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Evolution of internal stress variables during cyclic deformation of copper
MaterialsScienceand Engineering, A128 (1990) 183-193
183
Evolution of Internal Stress Variables during Cyclic Deformation of Copper MUKESHJAIN Departmentof MechanicalEngineering, Universityof Manitoba, Winnipeg,Manitoba R3T 2N2 (Canada) (ReceivedJanuary 10, 1990)
Abstract
Continuum mechanical and microstructural theories of cyclic plasticity are analysed with regard to the development of internal stress variables and their techniques of measurement. Annealed polycrystalline commercial copper specimens are deformed in strain-controlled low cycle fatigue at various strain rates and strain amplitudes. The internal stresses are measured by unloading and reloading the specimens within the primary hysteresis loop by assuming that the strength of the material is composed of an isotropic stress component and a kinematic stress component. The evolution of internal stress variables as a function of strain rate, strain amplitude and number of cycles is determined. A number of features such as stress asymmetry in cyclic flow stress, back stress and the Bauschinger effect are observed and their dependence on strain rate and strain amplitude during cyclic hardening and saturation is determined. The results are discussed in terms of dislocation dynamics in the dislocation cell interior during fatigue. 1. Introduction
The concept of internal stresses has been employed both in viscoplastic models of continuum mechanics as well as in microscopic theories of deformation. In the continuum approach, the internal stresses are often based on the assumption of the existence of a yield surface. The two commonly used internal stress variables are kinematic back stress and isotropic drag stress [1]. These two quantities are chosen because of their usefulness when working with multiaxial states of stress and strain. The evolution of these variables along with flow law and kinetic equations constitute a complete set of equations defining a viscoplastic problem. In the microscopic approach, the two variables often chosen are microstructural 0921-5093/90/$3.50
back stress and effective stress. Although few predictive models based on the evolution of microstructural stress variables exist, the variables are empirically related to the microstructural quantities such as grain size, dislocation density, dislocation velocity and dislocation cell size. In the following, the continuum and microscopic variables are described with particular attention devoted to the terminology and the different methods of measurements of internal stress variables. Modification of a recent technique [2] utilized in the measurement of kinematic back stress and isotropic drag stress in the present work is described. The experimental results of these measurements during cyclic deformation at various strain amplitudes and strain rates are presented. The technique of measurement and the results are compared with the microstructural approaches of other workers. Qualitative similarities are discussed in relation to yield surface motion and dislocation dynamics within the cell interior during cyclic deformation.
1.1. Description of stress variables The physical basis of the continuum internal stress variables--kinematic back stress and isotropic drag stress--is macroscopic. No specific microstructural feature or quantity is usually emphasized with regard to the origin of these variables. Instead, existence of a yield surface (or elastic region) is hypothesized. The isotropic drag stress r accounts for isotropic expansion (or contraction) of the yield surface. Similarly, the translation of the yield surface (or kinematic hardening) is accounted for by kinematic back stress a. The shape of the yield surface is taken to be a sphere and the evolution of plastic state is considered in terms of the evolution of these two variables. In the uniaxial state, the centre and radius of the yield surface (or elastic region) are © Elsevier Sequoia/Printed in The Netherlands
184
compressive strain~ limit /
(tensile strain ~, limit /
( no strain) o-2 (-~yl
, (0~ =0"t ) deformation
(--0 i "0"c ) :
axis
( nc'Kc)--/-/ T
I
Ilsotmpic yield surface)
!
