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2013-03-08T16:59:24Z

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			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 [[tectonics\|tectonic]] observations in [[accretionary wedge]]s. Every wedge has a certain "critical angle", which depends on its material properties and the forces at work.<ref>Chapple (1978)</ref><ref>Davis ''et al.'' (1983)</ref> 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.

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			An important presumption is that the internal deformation of the wedge takes place by [[frictional sliding]] or [[brittle fracture\|brittle fracturing]] and is therefore independent of temperature.<ref>Davis ''et al.'' (1983)</ref>

			==Mechanical quantification==
			[[Image:Critical taper wedge.svg\|thumb\|350px\|Schematic representation of a wedge of sediments, pushed over a slope (for example a downward bending [[tectonic plate\|plate]]) by a force <math>\sigma</math><sub>x</sub>. At mechanic equilibrium, the resisting force parallel to the base slope (indicated by a red arrow) will equal the pushing force. The material properties and magnitude of the forces determine the critical angle <math>\alpha + \beta</math> of the wedge.]]
			The critical taper concept assumes mechanical equilibrium, which means the [[Compression (geology)\|compressional]] 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 [[friction]]al resistance at the base of the wedge (this is the [[shear strength]] at/of the base, <math>\tau_b</math>). Mechanical equilibrium thus means:

			:load of wedge + load of water + <math>\!\tau_b</math> = 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 (<math>\rho</math>) times the [[gravitational acceleration]] (g), working on a surface with dimensions dx and dy ([[unit vector]]s). This is multiplied by the [[sine]] of the angle of the base of the wedge (<math>\beta</math>) to get the component parallel to the base:

			:load of wedge = <math>\!\rho g H sin\beta</math>

			The second term (<math>\rho_w * g * D * sin(\alpha + \beta)</math>) is the resisting force of the load of an eventual water column on top of the wedge. Accretionary wedges in front of [[subduction zone]]s 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 <math>\alpha + \beta</math> (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 (<math>\rho_w</math>) and the gravitational acceleration (g):

			:load of water = <math>\!\rho_w g D sin(\alpha + \beta)</math>

			The third term (<math>\tau_b</math>, the shear strength at the base of the wedge) can be given by the [[Mohr-Coulomb theory\|criterion of Mohr-Coulomb]]:

			:<math>\!\tau_b = S_0 + \mu(\sigma_n - P_f)</math>

			In which S<sub>0</sub> is the [[Cohesion (chemistry)\|cohesion]] of the material at the base, <math>\mu</math> is a [[coefficient of internal friction]], <math>\sigma_n</math> is the [[normal stress]] and P<sub>f</sub> 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:

			:<math>\rho g H sin\beta + \rho_w g D sin(\alpha + \beta) + \tau_b = \frac{d}{dx}\int_{0}^{H}\sigma_x dz</math>

			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.

			==References==
			===Notes===
			{{reflist\|2}}

			===Sources===
			*{{aut\|Chapple, W.M.}}; '''1978''': ''Mechanics of Thin-Skinned Fold-and-Thrust Belts'', Geological Society of America Bulletin '''89''', pp 1189–1198.
			*{{aut\|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.
			*{{aut\|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.

			{{DEFAULTSORT:Critical Taper}}
			[[Category:Tectonics]]

en>Evilbu: added independence number (not so trivial, source is http://mathworld.wolfram.com/IndependentSet.html), perhaps should be in infobox?

2012-07-05T10:15:19Z

added independence number (not so trivial, source is http://mathworld.wolfram.com/IndependentSet.html), perhaps should be in infobox?

New page

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Frucht graph - Revision history

en>Addbot: Bot: Migrating 3 interwiki links, now provided by Wikidata on d:q2061755

en>Evilbu: added independence number (not so trivial, source is http://mathworld.wolfram.com/IndependentSet.html), perhaps should be in infobox?