In
continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such m ...
, stress is a
physical quantity
A physical quantity is a physical property of a material or system that can be quantified by measurement. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a ' Numerical value ' and a ' Unit '. For examp ...
. It is a quantity that describes the magnitude of forces that cause deformation. Stress is defined as ''force per unit area''. When an object is pulled apart by a force it will cause elongation which is also known as deformation, like the stretching of an elastic band, it is called tensile stress. But, when the forces result in the compression of an object, it is called compressive stress. It results when forces like
tension
Tension may refer to:
Science
* Psychological stress
* Tension (physics), a force related to the stretching of an object (the opposite of compression)
* Tension (geology), a stress which stretches rocks in two opposite directions
* Voltage or el ...
or
compression
Compression may refer to:
Physical science
*Compression (physics), size reduction due to forces
*Compression member, a structural element such as a column
*Compressibility, susceptibility to compression
* Gas compression
*Compression ratio, of a ...
act on a body. The greater this force and the smaller the cross-sectional area of the body on which it acts, the greater the stress. Therefore, stress is measured in newton per square meter (N/m
2) or pascal (Pa).
Stress expresses the internal
force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
s that neighbouring
particle
In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object which can be described by several physical property, physical or chemical property, chemical ...
s of a continuous material exert on each other, while
strain
Strain may refer to:
Science and technology
* Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes
* Strain (chemistry), a chemical stress of a molecule
* Strain (injury), an injury to a mu ...
is the measure of the deformation of the material. For example, when a
solid
Solid is one of the State of matter#Four fundamental states, four fundamental states of matter (the others being liquid, gas, and Plasma (physics), plasma). The molecules in a solid are closely packed together and contain the least amount o ...
vertical bar is supporting an overhead
weight
In science and engineering, the weight of an object is the force acting on the object due to gravity.
Some standard textbooks define weight as a Euclidean vector, vector quantity, the gravitational force acting on the object. Others define weigh ...
, each particle in the bar pushes on the particles immediately below it. When a
liquid
A liquid is a nearly incompressible fluid that conforms to the shape of its container but retains a (nearly) constant volume independent of pressure. As such, it is one of the four fundamental states of matter (the others being solid, gas, a ...
is in a closed container under
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
, each particle gets pushed against by all the surrounding particles. The container walls and the
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
-inducing surface (such as a piston) push against them in (Newtonian)
reaction
Reaction may refer to a process or to a response to an action, event, or exposure:
Physics and chemistry
*Chemical reaction
*Nuclear reaction
*Reaction (physics), as defined by Newton's third law
*Chain reaction (disambiguation).
Biology and me ...
. These macroscopic forces are actually the net result of a very large number of
intermolecular force
An intermolecular force (IMF) (or secondary force) is the force that mediates interaction between molecules, including the electromagnetic forces of attraction
or repulsion which act between atoms and other types of neighbouring particles, e.g. a ...
s and
collision
In physics, a collision is any event in which two or more bodies exert forces on each other in a relatively short time. Although the most common use of the word ''collision'' refers to incidents in which two or more objects collide with great fo ...
s between the particles in those
molecule
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioch ...
s. Stress is frequently represented by a lowercase Greek letter sigma (''σ'').
Strain inside a material may arise by various mechanisms, such as ''stress'' as applied by external forces to the bulk material (like
gravity
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
) or to its surface (like
contact force
A contact force is any force that occurs as a result of two objects making contact with each other. Contact forces are ubiquitous and are responsible for most visible interactions between macroscopic collections of matter. Pushing a car or kickin ...
s, external pressure, or
friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction:
*Dry friction is a force that opposes the relative lateral motion of t ...
). Any
strain (deformation) of a solid material generates an internal ''elastic stress'', analogous to the reaction force of a
spring
Spring(s) may refer to:
Common uses
* Spring (season)
Spring, also known as springtime, is one of the four temperate seasons, succeeding winter and preceding summer. There are various technical definitions of spring, but local usage of ...
, that tends to restore the material to its original non-deformed state. In liquids and
gas
Gas is one of the four fundamental states of matter (the others being solid, liquid, and plasma).
A pure gas may be made up of individual atoms (e.g. a noble gas like neon), elemental molecules made from one type of atom (e.g. oxygen), or ...
es, only deformations that change the volume generate persistent elastic stress. However, 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''.
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
Prestressed concrete is a form of concrete used in construction. It is substantially "prestressed" ( compressed) during production, in a manner that strengthens it against tensile forces which will exist when in service. Post-tensioned concreted i ...
and
tempered glass
Tempered or toughened glass is a type of safety glass processed by controlled thermal or chemical treatments to increase its strength compared with normal glass. Tempering puts the outer surfaces into compression and the interior into tension. ...
