Structural Geology and Rock Mechanics

Fold geometry and outcrop patterns

A fold is a bend in an originally planar surface, such as a bedding plane, and its geometry is described by standard elements. The hinge is the line of maximum curvature, the limbs are the relatively planar parts on either side of it, and the axial plane (axial surface) is the surface that joins the hinges of successive layers. The fold axis is the line about which the layers are folded, and it is defined by its trend and plunge. An antiform is a fold that arches upward, and a synform is a fold that sags downward; these terms describe shape without reference to the age of the beds. An anticline and a syncline are the stratigraphic equivalents, so they should be used only when the younging direction is known to lie toward or away from the core. The facing of a fold is the direction toward which the stratigraphic tops lie, and it is read from the sequence of beds across the structure. For an antiform whose younger units lie toward the west on the limb, the fold faces west even though its hinge is convex upward.

Folds are further classified by the attitude of the axial plane and the plunge of the axis. An upright fold has a vertical axial plane with its limbs dipping in opposite directions, a recumbent fold has a horizontal axial plane, and a vertical fold has a vertical axis. On a stereonet the fold axis is plotted as a point and the axial plane as a great circle. An axis plotted at the centre has a plunge of 90 degrees, and an axial plane plotted as a diameter is vertical, whereas one plotted on the primitive circle is horizontal. Chevron folds are a related type with straight limbs and sharp, angular hinges at both the antiformal and synformal positions.

Fold shape is also read from the outcrop pattern on a map. On flat topography, a closed, elongate pattern indicates a doubly plunging fold, and the plunge direction is marked by fold-axis arrows. If the arrows point inward toward the centre, the axis descends toward the middle, so the lowest point of the fold lies at the centre and the structure is a synform with an axial depression. If the arrows point outward, the axis rises toward the centre and the structure is an antiform with an axial culmination. A horizontal fold shows no closed outcrop pattern because its axis has zero plunge.

Ramsay's class 1B, the parallel fold, is defined by a constant orthogonal thickness of the folded layer. The inner arc of such a layer is more sharply curved than the outer arc, so the two arcs do not share the same curvature. In an antiform of this class, dip isogons (lines joining points of equal dip on successive surfaces) converge toward the core, so the layer appears thinner on the limbs when measured parallel to the axial plane.

Isotropic elastic constants and wave speeds

Within linear elasticity, stress and strain are proportional, and the constant of proportionality is an elastic modulus. For uniaxial loading, Hooke's law gives \sigma = E\varepsilon, where \sigma = F/A is the axial stress, \varepsilon = \Delta L/L is the axial strain and E is Young's modulus. Substituting these gives F = EA\,\Delta L/L, so the force is proportional to the cross-sectional area times the strain. The coefficient E is the intrinsic material constant, whereas the combination EA/L depends on the size and shape of the body. Young's modulus is the slope of the linear part of a stress-strain curve, E = \sigma/\varepsilon, and the specimen dimensions do not enter because stress and strain are normalized quantities. For example, a specimen that reaches 150 MPa at an axial strain of 0.0025 has E = 150/0.0025 = 6.0 \times 10^{4} MPa, or 60 GPa. Since \varepsilon = \sigma/E for a given stress, a rock that shows a smaller strain than others under the same stress has a larger elastic modulus and is stiffer, which means it resists deformation more strongly.

An isotropic linear elastic material needs only two independent constants, and the others follow from them. The common pairs are (\lambda,\mu), (E,\nu) and (K,\mu), where \mu is the shear modulus, \nu is Poisson's ratio and K is the bulk modulus. The relations are E = \mu(3\lambda+2\mu)/(\lambda+\mu), \nu = \lambda/[2(\lambda+\mu)] and K = \lambda + 2\mu/3. The wave speeds follow from the same constants: the P-wave speed is V_P = \sqrt{(\lambda+2\mu)/\rho} and the S-wave speed is V_S = \sqrt{\mu/\rho}, so V_P/V_S = \sqrt{(\lambda+2\mu)/\mu} is always greater than 1. Because \mu = 0 in a fluid, S-waves do not propagate through a fluid.

Ductile deformation products

Ductile deformation takes place at relatively high temperature and pressure, where rocks flow by crystal-plastic processes such as dislocation creep and dynamic recrystallization. Within shear zones it produces grain-size reduction and a strong foliation, and the typical product is mylonite, a fine-grained, strongly foliated rock. Fault rocks formed by brittle processes are different. Gouge is a fine, incohesive comminuted material, breccia consists of angular fragments set in a finer matrix, and cataclasite is a cohesive fault rock formed by fracturing and grain crushing. These are brittle products and are not the typical result of ductile flow.

