Fracture of brittle particles in a ductile matrix

Prerequisites for precipitate cracking in a yielding ductile matrix have been examined. A statistical model based on the fibre loading model combined with weakest link fracture theory is presented. With the model it is possible to estimate the effect of different variables on the particle fracture probability quantitatively. The predictions made are in excellent agreement with experimental results for a variety of different precipitate types. The result can be applied to calculate the probability of cleavage fracture for steels.

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Résumé On examine les conditions pour lesquelles un précipité fragile se rompt dans une matrice ductile en déformation plastique. On présente un modèle statistique basé sur les modèles de chargement d'une fibre et combiné avec la théorie de la rupture de la liaison la plus faible. Avec ce modèle, il est possible d'estimer quantitativement l'effet de diverses variables sur la probabilité de rupture d'une particule. Les prédictions qui sont avancées sont en excellent accord avec les résultats expérimentaux, pour une gamme de divers types de précipités. Les résultats peuvent être appliqués au calcul de la probabilité de rupture par clivage dans le cas des aciers.

     

Pull-out of a ductile fibre from a brittle matrix

The pull-out of a ductile fibre from a brittle matrix was analysed in Part I [1] using a shear-lag model. However, the analysis is formidable due to the consideration of Poisson's effect along the sliding length. This consideration is essential when the debonded fibre-matrix interface is subjected to Coulomb friction during fibre pull-out. To simplify the analysis, Poisson's effect is treated in an average sense in the present study, whereas it was treated pointwise in Part I. The present simplified solutions are in excellent agreement with the previous more rigorous and more complex solutions. The simplified model thus provides adequate solutions for the pull-out of a ductile fibre from a brittle matrix, and can be readily used for further applications.     

Pull-out of a ductile fibre from a brittle matrix

Pull-out of a ductile fibre from a brittle matrix has been analysed using a shear lag model. Debonding at the fibre-matrix interface and yielding of the fibre occurred during the pull-out process. Both Poisson's contraction of the fibre and Coulomb friction of the debonded interface were considered. The debond length, which consists of an elastic zone length and a plastic zone length, was also analysed. When the fibre has a finite embedded length, it was found that necking prior to full pull-out of the fibre was required to optimize the toughening of a brittle matrix due to plastic deformation of the fibres. The essential material properties to achieve this are addressed.     

Analysis and model of the crack bridging mechanisms in a ductile fiber reinforced ceramic matrix composite

The force resisting the opening of a crack in a brittle matrix composite that is bridged by ductile fibers was studied (Acta Mater. 46(18) (1998) 6381; Acta Mater. 45(9) (1997) 3609). to gain a generic understanding of the crack-bridging process by ductile reinforcements. The matrix was alumina, initially containing a parallel array of fine cylindrical holes. Molten Al was cast into the holes to produce the fibers in situ. A crack was gently introduced to traverse the specimen. The matrix halves were pulled apart in a controlled manner to open the crack. The resisting force increased proportionally to the crack opening over a wide range until a force plateau was reached. Thereafter the force diminished very gradually until failure intervened. Analysis of this counter-intuitive behavior indicated that the excellent adhesion between the fiber and the matrix in combination with the large thermal expansion mismatch must have led to extensive but spotty debonding already from the start of the start of the crack opening. In spite of the well-known ductility of the fibers, the bridging showed quasi-elastic behavior over much of the crack opening. Necking appeared to be suppressed until the separation approached failure. Detailed modeling is offered to provide interpretation of this observed behavior.     

Asymptotic analysis of crack bridging by ductile fibres

The problem that is addressed relates to a brittle matrix — typically, a ceramic — reinforced by ductile fibres. If a crack develops in the matrix, its opening is restrained by the fibres and toughness is enhanced. The problem can be formulated as an integral equation which can be solved numerically, but one limiting situation, of some practical relevance, can be treated analytically. This is the case of a matrix that is sufficiently brittle to allow the crack to extend relatively far before the fibres begin to fail. Analysis so far performed has provided an analytic expression for failure load versus crack length prior to fibre failure, plus an estimate for crack length at first fibre failure, when the physical parameters allow this sequence. Current work is devoted to the analysis of the progressive failure of fibres.     

