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99-734: The Chaman Fault is a major, active geological fault in Pakistan and Afghanistan that runs for over 850 km. Tectonically , it is actually a system of related geologic faults that separates the Eurasian plate from the Indo-Australian plate . It is a terrestrial, primarily transform , left-lateral strike-slip fault. The slippage rate along the Chaman fault system as the Indo-Australian plate moves northward (relative to

198-420: A flow of viscous liquid , the force F may not be perpendicular to S ; hence the stress across a surface must be regarded a vector quantity, not a scalar. Moreover, the direction and magnitude generally depend on the orientation of S . Thus the stress state of the material must be described by a tensor , called the (Cauchy) stress tensor ; which is a linear function that relates the normal vector n of

297-439: A "particle" as being an infinitesimal patch of the plate's surface, so that the boundary between adjacent particles becomes an infinitesimal line element; both are implicitly extended in the third dimension, normal to (straight through) the plate. "Stress" is then redefined as being a measure of the internal forces between two adjacent "particles" across their common line element, divided by the length of that line. Some components of

396-392: A coordinate system with axes e 1 , e 2 , e 3 {\displaystyle e_{1},e_{2},e_{3}} , the stress tensor is a diagonal matrix, and has only the three normal components λ 1 , λ 2 , λ 3 {\displaystyle \lambda _{1},\lambda _{2},\lambda _{3}}

495-457: A cylindrical bar such as a shaft is subjected to opposite torques at its ends. In that case, the shear stress on each cross-section is parallel to the cross-section, but oriented tangentially relative to the axis, and increases with distance from the axis. Significant shear stress occurs in the middle plate (the "web") of I-beams under bending loads, due to the web constraining the end plates ("flanges"). Another simple type of stress occurs when

594-454: A differential formula for friction forces (shear stress) in parallel laminar flow . Stress is defined as the force across a small boundary per unit area of that boundary, for all orientations of the boundary. Derived from a fundamental physical quantity (force) and a purely geometrical quantity (area), stress is also a fundamental quantity, like velocity, torque or energy , that can be quantified and analyzed without explicit consideration of

693-405: A fault as oblique requires both dip and strike components to be measurable and significant. Some oblique faults occur within transtensional and transpressional regimes, and others occur where the direction of extension or shortening changes during the deformation but the earlier formed faults remain active. The hade angle is defined as the complement of the dip angle; it is the angle between

792-582: A fault hosting valuable porphyry copper deposits is northern Chile's Domeyko Fault with deposits at Chuquicamata , Collahuasi , El Abra , El Salvador , La Escondida and Potrerillos . Further south in Chile Los Bronces and El Teniente porphyry copper deposit lie each at the intersection of two fault systems. Faults may not always act as conduits to surface. It has been proposed that deep-seated "misoriented" faults may instead be zones where magmas forming porphyry copper stagnate achieving

891-489: A fault plane, where it becomes locked, are called asperities . Stress builds up when a fault is locked, and when it reaches a level that exceeds the strength threshold, the fault ruptures and the accumulated strain energy is released in part as seismic waves , forming an earthquake . Strain occurs accumulatively or instantaneously, depending on the liquid state of the rock; the ductile lower crust and mantle accumulate deformation gradually via shearing , whereas

990-408: A fault's age by studying soil features seen in shallow excavations and geomorphology seen in aerial photographs. Subsurface clues include shears and their relationships to carbonate nodules , eroded clay, and iron oxide mineralization, in the case of older soil, and lack of such signs in the case of younger soil. Radiocarbon dating of organic material buried next to or over a fault shear

1089-427: A hanging wall or foot wall where a thrust fault formed along a relatively weak bedding plane is known as a flat and a section where the thrust fault cut upward through the stratigraphic sequence is known as a ramp . Typically, thrust faults move within formations by forming flats and climbing up sections with ramps. This results in the hanging wall flat (or a portion thereof) lying atop the foot wall ramp as shown in

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1188-484: A major fault. Synthetic faults dip in the same direction as the major fault while the antithetic faults dip in the opposite direction. These faults may be accompanied by rollover anticlines (e.g. the Niger Delta Structural Style). All faults have a measurable thickness, made up of deformed rock characteristic of the level in the crust where the faulting happened, of the rock types affected by

1287-400: A manner that creates multiple listric faults. The fault panes of listric faults can further flatten and evolve into a horizontal or near-horizontal plane, where slip progresses horizontally along a decollement . Extensional decollements can grow to great dimensions and form detachment faults , which are low-angle normal faults with regional tectonic significance. Due to the curvature of

