Transcript • The various engineering and true stress-strain properties

• The various engineering and true stress-strain properties
obtainable from a tension test are summarized by the
categorized listing of Table 1.1.
• Note that the engineering fracture strain ef and the %
elongation are only different ways of stating the same
quantity. Also, the %RA and ef can be calculated from each
other.
• Note that the strength coefficient H determines the
magnitude of the true stress in the large strain region of the
stress-strain curve, and so it is included as a measure of
strength.
• The strain hardening exponent n is a measure of the rate of
strain hardening.
Table 1.1 Materials Properties Obtainable from Tension Tests
Category
Engineering Property
True Stress-Strain
Property
Elastic
Constants
Elastic modulus, E
Poisson's ratio, 
Strength
Proportional limit, p
Yield strength, y
Ultimate tensile strength, 
Engineering fracture strength,
True fracture strength, f
Strength coefficient, H or K
Ductility
Percent elongation, 100
Reduction in area, %RA
True fracture strain, 
Energy Capacity Resilience, ur
Tensile toughness, ut
True toughness,u
Strain hardening Strain hardening
Ratio, /
Strain hardening
exponent, n
Modulus of Elasticity
•The slope of the initial portion of the stress-strain curve is
the modulus of elasticity, or Young’s Modulus. The
modulus of elasticity is a measure of the stiffness of the
material. It is an important design value.
•The modulus of elasticity is determined by the building
forces between atoms. It is only slightly affected by
alloying.
Measures of Yielding
• Yielding defines the point at which plastic deformation begins.
This point may be difficult to determine in some materials, which
have gradual transition from elastic to plastic behavior. Therefore,
various criteria (depends on the sensitivity of the strain
measurements) are used to define yielding.
1. Proportional Limit - This is the highest stress at which stress is
directly proportional to strain.
2. Elastic Limit - This is the greatest stress the material can
withstand without any measurable permanent strain remaining on
the complete release of the load.
3. Yield Strength - This is the stress required to produce a small
(0.2% strain) specified amount of plastic deformation.
(a)
(a)
(b)
Figure 1-13. (a) Typical stress-strain (type II) behavior for a metal showing
elastic and plastic deformations, the proportional limit P, and the yield
strength y, as determined using the 0.002 strain offset method.
(b) Representative stress-strain (type IV) behavior found for some steels
demonstrating the yield drop (point) phenomenon.
Poisson’s Ratio
•If the applied stress is uniaxial (only in the z direction), then ex = ey .
A parameter termed Poisson’s ratio v is defined as the ratio of the
lateral and axial strains, or
ey
ex
v

ez
ez
(1.8)
z
x
Figure 1-14.
l l  lo
ez  
l
lo
d
d  do
ez 

do
do
(1.9)
(1.10)
Measures of Ductility
• Ductility is a qualitative, subjective property of a material. It
usually indicates the extent to which a metal can be deformed
without fracture.
• Two methods one can obtain ductility from tension test are:
- the engineering strain at fracture, ef, known as elongation
where
L f  Lo
(1.11)
ef 
Lo
- the reduction in area at fracture, q
where
A A
q
o
Ao
f
(1.12)
• The two properties are obtained by putting the fractured
specimen back together, and taking measurements of Lf and Af.
• Both elongation and reduction of area are usually expressed as
a percentage.
• The value of ef will depend on the gage length Lo in necked
specimens. The reduction in area is a better method of
reporting elongation, especially for ductile materials.
Toughness
• The toughness of a material is its ability to absorb energy in the
plastic range. This property is particular desirable in parts such
as freight car couplings, gears, chains, and crane hooks.
• One way of looking at toughness is to consider it as the total
area under the stress- strain curve. This area is an indication of
the amount of work per unit volume which can be done on the
material without causing it to rupture.
•
Figure 1-15 shows the stress strain curve for high and low
toughness materials.
Figure 1-15. Comparison of stress-strain curves for
high and low toughness materials.
• The area under the curve for ductile metals (stress-strain curve is
like that of the structural steel) can be approximated by either of
the following equations:
(1.13)
U T  su e f
or
s o  su
UT 
ef
2
(1.14)
• The area under the curve for brittle materials (stress-strain curve
is sometimes assumed to be a parabola) can be given by:
2
U T  su e f
3
(1.15)
• All these relations are only approximately to the area under the
stress-strain curve.
Resilience
• The ability of a material to absorb energy when deformed
elastically is called resilience. Otherwise called modulus of
resilience, it is the strain energy per unit volume required to stress
the material from zero stress to the yield stress o. The strain
energy per unit volume for uniaxial tension is
1
U o   x ex
2
•
(1.16)
From the above definition the modulus of resilience is
1
1 so so
U R  so eo  so 
2
2 E 2E
(1.17)
Resilience Continued. . .
• The value can be obtained by integrating over the area under the
curve up to the yield point, and this is given as:
Ur 
•

ey
0
 de
(1.18a)
Assuming a linear elastic region,
1
U r   yey
2
(1.18b)
2
1
1  y   y
U r   y ey   y   
2
2  E  2E
(1.19)
True Stress-True Strain Curve
• The relationship between the true stress, , and engineering
stress, s, is given by:
P

