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.. Ac

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

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

Figure 4.30 Geometry of beam specimen of mixed mode I – II – III

fracture test

Table 4.1 Angles of inclination of beam groups

§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

cement mortat specimen

three stainless steel platesβ

Figure 4.31 Formation of groove and pre-crack by three steel plates

(a) Mould of test specimen

Figure 4.32 Mould of mixed mode I−II−III fracture test specimen

(b) Steel plates used to form groove in specimen

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 1 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 3 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

500 220

330 500

30 110

30

F 110 F

140 30

F

F

110 500

Note: dimensions are in mm

(Beam Groups B and C)

(Beam Groups B, C and D)

(Beam Groups A, B and C)

(Beam Groups A and D)

detailed view in Figure 4.33

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

F C 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

crack front

Figure 4.36

Figure 4.37 Distributions of stress intensity factors across crack fronts of

various beam groups

Figure 4.38 Typical load-stroke displacement curves for bending and shear

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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

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

(b) specimen of beam group B (a) specimen of beam group A

(d) specimen of beam group D Figure 4.39 Failure of mixed mode I – II – III fracture test specimens

(c) specimen of beam group C

Figure 4.40 Comparison of fracture criterion with mixed mode I – II – III

fracture test results

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