. Stress may also be imposed on a material without the application of
net forces, for example by
changes in temperature or
chemical
A chemical substance is a form of matter having constant chemical composition and characteristic properties. Some references add that chemical substance cannot be separated into its constituent elements by physical separation methods, i.e., wi ...
composition, or by external
electromagnetic field
An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical c ...
s (as in
piezoelectric
Piezoelectricity (, ) is the electric charge that accumulates in certain solid materials—such as crystals, certain ceramics, and biological matter such as bone, DNA, and various proteins—in response to applied Stress (mechanics), mechanical s ...
and
magnetostrictive
Magnetostriction (cf. electrostriction) is a property of magnetic materials that causes them to change their shape or dimensions during the process of magnetization. The variation of materials' magnetization due to the applied magnetic field chan ...
materials).
The relation between mechanical stress, deformation, and the
rate of change of deformation can be quite complicated, although a
linear approximation
In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function). They are widely used in the method of finite differences to produce first order methods for solving or a ...
may be adequate in practice if the quantities are sufficiently small. Stress that exceeds certain
strength limits of the material will result in permanent deformation (such as
plastic flow,
fracture
Fracture is the separation of an object or material into two or more pieces under the action of stress. The fracture of a solid usually occurs due to the development of certain displacement discontinuity surfaces within the solid. If a displa ...
,
cavitation
Cavitation is a phenomenon in which the static pressure of a liquid reduces to below the liquid's vapour pressure, leading to the formation of small vapor-filled cavities in the liquid. When subjected to higher pressure, these cavities, cal ...
) or even change its
crystal structure
In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystal, crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric pat ...
and
chemical composition
A chemical composition specifies the identity, arrangement, and ratio of the elements making up a compound.
Chemical formulas can be used to describe the relative amounts of elements present in a compound. For example, the chemical formula for ...
.
History
![Inca bridge](https://upload.wikimedia.org/wikipedia/commons/8/87/Inca_bridge.jpg)
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
A composite bow is a traditional bow made from horn, wood, and sinew laminated together, a form of laminated bow. The horn is on the belly, facing the archer, and sinew on the outer side of a wooden core. When the bow is drawn, the sinew (stret ...
and
glass blowing
Glassblowing is a glassforming technique that involves inflating molten glass into a bubble (or parison) with the aid of a blowpipe (or blow tube). A person who blows glass is called a ''glassblower'', ''glassmith'', or ''gaffer''. A '' lampworke ...
.
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
capitals,
arch
An arch is a vertical curved structure that spans an elevated space and may or may not support the weight above it, or in case of a horizontal arch like an arch dam, the hydrostatic pressure against it.
Arches may be synonymous with vaul ...
es,
cupola
In architecture, a cupola () is a relatively small, most often dome-like, tall structure on top of a building. Often used to provide a lookout or to admit light and air, it usually crowns a larger roof or dome.
The word derives, via Italian, from ...
s,
truss
A truss is an assembly of ''members'' such as beams, connected by ''nodes'', that creates a rigid structure.
In engineering, a truss is a structure that "consists of two-force members only, where the members are organized so that the assembl ...
es and the
flying buttress
The flying buttress (''arc-boutant'', arch buttress) is a specific form of buttress composed of an arch that extends from the upper portion of a wall to a pier of great mass, in order to convey lateral forces to the ground that are necessary to pu ...
es of
Gothic cathedrals
Gothic cathedrals and churches are religious buildings created in Europe between the mid-12th century and the beginning of the 16th century. The cathedrals are notable particularly for their great height and their extensive use of stained glass ...
.
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
Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He was ...
's rigorous
experimental method
An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into cause-and-effect by demonstrating what outcome occurs when a ...
,
René Descartes
René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Mathem ...
's
coordinates
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
and
analytic geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.
Analytic geometry is used in physics and engineerin ...
, and
Newton
Newton most commonly refers to:
* Isaac Newton (1642–1726/1727), English scientist
* Newton (unit), SI unit of force named after Isaac Newton
Newton may also refer to:
Arts and entertainment
* ''Newton'' (film), a 2017 Indian film
* Newton ( ...
's
laws of motion and equilibrium and
calculus of infinitesimals.
With those tools,
Augustin-Louis Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He ...
was able to give the first rigorous and general mathematical model of a deformed elastic body by introducing the notions of stress and strain. Cauchy observed that the force across an imaginary surface was a linear function of its normal vector; and, moreover, that it must be a symmetric function (with zero total momentum).
The understanding of stress in liquids started with Newton, who provided a differential formula for friction forces (shear stress) in parallel
laminar flow
In fluid dynamics, laminar flow is characterized by fluid particles following smooth paths in layers, with each layer moving smoothly past the adjacent layers with little or no mixing. At low velocities, the fluid tends to flow without lateral mi ...
.
Overview
Definition
Stress is defined as the force across a "small" boundary per unit area of that boundary, for all orientations of the boundary.
