Mixed mode i II III fracture criterion and its application to cement mortar 3

4.3 Mixed Mode I – III Fracture Testing 4.3.1 Geometry of Specimen and Loading Fixture To study the fracture behaviour of cement mortar specimens, under three-dimensional loading in the laboratory, special considerations would have to be taken into account, with regard to the ease of preparation of the test specimen, and the provision of an appropriate test set-up and procedure. The former would include the casting and de-moulding of the specimen, as well as the formation of its pre-crack and notch, while the latter would consist of the application of mixed mode loading, and the design of loading fixtures. Accordingly, the traditional compact tension specimen was adopted for mixed mode I – III testing, just as its respective pure mode fracture tests. The advantages of the corresponding test approach adopted will be discussed in following §4.3.2. Figure 4.15 illustrates the design of the mortar specimen. The overall dimensions of the specimen are 160mm × 160mm × 40mm. All the loading holes are of 30mm diameter, while the width and length of the pre-cast notch are 40mm and 80mm, respectively. A 0.5mm thick steel plate coated with mould oil was fixed to the mould before casting, to subsequently form a relatively sharp pre-crack, of 10mm or 12mm length, in the specimen. It was removed within hours of casting, so as not to adhere to the specimen. Figure 4.16 shows the mould used to cast the test specimen. Side grooves were cut into both faces of the specimen, along the self-similar 91 30 160 40 30 30 160 1.0 wide groove T=40 Plan View 80 pre-crack length=10 or 22 20 30 30 30 Sectional View throat thickness t Note: dimensions are in mm. Figure 4.15 Geometrical configuration of mixed mode I – III test specimen 92 Figure 4.16 Specimen mould for mixed mode I – III fracture test 93 direction of the pre-crack, after the specimen had been cured for 28 days in the fog room. In pure mode III loading, a crack would extend in the self-similar direction. However, this is limited to a thin specimen. As shown by Rosenfield and Duckworth (1987), in a thick specimen, pure mode III loading would cause the crack plane to twist. This observation was also found prevalent in mixed mode I-III loading. In both cases, the crack plane would tend to twist into an orientation which is normal to that of the applied tensile loading. This is because, since the bending stresses across the specimen thickness induced by mode III loading would be in the opposite sense, across the crack-face, varying magnitudes of the mode II stress intensity factor KII0 would develop along the crack-front. These would be greatest at the specimen surfaces and zero at the neutral axis – which, in this case, would be at the mid-section of the specimen. As a result, the crack front would deviate from the self-similar direction, in such a way as to reflect the variation of KII0. Such a re-orientation could give rise to error in analysis, since the loading, crack front and specimen planes would no longer be orthogonal. On the other hand, a reduction in the throat thickness would reduce the longitudinal, tensile and compressive stresses, which would, in turn, mitigate such a re-orientation. Accordingly, a relatively more uniform stress distribution would thus be obtained across the crack front. In addition, a groove would reduce the zone of plastic deformation in front of the crack tip so that brittle fracture would occur more readily. Also, a grooved specimen would have less resistance to fracture, and thus require a lower failure load than for a smooth one. Indeed, it would, in principle, ensure that the specimen would not fail at locations other than the throat. In this connection, a ratio of throat to smooth specimen thickness of 0.5 was, based on finite element analyses, estimated to be sufficient for pure mode I and mixed mode I – III fracture testing. Since 94 the pure mode III fracture toughness was expected to be higher than that of pure mode I, the failure load was expected to be correspondingly higher in pure mode III testing. Accordingly, a thickness ratio of 0.3 was adopted for such testing, based on preliminary numerical analysis. 