Mechanics and micromechanisms of fatigue crack growth in brittle solids

This paper is concerned with the mechanics and micromechanisms of stable mode I crack growth in brittle solids subjected to compression-compression fatigue and tension-tension fatigue loads. Constitutive models, results of finite element analyses, and experimental observations are described for monolithic ceramics and ceramic-matrix composites, plain concrete, and a transformation-toughened ceramic in an attempt to deduce a general theory on the origin of mode I fracture in notched plates under uniaxial cyclic compression at room temperature. An analysis of the residual stress field which develops at elevated temperatures in response to power law creep and far-field compressive cyclic loads is also presented. The principal driving force for mode I fracture in cyclic compression is the generation of a near-tip zone of residual tension, when the deformation at the notch-tip leaves permanent strains upon unloading from the far-field compressive stress. The results indicated that materials with very different microscopic deformation mechanisms, i.e., microcracking, dislocation plasticity, martensitic transformation, interfacial debonding/slip, or creep, exhibit a macroscopically similar, stable fracture under far-field cyclic compression because the zone of residual tension is embedded in material which is elastically strained in compression. It is shown that cyclic compression loading offers a unique method for fatigue precracking notched specimens of brittle solids prior to tensile fracture testing, whereby an unambiguous interpretation of the critical stress intensity factors for crack initiation and growth can be achieved. Fatigue crack growth characteristics of a transformation-toughened ceramic and a creeping ceramic composite under tension-tension fatigue loads are also discussed.     

A model for the porosity dependence of Young's modulus in brittle solids based on crack opening displacement

A crack opening displacement concept has been introduced to model the porosity dependence of Young's modulus in polycrystalline and single phase solids. In developing the theoretical model, it is assumed that each cylindrical cavity possesses radial cracks and spherical pores possess annular flaws. When an external stress is applied on such a solid, its elastic response is shown to be governed by the pore size, the width of an annular flaw, the number of pores (or pore volume fraction) and the flaw to pore size ratio. The validity of the present approach is tested against a number of experimental data.[/p]     

Local stress and strain during crack growth by steady state creep

Experimental methods are used to measure the distribution of plastic strain ahead of a crack propagating under steady state creep conditions. Using these strains the local strain-rates are known, and from these the steady state stress distribution is deduced assuming power law behaviour. The resulting information indicates that the stress distribution is closer to _ss X ^–1/(m+1) than to _ss X ^–1/(2m). It is shown that for low values of the exponent,m, in the power law, that creep crack growth should correlate with the elastic stress intensity factor, whereas at largem values a better correlation is expected with the net stress.     

Effect of microstructure degradation on creep crack growth

Creep crack growth by grain boundary cavitation is analysed numerically for center cracked and edge cracked panels. Creep acceleration induced by microstructure degradation is incorporated in the material model that describes the nucleation and growth of cavities in the grain boundary, including the effect of diffusion, dislocation creep and grain boundary sliding. It is found that the creep acceleration significantly reduces the notch sensitivity of the material.

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Résumé On analyse par voie numérique la croissance d'une fissure de fluage par cavitations aux joints de grain dans le cas de plaques fissurées en leur centre ou sur leurs bords. Un modèle du matériau, qui incorpore l'accélération du fluage due à sa dégradation, décrit la naissance et la croissance des cavités aux joints de grains, en tenant compte de l'effet de la diffusion, du fluage associé aux dislocations et du glissement des joints de grains.On trouve que l'accélération du fluage réduit de manière significative la sensibilité à l'entaille du matériau.

     

Prediction of elastic moduli of solids with oriented porosity

There have been no simple equations available to predict the effects of arbitrarily shaped voids over the entire range of porosity encountered in real materials. Empirical expressions have been proposed, but they agree neither with appropriate theoretical analyses nor with extensive experimental data. Theoretical predictions have been linear in porosity and thus predict an insufficient reduction. Only a few analyses account for void shapes other than spherical. The present work represents a semi-empirical approach to fill these information deficiences.     

Experimental investigations of crack trapping in brittle heterogeneous solids

Over the past two decades many mechanisms of toughening have been considered for brittle solids. Some of the most prominent ones, applicable to either monolithic materials or fiber-reinforced composites, include deformation-induced local transformations, microcracking, crack trapping, crack bridging, and fiber pull-out. Few, if any, of these have been studied in the past in a manner which permitted evaluation of the effects of individual mechanisms in the absence of other interacting mechanisms. Here, we present an experimental study of toughening by the process of crack trapping by second-phase particles (spheres and fibers) of such toughness that make them impenetrable by probing cracks, forcing the cracks to bow around the obstacles with increasing applied load.

