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Anderson's theory of faulting

Anderson's theory of faulting is a seminal framework in structural geology that explains the initiation, orientation, and mechanics of faults in the brittle upper crust as a function of the regional stress field. Developed by Scottish geologist Ernest Masson Anderson, the theory posits that fault planes form at optimal angles to the principal stress directions to accommodate shear failure under varying tectonic conditions.^[1] The theory rests on two key assumptions applicable to shallow crustal levels: the Earth's surface represents a free boundary with no shear stress, implying that the principal stresses align with the vertical and horizontal directions; and the vertical stress arises primarily from the lithostatic load of the overlying rock column.^[2] These assumptions lead to three fundamental tectonic regimes defined by the relative magnitudes of the principal stresses (denoted σ₁ as the maximum compressive, σ₂ intermediate, and σ₃ minimum): the gravity regime where σ₁ is vertical, favoring extensional normal faulting; the thrust regime where σ₃ is vertical, promoting compressional reverse faulting; and the wrench regime where σ₂ is vertical, resulting in strike-slip faulting.^[1] Building on the Coulomb-Mohr failure criterion, Anderson's model predicts conjugate fault pairs inclined at approximately 30°–45° to the maximum compressive stress direction, yielding characteristic dips of about 60° for normal faults, 30° for low-angle reverse (thrust) faults, and near-vertical planes for strike-slip faults, assuming a typical rock friction angle of 30°.^[2] First outlined in Anderson's 1905 address to the Edinburgh Geological Society and elaborated in his 1942 monograph The Dynamics of Faulting and Dyke Formation with Applications to Britain, the theory has profoundly influenced modern tectonics, seismology, and rock mechanics by providing a basis for inferring paleostress orientations from fault geometries observed in the field.^[1]^[3]

Introduction

Historical development

Ernest Masson Anderson, a Scottish geologist trained at the University of Edinburgh, first proposed his theory of faulting in 1905 during a presentation to the Edinburgh Geological Society. In his paper "The dynamics of faulting," Anderson introduced a classification of faults based on the orientation of the three principal stresses, assuming that one principal stress acts vertically due to the influence of the Earth's free surface. This work marked a pioneering application of stress analysis to geological structures, predating widespread recognition of strike-slip faulting, as evidenced by the 1906 San Francisco earthquake.^[4] Anderson's ideas drew on late 19th-century foundations in rock mechanics. His development occurred in the context of early 20th-century advances, including his appointment to the Geological Survey of Great Britain in 1903, which provided field opportunities in Scotland to observe fault patterns, and growing interest in tectonic processes following continental drift hypotheses.^[4] These influences allowed Anderson to integrate continuum mechanics with empirical observations, establishing a theoretical basis for fault initiation under varying stress regimes. Anderson expanded and formalized his framework in the book The Dynamics of Faulting and Dyke Formation with Applications to Britain, first published in 1942 and revised in a second edition in 1951. This seminal text elaborated on the mechanics of faulting, incorporating the Coulomb failure criterion and applications to British geology, solidifying the theory's role in structural geology.^[5] The centennial of Anderson's 1905 paper was marked by reflections on its lasting influence, particularly in a 2008 review that analyzed global seismicity data to affirm the prevalence of Andersonian fault types in earthquake focal mechanisms. This recognition underscored the theory's foundational status, with over a century of refinements in rock mechanics and tectonics continuing to build upon its core assumptions.^[6]

Core principles

Anderson's theory of faulting establishes that shear failure in the brittle crust occurs on planes inclined at approximately 30° to 45° relative to the maximum principal compressive stress direction (σ₁), positioning the fault roughly perpendicular to the minimum principal stress (σ₃). This orientation minimizes the shear stress required for failure while maximizing resistance due to normal stress on the plane, assuming frictional sliding as the dominant mechanism. The specific angle depends on the internal friction angle of the rock, typically around 30° for common crustal materials with a friction coefficient of about 0.6.^[7]^[8] Central to the theory is the assumption that the Earth's free surface imposes a boundary condition where no shear stress acts on horizontal planes, making the vertical direction one of the principal stress axes. Consequently, the vertical stress (σᵥ) aligns with either σ₁, σ₂, or σ₃, simplifying the analysis of fault orientations in a three-dimensional stress field. This vertical principal stress arises from the overburden pressure and gravitational loading, influencing how horizontal tectonic forces interact with the crust.^[7]^[8] The theory classifies faults into three primary types based on which principal stress is vertical, directly linking stress regimes to fault geometry:

Normal faults, formed when σ₁ is vertical (overburden dominates) and σ₃ is horizontal, accommodating crustal extension with typical dips of around 60°.
Reverse and thrust faults, occurring when σ₃ is vertical and σ₁ is horizontal, reflecting horizontal contraction with shallower dips of about 30°.
Strike-slip faults, developing when σ₂ is vertical and both σ₁ and σ₃ are horizontal, producing near-vertical fault planes for lateral shear.^[7]^[8]

Conjugate fault pairs represent a key prediction, consisting of two symmetric sets of faults mirrored across the principal stress axes, enabling balanced shear displacement in opposite senses while maintaining overall stress equilibrium. These pairs form at angles of 45° ± (friction angle)/2 to σ₁, providing insight into the bisected nature of the failure envelope.^[7]^[9]

Stress Fundamentals

Stress and strain in rocks

Stress is defined as the force applied per unit area on a rock body, quantified in units such as pascals (Pa) or megapascals (MPa).^[10] In the Earth's crust, rocks experience two primary categories of stress: lithostatic stress, which arises from the uniform overburden pressure of overlying rock layers and acts isotropically in all directions, and tectonic stress, which results from plate movements and is typically anisotropic, leading to differential loading. Lithostatic stress increases linearly with depth due to the weight of the rock column above, while tectonic stress superimposes directional components that drive deformation.^[11] Rocks are subjected to three fundamental types of stress based on the direction of force application: compressive stress, which shortens the rock; tensile stress, which elongates it; and shear stress, which causes sliding or distortion parallel to a plane.^[12] These can be decomposed into hydrostatic (or mean) stress, representing the isotropic component equal in all directions, and deviatoric stress, capturing the differential part that promotes shape change without volume alteration.^[10] In crustal settings, hydrostatic stress dominates under lithostatic conditions, but deviatoric tectonic stress is crucial for initiating faulting by amplifying shear components on potential failure planes. Strain represents the response of a rock to applied stress, manifesting as a change in volume or shape, often expressed as a dimensionless ratio.^[10] Rocks exhibit elastic strain at low stress levels, where deformation is reversible and proportional to stress, following Hooke's law; plastic (ductile) strain involves permanent deformation through mechanisms like crystal plasticity; and brittle failure occurs via fracturing when stress exceeds strength.^[13] The dominant response varies with environmental conditions: brittle failure prevails at shallow depths (<10-15 km) and low temperatures (<300-400°C) under high strain rates, transitioning to plastic flow at greater depths and higher temperatures where pressure suppresses fracturing.^[14] This transition influences fault susceptibility, as brittle regimes favor discrete fault formation.^[15] Key rock properties governing fault susceptibility include cohesion, the inherent tensile strength resisting initial failure, and the internal friction angle, which quantifies resistance to shear sliding along planes, typically ranging from 20° to 40° for crustal rocks depending on mineralogy and texture. Higher cohesion enhances resistance to brittle failure, while lower friction angles, often seen in clay-rich or weathered rocks, promote easier slip and fault reactivation.^[16] These parameters, derived from laboratory triaxial tests, are integral to assessing how rocks deform under combined stress regimes.^[17] To evaluate stress orientation on arbitrary planes within a rock, the normal stress \sigma_n and shear stress \tau on a plane inclined at angle \theta to the maximum principal stress \sigma_1 (with minimum principal stress \sigma_3) are given by:

\sigma_n = \frac{\sigma_1 + \sigma_3}{2} + \frac{\sigma_1 - \sigma_3}{2} \cos(2\theta)

\tau = \frac{\sigma_1 - \sigma_3}{2} \sin(2\theta)

These equations highlight how shear stress maximizes at \theta = 45^\circ, providing a foundation for understanding potential fault orientations under differential loading.^[18]

Principal stresses and Mohr circle

In rock mechanics, principal stresses are defined as the maximum (σ₁), intermediate (σ₂), and minimum (σ₃) normal stresses acting on mutually perpendicular planes where shear stress vanishes, with the convention σ₁ ≥ σ₂ ≥ σ₃ assuming compression positive.^[19] These stresses represent the eigenvalues of the three-dimensional stress tensor, which encapsulates the full state of stress at a point in a continuum. The corresponding principal directions are the eigenvectors of this tensor, determined qualitatively through diagonalization—a process that transforms the coordinate system to align with axes where the tensor becomes diagonal, eliminating off-diagonal shear components. In the geological context of faulting, σ₁ denotes the axis of maximum compressive stress, while σ₃ indicates the axis of minimum compressive stress (or maximum extension), with σ₂ being the intermediate value.^[6] This orientation framework is fundamental to analyzing how applied stresses lead to deformation in the Earth's crust, as the principal axes dictate the planes of pure normal loading without superimposed shear.^[20] The Mohr circle offers a graphical tool to visualize and compute the stress components on arbitrary planes within a given stress state. For a two-dimensional analysis in the plane normal to σ₂, it is plotted with normal stress σ_n on the abscissa and shear stress τ on the ordinate; the circle's center lies at the mean stress \frac{\sigma_1 + \sigma_3}{2}, and its radius is \frac{\sigma_1 - \sigma_3}{2}, yielding a diameter equal to the differential stress σ₁ - σ₃.^[19]

\begin{aligned} \text{Center: } & \quad \left( \frac{\sigma_1 + \sigma_3}{2}, 0 \right) \\ \text{Radius: } & \quad \frac{\sigma_1 - \sigma_3}{2} \end{aligned}