4.3.2 Laboratory Set-up and Test Procedure The fracture tests were conducted on an INSTRON 1334 servo-hydraulic testing machine. A schematic diagram of the test components is shown in Figure 4.17. The specimen was mounted on a steel loading frame, in two halves, by bolting, and subjected to pure modes I and III, as well as mixed mode I–III, loading. Two purpose-built couplings were fabricated, and connected each half of the loading frame to the INSTRON machine, respectively. This was to prevent the development of moments at the support points, and thus justify a fully-pinned condition at each point, in the analysis. The geometrical configuration of the loading fixture is depicted by Figure 4.18. Differing combinations of modes I and III loading may be achieved by changing the loading angle α. The five sets of holes on the loading frame correspond to five loading orientations, namely α=0°, α=22.5°, α=45°, α=67.5° and α=90°. The first and last cases refer to pure modes III and I loading, respectively, while the rest would be those of mixed mode I-III loading. A piece of rubber pad was placed between the loading frame and specimen to mitigate the development of stress concentration. Figure 4.19 shows the application of mixed mode I-III loading at an angle of α=45°. The main advantage of the test configuration is that the entire mixed mode I-III 95 F α gle n a ing load coupling loading frame rubber pad specimen F Figure 4.17 Schematic diagram of mixed mode I – III test components 96 97 Figure 4.18 Geometrical configuration of partial loading fixture and coupling Figure 4.19 Mixed mode I – III fracture test arrangement (angle of loading of α=45°) 98 spectrum of loading, including pure modes I and III, respectively, may be readily achieved by the same arrangement of the specimen under co-linear tensile loading. Accordingly, the coupling is simply connected to the corresponding hole of the loading frame. Other advantages include the relative ease of specimen preparation, its compactness, and the ease of setting up and loading. The load was applied monotonically at a rate of 0.1mm/min, until failure took place in the specimen. The applied force, and corresponding stroke displacement of the cross-head of the testing machine, were automatically recorded throughout the test. 4.3.3 Determination of Stress Intensity Factors by Finite Element Analysis Three-dimensional finite element models representing the experimental set-up were generated using PATRAN Version 8.5 (The MacNeal-Schwendler Corporation, 1999). Generally, 20-noded, second-order isoparametric, quadratic brick elements were used in the model. Around the crack front, however, quarter-point, triangular prismatic elements were used to simulate the strain and stress singularity, as specified in foregoing §3.1.2. The throat was modelled by 20 layers of elements near the crack front. A typical FE model of a specimen with a ratio of throat-to-smooth-specimen thickness of t/T=0.5 is depicted in Figures (4.20) – (4.23). Each half of the loading fixture and coupling were meshed separately with 20-noded brick elements, to which the loading pins were connected (Figure 4.24). The test specimen was attached to the loading fixture via common nodes with the pins. Five cases of loading orientation, namely α=0°, 22.5°, 45°, 67.5° and 90°, were modelled. Figure 4.25 shows the entire assembly for a loading angle of α=45°, which consisted of 99 100 bottom of groove Figure 4.20 Finite element model of mixed mode I – III fracture test specimen groove throat groove Note: Applying of load is shown in Figure 4.23 101 Figure 4.21 Mid-sectional view of left half of test specimen detailed view in Figure 4.20 102 crack front pre-crack face Figure 4.22 Mid-sectional view showing details at crack front (left half of test specimen) half groove width element at bottom of groove 103 crack front twenty layers of quarter-point elements Figure 4.23 Sectional view of twenty layers of quarter-point triangular prismatic elements around crack front with cut-out view (left half of test specimen) block removed for cut-out view half groove width element at bottom of groove rubber pad bolt (a) Left half of loading fixture bolt (b) Coupling Figure 4.24 Finite element modelling of left half of loading fixture and coupling 104 F α F Figure 4.25 Finite element model of mixed mode I – III loading (α=45°) 105 3608 elements and 18367 nodes. In each loading case, the numerical analyses were carried out using the ABAQUS Version 5.8 (Hibbitt, Karlsson and Sorensen, Inc., 1998) software, and for each layer of elements across the throat, the stress intensity factors KI0, KII0 and KIII0, due to unit loading, were obtained from the nodal displacements at the crack face, according to equations (3.11), (3.12) and (3.15) respectively. The distributions of stress intensity factors along the crack front were then obtained. As shown in Figure 4.26, the distributions of KI0 and KIII0 were symmetric and relatively uniform in all loading cases, except near the surface. On the other hand, the values of KII0 in loading cases to were distributed anti-symmetrically. It should be emphasized that