Mixed mode i II III fracture criterion and its application to cement mortar 4

4.4 Mixed Mode I – II – III Fracture Testing 4.4.1 Geometry of Specimen Unlike other mixed mode fracture tests, mixed mode I–II–III fracture tests are rarely found in the literature. This could be due to the complexity of its specimen geometry and loading configuration. According to the literature review of foregoing §1.1.2.4, three different geometries of test specimen have been designed for fracture under mixed mode I–II–III loading. The experimental set-up proposed by Richard and Kuna (1990) was designed for plexiglas and aluminium specimens. However, the loading device would be too clumsy to handle if it were adapted to cement mortar specimens, since the size of latter would have to be much larger than those of the former. The specimen designed by Arslan et al (1991) had four failure surfaces in different zones. However, it would be difficult to control the test in such a way that all the individual failure surfaces would occur simultaneously, which would be a requirement of the numerical modelling. The axisymmetric bar-type specimen of Hyde and Aksogan (1994) is more suited for metal testing and, therefore, not considered in this application. Furthermore, in spite of the fact that all three stress intensity factors, KI0, KII0 and KIII0, were induced at the specimen crack tip under mixed mode I–II–III loading, none of the abovementioned test set-ups actually achieved a true mixed mode I–II–III fracture, since the specimens were of uniform thickness, whereas, according to the unified model (Lo et al., 1996a), mixed mode loading does not necessarily lead to 114 mixed mode fracture, unless an artifice is provided for weakening the potential fracture plane. In view of the preceding considerations, a novel geometry of the cement mortar specimen was designed for the present study of mixed mode I–II–III fracture, which would be easy to prepare and simple to load. Using the conventional notched beam specimen for conventional mixed mode I-II fracture testing as the basis, a modified beam specimen with an inclined groove ligament was found, from an analytical study, to be optimal in the sense that all the three modes of deformation could be effectively applied at the crack front. Accordingly, instead of having a vertical notch, the adopted specimen would have a groove, which was rotated vertically, as well as horizontally, with respect to the beam section. Figure 4.30 illustrates the final design of the beam specimen. The overall dimensions of the specimen were 500mm (length) × 100mm (depth) × 80mm (width). A 2mm-wide groove which was rotated, both vertically through an angle of α and horizontally through an angle of β, was formed in the specimen, leaving a V-shaped throat segment, so that a true mixed mode I–II–III fracture would be generated. The ligament was located such that the middle of the crack front coincided with the centre of the specimen. Two values were chosen for α and β, namely 26.56° and 45° respectively, which, as indicated by the subsequent numerical analyses under differing load cases, would give rise to various combination of mixed mode deformation at crack front, within the practical limit of preparing the pre-crack plane and groove ligament. Hence, four groups of beam configuration, with differing combinations of α and β values, were prepared for testing, as listed in Table 4.1. 115 80 500 β 100 2mm wide groove α Isometric View V-shape throat segment 50 crack front 50 10mm long pre-crack groove 20 100 Sectional View Note: units in mm where not stated Figure 4.30 Geometry of beam specimen of mixed mode I – II – III fracture test 116 Table 4.1 Angles of inclination of beam groups Beam Group A Beam Group B Beam Group C Beam Group C α 26.65° 45° 45° 26.65° β 26.65° 26.65° 45° 45° It is apparent, from Figure 4.30, that the groove segment is inclined three-dimensionally and has an awkward shape due to the inclination angles α and β. Therefore, it would be difficult to achieve by cutting, after the specimen had been cured for 28 days, unlike the cases of the beam and compact tension specimens, for pure modes I and II, as well as mixed mode I–III, fracture testing, as dealt with in §4.2.1 and §4.3.1 respectively. Hence, the grooved segment would have to be formed during casting of the specimen. Accordingly, three