-~(
j
+a(
~'~'-'~-- (Or, Kt) I
z_ Expandedand translated yield surfaces
Fig. 1. Yield surface description of stress states in a symmetrical strain-controlled cyclic test: t, tension; c, compression. Ot = tzt + Kt; t r C = Ore + K c.
often taken as kinematic back stress and isotropic drag stress variables respectively [2, 3]. The isotropic stress variable associated with the plastic strain rate tensor has its end points lying on the yield surface (Fig. 1). The kinematic stress variable relates the centre of the yield surface, at a certain cyclic plastic strain, to the initial state of the material where the origin of the coordinate system is assumed. Therefore, in general, one can follow the evolution of two precisely defined stress variables which terminate at the centre and radius of a continuously evolving yield surface. The applied stress is taken to be the sum of kinematic back stress and isotropic drag stress. As the applied stress attains a peak tensile value, corresponding to the tensile strain limit, the yield surface expands and translates in the positive tensile direction. On stress reversal, it translates in the reverse direction and contracts to a smaller value during unloading to zero load. It expands again to the maximum value at the peak compressive stress corresponding to the compressive strain limit. This to-and-fro motion of the yield surface and its contraction and expansion within the loop bring the material to a steady state. In the steady state, the size of the yield surface becomes constant at any given position along the hysteresis loop. The microstructural variables--back stress and effective stress--are defined on the basis of dislocation behaviour during plastic deformation. Orowan [4] proposed that plastic strain rate is related to the mobile dislocation density Pm and velocity vm by a relationship of the form, ~p -- ~bPm'ffm, where ~ is a geometrical factor and b the Burgers vector of the dislocations. The empirical relationship between vm and the stress oe acting on a dislocation proposed by Johnston and Gilman [5] is expressed by v m= Bo~m, where
1 w ill
O"
n
0n
n
!o o .o I
Distance Fig. 2. A schematic representation of the microstructural internal stress distribution in a material with a well-developed cell structure. (From ref. 7.)
B and m are constants inherent to the material. The stress o e acting on the dislocations, commonly referred to as the effective stress, is rarely equivalent to the applied stress o and a longrange back stress, oh, resists the motion of dislocations (or frictional stress). The back stress o b opposing the movement of dislocations is generally referred to as the microstructural back stress but other terms such as internal stress and threshold stress have been mentioned in the literature [6]. In the present work, two variables--microstructural back stress and microstructural effective stress--are used based on their microstructural origin. These variables evolve during plastic deformation as various mobile dislocations, and immobile dislocation configurations such as tangles, forests and cells interact with one another by processes of multiplication, annihilation, immobilization and remobilization. A schematic diagram of the microstructural back stress and effective stress distribution in a material with a well-developed cell structure is shown in Fig. 2 [7].
185
2. M e t h o d s o f m e a s u r e m e n t o f internal state variables in cyclic state 2.1. Microstructural internal stresses
Different experimental techniques have been used for measuring the microstructural internal stresses during uniaxial cyclic deformation [8-13]. The measurements are usually made of only one of the two internal stress variables and the other is determined from the assumption that their sum is equal to the applied cyclic stress, i.e. a=a+
~c
(1)
One of the older techniques of measurement of microstructural effective stress involves short strain rate change tests within the primary hysteresis loop [8]. The strain rate is decreased by a certain fraction of the outer loop strain rate for a short time until a steady state value of the flow stress is achieved. The strain rate is again increased to the base value and the cyclic deformation is continued. The effective stress is determined by the following equation: (~b/~o)l/m* o ~ - . . . . . 1/m*-1 A o ~eb/en)
(2)
where Ao, eb and gn are the stress change during the strain rate jump, the base strain rate, and the jump strain rate respectively. The constant m* is taken to be the dislocation velocity-strain exponent. The microstructural back stress ob is determined by substituting the effective stress from the flow stress. However, the procedure involves considerable uncertainty in the determination of m*. A more commonly used technique for determining the microstructural back stress is to perform a stress relaxation test to exhaustion [9]. Since the plastic strain rate is directly proportional to the stress rate during a stress relaxation test, a zero stress rate corresponds to a zero plastic strain rate and the microstructural back stress is taken as the measured rate of exhaustion. A more recent method propOsed by Polak et al. [10] is the stress dip technique where no extrapolation to zero strain rate is needed. At the saturation point on the loop when the straining is reversed, the stress is decreased by a constant stress value and is held constant. During this time the strain vs. time dependence is followed and the average strain rate is evaluated. The plot of mean relaxation rate vs. stress decrement is obtained
