Plastic Deformation of Metals
The classical theory of plasticity is based on the concept of a yield criterion f(б ij )= k , where б ij is the stress tensor and k is a material constant, which is in general a function of the previous strain history. For an isotropic material, the funct
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The classical theory of plasticity is based on the concept of a yield criterion f(G;.J ):= k, where
6i.j
is the
stress tensor and k is a material constant, which is in general a function of the previous strain history. For an isotropic material, the function f may be expressed in terms of the three invariants 11
•
12 • 13
•
of the stress tensor. Experimentally, it is
found that the hydrostatic component of stress does not affect plastic flow, so that the stress may be replaced by the stress deviation
s."I. - cs"I .. - ..!..3 s a"i.. where ..
i)}
(1.1.7)
and where
K is
the bulk modulus. For a test nnder uniaxial tension 5, (1.1·7) becomes (1.1.8)
where
e is
the strain and
I'Jo- V3
k the uniaxial tensile yield
stress. The theory thus predicts that at constant plastic strain
8
Chap. 1: Plastic Deformation of Metals
rate the stress at a given strain will be a multiple of the static yield stress. It follows that the observed work-hardening rate d_ ~ ~~-
Gl~
r--
1-
I.:a..
1-'I-I-~---
I- ~~
t-"
I- f-'"
1-'
--
-r- ~
- ...4
~
1-'"'
E "0.15
I
~~
(-1,
10
-1--1-
s 0
,...-
G~
_p;-i--1-
~
1-- ~
'"
~
~
-
1-- P'"
1- ~ tr"
1- f.- I-
f-1~
-
-'-'
!A-
(. 0.10
I
~I-"'
~
.. t
00
jtl
~)
10
0
f:
1-
~~- ~
15
~
lA
~ 1-' ~~-
t.
p.
1-- i-
j..-oo-!!.....
A
Ui
_...
15
f:
00
294°K 399°K 533°K 672°K
~.
t
..
'()
[3o
-
-
0 20
00 ::.:: '-'
G-
5
d
-
,.!
rs.
,3;
~
-~
fl..
.A \""'
-~
t3-.
-
-
~
I--
1---:::: -r-
-6 .....
lil"'r"
E
10
Strain
-
IE,!.
= 0.05
l
rate, e(sec" 1 )
Fig. 10. Stress vs. strain-rate at constant temperature and strain. [4]
20
Chap. 1: Plastic Deformation of Metals In the dislocation model discussed above, the as-
sumption that the work done by the applied stress increases linearly with stress implies that the 'barrier shape' is rectangular. This assumption leads to a linear relation between the applied stress and the logarithm of the plastic strain rate, at constant temperature, or between the applied stress and temperature, at constant strain rate. If a different barrier shape is assumed, corresponding to a stress-dependent activation volume, these relations become non-linear. For B.C.C. metals, the mechanism governing the motion of dislocations is believed to be that corresponding to the overcoming of the Peierls-Nabarro force, which is due to the periodic stress field of the lattice itself. Experimental data (15], [16] show that the variation of flow stress with temperature and strain rate cannot be explained by a rate equation such as (1.2.5) unless the activation volume varies with temperature. This is indicated by the fact that for any barrier shape the theoretical curves of flow stress at constant temperature all converge towards the point
't = "tb,
'fp = NAb'17o, where
"tb
is the value of
stress at which the barrier can be surmounted in the absence of thermal activation; the experimental results, on the other hand, show either parallel or divergent curves (Figs.11,12,13, see pp. 21 and 22).
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