Section 4.2
Solid Mechanics Part I
Kelly
97
4.2 The Linear Elastic Model
4.2.1
The Response of Real Materials
The Tension Test
Consider the following key experiment, the
tensile test
, in which a small, usually
cylindrical, specimen is gripped and stretched, usually at some given rate of stretching.
The force required to hold the specimen at a given displacement/stretch is recorded, Fig.
4.2.1.
For many engineering materials, for example steel, rocks and concrete, it is found that the
force is proportional to displacement as with the portion
OA
in Fig. 4.2.1.
The following
observations will also be made:
(1)
if the material is unloaded, the force/displacement curve will trace back along the
line
OA
down to zero force and zero displacement; further loading and unloading
will again be up and down
OA
(2)
the forcedisplacement curve will be more or less the same regardless of the rate at
which the specimen is stretched (at least at moderate temperatures).
(3)
the loading curve remains linear up to a certain force level, the
yield point
or
elastic
limit
of the material (point
A
).
Beyond this point,
permanent deformations
are
induced; on unloading to zero force, the specimen will have a permanent elongation.
(4)
the strains up to the elastic limit are small
Figure 4.2.1: force/displacement curve for the tension test
StressStrain Curve
There are two definitions of stress used to describe the tension test.
First, there is the
force divided by the
original
cross sectional area of the specimen
0
A
; this is the
nominal
stress
or
engineering stress
,
0
A
F
n
=
σ
(4.2.1)
0
A
Force
Displacement
elastic
limit
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Section 4.2
Solid Mechanics Part I
Kelly
98
Alternatively, one can evaluate the force divided by the (smaller)
current
crosssectional
area
A
, leading to the
true stress
A
F
=
σ
(4.2.2)
in which
F
and
A
are both changing with time.
For small elongations, within the linear
range
OA
, the crosssectional area of the material undergoes negligible change and both
definitions of stress are more or less equivalent.
Similarly, one can describe the deformation in two alternative ways.
Denoting the
original specimen length by
0
l
and the current length by
l
, one has the
engineering strain
0
0
l
l
l
−
=
ε
(4.2.3)
Alternatively, the
true strain
accounts for the fact that the “original length” is continually
changing; a small change in length
dl
leads to a
strain increment
l
dl
d
/
=
ε
and the true
strain is defined as the accumulation of these increments:
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
=
∫
0
ln
0
l
l
l
dl
l
l
t
ε
(4.2.4)
The true strain is also called the
logarithmic strain
.
Again, at small deformations, the
difference between these two strain measures is negligible.
The stressstrain diagram for a tension test can now be described using the true
stress/strain or nominal stress/strain definitions, as in Fig. 4.2.2.
The stress value at the
elastic limit is called the
yield stress
Y
.
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 Fall '11
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