Static and energetic fracture parameters for rocks and concretes

The object of the paper is to determine the fracture toughness parameters K_1C,G _1C and J_1C for some aggregative materials. Values of the J-integral are calculated from load-displacement curves, following the procedure suggested by Begley and Landes for steel alloys. Some recurring experimental incoherences are explained applying Buckingham's Theorem for physical similitude and scale modeling to Fracture Mechanics. Thus a non-dimensional parameter can be defined (the test brittleness number), which governs the fracture-sensitivity phenomenon. The fracture parameters K_1C and J_1C are connected by a fictitious Young's modulus E^*, which is lower than the real modulus E and represents the stiffness of the damaged material near the crack tip before the extension. When the specimen sizes are so small that the material becomes fracture insensitive, then E^* appears higher than E.     

Carbon fibre composites with rubber toughened matrices

In carbon fibre reinforced plastics (CFRP), the initial resistance to crack propagation parallel to fibres is determined largely by the matrix toughness. The fracture toughness (G _IC) of an epoxide resin can be increased considerably by the addition of butadieneacrylonitrile co-polymers (CTBN). These cause the precipitation of small spheres of a second phase and, for example, increaseG _IC from ～ 300 to ～ 3000 J m^?2 on the addition of 9 wt% CTBN. The large increases obtained in bulk resins are not obtained in CFRP, instead significant but modest increases are achieved. The suppression of toughness is related to the thickness of resin film through which the crack propagates.     

The toughening of cyanate-ester polymers Part I Physical modification using particles, fibres and woven-mats

A fracture mechanics approach has been used to investigate the effects of the addition of physical modifiers on the fracture energy, G _c, of brittle cyanate-ester polymers. Tests were performed using adhesive joint specimens at –55, 21 and 150°C, with all the specimens exhibiting cohesive failure in the cyanate-ester adhesive layer. The fracture energies of systems modified using a range of inorganic and thermoplastic particles, fibres and woven-mats have been measured, and scanning electron microscopy has been used to determine the toughening micromechanisms involved. Firstly, it is shown that the addition of 10% by weight of particulate modifiers can increase the fracture energy of the cyanate-ester polymer by 100%, due to a combination of toughening micromechanisms such as crack deflection, pinning and matrix cavitation around the second-phase particles. These experimental data have been compared to predictions from an analytical model. Secondly, it is demonstrated that the use of long fibres or woven-mats can give an a major increase in the value of the fracture energy, G _c, at initiation, and a further increase with increasing crack length, i.e. a significant R-curve effect is observed. At relatively long crack lengths, the measured fracture energy may be six times greater than that of the unmodified polymer value, due to fibres debonding and bridging across the fracture surfaces. Finally, it is shown that several of the physically-modified polymers developed in the present work have fracture energies that are greater than a typical commercially-available cyanate-ester film adhesive.     

Fracture studies on polypropylene

Measurements of fracture surface energy have been made on polypropylene in the undrawn state and at different states of orientation over the temperature range ?60 to 60° C. Tear tests were employed and it was found that the fracture surface energy of unoriented material was of the order of 10⁴ to 10⁵ J m^?2. As orientation (represented by birefringence) increased, the fracture surface energy decreased by a factor of approximately 100 at room temperature but this factor was found to decrease with decreasing temperature. For all degrees of orientation, the fracture surface energy increased with increasing temperature in the range ?60 to 60°C, Scanning electron microscope studies showed a direct relation between the crack tip diameter and the fracture surface energy of unoriented specimens. From comparable studies on the tearing of rubber, Thomas has interpreted such a relationship as implying that the high values of fracture surface energy arise from the energy required to deform the material in the crack tip up to the breaking point. On this basis the reduction in fracture surface energy with increase in orientation may be regarded as being due to the associated diminution of the crack tip diameter. This interpretation is substantiated by direct measurements of crack tip diameter for specimens of intermediate and high orientation. Further microscopic studies of fracture surfaces indicate three modes of fracture which have been correlated with the appearance of the crack tip and tend to occur in certain ranges of birefringence.     