and the critical stress drop for which zero strain rate results is evaluated. Long times are, however, required to obtain an accurate estimate of the back stresses, in the stress relaxation and stress dip tests. Another method for determining the microstructural back stress is the strain transient dip test [11]. In this method, the load on a creeping specimen is rapidly reduced and the subsequent strain rate behaviour is monitored. The microstructural back stress is equal to the reduced stress that results in a zero creep rate immediately after the load reduction. The technique is difficult to apply at room temperature owing to limited resolution at low strain rates. Also, there is recovery of the microstructure if the tests are conducted at high temperatures. In addition, a large number of tests are required to determine the evolution of internal stresses as a function of number of cycles. A general drawback of the stress relaxation, stress dip technique and strain transient dip tests is the difficulty in the interpretation of the transient strain rate behaviour. A relatively simple technique for the measurement of microstructural back stress and effective stress which consists of analysing the recorded cyclic stress-strain hysteresis loops is the method of Kuhlmann-Wilsdorf and Laird (KWL) [12]. At the start of plastic tensile deformation the back stress which was generated in the preceding compressive half-cycle acts in the same direction as the applied tensile stress. Therefore in the presence of the effective stress, as, can be written as as, = Oe,- Obo
(3)
and is shown in Fig. 3. At the end of the forward O"
/ ~
/ Compression
% "
~"
,,apt
_
•
%t + ~ c
l
|
"1
Fig. 3. Schematic diagram of a typical cyclic stress-strain hysteresis loop.
186
cycle, the microstructural back stress Oh, generated in this tensile deformation will be in a direction opposite to the tensile deformation (the subsubscripts t and c refer to tension and compression respectively). Hence, the maximum applied tensile stress Op, becomes ap = oe,+ Ob,
(4)
Similarly, for the compressive half-cycle the appropriate equations are osc = aeo- oh,
(5)
and ape = o~o+ Oh0
(6)
Equations (3)-(6) under the conditions of symmetry of stresses in tension and compression (i.e. Oe=Oeo=oe; O'bt=~be=O'b; ~¢pt=O'pc = Otp; as, = as0 = as) yield Oe= (Op + as)/2
(7)
a b = (Op - os)/2
(8)
The above technique, although utilized by many in recent years, however, remains unsuitable for the measurement of internal stresses within the hysteresis loop. In addition, the assumptions with regard to the symmetry will not hold for the materials exhibiting the Bauschinger effect. Also, as pointed out by Dickson et al. [13], no account is taken of the stress relaxation at the tensile and compressive peak stresses. Dickson et al. [13] suggested a variation of the previous technique involving a stress dip to zero stress relaxation without performing the actual relaxation. The portion of the hysteresis loop where the slope agrees with the experimental elastic modulus is determined by taking the midpoint of this portion and then measuring the effective stress as equal to the difference between the preceding peak stress and the midpoint of the elastic region. This technique, however, accounts for the stress relaxation at the tensile and compressive strain limits but suffers from the other limitations of the KWL [12] method. TABLE 1
2.2. Measurement techniques for continuum mechanical stress variables Several different techniques have been employed for the measurement of kinematic back stress and isotropic drag stress during cyclic deformation [2, 3, 14]. Cescotto and Leckie [14] have utilized creep tests at several stress levels, monotonic tension tests at high strain rates and the stable hysteresis loops from completely reversed cyclic tests at several strain amplitudes to determine the changes in kinematic back stress and isotropic drag stress. A large number of tests are required to determine the evolution of a and r. Controlled unloading within the primary loop has been recently utilized to determine the elastic range in real time [3, 15]. The technique, first suggested by Onat [3], involves measurement of yield strengths in tension and compression by unloading the specimen at predetermined strain points within the primary hysteresis loop. At a specified strain offset during elastic unloading, as determined in real time, the loading is stopped and the material is reloaded (Fig. 4). In this way, small load reversals are used to locate continuously the elastic range of a material during cyclic plastic deformation. The kinematic back stress and isotropic drag stress are then taken as the centre and radius of the elastic range. This method overcomes the limitations of the techo-
I Fig. 4. Schematic of unloading-reloading procedure within the primary hysteresis loop.