Fracture mechanics of brittle matrix ductile fiber composites

A model to predict the increase in critical flaw size or stable crack growth potential which can occur by the inclusion of ductile fibers in a brittle matrix is considered. The model is based upon the super-position of two known stress intensity solutions; one for the crack opening mode resulting from a remotely applied stress and the second, an opposing stress intensity that results from a crack closing force exerted by unbroken fibers spanning the crack surfaces. The extent of stable growth possible is computed at the ultimate stress of the brittle phase as functions of fiber strength and of volume fraction for various amounts of fiber rupture. A hot pressed beryllium matrix is used as an example. The crack surface displacement over which a given fiber is capable of deforming without rupture is found to be sensitive to the fiber-matrix interface strength. The factors leading to maximum crack surface displacement without rupture are a high strain hardening capability of the fiber and an interface designed to fail at fiber stresses between yield and ultimate strengths.     

Work-of-fracture of brittle materials with microcracking and crack bridging

The fracture mechanics basis of fracture energies is considered through micromechanical phenomena in the crack tip frontal process zone, the following crack wake and crack bridging regions of brittle materials, such as cement-based materials, rocks, ceramics, and ceramic composites. The discussion is mainly focused on the work-of-fracture parameter ({ie65-1}) as a material characteristic for representing the resistance to crack extension. Theoretical considerations of the dependence of {ie65-2} on the unnotched remaining ligament length of the fracture test specimen lead to the concept of the essential work-of-fracture, {ie65-3}. The experimental results obtained for three different types of ceramic materials support the theoretical predictions.     

The debonding and pull-out of ductile wires from a brittle matrix

An experimental investigation into the debonding and pull-out of nickel wires from epoxy resin and cement paste matrices has been carried out. Above a critical embedded length both the debonding and pull-out stresses attain limiting values. A theory based on the model of a yielded zone travelling up the wire behind a debonding front was shown to describe the observed dependence of the limiting debonding stress on the yield stress, diameter and surface roughness of the wire. Pull-out behaviour subsequent to debonding was explained using this model in terms of an unyielded plug at the end of the wire. Orientation of the wire to the loading direction was found to raise the limiting debonding and pull-out stresses due to enhanced friction at the wire exit point.     

Stable crack growth in ductile polymers

Growth of a crack in ductile polymers (polyarylate, PTFE, and PU/PMMA blends) was studied. The crack growth was described assuming that local yielding in the crack tip is similar to large-scale shear yielding in rigid-plastic materials. Crack growth was stable, and a wedge shaped crack tip was formed. The crack tip opening displacement and the crack extension in the initial stage of the crack growth were proportional to the square of the strain up to 11% elongation. Dugdale’s equation was modified to describe the magnitude of the crack tip opening. In PA, a yield subzone near the crack tip was observed. This revised version was published online in November 2006 with corrections to the Cover Date.     

Creep crack growth in brittle materials

During steady state crack growth by diffusive cavitation at grain boundaries, crack tip fields are relaxed due to the presence of a cavitation zone. In the present analysis, analytic solutions for the actual crack tip stress fields and the crack velocity in the presence of cavitation zone consisting of continuously distributed cavities ahead of the crack tip are derived using the smeared volume concept. Results indicate that the r ^–1/2 singularity is now attenuated to r ^{–1/2 +} (0<<1/2) singularity. The singularity attenuation parameter is a function of the crack velocity and material parameters. The crack growth rate is related to the mode I stress intensity factor K by K ² at relatively high load, K ⁿ at intermediate load, and approaches zero at small load near K _th. Meanwhile, the cavitation zone extends further into the material due to the stress relaxation at the crack tip and the subsequent stress redistribution. Such relaxation effects become very distinct at low crack velocity and low applied load. Key words: Creep crack growth, brittle material, diffusive cavity growth, sintering stress, crack tip stress field.     

Modeling of crack bridging in a unidirectional metal matrix composite

The effective fatigue crack driving force and crack opening profiles were determined analytically for fatigue tested unidirectional composite specimens exhibiting fiber bridging. The crack closure pressure due to bridging was modeled using two approaches; the fiber pressure model and the shear lag model. For both closure models, the Bueckner weight function method and the finite element method were used to calculate crack opening displacements and the crack driving force. The predicted near crack tip opening profile agreed well with the experimentally measured profiles for single edge notch SCS-6/Ti-15-3 metal matrix composite specimens. The numerically determined effective crack driving force, K _eff, was calculated using both models to correlate the measured crack growth rate in the composite. The calculated K _eff from both models accounted for the crack bridging by showing a good agreement between the measured fatigue crack growth rates of the bridged composite and that of unreinforced, unbridged titanium matrix alloy specimens.     