Size effects in concrete fracture – Part II: Analysis of test results

This paper presents an analysis of the extensive experimental program aimed at assessing the influence of maximum aggregate size and specimen size on the fracture properties of concrete. Concrete specimens used were prepared with varying aggregate sizes of 4.75, 9.5, 19, 38, and 76mm. Approximately 250 specimens varying in dimension and maximum aggregate size were tested to accomplish the objectives of the study. Every specimen was subjected to the quasi-static cyclic loading at a rate of 0.125mm/min (0.005in./min) leading to a controlled crack growth. The test results were presented in the form of load-crack mouth opening displacement curves, compliance data, surface measured crack length and crack trajectories as well as calculated crack length, critical energy release rate, and fracture toughness (G ₁). There is a well pronounced general trend observed: G ₁ increases with crack length (R-curve behavior). For geometrically similar specimens, where the shape and all dimensionless parameters are the same, the R-curve for the larger specimens is noticeably higher than that for the smaller ones. For a fixed specimen size, G ₁ increases with an increase in the aggregate size (fracture surface roughness). For the same maximum aggregate size specimens, the apparent toughness increases with specimen size. It was clear that the rate of increase in G ₁, with respect to an increase of the dimensionless crack length (the crack length normalized by the specimen width), increases with both specimen size and maximum aggregate size increase. The crack trajectory deviates from the rectilinear path more in the specimens with larger aggregate sizes. Fracture surfaces in concrete with larger aggregate size exhibit higher roughness than that for smaller aggregate sizes. For completely similar specimens, the crack tortuosity is greater for the larger size specimens. The crack path is random, i.e., there are no two identical specimens that exhibit the same fracture path, however, there are distinct and well reproducible statistical features of crack trajectories in similar specimens. Bridging and other forms of crack face interactions that are the most probable causes of high toughness, were more pronounced in the specimens with larger maximum size aggregates.     

A compliance-based approach to measure fracture resistance curve for surface cracked steel plates

This paper proposes a compliance-based approach to determine the fracture resistance $J$ – $R$ curve for surface cracks in high-strength steel (S690) plate specimens in a four-point-bend set up. This study extends the $\eta $ -approach used in the fracture test for the typical specimens with a through-thickness crack, to the surface cracks in plate specimens in calculating the energy release rate from the area below the measured moment versus the crack-plane rotation. The energy release rate, computed from the detailed finite element models using the domain integral approach, confirms a constant $\eta $ value for surface cracked steel plates. Coupled with the post-test sectioning, the unloading compliance method quantifies the extended crack-front profiles ahead of the fatigue pre-cracked surface notch, using the crack-size versus the compliance relationship derived from linear-elastic finite element analyses. The fracture resistance curves thus obtained remain similar at different locations along the crack front and comparable with the fracture resistance measured using a standard side-grooved compact tension specimen at a finite crack extension.     

Relationship between fracture parameters from two parameter fracture model and from size effect model

This paper studies the relationship between the two parameter fracture model and the size effect model. An equivalency between two models is first established based on infinitely large size specimens. Based on this equivalency, relationships between material fracture parameters (K _Ic ^s , CTOD_c) and (G _f, c_f) are derived. Using these relationships, values of (K _Ic ^s , CTOD_c) and (G _f, c_f) can be predicted from each other. It is found that the relationship betweenCTOD _c andc _f theoretically depends on both specimen geometry and initial crack length. However this dependency is numerically insignificant, except for tensile plate with a short center notch. The obtained results may explain why both the two parameter fracture model and the size effect model can reasonably predict fracture behavior of quasi-brittle materials.     

