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.     

A MICROCRACK GROWTH LAW FOR MULTIAXIAL FATIGUE

Within the past decade, critical plane approaches have gained increasing support based on correlation of experimentally observed fatigue lives and microcrack orientations under predominately low cycle fatigue (LCF) conditions for various stress states. In this paper, we further develop an engineering model for microcrack propagation consistent with critical plane concepts for correlation of both LCF and high cycle fatigue (HCF) behavior, including multiple regimes of small crack growth. The critical plane microcrack propagation approach of McDowell and Berard serves as a starting point to incorporate multiple regimes of crack nucleation, shear growth under the influence of microstructural barriers, and transition to linear crack length-dependent growth related to elastic-plastic fracture mechanics (EPFM) concepts. Microcrack iso-length data from uniaxial and torsional fatigue tests of 1045 steel and IN 718 are examined and correlated by introducing a transition crack length which governs the shift from nonlinear to linear crack length dependence of da/dN. This transition is related to the shift from strong microstructural influence to weak influence on the propagation of microcracks. Simple forms are introduced for both the transition crack length and the crack length-dependence of crack growth rate within the microcrack propagation framework (introduced previously by McDowell and Berard) and are employed to fit the 1045 steel and IN 718 microcrack iso-length data, assuming preexisting sub-grain size cracks. The nonlinear evolution of crack length with normalized cycles is then predicted over a range of stress amplitudes in uniaxial and torsional fatigue. The microcrack growth law is shown to have potential to correlate microcrack propagation behavior as well as damage accumulation for HCF-LCF loading sequences and sequences of applied stress states.     

Path-independent integral analyses for microcrack damage in dissimilar anisotropic materials

Summary This paper deals with microcrack damage below an interface in a dissimilar anisotropic material. The interaction problem among arbitrarily oriented and located subinterface microcracks is studied in detail by solving the associated singular integral equations derived from the dislocation technique and superimposing technique. The path-independent integral analyses are performed by adopting theJ _k -integral vector and theM-integral, which are evaluated, respectively, along specially introduced contours. It is found that the total contributions of the subinterface microcracks to the first component of theJ _k -vector vanish, provided that the chosen contour encloses all the microcracks. This leads to a conservation law of theJ ₁-integral in the microcrack damage problem, from which a consistency check or a necessary condition for this kind of problem is presented. Similar conclusions are also given for the second component of the vector under the more strict assumption that the closed contour chosen to calculate theJ ₂-integral is infinite large and encloses not only all the microcracks, but also the whole interface. Therefore, theM-integral analysis could be performed, which is proved to be divided into two parts. One of them, called the net part, is the simple summation of the distinct contribution induced from the stress intensity factors at all microcrack tips, while the other part, called the additional part, is induced from the global coordinates of each microcrack center and theJ _k -vectorevaluated along a specially introduced contour surrounding only one single microcrack completely. Numerical results are given for a particular dissimilar anistropic material whose upper half part shows anisotropy (the fiber direction is parallel to the interface) and the lower half part shows isotropic (the fiber direction is perpendicular to the global coordinate plane). It is concluded that the numerical results do actually meet the consistency check. Moreover, it is found that the values of theM-integral for the present dissimilar material is always smaller than those for the corresponding isotropic material, no matter how the subinterface matrix microcracks are located and oriented. It is concluded that theM-integral plays an important role in the present microcrack damage problem and that any aray of matrix microcracks below the interface shows less unstable nature than the same array in the corresponding isotropic material.     

Analysis of fatigue damage evolution and associated anisotropic elastic property degradation in random short-fiber composite

