Mixed mode i II III fracture criterion and its application to cement mortar 2

CHAPTER EXPERIMENTAL VERIFICATION OF FRACTURE CRITERION In this chapter, the results of three-dimensional mixed mode fracture tests, on cement mortar specimens, will be reported. A finite element model will be generated to idealize the laboratory set-up, and the respective pure mode stress intensity factors determined. As a result of the analyses and laboratory measurements, the proposed three-dimensional fracture criterion of §2.4, in Chapter 2, will be confirmed. In the above connection, the choice of cement mortar as the test material will first be justified. This will be followed by an introduction to the test set-up. The fracture tests on cement mortar specimens will then follow. 4.1 4.1.1 Test Material and Set-up Test Material Based on their constituents, cementitious materials may be classified as paste, mortar, or concrete. Paste is a mixture of cement and water. Mortar is a mixture of fine aggregate (usually sand), cement, and water. Concrete is a composite consisting of cement, fine aggregate, coarse aggregate, and water. The properties of these materials are influenced by the chemical composition of the constituents, and their micro- and macro-structures, which are characterized by the amount and distribution of internal 68 pores and cracks. The hierarchy of fracture processes in cement-based material is as presented in Figure 4.1 (Shah et al, 1995). Since, as shown in Figure 4.1(a), the microstructure of cement paste is on a scale of nanometres, fracture processes in cement paste would be influenced by particles and voids on the same scale. It was found that cement mortar is a highly brittle material (Li and Maalej, 1996) and can be adequately analyzed by the concept of LEFM (Mai, 1984). The internal structure of mortar is shown in Figure 4.1(b). The use of sands or fine aggregates results in voids in mortar of up to the scale of a micrometre. As a result, fracture processes in mortars would, also, primarily involve extension of internal voids on the scale of a micrometre. From this point of view, it would be valid to consider cement mortar as an isotropic and homogeneous material. With coarse aggregates, concrete is physically a two-phase material consisting of the mortar matrix and coarse aggregate. They are bonded together at the interface. Interfacial cracks and weak interfacial zones (Shah and McGarry, 1971; Jenq and Shah, 1985b) between matrix and aggregate, on a scale of a millimetre, are major defects of concrete, as suggested by Figure 4.1(c). Therefore, concrete is an anisotropic and heterogeneous material, and fracture processes in concrete may depend primarily on the stability of interfacial cracks and weak interfacial zones. According to LEFM, the stress at a crack tip would approach infinity. However, infinite stress cannot develop in a real material, hence, a certain range of inelastic zone must exist in the vicinity of the crack tip. Since concrete is a heterogeneous material consisting of different phases, the inelastic zone, which is dominated by complicated mechanisms, is termed the fracture process zone. By using 69 10µm (a) cement paste 50µm (b) mortar 50mm (c) concrete Figure 4.1 Hierarchy of fracture process in cement-based materials 70 a scanning electron microscope, Mindess and Diamond (1982) reported that the crack surface in concrete is tortuous, and the crack process zone is complicated. The fracture behaviour of concrete would, therefore, be greatly influenced by the presence of the fracture process zone. Experimental results have indicated that the strength of concrete usually decreases with increasing size of structures, and then remains constant. This is known as size effect of concrete, which may be primarily explained by the fracture process zone (Bazant et al., 1991). It follows that when a concrete structure is loaded, some of the strain energy produced by the applied load is converted to the energy consumed to create new fracture surfaces and the energy absorbed in the fracture process zone. For a large-sized structure, the latter is negligible compared to the former, whereas for a small-sized structure, these can be comparable. Therefore, the