1628:
1171:
1079:
4490:, any external forces being applied to such 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 the external forces that are acting on the system. The latter may be
4430:
2444:
156:
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50:
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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 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
4588:
2023:
1649:, and electromagnetic fields) act on 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
3168:
2998:
1695:) 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 is given in the article on
3927:
4596:
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 a two-dimensional one, and/or replace the general stress and strain tensors by simpler models like uniaxial tension/compression, simple shear, etc.
3579:{\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}}}
4441:
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,
4404:
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
4595:
Stress analysis is simplified when 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
1337:
effects and the detailed motions of molecules. Thus, the force between two particles 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,
4465:
that covers the determination of the internal distribution of internal forces in solid objects. It 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
1296:
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
4583:
for continuous media); that is, 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
2324:
Another simple type of stress occurs when 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
4599:
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
1847:
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 the two halves across the cut. This type of stress may be called (simple) normal stress or uniaxial stress; specifically, (uniaxial, simple, etc.) tensile
4514:
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 is often used for safety certification and monitoring. Most stress is analysed by mathematical
4408:
In that view, one redefines 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
2143:
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
2030:
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 the layer; or a section of a soft metal bar that is being cut by the jaws of a
1196:
3003:
1259:
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:
2860:
1309:
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,
4501:
In stress analysis one normally disregards 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
2651:
4543:
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
3771:
2460:
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.
2455:
Illustration of typical stresses (arrows) across various surface elements on the boundary of a particle (sphere), in a homogeneous material under uniform (but not isotropic) triaxial stress. The normal stresses on the principal axes are +5, +2, and −3
1231:
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 the
2325:
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 the effect of gravity and other external forces can be neglected.
1219:
Humans have known about stress inside materials since ancient times. Until the 17th century, this understanding was largely intuitive and empirical, though this did not prevent the development of relatively advanced technologies like the
1937:
This analysis assumes 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
2328:
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
4442:
and redefine a "particle" as being a piece of wire with infinitesimal length between two such cross sections. The ordinary stress is then reduced to a scalar (tension or compression of the bar), but one must take into account also a
2001:
will still be normal (perpendicular to 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
1338:
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 the grains of a
2573:
297:
4254:
3636:
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 (
1997:
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
4361:
4199:
1325:
The stress across a surface element (yellow disk) is the force that the material on one side (top ball) exerts on the material on the other side (bottom ball), divided by the area of the surface.
3980:
1086:
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
4046:
3323:
3163:{\displaystyle {\begin{bmatrix}\sigma _{xx}&\sigma _{xy}&\sigma _{xz}\\\sigma _{yx}&\sigma _{yy}&\sigma _{yz}\\\sigma _{zx}&\sigma _{zy}&\sigma _{zz}\\\end{bmatrix}}}
1333:
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
4380:
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
2993:{\displaystyle {\begin{bmatrix}\sigma _{11}&\sigma _{12}&\sigma _{13}\\\sigma _{21}&\sigma _{22}&\sigma _{23}\\\sigma _{31}&\sigma _{32}&\sigma _{33}\end{bmatrix}}}
3205:
2546:
1683:
in the material, even if it 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
365:
3766:
3726:
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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,
4087:
2759:
2725:
2568:
1845:
2139:
will only be an average ("nominal", "engineering") stress. That average is often sufficient for practical purposes. Shear stress is observed also when a cylindrical bar such as a
2114:
1806:. Therefore, 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,
4307:
4143:
3289:
2823:
1641:
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
3922:{\displaystyle {\begin{bmatrix}\sigma _{x}&\tau _{xy}&\tau _{xz}\\\tau _{xy}&\sigma _{y}&\tau _{yz}\\\tau _{xz}&\tau _{yz}&\sigma _{z}\end{bmatrix}}}
2703:
2306:
2236:
1297:(with zero total momentum). The understanding of stress in liquids started with Newton, who provided a differential formula for friction forces (shear stress) in parallel
2271:
2201:
1915:
1067:, 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
4060:
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
1956:
1874:
2345:. Gases by definition cannot withstand tensile stresses, but some liquids may withstand very large amounts of isotropic tensile stress under some circumstances. see
2855:
2073:
1403:; 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
963:
stress and may undergo shortening. The greater the force and the smaller the cross-sectional area of the body on which it acts, the greater the stress. Stress has
4563:. When the applied loads cause permanent deformation, one must use more complicated constitutive equations, that can account for the physical processes involved (
3768:. Therefore, 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
2677:
3626:
3606:
3228:
2509:
2482:
2419:
2399:
1792:
914:
1738:
In some situations, 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
5474:
5075:
Marchetti, M. C.; Joanny, J. F.; Ramaswamy, S.; Liverpool, T. B.; Prost, J.; Rao, Madan; Simha, R. Aditi (2013). "Hydrodynamics of soft active matter".
2361:, such as wheels, axles, pipes, and pillars, are very common in engineering. Often the stress patterns that occur in such parts have rotational or even
2731:, completely describes the stress state of a uniformly stressed body. (Today, any linear connection between two physical vector quantities is called a
1627:
1450:
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
4494:(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
1702:
The relation between stress and its effects and causes, including deformation and rate of change of deformation, can be quite complicated (although a
4372:
compression or tension, always perpendicular to any surface, there is no shear stress, and the tensor is a diagonal matrix in any coordinate frame.
3330:
3231:
1657:, self-propulsion of microscopic particles generates macroscopic stress profiles. In general, the stress distribution in a body is expressed as a
2131:
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
5393:
5204:
4879:
4649:
4498:, 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.
67:
244:
4854:
1667:
Conversely, stress is usually correlated with various effects on the material, possibly including changes in physical properties like
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5505:
5484:
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particle of the medium surrounding that point, and taking the average stresses in that particle as being the stresses at the point.
4204:
1170:
907:
133:
5127:
Ronald L. Huston and Harold
Josephs (2009), "Practical Stress Analysis in Engineering Design". 3rd edition, CRC Press, 634 pages.
5059:
3641:
1770:
A common situation with a simple stress pattern is when a straight rod, with uniform material and cross section, is subjected to
4959:
2381:. In normal and shear stress, the magnitude of the stress is maximum for surfaces that are perpendicular to a certain direction
3629:
114:
2421:. When the shear stress is zero only across surfaces that are perpendicular to one particular direction, the stress is called
1318:, that can be quantified and analyzed without explicit consideration of the nature of the material or of its physical causes.
4940:
4591:
Simplified model of a truss for stress analysis, assuming unidimensional elements under uniform axial tension or compression.
4536:
1619:. Because mechanical stresses easily exceed a million Pascals, MPa, which stands for megapascal, is a common unit of stress.
954:
880:
86:
71:
5221:
4313:
4151:
2646:{\displaystyle {\boldsymbol {\sigma }}(\alpha u+\beta v)=\alpha {\boldsymbol {\sigma }}(u)+\beta {\boldsymbol {\sigma }}(v)}
1817:
1637:
effect. The cracks are the result of brief but intense stress created when the semi-molten piece is briefly dipped in water.
581:
418:
5109:
Sharma, B and Kumar, R "Estimation of bulk viscosity of dilute gases using a nonequilibrium molecular dynamics approach.",
4746:
4736:
4532:
2086:
1676:
1604:
3932:
4146:
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900:
621:
507:
93:
576:
5540:
History of strength of materials: with a brief account of the history of theory of elasticity and theory of structures
4560:
3989:
3294:
2769:
485:
1991:
368:
2160:
Isotropic tensile stress. Top left: Each face of a cube of homogeneous material is pulled by a force with magnitude
4801:
4771:
4520:
4487:
1099:
4575:, etc.). Engineered structures are usually designed so the maximum expected stresses are well within the range of
3173:
2514:
1986:, then the stress can be assumed to be uniformly distributed over any cross-section that is more than a few times
100:
60:
492:
321:
4786:
5568:
4605:
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787:
782:
397:
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1078:
571:
564:
4906:
4070:
2742:
2708:
2551:
82:
4731:
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Stress analysis is generally concerned with objects and structures that can be assumed to be in macroscopic
4429:
2007:
1680:
1052:
989:
850:
845:
514:
38:
2172:
of the cube must balance the forces applied below the section. In the three sections shown, the forces are
4545:
4503:
3235:
1849:
1795:
1504:
1463:
1233:
402:
30:
This article is about stresses in classical (continuum) mechanics. For stresses in material science, see
4682:
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1707:
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1111:
1021:
958:
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443:
202:
31:
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2043:
be the midplane of that layer. Just as in the normal stress case, the part of the layer on one side of
155:
2369:
can take advantage of the symmetry to reduce the dimension of the domain and/or of the stress tensor.
2120:
is directed parallel to the cross-section considered, rather than perpendicular to it. For any plane
5084:
4796:
4726:
4564:
4556:
4552:
4495:
3334:
2728:
2438:
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1983:
1711:
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663:
480:
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448:
392:
237:
5406:
Theoretical soil mechanics: with practical applications to soil mechanics and foundation engineering
4584:
loads, too. For small enough stresses, even non-linear systems can usually be assumed to be linear.