Size-scale transition from ductile to brittle failure: structural response vs. crack growth resistance curve

The concept of brittleness number is revised emphasizing the distinction between a stress-intensification treatment and an energy treatment. In the case of power-law hardening materials the relation between plastic stress-intensity factor and J-integral is translated into a relation between stress brittleness number and energy brittleness number. The structural response generally depends on both such numbers and, therefore, is not physically similar when varying the size-scale of the body. A connection is then established between structural response and crack growth resistance curve, the relevant parameters of a scale-invariant J-resistance curve being related to the energy brittleness number.     

Simulation of crack growth in ductile materials

During crack growth of real materials, the total energy released can be partitioned into elastic and dissipative terms. By analyzing material models with mechanisms for dissipating energy and tracking all energy terms during crack growth, it is proposed that computer simulations of fracture can model crack growth by a total energy balance condition. One approach for developing fracture simulations is illustrated by analysis of elastic-plastic fracture. General equations were derived to predict crack growth and crack stability in terms of global energy release rate and irreversible energy effects. To distinguish plastic fracture from non-linear elastic fracture, it was necessary to imply an extra irreversible energy term. A key component of fracture simulations is to model this extra work. A model used here was to assume that the extra irreversible energy is proportional to the plastic work in a plastic-flow analysis. This idea was used to develop a virtual material based on Dugdale yield zones at the crack tips. A Dugdale virtual material was subjected to computer fracture experiments that showed it has many fracture properties in common with real ductile materials. A Dugdale material can serve as a model material for new simulations with the goal of studying the role of structure in the fracture properties of composites. One sample calculation showed that the toughness of a Dugdale material in an adhesive joint mimics the effect of joint thickness on the toughness of real adhesives. It is expected, however, that better virtual materials will be required before fracture simulations will be a viable approach to studying composite fracture. The approach of this paper is extensible to more advanced plasticity models and therefore to the development of better virtual materials.     

The propagation of cracks in composites consisting of ductile wires in a brittle matrix

Two types of fracture toughness specimens, double cantilever beam and single-edge notch, were constructed from nickel wires in an epoxy resin matrix. The critical stress intensity factor to cause propagation of an unbridged matrix crack arrested at a wire was found to be greater than that for plain resin. Subsequent crack propagation could be described by taking account of the stress intensity factor due to known forces in the crack-bridging wires which subtracted from that due to the applied forces. The debonding and pull-out behaviour observed previously in single-wire specimens was confirmed in the multi-wire fracture toughness specimens.     

Numerical analysis of fibre bridging and fatigue crack growth in metal matrix composite materials

The analysis of bridged crack configurations in unidirectional fibre-reinforced composites is relevant to a variety of crack growth problems, including the fatigue of metal matrix composites and the study of fibre failure in the wake of a bridged matrix crack. Details of numerical procedures for predicting fibre stresses and their effect on crack tip stress intensity factors are presented here to provide a useful overview of how standard bridging calculations are done. Results are presented and discussed in the context of predicting fatigue crack growth with fibre failure in metal matrix composites.     

Estimating the toughening effect of crack-bridging particles in a brittle matrix

Several toughening mechanisms for cracks in brittle solids depend on the restraining effects of material elements that bridge the faces of a crack. A standard way of quantifying the toughening effect is to smear-out the restraining stresses provided by the individual bridging elements. The present paper, by using a very simple simulation model which allows for the discreteness of the system, together with a general force-law for the behaviour of an individual bridging element, shows that the smearing-out procedure is valid if the bridging elements are ductile; however, it can underestimate the toughening effect when the elements are brittle.     

Modeling of crack growth through particulate clusters in brittle matrix by symmetric-Galerkin boundary element method

The interaction of a crack with perfectly bonded rigid isolated inclusions and clusters of inclusions in a brittle matrix is investigated using numerical simulations. Of particular interest is the role inclusions play on crack paths, stress intensity factors (SIFs) and the energy release rates with potential implications to the fracture behavior of particulate composites. The effects of particle size and eccentricity relative to the initial crack orientation are examined first as a precursor to the study of particle clusters. Simulations are accomplished using a new quasi-static crack-growth prediction tool based on the symmetric-Galerkin boundary element method, a modified quarter-point crack-tip element, the displacement correlation technique for evaluating SIFs, and the maximum principal stress criterion for crack-growth direction prediction. The numerical simulations demonstrate a complex interplay of crack-tip shielding and amplification mechanisms leading to significant toughening of the material.