A fracture evolution procedure for cohesive materials

The present paper deals with the problem of the evaluation of the softening mechanical response of cohesive materials under tensile loading. A nonlinear fracture mechanics approach is adopted. A new numerical procedure is developed to study the evolution of the crack processes for 2D solids. The proposed algorithm is based on the derivation and use of the fracture resistance curve, i.e., the R-curve, and it takes into account the presence of the process zone at the crack tip. In fact, assuming a nonlinear constitutive law for the cohesive interface, the procedure is able to determine the R-curve, the process zone length and hence the mechanical response of any material and structure. Numerical applications are developed for studying the damage behavior of a infinite solid with a periodic crack distribution. Size effects are investigated and the ductile-brittle transition behavior for materials characterized by the same crack density is studied. The results obtained adopting the proposed procedure are in good accordance with the results recovered through nonlinear step by step finite element analyses. Moreover, the developed computations demonstrate that the procedure is simple and efficient.     

Size dependence of concrete fracture energy determined by RILEM work-of-fracture method

The paper analyzes the size dependence of the fracture energy of concrete obtained according to the existing RILEM recommendation proposed by Hillerborg and based on the work-of-fracture method of Nakayama, Tattersal and Tappin, in which the energy dissipated at the fracture front is evaluated from the measured load-displacement curve. The analysis is based on the size effect law proposed by Baant, which has been shown to be applicable to the size ranges up to about 1:20 and apply in the same form for all specimen geometries. The analysis utilizes the previously developed method for calculating the R-curve from the size effect, and the load-deflection curve from the R-curve. The R-curve is dependent on the geometry of the specimen. The results show that the fracture energy according to the existing RILEM recommendation is not size-independent, as desired, but depends strongly on the specimen size. This dependence is even stronger than that of the R-curve. When the specimen size is extrapolated to infinity, the fracture energy according to the RILEM recommendation coincides with the fracture energy obtained by the size effect method. It is also found that, in fracture specimens of usual sizes, the pre-peak contribution of the work of the load to the fracture energy is relatively small. Finally, as a by-product, the analysis also verifies the fact that, in three-point bend fracture specimens, the fracture energy according to the RILEM definition is dependent on the notch depth.     

R-curve measurement of silicon nitride ceramics using single-edge notched beam specimens

R-curve behaviour of three kinds of silicon nitride-based ceramics has been studied using the single-edge notched beam (SENB) technique. If the notch is deep enough, the specimen shows stable fracture during the bending test, even when the sample is a brittle material. The conditions required to obtain stable fracture in the bending test are clarified by the analysis. The crack length of the specimen was also calculated from the changing load during the fracture test. In this study, coarse-grained silicon nitride shows a large increase of theR-curve. On the other hand, silicon nitride with silicon carbide whiskers shows noR-curve increase. The rise of theR-curve should be related to the microstructure of the ceramics, and especially to the grain size of the specimen, because silicon carbide whiskers are not large compared to the silicon nitride grains, and silicon carbide can reduce the grain growth of silicon nitride during sintering.     

Subcritical interlaminar crack growth in fibre composites exhibiting a risingR-curve

Subcritical crack growth behaviour has been evaluated in composite laminates based on uniaxial carbon fibres in poly(ether-ether ketone) matrices. Double cantilever beam (DCB) specimens have been employed to give mode I loading and it is first shown that the materials exhibit a risingR-curve, i.e. the value of the interlaminar fracture energy,G _IC, increases as the crack propagates through the specimens. Secondly, when a DCB specimen is held at a constant displacement, subcritical crack growth is found to occur. The velocity of the subcritical crack growth,v, has been measured using a load-relaxation technique. Hence, values of the crack velocity,v, have been obtained as a function of the strain-energy release rate,G _I applied during subcritical crack growth. Owing to the presence of theR-curve, these data have been measured at various stages during the development of theR-curve. The relationships betweenv andG _I are modelled using power-law expressions. Finally, it is considered that theR-curve behaviour is most likely caused by the fibre bridging which develops behind the crack tip as the delamination propagates through the specimen. Fibre bridging allows stress to be transferred across the crack faces, behind the advancing crack tip, and so results in a shielding of the stress field at the crack tip from the applied stress. Therefore, the expression ascertained for the relationship between the velocity,v, of subcritical crack growth and the corresponding value ofG _I has been further refined and modelled to account for the presence of fibre bridging.     