2682:
4821:
2358:
2279:
2209:
1661:
1265:
1253:
1008:, each particle gets pushed against by all the surrounding particles. The container walls and the
964:
928:
865:
713:
606:
312:
2377:
Often, mechanical bodies experience more than one type of stress at the same time; this is called
1133:
may be adequate in practice if the quantities are sufficiently small. Stress that exceeds certain
5515:
4781:
4702:
4631:
4483:
4417:
of the plate. These simplifications may not hold at welds, at sharp bends and creases (where the
4365:
3633:
2425:, and can be viewed as the sum of two normal or shear stresses. In the most general case, called
2252:
2182:
1886:
983:
885:
519:
475:
5544:
5538:
5523:
5501:
5497:
Basic
Engineering Plasticity – An Introduction with Engineering and Manufacturing Applications
5480:
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5455:
5435:
5410:
5389:
5368:
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5311:
5290:
5271:
5250:
5227:
5200:
5196:
5180:
5160:
5128:
5043:
5019:
4999:
4967:
4936:
4576:
4446:(that tries to change the bar's curvature, in some direction perpendicular to the axis) and a
1917:
may be only an average stress. The stress may be unevenly distributed over the cross section (
1771:
1723:
1703:
1687:, tending to restore the material to its original undeformed state. Fluid materials (liquids,
1650:
1646:
1508:
1498:
1334:
1277:
1269:
1150:
1130:
1103:
948:
936:
502:
453:
107:
5383:
5326:
5265:
5176:
5157:
5039:
5015:
4995:
1941:
1859:
1706:
may be adequate in practice if the quantities are small enough). Stress that exceeds certain
947:. For example, an object being pulled apart, such as a stretched elastic band, is subject to
5573:
5495:
5305:
5092:
4806:
4751:
4741:
4572:
4528:
4061:
3645:
2762:
2735:, reflecting Cauchy's original use to describe the "tensions" (stresses) in a material.) In
1692:
1443:
1285:
1119:
1115:
840:
815:
728:
703:
698:
653:
4580:
2828:
2058:
4963:
4756:
4707:
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4524:
4515:
methods, especially during design. The basic stress analysis problem can be formulated by
4467:
4462:
4458:
2736:
2485:
2366:
2011:
1684:
1261:
1249:
1071:, opposing that change. Elastic and viscous stresses are usually combined under the name
1060:
830:
754:
668:
599:
533:
435:
164:
160:
4516:
2656:
1063:, that tends to restore the material to its original non-deformed state. In liquids and
718:
588:
5088:
2768:
Like any linear map between vectors, the stress tensor can be represented in any chosen
2451:
1321:
5243:
4831:
4811:
4761:
3611:
3591:
3213:
2494:
2489:
2467:
2404:
2384:
2156:
1777:
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1616:
1596:
1420:
1388:
1205:
1095:
1013:
835:
693:
658:
559:
465:
1852:
on the bar, rather than stretching it, the analysis is the same except that the force
5562:
5448:
4956:
4826:
4816:
4776:
4540:
4535:, together with the appropriate constitutive equations. Thus one obtains a system of
4381:
3208:
1668:
1654:
1592:
1533:
1225:
1221:
1044:
972:
875:
708:
190:
1591:, and therefore its coordinates are measured in the same units as pressure: namely,
5367:. Prentice-Hall civil engineering and engineering mechanics series. Prentice-Hall.
4855:"12.3 Stress, Strain, and Elastic Modulus - University Physics Volume 1 | OpenStax"
4766:
4434:
4392:
4309:, the stress tensor is a diagonal matrix, and has only the three normal components
2772:
by a 3×3 matrix of real numbers. Depending on whether the coordinates are numbered
1600:
1513:
1467:
1451:
1298:
1281:
1176:
860:
855:
820:
552:
214:
1762:
1532:) is assumed fixed, the normal component can be expressed by a single number, the
5404:
5362:
5144:
1016:. These macroscopic forces are actually the net result of a very large number of
4690:
1642:
1633:
1612:
1608:
1447:
1330:
1180:
1126:
1000:, each particle in the bar pushes on the particles immediately below it. When a
870:
773:
49:
5096:
4491:
1719:
1416:
1367:
between adjacent parts of the material across an imaginary separating surface
1146:
792:
688:
206:
2148:
under bending loads, due to the web constraining the end plates ("flanges").
17:
5158:"An Introduction to Continuum Mechanics after Truesdell and Noll". Springer.
4471:
4414:
4369:
3326:
1727:
1696:
1658:
1154:
1107:
957:. An object being pushed together, such as a crumpled sponge, is subject to
764:
759:
593:
5065:
Lamberts Glashütte (LambertsGlas) product brochure. Accessed on 2013-02-08.
4983:
https://archive.org/details/historyofstrengt0000timo_k8r2/page/110/mode/2up
1879:
4587:
2022:
4662:
4568:
4466:
many other disciplines; for example, in geology, to study phenomena like
4397:
1715:
1588:
1289:
1142:
1048:
1025:
1009:
1005:
979:
968:
743:
648:
628:
614:
222:
27:
Physical quantity that expresses internal forces in a continuous material
5241:
Beer, Ferdinand Pierre; Elwood
Russell Johnston; John T. DeWolf (1992).
2051:. Assuming that the direction of the forces is known, the stress across
1195:
4721:
4695:
1379:
at rest the force is perpendicular to the surface, and is the familiar
1040:
497:
185:
1742:
situations, that are often encountered in engineering design, are the
1699:. The same for normal viscous stresses can be found in Sharma (2019).
1098:. Stress may also be imposed on a material without the application of
1028:. Stress is frequently represented by a lowercase Greek letter sigma (
1012:-inducing surface (such as a piston) push against them in (Newtonian)
2732:
2346:
2145:
1679:. The imposition of stress by an external agent usually creates some
1408:
1392:
1315:
1311:
1241:
1001:
997:
638:
230:
4586:
4428:
4405:
as two-dimensional surfaces rather than three-dimensional bodies.
4391:
2450:
2442:
2155:
2124:
that is perpendicular to the layer, the net internal force across
2032:
2021:
1878:
1761:
1626:
1384:
1376:
1343:
1339:
1320:
1245:
1077:
1035:
Strain inside a material may arise by various mechanisms, such as
993:
940:
542:
226:
5310:. Dover books on engineering. Dover Publications. pp. 1–33.
1329:
Following the basic premises of continuum mechanics, stress is a
292:{\displaystyle {\mathsf {L}}^{-1}{\mathsf {M}}{\mathsf {T}}^{-2}}
2511:, the unit-length vector that is perpendicular to it. That is,
2140:
1407:. Thus the stress state of the material must be described by a
1347:
1237:
1201:
4474:; and in biology, to understand the anatomy of living beings.
4249:{\displaystyle {\boldsymbol {\sigma }}e_{i}=\lambda _{i}e_{i}}
1766:
Idealized stress in a straight bar with uniform cross-section.
1710:
of the material will result in permanent deformation (such as
1688:
1137:
of the material will result in permanent deformation (such as
1064:
678:
43:
5270:(Third ed.). Kluwer Academic Publisher. pp. 17–29.
2026:
Shear stress in a horizontal bar loaded by two offset blocks.
1561:(compressive stress) The shear component is then the vector
4935:(2. Da Capo Press ed.). Cambridge, MA: Da Capo Press.
4626:
Other useful stress measures include the first and second
1876:
change sign, and the stress is called compressive stress.
1039:
as applied by external forces to the bulk material (like
2075:, calculated simply with the magnitude of those forces,
1125:
The relation between mechanical stress, strain, and the
4880:"Class Physical-Quantity in theory Physical-Quantities"
4368:. If the three eigenvalues are equal, the stress is an
4311:
978:
Stress expresses the internal forces that neighbouring
5522:(Third ed.). McGraw-Hill International Editions.
5307:
Elasticity: tensor, dyadic, and engineering approaches
4356:{\displaystyle \lambda _{1},\lambda _{2},\lambda _{3}}
4194:{\displaystyle \lambda _{1},\lambda _{2},\lambda _{3}}
3780:
3455:
3402:
3349:
3012:
2869:
2429:, the stress is nonzero across every surface element.
5226:(3 ed.). Butterworth-Heinemann. pp. 17–32.
4450:(that tries to twist or un-twist it about its axis).
4316:
4262:
4207:
4154:
4098:
4073:
3992:
3935:
3774:
3734:
3694:
3654:
3614:
3594:
3343:
3297:
3244:
3216:
3176:
3006:
2863:
2831:
2778:
2745:
2711:
2685:
2659:
2576:
2554:
2517:
2497:
2470:
2407:
2387:
2282:
2255:
2212:
2185:
2089:
2061:
1944:
1889:
1862:
1820:
1802:
with continuity through the full cross-sectional area
1780:
324:
247:
2401:, and zero across any surfaces that are parallel to
2164:, applied evenly over the entire face whose area is
982:
of a continuous material exert on each other, while
5479:(2 ed.). Taylor & Francis. pp. 1–30.