overlapping stainless steel plates, of 2mm thickness overall, and coated with mould oil, were fixed to the mould before casting. In order to form the pre-crack face, the centre plate had a deeper embedment, by 10mm, than the other two, which were geometrically identical (Figure 4.31). To prevent the plates from getting stuck in the specimen, due to the drying shrinkage of the cement mortar, the centre-piece was removed within three hours after casting, while the remaining two pieces were left to prevent the cement mortar from caving in, and removed only during de-moulding. Figure 4.32 shows the design of the mould used to cast the test specimen. 117 β three stainless steel plates length of pre-crack face cement mortat specimen Figure 4.31 Formation of groove and pre-crack by three steel plates 118 (a) (b) Mould of test specimen Steel plates used to form groove in specimen Figure 4.32 Mould of mixed mode I−II−III fracture test specimen 119 4.4.2 Laboratory Set-up and Test Procedure The fracture tests were conducted on an INSTRON 1334 servo-hydraulic testing machine. In order to achieve differing mode I/mode II and mode I/mode III loading ratios, each beam group (according to §4.4.1) was subjected to various loading cases, as depicted in Figure 4.33. In all cases, the specimen was simply-supported. For cases and 2, the load was applied to the steel I-beam, which distributed the load to the specimen at two points, in such a way that pure bending and shear would be obtained at mid-section, under four-point bending and shear, respectively. On the other hand, in cases and 4, the load was applied directly to the specimen, and a combination of tensile and shear stresses would, thereby, be produced. The load was applied monotonically at a rate of 0.1mm/min, until the specimen failed. The force applied, and corresponding stroke displacement of the cross-head of the testing machine, were recorded automatically throughout the test. 4.4.3 Determination of Stress Intensity Factors by Finite Element Analysis Four three-dimensional finite element models, representing the respective beam groups, 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, quarter-point triangular prismatic elements were used to simulate the strain and stress singularity, as specified in foregoing §3.1.2. The crack front was modelled by twenty layers of elements. Figure 4.34 shows the overall assembly for beam group A, which consists of 5944 elements 120 Steel I-beam F Case (Beam Groups A and D) 30 110 110 110 110 30 500 Steel I-beam F Case (Beam Groups B and C) 30 110 110 110 110 30 500 F Case (Beam Groups B, C and D) 30 110 30 330 500 F Case (Beam Groups A, B and C) 30 110 220 140 500 Note: dimensions are in mm. Figure 4.33 Loading cases used in mixed mode I–II–III fracture tests 121 122 F groove V-shaped throat segment F detailed view in Figure 4.33 Figure 4.34 Finite element model for beam group A of mixed mode I – II – III fracture test F F and 27019 nodes, while Figures 4.35 – 4.36 illustrate the details of the modelling near the crack front. Next, numerical analyses were carried out by ABAQUS Version 5.8 (Hibbitt, Karlsson and Sorensen, Inc., 1998), and the stress intensity factors, KI0, KII0 and KIII0, for each layer of elements across the throat, and in each case of unit loading, were obtained from corresponding nodal displacement at the crack face, according to equations (3.11), (3.12) and (3.15). The distributions of stress intensity factors along the crack front were then obtained. Figure 4.37 shows the distributions of stress intensity factors of each beam group. 4.4.4 Comparison of Analytical and Experimental Results For each of the beam groups, three specimens were tested under each category of loading case. Accordingly, thirty beam specimens were tested in all. In every case, the load was found to rise with stroke displacement of the cross-head of the test machine, initially. Crack extension started when the load reached its critical value of FC. As illustrated in Figure 4.38, the value of FC under four-point shear loading was significantly higher than those under three- and four-point bending. In the two cases of bending, the load decreased gradually after FC, until failure occurred in the specimen. This would imply that additional energy was required to maintain crack extension. In contrast, in the cases of four-point shear, the peak load dropped suddenly, as the specimen underwent sudden failure. The cracks were found to extend along the groove initially, but after a certain 123 124 half groove width element Figure 4.35 Sectional view showing details of crack front (right half of test specimen) pre-crack face crack front 125 block removed for cut-off view Figure 4.36 Sectional view of twenty layers of quarter-point triangular prismatic elements around crack front with cut-out view (right half of taest specimen) crack front twenty layers of quarter-point elements ) -3/2 Load Case 1. 