by McDowell, who introduced the concept of the cycle nonproportionality effect and ... 1. Cyclic strain paths. where N is the unit vector in the direction of the ...
Strength of Materials, Vol. 27, Nos. 5-6, 1995
DETERMINATION
OF CYCLE NONPROPORTIONALITY
COEFFICIENT M. V. Borodii
UDC 539.376
We propose a cycle nonproportionality coefficient for a broad class of complex cyclic paths, allowing us to establish a clear relationship between the geometry of the strain path and the maximum hardening level attainable in the material in the steady state for a fixed value of the maximum plastic or total strain range. We discuss the effectiveness of using the proposed cycle nonproportionality coefficient compared with analogous parameters familiar from the literature. Based on a previously developed version of the endochronic theory of plasticity, we have numerically modeled the behavior of a material for cyclic loading along different planar nonproportional paths.
Cyclic loadings encountered in technology in most cases are multiaxial and vary in a nonproportional manner. Accordingly, work devoted to investigation of the behavior of materials under complex cyclic loading conditions has become very important today. Appreciable progress has been made in the area of nonproportional cyclic plasticity due to a number of experimental investigations determining the important effects which arise under complex cyclic loading and often are not observed under simple loading of variable sign [1-4]. Many papers have been devoted to the study of the effect of the loading history, the amplitude and shape of curvilinear or piecewise-linear cyclic strain paths on the deformation properties and the useful life of construction materials. The major results of these investigations are summarized as follows. Under complex cyclic deformation, the material is hardened much more than for proportional deformation; in this case, the degree of hardening increases with an increase in the complexity of the path (for the same maximum deformation amplitude) and may be 1.5-3 times greater than for simpler deformation. At the same time, a significantly smaller change in the stress state often leads to a change in the useful life by an order of magnitude or more. Such appreciable strain hardening does not correlate with simple measures of the deformation such as the length of the arc of the strain path in a cycle, the length of the arc of plastic strain, the work of plastic deformation, etc. [5]. This circumstance has stimulated work studying the effect of the cycle geometry on the strain hardening. Two different approaches have been formulated for solution of this problem. According to the ftrst approach, the equations of state in the theory of plasticity should take into account the complexity of the loading process: the curvature of the path, the points of inflection and discontinuity, their effect on the deformation process [6]. The second approachproposes use ofanonproportionalitycoefficient, anintegral characteristic of the cycle geometry [7, 8]. One of the first attempts to characterize the shape of the cycle and to take into account the effect of the direction of loading in order to account for the additional hardening arising upon deformation along a circular path should be considered introduction of the rotation coefficient in [2]. Further development of investigations in this direction is connected with work by McDowell, who introduced the concept of the cycle nonproportionality effect and proposed a continuous (instantaneous) measure of the nonproportionality X as applied to the strain rate tensor [9] and subsequently to the plastic strain tensor [7]:
y i=
aeP
i fNf ,
(1)
Institute of Problems of Strength, National Academy of Science of Ukraine, Kiev. Translated from Problemy Prochnosti, Nos. 5-6, pp. 29-38, May-June, 1995. Original article submitted March 28, 1994. 0039-2316/95/2756-0265512.50
9
Plenum Publishing Corporation
265
TABLE 1. Values of the Nonproportionality Coefficients and Maximum Equivalent Stresses for Different Cyclic Plastic Strain Paths No. of cycle shape
Shape of cycle
e~, %
r [71
!"
O"
0,2
0
2
G
0,2
3
~
4 5"
~ (7)
calc
am
exp
a,m
,MPa
.,MPa
0
(11) 290
290
0,624
0,6366
417
410
0,2
0,926
0,810
458
44O
O
0,2
0,850
0,890
467
465:
Q
0,2
1
1
490
-i
490
*Here and in Table 2, the data from the baseline experiments are marked with an asterisk (*).
L,
Fig. 1. Cyclic strain paths. where N is the unit vector in the direction of the maximum plastic strain range; ~i and (f are the accumulated plastic strain at the beginning and at the end of the loading block. According to [7], the nonproportionality coefficient is defined by ',he expression (2)
Despite the fact that a nonproportionality coefficient in the form of (2) has been used successfully in [7, 10], some negative aspects of this parameter have been noted in [11]. Simultaneously the authors of [8], in analysis of the experimental results, proposed defining the degree of nonproportionality of the cycle as a quantity equal to the angle between the vectors for the rate of variation in the stresses and the rate of variation in the plastic strain: A -- 1 - cos20,
(3)
where
(11 11 I1 11)
(4)
is the projection of the component of the normalized vector eP in the direction perpendicular to the normalized vector s. Then this parameter was used in [ 12]. However, the idea remained attractive that the accuracy of the calculated values of the harden266
TABLE 2. Values of the Nonproportionality Coefficients and Maximum Equivalent Stresses for Different Cyclic Strain Paths No. of Shape of shape cycle cycle
em. %
(I) (7)
exp
0 0 0 0,0146
340 350 370 344
340 350 370 350
am
,
MPa (I 1)
a~ale
,
MPa
7
Q
0,5 0,53 0,585 0,5
8
Q
0,5
0,0397
350
360
9
Q
0,5
0,1925
386
390
10
Q
0,5
0,3354
420
420
11
Q
0,5
0,5760
478
490
12
O
0,5
0,89
554
560
13
G
0,5
0,6366
493
500
14
~
0,5
0,810
534
520
15
(~)
0,53
0,11
376
375
16
(~
0,53
0,823
547
545
17
Q
0,585
0,203
428
'l
0,5 0,53
1 1
580 590
580 590
I
0,585
1
66O
660
6" Q
18" Q
[4]
420
Note, For cycle shape No. 7, the number of steps is 128; for No. 8 - 32; for No. 9 - 4; for No. 10 - 2; for No. 11 - I; for No. 15, p = 0.25, r = 33~ for No. 17, p = 0 . 5 , r
= 3 3 ~.
ing level may be increased if the nonproportionality parameter is calculated on the basis of data on plastic deformation rather than on the stresses. Based on McDowell's work, Doong and Socie [13] developed a more complex expression for def'mition
of the nonproportionality coefficient: 2
2
~F(I left I)"
cycle
(5)
O=
~,F(I le~ I)"
1-
e'~
9
de~
aw,~
where F is the weighting function; dwP = (s.d@). Furthermore, the authors proposed an equation for evolution of the nonproportionality coefficient: dq)' = (q) - (~') ~ d~'=0
for
q) > O';
for
q)-.
