Critical wedges in three dimensions: Analytical expressions from Mohr‐Coulomb constrained perturbation analysis

Abstract

Abstract. We present simple analytical expressions for evaluating three-dimensional stress fields in critical wedges (Coulomb failure throughout) that are deforming within an oblique convergent zone. We assume that the load and geometry variations in the lateral direction, and the topographic slope, are relatively small. The stress state of a two dimensional critical wedge is perturbed by admitting a small lateral shear, which is accommodated (to maintain criticality) by a reduction in the dominant normal compressive stress. Cohesionless cases are further simplified with Taylor series expansions in the topographic slope. The analytical expressions distinguish the behavior of strong base wedges associated with steep slopes from weak base wedges with more modest slopes, and are reliable for lateral shear values as large as half the lithostatic load. In the weak base case where the vertical shear is small, the orientations of the principal stresses are very sensitive to the perturbation. As the lateral shear increases through relatively small values, the roles of the lateral and vertical coordinate axes are interchanged, with a resultant transition from predominantly ttu'ust failure to strike slip failure. Transition to strike slip in the strong base case is more gradual. The particular value of horizontal shear for which transition occurs also depends on the difference of the minor normal stresses. We produce an analytical expression predicting displacement patterns compatible with those observed in natural and analog orogens. Displacement partitioning is predicted for weak base wedges due to minor changes in lateral boundary conditions associated with variable orogen geometry.

Mechanical links between erosion and metamorphism in Nanga Parbat, Pakistan Himalaya

“…In an attempt to provide approximations of the geomorphic effect, we also assume a constant pore pressure ratio (Hubbert and Rubey, 1959), recognizing that an infinite variety of possible permutations of fluid pressure, rock compositions and strain-related rheologies may exist. Unless the intermediate principal stress ( 2 ) is close to the coordinate normal stress associated with either major or minor principal stresses, the topographic effect is generally insufficient to cause switching of the principal stresses and resultant displacement partitioning (Liu and Zoback, 1992;Enlow and Koons, 1998). For most of the examples considered here in this dominantly convergent part of the margin, the transition value of stress, , required to switch the sense of displacements from reverse to strike slip, given by: (Enlow and Koons, 1998) is not approached.…”

“…Unless the intermediate principal stress ( 2 ) is close to the coordinate normal stress associated with either major or minor principal stresses, the topographic effect is generally insufficient to cause switching of the principal stresses and resultant displacement partitioning (Liu and Zoback, 1992;Enlow and Koons, 1998). For most of the examples considered here in this dominantly convergent part of the margin, the transition value of stress, , required to switch the sense of displacements from reverse to strike slip, given by: (Enlow and Koons, 1998) is not approached. The influence of positive topographic loads on stress localization has been discussed elsewhere and arises from the combination of increased normal stress beneath the load, and shear stress concentration beneath the slopes and edges of the load (McTique and Mei, 1981;Savage and Swolfs, 1986;Liu and Zoback, 1992;Enlow and Koons, 1998).…”

“…For most of the examples considered here in this dominantly convergent part of the margin, the transition value of stress, , required to switch the sense of displacements from reverse to strike slip, given by: (Enlow and Koons, 1998) is not approached. The influence of positive topographic loads on stress localization has been discussed elsewhere and arises from the combination of increased normal stress beneath the load, and shear stress concentration beneath the slopes and edges of the load (McTique and Mei, 1981;Savage and Swolfs, 1986;Liu and Zoback, 1992;Enlow and Koons, 1998). The regional strain effects of a plateau-like topographic load, elongate ridge and isolated massif are similar in that each tends to deflect deformation away from the load to the region at the base of the confining slopes (England and Searle, 1986).…”

Upper crustal structure beneath the eastern Southern Alps and the Mackenzie Basin, New Zealand, derived from seismic reflection data

“…Some consist of finite-element solutions using complex but realistic rheologies (Braun & Beaumont 1995;Beaumont et al 1996;Batt & Braun 1997, 1999Koons et al 1998), while others use analytical and numerical applications of critical-wedge theory (Koons 1990(Koons , 1994Enlow & Koons 1998), and others are experimental sandbox analogues (Koons & Henderson 1995). The models are able to reproduce the mean topographic shape of the Southern Alps and provide a context for the interpretation of many of the observed structural features using relatively simplistic boundary conditions.…”

Viscous flow and deformation of regional metamorphic belts at convergent plate boundaries

“…[5] Another type of the models assumes a wedge of Coulomb or perfectly plastic rheology with the yield stress [e.g., Chapple, 1978;Davis et al, 1983;Stockmal, 1983;Enlow and Koons, 1998]. A diagnostic feature of this type of rheology comes from the relationship between the obliquity of convergence and the slip vectors of thrust earthquakes at subduction zones.…”