Critical taper
Encyclopedia
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.
In geodynamics the concept is used to explain tectonic
observations in accretionary wedge
s. Every wedge has a certain "critical angle", which depends on its material properties and the forces at work. This angle is determined by the ease by which internal deformation versus slip along the basal fault (décollemont) 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.
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.
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
.
force (the tectonic push) that created the wedge will be equal to the resisting forces inside the wedge.
al resistance at the base of the wedge (this is the shear strength
at/of the base,
). Mechanical equilibrium thus means:
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 (
) 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 (
) to get the component parallel to the base:
The second term (
) 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 ocean
s 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
(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 (
) and the gravitational acceleration (g):
The third term (
, the shear strength at the base of the wedge) can be given by the criterion of Mohr-Coulomb
:
In which S0 is the cohesion
of the material at the base,
is a coefficient of internal friction, 
is the normal stress and Pf the pore fluid pressure. These parameters determine the resistance to shear at the base.
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.
Mechanics
Mechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment....
and geodynamics
Geodynamics
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...
, 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
Decollement
Décollement is a gliding plane between two rock masses. In French, "décoller" means "to detach from" or "to rip off" and was first used by geologists studying the structure of the Swiss Jura Mountains, but is also known as a detachment zone. This is a structure of strata owing to deformation,...
) along the base of the wedge.
In geodynamics the concept is used to explain tectonic
Tectonics
Tectonics is a field of study within geology concerned generally with the structures within the lithosphere of the Earth and particularly with the forces and movements that have operated in a region to create these structures.Tectonics is concerned with the orogenies and tectonic development of...
observations in accretionary wedge
Accretionary wedge
An accretionary wedge or accretionary prism is formed from sediments that are accreted onto the non-subducting tectonic plate at a convergent plate boundary...
s. Every wedge has a certain "critical angle", which depends on its material properties and the forces at work. This angle is determined by the ease by which internal deformation versus slip along the basal fault (décollemont) 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.
When natural processes (such as erosion
Erosion
Erosion is when materials are removed from the surface and changed into something else. It only works by hydraulic actions and transport of solids in the natural environment, and leads to the deposition of these materials elsewhere...
, or an increase in load on the wedge due to emplacement of a sea or ice cap
Ice cap
An ice cap is an ice mass that covers less than 50 000 km² of land area . Masses of ice covering more than 50 000 km² are termed an ice sheet....
) 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.
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
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
.
Mechanical quantification
The critical taper concept assumes mechanical equilibrium, which means the compressionalCompression (geology)
In geology the term compression refers to a set of stresses 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...
force (the tectonic push) that created the wedge will be equal to the resisting forces inside the wedge.
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 frictionFriction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of friction:...
al resistance at the base of the wedge (this is the shear strength
Shear strength
Shear strength in engineering is a term used to describe the strength of a material or component against the type of yield or structural failure where 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...
at/of the base,
). Mechanical equilibrium thus means:- load of wedge + load of water +
= 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
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
of the wedge material (
) times the gravitational accelerationGravitational acceleration
In physics, gravitational acceleration is the acceleration on an object caused by gravity. Neglecting friction such as air resistance, all small bodies accelerate in a gravitational field at the same rate relative to the center of mass....
(g), working on a surface with dimensions dx and dy (unit vectors). This is multiplied by the sine
Sine
In mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....
of the angle of the base of the wedge (
) to get the component parallel to the base:- load of wedge =
The second term (
) 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 oceanOcean
An ocean is a major body of saline water, and a principal component of the hydrosphere. Approximately 71% of the Earth's surface is covered by ocean, a continuous body of water that is customarily divided into several principal oceans and smaller seas.More than half of this area is over 3,000...
s 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
(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 () and the gravitational acceleration (g):- load of water =
The third term (
, the shear strength at the base of the wedge) can be given by the criterion of Mohr-CoulombMohr-Coulomb theory
Mohr–Coulomb theory is a mathematical model describing the response of brittle materials such as concrete, or rubble piles, to shear stress as well as normal stress. Most of the classical engineering materials somehow follow this rule in at least a portion of their shear failure envelope...
:
In which S0 is the cohesion
Cohesion
Cohesion may refer to:* Cohesion , the intermolecular attraction between like-molecules* Cohesion , a measure of how well the lines of source code within a module work together...
of the material at the base,
is a coefficient of internal friction, is the normal stress and Pf 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:The pushing force is here assumed to be working on the total height of the wedge. Therefore it is written as the integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
of the force over the wedge height, where z is the direction perpendicular to the base of the wedge and parallel to vector H.