Being derived from a fundamental physical quantity (force) and a purely geometrical quantity (area), stress is also a fundamental quantity, like velocity,
torque
In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
or
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
, that can be quantified and analyzed without explicit consideration of the nature of the material or of its physical causes.
![Cmec stress defn f02 t6](https://upload.wikimedia.org/wikipedia/commons/d/d6/Cmec_stress_defn_f02_t6.png)
Following the basic premises of continuum mechanics, stress is a
macroscopic
The macroscopic scale is the length scale on which objects or phenomena are large enough to be visible with the naked eye, without magnifying optical instruments. It is the opposite of microscopic.
Overview
When applied to physical phenomena an ...
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
In physics, a quantum (plural quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the hypothesis of quantizati ...
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.
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
A metal (from Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appearance, and conducts electricity and heat relatively well. Metals are typicall ...
rod or the
fiber
Fiber or fibre (from la, fibra, links=no) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often incorporate ...
s of a piece of
wood
Wood is a porous and fibrous structural tissue found in the stems and roots of trees and other woody plants. It is an organic materiala natural composite of cellulose fibers that are strong in tension and embedded in a matrix of lignin th ...
.
Quantitatively, the stress is expressed by the ''Cauchy traction vector'' ''T'' defined as the traction force ''F'' between adjacent parts of the material across an imaginary separating surface ''S'', divided by the area of ''S''.
In a
fluid
In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
at rest the force is perpendicular to the surface, and is the familiar
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
. In a
solid
Solid is one of the State of matter#Four fundamental states, four fundamental states of matter (the others being liquid, gas, and Plasma (physics), plasma). The molecules in a solid are closely packed together and contain the least amount o ...
, or in a
flow of viscous
liquid
A liquid is a nearly incompressible fluid that conforms to the shape of its container but retains a (nearly) constant volume independent of pressure. As such, it is one of the four fundamental states of matter (the others being solid, gas, a ...
, 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
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
, called the
(Cauchy) stress tensor; which is a
linear function
In mathematics, the term linear function refers to two distinct but related notions:
* In calculus and related areas, a linear function is a function (mathematics), function whose graph of a function, graph is a straight line, that is, a polynomia ...
that relates the
normal vector
In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at ...
''n'' of a surface ''S'' to the traction vector ''T'' across ''S''. With respect to any chosen
coordinate system
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
, the Cauchy stress tensor can be represented as a
symmetric matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,
Because equal matrices have equal dimensions, only square matrices can be symmetric.
The entries of a symmetric matrix are symmetric with re ...
of 3×3 real numbers. Even within a
homogeneous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
body, the stress tensor may vary from place to place, and may change over time; therefore, the stress within a material is, in general, a time-varying
tensor field
In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...
.
Normal and 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 Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
stress'' (
compression
Compression may refer to:
Physical science
*Compression (physics), size reduction due to forces
*Compression member, a structural element such as a column
*Compressibility, susceptibility to compression
* Gas compression
*Compression ratio, of a ...
or
tension
Tension may refer to:
Science
* Psychological stress
* Tension (physics), a force related to the stretching of an object (the opposite of compression)
* Tension (geology), a stress which stretches rocks in two opposite directions
* Voltage or el ...
) perpendicular to the surface, and the ''
shear stress
Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the ot ...
'' 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
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
. 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 .
Units
The dimension of stress is that of
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
, and therefore its coordinates are commonly measured in the same units as pressure: namely,
pascal
Pascal, Pascal's or PASCAL may refer to:
People and fictional characters
* Pascal (given name), including a list of people with the name
* Pascal (surname), including a list of people and fictional characters with the name
** Blaise Pascal, Fren ...
s (Pa, that is,
newton
Newton most commonly refers to:
* Isaac Newton (1642–1726/1727), English scientist
* Newton (unit), SI unit of force named after Isaac Newton
Newton may also refer to:
Arts and entertainment
* ''Newton'' (film), a 2017 Indian film
* Newton ( ...
s per
square metre
The square metre ( international spelling as used by the International Bureau of Weights and Measures) or square meter (American spelling) is the unit of area in the International System of Units (SI) with symbol m2. It is the area of a square w ...
) in the
International System, or
pounds per
square inch
A square inch (plural: square inches) is a unit of area, equal to the area of a square with sides of one inch. The following symbols are used to denote square inches:
*square in
*sq inches, sq inch, sq in
*inches/-2, inch/-2, in/-2
*inches^2, inc ...
(psi) in the
Imperial system
The imperial system of units, imperial system or imperial units (also known as British Imperial or Exchequer Standards of 1826) is the system of units first defined in the British Weights and Measures Act 1824 and continued to be developed th ...
. Because mechanical stresses easily exceed a million Pascals, MPa, which stands for megapascal, is a common unit of stress.