12.0 Note: For load cases, refer to Figure 4.31. -3 Stress intensity factor (×10 mm 15.0 9.0 6.0 3.0 0.0 KI0 -3.0 -1.0 -0.5 0.0 0.5 KII0 1.0 2z/t Load Case 4. 12.0 -3 Stress intensity factor (×10 mm -3/2 ) KIII0 15.0 9.0 x throat segment 6.0 crack front 3.0 0.0 t z -3.0 -1.0 -0.5 0.0 0.5 1.0 2z/t (a) beam group A 126 ) -3/2 ) 4.0 Load Case 3. 2.0 -3 0.8 0.6 0.4 0.2 0.0 -1.0 -0.5 0.0 0.5 1.0 0.0 -2.0 -4.0 -6.0 -8.0 -10.0 -1.0 -0.5 2z/t Load Case 4. KI0 4.0 2.0 0.0 2z/t 0.5 1.0 KII0 -3 Stress intensity factor (×10 mm -3/2 Load Case 2. Stress intensity factor (×10 mm ) -3/2 -3 Stress intensity factor (×10 mm 1.0 0.0 KIII0 -2.0 -4.0 -6.0 throat segment x -8.0 crack front z -10.0 -1.0 -0.5 0.0 0.5 1.0 2z/t t (b) beam group B 127 ) -3/2 Load Case 2. Stress intensity factor (×10 mm 1.0 0.8 0.6 0.4 0.2 -1.0 10.0 8.0 Load Case 3. -3 ) -3/2 -3 Stress intensity factor (×10 mm 1.2 -0.5 0.0 0.5 1.0 6.0 4.0 2.0 0.0 -2.0 -4.0 -1.0 -0.5 -3 Stress intensity factor (×10 mm -3/2 ) 2z/t 0.0 0.5 1.0 2z/t KI0 6.0 Load Case 4. KII0 4.0 KIII0 2.0 throat segment x 0.0 crack front -2.0 -1.0 -0.5 0.0 0.5 1.0 t z 2z/t (c) beam group C 128 ) -3/2 -3 Stress intensity factor (×10 mm 16.0 Load Case 1. 12.0 8.0 4.0 0.0 KI0 -4.0 -1.0 -0.5 0.0 0.5 1.0 KII0 -3 Stress intensity factor (×10 mm -3/2 ) 2z/t KIII0 12.0 Load Case 3. 8.0 throat segment x 4.0 crack front z 0.0 t -4.0 -1.0 -0.5 0.0 0.5 1.0 2z/t (d) beam group D Figure 4.37 Distributions of stress intensity factors across crack fronts of various beam groups 129 14 Fcs 12 shear Load (kN) 10 4-point-bending 0.0 FCB4 0.2 3-point-bending FCB3 0.4 0.6 0.8 Stroke displacement (mm) Figure 4.38 Typical load-stroke displacement curves for bending and shear loadings 130 stage, deviated from the throat segment to extend vertically upwards (Figure 4.39). The reason for the deviation was that, as the crack approached the top face of the specimen, the length of the crack front increased to the extent that the effect of grooving was insufficient to guide the crack to extend along the throat segment, with the result that the crack extended along the most critical direction, which was upwards. The fracture toughness in pure deformation modes I, II and III have been discussed in foregoing §4.2 and §4.3, as being 0.468MPa√m, 0.759MPa√m and 1.12MPa√m, respectively. For each of the mixed mode fracture cases, on the other hand, KIθ, KIIθ and KIIIθ (where θ = 0) were evaluated as K Iθ = K I0 ⋅ FC , (4.7) K IIθ = K II0 ⋅ FC (4.8) K IIIθ = K III0 ⋅ FC , (4.9) and where KI0, KII0 and KIII0 are the stress intensity factors, due to unit loading, obtained according to the numerical analysis of the foregoing §4.4.3, and FC the critical load measured in corresponding laboratory tests, as listed in the appendix of §A.3. The test results have been plotted in Figure 4.40, upon which the unified fracture envelope of preceding equation (2.83), as well as tests results of pure mode fracture tests, have been superimposed. Accordingly, there is reasonably good agreement between the unified prediction and experimental results, that is, to within 10%. 131 (a) specimen of beam group A (b) specimen of beam group B 132 (c) specimen of beam group C (d) specimen of beam group D Figure 4.39 Failure of mixed mode I – II – III fracture test specimens 133 IIθ ) + ( KKIIIθ )=1 ( KKICIθ )2+ ( KKIIC IIIIC √(K II θ / KIIC)2 + (KIIIθ / KIIIC) 1.0 0.8 0.6 0.4 0.2 Test results 0.0 0.0 0.2 0.4 0.6 0.8 1.0 KI θ / KIC Figure 4.40 Comparison of fracture criterion with mixed mode I – II – III fracture test results 134 [...]... critical direction, which was upwards The fracture toughness in pure deformation modes I, II and III have been discussed in foregoing 4. 2 and 4. 