View source for Stress (mechanics)

{{Short description|Physical quantity that expresses internal forces in a continuous material}}
{{About|stresses in classical (continuum) mechanics|stresses in material science|Strength of materials}}
{{Other uses|Stress (disambiguation){{!}}Stress}}
{{More citations needed|date=August 2021}}
{{Infobox physical quantity
| name = Stress
| image = Plastic Protractor Polarized 05375.jpg
| caption = [[Residual stress]]es inside a plastic protractor are revealed by [[photoelasticity|polarized light]].
| unit = [[Pascal (unit)|pascal]]
| otherunits = [[Pound per square inch|psi]], [[Bar (unit)|bar]]
| symbols =  ''σ''
| baseunits = Pa = [[Kilogram|kg]]&sdot;[[metre|m]]<sup>−1</sup>&sdot;[[second|s]]<sup>−2</sup>
| dimension = wikidata
}}
{{Continuum mechanics|solid}}

In [[continuum mechanics]], '''stress''' is a [[physical quantity]] that describes [[Force|forces]] present during [[Deformation (physics)|deformation]]. For example, an object being pulled apart, such as a stretched elastic band, is subject to [[Tension (physics)|''tensile'']] stress and may undergo [[Elongation (materials science)|elongation]]. An object being pushed together, such as a crumpled sponge, is subject to [[Compression (physics)|''compressive'']] stress and may undergo shortening.<ref name=":0">{{Cite web |title=12.3 Stress, Strain, and Elastic Modulus - University Physics Volume 1 {{!}} OpenStax |url=https://openstax.org/books/university-physics-volume-1/pages/12-3-stress-strain-and-elastic-modulus |access-date=2022-11-02 |website=openstax.org |date=19 September 2016 |language=en}}</ref><ref>{{Cite web |title=Class Physical-Quantity in theory Physical-Quantities |url=http://www-ksl.stanford.edu/knowledge-sharing/ontologies/html/physical-quantities/PHYSICAL-QUANTITY.html |access-date=2022-11-02 |website=www-ksl.stanford.edu}}</ref> The greater the force and the smaller the cross-sectional area of the body on which it acts, the greater the stress. Stress has [[Dimension (physics)|dimension]] of force per area, with [[SI Units|SI units]] of newtons per square meter (N/m<sup>2</sup>) or [[Pascal (unit)|pascal]] (Pa).<ref name=":0"/>

Stress expresses the internal forces that neighbouring [[particle]]s of a continuous material exert on each other, while [[Strain (mechanics)|''strain'']] is the measure of the relative [[deformation (mechanics)|deformation]] of the material.<ref name=":1">{{Cite web |date=2020-07-31 |title=What is Shear Stress - Materials - Definition |url=https://material-properties.org/what-is-shear-stress-materials-definition/ |access-date=2022-11-02 |website=Material Properties |language=en-US}}</ref> For example, when a [[solid]] vertical bar 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 force|reaction]]. These macroscopic forces are actually the net result of a very large number of [[intermolecular force]]s and [[statistical mechanics|collision]]s between the particles in those [[molecule]]s. Stress is frequently represented by a lowercase Greek letter sigma (''σ'').<ref name=":1" />

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 force]]s, external pressure, or [[friction]]).  Any [[deformation (mechanics)|strain (deformation)]] of a solid material generates an internal ''elastic stress'', analogous to the reaction force of a [[spring (device)|spring]], that tends to restore the material to its original non-deformed state.  In liquids and [[gas]]es, 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 the name ''mechanical stress''.
[[File:Different-types-of-mechanical-stress_EN.svg#%7B%7Bint%3Afiledesc%7D%7D|thumb|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 [[Newton's laws of motion#Newton's second law|net forces]], for example by [[thermal expansion|changes in temperature]] or [[chemistry|chemical]] composition, or by external [[electromagnetic field]]s (as in [[piezoelectricity|piezoelectric]] and [[magnetostriction|magnetostrictive]] materials).

The relation between mechanical stress, strain, and the [[strain rate]] can be quite complicated, although a [[linear elasticity|linear approximation]] may be adequate in practice if the quantities are sufficiently small. Stress that exceeds certain [[strength of materials|strength limits]] of the material will result in permanent deformation (such as [[plasticity (physics)|plastic flow]], [[fracture]], [[cavitation]]) or even change its [[crystal structure]]  and [[chemistry|chemical composition]].

