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.     

Size effects in concrete fracture: Part I, experimental setup and observations

This paper presents an experimental investigation on the influence of microstructural parameters, such as aggregate size, and macroscopic parameters, such as specimen dimensions, on brittle fracture. Maximum aggregate size was used as a representative parameter of aggregate distribution in agreement with ASTM C136 standards. Six groups of geometrically similar concrete specimens with various dimensions and aggregate sizes were prepared. Similarity of the specimens was strictly maintained by scaling the specimen dimensions from one group to another by a factor of two starting from a specimen size of (width × total depth × thickness) 105×105×12.5mm to 1680×1680×200mm. Two separate sets of removable pre-cast notches were designed to determine the effect of initial notch size. A considerable effort was devoted to the design of the loading fixture to have a reproducible crack initiation and controlled crack growth. Several loading fixtures were evaluated prior to selection of the one used in the experimental program. Quasi-static splitting cyclic loading in edge cleavage configuration was applied. A servo-hydraulic Instron machine was used for testing. The fracture process was monitored by optical and acoustic imaging techniques. Three forms of comparisons of the test results with respect to the specimen and aggregate sizes were adopted. The first corresponded to the various specimen sizes cast with the same maximum aggregate size. The second comparison was based on the geometrically identical specimens cast with various maximum aggregate sizes. The third form of comparison dealt with complete geometrical similarity, i.e., all dimensionless geometrical characteristics including specimen thickness to maximum aggregate size ratio were identical. Results from this study indicated that as the specimen size decreases, the envelope becomes larger within the first and third forms of comparison. In the second form of comparison, i.e., geometrically identical specimens cast with various maximum aggregate sizes, the area under the envelope was greater as the maximum aggregate size increased. The existence of a trend in dimensionless critical load-CMOD envelopes despite the apparent geometrical and physical similarity of the test conditions is the direct indication of a scale effect, i.e., the modified fracture energy, indicates the existence of a strong scale effect: increases with the specimen dimensions as well as maximum aggregate size.     

Structural analysis of Sm doped nanocrystalline Mg–Cd ferrites prepared by oxalate co-precipitation method

The structural properties of polycrystalline Sm³⁺ doped Mg_1 − xCd_x Fe₂O₄ (x = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0) have been investigated by oxalate co-precipitation method from high purity sulphates. The samples were sintered at 1050 °C for a duration of 5 h. The X-ray diffraction measurements confirmed the formation of a cubic spinel structure. The different parameters like lattice constant, X-ray density, physical density, porosity, crystallite size, site radii and bond length on tetrahedral and octahedral sites have been calculated. The lattice constant increases with an increase in Cd²⁺ content and shows non linear behavior. The crystallite size was calculated using Scherrer formula and varies from 28.69 to 32.05 nm. Physical densities were obtained by Archimedes principle. The surface morphology studied by scanning electron microscope shows that the grain size of the samples increases with an increase in Cd²⁺ content. The IR spectra show two strong absorption bands around 5.87 × 10⁴ m^− 1 and 4.27 × 10⁴ m^− 1 on the tetrahedral and octahedral sites respectively. IR spectra also show that Sm³⁺ occupies the octahedral B-site.     

Effects of thickness and environmental temperature on fracture behaviour of polyetherimide (PEI)

The fracture behaviour of a polyetherimide (PEI) thermoplastic polymer was studied using compact tension (CT) specimens with a special emphasis on effects of specimen thickness and testing temperatures on the plane strain fracture toughness. The results show that the valid fracture toughness of the critical stress intensity factor, K _IC, and strain energy release rate, G _IC, is independent of the specimen thickness when it is larger than 5 mm at ambient temperature. On the other hand, the fracture toughness is relatively sensitive to testing temperatures. The K _IC value remains almost constant, 3.5 MPa in a temperature range from 25 to 130°C, but the G _IC value slightly increases due to the decrease in Young's modulus and yield stress with increasing temperature. The temperature dependence of the fracture toughness, G _IC, was explained in terms of a plastic deformation zone around the crack tip and fracture surface morphology. It was identified that the larger plastic zone and extensive plastic deformation in the crack initiation region were associated with the enhanced G _IC at elevated temperatures.     

