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
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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
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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
(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.
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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
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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
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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
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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
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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
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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
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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
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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,
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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.
Stress (mechanics) - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Stress_(physics)
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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
Stress (mechanics) - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Stress_(physics)
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http://books.google.ca/http://books.google.ca/books?id=s0BaKxL11KsC&lpg=PP1&pg=PA18#v=onepage&http://books.google.com/books?id=9-pJ7Kg5XmAC&lpg=PP1&http://books.google.ca/books?id=4Z11rZaUn1UC&lpg=PP1&pg=PA16#v=onepage&http://books.google.ca/books?id=yYkYAQAAIAAJ&http://en.wikipedia.org/wiki/Stress_http://www.pdffactory.com/http://www.pdffactory.com/http://en.wikipedia.org/wiki/Stress_http://books.google.ca/books?id=yYkYAQAAIAAJ&http://books.google.ca/books?id=4Z11rZaUn1UC&lpg=PP1&pg=PA16#v=onepage&http://books.google.com/books?id=9-pJ7Kg5XmAC&lpg=PP1&http://books.google.ca/books?id=s0BaKxL11KsC&lpg=PP1&pg=PA18#v=onepage&http://books.google.ca/ -
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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.
References
^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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