Causes and effects
![Vase-craquele-Emile-Galle-vers-1880-decor-mante-religieuse-cigale-1301](https://upload.wikimedia.org/wikipedia/commons/2/22/Vase-craquele-Emile-Galle-vers-1880-decor-mante-religieuse-cigale-1301.jpg)
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
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
and
phase
Phase or phases may refer to:
Science
*State of matter, or phase, one of the distinct forms in which matter can exist
*Phase (matter), a region of space throughout which all physical properties are essentially uniform
* Phase space, a mathematic ...
, 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
Impulse or Impulsive may refer to:
Science
* Impulse (physics), in mechanics, the change of momentum of an object; the integral of a force with respect to time
* Impulse noise (disambiguation)
* Specific impulse, the change in momentum per uni ...
s due to collisions). In
active matter
Active matter is matter composed of large numbers of active "agents", each of which consumes energy in order to move or to exert mechanical forces. Such systems are intrinsically out of thermal equilibrium. Unlike thermal systems relaxing towa ...
, self-propulsion of microscopic particles generates macroscopic stress profiles.
In general, the stress distribution in a body is expressed as a
piecewise
In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. Pi ...
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
of space and time.
Conversely, stress is usually correlated with various effects on the material, possibly including changes in physical properties like
birefringence
Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light. These optically anisotropic materials are said to be birefringent (or birefractive). The birefring ...
,
polarization, and
permeability. The imposition of stress by an external agent usually creates some
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
Spring(s) may refer to:
Common uses
* Spring (season)
Spring, also known as springtime, is one of the four temperate seasons, succeeding winter and preceding summer. There are various technical definitions of spring, but local usage of ...
, tending to restore the material to its original undeformed state. Fluid materials (liquids,
gas
Gas is one of the four fundamental states of matter (the others being solid, liquid, and plasma).
A pure gas may be made up of individual atoms (e.g. a noble gas like neon), elemental molecules made from one type of atom (e.g. oxygen), or ...
es and
plasma
Plasma or plasm may refer to:
Science
* Plasma (physics), one of the four fundamental states of matter
* Plasma (mineral), a green translucent silica mineral
* Quark–gluon plasma, a state of matter in quantum chromodynamics
Biology
* Blood pla ...
s) by definition can only oppose deformations that would change their volume. However, if the deformation is changing 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 viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inte ...
. The same for normal viscous stresses can be found in Sharma (2019).
[Sharma, B and Kumar, R "Estimation of bulk viscosity of dilute gases using a nonequilibrium molecular dynamics approach.", ''Physical Review E'',100, 013309 (2019)]
The relation between stress and its effects and causes, including deformation and rate of change of deformation, can be quite complicated (although a
linear approximation
In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function). They are widely used in the method of finite differences to produce first order methods for solving or a ...
may be adequate in practice if the quantities are small enough). Stress that exceeds certain
strength limits of the material will result in permanent deformation (such as
plastic flow,
fracture
Fracture is the separation of an object or material into two or more pieces under the action of stress. The fracture of a solid usually occurs due to the development of certain displacement discontinuity surfaces within the solid. If a displa ...
,
cavitation
Cavitation is a phenomenon in which the static pressure of a liquid reduces to below the liquid's vapour pressure, leading to the formation of small vapor-filled cavities in the liquid. When subjected to higher pressure, these cavities, cal ...
) or even change its
crystal structure
In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystal, crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric pat ...
and
chemical composition
A chemical composition specifies the identity, arrangement, and ratio of the elements making up a compound.
Chemical formulas can be used to describe the relative amounts of elements present in a compound. For example, the chemical formula for ...
.
Simple stress
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''.
Uniaxial normal stress
![Axial stress noavg](https://upload.wikimedia.org/wikipedia/commons/0/0f/Axial_stress_noavg.svg)
A common situation with a simple stress pattern is when a straight rod, with uniform material and cross section, is subjected to
tension
Tension may refer to:
Science
* Psychological stress
* Tension (physics), a force related to the stretching of an object (the opposite of compression)
* Tension (geology), a stress which stretches rocks in two opposite directions
* Voltage or el ...
by opposite forces of magnitude
along its axis. If the system is in
equilibrium and not changing with time, and the weight of the bar can be neglected, then through each transversal section of the bar the top part must pull on the bottom part with the same force, ''F'' with continuity through the full cross-sectional area'', A''. Therefore, 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''.
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.
If the load is
compression
Compression may refer to:
Physical science
*Compression (physics), size reduction due to forces
*Compression member, a structural element such as a column
*Compressibility, susceptibility to compression
* Gas compression
*Compression ratio, of a ...
on the bar, rather than stretching it, the analysis is the same except that the force ''F'' and the stress
change sign, and the stress is called compressive stress.