Fracture criterion for kinking cracks in a tri-material adhesively bonded joint under mixed mode loading

The fracture behavior of a composite/adhesive/steel bonded joint was investigated by using double cantilever beam specimens. A starter crack is embedded at the steel/adhesive interface by inserting Teflon tape. The composite adherend is a random carbon fiber reinforced vinyl ester resin composite while the other adherend is cold rolled steel. The adhesive is a one-part epoxy that is heat cured. The Fernlund-Spelt mixed mode loading fixture was employed to generate five different mode mixities. Due to the dissimilar adherends, crack turning into the adhesive (or crack kinking) associated with joint failure, was observed. The bulk fracture toughness of the adhesive was measured separately by using standard compact tension specimens. The strain energy release rates for kinking cracks at the critical loads were calculated by a commercial finite element analysis software ABAQUS in conjunction with the virtual crack closure technique. Two fracture criteria related to strain energy release rates were examined. These are (1) maximum energy release rate criterion (G_max) and, (2) mode I facture criterion (G_II = 0). They are shown to be equivalent in this study. That is, crack kinking takes place at the angle close to maximum G or G_I (also minimum G_II, with a value that is approximately zero). The average value of G_IC obtained from bulk adhesive tests using compact tension specimens is shown to be an accurate indicator of the mode I fracture toughness of the kinking cracks within the adhesive layer. It is concluded that the crack in tri-material adhesively bonded joint tends to initiate into the adhesive along a path that promotes failure in pure mode I, locally.     

Isotactic polypropylene/EPDM blends: Effect of testing temperature and rubber content on fracture

Charpy impact tests in the temperature range ?100 to +20° C have been carried out on two isotactic polypropylenes (PP) having different molecular weight and their blends containing as rubbery phase an ethylene-propylene-diene terpolymer (EPDM). For fractures of brittle nature the impact data were analysed in terms of the linear elastic fracture mechanics andK _c andG _c were determined. This behaviour was observed for the homopolymers over the temperature range investigated, and for the blends only up to ?20° C. At higher temperatures such materials showed fracture of a semiductile type with visible evidence of craze whitening around the crack tip, followed by brittle type fracture. In this case the results were analysed in terms of a ductile contribution (energy required to form the crazed area) and of a brittle one (relative to the crack propagation area) from whichG _c could be derived according to a procedure proposed in the literature. Tentative interpretations of the results also on a molecular and structural basis have been given. A critical discussion of the elaboration of the semiductile fracture data proposed in the literature has also been provided.     

Adhesive fracture mechanics

Recently Williams, Malyshev and Salganik and others have applied the concepts of fracture mechanics to predict adhesive fracture. The specific adhesive fracture energy, γ_a, is defined as the energy released per unit of new surface created on separation of dissimilar materials. Williams has elucidated the similarity between adhesive and cohesive fracture from the standpoint of a Griffith energy balance analysis. One finds that for either cohesive or adhesive fracture, crack instability (where the crack has a characteristic dimension,a) is predicted by a general equation of the form \(\sigma _{cr} = k\sqrt {E\gamma /a}\) wherek=f [geometry and loading] and includes all loading and geometric factors,E and γ are Young's modulus and specific fracture energy, respectively, and σ_cr is the applied load at incipient failure. Within the fracture mechanics interpretation the adhesive fracture energy γ_a is viewed as a fundamental property of the adhesive system. It is important to note, however, that it may depend on surface preparation, curing conditions, absorbed monolayers, etc. It is, therefore, essential that if γ_a is used to predict adhesive fracture for different geometries, then the surface preparation must be identical with that for the test specimen. If γ_a is a system parameter, then it would be possible to predict fracture by conducting an energy balance analysis of the configuration, utilizing values of γ_a, Young's modulus, and Poisson's ratio as determined from separate simple test specimens. There is, however, a need to establish that γ_a is a system parameter which is independent of geometry. One can in principle perform a number of tests on several specimen configurations or, more effectively, several tests on a single specimen which changes configuration between successive tests. In this latter case the surface and dissimilar materials remain constant. A specimen which is suitable for our purpose was developed by Williams and Jones by incorporating aspects of tests first suggested by Dannenburg and Salganik and Malyshev. The test method considers a disk or plate which has been bonded to a substrate material except for a central portion of radiusa. When pressure,p, is injected into the unbonded region, the plate lifts off the substrate and forms a blister whose radius stays fixed until a critical pressure,p _cr, is reached. At this critical value the radius of the blister increases in size, signifying an adhesive failure along the interface. An energy balance analysis is available for the circular blister specimen in the two limiting cases of a thick plate (Williams) or a very thick medium (Mossakovskii), each with an infinite outer radius. Having established the utility of this general test method, we have considered it necessary to extend the analysis and test calibration capability to other thicknesses for more general engineering applications, as for example, very thin membranes which are used in paint coatings. An axisymmetric finite element numerical analysis was, therefore, conducted for specimens of different thickness and debond radii to establish the energy balance for the various arbitrary thicknesses. A continuous curve for arbitrary specimen thickness was then produced on a dimensionless plot ofp ² a/Eγ versush/a whereh is the specimen thickness. The region ofh/a over which the limiting case equations are valid was also established within the accuracy of the numerical analysis. Since many, if not most, bond geometries are not readily analyzed in closed form, the numerical procedures for energy balance analysis is included and may be used for analyzing geometries other than the blister test specimens. Experiments were conducted over a broad range ofh/a and found to agree with the analytical elasticity solutions which assumed a constant (for given surface preparation, temperature, loading conditions, strain rate independence, etc.) value for γ_a. These experiments confirm that for this system at least γ_a is a constant independent of geometry.     