{{TOC limit|limit=2}}
==History==
{{Multi image
| direction         = vertical
| total_width       = 300
| image1            = Roman era stone arch bridge, Ticino, Switzerland cropped.JPG
| caption1          = [[Roman Empire|Roman]]-era bridge in [[Switzerland]]. The stone arches in the bridge are subject to ''compressive'' stresses.
| alt1              = Stone arch bridge spanning a river.
| image2            = Inca bridge.jpg
| alt2              = Rope bridge spanning a deep river valley.
| caption2          = [[Inca]] bridge on the [[Apurimac River]]. The rope in the bridge is subject to ''tensile'' stresses.
}}
<!--[[File:Pantheon-Kuppel von innen.JPG|right|thumb|Cupola of the [[Pantheon, Rome|Pantheon]], [[Rome]]]]-->
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 [[composite bow]] and [[glass blowing]].<ref>{{cite book|last1=Gordon|first1=J.E.|title=Structures, or, Why things don't fall down|date=2003|publisher=Da Capo Press|location=Cambridge, MA|isbn=0306812835|edition=2. Da Capo Press}}</ref>

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 [[capital (architecture)|capitals]], [[arch]]es, [[cupola]]s, [[truss]]es and the [[flying buttress]]es of [[Gothic architecture|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 [[Cartesian coordinates|coordinates]] and [[analytic geometry]], and [[Isaac Newton|Newton]]'s [[Newton's laws|laws of motion and equilibrium]] and [[calculus|calculus of infinitesimals]].<ref name="Lubliner" />  With those tools, [[Augustin-Louis Cauchy]] was able to give the first rigorous and general mathematical model of a deformed elastic body by introducing the notions of stress and strain.<ref>https://archive.org/details/historyofstrengt0000timo_k8r2/page/110/mode/2up, pp.107-110</ref>  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 a differential formula for friction forces (shear stress) in parallel [[laminar flow]].

== Definition ==
Stress is defined as the force across a small boundary per unit area of that boundary, for all orientations of the boundary.<ref name="Chen" /> Derived from a  physical quantity (force) and a purely geometrical quantity (area), stress is also a physical quantity, like velocity, [[torque]] or [[energy]], that can be quantified and analyzed without explicit consideration of the nature of the material or of its physical causes.
[[File:Cmec stress defn f02 t6.png|thumb|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.]]

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 mechanics|quantum]] 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, are assumed to be smoothly distributed over them.<ref name="Chadwick" />{{rp|pp=90–106}} 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 [[metal]] rod or the [[fiber]]s of a piece of [[wood]].

Quantitatively, the stress is expressed by the ''Cauchy traction vector'' ''T'' defined as the [[Traction (mechanics)|traction force]] ''F'' between adjacent parts of the material across an imaginary separating surface ''S'', divided by the area of ''S''.<ref name="Liu" />{{rp|pp=41–50}} In a [[fluid]] at rest the force is perpendicular to the surface, and is the familiar [[hydrostatic pressure|pressure]].  In a [[solid]], or in a [[Fluid dynamics|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|(Cauchy) stress tensor]]; which is a [[linear map|linear function]] that relates the [[surface normal|normal vector]] ''n'' of a surface ''S'' to the traction vector ''T'' across ''S''.  With respect to any chosen [[Cartesian coordinates|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]].

=== Normal and shear ===
{{Further|Compression (physical)|Shear stress}}

In general, 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 (geometry)|normal]] stress'' ([[Compression (physical)|compression]] or [[Tension (physics)|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'') is assumed fixed, the normal component can be expressed by a single number, the [[inner product|dot product]] {{nowrap|''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 {{nowrap|''T'' − (''T'' · ''n'')''n''}}.

== Units ==
The dimension of stress is that of [[pressure]], and therefore its coordinates are measured in the same units as pressure: namely, [[pascal (unit)|pascal]]s (Pa, that is, [[newton (force)|newton]]s per [[square metre]]) in the [[International System of Units|International System]], or [[pound-force|pounds]] per [[square inch]] (psi) in the [[Imperial units|Imperial system]]. Because mechanical stresses easily exceed a million Pascals, MPa, which stands for megapascal, is a common unit of stress.