![Normal stress](https://upload.wikimedia.org/wikipedia/commons/4/4f/Normal_stress.svg)
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
= ''F''/''A'' will be only the average stress, called ''engineering stress'' or ''nominal stress''. However, if the bar's length ''L'' is many times its diameter ''D'', and it has no gross defects or
built-in stress
Built-in, builtin, or built in may refer to:
Computing
* Shell builtin, a command or a function executed directly in the shell itself
* Builtin function, in computer software and compiler theory
Other uses
* Built-in behavior, of a living organ ...
, 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 Saint-Venant's principle, named after Adhémar Jean Claude Barré de Saint-Venant, a French elasticity theorist, may be expressed as follows:
The original statement was published in French by Saint-Venant in 1855. Although this informal stateme ...
).
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
Pipe(s), PIPE(S) or piping may refer to:
Objects
* Pipe (fluid conveyance), a hollow cylinder following certain dimension rules
** Piping, the use of pipes in industry
* Smoking pipe
** Tobacco pipe
* Half-pipe and quarter pipe, semi-circula ...
or
vessel filled with pressurized fluid.
Simple shear stress
![Shear stress](https://upload.wikimedia.org/wikipedia/commons/d/df/Shear_stress.svg)
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
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
, calculated simply with the magnitude of those forces, ''F'' and the cross sectional area, ''A''.
However, unlike normal stress, this ''simple shear stress'' is directed parallel to the cross-section considered, rather than perpendicular to it.
For any plane ''S'' that 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. However, that average is often sufficient for practical purposes.
Shear stress is observed also when a cylindrical bar such as a
shaft is subjected to opposite torques at its ends. In that case, the shear stress on each cross-section is parallel to the cross-section, but oriented tangentially relative to the axis, and increases with distance from the axis. Significant shear stress occurs in the middle plate (the "web") of
I-beam
An I-beam, also known as H-beam (for universal column, UC), w-beam (for "wide flange"), universal beam (UB), rolled steel joist (RSJ), or double-T (especially in Polish language, Polish, Bulgarian language, Bulgarian, Spanish language, Spanish ...
s under bending loads, due to the web constraining the end plates ("flanges").
Isotropic stress
![Isotropic stress noavg](https://upload.wikimedia.org/wikipedia/commons/4/4d/Isotropic_stress_noavg.svg)
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
The Z-tube is an experimental apparatus for measuring the tensile strength of a liquid.
It consists of a Z-shaped tube with open ends, filled with a liquid, and set on top of a spinning table. If the tube were straight, the liquid would immediate ...
.
Cylinder stresses
Parts with
rotational symmetry
Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which i ...
, 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
In geometry, circular symmetry is a type of continuous symmetry for a planar object that can be rotated by any arbitrary angle and map onto itself.
Rotational circular symmetry is isomorphic with the circle group in the complex plane, or th ...
. The analysis of such
cylinder stress
In mechanics, a cylinder stress is a stress distribution with rotational symmetry; that is, which remains unchanged if the stressed object is rotated about some fixed axis.
Cylinder stress patterns include:
* circumferential stress, or hoop stres ...
es can take advantage of the symmetry to reduce the dimension of the domain and/or of the stress tensor.
General stress
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
, and zero across any surfaces that are parallel to
. 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.
The Cauchy stress tensor
![Cmec stress ball f02 t6](https://upload.wikimedia.org/wikipedia/commons/3/3e/Cmec_stress_ball_f02_t6.png)
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.
However, Cauchy observed that the stress vector
across a surface will always be a
linear function
In mathematics, the term linear function refers to two distinct but related notions:
* In calculus and related areas, a linear function is a function (mathematics), function whose graph of a function, graph is a straight line, that is, a polynomia ...
of the surface's
normal vector
In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at ...
, the unit-length vector that is perpendicular to it. That is,
, where the function
satisfies
:
for any vectors
and any real numbers
.
The function
, now called the
(Cauchy) stress tensor, completely describes the stress state of a uniformly stressed body. (Today, any linear connection between two physical vector quantities is called a
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
, reflecting Cauchy's original use to describe the "tensions" (stresses) in a material.) In
tensor calculus
In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. in spacetime).
Developed by Gregorio Ricci-Curbastro and his student Tullio Levi ...
,
is classified as second-order tensor of
type (0,2).
Like any linear map between vectors, the stress tensor can be represented in any chosen
Cartesian coordinate system
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
by a 3×3 matrix of real numbers. Depending on whether the coordinates are numbered
or named
, the matrix may be written as
or
The stress vector
across a surface with
normal vector
In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at ...
(which is
covariant -
"row; horizontal" - vector) with coordinates
is then a matrix product
(where T in upper index is
transposition, and as a result we get
covariant (row) vector ) (look on
Cauchy stress tensor
In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that complete ...
