Critical taper

Last updated March 22, 2019

In mechanics and geodynamics, a critical taper is the equilibrium angle made by the far end of a wedge-shaped agglomeration of material that is being pushed by the near end. The angle of the critical taper is a function of the material properties within the wedge, pore fluid pressure, and strength of the fault (or décollement) along the base of the wedge.

Mechanics is that area of science concerned with the behaviour of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment. The scientific discipline has its origins in Ancient Greece with the writings of Aristotle and Archimedes. During the early modern period, scientists such as Galileo, Kepler, and Newton laid the foundation for what is now known as classical mechanics. It is a branch of classical physics that deals with particles that are either at rest or are moving with velocities significantly less than the speed of light. It can also be defined as a branch of science which deals with the motion of and forces on objects. The field is yet less widely understood in terms of quantum theory.

Geodynamics Study of dynamics of the Earth

Geodynamics is a subfield of geophysics dealing with dynamics of the Earth. It applies physics, chemistry and mathematics to the understanding of how mantle convection leads to plate tectonics and geologic phenomena such as seafloor spreading, mountain building, volcanoes, earthquakes, faulting and so on. It also attempts to probe the internal activity by measuring magnetic fields, gravity, and seismic waves, as well as the mineralogy of rocks and their isotopic composition. Methods of geodynamics are also applied to exploration of other planets.

Décollement is a gliding plane between two rock masses, also known as a basal detachment fault. Décollements are a deformational structure, resulting in independent styles of deformation in the rocks above and below the fault. They are associated with both compressional settings and extensional settings.

In geodynamics the concept is used to explain tectonic observations in accretionary wedges. Every wedge has a certain "critical angle", which depends on its material properties and the forces at work.^[1]^[2] This angle is determined by the ease by which internal deformation versus slip along the basal fault (décollement) occurs. If the wedge deforms more easily internally than along the décollement, material will pile up and the wedge will reach a steeper critical taper until such a point as the high angle of the taper makes internal deformation more difficult than sliding along the base. If the basal décollement deforms more easily than the material does internally, the opposite will occur. The result of these feedbacks is the stable angle of the wedge known as the critical taper.

Tectonics is the process that controls the structure and properties of the Earth's crust and its evolution through time. In particular, it describes the processes of mountain building, the growth and behavior of the strong, old cores of continents known as cratons, and the ways in which the relatively rigid plates that constitute the Earth's outer shell interact with each other. Tectonics also provides a framework for understanding the earthquake and volcanic belts that directly affect much of the global population. Tectonic studies are important as guides for economic geologists searching for fossil fuels and ore deposits of metallic and nonmetallic resources. An understanding of tectonic principles is essential to geomorphologists to explain erosion patterns and other Earth surface features.

Accretionary wedge The sediments accreted onto the non-subducting tectonic plate at a convergent plate boundary

An accretionary wedge or accretionary prism forms from sediments accreted onto the non-subducting tectonic plate at a convergent plate boundary. Most of the material in the accretionary wedge consists of marine sediments scraped off from the downgoing slab of oceanic crust, but in some cases the wedge includes the erosional products of volcanic island arcs formed on the overriding plate.

When natural processes (such as erosion, or an increase in load on the wedge due to emplacement of a sea or ice cap) change the shape of the wedge, the wedge will react by internally deforming to return to a critically tapered wedge shape. The critical taper concept can thus explain and predict phases and styles of tectonics in wedges.

In earth science, erosion is the action of surface processes that removes soil, rock, or dissolved material from one location on the Earth's crust, and then transports it to another location. This natural process is caused by the dynamic activity of erosive agents, that is, water, ice (glaciers), snow, air (wind), plants, animals, and humans. In accordance with these agents, erosion is sometimes divided into water erosion, glacial erosion, snow erosion, wind (aeolic) erosion, zoogenic erosion, and anthropogenic erosion. The particulate breakdown of rock or soil into clastic sediment is referred to as physical or mechanical erosion; this contrasts with chemical erosion, where soil or rock material is removed from an area by its dissolving into a solvent, followed by the flow away of that solution. Eroded sediment or solutes may be transported just a few millimetres, or for thousands of kilometres.

Ice cap ice mass that covers less than 50,000 km² of land area

An ice cap is a mass of ice that covers less than 50,000 km² of land area. Larger ice masses covering more than 50,000 km² are termed ice sheets.

In materials science, deformation refers to any changes in the shape or size of an object due to

An important presumption is that the internal deformation of the wedge takes place by frictional sliding or brittle fracturing and is therefore independent of temperature.^[3]

Mechanical quantification

The critical taper concept assumes mechanical equilibrium, which means the compressional force (the tectonic push) that created the wedge will be equal to the resisting forces inside the wedge.