1386-421: A non-vertical fault are known as the hanging wall and footwall . The hanging wall occurs above the fault plane and the footwall occurs below it. This terminology comes from mining: when working a tabular ore body, the miner stood with the footwall under his feet and with the hanging wall above him. These terms are important for distinguishing different dip-slip fault types: reverse faults and normal faults. In

1485-464: A reverse fault, the hanging wall displaces upward, while in a normal fault the hanging wall displaces downward. Distinguishing between these two fault types is important for determining the stress regime of the fault movement. Faults are mainly classified in terms of the angle that the fault plane makes with the Earth's surface, known as the dip , and the direction of slip along the fault plane. Based on

1584-463: A solid material generates an internal elastic stress , analogous to the reaction force of a spring , that tends to restore the material to its original non-deformed state. In liquids and gases , only deformations that change the volume generate persistent elastic stress. If the deformation changes gradually with time, even in fluids there will usually be some viscous stress , opposing that change. Elastic and viscous stresses are usually combined under

1683-452: A surface S to the traction vector T across S . With respect to any chosen coordinate system , the Cauchy stress tensor can be represented as a symmetric matrix of 3×3 real numbers. Even within a homogeneous body, the stress tensor may vary from place to place, and may change over time; therefore, the stress within a material is, in general, a time-varying tensor field . In general,

1782-1007: A surface will always be a linear function of the surface's normal vector n {\displaystyle n} , the unit-length vector that is perpendicular to it. That is, T = σ ( n ) {\displaystyle T={\boldsymbol {\sigma }}(n)} , where the function σ {\displaystyle {\boldsymbol {\sigma }}} satisfies σ ( α u + β v ) = α σ ( u ) + β σ ( v ) {\displaystyle {\boldsymbol {\sigma }}(\alpha u+\beta v)=\alpha {\boldsymbol {\sigma }}(u)+\beta {\boldsymbol {\sigma }}(v)} for any vectors u , v {\displaystyle u,v} and any real numbers α , β {\displaystyle \alpha ,\beta } . The function σ {\displaystyle {\boldsymbol {\sigma }}} , now called

1881-434: A surface with normal vector n {\displaystyle n} (which is covariant - "row; horizontal" - vector) with coordinates n 1 , n 2 , n 3 {\displaystyle n_{1},n_{2},n_{3}} is then a matrix product T = n ⋅ σ {\displaystyle T=n\cdot {\boldsymbol {\sigma }}} (where T in upper index

1980-413: A system must be balanced by internal reaction forces, which are almost always surface contact forces between adjacent particles — that is, as stress. Since every particle needs to be in equilibrium, this reaction stress will generally propagate from particle to particle, creating a stress distribution throughout the body. The typical problem in stress analysis is to determine these internal stresses, given

2079-434: A system of partial differential equations involving the stress tensor field and the strain tensor field, as unknown functions to be determined. The external body forces appear as the independent ("right-hand side") term in the differential equations, while the concentrated forces appear as boundary conditions. The basic stress analysis problem is therefore a boundary-value problem . Stress analysis for elastic structures

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2178-489: A two-dimensional one, and/or replace the general stress and strain tensors by simpler models like uniaxial tension/compression, simple shear, etc. Still, for two- or three-dimensional cases one must solve a partial differential equation problem. Analytical or closed-form solutions to the differential equations can be obtained when the geometry, constitutive relations, and boundary conditions are simple enough. Otherwise one must generally resort to numerical approximations such as

2277-435: A very large number of intermolecular forces and collisions between the particles in those molecules . Stress is frequently represented by a lowercase Greek letter sigma ( σ ). Strain inside a material may arise by various mechanisms, such as stress as applied by external forces to the bulk material (like gravity ) or to its surface (like contact forces , external pressure, or friction ). Any strain (deformation) of

2376-1092: Is transposition , and as a result we get covariant (row) vector) (look on Cauchy stress tensor ), that is [ T 1 T 2 T 3 ] = [ n 1 n 2 n 3 ] ⋅ [ σ 11 σ 21 σ 31 σ 12 σ 22 σ 32 σ 13 σ 23 σ 33 ] {\displaystyle {\begin{bmatrix}T_{1}&T_{2}&T_{3}\end{bmatrix}}={\begin{bmatrix}n_{1}&n_{2}&n_{3}\end{bmatrix}}\cdot {\begin{bmatrix}\sigma _{11}&\sigma _{21}&\sigma _{31}\\\sigma _{12}&\sigma _{22}&\sigma _{32}\\\sigma _{13}&\sigma _{23}&\sigma _{33}\end{bmatrix}}} The linear relation between T {\displaystyle T} and n {\displaystyle n} follows from

2475-422: Is a horst . A sequence of grabens and horsts on the surface of the Earth produces a characteristic basin and range topography . Normal faults can evolve into listric faults, with their plane dip being steeper near the surface, then shallower with increased depth, with the fault plane curving into the Earth. They can also form where the hanging wall is absent (such as on a cliff), where the footwall may slump in