(e  1)  s(e  1)
A
o
(1.20)
where P is the Load, and Ao is the original length
• The derivation of Eq. (1.20) assumes both constancy of volume
and a homogenous distribution of strain along the gage length of
the tension specimen. Thus, Eq. (1.20) should only be used
until the onset of necking.
• It must be emphasized that the engineering stress-strain curve
does not give a true indication of the deformation characteristics
of a metal because it is based on the original dimensions of the
specimen.
• In actuality, ductile materials continue to strain-harden up to
fracture, but engineering stress-strain curve gives a different
picture. The occurrence of necking in ductile materials leads to a
drop in load and engineering stress required to continue
deformation, once the maximum load is exceeded.
• An assessment of the true stress-true strain curve provides a
realistic characteristic of the material.
• Beyond maximum load the true stress should be determined
from actual measurements of load and cross-sectional area.
 
P
A
(1.21)
• The true strain e may be determined from the engineering or
conventional strain e by
e  ln( e  1)
(1.22)
• This equation is applicable only to the onset of necking for the
reasons discussed above.
• Beyond maximum load the true strain should be based on
actual area or diameter measurements.
2
Ao
Do
( pi / 4) Do
e  ln
 ln

2
ln
A
( pi / 4) D 2
D
(1.23)
• Figure 1-16 compares the true-stress true-strain curve for AISI
4140 hot-rolled steel with its corresponding engineering
stress-strain curve.
Figure 1-17. True stress-strain and engineering stress-strain
curves for AISI 4140 hot-rolled steel
• The annealed structure is ductile, but has low yield stress.
The ultimate tensile stresses (the maximum engineering
stresses) are marked by arrows. After these points, plastic
deformation becomes localized (called necking), and the
engineering stresses drop because of the localized
reduction in cross-sectional area.
• However, the true stress continues to rise because the
cross-sectional area decreases and the material workhardens in the neck region. The true-stress-true-strain
curves are obtained by converting the tensile stress and its
corresponding strain into true values and extending the
curve.
Instability in Tension
Necking or localized deformation begins at maximum load, where
the increase in stress due to decrease in the cross-sectional area of
the specimen becomes greater than the increase in the load-carrying
ability of the metal due to strain hardening. This conditions of
instability leading to localized deformation is defined by the
condition dP = 0.
P  A
dP  dA  Ad  0
(1.24)
From the constancy-of-volume relationship,
V  Ao Lo  AL
dL
dA
   de
L
A
(1.25)
From the instability condition,

dA d

A

(1.26a)
so that at a point of tensile instability
d

de
(1.26b)
The necking criterion can be expressed more explicitly if engineering
strain is used. Starting with Eq. (1.26b )
d d de d dL / Lo d



( L  e)  
de de de de dL / L
de
d
de


1 e
(1.27)
We know that the volume V is constant in plastic deformation:
V  Ao Lo  AL
Consequently,
A
Ao Lo
L
In what follows, we use the subscripts e and e for engineering
(nominal) and true stresses and strains, respectively. We have
L  Lo
Ao
ee 

1
Lo
A
e
P Ao
Ao

*

 1  ee
e
A P
A
 e  1  e e   e
(1.28)
• On the other hand, the incremental longitudinal true strain is
defined as
de e 
•
dL
(1.29)
L
For extended deformation, integration is required:
ee 
L
L
o
dL
L
 ln
L
Lo
(1.30)
exp e e 
L

Lo
(1.31)
• On substituting, we get
P
e 
exp e e 
Ao
(1.32)
True Stress at Maximum Load
su 
Pmax
Ao
and
u
Pmax

Au
Ao
e u  ln
Au
Eliminating Pmax yields
Ao
 u  su
Au
and
 u  su e
eu
(1.29)
True Fracture Strain
• The true fracture strain, ef , is the true strain based of the original
area Ao and the area after fracture Af.
e f  ln
Ao
Af
(1.30)
• For cylindrical tensile specimens the reduction of area q is related
to the true fracture strain by the relationship
ef
1
 ln
1 q
(1.31)