Interlaminar fracture toughness and associated fracture behaviour of bead-filled epoxy/glass fibre hybrid composites

To investigate enhancement of matrix-dominated properties (such as interlaminar fracture toughness) of a composite laminate, two different bead-filled epoxies were used as matrices for the bead-filled epoxy/glass fibre hybrid composites. The plane strain fracture toughness of two different bead-filled epoxies have been measured using compact tension specimens. Significant increases in toughness were observed. Based on these results the interlaminar fracture toughness and fracture behaviour of hybrid composites, fabricated using bead-filled epoxy matrices, have been investigated using double cantilever beam and end notch flexure specimens for Mode I and Mode II tests, respectively. The hybrid composites based on carbon bead-filled matrix shows an increase in both G _IC initiation and G _IIC values as compared to a glass fibre reinforced plastic laminate with unmodified epoxy matrix. The optimum bead volume fraction for the hybrid composite is between 15% and 20%. However, the unmodified epoxy glass-fibre composite shows a higher G _IC propagation value than that of hybrid composites, due to fibre bridging, which is less pronounced in the hybrids as the presence of the beads results in a matrix-rich interply region.     

Specimen Size Requirements for Two-parameter Fracture Toughness Testing

Using a single parameter fracture mechanics theory, a minimum specimen size requirement of min(a, b, B) >200J/σ₀ in tension and min(a, b, B) >25J/σ₀ in bending, where B is the thickness, b the remaining ligament and a is the crack length of the specimen, were derived [Shih and German (1981), International Journal of fracture 17, 27–43] which have provided the basis for modern fracture toughness testing procedures. Two-parameter fracture toughness testing including the constraint, on the other hand, is desirable since it offers a solution to the transferability issue. A size requirement for a valid two-parameter fracture toughness testing based on the J-A₂ three-term solution was determined as min(a, b, B) > 11J/σ₀ [Chao and Zhu (1998), International Journal of fracture 89, 285–307] in which the limiting case is bend specimens under large scale yielding (LSY). Recent work by Chao et al. (2004, International Journal of fracture, 27, 283–302) has shown that the J-A₂ dominance at a crack tip can be significantly enhanced for bending specimens under LSY if a modified J-A₂ solution is adopted. This current paper further studies the size of the J-A₂ dominant zone using the modified J-A₂ solution for deep bend specimens with hardening from low to high and loading from SSY to LSY using finite element analysis. Based on the results, a rather relaxed specimen size requirement min(a, b, B) >6J/σ₀ is developed and recommended for a valid two-parameter fracture toughness testing using the J-A₂ fracture criterion. Validity of the size requirement is demonstrated by using the experimental J-R curves from non-standard bending specimens for A285 steel.     

Determination of double-K criterion for crack propagation in quasi-brittle fracture, Part III: Compact tension specimens and wedge splitting specimens

This paper shows how the double-K fracture parameters K _Ic ⁱⁿⁱ and K _Ic ^un can be determined for concrete using CT-specimens and wedge splitting specimens. The experimental results collected from the fracture tests the very large size CT-specimens and small size wedge splitting specimens carried out by many researchers are utilized to investigate the characters of the obtained double-K fracture parameters K _Ic ⁱⁿⁱ and K _Ic ^un . It was found that the double-K fracture parameters K _Ic ⁱⁿⁱ and K _Ic ^un determined from fracture tests on the large size CT-specimens are size-independent. And the values of K _Ic ⁱⁿⁱ and K _Ic ^un determined from small size wedge splitting specimens with same dimensions are independent of the relative preformed notch length a₀/D. However, when the dimensions of small size wedge splitting specimens change from 150×150×150 mm³ to 450×450×450 mm³, the values of K _Ic ⁱⁿⁱ and K _Ic ^un slightly depend on the heights of the specimens and do not depend on the thickness of the specimens.     

Effect of particles on interlaminar crack growth in cross-plied carbon-fibre/epoxy laminates

This paper discusses the effect of particulate additions on the mode-I and mode-II interlaminar fracture toughness of a cross-plied, carbon-fibre-reinforced, epoxy-resin laminate. Particles of graphite, silicon carbide and polyethylene were mixed with the epoxy resin prior to laminating with woven carbon-fibre cloth. Tests have been performed on double cantilever beam (DCB) and end-notched-flexure (ENF) specimens to obtain the critical-strain energy-release rates,G _IC andG _IIC, for the laminates with and without particulate additions. The dependences of the values ofG _IC andG _IIC on the crack length are also considered. The results indicate that the interlaminar-fracture-toughness (mode-I and mode-II) values of the CFRP laminate increase with increases in the particle content up to about 3%, and thereafter they decrease with further increases in the particle content. This was thought to be due to an increase in multiple-crack formation in the middle region of the cracked-plate samples. Furthermore, mode-I tests indicate that the propagation values ofG _IC are dependent on the crack length.     

