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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 (bottomball), divided by the area of the surface.

In continuum mechanics, stress is a physical quantitythat expresses the internal forces that

neighboring particlesof a continuous material exert on each other. For example, when a solid vertical bar is

supporting aweight, each particle in the bar pulls on the particles immediately above and below it. When a liquid is

underpressure, each particle gets pushed inwards by all the surrounding particles, and, in reaction, pushes them

outwards. These forces are actually the average of a very large number ofintermolecular

forcesand collisionsbetween themolecules in those particles.

Stress inside a body may arise by various mechanisms, such as reaction to external forces applied to the bulk

material (likegravity) or to its surface (like contact forces, external pressure, orfriction). Any strain (deformation) of

a solid material generates an internalelastic stress, analogous to the reaction force of aspring, that tends to

restore the material to its original undeformed state. In liquids and gases, only deformations that change the

volume generate persistent elastic stress. However, if the deformation is gradually changing with time, even in

fluids there will usually be some viscous stress, opposing that change. Elastic and viscous stresses are usually

combined under the namemechanical stress.

Significant stress may exist even when deformation is negligible (a common assumption when modeling the flow

of water) or non-existent. Stress may exist in the absence of external forces; suchbuilt-in stress is important, for

example, inprestressed concreteand tempered glass. Stress may also be imposed on a material without the

application of net forces, for example by changes in temperature orchemicalcomposition, or byexternalelectromagnetic fields (as inpiezoelectric andmagnetostrictivematerials).

Quantitatively, the stress is expressed by the Cauchy traction

vectorTdefined as the traction force Fbetween adjacent parts of the

material across an imaginary separating surface S, divided by the area

ofS.[1]:p.4150 In a fluidat rest the force is perpendicular to the surface, and

is the familiarpressure. In asolid, or in a flow of viscous liquid, the

force Fmay 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 ofS. Thus the stress

state of the material must be described by tensor, called the(Cauchy)

stress tensor; which is a linear function that relates the normal vectorn of

a surfaceS to the stress Tacross S. With respect to any

chosen coordinate system, the Cauchy stress tensor can be represented

as asymmetricmatrix of 3x3 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.

The relation between mechanical stress, deformation, and therate of

change of deformation can be quite complicated, although a linearapproximation may be adequate in practice if the quantities are small

enough. Stress that exceeds certainstrength limits of the material will

result in permanent deformation (such asplastic flow, fracture,cavitation) or even change its crystal

structure andchemical composition.

In some branches ofengineering, the termstress is occasionally used in a looser sense as a synonym of "internal

force". For example, in the analysis oftrusses, it may refer to the total traction or compression force acting on a

beam, rather than the force divided by the area of its cross-section.

Contents [hide]

1 History

2 Overview

3 Simple stresses

4 General stress
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A Roman-era bridge in Switzerland.

Incasuspension bridge on theApurimac

River.

5 Stress analysis

6 Theoretical background

7 Alternative measures of stress

8 See also

9 Further reading

10 References

HistorySince ancient times humans have been consciously aware of

stress inside materials. Until the 17th century the understanding of

stress was largely intuitive and empirical; and yet it resulted in

some surprisingly sophisticated technology, like thecomposite

bow andglass blowing.

Over several millennia, architects and builders, in particular, learned

how to put together carefully shaped wood beams and stone blocks

to withstand, transmit, and distribute stress in the most effective

manner, with ingenious devices such as

the capitals,arches, cupolas,trusses and the flying

buttresses ofgothic 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 the necessary tools were invented in the

17th and 18th centuries: Galileo's rigorous experimental

method, Descartes'scoordinates andanalytic geometry,

and Newton's laws of motion and equilibrium andcalculus ofinfinitesimals. With those tools, Cauchywas able to give the first

rigorous and general mathematical model for stress in a

homogeneous medium. 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 himself, who provided a differential formula for friction

forces (shear stress) in laminar parallel flow.

Overview

Definition

Stress is defined as the average force per unit area that some particle of a body exerts on an adjacent particle,

across an imaginary surface that separates them.[2]:p.4671

Being derived from a fundamental physical quantity (force) and a purely geometrical quantity (area), stress is also

a fundamental quantity, like velocity,torque orenergy, that can be quantified and analyzed without explicit

consideration of the nature of the material or of its physical causes.

