stress (mechanics)

8/12/2019 Stress (Mechanics)

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Built-in stress inside a plastic

protractor, revealed by its

effect on polarized light.

Stress (mechanics)From Wikipedia, the free encyclopedia

(Redirected from Stress (physics))

In continuum mechanics, stressis a physical quantity that

expresses the internal forces that neighbouring particlesof a continuous material exert on each other. For example,

when a solid vertical bar is supporting a weight, each

particle in the bar pulls on the particles immediately above

and below it. When a liquid is under pressure, each

particle gets pushed inwards by all the surrounding

particles, and, in reaction, pushes them outwards. These

macroscopic forces are actually the average of a very

large number of intermolecular forces and collisions

between the particles in those molecules.

Stress inside a body may arise by various mechanisms,such as reaction to external forces applied to the bulk material (like gravity) or to its surface

(like contact forces, external pressure, or friction). Any strain (deformation) of a solid

material generates an internal elastic stress, analogous to the reaction force of a spring,

that tends to restore the material to its original 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 name mechanical stress.

Significant stress may exist even when deformation is negligible or non-existent (acommon assumption when modeling the flow of water). Stress may exist in the absence of

external forces; such built-in stressis important, for example, in prestressed concrete and

tempered glass. Stress may also be imposed on a material without the application of net

forces, for example by changes in temperature or chemical composition, or by external

electromagnetic fields (as in piezoelectric and magnetostrictive materials).

The relation between mechanical stress, deformation, and the rate of change of

deformation can be quite complicated, although a linear approximation may be adequate

in practice if the quantities are small enough. Stress that exceeds certain strength limits of

the material will result in permanent deformation (such as plastic flow, fracture, cavitation)

or even change its crystal structure and chemical composition.

In some branches of engineering, the term stressis occasionally used in a looser sense

as a synonym of "internal force". For example, in the analysis of trusses, 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

1 History2 Overview

Stress (mechanics) - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Stress_(physics)

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2/15

Roman-era bridge in Switzerland

Inca bridge on the Apurimac River

3 Simple stresses

4 General stress5 Stress analysis6 Theoretical background7 Alternative measures of stress8 See also

9 Further reading10 References

History

Since 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 the composite bow and

glass blowing.

Over several millennia, architects and builders, in

particular, learned how to put together carefully shaped

wood beams and stone blocks to withstand, transmit,

and distribute stress in the most effective manner, with

ingenious devices such as the capitals, arches,

cupolas, trusses and the flying buttresses of Gothic

cathedrals.

Ancient and medieval architects did develop some

geometrical methods and simple formulas to compute

the proper sizes of pillars and beams, but the scientific

understanding of stress became possible only after the

necessary tools were invented in the 17th and 18th centuries: Galileo's rigorous

experimental method, Descartes's coordinates and analytic geometry, and Newton's laws

of motion and equilibrium and calculus of infinitesimals. With those tools, Cauchy was 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 parallel laminar flow.

Overview

Definition

Stress is defined as the average force per unit area that some particle of a body exerts onan adjacent particle, across an imaginary surface that separates them. [1]:p.4671




3/15

The stress across a surface

element (yellow disk) is the

force that the material on one

side (top ball) exerts on the

material on the other side

(bottom ball), divided by the

area of the surface.

Being derived from a fundamental physical quantity (force) and a purely geometrical

quantity (area), stress is also a fundamental quantity, like velocity, torque or energy, that

can be quantified and analyzed without explicit consideration of the nature of the material

or of its physical causes.

Following the basic premises of continuum mechanics, stress is a macroscopic concept.

Namely, the particles considered in its definition and analysis should be just small enoughto be treated as homogeneous in composition and state, but still large enough to ignore

quantum effects and the detailed motions of molecules. Thus, the force between two

particles 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.[2]:p.90106

Depending on the context, one may also assume that the particles are

large enough to allow the averaging out of other microscopic features, like the grains of a

metal rod or the fibers of a piece of wood.