The grain size dependence of fracture toughness in an Al-6.0% Zn-2.5% Mg alloy

The grain size dependence of the fracture toughness (K _IC) of an aged Al-6.0% Zn-2.5% Mg alloy was studied experimentally. K _IC depended strongly upon grain size (L _G) in two ways. In the small grain size region K _IC decreased with increasing average grain size. In contrast, K _IC increased with increasing average grain size for large grain sizes. The increase in K _IC with increasing grain size arose as a result of the presence of abnormally large grains compared to the average grain size in the large-grained specimens.     

The Transfer of Matrix Toughness to Composite Mode I Interlaminar Fracture Toughness in Glass-Fibre/vinyl Ester Composites

The transfer of matrix toughness to composite mode I interlaminar fracture toughness (G _Ic ) has been investigated in unidirectional glass-fibre reinforced composites with brittle and rubber-toughened vinyl ester matrices. Single-edge-notch bend (SENB) and double cantilever beam (DCB) specimens were used for matrix and composite G _Ic characteristion, respectively. The initial crack opening displacement rate was used as the parameter for comparison of G _Ic results. Matrix G _Ic was completely transferred to composite G _Ic for crack initiation (G _Ic-init) in the brittle-matrix composites, but in the toughened composites transfer was only partial due to the presence of fibres. The conclusion is that the maximum contribution to energy absorption by the matrix is more accurately reflected by G _Ic-init, and should be used for further assessment of the enhancing effect of fibre bridging during steady-state crack propagation, instead of matrix G _Ic . A plot of composite G _Ic for steady-state crack propagation, G _Ic-prop versus G _Ic-init indicates that the enhancing effect of fibre bridging is greater in the toughened composites. This enhancement is related to a larger deformation zone size in the toughened matrices.     

Thermodynamical approach to the brittle fracture of dry plasters

The evolution of the fracture toughness, K _lc, and fracture energy, G _lc, of set plasters was determined on notched beams as a function of sample porosity, P, and characteristic size, W. Toughness was found to decrease with decreasing crack width. For set plasters of 57.7% porosity, the lowest toughness measured was K _lc=0.13 MPa m^1/2 for a crack width of 0.2 mm. For this crack width, fracture toughness and fracture energy linearly changed with porosity: K _lc=0.5 1–1.3 P) MPa m^1/2 and G _lc= 13.47 (1–1.12 P) Jm^–2. Dense plasters were more difficult to break than porous ones. The fracture energies were affected by the velocity of the fracture propagation, which induces damaging and multicracking of the material, so that the roughly calculated chemical surface energy of set plaster was too high. After correction it was estimated to be 0.4 J m ^–2. Finally, because toughness increased with increasing sample size, it was concluded that fracture toughness and energy were not intrinsic parameters of the material. On the other hand, for our sample porosities and sizes, the reduced rupture force, F _rupt W ^–0.65 is a constant and seems to be a characteristic parameter of the mechanical resistance of set plaster beams.     

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.     

Effect of rubber particle size on the impact tensile fracture behavior of MBS resin with a bimodal particle size distribution

We performed impact tensile fracture experiments on methylmethacrylate–butadiene–styrene (MBS) resin with small and large particles in a bimodal size distribution, and examined the effects of particle size on fracture behavior by fixing the total rubber content (28 wt%) and the small particle size (about 140 nm), and varying the size of large particles (about 490 nm or 670 nm). Dynamic load P′ and displacement δ′ of single-edge-cracked specimens were measured using a Piezo sensor and a high-speed extensometer, respectively. A P′−δ′ diagram was used to determine external work U _ex applied to the specimen, elastic energy E _e stored in the specimen, and fracture energy E _f for creating a new fracture surface A _s. Energy release rate was then estimated using G _f = E _f/A _s. Values of G _f were correlated with fracture loads and mean crack velocity v _m determined from load and time relationships. We then examined the effect of particle size on G _f and v _m, and results indicated that particle size plays an important role in changing the values of G _f and v _m.     

