4.2 Tensile failure

4.2.1 Direct tension

Application of tensile stresses (with negative sign according to our geomechanics convention) on a metal bar results in tensile strains (negative too). In this example the state of stress is relatively simple with tensile stress in the axial direction and zero-stress in any direction perpendicular to the axis of the bar (Figure 4.4). The maximum tensile stress taken by the bar is called tensile strength. Metals are usually “ductile” and deform after reaching a peak stress. When unstressed, the bar in the example figure does not recover its original length but remains with “plastic deformation”.

Figure 4.4: Tensile strength of a ductile metal bar.

The one-dimensional tensile strength test for metals (Fig. 4.4) is not easy to implement in rocks. You would have to grab the rock on its sides or glue it on the ends to perform such tests. Even in that case, your rock may break at the those “grabbing” points. One alternative is to “machine” the rock to a convenient shape, so that, you can pull it without using glues or grabbing jaws (shown in Figure 4.5). However, rocks are not easy to “machine” in general, and thus this test becomes impractical in many situations. Rock failure in simple tension usually displays “brittle” failure, no plastic strains follow after reaching tensile strength. It just breaks quickly.

Figure 4.5: Direct tension test on brittle rock. Correct interpretation depends on rock micro-structure and possible existence of pre-existent flaws.

4.2.2 The Brazilian test

The Brazilian test is a convenient method to measure tensile strength. It uses short cylindrical samples and takes advantage of the shape of the rock specimen to create tensile stresses with application of a compressive force along the sample diameter (Figure 4.6). A solution of the state of stress within the rock (assuming a linear elastic homogeneous material) yields the tensile strength value $T_S$ equal to

$$T_S = \frac{P_B}{\pi L R}$$ (4.1)

where $P_B$ is the peak compressive force, $L$ is the specimen length, and $R$ is the specimen radius. Notice that you have a combined state of stress with compression in the direction of the compressive load and tension in the direction perpendicular to the load along the diameter.

Figure 4.6: The Brazilian test: sample geometry and example.

PROBLEM 4.1: Determine the tensile strength of the shale sample shown in Fig. 4.6. The sample diameter is 1.00 in and the length is 1.00 in.

SOLUTION
The sample dimensions are

$$R = \frac{1}{2}$$    in$$= 0.0127$$    m

and

$$L = 1$$    in$$= 0.0254$$    m

Hence, the tensile strength is

$$T_S = \frac{2084 \text { N}}{\pi (0.0254 \text{ m}) (0.0127 \text{ m})} = 2.06 \times 10^6 \text{ Pa} = 2.06 \text{ MPa} \: \: \blacksquare$$

Typical values of tensile strength for cemented sedimentary rocks range from 0.5 MPa to 10 MPa. Uncemented sediments -very common in sedimentary basins- have zero tensile strength. Figure 4.7 summarizes typical values of tensile strength for rocks.

Figure 4.7: Values of tensile strength in a set of cemented rocks measured with direct tension tests (DTS) and Brazilian tension tests (BTS)[Data from Geotech. Geol. Engineering (2014), 32]. The value of the igneous set is an average of granite, latite, meta-pegmatite and peridotite. The tensile strength of the metamorphic set is an average of gneiss, marble, quartzite and slate. Compare these values to tensile strength of fused silica: 48 MPa, 304 stainless steel: 505 MPa, and titanium: 1860 MPa.