Influence of grain size and degree of crystallization of intergranular glassy phase on the mechanical behaviour of a debased alumina

The influence of microstructure on the crack resistance (R-curve) behaviour of a commercial debased alumina containing large amounts of glassy phase (28 vol%) has been studied by strength measurements at controlled flaw sizes produced by indentation. Both the individual and combined effects of (a) grain size, and (b) intergranular second phase (glassy or crystalline) were evaluated. Enhancement of theR-curve behaviour was observed when the average grain size was increased from 3–18 μm by thermal treatment. However, no effect of the degree of crystallinity of the intergranular second phase on theR-curve behaviour, in either small or large-grained materials, was observed. These results are discussed with reference to the influence of grain-boundary residual stresses on grain bridging across the crack interface.     

R-curve modeling of rate and size effects in quasibrittle fracture

The equivalent linear elastic fracture model based on an R-curve (a curve characterizing the variation of the critical energy release rate with the crack propagation length) is generalized to describe both the rate effect and size effect observed in concrete, rock or other quasibrittle materials. It is assumed that the crack propagation velocity depends on the ratio of the stress intensity factor to its critical value based on the R-curve and that this dependence has the form of a power function with an exponent much larger than 1. The shape of the R-curve is determined as the envelope of the fracture equilibrium curves corresponding to the maximum load values for geometrically similar specimens of different sizes. The creep in the bulk of a concrete specimen must be taken into account, which is done by replacing the elastic constants in the linear elastic fracture mechanics (LEFM) formulas with a linear viscoelastic operator in time (for rocks, which do not creep, this is omitted). The experimental observation that the brittleness of concrete increases as the loading rate decreases (i.e. the response shifts in the size effect plot closer to LEFM) can be approximately described by assuming that stress relaxation causes the effective process zone lenght in the R-curve expression to decrease with a decreasing loading rate. Another power function is used to describe this. Good fits of test data for which the times to peak range from 1 sec to 250000 sec are demonstrated. Furthermore, the theory also describes the recently conducted relaxation tests, as well as the recently observed response to a sudden change of loading rate (both increase and decrease), and particularly the fact that a sufficient rate increase in the post-peak range can produce a load-displacement response of positive slope leading to a second peak.     

The resistance to crack growth of asbestos cement

The crack resistance of sheet asbestos cement has been characterized in terms of anR-curve which can accomodate effects which often influence the measurement of the critical stress intensity factorK _c. The detection and location of the acoustic emission (AE) obtained from the asbestos cement has shown that it originates from microcracks in a zone just in front of the crack. The size of this zone increases to a maximum during slow propagation of the major crack and afterwards remains of constant size during the final crack growth. The form of theR-curve has been explained in terms of the mechanisms of fracture with the aid of AE and fractography studies. An analytical study has related the experimentalR-curve to a theoreticalR-curve and, hence, to the volume fraction, fibre aspect ratio and the strength of the fibre—matrix interface. It has been shown that the microcracking zone can be considered as a theoretical extension, of about one third of the zone length, to the real crack length.     

A load-cycling technique for R-curve behaviour: application to a low cement refractory

The data presented below for a low cement refractory shows that the material has strong R-curve behaviour for certain specimen sizes. The superposition method proposed by Sakai and Bradt [1] was coupled with the effective crack model developed by Karihaloo and Nallathambi [2] and used to investigate this R-curve behaviour. The technique that was developed involves load cycling on one specimen to evaluate K _IC values with crack extension, and was shown to give favourable results for this material.     

Boundary effect on the softening curve of concrete

The apparent fracture energy of concrete experimentally determined on the basis of the work of fracture in bending or wedge splitting tests becomes larger with increasing specimen dimensions. This experimental observation may be attributed to the varying local fracture energy along the crack path. When the crack tip approaches the specimen boundary, the size of the fracture process zone will be reduced and, consequently, only a portion of the fracture energy is activated; i.e., the local fracture energy is getting smaller. The influence of this boundary effect diminishes with increasing specimen size resulting in the size dependence of the apparent fracture energy determined by the work-of-fracture method as an average value in the ligament. With varying local fracture energy, the local softening curve will also show variations. The latter are subject of the present study. Wedge splitting tests with different specimen sizes as well as inverse analyses of these experiments were carried out. For the inverse analyses, the cohesive crack model was adopted and an evolutionary optimization algorithm has been used. The boundary effect on the local fracture properties was taken into account and, as a result, the variation of the softening curve along the crack path could be determined. It was found that the tail of the softening curve is shortened and lowered due to the boundary effect whereas the initial slope of this curve appears to be not affected.     