Energy rates and crack stability in small scale yielding

Crack stability in small scale yielding is traditionally analysed using the R-curve approach with toughness indexed by either of the linear elastic fracture mechanics parameters K or G. In ductile materials stable tearing commences well before crack instability and progresses under increasing G_R. This is often assumed to mean that toughness is increasing with crack growth. It is shown in this paper that a rising G_R curve is generated even when a crack propagates with constant toughness (constant energy dissipation rate). The paper demonstrates that this apparent anomaly occurs because G does not represent the energy input rate for a crack advancing under increasing load in an elastic-plastic material. The constant energy dissipation rate model is consistent with a size independent G_R curve; also crack instability predictions are identical with both theories. The G_R curve approach has practical advantages, but use of energy dissipation rate offers better physical insight and greater versatility when analysing tough materials.     

Measurement of mode I and mode II rock dynamic fracture toughness with cracked straight through flattened Brazilian disc impacted by split Hopkinson pressure bar

In order to find an effective and convincing method to measure rock dynamic fracture toughness for mode I and mode II, cracked straight through flattened Brazilian disc specimens of marble, which were geometrically similar for three size, were diametrically impacted by split Hopkinson pressure bar on the flat end of the specimen with three load angle respectively. History of stress intensity factors (K_I(t) for opening mode I, and K_II(t) for sliding mode II), mode mixture ratio (K_I(t)/K_II(t)), as well as mode I and mode II dynamic fracture toughness at crack initiation (K_Id and K_IId) were determined with the experimental–numerical method. It is found that there is a unique size effect for dynamic fracture test with the specimens, the mode mixture ratio is not solely determined by load angle (the angle between load direction and crack line) as in the static loading; the pure mode II load angle is 19° for the ?50 mm specimen, however it is 10° for the ?130 mm and ?200 mm specimens; the mode II load angle decreases with increment of specimen size. Realization of pure mode II is justified by the mode mixture ratio approaching zero, it can be realized under certain load angle and loading rate for the specimen of specified size. K_IId is generally greater than K_Id. Both K_Id and K_IId increase with increment of specimen size, and this trend for K_IId is more remarkable than that for K_Id.     

A practical testing approach to determine mode II fracture energy GIIF for concrete

Continuing the experiments on the double-edge notched specimens on which the mode II fracture toughness K _IIc of concrete was measured, a practical testing approach to determine mode II fracture energy G _IIF is studied using the same geometry.     

Mode II Interlaminar Fracture of Filament Wound Angle-ply Specimens

This paper presents a study of the interlaminar fracture of filament wound composites. Mode II end notched flexure (ENF) tests were performed on flat glass/polyester specimens. The tested specimens had asymmetric [±]₄ angle-ply stacking-sequences, with values from 0.8° (hoop winding) to 30°. Due to the low stiffness and probable high toughness, it was not possible to propagate the crack in [±60°]₄ specimens before they suffered considerable permanent deformations. A short support span had to be employed for [±30°]₄ specimens in order to obtain crack propagation. Nevertheless, the results for those specimens should be viewed with caution, as some nonlinear behaviour and small permanent deformations were detected. No other unusual features, such as crack jumping to other interface, were observed. The scatter in the critical strain energy release rate values (G _IIc ) was higher for [±10°]₄ and [±30°]₄ specimens than for the quasi-unidirectional ones. G _IIc values from the insert were generally lower than those from mode II pre-cracks, except for quasi-unidirectional specimens. A plot of average G _IIc values against showed a minimum at =5°.     

Interlaminar fracture properties of weft-knitted/woven fabric interply hybrid composite materials

An experimental study has been undertaken to characterize the delamination behavior and tensile properties of interply hybrid laminated composites reinforced by interlock weft-knitted and woven glass fiber preform fabrics. The hybrid composites, comprising the alternate layers of interlock and uniweave fabrics, were compared to interlock knitted (only) and uniweave (only) composites with respect to delamination and tensile performances. Mode-I double cantilever beam and mode-II end-notched flexure tests were carried out to assess the interlaminar fracture toughness using aluminum-strip stiffened specimens. The mode-I and mode-II interlaminar fracture toughness values, G _IC and G _IIC, for the hybrid composite were about three and two times higher than that for the uniweave composite, respectively. The tensile strength and modulus of the hybrid composite were 315 MPa and 12.8 GPa in the wale direction, respectively, demonstrating that the strength and modulus were found to be slightly lower than those of the uniweave composite, and significantly improved in comparison with the interlock knitted composites.