2574-423: Is a zone of folding close to a fault that likely arises from frictional resistance to movement on the fault. The direction and magnitude of heave and throw can be measured only by finding common intersection points on either side of the fault (called a piercing point ). In practice, it is usually only possible to find the slip direction of faults, and an approximation of the heave and throw vector. The two sides of

2673-410: Is actually the average of a very large number of atomic forces between their molecules; and physical quantities like mass, velocity, and forces that act through the bulk of three-dimensional bodies, like gravity, are assumed to be smoothly distributed over them. Depending on the context, one may also assume that the particles are large enough to allow the averaging out of other microscopic features, like

2772-583: Is an essential tool in engineering for the study and design of structures such as tunnels, dams, mechanical parts, and structural frames, under prescribed or expected loads. It is also important in many other disciplines; for example, in geology, to study phenomena like plate tectonics , vulcanism and avalanches ; and in biology, to understand the anatomy of living beings. Stress analysis is generally concerned with objects and structures that can be assumed to be in macroscopic static equilibrium . By Newton's laws of motion , any external forces being applied to such

2871-406: Is assumed fixed, the normal component can be expressed by a single number, the dot product T · n . This number will be positive if P is "pulling" on Q (tensile stress), and negative if P is "pushing" against Q (compressive stress). The shear component is then the vector T − ( T · n ) n . The dimension of stress is that of pressure , and therefore its coordinates are measured in

2970-478: Is based on the theory of elasticity and infinitesimal strain theory . When the applied loads cause permanent deformation, one must use more complicated constitutive equations, that can account for the physical processes involved ( plastic flow , fracture , phase change , etc.). Engineered structures are usually designed so the maximum expected stresses are well within the range of linear elasticity (the generalization of Hooke's law for continuous media); that is,

3069-544: Is defined by the direction of movement of the ground as would be seen by an observer on the opposite side of the fault. A special class of strike-slip fault is the transform fault when it forms a plate boundary. This class is related to an offset in a spreading center , such as a mid-ocean ridge , or, less common, within continental lithosphere , such as the Dead Sea Transform in the Middle East or

Chaman Fault - Misplaced Pages Continue

3168-641: Is given in the article on viscosity . The same for normal viscous stresses can be found in Sharma (2019). The relation between stress and its effects and causes, including deformation and rate of change of deformation, can be quite complicated (although a linear approximation may be adequate in practice if the quantities are small enough). Stress that exceeds certain strength limits of the material will result in permanent deformation (such as plastic flow , fracture , cavitation ) or even change its crystal structure and chemical composition . In some situations,

3267-543: Is often critical in distinguishing active from inactive faults. From such relationships, paleoseismologists can estimate the sizes of past earthquakes over the past several hundred years, and develop rough projections of future fault activity. Many ore deposits lie on or are associated with faults. This is because the fractured rock associated with fault zones allow for magma ascent or the circulation of mineral-bearing fluids. Intersections of near-vertical faults are often locations of significant ore deposits. An example of

3366-513: Is often used for safety certification and monitoring. Most stress is analysed by mathematical methods, especially during design. The basic stress analysis problem can be formulated by Euler's equations of motion for continuous bodies (which are consequences of Newton's laws for conservation of linear momentum and angular momentum ) and the Euler-Cauchy stress principle , together with the appropriate constitutive equations. Thus one obtains

3465-492: Is particularly clear in the case of detachment faults and major thrust faults . The main types of fault rock include: In geotechnical engineering , a fault often forms a discontinuity that may have a large influence on the mechanical behavior (strength, deformation, etc.) of soil and rock masses in, for example, tunnel , foundation , or slope construction. The level of a fault's activity can be critical for (1) locating buildings, tanks, and pipelines and (2) assessing

3564-408: Is perpendicular to the layer, the net internal force across S , and hence the stress, will be zero. As in the case of an axially loaded bar, in practice the shear stress may not be uniformly distributed over the layer; so, as before, the ratio F / A will only be an average ("nominal", "engineering") stress. That average is often sufficient for practical purposes. Shear stress is observed also when

3663-412: Is subjected to tension by opposite forces of magnitude F {\displaystyle F} along its axis. If the system is in equilibrium and not changing with time, and the weight of the bar can be neglected, then through each transversal section of the bar the top part must pull on the bottom part with the same force, F with continuity through the full cross-sectional area , A . Therefore,

3762-410: Is supporting an overhead weight , each particle in the bar pushes on the particles immediately below it. When a liquid is in a closed container under pressure , each particle gets pushed against by all the surrounding particles. The container walls and the pressure -inducing surface (such as a piston) push against them in (Newtonian) reaction . These macroscopic forces are actually the net result of

