Stress-strain curve

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Stress
-
strain curve

A
stress
-
strain

curve is a graph derived from measuring load (
stress

-

σ) versus
extension (
strain

-

ε) for a sample of a material. The nature of the curve varies from
material to material. The following diagrams illustrate the stress
-
strain behaviour of
typical materials in terms of the
enginee
ring stress

and
engineering strain

where the
stress and strain are calculated based on the original dimensions of the sample and not
the instantaneous values. In each case the samples are loaded in
tension

although in
many cases similar behaviour is observed in
compression
.

Ductile materials



Fig 1. A stress
-
strain curve typical of structural
steel

1. Ultimate Strength

2.
Yield Strength

3. Rupture

4.
Strain hardening

region

5. Necking region.

Steel generally exhibits a very linear stress
-
strain relationship up to a well defined
yield point

(figure 1). The linear portion of the curve is the
elastic

region and the
slope is the
modulus of elasticity

or
Young's Modulus
. After the yield point the curve
typically

decreases slightly due to
dislocations

escaping from
Cottrell atmospheres
.
As deformation

continues the stress increases due to
strain hardening

until it reaches
the
ultimate s
trength
. Until this point the cross
-
sectional area decreases uniformly due
to
Poisson contractions
. However, beyond this point a
neck

forms where the local
cross
-
sectional

area decreases more quickly than the rest of the sample resulting in an
increase in the
true stress
. On an engineering stress
-
strain curve this is seen as a
decrease i
n the stress. Conversely, if the curve is plotted in terms of
true stress

and
true strain

the stress will continue to rise until failure. Eventually the neck becomes
unstable and the specimen ruptures (
fractures
).

Most
ductile

metals

other than steel do not have a well
-
defined yield point (figure 2).
For thes
e materials the yield strength is typically determined by the "offset yield
method", by which a line is drawn parallel to the linear elastic portion of the curve
and intersecting the abscissa at some arbitrary value (most commonly .2%). The
intersection of

this line and the stress
-
strain curve is reported as the yield point.

Brittle materials

Brittle

materials such as
concrete

or
ceramics

do not have a yield point. For these
materials the rupture strength and the ultimate strength are the same.

Properties

The area underneath the stress
-
strain curve is the
toughness

of the material
-

i.e. the
energy the material can absorb prior to rupture.........

The
resilience

of the mater
ial is the triangular area underneath the
elastic

region of the
curve.

Yield (engineering)

Yield strength
, or the yield point, is defined in
engineering

and
materials science

as
the
stress

a
t which a
material

begins to
plastically deform
. Prior to the yield point the
material will
deform
elastically

and will return to its original shape when the applied
stress is removed. Once the yield point is passed some fraction of the deformation will
be permanent

and non
-
reversible. Knowledge of the yield point is vital when
designing a component since it generally represents an upper limit to the load that can
be applied. It is also important for the control of many materials production
techniques such as
forging
,
rolling
, or
pressing

In structural engineering,
yield

is the permanent
plastic deformation

of a structural
member under
stress
. This is a soft failure mode which does not normally cause
catastrophic failure

unless it accelerates
buckling
.

In 3D space of principal
stresses


1

2

3
), an infinite number of yield points form
together a
yield surface
.

Definition

It is often difficult to precisely define yield due to the wide variety of
stress
-
strain
behaviours

exhibited by real materials. In addition there are several possible ways to
define the yield point in a given material:



The point at which
dislocations

first begin to move. Given that dislocations
begin to move at very low stresses, and the difficulty in detecting such
movement, this definition is rarely used.



Elastic Limit

-

The lowest stress at which permanent deformation can be
measured. This requires
a complex iteractive load
-
unload procedure and is
critically dependent on the accuracy of the equipment and the skill of the
operator.



Proportional Limit

-

The point at which the
stress
-
strain curve

becomes non
-
linear. In most metallic materials the
elastic limit

and
proportional limit

are
essentially the same.