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 of S.[3]:p.4150In

a fluid at rest the force is perpendicular to the surface,

and is the familiar pressure. In a solid, 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 of S. Thus

the stress state of the material must be described by a

tensor, called the (Cauchy) stress tensor; which is a linear

function that relates the normal vector nof a surface Sto

the stress Tacross S. With respect to any chosen

coordinate system, the Cauchy stress tensor can be

represented as a symmetric matrix 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.

Normal and shear stress

Further information: compression (physical) and Shear stress

In general, the stress Tthat a particle Papplies on another particle Qacross a surface S

can have any direction relative to S. The vector Tmay be regarded as the sum of two

components: the normal stress(Compression or Tension) perpendicular to the surface,

and the shear stressthat is parallel to the surface.

If the normal unit vector nof the surface (pointing from Qtowards P) is assumed fixed, the

normal component can be expressed by a single number, the dot product Tn. Thisnumber will be positive if Pis "pulling" on Q(tensile stress), and negative if Pis "pushing"

against Q(compressive stress) The shear component is then the vector T - (Tn)n.




4/15

Glass vase with the

craqueleffect. The

cracks are the result of

brief but intense stress

created when thesemi-molten piece is

briefly dipped in water.[4]

Units

The dimension of stress is that of pressure, and therefore its coordinates are commonly

measured in the same units as pressure: namely, pascals (Pa, that is, newtons per square

metre) in the International System, or pounds per square inch (psi) in the Imperial system.

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 in

temperature and phase, and electromagnetic fields) act on the

bulk of the material, varying continuously with position and

time. Other agents (like external loads and friction, ambient

pressure, and contact forces) may create stresses and forces

that are concentrated on certain surfaces, lines, or points; and

possibly also on very short time intervals (as in the impulsesdue to collisions). In general, the stress distribution in the

body is expressed as a piecewise continuous function of

space and time.

Conversely, stress is usually correlated with various effects on

the material, possibly including changes in physical properties

like birefringence, polarization, and permeability. The

imposition of stress by an external agent usually creates some

strain (deformation) in the material, even if it is too small to be

detected. In a solid material, such strain will in turn generate

an internal elastic stress, analogous to the reaction force of a

stretched spring, tending to restore the material to its original

undeformed state. Fluid materials (liquids, gases and

plasmas) by definition can only oppose deformations that

would change their volume. 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 certain

strength limits of the material will result in permanent deformation (such as plastic flow,fracture, cavitation) or even change its crystal structure and chemical composition.

Simple stresses

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.[5]

Uniaxial normal stress

A common situation with a simple stress pattern is when a straight rod, with uniform




5/15

Idealized stress in a straight bar with

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

attachment points (nn).

material and cross section, is subjected to tension

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. Therefore the stress throughout the bar,across any horizontalsurface, can be described by

the number = F/A, whereAis 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 stressor uniaxial 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 force Fand the stress change

sign, and the stress is called compressive stress.

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/Awill be

only the average stress, called engineering stress

or nominal stress. However, if the bar's length Lismany times its diameter D, and it has no gross

defects or built-in stress, then the stress can be

assumed to be uniformly distributed over any cross-

section that is more than a few times 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 bentin one of its planes of symmetry, the resulting

bending stresswill 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 stressthat occurs on the walls of a cylindrical pipe or vessel filled with pressurized

fluid.

Simple shear stress

Another simple type of stress occurs when a uniformly thick layer of elastic material likeglue 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




6/15

Shear stress in a horizontal bar

loaded by two offset blocks.

scissors-like tool. Let Fbe 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 of Mmust 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 ofthose forces andAis the area of the layer.

However, unlike normal stress, this simple 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 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/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 cylindrical bar such as a shaft is subjected

to opposite torques at its ends. In that case, the shear stress on each cross-section is

parallel to the cross-section, but oriented tangentially relative to the axis, and increases

with distance from the axis. Significant shear stress occurs in the middle plate (the "web")

of I-beams under bending loads, due to the web constraining the end plates ("flanges").

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 atrest, 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 normalor just isotropic;

if it is compressive, it is called hydrostatic pressureor just pressure. 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 such cylinder stresses can take advantage of the

symmetry to reduce the dimension of the domain and/or of the stress tensor.