), that is
:
The linear relation between
and
follows from the fundamental laws of
conservation of linear momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass and ...
and
static equilibrium of forces, and is therefore mathematically exact, for any material and any stress situation. The components of the Cauchy stress tensor at every point in a material satisfy the equilibrium equations (
Cauchy’s equations of motion for zero acceleration). Moreover, the principle of
conservation of angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed system ...
implies that the stress tensor is
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
, that is
,
, and
. 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
:
where the elements
are called the ''orthogonal normal stresses'' (relative to the chosen coordinate system), and
the ''orthogonal shear stresses''.
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
Mohr's circle is a two-dimensional graphical representation of the transformation law for the Cauchy stress tensor.
Mohr's circle is often used in calculations relating to mechanical engineering for materials' strength, geotechnical engineer ...
of stress distribution.
As a symmetric 3×3 real matrix, the stress tensor
has three mutually orthogonal unit-length
eigenvector
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s
and three real
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s
, such that
. Therefore, in a coordinate system with axes
, the stress tensor is a diagonal matrix, and has only the three normal components
the
principal stresses. If the three eigenvalues are equal, the stress is an
isotropic
Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
compression or tension, always perpendicular to any surface, there is no shear stress, and the tensor is a diagonal matrix in any coordinate frame.
Stress as a 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
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
particle of the medium surrounding that point, and taking the average stresses in that particle as being the stresses at the point.
Stress in thin plates
![W39504 stat Nbk2007](https://upload.wikimedia.org/wikipedia/commons/6/6d/W39504_stat_Nbk2007.jpg)
Man-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
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonic ...
of the plate. However, these simplifications may not hold at welds, at sharp bends and creases (where the
radius of curvature
In differential geometry, the radius of curvature, , is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius o ...
is comparable to the thickness of the plate).
Stress in thin beams
![Sandy Hook NJ beach fisherman's pole](https://upload.wikimedia.org/wikipedia/commons/0/02/Sandy_Hook_NJ_beach_fisherman%27s_pole.jpg)
The analysis of stress can be considerably simplified also for thin bars,
beam
Beam may refer to:
Streams of particles or energy
*Light beam, or beam of light, a directional projection of light energy
**Laser beam
*Particle beam, a stream of charged or neutral particles
**Charged particle beam, a spatially localized grou ...
s 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).
Other descriptions of stress
![CarWindowPolarization](https://upload.wikimedia.org/wikipedia/commons/b/b2/CarWindowPolarization.jpg)
The Cauchy stress tensor is used for stress analysis of material bodies experiencing
small deformations where the differences in stress distribution in most cases can be neglected. For large deformations, also called
finite deformations, other measures of stress, such as the
first and second Piola–Kirchhoff stress tensors, the
Biot stress tensor, and the
Kirchhoff stress tensor, are required.
Solids, liquids, and gases have
stress field
A stress field is the distribution of internal forces in a body that balance a given set of external forces. Stress fields are widely used in fluid dynamics and materials science.
Consider that one can picture the stress fields as the stress cre ...
s. Static fluids support normal stress but will flow under
shear stress
Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the ot ...
. Moving
viscous fluids can support shear stress (dynamic pressure). Solids can support both shear and normal stress, with
ductile
Ductility is a mechanical property commonly described as a material's amenability to drawing (e.g. into wire). In materials science, ductility is defined by the degree to which a material can sustain plastic deformation under tensile stres ...
materials failing under shear and
brittle
A material is brittle if, when subjected to stress, it fractures with little elastic deformation and without significant plastic deformation. Brittle materials absorb relatively little energy prior to fracture, even those of high strength. Bre ...
materials failing under normal stress. All materials have temperature dependent variations in stress-related properties, and
non-Newtonian materials have rate-dependent variations.
Stress analysis
Stress analysis
Stress may refer to:
Science and medicine
* Stress (biology), an organism's response to a stressor such as an environmental condition
* Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
is a branch of
applied physics
Applied physics is the application of physics to solve scientific or engineering problems. It is usually considered to be a bridge or a connection between physics and engineering.
"Applied" is distinguished from "pure" by a subtle combination ...
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
Plate tectonics (from the la, label=Late Latin, tectonicus, from the grc, τεκτονικός, lit=pertaining to building) is the generally accepted scientific theory that considers the Earth's lithosphere to comprise a number of large ...
, vulcanism and
avalanche
An avalanche is a rapid flow of snow down a slope, such as a hill or mountain.
Avalanches can be set off spontaneously, by such factors as increased precipitation or snowpack weakening, or by external means such as humans, animals, and earth ...
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
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in moti ...
, 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
body force
In physics, a body force is a force that acts throughout the volume of a body.
Springer site - Book 'Solid mechanics'preview paragraph 'Body forces'./ref>
Forces due to gravity, electric fields and magnetic fields are examples of body forces. Bo ...
s (such as gravity or magnetic attraction), that act throughout the volume of a material;
or concentrated loads (such as friction between an axle and a
bearing, or the weight of a train wheel on a rail), that are imagined to act over a two-dimensional area, or along a line, or at single point.