Offset Yield Point (proof stress)

-

Due to the lack of a clear border between
the elastic and plastic regions

in many materials, the yield point is often
defined as the stress at some arbitrary plastic strain (typically 0.2%
[1]
). Thi
s
is determined by the intersection of a line offset from the linear region by the
required strain. In some materials there is essentially no linear region and so a
certain value of plastic strain is defined instead. Although somewhat arbitrary
this method

does allow for a consistent comparison of materials and is the
most common.

Yield criterion

A yield criterion, often expressed as
yield surface
, is an hypothesis concerning the
limit of elasticity under any combination of stresses. There are two interpretations of
yield criterion: one is purely mathematical in taking a statistical approach while other
models attempt to provide a justification based on established physical princip
les.
Since stress and strain are
tensor

qualities they can be described on the basis of three
principal directions, in the case of stress these are denoted by
,
and
.

The following represent the most common yield criterion as applied to an isotropic
material (uniform properties in all directions). Other equations have been proposed or
are used in spec
ialist situations.

Maximum Principal Stress Theory

-

Yield occurs when the largest principal stress
exceeds the uniaxial tensile yield strength. Although this criterion allows for a quick
and easy comparison with experimental data it is rarely suitable for

design purposes.


Maximum Principal Strain Theory

-

Yield occurs when the maximum principal
strain

reaches the strain corresponding to the yield point during a simple tensile test.
In terms of the principal stresses this is determined by the equation:


Maximum Shear Stress Theory

-

Also known as the Tresca criterion, after the
French scientist
Henr
i Tresca
. This assumes that yield occurs when the shear stress
exceeds the shear yield strength
:


Total Strain Energy Theory

-

This theory assumes that the sto
red energy associated
with elastic deformation at the point of yield is independent of the specific stress
tensor. Thus yield occurs when the strain energy per unit volume is greater than the
strain energy at the elastic limit in simple tension. For a 3
-
di
mensional stress state this
is given by:


Distortion Energy Theory

-

This theory proposes that the total strain energy can be
separated int
o two components: the
volumetric

(
hydrostatic
) strain energy and the
shape

(distortion or
shear
) strain energy. It is pr
oposed that yield occurs when the
distortion component exceeds that at the yield point for a simple tensile test. This is
generally referred to as the
Von Mises criterion

a
nd is expressed as:


Based on a different theoretical underpinning this expression is also referred to as
octahedral shear stress theory
.

F
actors influencing yield stress

The stress at which yield occurs is dependent on both the rate of deformation (strain
rate) and, more significantly, the temperature at which the deformation occurs. Early
work by Alder and Philips in 1954 found that the rel
ationship between yield stress and
strain rate (at constant temperature) was best described by a power law relationship of
the form


where
C is a constant and m is the strain rate sensitivity. The latter generally increases
with temperature, and materials where m reaches a value greater than ~0.5 tend to
exhibit
super plastic behaviour
.

Later, more complex equations were proposed that simultaneously dealt with both
temperature and strain rate:


where α and A are constants and Z is the temperature
-
compensated strain
-
rate
-

often
described by the Zener
-
Hollomon parameter:


where Q
HW

is the activation energy for hot deformation and T is the absolute
temperature.

Implications for structural engineering

Yielded structures have a lower and less constant modulus of elasticity, so deflections
increase and buckling strength decreases, and bo
th become more difficult to predict.
When load is removed, the structure will remain permanently bent, and may have
residual pre
-
stress. If buckling is avoided, structures have a tendency to adapt a more
efficient shape that will be better able to sustain
(or avoid) the loads that bent it.
Because of this, highly engineered structures rely on yielding as a graceful failure
mode which allows fail
-
safe operation. In aerospace engineering, for example, no
safety factor is needed when comparing limit loads (the

highest loads expected during
normal operation) to yield criteria. Safety factors are only required when comparing
limit loads to ultimate failure criteria, (buckling or rupture.) In other words, a plane
which undergoes extraordinary loading beyond its op
erational envelope may bend a
wing slightly, but this is considered to be a fail
-
safe failure mode which will not
prevent it from making an emergency landing.