^[21] Points on the circle represent the σ_n and τ for planes at various orientations, where the angular position on the circle is twice the physical angle from the principal direction due to the quadratic nature of stress transformations.^[21] This construction facilitates rapid determination of stresses on any plane and highlights the principal planes as the intercepts on the σ_n axis, where τ = 0.^[19] In three dimensions, multiple Mohr circles describe the full stress state, with the largest circle (between σ₁ and σ₃) governing the maximum shear stresses relevant to deformation analysis.^[22]

Theoretical Framework

Key assumptions

Anderson's theory of faulting relies on a set of simplifying assumptions to predict the orientation and type of faults based on the tectonic stress regime. These premises idealize the brittle upper crust as a medium where failure occurs predictably under differential stress, drawing from the Coulomb failure criterion while adapting it to geological conditions. By assuming a uniform stress field and specific boundary conditions, the model enables the classification of normal, reverse, and strike-slip faults without accounting for complications like fluid effects or material heterogeneities in its foundational form.^[6] A central assumption is that one of the principal stresses—one of \sigma_1 (maximum), \sigma_2 (intermediate), or \sigma_3 (minimum)—is oriented vertically, arising from the traction-free nature of the Earth's surface, which imposes no shear traction at shallow depths. This vertical principal stress aligns with the direction of gravity and lithostatic pressure, simplifying the three-dimensional stress tensor to cases where the overburden stress is one of the extrema.^[6] The theory further posits that faults initiate as shear fractures oriented at the angle dictated by the rock's internal friction, commonly taken as approximately 30° relative to the maximum principal stress \sigma_1. This orientation emerges from the geometry of shear failure in brittle materials, where the fracture plane bisects the angle between \sigma_1 and \sigma_3 adjusted for frictional resistance. Additionally, the model assumes a homogeneous and isotropic brittle crust with a uniform stress field at the depths where faulting occurs, treating the rock mass as mechanically uniform without significant variations in strength or preexisting weaknesses.^[6] Initially, the framework neglects the effects of pore fluid pressure and material anisotropy, focusing on effective stress in dry or undrained conditions, although Anderson later acknowledged their roles in modifying fault behavior. These assumptions collectively imply characteristic fault dips: approximately 60° for normal faults (where \sigma_1 is vertical and \sigma_3 is horizontal), 30° for reverse faults (where \sigma_1 is horizontal and \sigma_3 is vertical), and near-vertical (90°) planes for strike-slip faults (where \sigma_2 is vertical).^[6]^[9]