3861-425: Is the cause of most earthquakes . Faults may also displace slowly, by aseismic creep . A fault plane is the plane that represents the fracture surface of a fault. A fault trace or fault line is a place where the fault can be seen or mapped on the surface. A fault trace is also the line commonly plotted on geologic maps to represent a fault. A fault zone is a cluster of parallel faults. However,

3960-437: Is then reduced to a scalar (tension or compression of the bar), but one must take into account also a bending stress (that tries to change the bar's curvature, in some direction perpendicular to the axis) and a torsional stress (that tries to twist or un-twist it about its axis). Stress analysis is a branch of applied physics that covers the determination of the internal distribution of internal forces in solid objects. It

4059-576: Is too small to be detected. In a solid material, such strain will in turn generate an internal elastic stress, analogous to the reaction force of a stretched spring , tending to restore the material to its original undeformed state. Fluid materials (liquids, gases and plasmas ) by definition can only oppose deformations that would change their volume. If the deformation changes with time, even in fluids there will usually be some viscous stress, opposing that change. Such stresses can be either shear or normal in nature. Molecular origin of shear stresses in fluids

Chaman Fault - Misplaced Pages Continue

4158-567: Is west of these ranges. Fault (geology) In geology , a fault is a planar fracture or discontinuity in a volume of rock across which there has been significant displacement as a result of rock-mass movements. Large faults within Earth 's crust result from the action of plate tectonic forces, with the largest forming the boundaries between the plates, such as the megathrust faults of subduction zones or transform faults . Energy release associated with rapid movement on active faults

4257-505: The (Cauchy) stress tensor , completely describes the stress state of a uniformly stressed body. (Today, any linear connection between two physical vector quantities is called a tensor , reflecting Cauchy's original use to describe the "tensions" (stresses) in a material.) In tensor calculus , σ {\displaystyle {\boldsymbol {\sigma }}} is classified as a second-order tensor of type (0,2) or (1,1) depending on convention. Like any linear map between vectors,

4356-762: The Alpine Fault in New Zealand. Transform faults are also referred to as "conservative" plate boundaries since the lithosphere is neither created nor destroyed. Dip-slip faults can be either normal (" extensional ") or reverse . The terminology of "normal" and "reverse" comes from coal mining in England, where normal faults are the most common. With the passage of time, a regional reversal between tensional and compressional stresses (or vice-versa) might occur, and faults may be reactivated with their relative block movement inverted in opposite directions to

4455-529: The Indo-Australian plate meet, which is just off the Makran Coast of Pakistan. The fault tracks northeast across Balochistan and then north-northeast into Afghanistan, runs just to the west of Kabul , and then northeastward across the right-lateral-slip Herat fault, up to where it merges with the Pamir fault system north of the 38° parallel. The Ghazaband and Ornach-Nal faults are often included as part of

4554-610: The capitals , arches , cupolas , trusses and the flying buttresses of Gothic cathedrals . Ancient and medieval architects did develop some geometrical methods and simple formulas to compute the proper sizes of pillars and beams, but the scientific understanding of stress became possible only after the necessary tools were invented in the 17th and 18th centuries: Galileo Galilei 's rigorous experimental method , René Descartes 's coordinates and analytic geometry , and Newton 's laws of motion and equilibrium and calculus of infinitesimals . With those tools, Augustin-Louis Cauchy

4653-993: The orthogonal shear stresses . The Cauchy stress tensor obeys the tensor transformation law under a change in the system of coordinates. A graphical representation of this transformation law is the Mohr's circle of stress distribution. As a symmetric 3×3 real matrix, the stress tensor σ {\displaystyle {\boldsymbol {\sigma }}} has three mutually orthogonal unit-length eigenvectors e 1 , e 2 , e 3 {\displaystyle e_{1},e_{2},e_{3}} and three real eigenvalues λ 1 , λ 2 , λ 3 {\displaystyle \lambda _{1},\lambda _{2},\lambda _{3}} , such that σ e i = λ i e i {\displaystyle {\boldsymbol {\sigma }}e_{i}=\lambda _{i}e_{i}} . Therefore, in

4752-457: The principal stresses . If the three eigenvalues are equal, the stress is an isotropic compression or tension, always perpendicular to any surface, there is no shear stress, and the tensor is a diagonal matrix in any coordinate frame. In general, stress is not uniformly distributed over a material body, and may vary with time. Therefore, the stress tensor must be defined for each point and each moment, by considering an infinitesimal particle of