Following the basic premises of continuum mechanics, stress is a macroscopicconcept. 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 ignorequantum 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.[3]:p.90106Depending on the context, one

may also assume that the particles are large enough to allow the averaging out of other microscopic features, like
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Glass vase w ith

the craqueleffect. The cracks

are the result of brief but

intense stress created w hen

the semi-molten piece is briefly

dipped in water.[4]

the grains of ametal rod or the fibersof a piece ofwood.

Normal and shear stress

Further information:compression (physical) andShear stress

In general, the stressTthat a particle Papplies on another particle Q across a surface S can have any direction

relative to S. The vectorTmay be regarded as the sum of two components: thenormal

stress(Compression orTension) perpendicular to the surface, and theshear stress that is parallel to it.

If the normal unit vectorn of the surface (pointing fromQ towards P) is assumed fixed, the normal component can

be expressed by a single number, thedot productTn. This number will be positive ifPis "pulling" on Q(tensile

stress), and negative ifPis "pushing" against Q(compressive stress) The shear component is then the vectorT -

(Tn)n.

Units

The dimension of stress is that ofpressure, and therefore its coordinates are commonly measured in the same

units as pressure: namely,pascals (Pa, that is,newtons persquare metre) in theInternational System,

orpounds persquare inch (psi) in theImperial system.

Causes and effects

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 intemperature andphase, 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 impulsesdue to

collisions). In general, the stress distribution in the body is expressed as

a piecewisecontinuous function of space and time.

Conversely, stress is usually correlated with various effects on the material,

possibly including changes in physical properties like birefringence,polarization,

andpermeability. The imposition of stress by an external agent usually creates

somestrain (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, tending to restore the

material to its original undeformed state. Fluid materials

(liquids, gases andplasmas) 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.

The relation between stress and its effects and causes, including deformation and rate of change of deformation,

can be quite complicated (although a linear approximation may be adequate in practice if the quantities are small

enough). Stress that exceeds certainstrength limits of the material will result in permanent deformation (such

asplastic flow, fracture,cavitation) or even change its crystal structure andchemical composition.

Simple stresses

In some situations, the stress within a body may adequately described by a single number, or by a single vector

(a number and a direction). Three suchsimple stresssituations, that are often encountered in engineering design,

are the uniaxial normal stress, thesimple shear stress, and the isotropic normal stress.[5]

Uniaxial normal stress
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Idealized stress in a straight bar w ith uniform

cross-section.

The ratio may be only an

average stress. The stress may be unevenly

distributed over the cross section (mm),

especially near the the attachment points (nn).

Shear stress in a horizontal bar loaded by

tw o offset blocks.

A common situation with a simple stress pattern is when a straight

rod, with uniform material and cross section, is subjected

to tensionby 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 FTherefore the stress throughout the bar, across

any horizontalsurface, can be described by the number = F/A,

whereA is the area of the cross-section.

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 oruniaxial stress; specifically, (uniaxial, simple, etc.) tensile stress.[5] If the load is compression on

the bar, rather than stretching it, the analysis is the same except that the forceFand the stress change sign,

and the stress is calledcompressive stress.

This analysis assumes the stress is evenly distributed over theentire 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, calledengineering stressornominal

stress. However, if the bar's length L is many times its diameterD,

and it has no gross defects orbuilt-in stress, then the stress can

be assumed to be uniformly distributed over any cross-section that

is more than a few times Dfrom 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 thehoop

stress that occurs on the walls of a cylindrical pipe orvessel filled with pressurized fluid.

Simple shear stress

Another simple type of stress occurs when an 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 ascissors-like tool. LetFbe the magnitude of those forces,

and Mbe the midplane of that layer. Just as in the normal stress

case, the part of the layer on one side ofMmust pull the other part

with the same force F. Assuming that the direction of the forces is

known, the stress across Mcan be expressed by the single

number = F/A, where Fis the magnitude of those forces andA isthe area of the layer.