General stressOften, mechanical bodies experience more than one type of stress at the same time; this is




7/15

Isotropic tensile stress. Top left: Each

face of a cube of homogeneous

material is pulled by a force with

magnitude F, applied evenly over theentire face whose area isA. The

force across any section Sof the

cube must balance the forces applied

below the section. In the three

sections shown, the forces are F(top

right), F (bottom left), and F

(bottom right); and the area of

SisA,A andA ,

respectively. So the stress across S

is F/Ain all three cases.Components of stress in three dimensions

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 stress is zero only across surfaces that

are perpendicular to one particular direction, the

stress is called biaxial, and can be viewed as thesum 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

Main article: Cauchy stress tensor

Combined

stressescannot 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 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 a tensor, reflecting

Cauchy's original use to describe the "tensions" (stresses) in a material.) In tensor

calculus, 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 by a 3"3 matrix of real numbers. Depending on whether the

coordinates are numbered or named , the matrix may be written as




8/15

Illustration of typical stresses

(arrows) across various surface

elements on the boundary of a

particle (sphere), in a homogeneousmaterial under uniform (but not

isotropic) triaxial stress. The normal

stresses on the principal axes are +5,

+2, and #3 units.

or

The stress vector across a surface with

normal vector with coordinates is thena matrix product , that is

The linear relation between and follows from

the fundamental laws of conservation of linear

momentum and static equilibrium of forces, and is

therefore mathematically exact, for any material andany stress situation. The components of the Cauchy

stress tensor at every point in a material satisfy the

equilibrium equations (Cauchy$s equations of motion

for zero acceleration). Moreover, the principle of

conservation of angular momentum implies that 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 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 thesystem of coordinates. A graphical representation of this transformation law is the Mohr's

circle of stress distribution.

As a symmetric 3"3 real matrix, the stress tensor has three mutually orthogonal

unit-length eigenvectors and three real eigenvalues , 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 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




9/15

A tank car made from bent andwelded steel plates.

In general, stress is not uniformly distributed over a material body, and may vary with time.

Therefore the stress tensor must be defined for each point and each moment, by

considering an infinitesimal particle of the medium surrounding that point, and taking the

average stresses in that particle as being the stresses at the point.

Stress in thin plates

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, 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 the radius

of curvature is comparable to the thickness of the plate).

Stress in thin beams

The analysis of stress can be considerably simplified also for thin bars, beams or wires of

uniform (or smoothly varying) composition and cross-section that are subjected to

moderate bending and twisting. For those bodies 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 perpendicularto 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 the

Kirchhoff stress tensor, are required.

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

flow under shear stress. Moving viscous fluids can support shear stress (dynamic




10/15

For stress

modeling, a fishing

pole may be

considered

one-dimensional.

pressure). Solids can support both shear and normal stress, with

ductile materials failing under shear and 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

Stress analysis is a branch of applied physics that covers the

determination of the internal distribution of stresses in solid objects. It

is an essential tool in engineering for the study and design of

structures such as tunnels, dams, mechanical parts, and structural

frames, under prescribed or expected loads. It is also important in

many other disciplines; for example, in geology, to study phenomena

like plate tectonics, vulcanism and avalanches; and in biology, to

understand the anatomy of living beings.

Goals and assumptions

Stress analysis is generally concerned with objects and structures

that can be assumed to be in macroscopic static equilibrium. By Newton's laws of motion,

any external forces 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.[3]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.4281or

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 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 by Euler's equations of motion forcontinuous bodies (which are consequences of Newton's laws for conservation of linear

momentum and angular momentum) and the Euler-Cauchy stress principle, together with




11/15

Simplified model of a truss for stressanalysis, assuming unidimensional

elements under uniform axial tension

or compression.

the appropriate constitutive equations. Thus one obtains a system of partial differential

equations involving the stress tensor field and the strain tensor field, as unknown functions

to be determined. The external body forces appear as the independent ("right-hand side")

term in the differential equations, while the concentrated forces appear as boundary

conditions. The basic stress analysis problem is therefore a boundary-value problem.

