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 engineering 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 strength. 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 in 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 these 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 material 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 at 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 (σ1,σ2,σ3), 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 nonlinear. 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]). This
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 principles.
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 specialist 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 Henri Tresca. This assumes that yield occurs when the shear stress
exceeds the shear yield strength
:
Total Strain Energy Theory - This theory assumes that the stored 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-dimensional stress state this
is given by:
Distortion Energy Theory - This theory proposes that the total strain energy can be
separated into two components: the volumetric (hydrostatic) strain energy and the
shape (distortion or shear) strain energy. It is proposed 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 and is expressed as:
Based on a different theoretical underpinning this expression is also referred to as
octahedral shear stress theory.
Factors 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 relationship 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 QHW 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 both 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 operational 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 yield 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. The three primary ones are

Young's modulus (E) describes tensile elasticity, or the tendency of an object
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 modulus 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
modulus.
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. Given 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 shear 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 elasticity, 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 stressstrain 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 gigapascals.
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, are 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 concrete'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 modulus, 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;
A0 is the original cross-sectional area through which the force is applied;
ΔL is the amount by which the length of the object changes;
L0 is the original length of the object.
Force exerted by stretched 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 Ue 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 expressed 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 various solids
Young's modulus Young's modulus (E) in
Material
(E) in GPa
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
Bacteriophage 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
30
4,350,000
Magnesium metal (Mg)
45
6,500,000
Aluminium alloy
69
10,000,000
High-strength
concrete
compression)
(under
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
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
Carbon
fiber
reinforced
(unidirectional, along grain)
plastic
150,000,000175,000,000