Elastic modulus

An
elastic modulus
, or
modulus of elasticity
, is the mathematical description of
an
object or substance's tendency to be deformed elastically (i.e. non
-
permanently) when
a
force

is applied to it. The elastic modulus of an object is defined as the
slope

of its
stress
-
strain curve

in the elastic deformation region:


where
λ

is the elastic modulus;
stress

is the force causing the deformation divided by
the area to which the force is applied; and
strain

is the ratio of the change caused by
the stress to the original state of the object. Because stress is measured in
pascals

and
strain is a unitless ratio, the units of
λ

are therefore pascals as well. An alternative
definition is that the elastic modulus is the stress required to cause a sample of the
material to double in length. This is not literally true for most materials because the
value is far greater than the yie
ld stress of the material or the point where elongation
becomes nonlinear but some may find this definition more intuitive.

Specifying how stress and strain are to be measured, including directions, allows for
many types of elastic moduli to be defined. Th
e three primary ones are



Young's modulus

(
E
) describes tensile
elasticity
, or the tendency of an obj
ect
to deform along an axis when opposing forces are applied along that axis; it is
defined as the ratio of
tensile stress

to
tensile

strain
. It is often referred to
simply as the
elastic modulus
.



The
shear modulus

or
modulu
s of rigidity

(
G

or μ) describes an object's
tendency to shear (the deformation of shape at constant volume) when acted
upon by opposing forces; it is defined as
shear stress

over
shear strain
. The
shear modulus is part of the derivation of
viscosity
.



The
bulk modulus

(
K
) describes volumetric elasticity, or the tendency of an
object's volume to deform when under pressure; it is defined as volumetric
stress over volumetric strain, and is the inverse of
compressibility
. The bulk
modulus is an extension of Young's modulus to three dimensions.

Three other elastic moduli are
Poisson's ratio
,
Lamé's first parameter
, and
P
-
wave
modul
us
.

Homogeneous and
isotropic

(similar in all directions) materials (solids) have their
(linear) elastic properties fully described by two elastic moduli, and one may choose
any pair. G
iven a pair of elastic moduli, all other elastic moduli can be calculated
according to formulas in the table below.

Inviscid fluids

are special in that they can not support s
hear stress, meaning that the
shear modulus is always zero. This also implies that
Young's modulus

is always zero.

Young's modulus

In
solid mechanics
,
Young's modulus (E)

is a measure of the
stiffness

of a given
material. It is also known as the
Young modulus
,
modulus of elas
ticity
,
elastic
modulus

or tensile modulus (the
bulk modulus

and
shear modulus

are different types
of
elastic modulus
). It is defined as the ratio, for small strains, of the rate of change of
stress

with
strain
.
[1]

This can be experimentally determined from the
slope

of a
stre
ss
-
strain curve

created during
tensile tests

conducted on a sample of the material.
Young's modulus is named after
Thomas Young
, the 18th Century British scientist.

Units

The
SI

unit of modulus of elasticity (E, or less commonly Y) is the
pascal
. Given the
large values typical of many common materials, figures are usually quoted in
megapascals or gigapascals. Some use an alternative unit form, kN/mm², which gives
the same numeric value as gigapa
scals.

The modulus of elasticity can also be measured in other units of pressure, for example
pounds per square inch
.

Usage

The Young's modulus allows

the behavior of a material under load to be calculated.
For instance, it can be used to predict the amount a wire will extend under tension, or
to predict the load at which a thin column will
buckle

under compression. Some
calculations also require the use of other material properties, such as the
shear
modulus
,
density
, or
Poisson's ratio
.

Linear vs non
-
linear

For many materials, Young's modulus is a constant over a range of strains. Such
materials are called
linear
, and are said to obey
Hooke's law
. Examples of linear
materials include
steel
,
carbon fiber
, and
glass
.
Rubber

and
soil

(except at very low
strains
) are
non
-
linear

materials.