Stress relaxation behaviour of some aromatic compounds and their alloys with linear polyethylene

The paper reports on stress relaxation measurements on piperonal, camphor and dimethyl terephthalate (DMTP), and on alloys of high density polyethylene (HDPE) with camphor, DMTP and thymol. The pure compounds were measured in compression, the HDPE-alloys in tension. The course of the relaxation curves is in certain cases complicated by recrystallization of the low molecular weight substances and also by sublimation stresses. When these effects are accounted for, the previously found general relationship between the slopeF of the curves, (dσ/d Int)_max, and the initial effective stress, σ ₀ ^* , i.e. $$F = (0.1 \pm 0.01)\sigma _0^* $$ is confirmed. The results thus extend the validity of this structure-independent relation to low molecular weight aromatic substances and their alloys with a crystalline polymer.     

Temperature dependence of cohesive laws for an epoxy adhesive in Mode I and Mode II loading

The influence of the temperature on the cohesive laws for an epoxy adhesive is studied in the glassy region, i.e. below the glass transition temperature. Cohesive laws are derived in both Mode I and Mode II under quasi-static loading conditions in the temperature range $-30\le T \le 80^{\,\circ }$ C. Three parameters of the cohesive laws are studied in detail: the elastic stiffness, the peak stress and the fracture energy. Methods for determining the elastic stiffness in Mode I and Mode II are derived and evaluated. Simplified bi-linear cohesive laws to be used at any temperature within the studied temperature range are derived for each loading mode. All parameters of the cohesive laws are measured experimentally using only two types of specimens. The adhesive has a nominal layer thickness of 0.3 mm and the crack tip opening displacement is measured over the adhesive thickness. The derived cohesive laws thus represent the entire adhesive layer as having the present layer thickness. It is shown that all parameters, except the Mode I fracture energy, decrease with an increasing temperature in both loading modes. The Mode I fracture energy is shown to be independent of the temperature within the evaluated temperature span. At $80^{\,\circ }$ C the Mode II fracture energy is decreased to about 2/3 of the fracture energy at $-30^{\,\circ }$ C. The experimental results are verified by finite element analyses.     

Fracture parameters for interfacial cracks: an experimental-finite element study of crack tip fields and crack initiation toughness

Crack tip measurements and analysis of interfacial parameters for PMMA-aluminum bimaterial system are presented. A variety of crack tip mode-mixities are obtained by subjecting asymmetric four-point-bend specimens to different boundary loads. The crack tip fields are mapped using the optical method of Coherent Gradient Sensing (CGS). The complex stress intensity factors and the associated crack tip mixities () are measured from CGS fringe patterns. The asymptotic expansion field for interface cracks is used for extracting fracture parameters by accounting for higher order contributions to the experimental data. The measurements are compared with complementary finite element computations. A linear relationship between crack tip mixity and the applied load mixity is experimentally demonstrated in this large elastic mismatch system. The fracture load and hence the energy release rate G _cr () at crack initiation is measured as applied load mixities are varied. Limited discussion on the influence of surface roughness prior to bonding on the fracture toughness is included. Positive and negative shear on the crack plane produce different failure responses in this bimaterial system and the observed asymmetry is akin to the one predicted by the T&H model that includes crack tip nonlinearty.     