4851-657: The seismic shaking and tsunami hazard to infrastructure and people in the vicinity. In California, for example, new building construction has been prohibited directly on or near faults that have moved within the Holocene Epoch (the last 11,700 years) of the Earth's geological history. Also, faults that have shown movement during the Holocene plus Pleistocene Epochs (the last 2.6 million years) may receive consideration, especially for critical structures such as power plants, dams, hospitals, and schools. Geologists assess

4950-442: The strain rate can be quite complicated, although a linear approximation may be adequate in practice if the quantities are sufficiently small. Stress that exceeds certain strength limits of the material will result in permanent deformation (such as plastic flow , fracture , cavitation ) or even change its crystal structure and chemical composition . Humans have known about stress inside materials since ancient times. Until

5049-445: The 17th century, this understanding was largely intuitive and empirical, though this did not prevent the development of relatively advanced technologies like the composite bow and glass blowing . Over several millennia, architects and builders in particular, learned how to put together carefully shaped wood beams and stone blocks to withstand, transmit, and distribute stress in the most effective manner, with ingenious devices such as

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5148-585: The Chaman fault system. South of the triple junction, where the fault zone lies undersea and extends southwest to approximately 10°N 57°E, it is known as the Owen Fracture Zone . While there is general agreement that the fault is slipping at a rate of at least 10 mm/yr, there is a report of volcanic rocks in Pakistan dated to 2 m.y. BP which have been offset such as to indicate a slip rate of 25–35 mm/yr. Offsets have been described throughout

5247-583: The Eurasian plate) has been estimated at 10 mm/yr or more. In addition to its primary transform aspect, the Chaman fault system has a compressional component as the Indian plate is colliding with the Eurasian plate. This type of plate boundary is sometimes called a transpressional boundary. From the south, the Chaman fault starts at the triple junction where the Arabian plate , the Eurasian plate and

5346-403: The brittle upper crust reacts by fracture – instantaneous stress release – resulting in motion along the fault. A fault in ductile rocks can also release instantaneously when the strain rate is too great. Slip is defined as the relative movement of geological features present on either side of a fault plane. A fault's sense of slip is defined as the relative motion of the rock on each side of

5445-449: The bulk of the material, varying continuously with position and time. Other agents (like external loads and friction, ambient pressure, and contact forces) may create stresses and forces that are concentrated on certain surfaces, lines or points; and possibly also on very short time intervals (as in the impulses due to collisions). In active matter , self-propulsion of microscopic particles generates macroscopic stress profiles. In general,

5544-498: The cross-section), but will vary over the cross section: the outer part will be under tensile stress, while the inner part will be compressed. Another variant of normal stress is the hoop stress that occurs on the walls of a cylindrical pipe or vessel filled with pressurized fluid. Another simple type of stress occurs when a uniformly thick layer of elastic material like glue or rubber is firmly attached to two stiff bodies that are pulled in opposite directions by forces parallel to

5643-402: The deformations caused by internal stresses are linearly related to them. In this case the differential equations that define the stress tensor are linear, and the problem becomes much easier. For one thing, the stress at any point will be a linear function of the loads, too. For small enough stresses, even non-linear systems can usually be assumed to be linear. Stress analysis is simplified when

5742-436: The direction of slip, faults can be categorized as: In a strike-slip fault (also known as a wrench fault , tear fault or transcurrent fault ), the fault surface (plane) is usually near vertical, and the footwall moves laterally either left or right with very little vertical motion. Strike-slip faults with left-lateral motion are also known as sinistral faults and those with right-lateral motion as dextral faults. Each

5841-709: The effect of gravity and other external forces can be neglected. In these situations, the stress across any imaginary internal surface turns out to be equal in magnitude and always directed perpendicularly to the surface independently of the surface's orientation. This type of stress may be called isotropic normal or just isotropic ; if it is compressive, it is called hydrostatic pressure or just pressure . Gases by definition cannot withstand tensile stresses, but some liquids may withstand very large amounts of isotropic tensile stress under some circumstances. see Z-tube . Parts with rotational symmetry , such as wheels, axles, pipes, and pillars, are very common in engineering. Often

5940-434: The elements σ x , σ y , σ z {\displaystyle \sigma _{x},\sigma _{y},\sigma _{z}} are called the orthogonal normal stresses (relative to the chosen coordinate system), and τ x y , τ x z , τ y z {\displaystyle \tau _{xy},\tau _{xz},\tau _{yz}}

6039-424: The external forces that are acting on the system. The latter may be body forces (such as gravity or magnetic attraction), that act throughout the volume of a material; or concentrated loads (such as friction between an axle and a bearing , or the weight of a train wheel on a rail), that are imagined to act over a two-dimensional area, or along a line, or at single point. In stress analysis one normally disregards