3975:{\displaystyle \sigma _{x},\sigma _{y},\sigma _{z}}
3640:for zero acceleration). Moreover, the principle of
1442:, the Cauchy stress tensor can be represented as a
236:
212:
196:
184:
172:
148:
74:. Unsourced material may be challenged and removed.
5447:
5242:
5016:"Continuum Mechanics: Concise Theory and Problems"
4355:
4301:
4248:
4193:
4137:
4081:
4067:As a symmetric 3×3 real matrix, the stress tensor
4040:
3974:
3921:
3760:
3720:
3680:
3620:
3600:
3578:
3317:
3283:
3222:
3199:
3162:
2992:
2849:
2817:
2753:
2719:
2697:
2671:
2645:
2562:
2540:
2503:
2476:
2413:
2393:
2300:
2265:
2230:
2195:
2108:
2067:
1990:from both ends. (This observation is known as the
1950:
1909:
1868:
1839:
1786:
1496:may be regarded as the sum of two components: the
359:
291:
5432:Treatise on the Mathematical Theory of Elasticity
5018:. Dover Publications, series "Books on Physics".
4519:for continuous bodies (which are consequences of
4041:{\displaystyle \tau _{xy},\tau _{xz},\tau _{yz}}
3986:(relative to the chosen coordinate system), and
3318:{\displaystyle T=n\cdot {\boldsymbol {\sigma }}}
1183:. The stone arches in the bridge are subject to
5156:Donald Ray Smith and Clifford Truesdell (1993)
4907:"What is Shear Stress - Materials - Definition"
4600:resort to numerical approximations such as the
1353:Quantitatively, the stress is expressed by the
5543:. Dover Books on Physics. Dover Publications.
5331:. Cambridge University Press. pp. 16–26.
4421:is comparable to the thickness of the plate).
4256:. Therefore, in a coordinate system with axes
5361:Holtz, Robert D.; Kovacs, William D. (1981).
2055:can be expressed simply by the single number
2047:must pull the other part with the same force
908:
8:
3200:{\displaystyle T={\boldsymbol {\sigma }}(n)}
2541:{\displaystyle T={\boldsymbol {\sigma }}(n)}
163:inside a plastic protractor are revealed by
5388:. Bull Ridge Corporation. pp. 95–112.
5364:An introduction to geotechnical engineering
5325:Davis, R. O.; Selvadurai. A. P. S. (1996).
360:{\displaystyle J=-D{\frac {d\varphi }{dx}}}
5476:Mohr circles, stress paths and geotechnics
5287:Soil Plasticity, Theory and Implementation
5143:Walter D. Pilkey, Orrin H. Pilkey (1974),
4933:Structures, or, Why things don't fall down
4645:
4539:involving the stress tensor field and the
4089:has three mutually orthogonal unit-length
2761:is classified as a second-order tensor of
2006:that occurs on the walls of a cylindrical
1925:), especially near the attachment points (
1055:of a solid material generates an internal
915:
901:
748:
538:
381:
303:
154:
5446:Marsden, J. E.; Hughes, T. J. R. (1994).
5123:
5121:
5119:
4347:
4334:
4321:
4315:
4293:
4280:
4267:
4261:
4240:
4230:
4217:
4208:
4206:
4185:
4172:
4159:
4153:
4129:
4116:
4103:
4097:
4074:
4072:
4029:
4013:
3997:
3991:
3966:
3953:
3940:
3934:
3905:
3890:
3875:
3858:
3846:
3831:
3814:
3799:
3787:
3775:
3773:
3761:{\displaystyle \sigma _{23}=\sigma _{32}}
3752:
3739:
3733:
3721:{\displaystyle \sigma _{13}=\sigma _{31}}
3712:
3699:
3693:
3681:{\displaystyle \sigma _{12}=\sigma _{21}}
3672:
3659:
3653:
3613:
3593:
3562:
3550:
3538:
3524:
3512:
3500:
3486:
3474:
3462:
3450:
3433:
3421:
3409:
3397:
3380:
3368:
3356:
3344:
3342:
3310:
3296:
3275:
3262:
3249:
3243:
3215:
3183:
3175:
3143:
3128:
3113:
3096:
3081:
3066:
3049:
3034:
3019:
3007:
3005:
2976:
2964:
2952:
2938:
2926:
2914:
2900:
2888:
2876:
2864:
2862:
2830:
2809:
2796:
2783:
2777:
2746:
2744:
2712:
2710:
2684:
2658:
2629:
2609:
2577:
2575:
2555:
2553:
2524:
2516:
2496:
2469:
2406:
2386:
2290:
2283:
2281:
2256:
2254:
2220:
2213:
2211:
2186:
2184:
2096:
2088:
2060:
1943:
1899:
1888:
1861:
1827:
1819:
1779:
337:
323:
280:
274:
273:
266:
265:
256:
250:
249:
246:
134:Learn how and when to remove this message
5500:. Butterworth-Heinemann. pp. 1–32.
4658:
2765:(0,2) or (1,1) depending on convention.
2447:Components of stress in three dimensions
1966:will be only the average stress, called
1511:) perpendicular to the surface, and the
5423:Landau, L.D. and E.M.Lifshitz. (1959).
4966:(revised edition). Dover Publications.
4843:
4400:made from bent and welded steel plates.
4209:
4082:{\displaystyle {\boldsymbol {\sigma }}}
4075:
3311:
3184:
2754:{\displaystyle {\boldsymbol {\sigma }}}
2747:
2720:{\displaystyle {\boldsymbol {\sigma }}}
2713:
2630:
2610:
2578:
2563:{\displaystyle {\boldsymbol {\sigma }}}
2556:
2525:
2464:Cauchy observed that the stress vector
1208:. The rope in the bridge is subject to
1059:, analogous to the reaction force of a
996:vertical bar is supporting an overhead
772:
727:
677:
637:
541:
410:
384:
311:
5450:Mathematical Foundations of Elasticity
2128:, and hence the stress, will be zero.
2039:be the magnitude of those forces, and
1840:{\displaystyle \sigma ={\frac {F}{A}}}
275:
267:
251:
145:
5267:Rock Mechanics For Underground Mining
5197:"The Linearized Theory of Elasticity"
5034:
5032:
4996:"Plasticity for Structural Engineers"
4994:Wai-Fah Chen and Da-Jian Han (2007),
3628:follows from the fundamental laws of
2337:; if it is compressive, it is called
2308:, respectively. So the stress across
1129:can be quite complicated, although a
992:of the material. For example, when a
971:of newtons per square meter (N/m) or
7:
5304:Chou, Pei Chi; Pagano, N.J. (1992).
5285:Chen, Wai-Fah; Baladi, G.Y. (1985).
4901:
4899:
4849:
4847:
2109:{\displaystyle \tau ={\frac {F}{A}}}
1794:along its axis. If the system is in
72:adding citations to reliable sources
5473:Parry, Richard Hawley Grey (2004).
1587:The dimension of stress is that of
1545:. This number will be positive if
1488:can have any direction relative to
1446:of 3×3 real numbers. Even within a
5264:Brady, B.H.G.; E.T. Brown (1993).
4437:may be considered one-dimensional.
3644:implies that the stress tensor is
2484:across a surface will always be a
1553:(tensile stress), and negative if
25:
4302:{\displaystyle e_{1},e_{2},e_{3}}
4138:{\displaystyle e_{1},e_{2},e_{3}}
3284:{\displaystyle n_{1},n_{2},n_{3}}
2818:{\displaystyle x_{1},x_{2},x_{3}}
1982:, and it has no gross defects or
1517:that is parallel to the surface.
5434:. New York: Dover Publications.
5430:Love, A. E. H. (4 ed.). (1944).
5385:Deformation Theory of Plasticity
3642:conservation of angular momentum
2238:(bottom right); and the area of
2168:. The force across any section
1774:by opposite forces of magnitude
1194:
1169:
48:
5537:Timoshenko, Stephen P. (1983).
5518:; James Norman Goodier (1970).
5454:. Dover Publications. pp.
5345:Dieter, G. E. (3 ed.). (1989).
3630:conservation of linear momentum
2857:, the matrix may be written as
2014:filled with pressurized fluid.
1024:between the particles in those
1004:is in a closed container under
988:is the measure of the relative
59:needs additional citations for
5382:Jones, Robert Millard (2008).
4628:Piola–Kirchhoff stress tensors
4537:partial differential equations
3194:
3188:
2698:{\displaystyle \alpha ,\beta }
2640:
2634:
2620:
2614:
2600:
2582:
2535:
2529:
2079:and the cross sectional area,
1524:of the surface (pointing from
1438:. With respect to any chosen
1286:laws of motion and equilibrium
1090:is important, for example, in
1:
5195:William S. Slaughter (2012),
4737:Kelvin probe force microscope
4533:Euler-Cauchy stress principle
2301:{\displaystyle {\sqrt {3}}/2}
2231:{\displaystyle {\sqrt {3}}/2}
5409:. Van Nostrand Reinhold Co.