Coulomb failure criterion and fault initiation

The Coulomb failure criterion describes the conditions under which rocks undergo shear failure, forming the mechanical basis for fault initiation in Anderson's theory. This criterion states that failure occurs on a plane when the shear stress \tau acting parallel to it exceeds the rock's shear strength, expressed as \tau = C + \mu \sigma_n, where C is the cohesion (intrinsic tensile strength), \mu is the coefficient of internal friction, and \sigma_n is the effective normal stress perpendicular to the plane.^[9] In rocks, \mu typically ranges from 0.6 to 0.85, equivalent to a friction angle \phi = \tan^{-1} \mu \approx 30^\circ to 40^\circ, reflecting the frictional resistance derived from mineral grain interactions and surface roughness. Under a general triaxial stress state with principal stresses \sigma_1 > \sigma_2 > \sigma_3 (where \sigma_1 is the maximum compressive stress), failure initiates on the plane where the ratio \tau / \sigma_n is maximized relative to the material's strength. This optimal plane is oriented at an angle \theta = 45^\circ - \phi/2 to \sigma_1, ensuring the shear stress is sufficient to overcome frictional locking while minimizing the stabilizing normal stress.^[9] For \phi \approx 30^\circ, \theta \approx 30^\circ, meaning incipient faults form at shallow angles to the direction of maximum compression. The Mohr circle representation illustrates this by showing how the stress trajectory tangentially intersects the linear failure envelope defined by the Coulomb criterion.^[23] In Anderson's framework, this criterion integrates with the assumption that one principal stress (\sigma_v, the vertical lithostatic stress) remains fixed by gravitational loading near Earth's surface, constraining the possible orientations and types of faulting. Conjugate shear pairs emerge symmetrically about \sigma_1, separated by 2\theta \approx 60^\circ, as both planes reach the failure threshold simultaneously under pure shear conditions.^[24] The vertical \sigma_v then dictates the tectonic regime: if \sigma_v = \sigma_1 (maximum), extensional normal faulting with horizontal \sigma_3 (minimum); if \sigma_v = \sigma_3 (minimum), compressional reverse faulting with horizontal \sigma_1 (maximum); if \sigma_v = \sigma_2 (intermediate), strike-slip faulting with horizontal \sigma_1 and \sigma_3. This setup predicts that fault initiation requires the horizontal stress components to build until the Coulomb threshold is met on the conjugate planes aligned with the regime.^[6] Faults begin as Mode II (in-plane shear) fractures, where displacement is parallel to the fracture surface and perpendicular to the propagation direction, consistent with the Coulomb mechanism under differential compression.^[24] As slip accumulates, these initial fractures propagate and evolve, potentially developing listric geometries (concave-upward curvature) in ductile layers or remaining planar in brittle ones, influenced by rock anisotropy and evolving stress fields.^[25] The critical differential stress required for failure under the Coulomb criterion, assuming the optimal plane orientation, is \sigma_1 - \sigma_3 = \frac{2C \cos \phi}{1 - \sin \phi} + \frac{2 \sigma_3 \sin \phi}{1 - \sin \phi}.^[26] In the simplified cohesionless case (C = 0), this reduces to \sigma_1 - \sigma_3 = \sigma_3 \frac{2 \sin \phi}{1 - \sin \phi}, or equivalently \frac{\sigma_1}{\sigma_3} = \frac{1 + \sin \phi}{1 - \sin \phi}, highlighting the role of confining pressure \sigma_3 in enabling frictional sliding.^[26] For \phi \approx 30^\circ, the stress ratio is approximately 3, meaning \sigma_1 must exceed \sigma_3 by at least threefold under low cohesion to initiate faults.

Fault Classifications

Normal faults

In Anderson's theory of faulting, normal faults form under an extensional stress regime where the maximum principal compressive stress (σ₁) acts vertically due to lithostatic load, while the minimum principal compressive stress (σ₃) is oriented horizontally in the direction of extension, and the intermediate stress (σ₂) is the other horizontal direction.^[27] This configuration, often termed the "gravity regime," promotes failure along planes inclined at approximately 60° to the horizontal near the surface, as predicted by the Coulomb failure criterion assuming typical rock friction angles around 30°.^[9] The theory posits that such faults initiate as planar shear fractures but may evolve into listric geometries that flatten with depth, accommodating greater displacement in the subsurface.^[27] Geometrically, normal faults exhibit a hanging wall that moves downward relative to the footwall, creating characteristic structures such as horst and graben systems. Conjugate pairs of normal faults typically form symmetrically about the σ₁ direction (vertical), with antithetic faults dipping opposite to the main synthetic set, both striking perpendicular to the σ₃ direction to maximize shear stress resolution.^[9] This symmetry arises from the theory's assumption of a homogeneous, isotropic medium under uniform stress, leading to fault planes oriented at about 30° to σ₁ for optimal failure.^[27] Representative examples of normal faulting aligned with Anderson's model include the Basin and Range province in the western United States, where high-angle normal faults accommodate widespread horizontal extension and crustal thinning over millions of years.^[27] Similarly, rift zones like the East African Rift demonstrate this regime, with fault arrays forming half-grabens and promoting continental breakup.^[27] Kinematic indicators on normal faults include down-dip slickensides, which record the direction of hanging wall downdrop, and earthquake fault-plane solutions that reveal extensional mechanisms with P-axes aligned vertically (parallel to σ₁) and T-axes horizontal (parallel to σ₃).^[6] These features confirm the slip vector's consistency with the predicted stress field. The theory predicts that normal faults will strike perpendicular to the σ₃ direction, effectively accommodating horizontal stretching and vertical thinning of the crust without significant strike-slip component.^[27] This orientation facilitates efficient extensional deformation, as seen in regions where fault reactivation aligns with ongoing tectonic extension.^[9]