larger the structural size, the lower the nominal strength. However, the concrete strength approaches a constant when the size of the concrete structure becomes sufficiently large. It is apparent from the preceding discussion that the presence of fracture process zone generally deters the direct application of LEFM to concrete. One needs to use nonlinear fracture mechanics to simulate the mechanism and process of the fracture process zone, or alternatively, to adopt a relatively large specimen to minimize the size effect, so that LEFM would be applicable. In the cement mortar matrix, on the contrary, since the size of fine aggregate is much smaller compared to the coarse aggregate in concrete, the effect of the interfacial crack and non planar crack propagation in the fracture process zone is insignificant. The typical load-displacement data of the fracture tests, conducted on various cement mortar specimens with fine aggregate size less than 1-2mm, showed that a linear elastic relationship would be 71 predominant, and the short non-linear region just before the maximum load was negligible, such that LEFM would be reasonably applicable to study their fracture behaviour (Nallathambi et al., 1984; Dasgupta et al., 1998). Therefore, to start with the investigation of the fracture of concrete, the matrix (that is, the cement mortar) has been chosen as the test material for the present study. This would require a relatively small-sized specimen. Ordinary Portland cement and natural fine-graded river sand, with a specific grading of between No.100 (150µm) and No.16 (1.18mm) sieves, as defined in ASTM E11-01 (American Society of Testing and Materials, 2001), were used to cast the cement mortar specimens. The purpose of using fine-graded sand was to effectively eliminate the effect of aggregate size and reduce the size of fracture process zone, so that LEFM could reasonably be applied (Mai, 1984). The mix proportions of cement : water : sand was 1.0 : 0.31 : 0.8. The materials were mixed in a drum mixer for a period of not less than 10 minutes to ensure the uniformity of the specimen. Stainless steel moulds were used to prepare the specimens, which were cured in the fog room for 28 days. The procedures for mixing and testing of samples were in accordance to BS 1881: Parts 125 (British Standards Institution, 1986), 116 (British Standards Institution, 1983a) and 121 (British Standards Institution, 1983b), respectively. The mean compressive cube strength, Young’s modulus and Poisson’s ratio were measured as 68MPa, 28GPa and 0.2, respectively. The typical compressive load – axial strain curves for evaluating Young’s modulus and compressive load - axial and transversal strain curves for Poisson’s ratio are shown in Figures 4.2 and 4.3, respectively. 72 160 Compressive load (kN) 120 80 40 0 0.02 0.04 -6 Axial strain (10 ) 0.06 0.08 Figure 4.2 Typical compressive load - axial strain curves for evaluation of Young’s modulus 73 160 Transversal strain Compressive load (kN) 120 80 Axial strain 40 0 200 400 -6 Microstrain (10 ) 600 800 Figure 4.3 Typical compressive load - axial and transversal strain curves for evaluation of Poisson’s ratio 74 4.1.2 Test Set-up In the following fracture testing, the specimens would have a sharp pre-crack and would be subject to unstable crack propagation, where the load would drop suddenly once the crack started to propagate. Therefore, the open loop test would be applied. The laboratory tests were conducted on an INSTRON 1334 servo-hydraulic testing machine, with a maximum 500kN load capacity, and 75mm stroke displacement of the cross-head (Figure 4.4). The loading rate was maintained at 0.1mm/min, so that inertial effects would be insignificant. The force applied, and corresponding stroke displacement of the cross-head of the testing machine, were recorded automatically throughout the test. In the following discussion, various fracture tests will be found to require the application of a groove in the specimen, so that proposed pure or mixed mode fracture may be guided along the desired direction. The cutting of the groove was carried out by a Norton Clipper Model ECW “Major” masonry bench saw (Figure 4.5). The maximum spindle speed of the rotating blade is 2700 rounds per minute. It is designed for cutting operations of a wide range of masonry, refractory and natural stone products. 