this development of KII0 is not caused by shear loading, but the longitudinal tensile and compressive stresses induced by bending, due to mode III loading, as specified in preceding §4.3.1. Nevertheless, the value of KII0, at mid-section, would be zero. In addition, based on the choice of stress intensity factors at mid-section of the specimen, corresponding experimental results indicated that the effects of KII0 would not be influential on the proposed fracture criterion of equation (2.83). Since it would be difficult to provide for an exact value of t/T ratio for the mortar specimen, the method of K-calibration had to be used to evaluate the stress intensity factors of each specimen. Accordingly, for each of the five above-mentioned loading cases, five values of t/T of specimens were subject to analysis, namely 0.3, 0.4, 0.5, 0.6 and 0.7. In each boundary value problem, the nodal displacements due to unit loading were determined by finite element analysis, and the values of KI0, KII0 and KIII0 then evaluated. A total of 25 different cases were thus analyzed, and the corresponding 106 6.0E-002 9.0E-002 5.0E-002 6.0E-002 KII0 (mm-3/2) KI0 (mm-3/2) 4.0E-002 3.0E-002 2.0E-002 1.0E-002 3.0E-002 0.0E+000 -3.0E-002 -6.0E-002 0.0E+000 -1.0E-002 -9.0E-002 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 2z/t 0.0 0.5 1.0 2z/t 5.0E-002 KIII0 (mm-3/2) 4.0E-002 α = 0° 3.0E-002 α = 22.5° 2.0E-002 α = 45° α = 67.5° 1.0E-002 α = 90° 0.0E+000 -1.0E-002 -1.0 -0.5 0.0 0.5 1.0 2z/t Figure 4.26 Distributions of stress intensity factors across throat of specimen (t/T = 0.5) 107 K-calibration graphs of stress intensity factors, at mid-section of the specimen, versus t/T, plotted as shown in Figure 4.27. 4.3.4 Comparison of Analytical and Experimental Results Twenty-nine cement mortar specimens have been tested, among which eight were subject to pure mode I loading, four to pure mode III loading, and the rest to mixed mode I-III loading. In all specimens, the load increased monotonically with increase of stoke displacement of the test machine, and a sudden failure occurred when it reached its peak value. Fracture was observed to take place along the self-similar direction, that is θC = 0, as shown in Figure 4.28. The modes I and III fracture toughness, KIC and KIIIC, were determined from numerical analysis and laboratory testing under pure mode loading (in which α=90° and α=0, respectively), as K IC = K I0 ⋅ FIC (4.3) K IIIC = K III0 ⋅ FIIIC . (4.4) and The respective stress intensity factors, KI0 and KIII0, were obtained from the K-calibration curves shown in Figure 4.27, according to the actual throat thickness measured on the specimen at fracture, where FIC and FIIIC are the fracture loads recorded 108 109 KI0 (mm-3/2 ) 0.3 0.0E+000 2.0E-002 4.0E-002 6.0E-002 0.4 0.6 0.7 0.3 0.02 0.04 0.06 0.08 0.4 Figure 4.27 KI0 and KIII0 with degree of grooving 0.5 t/T KIII0 (mm-3/2) 0.5 t/T 0.6 α = 90° α = 67.5° α = 45° α = 22.5° α = 0° 0.7 Figure 4.28 Failure surface of mixed mode I–III fracture test specimen in 110 in the corresponding tests. The mean values of KIC and KIIIC, thus determined, were 0.468MPa√m and 1.12MPa√m, respectively. It is thus apparent that the KIC value is very close to that obtained from the four-point bending test reported in foregoing §4.24, that is, 0.479MPa√m. The fracture toughness, in mode III deformation, was thus significantly greater than that in mode I, the ratio of KIIIC/KIC being approximately equal to 2.4. This confirms the observation that cement mortar is more resistant to shear than tensile deformation. A similar relationship has been reported for other brittle materials, like hot-pressed silicon nitride and glass, for which the ratios were found to be 1.5 and 3.59 respectively (Petrovic, 1985). For each of the mixed mode loading cases, KIθ and KIIIθ (where θ=0) were determined as K Iθ = K I0 ⋅ FC (4.5) K IIIθ = K III0 ⋅ FC , (4.6) and where FC was the fracture load measured in the corresponding tests. The tests results are depicted in the plot between the normalized unified pure mode I and III stress intensity factors, KIθ /KIC and KIIIθ /KIIIC, respectively, of Figure 4.29. The figure also shows the unified fracture envelope, specified by equation (2.83), superimposed on the results of 111 the fracture tests. Accordingly, there is agreement - within 10% - between the predicted and experimental results. The experimental and numerical results of each tested specimen are presented in the appendix of §A.2. 112 (KKIθIC)2 + (KKIIIθ ) =1 IIIC 1.0 KIIIθ / KIIIC 0.8 0.6 0.4 0.2 Test results 0.0 0.0 0.2 0.4 0.6 0.8 KIθ / KIC 1.0 Figure 4.29 Comparison of unified fracture criterion with mixed mode I–III fracture test results 113 [...]