As-received chemical composition, mechanical properties and grain size of copper specimens
Composition (wt.%)
0.2% initial oy
Fe
Zn
Pb
Sn
P
Cu
(MPa)
0.002
0.0027
0.0024
<0.01
0.0008
>99.98
25.26
Ours (MPa)
(/tin)
218
55
Grain size
187
niques mentioned earlier. A single specimen provides accurate and consistent data on the evolution of kinematic back stress and isotropic drag stress variables as a function of number of cycles. A modification of the method [3, 15] was utilized in the present experiments and is discussed in more detail in the next section.
3. Experimental conditions 3.1. Specimen The chemical composition and initial mechanical properties of the as received commercially pure (99.99%) polycrystalline copper used in the present work are shown in Table 1. The material was received in the form of bars 12.7 mm in diameter. The test specimens were machined out from the bars according to ref. 16. The specimens were cylindrical with a uniform reduced test section of 25.5 mm length and 6.35 mm in diameter. The specimens were annealed in a muffle furnace in air atmosphere at a temperature of 375 + 2 °C for 1 h. As-received-and-annealed specimens were cold mounted, polished, etched and observed by optical microscopy. Also, foils were made and observed by transmission electron microscopy (TEM) to ensure complete recrystallization of grains. The resulting average grain size, determined by the linear intercept method of Smith and Guttman [17], was found to be 55 pm. The specimens were mechanically polished with 0.3 p m alumina particles suspended in water until bright smooth gauge sections free' of any machining marks were obtained. 3.2. Measurement of kinematic back stress and isotropic drag stress To evaluate the kinematic back stresses and isotropic drag stresses during cyclic straining, high precision is required in the recording of the hysteresis loops and in the measurement of the elastic modulus (experimental) applicable to each unloading-reloading loop within a specific hysteresis loop. This was accomplished by a computerized materials testing system. The specimens were subjected to strain-controlled uniaxial cyclic loading in a microprocessor-based closed-loop servo-controlled electrohydraulic material test machine of 20 kin s-1 capacity. An IBM PC AT was utilized for the real-time control of the test machine as well as for data acquisition, storage and analysis. The system controller con-
stantly compared the command from the computer with the feedback from the transducer and provided a correctional signal to a servo valve, permitting a precisely controlled test to match the desired command. The computer interface allowed close monitoring of the loading, unloading and reloading within a particular hysteresis loop. The experimental procedure for the measurement of state variables--kinematic back stress and isotropic drag stress--was as follows. The procedure involved measuring the dynamic yield stress Y~ in tension and the dynamic yield stress Yc in compression at various predetermined strain levels within a completely reversed uniaxial cyclic stress-strain loop after a certain number of specified cycles. At a predetermined strain, the cyclic strain history was interrupted and the specimen was unloaded until the material yielded in the reverse direction. In order for subsequent measurements on the same specimen to be valid, it was imperative that the unloading be stopped immediately after the yield criterion was reached. Thus the determination of reverse yielding Ychad to be made in real time. This was possible with the data acquisition and analysis system attached to the testing machine. As the unloading began, the computer determined the elastic unloading line. The elastic modulus was calculated at the start of the test using the least-squares procedure and the stressstrain data from the initial