Next, the present method demonstrates obtaining efficient and accurate solutions for the problems of many interacting elliptical cracks, which cannot be solved in a practical sense by the ordinary numerical methods such as the finite element method. Furthermore, damaged solids containing periodically distributed elliptical microcracks are analyzed by the present alternating method. The effective elastic moduli are evaluated for varying microcrack density. Detailed structures of the interactions in the damaged solids are visualized and clarified. This revised version was published online in July 2006 with corrections to the Cover Date.     

Strength failure and crack coalescence behavior of brittle sandstone samples containing a single fissure under uniaxial compression

Uniaxial compression experiments were performed for brittle sandstone samples containing a single fissure by a rock mechanics servo-controlled testing system. Based on the experimental results of axial stress-axial strain curves, the influence of single fissure geometry on the strength and deformation behavior of sandstone samples is analyzed in detail. Compared with the intact sandstone sample, the sandstone samples containing a single fissure show the localization deformation failure. The uniaxial compressive strength, Young’s modulus and peak axial strain of sandstone samples with pre-existing single fissure are all lower than that of intact sandstone sample, which is closely related to the fissure length and fissure angle. The crack coalescence was observed and characterized from tips of pre-existing single fissure in brittle sandstone sample. Nine different crack types are identified based on their geometry and crack propagation mechanism (tensile, shear, lateral crack, far-field crack and surface spalling) for single fissure, which can be used to analyze the failure mode and cracking process of sandstone sample containing a single fissure under uniaxial compression. To confirm the subsequence of crack coalescence in sandstone sample, the photographic monitoring and acoustic emission (AE) technique were adopted for uniaxial compression test. The real-time crack coalescence process of sandstone containing a single fissure was recorded during the whole loading. In the end, the influence of the crack coalescence on the strength and deformation failure behavior of brittle sandstone sample containing a single fissure is analyzed under uniaxial compression. The present research is helpful to understand the failure behavior and fracture mechanism of engineering rock mass (such as slope instability and underground rock burst).     

The Interaction between Microcapsules with Different Sizes and Propagating Cracks

The microcapsule-contained self-healing materials are appealing since they can heal the cracks automatically and be effective for a long time. Although many experiments have been carried out, the influence of the size of microcapsules on the self-healing effect is still not well investigated. This study uses the two-dimensional discrete element method (DEM) to investigate the interaction between one microcapsule and one microcrack. The influence of the size of microcapsules is considered. The potential healing time and the influence of the initial damage are studied. The results indicate that the coalescence crack is affected by the size of holes. The elastic modulus, the compressive strength and the coalescence stress decrease with the rising radius of holes. The initial damage in experiments should be greater than 95% of the compressive strength to enhance the self-healing effect. The large microcapsules require slight initial damage. Both a new type of displacement field near the crack and a new category of coalescence crack are observed. The influence of sizes of holes on the cracking behavior of concrete with a circular hole and a pre-existing crack is clarified.     

Characterisation and quantification of multiple crack growth during low cycle fatigue

The collective growth of multiple microcracks (or short cracks) during low cycle fatigue of polycrystalline Cu–30%Zn, 316L and Fe–26Cr–1Mo is investigated. Scanning electron microscopy is employed to study the evolution of surface microcrack populations, as well as the growth of individual cracks, on smooth specimens cyclically deformed at constant plastic strain amplitudes. The damage accumulation process is quantified by construction of so-called damage accumulation (DA) profiles, which reveal important information about crack growth mode, crack initiation rates, strain localisation and crack coalescence. In addition, experimentally measured microcrack growth is quantified in terms of general crack growth relations. The uniformity of these relations for different materials indicates that growth barriers dictate the main differences in fatigue microcrack growth.     

On Absence of Quantitative Correlations Between Strength and Stiffness in Microcracking Materials

We show, by numerical simulations on crack arrays, that there is no stable quantitative correlation between strength of a brittle microcracking material and its effective elastic stiffness. The reason is that fracture processes are controlled by “details” of microcrack field geometry – such as local clusters of closely spaced cracks – to which the stiffness, being a volume average quantity, is almost insensitive.     

Frequency-dependent tensile and compressive effective moduli of elastic solids with distributed penny-shaped microcracks

This paper develops micromechanics models to estimate the tensile and compressive elastic moduli of elastic solids containing randomly distributed penny-shaped microcracks. The crack faces are open under tension and closed under compression. When the crack faces are closed, they may slide against one another following Coulomb’s law of dry friction. The micromechanics models provide analytical expressions of the tensile and compressive moduli for both static and dynamic cases. It is found that the tensile and compressive moduli are different. Further, under dynamic loading, both compressive and tensile moduli are frequency dependent. As a by-product, the micromechanics models also predict wave attenuation in the dynamic case. Numerical simulations using the finite element method are conducted to validate the micromechanics models.