5403:Jumikis, Alfreds R. (1969).
5249:. McGraw-Hill Professional.
3638:Cauchy's equations of motion
3588:The linear relation between
1810:, and cross sectional area,
1480:applies on another particle
1399:may not be perpendicular to
1110:composition, or by external
5328:Elasticity and geomechanics
4561:infinitesimal strain theory
4555:structures is based on the
4517:Euler's equations of motion
3325:(where T in upper index is
3238:- vector) with coordinates
2770:Cartesian coordinate system
2266:{\displaystyle {\sqrt {2}}}
2196:{\displaystyle {\sqrt {2}}}
2116:Unlike normal stress, this
1978:is many times its diameter
1910:{\displaystyle \sigma =F/A}
5590:
5097:10.1103/RevModPhys.85.1143
4802:Transient friction loading
4619:
3984:orthogonal normal stresses
2436:
1520:If the normal unit vector
1461:
1290:calculus of infinitesimals
1043:) or to its surface (like
36:
29:
5349:. New York: McGraw-Hill.
5077:Reviews of Modern Physics
4413:that tends to change the
4050:orthogonal shear stresses
3329:, and as a result we get
3291:is then a matrix product
1371:, divided by the area of
153:
5220:Chakrabarty, J. (2006).
5175:Fridtjov Irgens (2008),
4606:finite difference method
4064:of stress distribution.
1992:Saint-Venant's principle
1848:stress. If the load is
1047:, external pressure, or
967:of force per area, with
419:Clausius–Duhem (entropy)
369:Fick's laws of diffusion
5061:The art of making glass
5014:Peter Chadwick (1999),
4955:Jacob Lubliner (2008).
4747:Lamé's stress ellipsoid
4732:Critical plane analysis
4636:Kirchhoff stress tensor
4610:boundary element method
4579:(the generalization of
4488:Newton's laws of motion
4433:For stress modeling, a
3333:(row) vector) (look on
2365:. The analysis of such
1951:{\displaystyle \sigma }
1869:{\displaystyle \sigma }
1752:isotropic normal stress
1653:due to collisions). In
1472:In general, the stress
1430:to the traction vector
953:stress and may undergo
577:Navier–Stokes equations
515:Material failure theory
83:"Stress" mechanics
5516:Timoshenko, Stephen P.
5245:Mechanics of Materials
4592:
4546:boundary-value problem
4504:constitutive equations
4438:
4401:
4357:
4303:
4250:
4195:
4139:
4083:
4042:
3976:
3923:
3762:
3722:
3682:
3622:
3602:
3580:
3319:
3285:
3224:
3207:across a surface with
3201:
3164:
2994:
2851:
2819:
2755:
2729:(Cauchy) stress tensor
2721:
2699:
2673:
2647:
2564:
2542:
2505:
2478:
2457:
2448:
2415:
2395:
2321:
2302:
2267:
2232:
2197:
2110:
2069:
2027:
1974:. If the bar's length
1952:
1934:
1911:
1870:
1841:
1788:
1767:
1744:uniaxial normal stress
1638:
1464:Compression (physical)
1413:(Cauchy) stress tensor
1355:Cauchy traction vector
1326:
1112:electromagnetic fields
1104:changes in temperature
1083:
361:
293:
5347:Mechanical Metallurgy
5177:"Continuum Mechanics"
5145:"Mechanics of solids"
5040:"Continuum Mechanics"
4998:. J. Ross Publishing
4931:Gordon, J.E. (2003).
4602:finite element method
4590:
4478:Goals and assumptions
4432:
4395:
4358:
4304:
4251:
4196:
4140:
4084:
4056:Change of coordinates
4043:
3977:
3924:
3763:
3723:
3683:
3623:
3603:
3581:
3320:
3286:
3225:
3202:
3165:
2995:
2852:
2850:{\displaystyle x,y,z}
2820:
2756:
2722:
2700:
2679:and any real numbers
2674:
2648:
2565:
2548:, where the function
2543:
2506:
2479:
2454:
2446:
2416:
2396:
2303:
2268:
2233:
2198:
2159:
2111:
2070:
2068:{\displaystyle \tau }
2025:
1953:
1912:
1882:
1871:
1842:
1789:
1765:
1722:) or even change its
1630:
1557:is "pushing" against
1462:Further information:
1324:
1294:Augustin-Louis Cauchy
1292:. With those tools,
1149:) or even change its
1081:
1018:intermolecular forces
572:Bernoulli's principle
565:Archimedes' principle
362:
294:
32:Strength of materials
5520:Theory of Elasticity
5494:Rees, David (2006).
5425:Theory of Elasticity
5223:Theory of plasticity
4884:www-ksl.stanford.edu
4797:Stress concentration
4787:Stress–energy tensor
4727:Compressive strength
4557:theory of elasticity
4551:Stress analysis for
4523:for conservation of
4314:
4260:
4205:
4152:
4096:
4071:
3990:
3933:
3772:
3732:
3692:
3652:
3612:
3592:
3341:
3335:Cauchy stress tensor
3295:
3242:
3214:
3174:
3004:
2861:
2829:
2776:
2743:
2709:
2683:
2657:
2574:
2552:
2515:
2495:
2468:
2439:Cauchy stress tensor
2405:
2385:
2363:cylindrical symmetry
2339:hydrostatic pressure
2280:
2253:
2210:
2183:
2087:
2059:
1942:
1887:
1860:
1818:
1778:
1728:chemical composition
1704:linear approximation
1681:strain (deformation)
1631:Glass vase with the
1605:International System
1155:chemical composition
1131:linear approximation
1092:prestressed concrete
1053:strain (deformation)
664:Cohesion (chemistry)
486:Infinitesimal strain
322:
245:
68:improve this article
37:For other uses, see
5199:. Birkhäuser Basel
5113:,100, 013309 (2019)
5089:2013RvMP...85.1143M
5038:I-Shih Liu (2002),
4957:"Plasticity Theory"
4911:Material Properties
4861:. 19 September 2016
4822:Yield (engineering)
4792:Stress–strain curve
4650:Conjugate variables
4419:radius of curvature
3929:where the elements
2672:{\displaystyle u,v}
2359:rotational symmetry
2320:in all three cases.
2203:(bottom left), and
2118:simple shear stress
1748:simple shear stress
1664:of space and time.
1662:continuous function
1266:experimental method
929:continuum mechanics
582:Poiseuille equation
313:Continuum mechanics
307:Part of a series on
4962:2010-03-31 at the
4782:Strain rate tensor
4703:Chemical potential
4632:Biot stress tensor
4593:
4484:static equilibrium
4439:
4402:
4366:principal stresses
4353:
4299:
4246:
4191:
4135:
4079:
4038:
3972:
3919:
3913:
3758:
3718:
3678:
3634:static equilibrium
3618:
3598:
3576:
3570:
3441:
3388:
3315:
3281:
3220:
3197:
3170:The stress vector
3160:
3154:
2990:
2984:
2847:
2815:
2751:
2717:
2695:
2669:
2643:
2560:
2538:
2501:
2474:
2458:
2449:
2411:
2391:
2322:
2298:
2263:
2228:
2193:
2106:
2065:
2033:scissors-like tool
2028:
1968:engineering stress
1948:
1935:
1907:
1866:
1837:
1784:
1768:
1639:
1623:Causes and effects
1327:
1084:
788:Magnetorheological
783:Electrorheological
520:Fracture mechanics
357:
289:
216:SI base units
5395:978-0-9787223-1-9
5205:978-0-8176-4117-7
5111:Physical Review E
4717:
4716:
4713:
4712:
4652:of thermodynamics
4577:linear elasticity
3621:{\displaystyle n}
3601:{\displaystyle T}
3236:"row; horizontal"
3223:{\displaystyle n}
2727:, now called the
2504:{\displaystyle n}
2488:of the surface's
2477:{\displaystyle T}
2414:{\displaystyle d}
2394:{\displaystyle d}
2367:cylinder stresses
2288:
2261:
2218:
2191:
2104:
1835:
1787:{\displaystyle F}
1724:crystal structure
1484:across a surface
1440:coordinate system
1419:that relates the
1278:analytic geometry
1254:Gothic cathedrals
1250:flying buttresses
1151:crystal structure
1102:, for example by
1082:Mechanical stress
1073:mechanical stress
937:physical quantity
925:
924:
800:
799:
734:
733:
503:Contact mechanics
426:
425:
355:
302:
301:
161:Residual stresses
144:
143:
136:
118:
16:(Redirected from
5581:
5554:
5533:
5511:
5490:
5469:
5453:
5420:
5399:
5378:
5342:
5321:
5300:
5281:
5260:
5248:
5237:
5207:
5193:
5187:
5173:
5167:
5154:
5148:
5141:
5135:
5125:
5114:
5107:
5101:
5100:
5083:(3): 1143–1189.