FATIGUE CRACK GROWTH BEHAVIOUR AND FRACTURE RESISTANCE IN CERAMICS

Fatigue crack growth and the fracture resistance curve (R-curve) were investigated in a polycrystalline alumina (AD90) and a silicon carbide whisker-reinforced alumina composite (Al₂O₃-SiC_w) at room temperature in air using a combined loading technique for stabilizing crack growth, and a surface film technique for monitoring crack length. Fatigue crack growth was evaluated successfully with those experimental techniques. Load shedding tests were performed until the crack became dormant, in order to determine the threshold stress intensity factor K_th. Subsequently, the specimens were used for quasi-static crack growth tests under a monotonic loading condition. The R-curves were determined in this experiment; however, fracture resistance did not increase markedly with crack growth. Detailed observations of the crack growth behaviour revealed that the flat R-curve was attributed to the shielding effect of the fatigue crack tip wake. Thus, the fatigue precrack introduced by the load shedding test was not regarded as an ideal crack for determining the R-curve. Fractographic observations were performed to investigate the mechanistic difference between fatigue and quasi-static crack growth. It was found that the cyclic loading produced fretting damage in the wake region and it reduced the shielding effect of the fatigue cracks. Based on the experimental results, the relationship between the fatigue crack growth and the R-curve is discussed as is the significance of K_th as a material parameter.     

Effect of microstructure on elevated-temperature fracture behaviour of two-dimensional carbon/carbon composites

The fracture behaviour of two-dimensional carbon/carbon composites has been studied at temperatures upto 1650°C, using both chevron-and straight-notch single-edge notch beam (SENB) specimens. In all cases, the R-curve behaviour and fracture toughness variations with specimen orientation and temperature are characterized and correlated with the specific microstructure and failure micromechanisms. Higher crack growth resistance and fracture toughness of the longer fibre composite are attributed to the enhanced fibre pull-out and fibre bridging in the following wake region. The relative contribution from the frontal and following wake zone is determined experimentally by the use of renotching methods which demonstrate the effectiveness of the traction zone behind the crack tip. The temperature effects on the toughening mechanisms are examined in terms of crystal structure and fibre matrix interfacial characteristics.     

R-curve behavior and roughness development of fracture surfaces

We investigate the idea that the fractal geometry of fracture surfaces in quasibrittle materials such as concrete, rock, wood and various composites can be linked to the toughening mechanisms. Recently, the complete scaling analysis of fracture surfaces in quasibrittle materials has shown the anisotropy of the crack developments in longitudinal and transverse directions. The anomalous scaling law needed to describe accurately these particular crack developments emphasizes the insufficiency of the fractal dimension, usually used to characterize the morphology of fracture surfaces. It is shown that a fracture surface initiating from a straight notch, exhibits a first region where the amplitude of roughness increases as a function of the distance to the notch, and a second one where the roughness saturates at a value depending on the specimen size. Such a morphology is shown to be related to an R-curve behavior in the zone where the roughness develops. The post R-curve regime, associated with the saturation of the roughness, is characterized by a propagation at constant fracture resistance. Moreover, we show that the main consequence of this connection between anomalous roughening at the microscale and fracture characteristics at the macroscale is a material-dependent scaling law relative to the critical energy release rate. These results are confirmed by fracture experiments in Wood (Spruce and Pine).     

Size effect on tensile softening relation

Tensile softening essential to the well-known fictitious crack model should be independent of specimen size and test methods. Commonly observed size effect (SE) on tensile softening relations measured from direct tensile specimens with smooth surface needs to be explained in order to determine the size independent tensile softening behaviour. In this paper, SE on tensile softening from direct tensile tests is elucidated by considering a boundary region where the key tensile softening mechanisms such as aggregate interlocking and frictional pull-out activities are limited in comparison with the inner region where those crack-bridging activities can be fully developed. SE on the tensile softening relation and the closely related specific fracture energy G _f is inevitable if the boundary and inner regions are comparable. The same SE is gradually diminished with the increasing specimen size simply because the relative contribution from the boundary region is reduced in comparison with that from the increasing inner region. In principle, the size independent tensile softening relation and the size independent specific fracture energy G _F in the inner region can be obtained by separating the influence of the boundary region from the test results.     

Cohesive models for quasistatic cracking in inelastic solids

In has been shown that slow stable cracking process can be sustained by strain-softening materials such as cementitious composites, e.g., rock, concrete, mortars, ceramics and others. A mathematical model is proposed based on the theory of quasi-static crack extension governed by Wnuk's criterion of final stretch (equivalent to the CTOA condition). Requirement of self-similarity of the crack opening profile, maintained during the quasistatic growth phase, leads to a differential equation defining the material resistance to sustained crack extension, as a function of crack growth increment, material properties such as tearing modulus, strain-softening parameter, and the size of the process zone.The equations derived on the basis of the step-like distribution of the restraining stress which operates within a structured end zone associated with a moving crack, are compared against the Wnuk-Rice-Sorensen equation to ran R-curve in an ideal elasto-plastic material, and with the more recent results of Wnuk and Hunsacharoonroj [1] pertaining to strain-hardening materials which obey the Ramberg-Osgood power law.Present results suggest that the nature of constitutive equations substantially influence composite fracture resistance as measured by the R-curve. Ability to absorb energy and the ensuing resistance to crack growth can be significantly enhanced when the mechanisms of energy dissipation within the matrix are properly understood. This work establishes a bridge between the continuum mechanics and micromechanics of deformation and meture processes by providing a relation between material fracture toughness and its constitutive equations supplemented by certain microstructural characteristics.     