Reverse and thrust faults

In Anderson's theory of faulting, reverse and thrust faults develop in a compressional tectonic regime where the minimum principal stress (σ₃) is oriented vertically, governed primarily by the lithostatic overburden pressure, while the maximum principal stress (σ₁) acts horizontally in the direction of shortening, and the intermediate stress (σ₂) is the other horizontal direction.^[28] This stress configuration, combined with the Coulomb failure criterion and a typical internal friction angle (φ) of approximately 30° for rocks, predicts that optimally oriented conjugate fault planes will dip at about 30° to the horizontal.^[6] The theory assumes a homogeneous, isotropic brittle crust near the surface, where the minimum principal stress aligns with gravity (vertical), combined with horizontal tectonic compression making the maximum principal stress horizontal.^[9] Geometrically, these faults exhibit reverse motion, with the hanging wall displaced upward relative to the footwall, and can manifest as low-angle thrusts (dips <30°) or steeper high-angle reverse faults (dips up to 50°–70°), the latter often representing reactivated or non-optimal planes.^[28] In mature compressional settings, such as fold-thrust belts, reverse and thrust faults commonly arrange into imbricate fans—overlapping series of subparallel thrusts that stack crustal slices and promote wedge-shaped thickening of the sedimentary cover.^[29] Kinematic indicators include slickenlines raking near 90° on the fault plane, indicating pure dip-slip motion, as well as overturned strata and asymmetric folds adjacent to the fault, where the hanging wall sequences are inverted due to the upward transport.^[28] Prominent examples of Andersonian reverse and thrust faulting occur in foreland basins and orogenic wedges, such as the fold-thrust belt of the European Alps, where Miocene compression produced imbricate thrusts like the Pennine Front, accommodating Africa-Europe convergence through low-angle detachment and ramping structures.^[29] Similarly, the Appalachian Valley and Ridge province features thrust sheets that align with the predicted 30° dips in shallow crustal levels, illustrating the theory's application to continental collision zones.^[6] While the pure Andersonian model assumes constant fault orientation, real-world thrusts often exhibit listric geometry, flattening at depth due to increasing overburden and pore pressure effects, which reduce effective normal stress and allow deviations from the ideal 30° dip toward even lower angles (e.g., 15°–20°).^[28] This adaptation arises from non-hydrostatic stress gradients and anisotropic layering, slightly modifying the theory's predictions without invalidating its foundational principles for brittle fault initiation.^[9]

Strike-slip faults

In Anderson's theory of faulting, strike-slip faults develop in a wrench tectonic regime where the intermediate principal stress (σ₂) is vertical due to the influence of lithostatic pressure near Earth's surface, while the maximum (σ₁) and minimum (σ₃) principal stresses are horizontal and oriented at approximately 45° to the fault plane.^[2] This configuration arises from the Mohr-Coulomb failure criterion, which predicts shear failure along planes optimally oriented to the principal stress directions under these conditions.^[30] The geometry of strike-slip faults in this regime consists of near-vertical fault planes that accommodate purely horizontal relative displacement between the hanging wall and footwall, resulting in either right-lateral (dextral) or left-lateral (sinistral) motion depending on the sense of shear. In regions of restraining bends along these faults, where the fault trace curves to impede lateral motion, positive flower structures can form, characterized by upward-splaying subsidiary faults that root into a subvertical master fault and produce localized uplift.^[2] Prominent examples include the San Andreas Fault system in California, a right-lateral strike-slip fault that exemplifies Anderson's predictions for transform boundaries between tectonic plates, with horizontal slip rates averaging about 3–5 cm per year.^[31] Kinematic indicators on strike-slip faults typically include horizontal slickensides—polished and striated fault surfaces with lineations parallel to the slip direction—and en echelon arrays of subsidiary shears or tension gashes that form at low angles to the main fault trace, revealing the sense of lateral movement.^[32] The theory predicts that strike-slip faults will strike parallel to the σ₂ direction, maximizing shear stress on vertical planes while producing no net dip-slip component, though conjugate shear pairs may develop at acute angles to σ₁ for balanced accommodation of horizontal strain.^[30]