== Causes and effects ==
[[File:Vase-craquele-Emile-Galle-vers-1880-decor-mante-religieuse-cigale-1301.jpg|upright|thumb|Glass vase with the ''[[craquelé]]'' effect. The cracks are the result of brief but intense stress created when the semi-molten piece is briefly dipped in water.<ref name=lamglass/>]]
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 (chemistry)|phase]], 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 [[impulse (physics)|impulse]]s due to collisions). In [[active matter]], self-propulsion of microscopic particles generates macroscopic stress profiles.<ref>{{cite journal|doi=10.1103/RevModPhys.85.1143|title=Hydrodynamics of soft active matter|journal=Reviews of Modern Physics|volume=85|issue=3|pages=1143–1189|year=2013|last1=Marchetti|first1=M. C.|last2=Joanny|first2=J. F.|last3=Ramaswamy|first3=S.|last4=Liverpool|first4=T. B.|last5=Prost|first5=J.|last6=Rao|first6=Madan|last7=Simha|first7=R. Aditi|bibcode=2013RvMP...85.1143M}}</ref> In general, 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]], [[polarizability|polarization]], and [[permeability (earth sciences)|permeability]]. The imposition of stress by an external agent usually creates some [[deformation (mechanics)|strain (deformation)]] 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 [[spring (device)|spring]], tending to restore the material to its original undeformed state. Fluid materials (liquids, [[gas]]es and [[plasma (physics)|plasma]]s) 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 [[viscosity]]. The same for normal viscous stresses can be found in Sharma (2019).<ref name="sharma2019">Sharma, B and Kumar, R "Estimation of bulk viscosity of dilute gases using a nonequilibrium molecular dynamics approach.", ''Physical Review E'',100, 013309 (2019)</ref>

The relation between stress and its effects and causes, including deformation and rate of change of deformation, can be quite complicated (although a [[linear elasticity|linear approximation]] may be adequate in practice if the quantities are small enough). Stress that exceeds certain [[strength of materials|strength limits]] of the material will result in permanent deformation (such as [[plasticity (physics)|plastic flow]], [[fracture]], [[cavitation]]) or even change its [[crystal structure]]  and [[chemistry|chemical composition]].

==Simple types==

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 '''simple stress''' situations, that are often encountered in engineering design, are the ''uniaxial normal stress'', the ''simple shear stress'', and the ''isotropic normal stress''.<ref name="Huston" />

===Uniaxial normal===
[[File:Axial stress noavg.svg|240px|thumb|right|Idealized stress in a straight bar with uniform cross-section.]]
A common situation with a simple stress pattern is when a straight rod, with uniform material and cross section, is subjected to [[tension (physics)|tension]] by opposite forces of magnitude <math>F</math> along its axis. If the system is in [[mechanical equilibrium|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, 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''.<math display="block">\sigma=\frac{F}{A}</math> 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 stress.<ref name="Huston" />  If the load is [[compression (physical)|compression]] on the bar, rather than stretching it, the analysis is the same except that the force ''F'' and the stress <math>\sigma</math> change sign, and the stress is called compressive stress.

[[File:Normal stress.svg|240px|thumb|left|The ratio <math>\sigma = F/A</math> may be only an average stress. The stress may be unevenly distributed over the cross section (''m''–''m''), especially near the attachment points (''n''–''n'').]]
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 <math>\sigma</math> = ''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 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 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 (fluid conveyance)|pipe]]  or [[pressure vessel|vessel]] filled with pressurized fluid.
===Shear===

[[File:Shear stress.svg|240px|right|thumb|Shear stress in a horizontal bar loaded by two offset blocks.]]
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 [[snips|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 the single number <math>\tau</math>, calculated simply with the magnitude of those forces, ''F'' and the cross sectional area, ''A''.<math display="block">\tau=\frac{F}{A}</math>Unlike normal stress, this ''simple shear stress'' is directed parallel to the cross-section considered, rather than perpendicular to it.<ref name="Huston" />  For any plane ''S'' that 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.<ref name="Pilkey" />{{rp|p=292}} Shear stress is observed also when a cylindrical bar such as a [[axle|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-beam]]s under bending loads, due to the web constraining the end plates ("flanges").