Stress analysis for elastic structures is based on the theory of elasticity and infinitesimalstrain 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 of linear elasticity (the generalization of Hooke$s law for

continuous media); that is, the deformations caused by internal stresses are linearly

related to them. In this case the differential equations that define the stress tensor are

linear, and the problem becomes much easier. For one thing, the stress at any point will be

a linear function of the loads, too. For small enough stresses, even non-linear systems can

usually be assumed to be linear.

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-dimensionalproblem 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. 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 second PiolaKirchhoff stress tensors,




12/15

the Biot stress tensor, and the Kirchhoff stress tensor.

PiolaKirchhoff stress tensor

In the case of finite deformations, the PiolaKirchhoff stress tensorsexpress the stress

relative to the reference configuration. This is in contrast to the Cauchy stress tensor which

expresses the stress relative to the present configuration. For infinitesimal deformationsand rotations, the Cauchy and PiolaKirchhoff tensors are identical.

Whereas the Cauchy stress tensor, relates stresses in the current configuration, the

deformation gradient 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 the deformation gradient and is the Jacobian determinant.

In terms of components with respect to an orthonormal 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 of engineering 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 configuration, the 2nd PiolaKirchhoff stress tensor relates forces in the

reference configuration to areas in the reference configuration. The force in the referenceconfiguration is obtained via a mapping that preserves the relative relationship between the

force direction and the area normal in the reference configuration.




13/15

In index notation 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 PiolaKirchhoff stress tensor remain constant, irrespective of material orientation.

The 2nd PiolaKirchhoff stress tensor is energy conjugate to the GreenLagrange finite

strain tensor.

See also

BendingKelvin probe forcemicroscopeMohr's circleResidual stressShot peening

StrainStrain tensorStrain rate tensorStressenergy tensorStressstrain curveStress concentration

Transient frictionloadingVirial stressYield stressYield surfaceVirial theorem

Further reading

Chakrabarty, J. (2006). Theory of plasticity(http://books.google.ca/books?id=9CZsqgsfwEAC&lpg=PP1&dq=related%3AISBN0486435946&rview=1&pg=PA17#v=onepage&q=&f=false) (3 ed.). Butterworth-Heinemann. pp. 1732.ISBN 0-7506-6638-2.

Beer, Ferdinand Pierre; Elwood Russell Johnston, John T. DeWolf (1992). Mechanics

of Materials. McGraw-Hill Professional. ISBN 0-07-112939-1.

Brady, B.H.G.; E.T. Brown (1993). Rock Mechanics For Underground Mining(http://books.google.ca/books?id=s0BaKxL11KsC&lpg=PP1&pg=PA18#v=onepage&q=&f=false) (Third ed.). Kluwer Academic Publisher. pp. 1729. ISBN 0-412-47550-2.

Chen, Wai-Fah; Baladi, G.Y. (1985). Soil Plasticity, Theory and Implementation.

ISBN 0-444-42455-5.Chou, Pei Chi; Pagano, N.J. (1992). Elasticity: tensor, dyadic, and engineering

approaches(http://books.google.com/books?id=9-pJ7Kg5XmAC&lpg=PP1&pg=PA1#v=onepage&q=&f=false). Dover books on engineering. Dover Publications.pp. 133. ISBN 0-486-66958-0.

Davis, R. O.; Selvadurai. A. P. S. (1996). Elasticity and geomechanics(http://books.google.ca/books?id=4Z11rZaUn1UC&lpg=PP1&pg=PA16#v=onepage&q=&f=false). Cambridge University Press. pp. 1626. ISBN 0-521-49827-9.

Dieter, G. E. (3 ed.). (1989). Mechanical Metallurgy. New York: McGraw-Hill. ISBN0-07-100406-8.

Holtz, Robert D.; Kovacs, William D. (1981).An introduction to geotechnicalengineering(http://books.google.ca/books?id=yYkYAQAAIAAJ&dq=inauthor:%22William+D.+Kovacs%22&cd=1). Prentice-Hall civil engineering and


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engineering mechanics series. Prentice-Hall. ISBN 0-13-484394-0.