75 Figure 4.4 INSTRON 1334 servo-hydraulic testing machine 76 Figure 4.5 Norton Clipper Model ECW “Major” masonry bench saw 77 4.2 Pure Modes I and II Fracture Testing Before verification of the proposed mixed mode I–II–III fracture criterion, given by foregoing equation (2.80) of §2.4, it would be necessary to evaluate the pure mode I, II and III fracture toughness, KIC, KIIC and KIIIC, respectively, of the test material. The evaluation of KIC and KIIC will be dealt with in the following discussion, while that of KIIIC will be addressed in subsequent §4.3. 4.2.1 Geometry of Specimen Beam specimens with centre-notch or pre-crack have been widely used to study the fracture behaviour of cementitious material (Bazant and Pfeiffer, 1985; RILEM, 1985; Li and Ward, 1988; Swartz et al., 1988b; Carpinteri and Swartz, 1991). Accordingly, it has been adopted in the present study, for pure modes I and II fracture testing. As shown in Figure 4.6, the overall dimensions of the beam specimen are 500mm (length) × 100mm (depth) × 80mm (width). The length of the pre-crack was chosen to be a0 = 35mm. The pre-crack was formed by a masonry bench saw, after the specimen had been cured in the fog room for 28 days. In view of the criterion for pure mode II fracture by the unified model (§1.1.3), an additional pair of side grooves was cut from the pre-crack of the test specimen to its opposite end, along both faces of the specimen and in the direction of the pre-crack, in order to provide a throat segment of sufficient narrowness to activate and guide the crack extension along the θ0C plane. The thickness of the throat segment was chosen to be t = 15mm. 78 pre-crack thickness=1 D=100 a =35 crack front W =80 L=500 Section 1-1 (a) Test specimen for pure mode I fracture testing pre-crack thickness=1 a =35 throat groove thickness=1 D=100 t=15 crack front pre-crack face W =80 L=500 Section 2-2 (b) Test specimen for pure mode II fracture testing Figure 4.6 Geometry of beam specimens for pure modes I and II fracture tests 79 4.2.2 Laboratory Set-up and Test Procedure The mode I fracture toughness, KIC, was determined by the four-point bending test (Hashimoto, 1982; Dong, 1984; Li and Ward, 1988) for beam specimens, as shown in Figures 4.7 and 4.8. Each specimen was loaded symmetrically in the manner shown, so that only tensile stresses would be induced at mid-span, which would correspond to pure mode I loading. On the other hand, the mode II fracture toughness, KIIC, was determined by the four-point shear test, as shown in Figures 4.9 and 4.10. In order to obtain pure mode II loading at the crack tip, the beam specimen was loaded asymmetrically, so as only to give rise to shear stresses at the mid-span of the beam. In both cases, the beam specimen was simply-supported. The load was applied via a load cell of 50kN capacity to a steel I-beam, the latter then transferring the load to the specimen by means of two rollers. The loading rate was maintained at 0.1mm/min until the specimen failed, so that inertial effects would be insignificant. 4.2.3 Determination of Stress Intensity Factors by Finite Element Analysis The pure mode I fracture of a beam specimen may be modelled as a two-dimensional problem. Therefore, a two-dimensional plane strain finite element model was generated using PATRAN Version 8.5 (The MacNeal-Schwendler Corporation, 1999). Generally, eight-noded, isoparametric, quadratic quadrilateral elements were used in the model. Around the crack tip, however, eight collapsed quarter-point elements were used to simulate the singularity at the crack tip, as 80 P I-beam 0.5P 0.5P specimen A B crack 0.5P 30 110 0.5P 110 110 110 30 500 Shear force diagram 0.5P 0.5P Bending moment diagram 55P Note: dimensions are in mm. Figure 4.7 Schematic diagram of pure mode I fracture test 81 Figure 4.8 Experimental set-up of pure mode I fracture test 82 P I-beam 0.33P 0.67P specimen A B crack groove 0.33P 30 110 0.67P 110 110 110 30 500 Shear force diagram 0.33P 0.33P 0.33P Bending moment diagram 36.3P 36.3P Note: dimensions are in mm. Figure 4.9 Schematic diagram of pure