Certain microstructures help identify the deformation mechanism and are used in shear-sense and recrystallization studies:

• Core-mantle structure — a large, strained grain (core) surrounded by a rim of fine, recrystallized grains (mantle).
• Decussate texture — randomly oriented grains, often dominated by crystal faces, as in sheet silicates in a non-foliated aggregate.
• Spherulite — a radiating, fibrous aggregate, commonly of K-feldspar with or without quartz.
• Millipede structure — inclusion trails in a porphyroblast that curve into the matrix foliation, producing a concave-outward pattern.

Conjugate shear fracture stress orientation

Anderson's theory relates conjugate shear fractures to the principal compressive stresses, with \sigma_1 \geq \sigma_2 \geq \sigma_3 in the compressive convention. The two conjugate fractures make symmetric angles with \sigma_1, and their line of intersection is parallel to the intermediate stress \sigma_2. The acute bisector of the fracture pair contains \sigma_1, and under plane strain each fracture makes an angle of about \theta = 45^\circ - \phi/2 with \sigma_1, where \phi is the angle of internal friction. The fault regime is identified by which principal stress is vertical. Normal faulting occurs when \sigma_1 is vertical, strike-slip faulting when \sigma_2 is vertical, and reverse (thrust) faulting when \sigma_3 is vertical.

On a stereonet the centre represents a vertical direction and the primitive circle represents horizontal directions. A stress plotted at the centre is therefore vertical, and a stress plotted on the primitive is horizontal. A vertical \sigma_3 with horizontal \sigma_1 and \sigma_2 defines the reverse regime, whereas a vertical \sigma_2 would indicate strike-slip faulting.

Reclined fold geometry

A reclined fold is one whose fold axis lies within a steep axial plane, so the axis plunges steeply. The pitch of the axis within the axial plane is 90 degrees, which means the axis is parallel to the dip direction of the axial plane, and its plunge equals the dip of that plane. For example, an axial plane that strikes 040 degrees and dips 35 degrees to the south-east has a dip direction of 130 degrees, so the fold axis plunges 35 degrees toward 130 degrees. The dip direction is found by adding 90 degrees to the strike when the plane dips to the right of the strike line.

On a stereonet, a reclined fold is recognized by its steep axial plane and steep fold axis. This distinguishes it from the upright type (vertical axial plane with horizontal axis), the inclined type (inclined axial plane with horizontal axis) and the recumbent type (horizontal axial plane).

Mohr-Coulomb failure criterion

The Mohr-Coulomb criterion states that failure occurs when the shear stress on a plane reaches \tau = c + \sigma_n\tan\phi, where c is the cohesion, \phi is the angle of internal friction and \sigma_n is the normal stress on the plane. Its failure envelope is a straight line in the \tau-\sigma_n plane. Under uniaxial compression the strength is C_0 = 2c\cos\phi/(1-\sin\phi), so the cohesion can be recovered as c = C_0(1-\sin\phi)/(2\cos\phi). For a rock with C_0 = 40 MPa and \phi = 30^\circ, this gives c = 40 \times 0.5/(2 \times 0.866) \approx 11.5 MPa.

In a saturated rock, pore water pressure p_w reduces the normal stress across a plane, so failure is evaluated with effective stresses \sigma' = \sigma - p_w. When a criterion is given in terms of \sigma_1' and \sigma_3', substitute \sigma_1 - p_w and \sigma_3 - p_w and solve for the value of p_w that just satisfies the criterion. For instance, take \sigma_1 = 20 MPa, \sigma_3 = 6 MPa and the criterion \sigma_1' = 4 + 3\sigma_3' (MPa). Setting 20 - p_w = 4 + 3(6 - p_w) gives 2p_w = 2, so p_w = 1 MPa.

Mean and deviatoric stress

The stress tensor can be split into a mean (hydrostatic) part and a deviatoric part. The mean stress is \sigma_m = (\sigma_1+\sigma_2+\sigma_3)/3, and the deviatoric stress is obtained by subtracting it from each normal stress, \sigma_i' = \sigma_i - \sigma_m. The mean stress changes the volume without changing the shape, whereas the deviatoric stress controls distortion and failure. For principal stresses of 50, 30 and 10 MPa, the mean stress is 30 MPa and the deviatoric stresses are 20, 0 and -20 MPa.

On a two-dimensional Mohr circle, the centre lies at the mean normal stress (\sigma_1+\sigma_3)/2, and the radius equals the maximum shear stress (\sigma_1-\sigma_3)/2. The radius therefore represents the deviatoric state, while the centre represents the mean stress.