... self-similar direction, that is θC = 0, as shown in Figure 4.28 The modes I and III fracture toughness, KIC and KIIIC, were determined from numerical analysis and laboratory testing under pure mode loading (in which α=90° and α=0, respectively), as K IC = K I0 ⋅ FIC (4 .3) K IIIC = K III0 ⋅ FIIIC (4.4) and The respective stress intensity factors, KI0 and KIII0, were obtained from the K-calibration curves... KIθ and KIIIθ (where θ=0) were determined as K I = K I0 ⋅ FC (4.5) K III = K III0 ⋅ FC , (4.6) and where FC was the fracture load measured in the corresponding tests The tests results are depicted in the plot between the normalized unified pure mode I and III stress intensity factors, KIθ /KIC and KIIIθ /KIIIC, respectively, of Figure 4.29 The figure also shows the unified fracture envelope, specified... surface of mixed mode I III fracture test specimen in 110 in the corresponding tests The mean values of KIC and KIIIC, thus determined, were 0.468MPa√m and 1.12MPa√m, respectively It is thus apparent that the KIC value is very close to that obtained from the four-point bending test reported in foregoing §4.24, that is, 0.479MPa√m The fracture toughness, in mode III deformation, was thus significantly... and Experimental Results Twenty-nine cement mortar specimens have been tested, among which eight were subject to pure mode I loading, four to pure mode III loading, and the rest to mixed mode I- III loading In all specimens, the load increased monotonically with increase of stoke displacement of the test machine, and a sudden failure occurred when it reached its peak value Fracture was observed to take... that in mode I, the ratio of KIIIC/KIC being approximately equal to 2.4 This confirms the observation that cement mortar is more resistant to shear than tensile deformation A similar relationship has been reported for other brittle materials, like hot-pressed silicon nitride and glass, for which the ratios were found to be 1.5 and 3. 59 respectively (Petrovic, 1985) For each of the mixed mode loading... tensile and compressive stresses induced by bending, due to mode III loading, as specified in preceding §4 .3. 1 Nevertheless, the value of KII0, at mid-section, would be zero In addition, based on the choice of stress intensity factors at mid-section of the specimen, corresponding experimental results indicated that the effects of KII0 would not be influential on the proposed fracture criterion of equation... specified by equation (2. 83) , superimposed on the results of 111 the fracture tests Accordingly, there is agreement - within 10% - between the predicted and experimental results The experimental and numerical results of each tested specimen are presented in the appendix of §A.2 112 K (KIθ )2 + (KIIIθ )2 = 1 KIIIC IC 1.0 KIIIθ / KIIIC 0.8 0.6 0.4 0.2 Test results 0.0 0.0 0.2 0.4 0.6 0.8 KIθ / KIC 1.0 Figure... factors along the crack front were then obtained As shown in Figure 4.26, the distributions of KI0 and KIII0 were symmetric and relatively uniform in all loading cases, except near the surface On the other hand, the values of KII0 in loading cases 1 to 4 were distributed anti-symmetrically It should be emphasized that this development of KII0 is not caused by shear loading, but the longitudinal tensile... numerical analyses were carried out using the ABAQUS Version 5.8 (Hibbitt, Karlsson and Sorensen, Inc., 1998) software, and for each layer of elements across the throat, the stress intensity factors KI0, KII0 and KIII0, due to unit loading, were obtained from the nodal displacements at the crack face, according to equations (3. 11), (3. 12) and (3. 15) respectively The distributions of stress intensity... shown in Figure 4.27, according to the actual throat thickness measured on the specimen at fracture, where FIC and FIIIC are the fracture loads recorded 108 109 KI0 (mm -3/ 2 ) 0 .3 0.0E+000 2.0E-002 4.0E-002 6.0E-002 0.4 0.6 0.7 0 .3 0 0.02 0.04 0.06 0.08 0.4 Figure 4.27 KI0 and KIII0 with degree of grooving 0.5 t/T KIII0 (mm -3/ 2) 0.5 t/T 0.6 α = 90° α = 67.5° α = 45° α = 22.5° α = 0° 0.7 Figure 4.28 Failure . K IIIC , were determined from numerical analysis and laboratory testing under pure mode loading (in which α=90° and α=0, respectively), as CC F II 0I KK ⋅ = (4 .3) and CC F IIIIII 0III KK ⋅ = based on finite element analyses, estimated to be sufficient for pure mode I and mixed mode I – III fracture testing. Since 95 the pure mode III fracture toughness was expected to be higher. concentration. Figure 4.19 shows the application of mixed mode I- III loading at an angle of α=45°. The main advantage of the test configuration is that the entire mixed mode I- III 96 Figure