mode II fracture test 83 Figure 4.10 Experimental set-up of pure mode II fracture test 84 outlined in §3.1.1. As indicated by Figure 4.6(b), for pure mode II fracture testing, due to the presence of the side grooves, the problem is, in principle, one of three dimensions. However, it has been found in similar cases that an equivalent two-dimensional plane strain mesh, based on pro-rating the Young’s modulus to reflect the local thickness of the specimen, would provide a satisfactory representation (Tamilselvan, 1998). Therefore, a two-dimensional mesh was similarly generated in the case of the pure mode I fracture test specimen, except that the pro-rated Young’s modulus was adopted for those elements lying in the grooved area. Figure 4.11 shows the FE mesh, consisting of 262 elements and 867 nodes. The numerical analyses were carried out by ABAQUS Version 5.8 (Hibbitt, Karlsson and Sorensen, Inc., 1998) and the stress intensity factors were obtained from the nodal displacements of the nodes around the crack tip by equations (3.11) and (3.12), as outlined in §3.1. Since it would be difficult to preset a0 /D and t/W ratios exactly 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 pure modes I and II fracture testing meshes, three values of a0 /D, namely 0.3, 0.35 and 0.4, and three values of t/W for pure mode II fracture testing mesh, namely 0.125, 0.25 and 0.375, were so analyzed. As a result, three and nine different cases for pure modes I and II loading were analyzed, respectively. The variations of the stress intensity factors with a0 /D and t/W, plotted from these results, are shown in Figure 4.12. 85 groove (for pure mode II test only) view (b) extent of groove (a) FE model of specimen width of groove (for pure mode II test only) crack tip eight quarter-point elements around crack tip pre-crack (b) Detailed view around crack tip Figure 4.11 Finite element model of beam specimen 86 KI0 (×10-3 mm-3/2 ) 5.5 4.5 0.3 0.325 0.35 a0 /D 0.375 0.4 (a) Mode I stress intensity factor KI0 KII0 (×10-3 mm-3/2 ) t/W = 0.125 1.6 t/W = 0.25 1.2 t/W = 0.375 0.8 0.4 0.3 0.325 0.35 a0 /D 0.375 0.4 (b) Mode II stress intensity factor KII0 Figure 4.12 K-calibrations of stress intensity factors 87 4.2.4 Pure Modes I and II Fracture Toughness Six cement mortar beam specimens were tested, among which three were subject to pure mode I loading, and the others to pure mode II loading. In all specimens, the load was found to rise with stroke displacement of the cross-head of the testing machine, reaching its maximum value when failure occurred (Figure 4.13). Fracture took place along the self-similar direction, that is θC = 0, as shown in Figure 4.14. The modes I and II fracture toughness, KIC and KIIC, were determined from corresponding numerical analysis and laboratory testing. For pure mode I loading, K IC = K I0 ⋅ FIC , (4.1) K IIC = K II0 ⋅ FIIC , (4.2) while for pure mode II loading, where KI0 and KII0 are the respective stress intensity factors obtained from K-calibration curves shown in Figure 4.12, based on the actual a0 /D and t/W ratios measured on the specimen at fracture, and the fracture loads, FIC and FIIC, are measured in the corresponding tests (refer to the appendix of §A.1). The mean values of KIC and KIIC, thus determined, were 0.479MPa√m and 0.759MPa√m, respectively. The fracture toughness in the mode II of deformation was thus greater than that in mode I, the ratio of KIIC/KIC being approximately equal to 1.58. 88 18 Pure Mode II Fracture 16 FIIC 14 Load (kN) 12 10 FIC 0.0 Pure Mode I Fracture 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Stroke displacement (mm) 0.8 0.9 1.0 Figure 4.13 Typical load-stroke displacement curves for pure mode I and II fracture 89 (a) Mode I fracture test specimen (b) Mode II fracture test specimen Figure 4.14 Failure of modes I and II fracture test specimens 90 [...]...4 .2 Pure Modes I and II Fracture Testing Before verification of the proposed mixed mode I II III fracture criterion, given by foregoing equation (2. 80) of 2. 