Apparent dip in outcrop

The true dip of a plane is its maximum inclination, measured perpendicular to the strike. When the same plane is exposed on a surface that is not perpendicular to the strike, the angle it makes with the horizontal on that surface is the apparent dip. The apparent dip is never greater than the true dip, and it is zero when the section is parallel to the strike. The relation is \tan\alpha = \tan\delta\,\sin\beta, where \delta is the true dip, \alpha is the apparent dip and \beta is the angle between the section line and the strike. For a plane with a true dip of 45 degrees viewed along a section making 30 degrees with the strike, \tan\alpha = \tan 45^\circ \sin 30^\circ = 0.5, so the apparent dip is about 26.6 degrees.

The apparent dip should be distinguished from related terms. Rake (pitch) is the angle a line makes within a plane, measured from the strike, and attitude is the strike and dip taken together. Neither is the inclination seen on a surface cutting across the strike.

Slope factor of safety

The factor of safety (FOS) is the ratio of resisting forces or strengths to driving forces or stresses, so FOS less than 1 indicates instability and FOS greater than 1 indicates stability. For a planar slide of a block of weight W on a failure plane dipping at \theta, with area A, cohesion c and friction angle \phi, the dry factor of safety is \mathrm{FOS} = (cA + W\cos\theta\tan\phi)/(W\sin\theta). The normal force is W\cos\theta and the driving force is W\sin\theta, and both cohesion and friction add to the resistance. As a worked example, take A = 20 m^2, c = 50 kPa, \phi = 25^\circ, \theta = 45^\circ and W = 1000 kN. The resistance is 1000 + 1000 \times 0.7071 \times 0.4663 \approx 1330 kN and the driving force is 707 kN, so \mathrm{FOS} \approx 1.88.

For a long, shallow slope in which failure occurs on a plane parallel to the ground surface, the infinite-slope model applies. The driving shear stress on a plane at vertical depth z is \tau = \gamma z\sin\beta\cos\beta, and the normal stress is \sigma_n = \gamma z\cos^2\beta, where \gamma is the unit weight and \beta is the slope angle. The depth must be taken vertically, because the vertical column controls the weight. For \gamma = 18 kN/m^3, z = 3 m and \beta = 20^\circ, \tau \approx 17.4 kPa. In SI units, 1 Pa equals 1 kg·m^{-1}·s^{-2}, so a stress written in kg/(m·s^2) is still a pressure.

Fault displacement components

Fault displacement is described by components in the fault plane and in the vertical section perpendicular to the strike. The net slip is resolved into a dip-slip component d along the dip direction and a strike-slip component s along the strike. The dip-slip component is further resolved into heave H, the horizontal separation, and throw T, the vertical separation, both measured in the vertical plane perpendicular to the strike. The dip-slip vector lies in that section, so d = \sqrt{H^2+T^2}, and the true displacement is D = \sqrt{d^2+s^2}, because dip-slip and strike-slip are perpendicular components.

The net slip s_{net} with pitch \lambda on a fault of dip \delta gives d = s_{net}\sin\lambda, H = d\cos\delta and T = d\sin\delta. The strike-slip component is parallel to the strike and therefore does not contribute to heave or throw. For example, a net slip of 10 m with pitch 45 degrees on a fault dipping 60 degrees has a dip-slip of 7.07 m, a heave of 3.54 m and a throw of 6.12 m.

Finite strain from length change

Finite strain is measured from the change in length of a line. Longitudinal (engineering) strain is e = \Delta L/L_0 = (L-L_0)/L_0, where L_0 is the original length and L is the deformed length. Extension gives a positive e and shortening gives a negative e. Rearranging gives the deformed length as L = L_0(1+e). For example, a line of 12 cm stretched by e = 0.25 becomes 12 \times 1.25 = 15 cm.

The quantity 1+e is the stretch ratio. For large deformations, the natural (logarithmic) strain \ln(1+e) is often preferred, because natural strains from successive increments add up.

Synsedimentary deformation structures

Synsedimentary deformation structures form when unlithified, water-saturated sediment is disturbed shortly after deposition and before compaction and cementation are complete. Common triggers are liquefaction, fluid escape and slumping on a slope. Convolute bedding is the characteristic structure of this process, consisting of contorted, folded laminae within a sand or silt layer, which shows that the sediment was deformed while still soft. Other examples include load casts, flame structures and slump folds. By contrast, festoon bedding (trough cross-stratification formed by migrating bedforms), flaser bedding (ripple-laminated sand with mud drapes) and tabular bedding (planar, undeformed stratification) are primary depositional structures and record no post-depositional disturbance.