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6138-456: The fault and of the presence and nature of any mineralising fluids . Fault rocks are classified by their textures and the implied mechanism of deformation. A fault that passes through different levels of the lithosphere will have many different types of fault rock developed along its surface. Continued dip-slip displacement tends to juxtapose fault rocks characteristic of different crustal levels, with varying degrees of overprinting. This effect

6237-402: The fault concerning the other side. In measuring the horizontal or vertical separation, the throw of the fault is the vertical component of the separation and the heave of the fault is the horizontal component, as in "Throw up and heave out". The vector of slip can be qualitatively assessed by studying any drag folding of strata, which may be visible on either side of the fault. Drag folding

6336-744: The fault in Pakistan that are young enough that "only the alluvium of the bottom of active dry washes is not displaced". The parallel mountain ranges of eastern Balochistan, (east to west) the Kirthar Mountains , the Khude Mountains, the Zarro Mountains, the Pab Mountains and the Mor Mountains, are a result of the compressional plate boundary and are aligned parallel to the Chaman fault movement. The fault itself

6435-576: The fault plane and a vertical plane that strikes parallel to the fault. Ring faults , also known as caldera faults , are faults that occur within collapsed volcanic calderas and the sites of bolide strikes, such as the Chesapeake Bay impact crater . Ring faults are the result of a series of overlapping normal faults, forming a circular outline. Fractures created by ring faults may be filled by ring dikes . Synthetic and antithetic are terms used to describe minor faults associated with

6534-401: The fault plane, the horizontal extensional displacement on a listric fault implies a geometric "gap" between the hanging and footwalls of the fault forms when the slip motion occurs. To accommodate into the geometric gap, and depending on its rheology , the hanging wall might fold and slide downwards into the gap and produce rollover folding , or break into further faults and blocks which fil in

6633-427: The fault-bend fold diagram. Thrust faults form nappes and klippen in the large thrust belts. Subduction zones are a special class of thrusts that form the largest faults on Earth and give rise to the largest earthquakes. A fault which has a component of dip-slip and a component of strike-slip is termed an oblique-slip fault . Nearly all faults have some component of both dip-slip and strike-slip; hence, defining

6732-412: The fundamental laws of conservation of linear momentum and static equilibrium of forces, and is therefore mathematically exact, for any material and any stress situation. The components of the Cauchy stress tensor at every point in a material satisfy the equilibrium equations ( Cauchy's equations of motion for zero acceleration). Moreover, the principle of conservation of angular momentum implies that

6831-475: The gap. If faults form, imbrication fans or domino faulting may form. A reverse fault is the opposite of a normal fault—the hanging wall moves up relative to the footwall. Reverse faults indicate compressive shortening of the crust. A thrust fault has the same sense of motion as a reverse fault, but with the dip of the fault plane at less than 45°. Thrust faults typically form ramps, flats and fault-bend (hanging wall and footwall) folds. A section of

6930-461: The grains of a metal rod or the fibers of a piece of wood . Quantitatively, the stress is expressed by the Cauchy traction vector T defined as the traction force F between adjacent parts of the material across an imaginary separating surface S , divided by the area of S . In a fluid at rest the force is perpendicular to the surface, and is the familiar pressure . In a solid , or in

7029-408: The layer; or a section of a soft metal bar that is being cut by the jaws of a scissors-like tool . Let F be the magnitude of those forces, and M be the midplane of that layer. Just as in the normal stress case, the part of the layer on one side of M must pull the other part with the same force F . Assuming that the direction of the forces is known, the stress across M can be expressed simply by

7128-429: The material body is under equal compression or tension in all directions. This is the case, for example, in a portion of liquid or gas at rest, whether enclosed in some container or as part of a larger mass of fluid; or inside a cube of elastic material that is being pressed or pulled on all six faces by equal perpendicular forces — provided, in both cases, that the material is homogeneous, without built-in stress, and that

7227-519: The medium surrounding that point, and taking the average stresses in that particle as being the stresses at the point. Human-made objects are often made from stock plates of various materials by operations that do not change their essentially two-dimensional character, like cutting, drilling, gentle bending and welding along the edges. The description of stress in such bodies can be simplified by modeling those parts as two-dimensional surfaces rather than three-dimensional bodies. In that view, one redefines

7326-448: The most general case, called triaxial stress , the stress is nonzero across every surface element. Combined stresses cannot be described by a single vector. Even if the material is stressed in the same way throughout the volume of the body, the stress across any imaginary surface will depend on the orientation of that surface, in a non-trivial way. Cauchy observed that the stress vector T {\displaystyle T} across

