The various engineering and true stress-strain properties obtainable from a tension test are summarized by the categorized listing of Table 1.1.

1

• 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.

2
Table 1.1 Materials Properties Obtainable from
Tension Tests
3

• Modulus of Elasticity
• The slope of the initial portion of the
  stress-strain curve is the modulus of elasticity,
  or Youngs 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.

4

• 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.

5
(a)
(b)
(a)
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.
6

• Poissons Ratio
• If the applied stress is uniaxial (only in the z
  direction), then ?x ?y . A parameter termed
  Poissons ratio v is defined as the ratio of the
  lateral and axial strains, or

(1.8)
Figure 1-14.
(1.9)
(1.10)
7
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
• - the reduction in area at fracture, q
• where

(1.11)
(1.12)
8

• 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.

9

• 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.

10
Figure 1-15. Comparison of stress-strain curves
for high and low toughness materials.
11

• 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
• or
• The area under the curve for brittle materials
  (stress-strain curve is sometimes assumed to be a
  parabola) can be given by
• All these relations are only approximately to the
  area under the stress-strain curve.

(1.13)
(1.14)
(1.15)
12

• 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
  so. The strain energy per unit volume for
  uniaxial tension is
• From the above definition the modulus of
  resilience is

(1.16)
(1.17)
13

• 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

(1.18a)

• Assuming a linear elastic region,

(1.18b)
(1.19)
14
True Stress-True Strain Curve

• The relationship between the true stress, ?, and
  engineering stress, s, is given by
• 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.

(1.20)
15

• 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.

16

• Beyond maximum load the true stress should be
  determined from actual measurements of load and
  cross-sectional area.
• The true strain e may be determined from the
  engineering or conventional strain e by
• This equation is applicable only to the onset of
  necking for the reasons discussed above.

(1.21)
(1.22)
17

• Beyond maximum load the true strain should be
  based on actual area or diameter measurements.

(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.

18
Figure 1-17. True stress-strain and engineering
stress-strain curves for AISI 4140
hot-rolled steel
19

• 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 work-hardens 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.

20
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. From the constancy-of-volume
relationship,
(1.24)
(1.25)
21
From the instability condition, so that at a
point of tensile instability The necking
criterion can be expressed more explicitly if
engineering strain is used. Starting with Eq.
(1.26b )
(1.26a)
(1.26b)
(1.27)
22
We know that the volume V is constant in plastic
deformation Consequently, In what follows,
we use the subscripts e and e for engineering
(nominal) and true stresses and strains,
respectively. We have
23
(1.28)

• On the other hand, the incremental longitudinal
  true strain is defined as
• For extended deformation, integration is
  required

(1.29)
(1.30)
24
(1.31)

• On substituting, we get

(1.32)
25
True Stress at Maximum Load and Eliminati
ng Pmax yields and
(1.29)
26

• True Fracture Strain
• The true fracture strain, ef , is the true
  strain based of the original area Ao and the
  area after fracture Af.
• For cylindrical tensile specimens the reduction
  of area q is related to the true fracture strain
  by the relationship

(1.30)
(1.31)