Oblique faults

Oblique faults, also known as oblique-slip faults, exhibit hybrid movement combining both dip-slip and strike-slip components, resulting in slip vectors that are neither purely vertical nor horizontal. These faults represent an extension of Anderson's classical theory of faulting, which primarily predicts pure normal, reverse, or strike-slip motions under the assumption that one principal stress axis is vertical due to gravitational loading near the Earth's surface. By relaxing this assumption—allowing the maximum (σ₁) and minimum (σ₃) principal stresses to be inclined relative to the vertical—oblique faulting emerges as a natural outcome, particularly in regions where tectonic forces deviate from simple vertical-horizontal alignments. This inclination leads to non-optimal fault orientations, where slip occurs on planes that accommodate both shear components to minimize overall stress.^[6] In the stress regime for oblique faults, the obliquity of σ₁ and σ₃ to the surface produces faults with variable dips, typically moderate (around 45–60°), and rakes (slip angles) that deviate significantly from 0° (pure strike-slip) or 90° (pure dip-slip). Slickenside lineations on fault surfaces often trend obliquely, indicating the combined motion, and fault planes may form en echelon arrays to better align with the resolved shear stress. Such configurations arise because the rotated principal stresses cause the Coulomb failure criterion to favor hybrid slip directions on pre-existing or newly formed fractures, rather than the conjugate pairs predicted by Anderson's vertical-stress model. Oblique faulting is relatively uncommon in shallow crustal seismicity (less than 5% of well-constrained events shallower than 30 km), but it becomes more prevalent in deeper or tectonically complex settings where stress rotations occur.^[33]^[6] These faults are commonly observed in oblique tectonic environments, such as subduction zones with non-perpendicular convergence or pull-apart basins within strike-slip systems. For instance, the Alpine Fault in New Zealand serves as a continental transform boundary with significant oblique slip, accommodating both dextral strike-slip and reverse dip-slip components along a ~50° inclined plane, driven by the oblique relative motion between the Pacific and Australian plates. Similarly, in the Gulf of California, an oblique rift system, faulting includes hybrid structures along basin margins, where northwest-southeast extension combines with strike-slip to form moderate-dipping normal-oblique faults, facilitating the rapid propagation of the rift. These examples illustrate how oblique faults derive from Andersonian principles but adapt to inclined stress fields in real-world tectonics.^[34]^[35]

Applications in Geology

Tectonic regimes and stress indicators

Anderson's theory of faulting provides a foundational framework for interpreting tectonic regimes by linking the orientation of principal stresses to dominant fault types at or near the Earth's surface, where one principal stress is assumed to be vertical due to gravitational loading. In the extensional regime, characteristic of divergent plate boundaries like mid-ocean ridges and rift zones, the maximum principal stress (σ₁) is vertical, promoting normal faults that dip at approximately 60° and strike perpendicular to the minimum principal stress (σ₃). Compressional regimes, prevalent at convergent plate margins such as subduction zones, feature a vertical σ₃, resulting in low-angle reverse and thrust faults dipping at around 30° toward the direction of σ₁. The transcurrent or strike-slip regime, associated with transform boundaries like the San Andreas Fault system, has a vertical intermediate principal stress (σ₂), yielding near-vertical faults with horizontal slip parallel to σ₁ or σ₃. These associations enable geologists to infer plate boundary dynamics from observed fault patterns, as validated by global seismic catalogs showing dominance of these modes in shallow crust (0–30 km depth).^[6] Stress indicators derived from Anderson's theory allow reconstruction of both contemporary and ancient stress fields. Fault orientations serve as primary proxies, with slip directions aligning to maximize shear stress on optimally oriented planes, typically at 30° to σ₁ under Coulomb failure criteria. Earthquake focal mechanisms provide dynamic indicators, where the P-axis (axis of maximum compression) approximates σ₁ and the T-axis (maximum extension) aligns with σ₃; for instance, vertical P-axes in normal faulting events confirm extensional regimes, while vertical T-axes indicate compression. These axes, derived from moment tensor inversions of seismic data, reveal global stress patterns consistent with plate tectonics, such as σ₁ orientations parallel to convergence directions in orogenic belts.^[6] Paleostress analysis employs multiple fault sets to uncover stress evolution, particularly through Andersonian conjugate pairs—symmetrical faults intersecting at ~60° with slip senses revealing the bisected acute angle as σ₁ direction. Graphical methods, like the dihedral angle approach, superimpose compression and extension quadrants from fault-slip data on stereonets to constrain principal axes, assuming contemporaneous deformation and no significant rotation. Such analyses of conjugate sets in sedimentary basins, for example, have documented regime shifts from extension to compression during orogenic cycles, providing timelines of tectonic history.^[36] The World Stress Map (WSM) project exemplifies global application, compiling 100,842 indicators—including fault-slip data and focal mechanisms—to map horizontal stress orientations (SHmax) and regimes worldwide as of 2025.^[37] By integrating WSM data with GPS-derived plate motion models, studies show that ~70% of continental stress aligns with absolute plate velocities, reinforcing Anderson's predictions; for instance, extensional regimes dominate in the Basin and Range Province, corroborated by both fault patterns and seismic P-T axes. This synthesis highlights how Andersonian theory bridges geological observations with geodetic data for plate-scale stress modeling, with recent database expansions enhancing resolution and coverage.^[6] Quantitative inversion methods compute principal stresses from fault data under Andersonian assumptions, reducing the stress tensor to three parameters (two directions and a shape ratio). Techniques like Reches' (1987) least-squares inversion minimize misfit between observed slip vectors and those predicted by the stress tensor on fault planes, yielding σ₁/σ₃ orientations and ratios (e.g., R = (σ₂ - σ₃)/(σ₁ - σ₃)). Simpler graphical inversions on stereonets transform data to identify great-circle distributions normal to the reduced stress vector, enabling separation of heterogeneous datasets into regime-specific subsets, as applied to Alpine fault arrays. These methods assume frictional sliding (μ ≈ 0.6) and vertical principal stress, providing robust estimates for tectonic interpretation.^[38]^[39]