7425-618: The name mechanical stress . Significant stress may exist even when deformation is negligible or non-existent (a common assumption when modeling the flow of water). Stress may exist in the absence of external forces; such built-in stress is important, for example, in prestressed concrete and tempered glass . Stress may also be imposed on a material without the application of net forces , for example by changes in temperature or chemical composition, or by external electromagnetic fields (as in piezoelectric and magnetostrictive materials). The relation between mechanical stress, strain, and

7524-420: The nature of the material or of its physical causes. Following the basic premises of continuum mechanics, stress is a macroscopic concept. Namely, the particles considered in its definition and analysis should be just small enough to be treated as homogeneous in composition and state, but still large enough to ignore quantum effects and the detailed motions of molecules. Thus, the force between two particles

7623-500: The original movement (fault inversion). In such a way, a normal fault may therefore become a reverse fault and vice versa. In a normal fault, the hanging wall moves downward, relative to the footwall. The dip of most normal faults is at least 60 degrees but some normal faults dip at less than 45 degrees. A downthrown block between two normal faults dipping towards each other is a graben . A block stranded between two grabens, and therefore two normal faults dipping away from each other,

7722-452: The physical causes of the forces or the precise nature of the materials. Instead, one assumes that the stresses are related to deformation (and, in non-static problems, to the rate of deformation) of the material by known constitutive equations . Stress analysis may be carried out experimentally, by applying loads to the actual artifact or to scale model, and measuring the resulting stresses, by any of several available methods. This approach

7821-424: The physical dimensions and the distribution of loads allow the structure to be treated as one- or two-dimensional. In the analysis of trusses, for example, the stress field may be assumed to be uniform and uniaxial over each member. Then the differential equations reduce to a finite set of equations (usually linear) with finitely many unknowns. In other contexts one may be able to reduce the three-dimensional problem to

7920-445: The plate). The analysis of stress can be considerably simplified also for thin bars, beams or wires of uniform (or smoothly varying) composition and cross-section that are subjected to moderate bending and twisting. For those bodies, one may consider only cross-sections that are perpendicular to the bar's axis, and redefine a "particle" as being a piece of wire with infinitesimal length between two such cross sections. The ordinary stress

8019-412: The right time for—and type of— igneous differentiation . At a given time differentiated magmas would burst violently out of the fault-traps and head to shallower places in the crust where porphyry copper deposits would be formed. As faults are zones of weakness, they facilitate the interaction of water with the surrounding rock and enhance chemical weathering . The enhanced chemical weathering increases

8118-671: The same units as pressure: namely, pascals (Pa, that is, newtons per square metre ) in the International System , or pounds per square inch (psi) in the Imperial system . Because mechanical stresses easily exceed a million Pascals, MPa, which stands for megapascal, is a common unit of stress. Stress in a material body may be due to multiple physical causes, including external influences and internal physical processes. Some of these agents (like gravity, changes in temperature and phase , and electromagnetic fields) act on

8217-424: The single number τ {\displaystyle \tau } , calculated simply with the magnitude of those forces, F and the cross sectional area, A . τ = F A {\displaystyle \tau ={\frac {F}{A}}} Unlike normal stress, this simple shear stress is directed parallel to the cross-section considered, rather than perpendicular to it. For any plane S that

8316-412: The size of the weathered zone and hence creates more space for groundwater . Fault zones act as aquifers and also assist groundwater transport. Stress (mechanics) Stress expresses the internal forces that neighbouring particles of a continuous material exert on each other, while strain is the measure of the relative deformation of the material. For example, when a solid vertical bar

8415-407: The stress T that a particle P applies on another particle Q across a surface S can have any direction relative to S . The vector T may be regarded as the sum of two components: the normal stress ( compression or tension ) perpendicular to the surface, and the shear stress that is parallel to the surface. If the normal unit vector n of the surface (pointing from Q towards P )

8514-507: The stress can be assumed to be uniformly distributed over any cross-section that is more than a few times D from both ends. (This observation is known as the Saint-Venant's principle ). Normal stress occurs in many other situations besides axial tension and compression. If an elastic bar with uniform and symmetric cross-section is bent in one of its planes of symmetry, the resulting bending stress will still be normal (perpendicular to

8613-411: The stress distribution in a body is expressed as a piecewise continuous function of space and time. Conversely, stress is usually correlated with various effects on the material, possibly including changes in physical properties like birefringence , polarization , and permeability . The imposition of stress by an external agent usually creates some strain (deformation) in the material, even if it

8712-475: The stress is evenly distributed over the entire cross-section. In practice, depending on how the bar is attached at the ends and how it was manufactured, this assumption may not be valid. In that case, the value σ {\displaystyle \sigma } = F / A will be only the average stress, called engineering stress or nominal stress . If the bar's length L is many times its diameter D , and it has no gross defects or built-in stress , then