4, it would be necessary to evaluate the pure mode I, II and III fracture toughness, KIC, KIIC and KIIIC, respectively, of the test material The evaluation of KIC and KIIC will be dealt with in the following discussion, while that of KIIIC will... testing machine, reaching its maximum value when failure occurred (Figure 4.13) Fracture took place along the self-similar direction, that is θC = 0, as shown in Figure 4.14 The modes I and II fracture toughness, KIC and KIIC, were determined from corresponding numerical analysis and laboratory testing For pure mode I loading, K IC = K I0 ⋅ FIC , (4.1) K IIC = K II0 ⋅ FIIC , (4 .2) while for pure mode II. .. 0.8 0.4 0.3 0. 325 0.35 a0 /D 0.375 0.4 (b) Mode II stress intensity factor KII0 Figure 4. 12 K-calibrations of stress intensity factors 87 4 .2. 4 Pure Modes I and II Fracture Toughness Six cement mortar beam specimens were tested, among which three were subject to pure mode I loading, and the others to pure mode II loading In all specimens, the load was found to rise with stroke displacement of the cross-head... respectively The fracture toughness in the mode II of deformation was thus greater than that in mode I, the ratio of KIIC/KIC being approximately equal to 1.58 88 18 Pure Mode II Fracture 16 FIIC 14 Load (kN) 12 10 8 6 FIC 4 2 0 0.0 Pure Mode I Fracture 0.1 0 .2 0.3 0.4 0.5 0.6 0.7 Stroke displacement (mm) 0.8 0.9 1.0 Figure 4.13 Typical load-stroke displacement curves for pure mode I and II fracture. .. pure mode II loading, where KI0 and KII0 are the respective stress intensity factors obtained from K-calibration curves shown in Figure 4. 12, based on the actual a0 /D and t/W ratios measured on the specimen at fracture, and the fracture loads, FIC and FIIC, are measured in the corresponding tests (refer to the appendix of §A.1) The mean values of KIC and KIIC, thus determined, were 0.479MPa√m and 0.759MPa√m,... testing pre-crack thickness=1 a =35 throat groove thickness=1 D=100 t=15 2 crack front pre-crack face 2 W =80 L=500 Section 2- 2 (b) Test specimen for pure mode II fracture testing Figure 4.6 Geometry of beam specimens for pure modes I and II fracture tests 79 4 .2. 2 Laboratory Set-up and Test Procedure The mode I fracture toughness, KIC, was determined by the four-point bending test (Hashimoto, 19 82; ... Li and Ward, 1988) for beam specimens, as shown in Figures 4.7 and 4.8 Each specimen was loaded symmetrically in the manner shown, so that only tensile stresses would be induced at mid-span, which would correspond to pure mode I loading On the other hand, the mode II fracture toughness, KIIC, was determined by the four-point shear test, as shown in Figures 4.9 and 4.10 In order to obtain pure mode II. .. 0.3, 0.35 and 0.4, and three values of t/W for pure mode II fracture testing mesh, namely 0. 125 , 0 .25 and 0.375, were so analyzed As a result, three and nine different cases for pure modes I and II loading were analyzed, respectively The variations of the stress intensity factors with a0 /D and t/W, plotted from these results, are shown in Figure 4. 12 85 groove (for pure mode II test only) view (b) extent... intensity factors were obtained from the nodal displacements of the nodes around the crack tip by equations (3.11) and (3. 12) , as outlined in §3.1 Since it would be difficult to preset a0 /D and t/W ratios exactly 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 pure modes I and II fracture testing meshes,... groove (a) FE model of specimen width of groove (for pure mode II test only) crack tip eight quarter-point elements around crack tip pre-crack (b) Detailed view around crack tip Figure 4.11 Finite element model of beam specimen 86 KI0 (×10-3 mm-3 /2 ) 6 5.5 5 4.5 4 0.3 0. 325 0.35 a0 /D 0.375 0.4 (a) Mode I stress intensity factor KI0 2 KII0 (×10-3 mm-3 /2 ) t/W = 0. 125 1.6 t/W = 0 .25 1 .2 t/W = 0.375 . criterion, given by foregoing equation (2. 80) of 2. 4, it would be necessary to evaluate the pure mode I, II and III fracture toughness, K IC , K IIC and K IIIC , respectively, of the test material 77 Figure 4.5 Norton Clipper Model ECW “Major” masonry bench saw 78 4 .2 Pure Modes I and II Fracture Testing Before verification of the proposed mixed mode I II III fracture criterion, . evaluation of K IC and K IIC will be dealt with in the following discussion, while that of K IIIC will be addressed in subsequent §4.3. 4 .2. 1 Geometry of Specimen Beam specimens with centre-notch