The model fracture specimens employed here were wedge-loaded double cantilever beams, cast of a brittle epoxy, containing macroscopic (3 mm diameter) inclusions of Nylon or polycarbonate, having elastic properties similar to the matrix. The tests were performed at −60°C to achieve controlled, stable crack propagation. Images of the crack fronts, advancing at velocities of about 10⁻⁴m/s, were recorded with good resolution, providing a continuous record of crack-front shapes during the evolution of the crack-trapping process — from the initial pinning configuration through the transition to crack-flank bridging. Remarkable agreement between these images and crack-front shapes predicted by the numerical simulation of Bower and Ortiz is demonstrated. A parametric approach was adopted to study the influence of obstacle spacing, surface adhesion, and thermally induced residual stresses upon the observed crack-front behavior and enhanced stress intensity required to propagate the cracks past these obstacles. Analysis of the quantitative data has demonstrated that in brittle matrices containing particle volume fractions of approximately 0.2, toughness enhancement by over a factor of 2, relative to neat matrix values, may be achieved through the crack-trapping mechanism alone, provided that a high level of adhesion can be maintained between the matrix and the tough reinforcing particles.     

Effect of stress dependent creep ductility on creep crack growth behaviour of steels for wide range of C*

Abstract

In this work, the effect of stress dependent creep ductility on the creep crack growth (CCG) behaviour of steels has been investigated by finite element simulations based on ductility exhaustion damage model. The relationship between the transition region of creep ductility and the transition behaviour of CCG rate on da/dt-C* curves has been examined and the CCG life assessments of components and CCG resistance of materials for a wide range of C* were discussed. The results show that with increasing the transition region size of creep ductility, the transition C* region size on da/dt-C* curves increases. With moving transition region position of creep ductility to high stress region (increasing transition stress levels), the transition C* region on the da/dt-C* curves also moves to high C* region. Decreasing transition stress levels and transition region sizes of creep ductility and increasing the lower shelf and upper shelf creep ductility values can improve the CCG resistance of materials. If the extrapolation CCG rate data from the high C* region or from the transition C* region are used in life assessments of the components at low C* region, the non-conservative or excessive conservative results may be produced. Therefore, the CCG rate data should be obtained for a wide range of C* by long term laboratory tests or numerical predictions using the stress dependent creep ductility and model.     

Prediction of creep crack growth in the presence of residual stress

Abstract

The mechanisms of creep crack growth are presented and the relationship between creep crack growth rate and uniaxial creep properties identified. Cracking under primary, secondary and combined primary and secondary loading is considered. The concepts described are applied to cracking in compact tension specimens of Type 316H austenitic stainless steel and Type 347 weld metal, each of which had previously been subjected to pre-compression to generate a tensile residual stress adjacent to the crack tip. Examples of the residual stress distributions produced before and after creep relaxation are presented and used to predict the crack growth anticipated. Comparisons are made between the behaviour of the two steels. Sensitivity studies are included to determine the extent to which the predictions are affected by the choice of material properties and analysis employed. For secondary loading only, it is shown that the amount of cracking predicted is relatively insensitive to the initial residual stress present.     

To the theory of crack propagation in deformed brittle solids

We present elements of the theory of propagation of cracks in a deformed brittle body. The problem of propagation of cracks is formulated with regard for the forces of interaction between the opposite lips of the crack in the vicinity of its ends. The basic equations of the problem are obtained and their solutions are given for some special cases. Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, L'viv. Published is Fizko-Khimichna Mekhanika Materialiv, Vol. 32, No. 4, pp. 119–122, July–August, 1996.     

Failure of brittle polymers by slow crack growth

The variation of crack velocity (V) with stress intensity factor (K _I) at the tip of a crack has been measured for an epoxy resin containing 42% by volume of irregularly-shaped silica particles. It has been found that at crack velocities above 10^–5 m sec^–1 the crack propagates primarily through the silica particles, whereas at velocities below this value, failure occurs primarily by particle pull-out. This variation in fracture mode is accompanied by a corresponding change in slope of the V(K) curve. Using data obtained from creep rupture experiments and the derived V(K) relationship, it has been possible to estimate the size of the inherent flaw in the composite. This was found to be approximately twice the average particle diameter which is also equal to the size of the largest particles (140 m). Fracture of the unfilled epoxy resin and the effect of environment upon slow crack growth in the composite have also been investigated.     

Failure of brittle polymers by slow crack growth

The mechanical properties of a silica particle-filled epoxy resin composite system have been investigated in air as a function of volume fraction of particles for volume fractions ranging from 0 to 0.52. The Young's modulus and the compressive yield stress both increase as the volume fraction of silica particles is increased and various models of particle strengthening have been used to explain this behaviour. Slow crack growth in the various particulate composites has been studied using a fracture mechanics approach. The variation of crack velocity (V) with stress intensity factor (K _I) has been measured for each of the compositions investigated. In each case, a unique relationship between V and K _I has been found with K _I increasing with volume fraction of particles at a given value of V. The failure mechanisms and the variation of other fracture mechanics parameters, for example, crack opening displacement and plastic zone size with increasing particle volume fraction have been discussed.     