Application of Fourier transform infrared spectroscopy in cement Alkali quantification

Alkali in cement is responsible for the Alkali–silica-reaction phenomenon that manifests itself in the form of premature cracking in concrete structures such as bridge decks and concrete pavements. X-ray fluorescence spectroscopy (XRF) is commonly used for cement Alkali quantification but a simpler and faster analytical procedure based on Fourier transform infrared spectroscopy (FTIR) has been expanded for this purpose. An analytical absorption band at 750 cm^?1 in the FTIR spectra of cement samples belonging to Alkali solid solution of tricalcium aluminate [C₃A(ss)] is used for Alkali quantification. Regression analysis of a plot correlating FTIR absorption band area ratio (750/923 cm^?1) to equivalent Alkali Na₂O _e (Na₂O _e  = % Na₂O + 0.658 × % K₂O) measured by XRF shows a linear correlation coefficient, R ², of 0.97. High Alkali cement samples show a higher microstructural disorder coefficient, C _d, which is a reactivity criterion introduced by Bachiorrini and co-authors (Proceedings of the seventh international conference on concrete alkali-aggregate reactions? 1986) for ASR-susceptible aggregates. Results of this research indicate applicability of FTIR technique to quantitatively predict cement vulnerability to ASR through the \( A_{{750\,{\text{cm}}^{ - 1} }} \) to \( A_{{923\,{\text{cm}}^{ - 1} }} \) band area ratio and the magnitude of the disorder coefficient (C _d).     

Effect of residual stress on crack propagation in MDPE pipes

Crack propagation behaviour in single edge notched specimens prepared from medium-density polyethylene (MDPE) pipe is examined under creep condition. The crack grown from an exterior notch (inbound) initiated faster than that grown from an interior notch (outbound). Subsequently, the outbound crack propagated monotonically to ultimate failure. The inbound crack showed anomalous behaviour involving two arrest stages prior to ultimate failure. The pipe is found to possess substantial residual stresses. The energy release rate for each case was calculated taking into account the respective residual stream distribution. The fact that the rates of crack propagation are not a unique function of the energy release rate indicates that the fracture is also influenced by morphological gradients imposed by processing conditions.     

Correlation between resin material variables and transverse cracking in composites

In this paper, the correlation between the resin material variables and the transverse cracking in composites is established. A theoretical model based on the fracture mechanics principle is built to describe thein situ failure process of transverse cracking. The central concept of the model is that the fracture is controlled by the plastic zone developed at the crack tip. Based on an approximate crack tip stress distribution, a quantitative representation is found to relate the laminate transverse cracking fracture toughness,G _c(comp), to certain resin properties: fracture toughness,G _c(resin), yield stress, _y, Young's modulus,E, and residual stress build-up, _R.G _c(comp) values of several fibre-glass/epoxy laminate systems were measured using the double torsion technique. The experimental results are found to be interpreted reasonably well by the theory. As a result, a clear picture of transverse cracking emerges. It seems that _y ²/E plays a more dominant role thanG _c(resin) in controllingG _c(comp). The residual stress _R can weaken the laminate significantly when its level is high. It is also shown that the failure model discussed here can be readily applied to laminate delamination failure as well as adhesive bond fracture.     

Generalized fracture mechanics of ductile steels

The generalized fracture mechanics approach is applied to two ductile steels, namely mild steel and 18/8 stainless steel in plane stress. The theory defines a fracture parameter \(\mathcal{T}\) , which is a truly plastic analogue of theJ contour integral and, for an edge crack specimen, is given by $$\mathcal{T} = k_1 ( \in _0 )cW_{0_c } $$ wherek ₁ is an explicit function,c is the crack length andε ₀, W_0c are respectively the strain and input energy density at fracture, remote from the crack. The functionk ₁(ε _o) is derived experimentally and the constancy of \(\mathcal{T}\) with respect to crack length and applied load is demonstrated. The variation of \(\mathcal{T}\) with crack extension during slow growth is investigated, as is the rate dependence of \(\mathcal{T}\) in mild steel.