A theoretical analysis of cyclic fatigue damage and associated anisotropic property degradation in a random short-fiber composite is presented. The fatigue damage takes various forms of microcracking, originated from microscopic stress concentrators in the highly heterogeneous material system. A probabilistic treatment of the microcracks is introduced to evaluate the statistical nature of the microscopic fatigue damage. Damage evolution and accumulation are analyzed through the development of probabilistic density functions of microcrack length and orientation during the cyclic loading history. Constitutive equations for the damaged fiber composite are then derived on the basis of a self-consistent mechanics scheme in conjunction with a three-dimensional elliptic crack theory and the microcrack density functions. Cyclic fatigue degradation and associated damage-induced anisotropy of composite material properties are determined and checked against experiments. The tensorial nature of material damage and composite stiffness changes during fatigue are evaluated explicitly. A power-law relationship between the rate of damage growth and the fatigue loading cycle is obtained. The rate of fatigue damage growth is found to decrease exponentially with load cycles—a phenomenon unique to the random short-fiber composite. This study provides a comprehensive analytical treatment of the homogeneous fatigue damage problem for random short-fiber composites. The fundamental mechanics and mechanisms of fatigue damage evolution and associated anisotropic property degradation of the composite are elucidated.     

基于微缺陷成核序列的岩石微裂纹生长和损伤演化模型

岩石在承载之初,由于微缺陷无序成核和有限生长,在材料内部形成大量分布性微裂纹。在该文中,这种演化机制被归结为:微缺陷随机、孤立成核生成最小微裂纹和微缺陷无重叠聚集成核、排列生长形成大尺度微裂纹,裂纹尺度生长是微缺陷成核数的函数。利用微裂纹尺度-频数分布分形以及微裂纹粗糙表面分形,建立基于微缺陷累计成核数序列的裂纹尺度生长模型和损伤演化模型。通过对二维岩石试件破坏过程的微裂纹尺度统计和损伤测试表明,模型的预测结果与观测值符合较好。由于微缺陷成核与声发射源机制具有相似性以及微缺陷成核数序列与声发射数序列具有相似性,所以该模型可用于通过声发射参数序列跟踪微裂纹生长和损伤演化。裂纹尺度生长对于完整认识材料宏观力学性质演化和预测材料灾变具有重要意义。     

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.     

The J and M Integrals and Energy of Two Unequal Collinear Cracks

The energetics of two unequal-length collinear cracks for modes- I, II, III is considered. The analysis is performed for remote uniform load normal to the plane of cracks by applying the J and M path-independent integrals and the relationship between them. The material forces for each crack as well as the total crack energy of the system are evaluated. The exact closed form solution to this interaction problem is expressed as a function of cracks’ dimensions and spacing between them. The expression derived for the energy of interaction between a main crack and a nearby microcrack can be used for an estimate of the toughness degradation due to microcracks. Another application of the two-crack energy solution is related to the elastic compliance of cracked solids relevant for the cases when crack faces are in contact along certain areas.     

Critical assessment of the theory of brittle fracture

Cottrell's theory of stable growth of cleavage microcracks is critically assessed for mild steels below the embrittlement transition temperature. The theory is shown to correspond to experiments if one abandons the assumption that the number of crack dislocations is equal to the number of piled-up dislocations at the stress considered. Reasons are presented why the equation for dislocation pile-ups should not be applied to the growth of microcracks. We replace it by the experimental fact that the unstable crack length amounts to a few grain diameters. Then the idea of stable microcrack growth proves consistent with experimental results.     

Numerical analysis of interaction and coalescence of numerous microcracks

The deformation and failure behaviors of brittle or quasi-brittle solids are closely related to interaction and propagation of stochastically distributed microcracks. The influence of microcrack interaction and evolution on the mechanical properties of materials presents a problem of considerable interest, which has been extensively argued but has not been resolved as yet. In the present paper, a novel numerical method is used to calculate the effective elastic moduli and the tensile strength, and to simulate the failure process of brittle specimens containing numerous microcracks. The influences of some crack distribution parameters reflecting the non-uniform spatial concentration, size and orientation distributions are examined. The effective elastic moduli and the tensile strength of brittle materials exhibit different dependences on microcrack interaction. For example, two microcrack distributions that lead to the identical effective elastic moduli may cause a pronounced difference in the tensile strengths and failure behaviors of materials. By introducing two criteria for microcrack growth and coalescence in terms of Griffith’s energy release rate, the above numerical method is extended to simulate the coalescence process of microcracks that results in a fatal crack and the final rupture of a specimen.     