=== Isotropic ===
[[File:Isotropic stress noavg.svg|upright=1.1|thumb|Isotropic tensile stress. Top left: Each face of a cube of homogeneous material is pulled by a force with magnitude ''F'', applied evenly over the entire face whose area is ''A''.  The force across any section ''S'' of the cube must balance the forces applied below the section. In the three sections shown, the forces are ''F'' (top right), ''F''<math>\sqrt{2}</math> (bottom left), and ''F''<math>\sqrt{3}/2</math> (bottom right); and the area of ''S'' is ''A'', ''A''<math>\sqrt{2}</math> and ''A''<math>\sqrt{3}/2</math>, respectively. So the stress across ''S'' is ''F''/''A'' in all three cases.]]
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 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.

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]].

=== Cylinder ===
Parts with [[rotational symmetry]], 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 [[cylindrical symmetry]]. The analysis of such [[cylinder stress]]es can take advantage of the symmetry to reduce the dimension of the domain and/or of the stress tensor.

== General types ==
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 the stress is maximum for surfaces that are perpendicular to a certain direction <math>d</math>, and zero across any surfaces that are parallel to <math>d</math>. 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 the most general case, called ''triaxial stress'', the stress is nonzero across every surface element.

== Cauchy tensor ==
{{Main|Cauchy stress tensor}}
[[File:Components stress tensor cartesian.svg|left|thumb|Components of stress in three dimensions]]
[[File:Cmec stress ball f02 t6.png|thumb|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 units.]]
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 <math>T</math> across a surface will always be a [[linear function]] of the surface's [[surface normal|normal vector]] <math>n</math>, the unit-length vector that is perpendicular to it.  That is, <math>T = \boldsymbol{\sigma}(n)</math>, where the function <math>\boldsymbol{\sigma}</math> satisfies
<math display="block">\boldsymbol{\sigma}(\alpha u + \beta v) = \alpha\boldsymbol{\sigma}(u) + \beta\boldsymbol{\sigma}(v)</math>
for any vectors <math>u,v</math> and any real numbers <math>\alpha,\beta</math>.
The function <math>\boldsymbol{\sigma}</math>, now called the [[Cauchy stress tensor|(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]], <math>\boldsymbol{\sigma}</math> is classified as a second-order tensor of [[type of a tensor|type]] (0,2) or (1,1) depending on convention.

Like any linear map between vectors, 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 <math>x_1,x_2,x_3</math> or named <math>x,y,z</math>, the matrix may be written as
<math display="block">
\begin{bmatrix}
  \sigma _{11} & \sigma _{12} & \sigma _{13} \\
  \sigma _{21} & \sigma _{22} & \sigma _{23} \\
  \sigma _{31} & \sigma _{32} & \sigma _{33} 
\end{bmatrix}
</math>  or  <math display="block">
\begin{bmatrix}
  \sigma _{xx} & \sigma _{xy} & \sigma _{xz} \\
  \sigma _{yx} & \sigma _{yy} & \sigma _{yz} \\
  \sigma _{zx} & \sigma _{zy} & \sigma _{zz} \\
\end{bmatrix}
</math>
The stress vector <math>T = \boldsymbol{\sigma}(n)</math> across a surface with [[normal vector]] <math>n</math> (which is [[Covariance and contravariance of vectors#Covariant transformation|covariant]] - [[Row and column vectors|"row; horizontal"]] - vector) with coordinates <math>n_1,n_2,n_3</math> is then a matrix product <math>T = n\cdot\boldsymbol{\sigma}</math> (where T in upper index is [[transpose|transposition]], and as a result we get [[Covariance and contravariance of vectors#Covariant transformation|covariant]] (row) vector) (look on [[Cauchy stress tensor]]), that is
<math display="block"> 
  \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}
</math>

The linear relation between <math>T</math> and <math>n</math> follows from 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 momentum equation|Cauchy's equations of motion]] for zero acceleration). Moreover, the principle of [[conservation of angular momentum]] implies that the stress tensor is [[symmetric matrix|symmetric]], that is <math> \sigma_{12} = \sigma_{21}</math>, <math>\sigma_{13} = \sigma_{31}</math>, and <math>\sigma_{23} = \sigma_{32} </math>. 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
<math display="block">
  \begin{bmatrix}
    \sigma_x & \tau_{xy} & \tau_{xz} \\
    \tau_{xy} & \sigma_y & \tau_{yz} \\
    \tau_{xz} & \tau_{yz} & \sigma_z
  \end{bmatrix}
</math>
where the elements <math>\sigma_x,\sigma_y,\sigma_z</math> are called the ''orthogonal normal stresses'' (relative to the chosen coordinate system), and <math>\tau_{xy}, \tau_{xz},\tau_{yz}</math> the ''orthogonal shear stresses''.{{citation needed|date=October 2021}}