Mechanical equilibrium (in classical mechanics) a particle is in mechanical equilibrium if the net force on that particle is zero

In classical mechanics, a particle is in mechanical equilibrium if the net force on that particle is zero. By extension, a physical system made up of many parts is in mechanical equilibrium if the net force on each of its individual parts is zero.

In geology, the term compression refers to a set of stress directed toward the center of a rock mass. Compressive strength refers to the maximum compressive stress that can be applied to a material before failure occurs. When the maximum compressive stress is in a horizontal orientation, thrust faulting can occur, resulting in the shortening and thickening of that portion of the crust. When the maximum compressive stress is vertical, a section of rock will often fail in normal faults, horizontally extending and vertically thinning a given layer of rock. Compressive stresses can also result in folding of rocks. Because of the large magnitudes of lithostatic stress in tectonic plates, tectonic-scale deformation is always subjected to net compressive stress.

Resisting forces

These forces resisting the tectonic force are the load (weight) of the wedge itself, the eventual load of an overlying column of water and the frictional resistance at the base of the wedge (this is the shear strength at/of the base, $\tau _{b}$ ). Mechanical equilibrium thus means:

Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction:

Shear strength strength of a material or component against the type of yield or structural failure when the material or component fails in shear

In engineering, shear strength is the strength of a material or component against the type of yield or structural failure when the material or component fails in shear. A shear load is a force that tends to produce a sliding failure on a material along a plane that is parallel to the direction of the force. When a paper is cut with scissors, the paper fails in shear.

load of wedge + load of water +

\!\tau _{b}

= tectonic push

The first term in this formula stands for the resisting force of the load of the wedge along the base of the wedge. This force is the density of the wedge material ( $\rho$ ) times the gravitational acceleration (g), working on a surface with dimensions dx and dy (unit vectors). This is multiplied by the sine of the angle of the base of the wedge ( $\beta$ ) to get the component parallel to the base:

load of wedge =

\!\rho gHsin\beta

The second term ( $\rho _{w}*g*D*sin(\alpha +\beta )$ ) is the resisting force of the load of an eventual water column on top of the wedge. Accretionary wedges in front of subduction zones are normally covered by oceans and the weight of the seawater on top of the wedge can be significant. The load of the water column is the hydrostatic pressure of the water column, multiplied by a factor $\alpha +\beta$ (the angle between the top of the wedge and the base of the wedge) to get the component parallel to the base of the wedge. The hydrostatic pressure is calculated as the product of the density of water ( $\rho _{w}$ ) and the gravitational acceleration (g):

load of water =

\!\rho _{w}gDsin(\alpha +\beta )

The third term ( $\tau _{b}$ , the shear strength at the base of the wedge) can be given by the criterion of Mohr-Coulomb:

\!\tau _{b}=S_{0}+\mu (\sigma _{n}-P_{f})

In which S₀ is the cohesion of the material at the base, $\mu$ is a coefficient of internal friction, $\sigma _{n}$ is the normal stress and P_f the pore fluid pressure. These parameters determine the resistance to shear at the base.

Mechanical equilibrium

Mechanical equilibrium means the resisting forces equal the push. This can be written as:

\rho gHsin\beta +\rho _{w}gDsin(\alpha +\beta )+\tau _{b}={\frac {d}{dx}}\int _{0}^{H}\sigma _{x}dz

The pushing force is here assumed to be working on the total height of the wedge. Therefore, it is written as the integral of the force over the wedge height, where z is the direction perpendicular to the base of the wedge and parallel to vector H.

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References

Notes

↑ Chapple (1978)
↑ Davis et al. (1983)
↑ Davis et al. (1983)

Sources

Chapple, W.M.; 1978: Mechanics of Thin-Skinned Fold-and-Thrust Belts, Geological Society of America Bulletin 89, pp 1189–1198.
Davis, D.; Suppe, J. & Dahlen, F.A.; 1983: Mechanics of Fold-and-Thrust Belts and Accretionary Wedges, Journal of Geophysical Research 88(B2), pp 1153–1178.
Dahlen, F.A.; Suppe, J. & Davis, D.; 1984: Mechanics of Fold-and-Thrust Belts and Accretionary Wedges' Cohesive Coulomb Theory, Journal of Geophysical Research 89(B12), pp 10,087-10,101.

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[1] Chapple (1978)

[2] Davis et al. (1983)

[3] Davis et al. (1983)