8811-424: The stress is maximum for surfaces that are perpendicular to a certain direction d {\displaystyle d} , and zero across any surfaces that are parallel to d {\displaystyle d} . When the shear stress is zero only across surfaces that are perpendicular to one particular direction, the stress is called biaxial , and can be viewed as the sum of two normal or shear stresses. In

8910-399: The stress patterns that occur in such parts have rotational or even cylindrical symmetry . The analysis of such cylinder stresses can take advantage of the symmetry to reduce the dimension of the domain and/or of the stress tensor. Often, mechanical bodies experience more than one type of stress at the same time; this is called combined stress . In normal and shear stress, the magnitude of

9009-684: The stress state of the medium at any point and instant can be specified by only six independent parameters, rather than nine. These may be written [ σ x τ x y τ x z τ x y σ y τ y z τ x z τ y z σ z ] {\displaystyle {\begin{bmatrix}\sigma _{x}&\tau _{xy}&\tau _{xz}\\\tau _{xy}&\sigma _{y}&\tau _{yz}\\\tau _{xz}&\tau _{yz}&\sigma _{z}\end{bmatrix}}} where

9108-411: The stress tensor can be ignored, but since particles are not infinitesimal in the third dimension one can no longer ignore the torque that a particle applies on its neighbors. That torque is modeled as a bending stress that tends to change the curvature of the plate. These simplifications may not hold at welds, at sharp bends and creases (where the radius of curvature is comparable to the thickness of

9207-1620: The stress tensor can be represented in any chosen Cartesian coordinate system by a 3×3 matrix of real numbers. Depending on whether the coordinates are numbered x 1 , x 2 , x 3 {\displaystyle x_{1},x_{2},x_{3}} or named x , y , z {\displaystyle x,y,z} , the matrix may be written as [ σ 11 σ 12 σ 13 σ 21 σ 22 σ 23 σ 31 σ 32 σ 33 ] {\displaystyle {\begin{bmatrix}\sigma _{11}&\sigma _{12}&\sigma _{13}\\\sigma _{21}&\sigma _{22}&\sigma _{23}\\\sigma _{31}&\sigma _{32}&\sigma _{33}\end{bmatrix}}} or [ σ x x σ x y σ x z σ y x σ y y σ y z σ z x σ z y σ z z ] {\displaystyle {\begin{bmatrix}\sigma _{xx}&\sigma _{xy}&\sigma _{xz}\\\sigma _{yx}&\sigma _{yy}&\sigma _{yz}\\\sigma _{zx}&\sigma _{zy}&\sigma _{zz}\\\end{bmatrix}}} The stress vector T = σ ( n ) {\displaystyle T={\boldsymbol {\sigma }}(n)} across

9306-431: The stress tensor is symmetric , that is σ 12 = σ 21 {\displaystyle \sigma _{12}=\sigma _{21}} , σ 13 = σ 31 {\displaystyle \sigma _{13}=\sigma _{31}} , and σ 23 = σ 32 {\displaystyle \sigma _{23}=\sigma _{32}} . Therefore,

9405-423: The stress within a body may adequately be described by a single number, or by a single vector (a number and a direction). Three such simple stress situations, that are often encountered in engineering design, are the uniaxial normal stress , the simple shear stress , and the isotropic normal stress . A common situation with a simple stress pattern is when a straight rod, with uniform material and cross section,

9504-440: The stress σ throughout the bar, across any horizontal surface, can be expressed simply by the single number σ, calculated simply with the magnitude of those forces, F , and cross sectional area, A . σ = F A {\displaystyle \sigma ={\frac {F}{A}}} On the other hand, if one imagines the bar being cut along its length, parallel to the axis, there will be no force (hence no stress) between

9603-468: The term is also used for the zone of crushed rock along a single fault. Prolonged motion along closely spaced faults can blur the distinction, as the rock between the faults is converted to fault-bound lenses of rock and then progressively crushed. Due to friction and the rigidity of the constituent rocks, the two sides of a fault cannot always glide or flow past each other easily, and so occasionally all movement stops. The regions of higher friction along

9702-440: The two halves across the cut. This type of stress may be called (simple) normal stress or uniaxial stress; specifically, (uniaxial, simple, etc.) tensile stress. If the load is compression on the bar, rather than stretching it, the analysis is the same except that the force F and the stress σ {\displaystyle \sigma } change sign, and the stress is called compressive stress. This analysis assumes

9801-400: Was able to give the first rigorous and general mathematical model of a deformed elastic body by introducing the notions of stress and strain. Cauchy observed that the force across an imaginary surface was a linear function of its normal vector; and, moreover, that it must be a symmetric function (with zero total momentum). The understanding of stress in liquids started with Newton, who provided

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