===Change of coordinates===

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 <math>\boldsymbol{\sigma}</math> has three mutually orthogonal unit-length [[eigenvalues and eigenvectors|eigenvector]]s <math>e_1,e_2,e_3</math> and three real [[eigenvalues and eigenvectors|eigenvalue]]s <math>\lambda_1,\lambda_2,\lambda_3</math>, such that <math> \boldsymbol{\sigma} e_i = \lambda_i e_i</math>. Therefore, in a coordinate system with axes <math>e_1,e_2,e_3</math>, the stress tensor is a diagonal matrix, and has only the three normal components <math qid=Q112270193>\lambda_1,\lambda_2,\lambda_3</math> 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.

===Tensor field===
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 the medium surrounding that point, and taking the average stresses in that particle as being the stresses at the point.
=== Thin plates ===
[[File:W39504 stat Nbk2007.jpg|thumb|A [[tank car]] made from bent and welded steel plates.]]
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 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 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 (mathematics)|radius of curvature]] is comparable to the thickness of the plate).

=== Thin beams ===
[[File:Sandy Hook NJ beach fisherman's pole.jpg|thumb|upright=0.65|For stress modeling, a [[fishing pole]] may be considered one-dimensional.]]
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 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).

== Analysis ==
[[Stress analysis]] is a branch of [[applied physics]] 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 many other disciplines; for example, in geology, to study phenomena like [[plate tectonics]], vulcanism and [[avalanche]]s; and in biology, to understand the anatomy of living beings.
=== Goals and assumptions ===
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 a system must be balanced by internal reaction forces,<ref name="Smith" />{{rp|p=97}} which are almost always surface contact forces between adjacent particles — that is, as stress.<ref name="Liu" /> 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 [[body force]]s (such as gravity or magnetic attraction), that act throughout the volume of a material;<ref name="Irgens" />{{rp|pp=42–81}} or concentrated loads (such as friction between an axle and a [[bearing (mechanical)|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 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]].<ref name="Slaughter" />

=== Methods ===
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 methods, especially during design.
The basic stress analysis problem can be formulated by [[Euler's laws|Euler's equations of motion]] for continuous bodies (which are consequences of [[Newton's laws of motion|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 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 [[Elasticity (physics)|elastic]] structures 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 ([[plasticity (physics)|plastic flow]], [[fracture]], [[phase transition|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|Hooke's law]] 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 loads, too. For small enough stresses, even non-linear systems can usually be assumed to be linear.

[[File:Loaded truss.svg|240px|right|thumb|Simplified model of a truss for stress analysis, assuming unidimensional elements under uniform axial tension or compression.]]
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 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.

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 the [[finite element method]], the [[finite difference method]], and the [[boundary element method]].

==Measures==
{{Main|Stress measures}}
Other useful stress measures include the first and second [[Piola–Kirchhoff stress tensor]]s, the [[Biot stress tensor]], and the [[Kirchhoff stress tensor]].

==See also==
{{Conjugate variables (thermodynamics)}}{{Div col|colwidth=20em}}
* [[Bending]]
* [[Compressive strength]]
* [[Critical plane analysis]]
* [[Kelvin probe force microscope]]
* [[Mohr's circle]]
* [[Lamé's stress ellipsoid]]
* [[Reinforced solid]]
* [[Residual stress]]
* [[Shear strength]]
* [[Shot peening]]
* [[Strain (materials science)|Strain]]
* [[Strain tensor]]
* [[Strain rate tensor]]
* [[Stress–energy tensor]]
* [[Stress–strain curve]]
* [[Stress concentration]]
* [[Transient friction loading]]
* [[Tensile strength]]
* [[Thermal stress]]
* [[Virial stress]]
* [[Yield (engineering)]]
* [[Yield surface]]
* [[Virial theorem]]
{{Div col end}}
{{clear}}

