This chapter discusses the relationship between stress and the formation of rock fractures.

1 Elastic Deformation and Experimental Fracturing of Rocks

*Elastic Deformation*- measured by axial force applied through pistons on cylinder of rock and associated changes in dimensions of the sample. so it is strain measured due to applied stress

**elastic deformation**- brittle materials at stresses below fracture , changes in stress induce an instantaneous change in sample dimensions

**recoverable strain**- strain completely disappears when stress is removed

**extensional strain**- measurement of strain

*e _{n}* = D
L/L

for uniaxial stress, the magnitude of the elastic extension is directly proportional to the magnitude of stress

*e _{n}* = s

where E is **Young’s modulus** (negative because a tensile stress produces positive extension)

**Poisson’s ratio (n
)-** the absolute value of the ration given by extension normal to applied compressive stress, divided by extension parallel to the applied compression

n
º
ï
*e^
*/ *eô
ô
ê
*

= 0.5 perfectly incompressible

ranges from 0.25 to 0.33

**Poisson expansion** is expansion normal to an applied compression

-under confined compression, the magnitude of the axial extension depends on axial stress and on radial pressure

*Experimental Fracturing*-

**strength**- the stress at which failure occurs

**brittle failure, fracture**- a surface or zone across which the material loses cohesion

**ductile failure**- when material becomes permanently deformed without losing cohesion

Pressure chamber experiment

-sample placed in sealed impermeable jacket and surrounded in a pressure chamber by a fluid under pressure

**confining pressure**- is the pressure of surrounding fluid, and **pore fluid pressure** is the pressure of fluid in pore spaces

two different types of fracture-

**extension fractures- **fracture plane** **is perpendicular to the minimum principal stress s
_{3} (tensile) and parallel to the maximum principal stress s
_{1, }displacement is normal to fracture surface

-form by **longitudinal splitting** if s
_{3} is equal or close to 0, and s
_{1} is axial

**shear fractures**- form in confined compression at angles of less and 45°
to s
_{1}, the maximum compressive stress, displacement is parallel to the fracture surface

2 Fracture Criterion for Tension Fractures

T_{0} is the **tensile strength** of a material, fracturing occurs when tensile stresses exceed this

**tension fracture envelope**- is the boundary between stable and unstable states of tensile stress, given by s
_{n}^{*} = T_{o}

**fracture plane angle** a
_{f} - is angle between maximum principal stress and fracture plane

3 Coulomb Fracture Criterion for Confined Compression

initiation of fracturing depends on differential stress (s
_{1}-s
_{3})

shear fracture envelope separates stable from unstable states of stress and is given by the Coulomb fracture criterion

½
s
* _{s}^{*½
}= c + m
_{n }* where m

f
* = *slope angle of the line, angle of internal friction

s
* _{s}* =

m
* = *coefficient of internal friction, slope of line

*c* = cohesion, intercept of line

for any critical stress state, when s
_{n} and s
_{s} satisfy equation, there are two possible conjugate shear planes

on the Mohr circle the radius to the tangent point of line is perpendicular to the fracture envelope

Coulomb fracture criterion does not apply in the tensile part of the Mohr diagram

4 Effects of Confining Pressure on Fracturing and Frictional Sliding

as confining pressure increases, the fracture angle increases

starting from T_{0 } and increasing confining pressure- 1) tension fracture, 2) mixed mode- extension and shear displacement (now under Coulomb fracture criterion), 3) brittle fracture under Coulomb fracture, 3) brittle-ductile transition zone, fracture plane angle decreases 4) von Mises criterion- ductile shear failure, where ï
s
_{s}^{*ï
}= constant

Frictional Sliding (Byerlee’s Law)

- a fracture plane is a plane of weakness, since there is no cohesion across it

frictional sliding criterion-

ï
s
_{s}^{*ï
} = µs
_{n}

where ï
s
_{s}^{*ï
}is the magnitude of the critical shear stress

and µ is the coefficient of sliding friction

**stable sliding**- at low confining pressure, frictional sliding occurs as a smooth, continuous motion

**stick-slip**- at higher confining pressure, motion is characterized by interval of rapid sliding and no motion

5 Effects on Fracturing and Frictional Sliding

a.* Pore Fluid Pressure*

Pore fluid causes rock to behave as if the confining pressure were lower by the amount of the pore fluid pressure *p _{f}* . The normal stress is now the

Effects of pore fluid pressure-

- lowers differential stress required to cause failure and allows for fracture at depths where the rock would otherwise be stable or governed by ductile behavior
- shifts the conditions for frictional sliding from stick-slip to stable sliding
- can shift deformation from cataclastic flow to frictional sliding

sources of pore fluid-

water incorporated into sediment during deposition

fluid released by dehydration reactions during metamorphism

- Anisotropy

Rocks are **mechanically anisotropic**- their strength is different in different directions

Rocks break more easily along preferred planar alignment, or cleavage

Two fracture criteria (fig 9.15)

- normal Coulomb fracture that applies to fractures that develop across the plane of weakness, Mohr circle cannot cross this line
- line that crosses the Mohr circle applies only to the surface stress components acting on the cleavage plane

**shear strength**- differential stress at shear fracture, is the diameter of the critical Mohr circle

- Intermediate Principal Stress

- rock strength is highest when intermediate and maximum principal stresses are equal
- lowest when intermediate and minimum principle stresses are equal

- Temperature

- rise in temperature lowers the von Mises yield stress for ductile behavior, and lowers the pressure of the brittle-ductile transition and reduces the field of brittle behavior

- Scale

-larger scale of actual rocks gives lower strengths than from lab rock samples

- Griffith Theory of Fracture

AA Griffith came up with Griffith cracks (microscopic, randomly oriented cracks) in all solids to explain the lower experimental tensile strength or rock, which was about 2 orders of magnitude than those predicted theoretically using the strengths of atomic bonds.

**Griffith cracks** are small slitlike cracks that are modeled as flatten ellipsoids (fig 9.18). It reduces the strength since an applied stress produces a local high concentration of tensile stress near the crack tip. *Applied stresses* (force/ area) are applied to the surfaces of the both, and **local stress**es describe the state of stress immediately adjacent to a Griffith crack. The local concentration is because of the smaller radius of curvature at the crack tip of the ellipse.

*Formation of Tension Fractures*-

Griffith cracks are free surfaces (no shear stress, no normal tensile stress). So the crack must be a principal plane of the local stress and the maximum tensile stress is parallel to the ellipse surface.

So the orientation of the most critically stressed Griffith crack is perpendicular to the maximum applied tensile stress, and when the true strength of the material is exceeded at the crack tip, the crack propagates in a plane normal to the local tensile stress and the plane of propagation is parallel to the plane of the crack.

*Longitudinal Splitting* (fig. 9.19)

Under uniaxial compression, Griffith cracks that are not parallel to the compressive stress are closed by the component of normal stress across their surfaces.

*Shear Fractures *(fig. 9.20)

Shear along closed cracks results in maximum tensile stress near crack tip. Tensile crack open to accommodate shear on crack surface.