A shear lip analysis of concave R-curves for an AlMgZn alloy

A concave R-curve for an AlMgZn alloy covers the transition from fully flat to fully slant fracture in 25 mm thick specimens of varying geometry. The transition is associated with the onset of plane stress conditions near the crack front. The concave R-curve is analysed using the increasing width of the shear lips, as in the Krafft et al. model, which increased the plastic work parabolically. The elastic and plastic work rates are evaluated using parabolic curve fits to the R-curve data. The present study showed that the plastic work rate scales with the crack extension from fully flat to fully slant fracture, which is geometry dependent     

Examination of the 4-ENF test for measuring the mode III R-curve of wood

The four-point bend end-notched flexure (4-ENF) test, which was originally developed for measuring the mode II R-curve, is thought to be applicable for measuring the mode III R-curve. In this study, a 4-ENF fracture test of spruce was conducted for obtaining the mode III R-curve, and the test method was numerically and experimentally analyzed. In the numerical analysis, three-dimensional finite element calculations were conducted to determine the distribution of the strain energy release rate along the delamination front by the virtual crack closure technique (VCCT). In the experimental analysis, the mode III R-curve was examined by the modified beam theory and compliance calibration methods of data reduction, which have been conventionally used for analyzing the mode I or mode II R-curve. In addition to these conventional data reduction methods, the strain at each loading point was measured, as was the loading-line displacement and critical load for crack propagation, and the R-curve was obtained by the combination of loading-line compliance, load-longitudinal strain compliance, and critical load for crack propagation, which is named the “compliance combination method”. The finite element analyses suggested that the pure mode III fracture state existed in the mid-section of the specimen in spite of the existence of a small mode II component at the free edges of the delamination front, and the mode III strain energy release rate component calculated by the VCCT coincided well with those obtained by the data reduction methods examined here. The actual R-curve obtained by the compliance combination method coincided well with those by the modified beam theory and compliance calibration methods when the strain was appropriately measured. From these results, therefore, the 4-ENF fracture test is a promising means for obtaining the mode III R-curve of wood.     

Effect of loading condition,specimen geometry,size-effect and softening function on double-<Emphasis Type="BoldItalic">K</Emphasis> fracture parameters of concrete

This paper presents numerical investigation of the influence of the specimen geometry, loading condition, size-effect and softening function of concrete on double-K fracture parameters. The input data needed for computation of the double-K fracture parameters are obtained from the well-known version of Fictitious Crack Model (FCM). FCM is developed for three standard specimens: three-point bend test, compact tension specimen and four-point bend test of size range 100–600 mm at relative size of initial crack length 0.3. The analysis of numerical results shows some interesting behaviour of double-K fracture parameters.     

Analytical and numerical modeling of R curves for cracks with bridging zones

At the onset of fracture in materials with process zones, the fracture resistance, or R curve, rises as the process zone develops. After process zone development, crack propagation proceeds by steady state growth. By considering J integral contours inside and outside the process zone, the available energy can be partitioned into crack tip energy release rate and process zone energy. To model the rising R curve, however, required assumptions about damage mechanisms in the process zone and partitioning of its energy into released and recoverable energy. By considering process zones that are elastic fiber-bridging zones with softening regions caused by fiber breakage or damage, equations for rising R curves were derived as a function of crack tip toughness and bridging zone mechanics. The new methods were implemented into the Material Point Method for generalized numerical crack propagation simulations with bridging zones. The simulation method includes pure fracture mechanics and pure cohesive zone models as extreme special cases. The most realistic simulations for many materials will likely fall between these two extremes. The results guided comments on interpretation of experimental R curves.     

Fracture energy and fracture process zone

The fracture energy G_f can be determined following a RILEM recommendation. However, it has been found that fracture energy depends on both size and geometry of the test specimen. The underlying fictitious crack model postulates that fracture energy, tensile strength, the critical opening of the fictitious crack, and the shape of the softening curve (softening factor) are constants for a given type of concrete. Here it is shown that a local fracture energy ccan be introduced. This local fracture energy varies with the width of the fracture process zone. As the crack approaches the back end of a specimen the fracture process zone becomes more and more confined and hence the local fracture energy decreases. Theoretical predictions are compared with experimental results obtained with the wedge splitting technique described earlier. It is shown that a local variation of the fracture energy leads to a size dependence of the global specific fracture energy.