== References ==
{{Reflist|30em|refs=
<ref name="Lubliner">Jacob Lubliner (2008). [http://www.ce.berkeley.edu/~coby/plas/pdf/book.pdf "Plasticity Theory"] {{webarchive|url=https://web.archive.org/web/20100331022415/http://www.ce.berkeley.edu/~coby/plas/pdf/book.pdf |date=2010-03-31 }} (revised edition). Dover Publications. {{ISBN|0-486-46290-0}}
</ref>
<ref name="Chen">Wai-Fah Chen and Da-Jian Han (2007), [https://books.google.com/books?id=E8jptvNgADYC&pg=PA46 "Plasticity for Structural Engineers"]. J. Ross Publishing {{ISBN|1-932159-75-4}}
</ref>
<ref name="Slaughter">William S. Slaughter (2012), [https://books.google.com/books?id=TxvnBwAAQBAJ&pg=PA193 "The Linearized Theory of Elasticity"]. Birkhäuser Basel {{ISBN|978-0-8176-4117-7}}</ref>
<ref name="Irgens">Fridtjov Irgens (2008), [https://books.google.com/books?id=q5dB7Gf4bIoC&pg=PA46 "Continuum Mechanics"]. Springer. {{ISBN|3-540-74297-2}}
</ref>
<ref name="Liu">I-Shih Liu (2002), [https://books.google.com/books?id=-gWqM4uMV6wC&pg=PA43 "Continuum Mechanics"]. Springer {{ISBN|3-540-43019-9}}
</ref>
<ref name="Chadwick">Peter Chadwick (1999), [https://books.google.com/books?id=QSXIHQsus6UC&pg=PA95 "Continuum Mechanics: Concise Theory and Problems"]. Dover Publications, series "Books on Physics". {{ISBN|0-486-40180-4}}. pages
</ref>
<ref name="Smith">Donald Ray Smith and Clifford Truesdell (1993) [https://books.google.com/books?id=ZcWC7YVdb4wC&pg=PA97 "An Introduction to Continuum Mechanics after Truesdell and Noll". Springer.] {{ISBN|0-7923-2454-4}}</ref>
<ref name="lamglass">(2009) [http://www.lamberts.de/fileadmin/user_upload/service/downloads/lamberts_broschuere_englisch.pdf ''The art of making glass''.] Lamberts Glashütte (LambertsGlas) product brochure. Accessed on 2013-02-08.
</ref>
<ref name="Pilkey">Walter D. Pilkey, Orrin H. Pilkey (1974), [https://books.google.com/books?id=d7I8AAAAIAAJ "Mechanics of solids"] (book)
</ref>
<ref name="Huston">Ronald L. Huston and Harold Josephs (2009), "Practical Stress Analysis in Engineering Design". 3rd edition, CRC Press, 634 pages. {{ISBN|9781574447132}}
</ref>
}}

