## Brittle deformation

Brittle fracture is believed to be caused by progressive failure along a network of micro- and meso-scale cracks. The cracks weaken rock by producing local high concentrations of tensile stress near their tips. The crack orientations relative to the applied stress determine the location and magnitude of local stress maxima. Fracturing occurs where the local stress maxima exceed the strength of the rock.

This theory, known as the Griffith theory of fracture, works well under conditions of applied tensile stress or where one of the principal stresses is compressional. When the magnitude of the tensile stress exceeds the tensile strength of the material, cracks orthogonal to this stress fail first and an extension fracture occurs. Below a depth of a few hundred meters, where all principal stresses are usually compressional, the behavior of cracks is more complex. Cracks close under compression and are probably completely closed at depths of >5 km due to increasing overburden pressure. This implies that the compressive strength of a material is much greater than the tensile strength. For example, the compressive strength of granite at atmospheric pressure is 140 MPa, and its tensile strength only about 4 MPa.

Where all cracks are closed, fracturing depends upon the inherent strength of the material and the magnitude of the differential stress (Section 2.10.1). Experiments show that shear fractures, or faults, preferentially form at angles of <45° on either side of the maximum principal compressive stress when a critical shear stress on the planes is exceeded. This critical shear stress (as*) depends upon the normal stress (an) on planes of potential failure and the coefficient of internal friction (p.) on those planes, which resists relative motion across them.

This relationship, called the Mohr-Coulomb fracture criterion, is described by the following linear equation:

The cohesion (c) describes the resistance of the material to shear fracture on a plane of zero normal stress. Byerlee (1978) showed that many rock types have nearly the same coefficient of friction, within the range 0.6-0.8. The form of the equation, which is written using the absolute value of the critical shear stress, allows a pair of fractures to form that is symmetric about the axis of maximum principal compressive stress. Pore fluid pressure enhances fracturing by reducing the frictional coefficient and counteracting the normal stresses (On) across the fault. The effect of pore fluid pressure explains faulting at depth, which would otherwise appear to require very high shear stresses because of the high normal stresses.

Under this compressional closed crack regime, the type of faulting which results, according to the theory of Anderson (1951), depends upon which of the principal stresses is vertical (Fig. 2.21). Normal, strike-slip, and thrust faults occur depending on whether Oj, 02 or O3 respectively, is vertical. This theory is conceptually useful. However, it does not explain the occurrence of some faults, such as low-angle normal faults (Section 7.3), which display dips of <30°, flat thrust faults, or faults that develop in previously fractured, anisotropic rock.

The strength ofrock increases with the pressure ofthe surrounding rock, termed the confining pressure, but decreases with temperature. In the uppermost 10-15 km of the crust the former effect is dominant and rock strength tends to increase with depth. Confining pressure increases with depth at a rate of about 33 MPa km-1

depending on the density of the overlying rocks. Below 10-15 km the effect of temperature takes over, and rocks may progressively weaken downwards. However, this simple relationship can be complicated by local variations in temperature, fluid content, rock composition, and preexisting weaknesses.

The deformation of brittle solids can take the form of cataclasis (Fig. 2.22) (Ashby & Verrall, 1977). This

results from repeated shear fracturing, which acts to reduce the grain size of the rock, and by the sliding or rolling of grains over each other.

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