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

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 3 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

Figure 4.16 Specimen mould for mixed mode I – III fracture test

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

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

Figure 4.17 Schematic diagram of mixed mode I – III test components

loading framecoupling

Figure 4.18

Figure 4.19 Mixed mode I – III fracture test arrangement

(angle of loading of α=45°)

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

groove

Note: Applying of load is shown in Figure 4.23

Figure 4.21

detailed view in Figure 4.20

Mid-sectional view sh

Figure 4.23

rubber pad

(a) Left half of loading fixture bolt

bolt

Figure 4.25 Finite element model of mixed mode I – III loading (α=45°)

α

F

F

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 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 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

Figure 4.26 Distributions of stress intensity factors across throat of specimen (t/T = 0.5)

-6.0E-002 -3.0E-002 0.0E+000 3.0E-002 6.0E-002 9.0E-002

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

Figure 4.28 Failure surface of mixed mode I–III fracture test specimen

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

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

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

Figure 4.29 Comparison of unified fracture criterion with mixed mode

I–III fracture test results

KIθ / KIC0.0

0.20.40.60.81.0

KIIIC)2

= 1