==Further reading==
{{Refbegin}}
*{{Cite book
| last = Chakrabarty
| first = J.
| title = Theory of plasticity
| edition= 3
| publisher = Butterworth-Heinemann
| year = 2006
| pages = 17–32
| url= https://books.google.com/books?id=9CZsqgsfwEAC&q=related%3AISBN0486435946&pg=PA17
| isbn = 0-7506-6638-2 }}
*{{Cite book
| last = Beer
| first = Ferdinand Pierre
|author2=Elwood Russell Johnston |author3=John T. DeWolf
 | title = Mechanics of Materials
| url = https://archive.org/details/mechanicsofmater00ferd_1
| url-access = registration
| publisher = McGraw-Hill Professional
| year = 1992
| isbn = 0-07-112939-1 }}
*{{Cite book
| last = Brady
| first = B.H.G.
|author2=E.T. Brown
 | title = Rock Mechanics For Underground Mining
| publisher = Kluwer Academic Publisher
| year = 1993
| edition = Third
| pages = 17–29
| url=https://books.google.com/books?id=s0BaKxL11KsC&pg=PA18
| isbn = 0-412-47550-2}}
*{{Cite book
| last = Chen
| first = Wai-Fah
|author2=Baladi, G.Y.
 | title = Soil Plasticity, Theory and Implementation
| year = 1985
| publisher = Elsevier Science
| isbn = 0-444-42455-5}}
*{{Cite book
| last = Chou
| first = Pei Chi
|author2=Pagano, N.J.
 | title = Elasticity: tensor, dyadic, and engineering approaches
| series = Dover books on engineering
| publisher = Dover Publications
| year = 1992
| pages = 1–33
| url= https://books.google.com/books?id=9-pJ7Kg5XmAC&pg=PA1
| isbn = 0-486-66958-0}}
*{{Cite book
| last = Davis
| first = R. O.
| title = Elasticity and geomechanics
|author2=Selvadurai. A. P. S. 
| publisher = Cambridge University Press
| year = 1996
| pages = 16–26
| url= https://books.google.com/books?id=4Z11rZaUn1UC&pg=PA16
| isbn =0-521-49827-9 }}
* Dieter, G. E. (3 ed.). (1989). ''Mechanical Metallurgy''. New York: McGraw-Hill. {{ISBN|0-07-100406-8}}.
*{{Cite book
| last = Holtz
| first = Robert D.
|author2=Kovacs, William D.
 | title = An introduction to geotechnical engineering
| series = Prentice-Hall civil engineering and engineering mechanics series
| publisher = Prentice-Hall
| year = 1981
| url= https://books.google.com/books?id=yYkYAQAAIAAJ
| isbn = 0-13-484394-0}}
*{{Cite book
| last = Jones
| first = Robert Millard
| title = Deformation Theory of Plasticity
| publisher = Bull Ridge Corporation
| year = 2008
| pages = 95–112
| url= https://books.google.com/books?id=kiCVc3AJhVwC&pg=PA95
| isbn = 978-0-9787223-1-9}}
*{{Cite book
| last = Jumikis
| first = Alfreds R.
| title = Theoretical soil mechanics: with practical applications to soil mechanics and foundation engineering
| publisher = Van Nostrand Reinhold Co.
| year = 1969
| url= https://books.google.com/books?id=NPZRAAAAMAAJ
| isbn = 0-442-04199-3}}
* Landau, L.D. and E.M.Lifshitz. (1959). ''Theory of Elasticity''.
* Love, A. E. H. (4 ed.). (1944). ''Treatise on the Mathematical Theory of Elasticity''. New York: Dover Publications. {{ISBN|0-486-60174-9}}.
*{{Cite book
| last = Marsden
| first = J. E.
|author2=Hughes, T. J. R. 
| title = Mathematical Foundations of Elasticity
| publisher = Dover Publications
| year = 1994
| pages = [https://archive.org/details/mathematicalfoun00mars/page/132 132]–142
| url = https://archive.org/details/mathematicalfoun00mars
| url-access = registration
| isbn = 0-486-67865-2 }}
*{{Cite book
| last = Parry
| first = Richard Hawley Grey
| title = Mohr circles, stress paths and geotechnics
| edition=2
| publisher = Taylor & Francis
| year = 2004
| pages = 1–30
| url= https://books.google.com/books?id=u_rec9uQnLcC&q=mohr%20circles%2C%20sterss%20paths%20and%20geotechnics&pg=PA1
| isbn = 0-415-27297-1}}
*{{Cite book
| last = Rees
| first = David
| title = Basic Engineering Plasticity – An Introduction with Engineering and Manufacturing Applications
| publisher = Butterworth-Heinemann
| year = 2006
| pages = 1–32
| url = https://books.google.com/books?id=4KWbmn_1hcYC&pg=PA1
| isbn = 0-7506-8025-3}}
*{{Cite book
| last = Timoshenko
| author-link = Stephen Timoshenko
| first = Stephen P.
|author2=James Norman Goodier
 | title = Theory of Elasticity
| publisher = McGraw-Hill International Editions
| year = 1970
| edition = Third
| isbn = 0-07-085805-5}}
*{{Cite book
| last = Timoshenko
| first = Stephen P.
| series= Dover Books on Physics
| title = History of strength of materials: with a brief account of the history of theory of elasticity and theory of structures
| publisher = Dover Publications
| year = 1983
| isbn = 0-486-61187-6
| url = https://books.google.com/books?id=tkScQmyhsb8C&q=stress}}
{{Refend}}
{{Authority control}}

{{DEFAULTSORT:Stress (Mechanics)}}
[[Category:Solid mechanics]]
[[Category:Tensors]]