Jones, Robert Millard (2008). Deformation Theory of Plasticity(http://books.google.ca/books?id=kiCVc3AJhVwC&lpg=PP1&pg=PA95#v=onepage&q=&f=false). Bull RidgeCorporation. pp. 95112. ISBN 0-9787223-1-0.

Jumikis, Alfreds R. (1969). Theoretical soil mechanics: with practical applications to

soil mechanics and foundation engineering(http://books.google.ca

/books?id=NPZRAAAAMAAJ). Van Nostrand Reinhold Co. ISBN 0-442-04199-3.Landau, L.D. and E.M.Lifshitz. (1959). Theory of Elasticity.

Love, A. E. H. (4 ed.). (1944). Treatise on the Mathematical Theory of Elasticity. NewYork: Dover Publications. ISBN 0-486-60174-9.

Marsden, J. E.; Hughes, T. J. R. (1994). Mathematical Foundations of Elasticity(http://books.google.ca/books?id=RjzhDL5rLSoC&lpg=PR1&pg=PA133#v=onepage&q&f=false). Dover Publications. pp. 132142. ISBN 0-486-67865-2.

Parry, Richard Hawley Grey (2004). Mohr circles, stress paths and geotechnics(http://books.google.ca/books?id=u_rec9uQnLcC&lpg=PP1&dq=mohr%20circles%2C%20sterss%20paths%20and%20geotechnics&pg=PA1#v=onepage&q=&f=false)(2 ed.). Taylor & Francis. pp. 130. ISBN 0-415-27297-1.

Rees, David (2006). Basic Engineering Plasticity

An Introduction with Engineering

and Manufacturing Applications(http://books.google.ca/books?id=4KWbmn_1hcYC&lpg=PP1&pg=PA1#v=onepage&q=&f=false). Butterworth-Heinemann. pp. 132.ISBN 0-7506-8025-3.

Timoshenko, Stephen P.; James Norman Goodier (1970). Theory of Elasticity(Thirded.). McGraw-Hill International Editions. ISBN 0-07-085805-5.

Timoshenko, Stephen P. (1983). History of strength of materials: with a brief account

of the history of theory of elasticity and theory of structures. Dover Books on Physics.Dover Publications. ISBN 0-486-61187-6.

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^Wai-Fah Chen and Da-Jian Han (2007), "Plasticity for Structural Engineers"(http://books.google.com/books?id=E8jptvNgADYC&pg=PA46 ). J. Ross Publishing ISBN1-932159-75-4

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^Peter Chadwick (1999), "Continuum Mechanics: Concise Theory and Problems"(http://books.google.ca/books?id=QSXIHQsus6UC&pg=PA95 ). Dover Publications, series

"Books on Physics". ISBN 0-486-40180-4. pages

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^ abI-Shih Liu (2002), "Continuum Mechanics" (http://books.google.com/books?id=-gWqM4uMV6wC&pg=PA43). Springer ISBN 3-540-43019-9

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^(2009) The art of making glass. (http://www.lamberts.de/fileadmin/user_upload/service/downloads/lamberts_broschuere_englisch.pdf) Lamberts Glashtte (LambertsGlas) productbrochure. Accessed on 2013-02-08.

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^ abcRonald L. Huston and Harold Josephs (2009), "Practical Stress Analysis in EngineeringDesign". 3rd edition, CRC Press, 634 pages. ISBN 9781574447132

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^Walter D. Pilkey, Orrin H. Pilkey (1974), "Mechanics of solids" (http://books.google.com/books?id=d7I8AAAAIAAJ) (book)

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after Truesdell and Noll". Springer. ISBN 0-7923-2454-4 (http://books.google.com/books?id=ZcWC7YVdb4wC&pg=PA97)

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^Fridtjov Irgens (2008), "Continuum Mechanics" (http://books.google.com/books?id=q5dB7Gf4bIoC&pg=PA46). Springer. ISBN 3-540-74297-2

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^Slaughter9.

^Jacob Lubliner (2008). "Plasticity Theory" (http://www.ce.berkeley.edu/~coby/plas/pdf/book.pdf) (revised edition). Dover Publications. ISBN 0-486-46290-0

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