5072:
5066:
5056:
5050:
5036:
5027:
5012:
5006:
4992:
4986:
4980:
4974:
4953:
4947:
4946:
4928:
4922:
4921:
4919:
4918:
4903:
4894:
4893:
4891:
4890:
4876:
4870:
4869:
4867:
4866:
4851:
4807:Tensile strength
4752:Reinforced solid
4659:
4646:
4529:angular momentum
4470:, vulcanism and
4448:torsional stress
4363:
4360:
4359:
4354:
4352:
4351:
4339:
4338:
4326:
4325:
4308:
4306:
4305:
4300:
4298:
4297:
4285:
4284:
4272:
4271:
4255:
4253:
4252:
4247:
4245:
4244:
4235:
4234:
4222:
4221:
4212:
4200:
4198:
4197:
4192:
4190:
4189:
4177:
4176:
4164:
4163:
4144:
4142:
4141:
4136:
4134:
4133:
4121:
4120:
4108:
4107:
4088:
4086:
4085:
4080:
4078:
4047:
4045:
4044:
4039:
4037:
4036:
4021:
4020:
4005:
4004:
3981:
3979:
3978:
3973:
3971:
3970:
3958:
3957:
3945:
3944:
3928:
3926:
3925:
3920:
3918:
3917:
3910:
3909:
3898:
3897:
3883:
3882:
3866:
3865:
3851:
3850:
3839:
3838:
3822:
3821:
3807:
3806:
3792:
3791:
3767:
3765:
3764:
3759:
3757:
3756:
3744:
3743:
3727:
3725:
3724:
3719:
3717:
3716:
3704:
3703:
3687:
3685:
3684:
3679:
3677:
3676:
3664:
3663:
3627:
3625:
3624:
3619:
3607:
3605:
3604:
3599:
3585:
3583:
3582:
3577:
3575:
3574:
3567:
3566:
3555:
3554:
3543:
3542:
3529:
3528:
3517:
3516:
3505:
3504:
3491:
3490:
3479:
3478:
3467:
3466:
3446:
3445:
3438:
3437:
3426:
3425:
3414:
3413:
3393:
3392:
3385:
3384:
3373:
3372:
3361:
3360:
3324:
3322:
3321:
3316:
3314:
3290:
3288:
3287:
3282:
3280:
3279:
3267:
3266:
3254:
3253:
3229:
3227:
3226:
3221:
3206:
3204:
3203:
3198:
3187:
3169:
3167:
3166:
3161:
3159:
3158:
3151:
3150:
3136:
3135:
3121:
3120:
3104:
3103:
3089:
3088:
3074:
3073:
3057:
3056:
3042:
3041:
3027:
3026:
2999:
2997:
2996:
2991:
2989:
2988:
2981:
2980:
2969:
2968:
2957:
2956:
2943:
2942:
2931:
2930:
2919:
2918:
2905:
2904:
2893:
2892:
2881:
2880:
2856:
2854:
2853:
2848:
2824:
2822:
2821:
2816:
2814:
2813:
2801:
2800:
2788:
2787:
2760:
2758:
2757:
2752:
2750:
2726:
2724:
2723:
2718:
2716:
2704:
2702:
2701:
2696:
2678:
2676:
2675:
2670:
2653:for any vectors
2652:
2650:
2649:
2644:
2633:
2613:
2581:
2569:
2567:
2566:
2561:
2559:
2547:
2545:
2544:
2539:
2528:
2510:
2508:
2507:
2502:
2483:
2481:
2480:
2475:
2420:
2418:
2417:
2412:
2400:
2398:
2397:
2392:
2331:isotropic normal
2307:
2305:
2304:
2299:
2294:
2289:
2284:
2272:
2270:
2269:
2264:
2262:
2257:
2237:
2235:
2234:
2229:
2224:
2219:
2214:
2202:
2200:
2199:
2194:
2192:
2187:
2115:
2113:
2112:
2107:
2105:
2097:
2074:
2072:
2071:
2066:
1957:
1955:
1954:
1949:
1916:
1914:
1913:
1908:
1903:
1875:
1873:
1872:
1867:
1846:
1844:
1843:
1838:
1836:
1828:
1793:
1791:
1790:
1785:
1578:
1549:is "pulling" on
1544:
1476:that a particle
1458:Normal and shear
1444:symmetric matrix
1198:
1173:
1120:magnetostrictive
917:
910:
903:
749:
714:Gay-Lussac's law
704:Combined gas law
654:Capillary action
539:
382:
366:
364:
363:
358:
356:
354:
346:
338:
304:
298:
296:
295:
290:
288:
287:
279:
278:
271:
270:
264:
263:
255:
254:
217:
158:
146:
139:
132:
128:
125:
119:
117:
76:
52:
44:
21:
5589:
5588:
5584:
5583:
5582:
5580:
5579:
5578:
5569:Solid mechanics
5559:
5558:
5557:
5551:
5536:
5530:
5514:
5508:
5493:
5487:
5472:
5466:
5445:
5417:
5402:
5396:
5381:
5375:
5360:
5339:
5324:
5318:
5303:
5297:
5284:
5278:
5263:
5257:
5240:
5234:
5219:
5215:
5213:Further reading
5210:
5194:
5190:
5174:
5170:
5155:
5151:
5142:
5138:
5126:
5117:
5108:
5104:
5074:
5073:
5069:
5057:
5053:
5037:
5030:
5013:
5009:
4993:
4989:
4981:
4977:
4964:Wayback Machine
4954:
4950:
4943:
4930:
4929:
4925:
4916:
4914:
4905:
4904:
4897:
4888:
4886:
4878:
4877:
4873:
4864:
4862:
4853:
4852:
4845:
4841:
4836:
4757:Residual stress
4708:Particle number
4651:
4644:
4624:
4622:Stress measures
4618:
4525:linear momentum
4512:
4480:
4468:plate tectonics
4463:applied physics
4461:is a branch of
4459:Stress analysis
4456:
4427:
4390:
4378:
4343:
4330:
4317:
4312:
4310:
4289:
4276:
4263:
4258:
4257:
4236:
4226:
4213:
4203:
4202:
4181:
4168:
4155:
4150:
4149:
4145:and three real
4125:
4112:
4099:
4094:
4093:
4069:
4068:
4058:
4025:
4009:
3993:
3988:
3987:
3982:are called the
3962:
3949:
3936:
3931:
3930:
3912:
3911:
3901:
3899:
3886:
3884:
3871:
3868:
3867:
3854:
3852:
3842:
3840:
3827:
3824:
3823:
3810:
3808:
3795:
3793:
3783:
3776:
3770:
3769:
3748:
3735:
3730:
3729:
3708:
3695:
3690:
3689:
3668:
3655:
3650:
3649:
3610:
3609:
3590:
3589:
3569:
3568:
3558:
3556:
3546:
3544:
3534:
3531:
3530:
3520:
3518:
3508:
3506:
3496:
3493:
3492:
3482:
3480:
3470:
3468:
3458:
3451:
3440:
3439:
3429:
3427:
3417:
3415:
3405:
3398:
3387:
3386:
3376:
3374:
3364:
3362:
3352:
3345:
3339:
3338:
3293:
3292:
3271:
3258:
3245:
3240:
3239:
3212:
3211:
3172:
3171:
3153:
3152:
3139:
3137:
3124:
3122:
3109:
3106:
3105:
3092:
3090:
3077:
3075:
3062:
3059:
3058:
3045:
3043:
3030:
3028:
3015:
3008:
3002:
3001:
2983:
2982:
2972:
2970:
2960:
2958:
2948:
2945:
2944:
2934:
2932:
2922:
2920:
2910:
2907:
2906:
2896:
2894:
2884:
2882:
2872:
2865:
2859:
2858:
2827:
2826:
2805:
2792:
2779:
2774:
2773:
2741:
2740:
2737:tensor calculus
2707:
2706:
2705:. The function
2681:
2680:
2655:
2654:
2572:
2571:
2550:
2549:
2513:
2512:
2493:
2492:
2486:linear function
2466:
2465:
2441:
2435:
2427:triaxial stress
2403:
2402:
2383:
2382:
2379:combined stress
2375:
2355:
2278:
2277:
2251:
2250:
2208:
2207:
2181:
2180:
2154:
2085:
2084:
2057:
2056:
2020:
1984:built-in stress
1940:
1939:
1885:
1884:
1858:
1857:
1856:and the stress
1816:
1815:
1776:
1775:
1760:
1758:Uniaxial normal
1736:
1708:strength limits
1625:
1617:Imperial system
1585:
1562:
1536:
1470:
1460:
1417:linear function
1360:defined as the
1307:
1262:Galileo Galilei
1217:
1216:
1215:
1214:
1213:
1199:
1190:
1189:
1188:
1179:-era bridge in
1174:
1163:
1135:strength limits
1088:built-in stress
943:present during
939:that describes
921:
892:
891:
890:
810:
802:
801:
755:Viscoelasticity
746:
736:
735:
723:
673:
669:Surface tension
633:
536:
534:Fluid mechanics
526:
525:
524:
438:
436:Solid mechanics
428:
427:
379:
371:
347:
339:
320:
319:
272:
248:
243:
242:
215:
199:
175:
168:
165:polarized light
140:
129:
123:
120:
77:
75:
65:
53:
42:
35:
28:
23:
22:
15:
12:
11:
5:
5587:
5585:
5577:
5576:
5571:
5561:
5560:
5556:
5555:
5549:
5534:
5528:
5512:
5506:
5491:
5485:
5470:
5464:
5443:
5428:
5421:
5415:
5400:
5394:
5379:
5373:
5358:
5343:
5337:
5322:
5316:
5301:
5295:
5282:
5276:
5261:
5255:
5238:
5232:
5216:
5214:
5211:
5209:
5208:
5188:
5168:
5149:
5136:
5115:
5102:
5067:
5051:
5028:
5007:
4987:
4975:
4948:
4941:
4923:
4895:
4871:
4842:
4840:
4837:
4835:
4834:
4832:Virial theorem
4829:
4824:
4819:
4814:
4812:Thermal stress
4809:
4804:
4799:
4794:
4789:
4784:
4779:
4774:
4769:
4764:
4762:Shear strength
4759:
4754:
4749:
4744:
4739:
4734:
4729:
4724:
4718:
4715:
4714:
4711:
4710:
4705:
4699:
4698:
4693:
4687:
4686:
4679:
4671:
4670:
4665:
4655:
4654:
4643:
4640:
4620:Main article:
4617:
4614:
4511:
4508:
4479:
4476:
4455:
4452:
4444:bending stress
4426:
4423:
4411:bending stress
4389:
4386:
4377:
4374:
4350:
4346:
4342:
4337:
4333:
4329:
4324:
4320:
4296:
4292:
4288:
4283:
4279:
4275:
4270:
4266:
4243:
4239:
4233:
4229:
4225:
4220:
4216:
4211:
4188:
4184:
4180:
4175:
4171:
4167:
4162:
4158:
4132:
4128:
4124:
4119:
4115:
4111:
4106:
4102:
4077:
4057:
4054:
4035:
4032:
4028:
4024:
4019:
4016:
4012:
4008:
4003:
4000:
3996:
3969:
3965:
3961:
3956:
3952:
3948:
3943:
3939:
3916:
3908:
3904:
3900:
3896:
3893:
3889:
3885:
3881:
3878:
3874:
3870:
3869:
3864:
3861:
3857:
3853:
3849:
3845:
3841:
3837:
3834:
3830:
3826:
3825:
3820:
3817:
3813:
3809:
3805:
3802:
3798:
3794:
3790:
3786:
3782:
3781:
3779:
3755:
3751:
3747:
3742:
3738:
3715:
3711:
3707:
3702:
3698:
3675:
3671:
3667:
3662:
3658:
3617:
3597:
3573:
3565:
3561:
3557:
3553:
3549:
3545:
3541:
3537:
3533:
3532:
3527:
3523:
3519:
3515:
3511:
3507:
3503:
3499:
3495:
3494:
3489:
3485:
3481:
3477:
3473:
3469:
3465:
3461:
3457:
3456:
3454:
3449:
3444:
3436:
3432:
3428:
3424:
3420:
3416:
3412:
3408:
3404:
3403:
3401:
3396:
3391:
3383:
3379:
3375:
3371:
3367:
3363:
3359:
3355:
3351:
3350:
3348:
3313:
3309:
3306:
3303:
3300:
3278:
3274:
3270:
3265:
3261:
3257:
3252:
3248:
3219:
3196:
3193:
3190:
3186:
3182:
3179:
3157:
3149:
3146:
3142:
3138:
3134:
3131:
3127:
3123:
3119:
3116:
3112:
3108:
3107:
3102:
3099:
3095:
3091:
3087:
3084:
3080:
3076:
3072:
3069:
3065:
3061:
3060:
3055:
3052:
3048:
3044:
3040:
3037:
3033:
3029:
3025:
3022:
3018:
3014:
3013:
3011:
2987:
2979:
2975:
2971:
2967:
2963:
2959:
2955:
2951:
2947:
2946:
2941:
2937:
2933:
2929:
2925:
2921:
2917:
2913:
2909:
2908:
2903:
2899:
2895:
2891:
2887:
2883:
2879:
2875:
2871:
2870:
2868:
2846:
2843:
2840:
2837:
2834:
2812:
2808:
2804:
2799:
2795:
2791:
2786:
2782:
2749:
2715:
2694:
2691:
2688:
2668:
2665:
2662:
2642:
2639:
2636:
2632:
2628:
2625:
2622:
2619:
2616:
2612:
2608:
2605:
2602:
2599:
2596:
2593:
2590:
2587:
2584:
2580:
2558:
2537:
2534:
2531:
2527:
2523:
2520:
2500:
2473:
2437:Main article:
2434:
2431:
2410:
2390:
2374:
2371:
2354:
2351:
2297:
2293:
2287:
2260:
2227:
2223:
2217:
2190:
2153:
2150:
2103:
2100:
2095:
2092:
2064:
2019:
2016:
1999:bending stress
1972:nominal stress
1947:
1906:
1902:
1898:
1895:
1892:
1865:
1834:
1831:
1826:
1823:
1783:
1759:
1756:
1735:
1732:
1624:
1621:
1595:(Pa, that is,
1584:
1581:
1492:. The vector
1459:
1456:
1362:traction force
1346:of a piece of
1306:
1303:
1270:René Descartes
1206:Apurimac River
1204:bridge on the
1200:
1193:
1192:
1191:
1175:
1168:
1167:
1166:
1165:
1164:
1162:
1159:
1096:tempered glass
1069:viscous stress
1057:elastic stress
1045:contact forces
923:
922:
920:
919:
912:
905:
897:
894:
893:
889:
888:
883:
878:
873:
868:
863:
858:
853:
848:
843:
838:
833:
828:
823:
818:
812:
811:
808:
807:
804:
803:
798:
797:
796:
795:
790:
785:
777:
776:
770:
769:
768:
767:
762:
757:
747:
742:
741:
738:
737:
732:
731:
725:
724:
722:
721:
716:
711:
706:
701:
696:
691:
685:
682:
681:
675:
674:
672:
671:
666:
661:
659:Chromatography
656:
651:
645:
642:
641:
635:
634:
632:
631:
612:
611:
610:
591:
579:
574:
562:
549:
546:
545:
537:
532:
531:
528:
527:
523:
522:
517:
512:
511:
510:
500:
495:
490:
489:
488:
483:
473:
468:
463:
458:
457:
456:
446:
440:
439:
434:
433:
430:
429:
424:
423:
422:
421:
413:
412:
408:
407:
406:
405:
400:
395:
387:
386:
380:
377:
376:
373:
372:
367:
353:
350:
345:
342:
336:
333:
330:
327:
316:
315:
309:
308:
300:
299:
286:
283:
277:
269:
262:
259:
253:
240:
234:
233:
219:
210:
209:
200:
197:
194:
193:
188:
182:
181:
176:
174:Common symbols
173:
170:
169:
159:
151:
150:
142:
141:
56:
54:
47:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5586:
5575:
5572:
5570:
5567:
5566:
5564:
5552:
5550:0-486-61187-6
5546:
5542:
5541:
5535:
5531:
5529:0-07-085805-5
5525:
5521:
5517:
5513:
5509:
5507:0-7506-8025-3
5503:
5499:
5498:
5492:
5488:
5486:0-415-27297-1
5482:
5478:
5477:
5471:
5467:
5465:0-486-67865-2
5461:
5457:
5452:
5451:
5444:
5441:
5440:0-486-60174-9
5437:
5433:
5429:
5426:
5422:
5418:
5416:0-442-04199-3
5412:
5408:
5407:
5401:
5397:
5391:
5387:
5386:
5380:
5376:
5374:0-13-484394-0
5370:
5366:
5365:
5359:
5356:
5355:0-07-100406-8
5352:
5348:
5344:
5340:
5338:0-521-49827-9
5334:
5330:
5329:
5323:
5319:
5317:0-486-66958-0
5313:
5309:
5308:
5302:
5298:
5296:0-444-42455-5
5292:
5288:
5283:
5279:
5277:0-412-47550-2
5273:
5269:
5268:
5262:
5258:
5256:0-07-112939-1
5252:
5247:
5246:
5239:
5235:
5233:0-7506-6638-2
5229:
5225:
5224:
5218:
5217:
5212:
5206:
5202:
5198:
5192:
5189:
5186:
5185:3-540-74297-2
5182:
5178:
5172:
5169:
5166:
5165:0-7923-2454-4
5162:
5159:
5153:
5150:
5146:
5140:
5137:
5134:
5133:9781574447132
5130:
5124:
5122:
5120:
5116:
5112:
5106:
5103:
5098:
5094:
5090:
5086:
5082:
5078:
5071:
5068:
5064:
5062:
5055:
5052:
5049:
5048:3-540-43019-9
5045:
5041:
5035:
5033:
5029:
5025:
5024:0-486-40180-4
5021:
5017:
5011:
5008:
5005:
5004:1-932159-75-4
5001:
4997:
4991:
4988:
4984:
4979:
4976:
4973:
4972:0-486-46290-0
4969:
4965:
4961:
4958:
4952:
4949:
4944:
4938:
4934:
4927:
4924:
4912:
4908:
4902:
4900:
4896:
4885:
4881:
4875:
4872:
4860:
4856:
4850:
4848:
4844:
4838:
4833:
4830:
4828:
4827:Yield surface
4825:
4823:
4820:
4818:
4817:Virial stress
4815:
4813:
4810:
4808:
4805:
4803:
4800:
4798:
4795:
4793:
4790:
4788:
4785:
4783:
4780:
4778:
4777:Strain tensor
4775:
4773:
4770:
4768:
4765:
4763:
4760:
4758:
4755:
4753:
4750:
4748:
4745:
4743:
4742:Mohr's circle
4740:
4738:
4735:
4733:
4730:
4728:
4725:
4723:
4720:
4719:
4709:
4706:
4704:
4701:
4700:
4697:
4694:
4692:
4689:
4688:
4684:
4680:
4677:
4673:
4672:
4669:
4666:
4664:
4661:
4660:
4657:
4656:
4653:
4648:
4647:
4641:
4639:
4637:
4633:
4629:
4623:
4615:
4613:
4611:
4607:
4603:
4597:
4589:
4585:
4582:
4578:
4574:
4570:
4566:
4562:
4558:
4554:
4549:
4547:
4542:
4541:strain tensor
4538:
4534:
4530:
4526:
4522:
4521:Newton's laws
4518:
4509:
4507:
4505:
4499:
4497:
4493:
4489:
4485:
4477:
4475:
4473:
4469:
4464:
4460:
4453:
4451:
4449:
4445:
4436:
4431:
4424:
4422:
4420:
4416:
4412:
4406:
4399:
4394:
4387:
4385:
4383:
4382:infinitesimal
4375:
4373:
4371:
4367:
4362:
4348:
4344:
4340:
4335:
4331:
4327:
4322:
4318:
4294:
4290:
4286:
4281:
4277:
4273:
4268:
4264:
4241:
4237:
4231:
4227:
4223:
4218:
4214:
4186:
4182:
4178:
4173:
4169:
4165:
4160:
4156:
4148:
4130:
4126:
4122:
4117:
4113:
4109:
4104:
4100:
4092:
4065:
4063:
4062:Mohr's circle
4055:
4053:
4051:
4033:
4030:
4026:
4022:
4017:
4014:
4010:
4006:
4001:
3998:
3994:
3985:
3967:
3963:
3959:
3954:
3950:
3946:
3941:
3937:
3914:
3906:
3902:
3894:
3891:
3887:
3879:
3876:
3872:
3862:
3859:
3855:
3847:
3843:
3835:
3832:
3828:
3818:
3815:
3811:
3803:
3800:
3796:
3788:
3784:
3777:
3753:
3749:
3745:
3740:
3736:
3713:
3709:
3705:
3700:
3696:
3673:
3669:
3665:
3660:
3656:
3647:
3643:
3639:
3635:
3631:
3615:
3595:
3586:
3571:
3563:
3559:
3551:
3547:
3539:
3535:
3525:
3521:
3513:
3509:
3501:
3497:
3487:
3483:
3475:
3471:
3463:
3459:
3452:
3447:
3442:
3434:
3430:
3422:
3418:
3410:
3406:
3399:
3394:
3389:
3381:
3377:
3369:
3365:
3357:
3353:
3346:
3336:
3332:
3328:
3327:transposition
3307:
3304:
3301:
3298:
3276:
3272:
3268:
3263:
3259:
3255:
3250:
3246:
3237:
3233:
3217:
3210:
3209:normal vector
3191:
3180:
3177:
3155:
3147:
3144:
3140:
3132:
3129:
3125:
3117:
3114:
3110:
3100:
3097:
3093:
3085:
3082:
3078:
3070:
3067:
3063:
3053:
3050:
3046:
3038:
3035:
3031:
3023:
3020:
3016:
3009:
2985:
2977:
2973:
2965:
2961:
2953:
2949:
2939:
2935:
2927:
2923:
2915:
2911:
2901:
2897:
2889:
2885:
2877:
2873:
2866:
2844:
2841:
2838:
2835:
2832:
2810:
2806:
2802:
2797:
2793:
2789:
2784:
2780:
2771:
2766:
2764:
2738:
2734:
2730:
2692:
2689:
2686:
2666:
2663:
2660:
2637:
2626:
2623:
2617:
2606:
2603:
2597:
2594:
2591:
2588:
2585:
2532:
2521:
2518:
2498:
2491:
2490:normal vector
2487:
2471:
2462:
2453:
2445:
2440:
2433:Cauchy tensor
2432:
2430:
2428:
2424:
2408:
2388:
2380:
2373:General types
2372:
2370:
2368:
2364:
2360:
2352:
2350:
2348:
2344:
2340:
2336:
2332:
2326:
2319:
2315:
2311:
2295:
2291:
2285:
2276:
2258:
2249:
2245:
2241:
2225:
2221:
2215:
2206:
2188:
2179:
2176:(top right),
2175:
2171:
2167:
2163:
2158:
2151:
2149:
2147:
2142:
2138:
2134:
2129:
2127:
2123:
2119:
2101:
2098:
2093:
2090:
2082:
2078:
2062:
2054:
2050:
2046:
2042:
2038:
2034:
2024:
2017:
2015:
2013:
2009:
2005:
2000:
1995:
1993:
1989:
1985:
1981:
1977:
1973:
1969:
1965:
1961:
1945:
1932:
1928:
1924:
1920:
1904:
1900:
1896:
1893:
1890:
1881:
1877:
1863:
1855:
1851:
1832:
1829:
1824:
1821:
1813:
1809:
1805:
1801:
1797:
1781:
1773:
1764:
1757:
1755:
1753:
1749:
1745:
1741:
1740:simple stress
1733:
1731:
1729:
1725:
1721:
1717:
1713:
1709:
1705:
1700:
1698:
1694:
1690:
1686:
1682:
1678:
1674:
1670:
1669:birefringence
1665:
1663:
1660:
1656:
1655:active matter
1652:
1648:
1644:
1636:
1635:
1629:
1622:
1620:
1618:
1615:(psi) in the
1614:
1610:
1606:
1602:
1598:
1594:
1590:
1582:
1580:
1577:
1573:
1569:
1565:
1560:
1556:
1552:
1548:
1543:
1539:
1535:
1531:
1527:
1523:
1518:
1516:
1515:
1510:
1506:
1502:
1500:
1495:
1491:
1487:
1483:
1479:
1475:
1469:
1465:
1457:
1455:
1453:
1449:
1445:
1441:
1437:
1433:
1429:
1426:of a surface
1425:
1422:
1421:normal vector
1418:
1415:; which is a
1414:
1411:, called the
1410:
1406:
1402:
1398:
1394:
1390:
1386:
1382:
1378:
1374:
1370:
1366:
1363:
1359:
1356:
1351:
1349:
1345:
1341:
1336:
1332:
1323:
1319:
1317:
1313:
1304:
1302:
1300:
1295:
1291:
1287:
1283:
1279:
1275:
1271:
1267:
1263:
1257:
1255:
1251:
1247:
1243:
1239:
1235:
1229:
1227:
1226:glass blowing
1223:
1222:composite bow
1211:
1207:
1203:
1197:
1186:
1182:
1178:
1172:
1160:
1158:
1156:
1152:
1148:
1144:
1140:
1136:
1132:
1128:
1123:
1121:
1117:
1116:piezoelectric
1113:
1109:
1105:
1101:
1097:
1093:
1089:
1080:
1076:
1074:
1070:
1066:
1062:
1058:
1054:
1050:
1046:
1042:
1038:
1033:
1031:
1027:
1023:
1019:
1015:
1011:
1007:
1003:
999:
995:
991:
987:
986:
981:
976:
974:
970:
966:
962:
961:
956:
952:
951:
946:
942:
938:
934:
930:
918:
913:
911:
906:
904:
899:
898:
896:
895:
887:
884:
882:
879:
877:
874:
872:
869:
867:
864:
862:
859:
857:
854:
852:
849:
847:
844:
842:
839:
837:
834:
832:
829:
827:
824:
822:
819:
817:
814:
813:
806:
805:
794:
791:
789:
786:
784:
781:
780:
779:
778:
775:
771:
766:
763:
761:
758:
756:
753:
752:
751:
750:
745:
740:
739:
730:
726:
720:
717:
715:
712:
710:
707:
705:
702:
700:
699:Charles's law
697:
695:
692:
690:
687:
686:
684:
683:
680:
676:
670:
667:
665:
662:
660:
657:
655:
652:
650:
647:
646:
644:
643:
640:
636:
630:
627:
623:
620:
616:
613:
608:
607:non-Newtonian
605:
601:
597:
596:
595:
592:
590:
587:
583:
580:
578:
575:
573:
570:
566:
563:
561:
558:
554:
551:
550:
548:
547:
544:
540:
535:
530:
529:
521:
518:
516:
513:
509:
506:
505:
504:
501:
499:
496:
494:
493:Compatibility
491:
487:
484:
482:
481:Finite strain
479:
478:
477:
474:
472:
469:
467:
464:
462:
459:
455:
452:
451:
450:
447:
445:
442:
441:
437:
432:
431:
420:
417:
416:
415:
414:
409:
404:
401:
399:
396:
394:
391:
390:
389:
388:
385:Conservations
383:
375:
374:
370:
351:
348:
343:
340:
334:
331:
328:
325:
318:
317:
314:
310:
306:
305:
284:
281:
260:
257:
241:
239:
235:
232:
228:
224:
220:
218:
211:
208:
204:
201:
195:
192:
189:
187:
183:
180:
177:
171:
166:
162:
157:
152:
147:
138:
135:
127:
116:
113:
109:
106:
102:
99:
95:
92:
88:
85: –
84:
80:
79:Find sources:
73:
69:
63:
62:
57:This article
55:
51:
46:
45:
40:
33:
19:
18:Normal stress
5539:
5519:
5496:
5475:
5449:
5431:
5424:
5405:
5384:
5363:
5346:
5327:
5306:
5286:
5266:
5244:
5222:
5191:
5179:. Springer.