Seismicity and earthquake mechanisms

Anderson's theory of faulting provides a framework for understanding fault reactivation in seismicity, where pre-existing faults oriented according to the theory—typically at angles near 30° to 45° to the principal stress directions—fail when the resolved shear stress approaches the Coulomb failure limit, often requiring only a small increase in differential stress to trigger earthquakes.^[6] This reactivation is particularly relevant in the brittle upper crust, where frictional sliding on favorably oriented planes dominates seismic slip, as supported by analyses of borehole stress measurements and global earthquake catalogs.^[6] Focal mechanisms, visualized as "beach ball" diagrams, illustrate the orientations of principal stresses (σ₁ and σ₃) and slip directions consistent with Anderson's classifications, with the compressional (P) and tensional (T) axes aligning to show fault types.^[6] For instance, in extensional rifts, normal faulting mechanisms display a subvertical σ₁ (P axis near vertical) and horizontal σ₃, producing beach balls with white quadrants indicating extension on near-45° dipping planes, as observed in mid-oceanic ridge seismicity.^[6] Similarly, strike-slip events show horizontal σ₁ and σ₃ with vertical σ₂, yielding beach balls with alternating black and white quadrants on vertical nodal planes, while reverse mechanisms in convergent settings feature subhorizontal σ₁ and vertical σ₃.^[6] These patterns confirm the theory's prediction that one principal stress is vertical in shallow seismicity (<30 km depth), with dip-slip and strike-slip events comprising 32%–70% of well-constrained global data.^[6] Prominent examples demonstrate the theory's application to major earthquakes. The 1906 San Francisco earthquake (M_w 7.9) exemplified strike-slip faulting along the San Andreas fault, with right-lateral rupture over 430 km and average slip of ~4 m, aligning with Anderson's transcurrent regime where vertical stress is intermediate and horizontal maximum stress is subparallel to the fault trace.^[4] In contrast, the 1989 Loma Prieta earthquake (M_w 6.9) involved reverse-oblique slip on a steep (~70°) northeast-dipping plane within a restraining bend of the San Andreas system, representing a non-ideal Andersonian orientation that required higher differential stress or lower friction for failure compared to optimal ~30°–45° dips in pure compression.^[40]^[6] The theory informs seismic hazard assessment by enabling predictions of slip directions from inferred stress orientations and aftershock patterns through Coulomb stress modeling. In stress shadows—regions of decreased Coulomb failure stress following a mainshock—aftershocks are suppressed on nearby Andersonian faults, as seen in reduced seismicity rates post-1906 along parts of the San Andreas, allowing probabilistic forecasts of triggered versus inhibited events.^[41] Centennial analyses in 2008 validated these concepts using the Global Centroid Moment Tensor catalog (1976–2004), confirming that shallow crustal earthquakes predominantly follow Andersonian faulting, with rare oblique mechanisms (<10% of events) and nodal plane dips clustering around theoretical values (e.g., ~45° for normal faults), thus underscoring the theory's robustness for interpreting global seismicity patterns.^[6]