The stress dependence of the crack growth rate during creep

An equation is derived for the crack growth rate under creep conditions. In the model, the propagation of a grain boundary crack is controlled by the plastic growth of cavities located in the grain boundaries ahead of the crack. It is assumed that the cavities grow by power law creep in the elastic crack tip stress field. Hence, the stress dependence of the crack velocity is provided through the elastic stress intensity factor, i.e., dC/dt=BK _I ^p .The cavity spacing, , appears as an important factor in the coefficient,B ^–(p–2)/2. At large values of , corresponding to less severe creep damage in the grain boundaries, the above equation would predict very low values for the crack velocity. Under such conditions, we suggest that another mechanism, whose stress dependence is provided through the net section stress, becomes active, i.e., dC/dt=B _net ^p . Since increases with decreasing applied stress, one should observe the _net correlation at low stresses. The results of recent creep crack growth experiments which tend to support this hypothesis are presented.

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Résumé On dérive une équation décrivant la vitesse de propagation d'une fissure dans les conditions de fluage. Dans ce modèle, la propagation d'une fissure aux frontières de grains est contrôlée par la croissance dans le domaine plastique de cavités situées aux frontières de grains en avant de la fissure.On suppose que les cavités s'étendent dans le champ de contraintes élastiques situées à l'extrémité de la fissure en suivant une loi de fluage parabolique. Dès lors, la dépendance de la vitesse de la fissuration en fonction de la contrainte est fournie par un facteur d'intensité de contrainte élastique, c'est-à-dire dC/dt=BK _I ^p .L'espace entre les cavités, , apparaît être un facteur important dans les coefficientsB. Pour de grandes valeurs de , qui correspondent à un dommage moins sévère par fluage aux frontières des grains, l'équation ci-dessus permettrait de prédire des valeurs très faibles de la vitesse de fissuration.Sous ces conditions, il est suggèré qu'un autre mécanisme, dont la dépendance de la contrainte est fournie par la contrainte agissant sur la section droite, devient plus actif; on a alors dC/dt=B _nette ^p .Comme augmente lorsque la contrainte appliquée diminue, on devrait observer une corrélation de nette à basses contraintes.Les résultats d'essais de croissance de fissure sous des conditions de fluage effectués récemment tendent à supporter cette hypothèse et sont présentés.

     

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.     

Grain-size dependence of fracture stress in anisotropic brittle solids

The stress concentrations that occur at grain boundaries due to thermal expansion anisotropy and elastic stress concentration are discussed, and the stress intensity factor that results from these stresses is estimated. The procedure for the stress intensity factor calculation is based on the model in which a spherical crystal (grain) is forced into a cavity of equal size possessing annular or radial cracks emanating from the boundary. The stress intensity factor equation thus obtained is extended to include the effect of elastic stress concentration due to the presence of a cavity, and is subsequently used to predict the grain-size dependence of strength in anisotropic brittle ceramics. In assessing the degradation of strength with increasing grain size in non-cubic ceramics, it is shown that, in addition to grain size, the effect of pre-existing crack size must also be considered. Cubic ceramics, on the other hand, are known to exhibit no thermal expansion anisotropy and, based on the present model, their strength is predicted to be governed by the pre-existing flaw size, rather than the grain size.     

Crack propagation in brittle heterogeneous solids: Material disorder and crack dynamics

Crack propagation in a linear elastic material with weakly inhomogeneous failure properties is analyzed. An equation of motion for the crack is derived in the limit of slow velocity. Predictions of this equation on both the average crack growth velocity and its fluctuations are compared with recent experimental results performed on brittle heterogeneous materials (Ponson in Phys Rev Lett, 103, 055501; Måløy et al. in Phys Rev Lett, 96, 045501). They are found to reproduce quantitatively the main features of crack propagation in disordered systems. This theoretical framework provides new tools to predict life time and fracture energy of materials from their properties at the micro-scale.     

Cyclic crack growth in elastic plastic solids: a description in terms of dislocation theory

Whenever the plastic deformation is in the order of some Burgers vectors, it appears to be reasonable to describe crack tip plasticity by means of “mathematical” dislocations. A discrete dislocation model is presented for the simulation of mode I fatigue crack propagation. In order to take into account the crack face contact behind the crack tip a procedure was developed which enables the computation of the dislocation motion even when crack closure occurs.