However, unlike normal stress, thissimple shear stressis directed parallel to the cross-section considered, rather

than perpendicular to it.[5] For any plane Sthat is perpendicular to the layer, the net internal force acrossS, and
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Isotropic tensile stress. Top left: Each f ace of

a cube of homogeneous material is pulled by a

force w ith magnitude F, applied evenly over the

entire face w hose area isA. The force across

any section S of the cube must balance theforces applied below the section. In the three

sections show n, the forces are F(top right), F

(bottom left), and F (bottom right);

and the area ofS isA,A andA ,

respectively. So the stress across S isF/A in all

three cases.

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/Awill only be an average ("nominal", "engineering") stress. However, that average

is often sufficient for practical purposes. [6]:p.292Shear stress is observed also when a cyindrical 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") ofI-beamsunder bending loads, due to the

web constraining the end plates ("flanges").

Isotropic stress

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 calledisotropic normal or

just isotropic; if it is compressive, it is called hydrostatic

pressure or justpressure . Gases by definition cannot withstand

tensile stresses, but liquids may withstand very small amounts of

isotropic tensile stress.

Cylinder stresses

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 suchcylinder stresses 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 calledcombined

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 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

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 of the
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Illustration of typical stresses (arrow s)

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.

surface's normal vector , 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

atensor, reflecting Cauchy's original use to describe the "tensions"

(stresses) in a material.) Intensor calculus, is classified as

second-order tensor oftype (0,2).

Like any linear map between vectors, the stress tensor can be

represented in any chosen Cartesian coordinate system by a 33

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 with coordinates is then a

matrix product , that is

The linear relation between and follows from the fundamental laws ofconservation 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 (Cauchys equations of motionfor zero acceleration). Moreover, the principle ofconservation of angular

momentumimplies that the stress tensor is symmetric, 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 theorthogonal 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 of stress distribution.

As a symmetric 33 real matrix, the stress tensor has three mutually orthogonal unit-lengtheigenvectorsand three real eigenvalues , such that . Threfore, in a coordinate system

with axes , the stress tensor is a diagonal matrix, and has only the three normal components

theprincipal stresses. If the three eigenvalues are equal, the stress is an isotropiccompression or

tension, always perpendicular to any surface; there is no shear stress, and the tensor is a diagonal matrix in any
http://en.wikipedia.org/wiki/Isotropichttp://en.wikipedia.org/wiki/Stress_(mechanics)#Principal_stresses_and_stress_invariantshttp://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectorshttp://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectorshttp://en.wikipedia.org/wiki/Mohr's_circlehttp://en.wikipedia.org/wiki/Symmetric_matrixhttp://en.wikipedia.org/wiki/Conservation_of_angular_momentumhttp://en.wikipedia.org/wiki/Cauchy_momentum_equationhttp://en.wikipedia.org/wiki/Static_equilibriumhttp://en.wikipedia.org/wiki/Conservation_of_linear_momentumhttp://en.wikipedia.org/wiki/Cartesian_coordinateshttp://en.wikipedia.org/wiki/Type_of_a_tensorhttp://en.wikipedia.org/wiki/Tensor_calculushttp://en.wikipedia.org/wiki/Tensorhttp://en.wikipedia.org/wiki/Cauchy_stress_tensorhttp://en.wikipedia.org/wiki/Surface_normalhttp://en.wikipedia.org/wiki/File:Cmec_stress_ball_f02_t6.pnghttp://en.wikipedia.org/wiki/File:Cmec_stress_ball_f02_t6.pnghttp://en.wikipedia.org/w/index.php?title=Stress_(mechanics)&action=edit&section=14


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A tank carmade from bent and w elded steel

plates.

For stress modeling,

a f ishing pole may be

considered one-

dimensional.

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 aninfinitesimal 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 platesMan-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, 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 stressthat tends to change the curvature of the plate. However, these

simplifications may not hold at welds, at sharp bends and creases (where theradius of curvature is comparable to

the thickness of the plate).

Stress in thin beamsThe 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 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

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 PiolaKirchhoff stress tensors,

the Biot stress tensor, and theKirchhoff stress tensor, are required.

Solids, liquids, and gases have stress fields. Static fluids support normal stress but will

flow undershear stress. Movingviscous fluids can support shear stress (dynamic

pressure). Solids can support both shear and normal stress, withductile materials failing under shearand brittle 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
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Stress analysis is a branch ofapplied physics that covers the determination of the internal distribution of stresses

in solid objects. It is an essential tool inengineering for the study and design of structures such astunnels, 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 andavalanches; and

inbiology, to understand theanatomy of living beings.