Observations of microcracking in cement paste upon drying and rewetting by environmental scanning electron microscopy

This study reports some preliminary observations of microcracks in hardened high performance cement paste (i.e., cement paste of low water:binder ratio). Specimens were examined at high magnifications and at relative humidities ranging from 0 to 100% in an environmental scanning electron microscope (ESEM). Direct microcracks were observed in samples that had not been dried, indicating that microcracks are probably an intrinsic feature of high performance concrete. It was further observed that the microcracks widened upon drying and closed again upon rewetting. Some practical consequences of these findings are discussed briefly.     

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.     

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.     

Mikrorissentstehung und Mikrorisswachstum in Aluminiumlegierungen bei zyklischer Beanspruchung – Werkstoffliche Untersuchung und Simulation

Microcrack Initiation and Microcrack Growth in Aluminium Alloys Subjected to Cyclic Loading – Material Analysis and Simulation In this paper a model designed to simulate the growth of microcracks under the influence of cyclic loading is presented. Considering fatigue crack growth microstructural barriers as well as the state of stress play an essential role. The polycristalline metal was modelled as an aggregate of hexagonal grains with each of the grains showing a different crystallographic orientation. The crack growth is initially dominated by shear stresses leading to microstructurally short cracks (stage I). As the tip of the microcrack approaches a grain boundary the crack growth rate decreases. The transition from stage I to stage II crack growth is also considered in the model as the crack reaches a specific length and continues to grow under the influence of normal stresses (physically short cracks). The model is applied to tubular specimens of the aluminum alloy AlMgSi1 which are subjected to tension and torsion as well as to combined tension‐torsion loading cycles. In terms of the microcrack distribution as a function of their orientation the simulated crack growth behaviour reveals a close match with the experimental results.     

Microcrack initiation and growth in heat-resistant 15Kh2MFA steel under cyclic deformation

This paper presents the results of investigation of a nuclear reactor pressure vessel steel 15Kh2MFA of two strength levels under cyclic loading. The mechanism of microcrack formation on the surface and in the bulk of 15Kh2MFA steel under cyclic deformation was investigated. Analysis of the specimen surfaces has shown that microcracks are caused by cyclic sliding in grains most favourably oriented with respect to the direction of the maximum shear stresses. Transmission electron microscope investigations show that microcracks in the material inside the grains are formed mainly along the band‐type dislocation structure parallel to the dislocation subboundary. During cyclic deformation, the dislocation density on the subboundaries increases, in the local areas the dislocation density becomes limiting and it reaches the plasticity limit and causes microcrack formation. The interrelation of the average length of microcracks and their surface density with the energy density of inelastic deformation has been found.     

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.     

On the effect of the release of residual stresses due to near-tip microcracking

This article provides a comprehensive theoretical treatment of the interaction effects associated with the release of a general residual stress field as a result of the formation of microcracks in the vicinity of a main crack. The theoretical formulations are based upon the use of the complex potentials of Muskhelishvili and an appropriate superposition procedure. The induced stress intensity factor at the main crack is obtained in a series form and the leading order closed form solution was utilized to elucidate two interesting features of the study. The first is concerned with shielding and amplification effects associated with the release of a uniform residual stress field from a single microcrack. The second is concerned with shielding effects due to the formation of a microcracking zone in the vicinity of a stationary and a steadily growing main crack.The results for the case of a single microcrack reveal that, depending upon the location and orientation of the mirocrack, shielding and amplification effects may become prevalent. In the case of a microcracking zone, equivalence between the current interaction model and the contiuum mechanics approach was established based upon similar microcrack nucleation criteria.     

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.     

In-situ observation of microcrack evolution in a dual-phase steel during tensile straining

ABSTRACT

In this paper, the microcrack evolution in DP590 dual-phase steel was observed by in-situ straining in straining transmission electron microscopy. It was found that a void initiated ahead of a main crack. After the load was applied, a thinned area was nucleated ahead of the void tip, and it grew gradually into shallow nanovoid, penetrating nanovoid, and then new void. Meanwhile, the old void connected with the main crack. The repetitions of the procedures resulted in the continuous propagation of a crack. Interaction between microcracks was observed. The propagation direction of one microcrack may change, affected by other microcracks. Voids were observed between microcracks and have a significant effect on the coalescence of microcracks.