In stress analysis one normally disregards 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
In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and app ...
.
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. However, most stress analysis is done by mathematical methods, especially during design.
The basic stress analysis problem can be formulated by
Euler's equations of motion for continuous bodies (which are consequences of
Newton's laws
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in motion ...
for conservation of
linear momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass and ...
and
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
) and the
Euler-Cauchy stress principle
In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that completely ...
, together with the appropriate constitutive equations. Thus one obtains a system of
partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
involving the stress tensor field and the
strain tensor
In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimally ...
field, as unknown functions to be determined. The external body forces appear as the independent ("right-hand side") term in the differential equations, while the concentrated forces appear as boundary conditions. The basic stress analysis problem is therefore a
boundary-value problem.
Stress analysis for
elastic
Elastic is a word often used to describe or identify certain types of elastomer, elastic used in garments or stretchable fabrics.
Elastic may also refer to:
Alternative name
* Rubber band, ring-shaped band of rubber used to hold objects togeth ...
structures is based on the
theory of elasticity
Solid mechanics, also known as mechanics of solids, is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation under the action of forces, temperature changes, phase changes, and ot ...
and
infinitesimal strain theory
In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimally ...
. When the applied loads cause permanent deformation, one must use more complicated constitutive equations, that can account for the physical processes involved (
plastic flow,
fracture
Fracture is the separation of an object or material into two or more pieces under the action of stress. The fracture of a solid usually occurs due to the development of certain displacement discontinuity surfaces within the solid. If a displa ...
,
phase change
In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic State of ...
, etc.).
However, engineered structures are usually designed so that the maximum expected stresses are well within the range of
linear elasticity
Linear elasticity is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mech ...
(the generalization of
Hooke’s law
In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of th ...
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.
![Loaded truss](https://upload.wikimedia.org/wikipedia/commons/b/b3/Loaded_truss.svg)
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 element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
, the
finite difference method
In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time interval (if applicable) are di ...
, and the
boundary element method
The boundary element method (BEM) is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations (i.e. in ''boundary integral'' form), including fluid mechanics, acoustics, ele ...
.
Alternative measures of stress
Other useful stress measures include the first and second
Piola–Kirchhoff stress tensors, the
Biot stress tensor, and the
Kirchhoff stress tensor.
Piola–Kirchhoff stress tensor
In the case of
finite deformations, the ''Piola–Kirchhoff stress tensors'' express the stress relative to the reference configuration. This is in contrast to the
Cauchy stress tensor
In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that complete ...
which expresses the stress relative to the present configuration. For infinitesimal deformations and rotations, the Cauchy and Piola–Kirchhoff tensors are identical.
Whereas the Cauchy stress tensor
relates stresses in the current configuration, the deformation
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
and strain tensors are described by relating the motion to the reference configuration; thus not all tensors describing the state of the material are in either the reference or current configuration. Describing the stress, strain and deformation either in the reference or current configuration would make it easier to define constitutive models (for example, the Cauchy Stress tensor is variant to a pure rotation, while the deformation strain tensor is invariant; thus creating problems in defining a constitutive model that relates a varying tensor, in terms of an invariant one during pure rotation; as by definition constitutive models have to be invariant to pure rotations). The 1st Piola–Kirchhoff stress tensor,
is one possible solution to this problem. It defines a family of tensors, which describe the configuration of the body in either the current or the reference state.
The 1st Piola–Kirchhoff stress tensor,
relates forces in the ''present'' ("spatial") configuration with areas in the ''reference'' ("material") configuration.
:
where
is the
deformation gradient
In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strai ...
and
is the
Jacobian determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
.
In terms of components with respect to an
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
, the first Piola–Kirchhoff stress is given by
:
Because it relates different coordinate systems, the 1st Piola–Kirchhoff stress is a
two-point tensor
Two-point tensors, or double vectors, are tensor-like quantities which transform as Euclidean vectors with respect to each of their indices. They are used in continuum mechanics to transform between reference ("material") and present ("configurat ...
. In general, it is not symmetric. The 1st Piola–Kirchhoff stress is the 3D generalization of the 1D concept of
engineering stress
In engineering, deformation refers to the change in size or shape of an object. ''Displacements'' are the ''absolute'' change in position of a point on the object. Deflection is the relative change in external displacements on an object. Strai ...
.
If the material rotates without a change in stress state (rigid rotation), the components of the 1st Piola–Kirchhoff stress tensor will vary with material orientation.
The 1st Piola–Kirchhoff stress is energy conjugate to the deformation gradient.
2nd Piola–Kirchhoff stress tensor
Whereas the 1st Piola–Kirchhoff stress relates forces in the current configuration to areas in the reference configuration, the 2nd Piola–Kirchhoff stress tensor
relates forces in the reference configuration to areas in the reference configuration. The force in the reference configuration is obtained via a mapping that preserves the relative relationship between the force direction and the area normal in the reference configuration.