5171:
5152:
5139:
5110:
5105:
5080:
5076:
5070:
5060:
5054:
5010:
4990:
4985:, pp.107-110
4978:
4951:
4932:
4926:
4915:. Retrieved
4913:. 2020-07-31
4910:
4887:. Retrieved
4883:
4874:
4863:. Retrieved
4859:openstax.org
4858:
4767:Shot peening
4675:
4625:
4598:
4594:
4573:phase change
4565:plastic flow
4550:
4513:
4500:
4481:
4457:
4447:
4443:
4440:
4435:fishing pole
4410:
4407:
4403:
4379:
4376:Tensor field
4201:, such that
4091:eigenvectors
4066:
4059:
4049:
3983:
3587:
2767:
2463:
2459:
2426:
2422:
2378:
2376:
2356:
2342:
2338:
2334:
2330:
2327:
2323:
2317:
2313:
2309:
2274:
2247:
2243:
2239:
2204:
2177:
2173:
2169:
2165:
2161:
2136:
2132:
2130:
2125:
2121:
2117:
2080:
2076:
2052:
2048:
2044:
2040:
2036:
2029:
2003:
1998:
1996:
1987:
1979:
1975:
1971:
1967:
1963:
1959:
1936:
1930:
1926:
1922:
1918:
1853:
1811:
1807:
1803:
1799:
1769:
1751:
1747:
1743:
1739:
1737:
1734:Simple types
1712:plastic flow
1701:
1677:permeability
1673:polarization
1666:
1640:
1632:
1601:square metre
1586:
1575:
1571:
1567:
1563:
1558:
1554:
1550:
1546:
1541:
1537:
1529:
1525:
1521:
1519:
1514:shear stress
1512:
1497:
1493:
1489:
1485:
1481:
1477:
1473:
1471:
1468:Shear stress
1452:tensor field
1435:
1431:
1427:
1423:
1404:
1400:
1396:
1395:, the force
1372:
1368:
1364:
1357:
1354:
1352:
1328:
1308:
1299:laminar flow
1264:'s rigorous
1258:
1230:
1218:
1209:
1184:
1139:plastic flow
1124:
1122:materials).
1087:
1085:
1072:
1068:
1056:
1036:
1034:
1029:
984:
977:
959:
949:
932:
926:
774:Smart fluids
719:Graham's law
625:
618:
603:
589:Pascal's law
585:
568:
556:
470:
411:Inequalities
186:SI unit
178:
130:
121:
111:
104:
97:
90:
78:
66:Please help
61:verification
58:
5042:. Springer
4691:Temperature
4581:Hooke's law
4492:body forces
4388:Thin plates
4147:eigenvalues
3337:), that is
2357:Parts with
2004:hoop stress
1850:compression
1796:equilibrium
1643:temperature
1613:square inch
1534:dot product
1505:compression
1448:homogeneous
1391:of viscous
1342:rod or the
1331:macroscopic
1274:coordinates
1185:compressive
1181:Switzerland
1127:strain rate
990:deformation
960:compressive
945:deformation
793:Ferrofluids
694:Boyle's law
466:Hooke's law
444:Deformation
198:Other units
124:August 2021
5563:Categories
4942:0306812835
4917:2022-11-02
4889:2022-11-02
4865:2022-11-02
4839:References
4634:, and the
4608:, and the
4531:) and the
4472:avalanches
4425:Thin beams
3648:, that is
3230:(which is
2570:satisfies
1883:The ratio
1750:, and the
1720:cavitation
1387:, or in a
1305:Definition
1147:cavitation
1100:net forces
1022:collisions
955:elongation
846:Gay-Lussac
809:Scientists
709:Fick's law
689:Atmosphere
508:frictional
461:Plasticity
449:Elasticity
94:newspapers
4415:curvature
4370:isotropic
4345:λ
4332:λ
4319:λ
4228:λ
4210:σ
4183:λ
4170:λ
4157:λ
4076:σ
4027:τ
4011:τ
3995:τ
3964:σ
3951:σ
3938:σ
3903:σ
3888:τ
3873:τ
3856:τ
3844:σ
3829:τ
3812:τ
3797:τ
3785:σ
3750:σ
3737:σ
3710:σ
3697:σ
3670:σ
3657:σ
3646:symmetric
3560:σ
3548:σ
3536:σ
3522:σ
3510:σ
3498:σ
3484:σ
3472:σ
3460:σ
3448:⋅
3331:covariant
3312:σ
3308:⋅
3232:covariant
3185:σ
3141:σ
3126:σ
3111:σ
3094:σ
3079:σ
3064:σ
3047:σ
3032:σ
3017:σ
2974:σ
2962:σ
2950:σ
2936:σ
2924:σ
2912:σ
2898:σ
2886:σ
2874:σ
2825:or named
2748:σ
2714:σ
2693:β
2687:α
2631:σ
2627:β
2611:σ
2607:α
2595:β
2586:α
2579:σ
2557:σ
2526:σ
2335:isotropic
2152:Isotropic
2091:τ
2063:τ
1946:σ
1891:σ
1864:σ
1822:σ
1697:viscosity
1659:piecewise
1603:) in the
1212:stresses.
1187:stresses.
1026:molecules
980:particles
965:dimension
886:Truesdell
816:Bernoulli
765:Rheometer
760:Rheometry
600:Newtonian
594:Viscosity
344:φ
332:−
282:−
258:−
238:Dimension
4960:Archived
4663:Pressure
4642:See also
4616:Measures
4569:fracture
4454:Analysis
4398:tank car
2353:Cylinder
2343:pressure
2341:or just
2333:or just
1716:fracture
1651:impulses
1634:craquelé
1589:pressure
1528:towards
1383:. In a
1381:pressure
1248:and the
1234:capitals
1143:fracture
1108:chemical
1051:). Any
1049:friction
1014:reaction
1010:pressure
1006:pressure
969:SI units
744:Rheology
649:Adhesion
629:Pressure
615:Buoyancy
560:Dynamics
398:Momentum
5574:Tensors
5085:Bibcode
5058:(2009)
5026:. pages
4722:Bending
4696:Entropy
4553:elastic
4510:Methods
4496:bearing
2423:biaxial
2146:I-beams
1772:tension
1693:plasmas
1597:newtons
1593:pascals
1509:tension
1434:across
1375:. In a
1335:quantum
1246:trusses
1242:cupolas
1210:tensile
1161:History
1114:(as in
1041:gravity
950:tensile
831:Charles
639:Liquids
553:Statics
498:Bending
229:⋅
225:⋅
108:scholar
5547:
5526:
5504:
5483:
5462:
5458:–142.
5438:
5413:
5392:
5371:
5353:
5335:
5314:
5293:
5274:
5253:
5230:
5203:
5183:
5163:
5147:(book)
5131:
5046:
5022:
5002:
4970:
4939:
4772:Strain
4683:Strain
4676:Stress
4668:Volume
4630:, the
4604:, the
3728:, and
2733:tensor
2456:units.
2347:Z-tube
2035:. Let
2012:vessel
1746:, the
1685:spring
1675:, and
1609:pounds
1501:stress
1499:normal
1409:tensor
1393:liquid
1344:fibers
1316:energy
1312:torque
1282:Newton
1280:, and
1238:arches
1061:spring
1037:stress
1002:liquid
998:weight
985:strain
975:(Pa).
973:pascal
941:forces
933:stress
881:Stokes
876:Pascal
866:Navier
861:Newton
851:Graham
826:Cauchy
729:Plasma
624:
622:Mixing
617:
602:
584:
567:
555:
543:Fluids
476:Strain
471:Stress
454:linear
403:Energy
191:pascal
149:Stress
110:
103:
96:
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