Directional materials

Most metals and ceramics, along with many other materials, ar
e
isotropic

-

their
mechanical properties are the same in all directions, but metals and ceramics can be
treated to create different grain sizes and orientations. This treatment makes them

anisotropic, meaning that Young's modulus will change depending on which direction
the force is applied from. However, some materials, particularly those which are
composites of two or more ingredients have a "grain" or similar mechanical structure.
As a
result, these
anisotropic

materials have different mechanical properties when
load is applied in different directions. For example,
carbon fiber

is much stiffer
(higher Young's modulus) when loaded parallel to the fibers (along the grain). Other
such materials include
wood

and
reinforced concrete
. Engineers can use this
directional phenomonon to their advantage in creating various structures in our
environment. Concrete is commonly used to construct support columns
in buildings,
supporting huge loads under compression. However, when concrete is used in the
construction of bridges and is in tension, it needs to be reinforced with steel which has
a far higher value of Young's modulus in tension and compensates for conc
rete's low
value in tension. Copper is an excellent conductor of electricity and is used to transmit
electricity over long distance cables, however copper has a relatively low value for
Young's modulus at 130GPa and it tends to stretch in tension. When the

copper cable
is bound completely in steel wire around its outside this stretching can be prevented as
the steel (with a higher value of Young's modulus in tension) takes up the tension that
the copper would otherwise experience.

Calculation

Young's modulu
s,
E
, can be calculated by dividing the
tensile stress

by the
tensile
strain
:


where

E

is the Young's modulus (modulus of elasticity) measured in
pascals
;

F

is the force applied to the object;

A
0

is the original cross
-
sectional area through which the force is applied;

ΔL

is the amount by which the length of the object changes;

L
0

is the original length of the object.

Force exerted by stretc
hed or compressed material

The Young's modulus of a material can be used to calculate the force it exerts under a
specific strain.


where
F

is the force exerted by the material when compressed or stretched by
ΔL
.

From this formula can be derived
Hooke's law
, which describes the stiffness of an
ideal spring:


where



Elastic potential energy

The
elastic potential energy

stored is given by the integral of this expression with
respect to
L
:


where
U
e

is the elastic potential energy.

The elastic potential energy per unit volume is given by:

, where
is the strain in the material.

This formula can also be expr
essed as the integral of Hooke's law:



Approximate values

Young's modulus can vary considerably depending on the exact composition of the
material. For example, the value for most metals can vary by 5% or more, depending
on the precise composition of the alloy and any heat treatment applied during
manufacture. As such, many of the values here are approximate.

Approximate Young's moduli of va
rious solids

Material

Young's modulus
(E) in
G
Pa

Young's modulus (E) in
lbf/in²

(psi)

Rubber

(small strain)

0.01
-
0.1

1,500
-
15,000

Low density polyethylene

0.2

30,000

Polypropylene

1.5
-
2

217,000
-
290,000

Bacterio
phage capsids

1
-
3

150,000
-
435,000

Polyethylene terephthalate

2
-
2.5

290,000
-
360,000

Polystyrene

3
-
3.5

435,000
-
505,000

Nylon

3
-
7

290,000
-
580,000

Oak
wood

(along grain)

11

1,600,000

High
-
strength
concrete

(under
compression)

30

4,350,000

Magnesium

metal

(Mg)

45

6,
500,000

Aluminium alloy

69

10,000,000

Glass

(all types)

72

10,400,000

Brass

and
bronze

103
-
124

17,000,000

Titanium

(Ti)

105
-
120

15,000,000
-
17,500,000

Carbon fiber reinforced plastic

(unidirectional, along grain)

10
-

20

1,500,000
-

3,200,000

Wrought iron

and
steel

190
-
210

30,000,000

Tungsten

(W)

400
-
410

58,000,000
-
59,500,000

Silicon carbide

(SiC)

450

65,000,000

Tungsten carbide

(WC)

450
-
650

65,000,000
-
94,000,000

Single carbon nanotube

[1]

1,000+

145,000,000

Diamond

(C)

1,050
-
1,200

150,000,000
-
175,000,000