3, as being 0 .46 8MPa√m, 0.759MPa√m and 1.12MPa√m, respectively For each of the mixed mode fracture cases, on the other hand, KIθ, KIIθ and KIIIθ (where θ = 0) were evaluated as K I = K I0 ⋅ FC , (4. 7) K II = K II0 ⋅ FC (4. 8) K III = K III0 ⋅ FC , (4. 9) and. .. mixed mode I – II – III fracture test specimens 133 KIIθ KIθ KIIIθ ( KIC )2+ ( KIIC ) + ( KIIIIC )=1 2 2 √(K II θ / KIIC)2 + (KIIIθ / KIIIC) 2 1.0 0.8 0.6 0 .4 0.2 Test results 0.0 0.0 0.2 0 .4 0.6 0.8 1.0 KI θ / KIC Figure 4. 40 Comparison of fracture criterion with mixed mode I – II – III fracture test results 1 34 ... mode fracture tests, have been superimposed Accordingly, there is reasonably good agreement between the unified prediction and experimental results, that is, to within 10% 131 (a) specimen of beam group A (b) specimen of beam group B 132 (c) specimen of beam group C (d) specimen of beam group D Figure 4. 39 Failure of mixed mode I – II – III fracture test specimens 133 KIIθ KIθ KIIIθ ( KIC )2+ ( KIIC... FC , (4. 9) and where KI0, KII0 and KIII0 are the stress intensity factors, due to unit loading, obtained according to the numerical analysis of the foregoing 4. 4.3, and FC the critical load measured in corresponding laboratory tests, as listed in the appendix of §A.3 The test results have been plotted in Figure 4. 40, upon which the unified fracture envelope of preceding equation (2.83), as well as... intensity factor (×10 mm -3/2 ) 2z/t KIII0 12.0 Load Case 3 8.0 throat segment x 4. 0 crack front z 0.0 t -4. 0 -1.0 -0.5 0.0 0.5 1.0 2z/t (d) beam group D Figure 4. 37 Distributions of stress intensity factors across crack fronts of various beam groups 129 14 Fcs 12 shear Load (kN) 10 8 4- point-bending 6 4 2 0 0.0 FCB4 0.2 3-point-bending FCB3 0 .4 0.6 0.8 Stroke displacement (mm) Figure 4. 38 Typical load-stroke... Stress intensity factor (×10 mm 1.2 -0.5 0.0 0.5 1.0 6.0 4. 0 2.0 0.0 -2.0 -4. 0 -1.0 -0.5 -3 Stress intensity factor (×10 mm -3/2 ) 2z/t 0.0 0.5 1.0 2z/t KI0 6.0 Load Case 4 KII0 4. 0 KIII0 2.0 throat segment x 0.0 crack front -2.0 -1.0 -0.5 0.0 0.5 1.0 t z 2z/t (c) beam group C 128 ) -3/2 -3 Stress intensity factor (×10 mm 16.0 Load Case 1 12.0 8.0 4. 0 0.0 KI0 -4. 0 -1.0 -0.5 0.0 0.5 1.0 KII0 -3 Stress intensity... to Figure 4. 31 -3 Stress intensity factor (×10 mm 15.0 9.0 6.0 3.0 0.0 KI0 -3.0 -1.0 -0.5 0.0 0.5 KII0 1.0 2z/t Load Case 4 12.0 -3 Stress intensity factor (×10 mm -3/2 ) KIII0 15.0 9.0 x throat segment 6.0 crack front 3.0 0.0 t z -3.0 -1.0 -0.5 0.0 0.5 1.0 2z/t (a) beam group A 126 ) -3/2 Load Case 2 Stress intensity factor (×10 mm 0.8 0.6 0 .4 0.2 0.0 -1.0 4. 0 Load Case 3 2.0 -3 ) -3/2 -3 Stress intensity...1 24 half groove width element Figure 4. 35 Sectional view showing details of crack front (right half of test specimen) pre-crack face crack front 125 block removed for cut-off view Figure 4. 36 Sectional view of twenty layers of quarter-point triangular prismatic elements around crack front with cut-out view (right half of taest specimen) crack front twenty layers of quarter-point elements... load-stroke displacement curves for bending and shear loadings 130 stage, deviated from the throat segment to extend vertically upwards (Figure 4. 39) The reason for the deviation was that, as the crack approached the top face of the specimen, the length of the crack front increased to the extent that the effect of grooving was insufficient to guide the crack to extend along the throat segment, with the... ) -3/2 -3 Stress intensity factor (×10 mm 1.0 -0.5 0.0 0.5 1.0 0.0 -2.0 -4. 0 -6.0 -8.0 -10.0 -1.0 -0.5 0.0 Load Case 4 1.0 KI0 4. 0 2.0 0.5 2z/t KII0 -3 Stress intensity factor (×10 mm -3/2 ) 2z/t 0.0 KIII0 -2.0 -4. 0 -6.0 throat segment x -8.0 crack front z -10.0 -1.0 -0.5 0.0 0.5 1.0 2z/t t (b) beam group B 127 ) -3/2 Load Case 2 Stress intensity factor (×10 mm 1.0 0.8 0.6 0 .4 0.2 -1.0 10.0 8.0 Load . 1 14 4 .4 Mixed Mode I – II – III Fracture Testing 4. 4.1 Geometry of Specimen Unlike other mixed mode fracture tests, mixed mode I II III fracture tests are rarely found in the literature. This. C F ⋅ = III0 IIIθ KK , (4. 9) where K I0 , K II0 and K III0 are the stress intensity factors, due to unit loading, obtained according to the numerical analysis of the foregoing 4. 4.3, and F C . an INSTRON 13 34 servo-hydraulic testing machine. In order to achieve differing mode I /mode II and mode I /mode III loading ratios, each beam group (according to 4. 4.1) was subjected to various