Limitations and Extensions

Violations of assumptions

Anderson's theory of faulting assumes that one of the principal stresses is vertical, arising from the free surface boundary condition at the Earth's surface, which leads to predictions of fault dips around 60° for normal faults and 30° for reverse faults relative to the principal stress axes.^[6] This assumption of vertical principal stresses is violated in various geological settings, such as near major faults, salt domes, and subduction zone interfaces, where the maximum principal stress (σ1) can become tilted due to local perturbations from structural features or density contrasts.^[6] For instance, around salt domes, rising salt bodies induce radial and circumferential stress fields that rotate principal stress directions away from vertical, leading to non-Andersonian fault orientations in the overlying sediments.^[42] In subduction zones, the underthrusting of oceanic plates causes σ1 to tilt subhorizontally toward the trench, deviating from the vertical orientation expected in Anderson's model.^[6] Observational evidence for these tilted stresses comes from borehole breakouts and hydraulic fractures; in vertical boreholes, breakouts form perpendicular to the maximum horizontal stress, but when principal stresses are inclined, breakout orientations vary with depth and indicate non-vertical σ1, as documented in profiles near the San Andreas Fault where stress directions fluctuate and tilt.^[43] Similarly, hydraulic fractures propagate perpendicular to the minimum principal stress, revealing tilted axes in proximity to fault zones.^[44] Another key violation involves low-angle normal faults (LANFs), which dip at less than 30° and contradict the theory's prediction of ~60° dips for normal faults under vertical σ1 and horizontal extension.^[45] These structures are prominent in metamorphic core complexes, where large-magnitude extension exhumes deep crustal rocks along shallowly dipping detachments.^[46] A representative example is the Snake Range décollement in east-central Nevada, a Tertiary low-angle normal fault (~20-25° dip) that accommodated over 36 km of eastward displacement, separating ductilely deformed mid-crustal rocks from brittle supracrustal layers, as evidenced by geological mapping and geophysical constraints.^[46] Such LANFs pose a mechanical paradox because frictional sliding on low-angle planes under Andersonian stress states would require unrealistically high differential stresses or strength drops.^[45] Pore fluid pressure further violates the theory's assumptions by reducing effective normal stress on fault planes, enabling slip on low-angle thrusts that dip below the predicted 30°.^[47] Elevated pore pressures, often approaching lithostatic levels in geosynclinal sediments, lower the effective stress coefficient (λ = pore pressure / overburden pressure) toward 1, allowing overthrust sheets to move despite shallow dips, as originally proposed for large-scale thrusting.^[47] This mechanism explains anomalies like low-angle thrusts in fold-and-thrust belts where Anderson's dry-rock friction model predicts instability.^[47] In the San Andreas Fault system, deviations from Andersonian stress orientations, including rotated principal axes and suboptimal fault angles, are partly attributed to such fluid effects combined with local stress perturbations, as inferred from earthquake focal mechanisms and in-situ measurements showing horizontal compression at acute angles to the fault trace.^[48] Subsequent developments have refined Anderson's theory by incorporating more realistic frictional behaviors observed in laboratory experiments. Byerlee's law provides an empirical description of rock friction on fault surfaces, stating that the shear stress required for sliding is approximately 0.85 times the normal stress for stresses above 200 MPa, and about 50 MPa for lower stresses, largely independent of rock type.^[49] This law extends the Coulomb friction assumption in Anderson's model by specifying friction coefficients (typically 0.6–0.85) that better match natural fault strengths in the upper crust.^[6] Further refinements account for the role of pore fluid pressure in reducing effective stress on faults, following Terzaghi's principle. The effective normal stress is given by \sigma' = \sigma - P, where \sigma is the total normal stress and P is the pore pressure, which lowers the frictional resistance and can trigger slip at lower differential stresses than predicted by Anderson's hydrostatic assumptions.^[50] Elevated pore pressures, often due to fluid migration or tectonic loading, explain seismicity in otherwise stable regimes and are integrated into modern fault stability assessments.^[4] Three-dimensional extensions of Anderson's theory relax the assumption of vertical principal stresses, allowing for tilted stress orientations that arise from complex tectonic interactions. Numerical models, such as those implemented in the FLAC (Fast Lagrangian Analysis of Continua) software, simulate fault development in heterogeneous media by incorporating full 3D stress tensors and material nonlinearity, revealing how inclined principals lead to non-conjugate fault patterns.^[51] These models demonstrate that deviations from Andersonian geometries occur in regions with pre-existing weaknesses or lateral variations in crustal properties.^[52] Alternative approaches hybridize Anderson's conjugate fault predictions with the Mohr-Coulomb criterion to handle intermediate stress effects and curved failure envelopes. In these models, fault orientations are determined by the full Mohr circle, accommodating hybrid fractures that combine extension and shear modes when the stress path intersects non-linear envelopes.^[53] Elastic dislocation theory provides another alternative for modeling fault evolution, treating faults as dislocations in an elastic medium to simulate progressive growth through linkage and propagation.^[54] This framework captures displacement accumulation and segment interaction, explaining irregular fault shapes beyond simple planar assumptions.^[55] Integrations with plate tectonics have mapped global stress provinces, where Andersonian regimes align with boundary forces like ridge push and slab pull. Zoback's analysis delineates first-order stress patterns over scales >500 km, showing compressional provinces in continental interiors consistent with plate-scale driving mechanisms.^[56] Recent advancements employ machine learning for stress inversion from fault-slip data, using neural networks to classify polyphase deformation and infer paleostress orientations with higher accuracy than traditional methods.^[57] As of 2025, Anderson's theory remains foundational for interpreting fault kinematics but is augmented by geodetic observations from GPS and InSAR, which provide real-time validation of stress orientations and slip vectors. These techniques reveal subtle deviations from ideal Andersonian dips, such as in global fault datasets, while confirming the theory's predictions in many active margins.^[58]^[59]

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