Goals and assumptions

Stress analysis is generally concerned with objects and structures that can be assumed to be inmacroscopic static equilibrium. ByNewton's laws of motion, any external forces are being applied to such a

system must be balanced by internal reaction forces, [7]:p.97which are almost always surface contact forces

between adjacent particles that is, as stress.[1] Since every particle needs to be in equilibrium, this reaction

stress will generally propagate from 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 forces (such as gravity or magnetic attraction), that act throughout

the volume of a material;[8]:p.4281 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 assumed that the stresses are related to deformation (and, in non-static problems, to the

rate of deformation) of the material by knownconstitutive equations.[9]

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 byEuler's equations of motion for continuous bodies (which

are consequences ofNewton's laws for conservation oflinear momentum andangular momentum) and the Euler-

Cauchy s tress principle, together with the appropriate constitutive equations. Thus one obtains a system ofpartial

differential equations involving the stress tensor field and the strain tensorfield, 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 forelastic structures is based on the theory of elasticity andinfinitesimal strain theory. When the

applied loads cause permanent deformation, one must use more complicated constitutive equations, that can

account for the physical processes involved (plastic flow,fracture, phase change, etc.).

However, engineered structures are usually designed so that the maximum expected stresses are well within the

range oflinear elasticity (the generalization ofHookes 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 the problem becomes much easier. For one thing, the stress at any point will be a linear

function of the loads, too. For small enough stresses, even non-linear systems can usually be assumed to be

linear.

Stress analysis is simplified when 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 (usuallylinear) 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.
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Simplified model of a truss for stress

analysis, assuming unidimensional elements

under uniform axial tension or compression.

Still, for two- or there-dimensional cases one must solve a partial

differential equation problem. Anlytical 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.

Theoretical background

The mathematical description of stress is founded on Euler's

laws for the motion of continuous bodies. They can be derived from

Newton's laws, but may also be taken as axioms describing the motions of such bodies. [10]

Alternative measures of stress

Main article:Stress measures

Other useful stress measures include the first and secondPiolaKirchhoff stress tensors, theBiot stress tensor,

and the Kirchhoff stress tensor.

PiolaKirchhoff stress tensor

In the case offinite deformations, thePiolaKirchhoff stress tensorsexpress the stress relative to the reference

configuration. This is in contrast to theCauchy stress tensorwhich expresses the stress relative to the present

configuration. For infinitesimal deformations or rotations, the Cauchy and PiolaKirchhoff tensors are identical.

Whereas the Cauchy stress tensor, relates stresses in the current configuration, the deformationgradient 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 PiolaKirchhoff stress tensor, relates forces in thepresentconfiguration with areas in

the reference("material") configuration.

where is thedeformation gradientand is the Jacobiandeterminant.

In terms of components with respect to anorthonormal basis, the first PiolaKirchhoff stress is given by

Because it relates different coordinate systems, the 1st PiolaKirchhoff stress is a two-point tensor. In general, it

is not symmetric. The 1st PiolaKirchhoff stress is the 3D generalization of the 1D concept ofengineering stress.

If the material rotates without a change in stress state (rigid rotation), the components of the 1st PiolaKirchhoff

stress tensor will vary with material orientation.

The 1st PiolaKirchhoff stress is energy conjugate to the deformation gradient.

2nd PiolaKirchhoff stress tensor

Whereas the 1st PiolaKirchhoff stress relates forces in the current configuration to areas in the reference
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configuration, the 2nd PiolaKirchhoff stress tensor relates forces in the reference configuration to areas in the

current 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 current configuration.

In index notation with respect to an orthonormal basis,

This tensor is symmetric.

If the material rotates without a change in stress state (rigid rotation), the components of the 2nd PiolaKirchhoff

stress tensor remain constant, irrespective of material orientation.

The 2nd PiolaKirchhoff stress tensor is energy conjugate to theGreenLagrange finite strain tensor.

See also
http://en.wikipedia.org/wiki/Finite_strain_theory#Finite_strain_tensorshttp://en.wikipedia.org/wiki/Index_notationhttp://en.wikipedia.org/w/index.php?title=Stress_(mechanics)&action=edit&section=26

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