:
In
index notation
In mathematics and computer programming, index notation is used to specify the elements of an array of numbers. The formalism of how indices are used varies according to the subject. In particular, there are different methods for referring to th ...
with respect to an orthonormal basis,
:
This tensor, a one-point tensor, is symmetric.
If the material rotates without a change in stress state (rigid rotation), the components of the 2nd Piola–Kirchhoff stress tensor remain constant, irrespective of material orientation.
The 2nd Piola–Kirchhoff stress tensor is energy conjugate to the
Green–Lagrange finite strain tensor.
See also
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Bending
In applied mechanics, bending (also known as flexure) characterizes the behavior of a slender structural element subjected to an external load applied perpendicularly to a longitudinal axis of the element.
The structural element is assumed to ...
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Compressive strength
In mechanics, compressive strength or compression strength is the capacity of a material or structure to withstand loads tending to reduce size (as opposed to tensile strength which withstands loads tending to elongate). In other words, compre ...
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Critical plane analysis
Critical plane analysis refers to the analysis of stresses or strains as they are experienced by a particular plane in a material, as well as the identification of which plane is likely to experience the most extreme damage. Critical plane analys ...
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Kelvin probe force microscope
Kelvin probe force microscopy (KPFM), also known as surface potential microscopy, is a noncontact variant of atomic force microscopy (AFM). By raster scanning in the x,y plane the work function of the sample can be locally mapped for correlatio ...
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Mohr's circle
Mohr's circle is a two-dimensional graphical representation of the transformation law for the Cauchy stress tensor.
Mohr's circle is often used in calculations relating to mechanical engineering for materials' strength, geotechnical engineer ...
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Lamé's stress ellipsoid
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Residual stress
In materials science and solid mechanics, residual stresses are Stress (physics), stresses that remain in a solid material after the original cause of the stresses has been removed. Residual stress may be desirable or undesirable. For example, l ...
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Shear strength
In engineering, shear strength is the strength of a material or component against the type of yield or structural failure when the material or component fails in shear. A shear load is a force that tends to produce a sliding failure on a materia ...
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Shot peening
Shot peening is a cold working process used to produce a compressive residual stress layer and modify the mechanical properties of metals and composites. It entails striking a surface with shot (round metallic, glass, or ceramic particles) with ...
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Strain
Strain may refer to:
Science and technology
* Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes
* Strain (chemistry), a chemical stress of a molecule
* Strain (injury), an injury to a mu ...
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Strain tensor
In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimally ...
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Strain rate tensor
In continuum mechanics, the strain-rate tensor or rate-of-strain tensor is a physical quantity that describes the rate of change of the deformation of a material in the neighborhood of a certain point, at a certain moment of time. It can be defi ...
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Stress–energy tensor
The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress ...
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Stress–strain curve
In engineering and materials science, a stress–strain curve for a material gives the relationship between stress and strain. It is obtained by gradually applying load to a test coupon and measuring the deformation, from which the stress and ...
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Stress concentration
In solid mechanics, a stress concentration (also called a stress raiser or a stress riser) is a location in an object where the stress is significantly greater than the surrounding region. Stress concentrations occur when there are irregularitie ...
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Transient friction loading Transient friction loading is the mechanical stress induced on an object due to transient or vibrational frictional forces.
Examples
A classic example of transient friction loading is the wooden block sliding over an unlevel, non-planar surface. Du ...
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Tensile strength
Ultimate tensile strength (UTS), often shortened to tensile strength (TS), ultimate strength, or F_\text within equations, is the maximum stress that a material can withstand while being stretched or pulled before breaking. In brittle materials t ...
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Thermal stress
In mechanics and thermodynamics, thermal stress is mechanical stress created by any change in temperature of a material. These stresses can lead to fracturing or plastic deformation depending on the other variables of heating, which include mater ...
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Virial stress
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Yield (engineering)
In materials science and engineering, the yield point is the point on a stress-strain curve that indicates the limit of elastic behavior and the beginning of plastic behavior. Below the yield point, a material will deform elastically and wi ...
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Yield surface
A yield surface is a five-dimensional surface in the six-dimensional space of stresses. The yield surface is usually convex and the state of stress of ''inside'' the yield surface is elastic. When the stress state lies on the surface the materi ...
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Virial theorem
In mechanics, the virial theorem provides a general equation that relates the average over time of the total kinetic energy of a stable system of discrete particles, bound by potential forces, with that of the total potential energy of the system. ...
References
Further reading
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* Dieter, G. E. (3 ed.). (1989). ''Mechanical Metallurgy''. New York: McGraw-Hill. .
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* 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. .
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{{DEFAULTSORT:Stress (Mechanics)
Solid mechanics
Tensors