loading. As the unloading continued, the computer calculated the strain offset from the elastic line for pairs of stress-strain values collected during the unloading. When the critical strain offset of 0.05% was met or exceeded, the unloading was stopped and the specimen reloaded until the 0.05% offset yield strength was once again reached. The cyclic strain history was then resumed. The above procedure was repeated at the next strain where the yield stress was to be measured. The kinematic back stress and isotropic drag stress variables were measured at the end of each strain increment. For uniaxial loading, kinematic back stress is simply given by ( Yt + Yc)/2 while isotropic drag stress is equal to ( Yt - Yc)/2. The test history of any specimen consisted primarily of three basic sets as shown schematically in Fig. 5. The first set consisted of the very first cycle of tensile straining and compressive reversal (Fig. 5(a)). The small unloading-reloading loops within the first cycle indicate the addi-
188
-
I
-0.005 -0.003
0.0
ODO~
0.005
Strain (a)
I00
50 Stress (MPo) o
~:/~
//////7//
//[[///I///I/
-50 -IOC I
I
-o Dos- 0.003
I
0.0
I
0.003 0.005
I I -0.0(35-0.003
0.0
Strain
Strain
(b)
(c)
I t 0.003 0 . 0 0 5
Fig. 5. Experimental test history showing unloading-reloading loops used for the measurement of kinematic back stress and isotropic drag stress: (a) first tensile quarter-cycle and first compressive half-cycle; (b) completely reversed second or a subsequent loop where the measurements were made; (c) completely reversed loop without measurements.
TABLE 2 tests
Summary of experimental conditions for cyclic
Strain amplitude Ae/2
(s- 1)
Strain rate ~
Number of cycles N
0.003 0.003 0.003 0.003 0.005 0.005 0.005
0.005 0.01 0.01 0.1 0.005 0.01 0.1
509 20 515 515 509 34 509
tional path required to estimate the yield stresses in tension and compression at the unloading point. Subsequent cycles were divided into two sets depending on whether or not the measurements of yield stress were made within the hysteresis loop (Figs. 5(b) and 5(c)). The tensile part of the first block of history, which was smaller, was divided into five increments. The strain range for subsequent measurement cycles was divided into seven increments. Measurements of Yt and Y~ were made on the first four cycles of the history and thereafter about ten measurements were made in the fatigue-hardening region and saturation region. The test was interrupted in
some cases during cyclic hardening or saturation and the specimen was removed to observe the dislocation struciures by T E M [18]. The entire test was conducted at a constant total strain amplitude and a constant strain rate with the exception of a usually fast strain rate (~=0.2 s -1) for the small unloading-reloading loops within a completely reversed large loop. The loading waveform was a fully reversed triangular wave for the large outer loop. 4. Results
4.1. Cycle-to-cycle data Table 2 summarizes the various specimen details, and experimental conditions. The specimens were cycled in the range 1-515 cycles at strain amplitudes Ae/2 of 0.3% and 0.5% and strain rates of 0.005, 0.01 and 0.1 s -1. All the specimens exhibited initial hardening and a tendency to saturation during which the hardening rate approached zero. Figure 6 shows the experimental plot of loading-unloading loops within the cyclic strain interval at a constant total strain amplitude Ae/2 of 0.3% at different numbers of cycles (Figs. 6(a)-(c)). The stress ampli-
189 200 I00 Stress
-
0
'¢'~'~" ~ ' ~ "
-100 -
200 I I - 0 . 0 0 4 - 0 . 0 0 2 0.0 0 0 0 2 0004 Stroin
(a) Cycle No. 2
200
200
I00 Stress (MPa)
o
IOO -
~
-
~//
-,oo
-I00
-200 I I - 0 . 0 0 4 - 0 0 0 2 0.0 0.002 0.004 Strain ( b ) Cycle No.6
- 200
I
1
- 0 . 0 0 4 - 0 0 0 2 0.0 0.002 0 . 0 0 4
Strain (c) Cycle No. II
Fig. 6. Experimental plot of tensile half-cycle at three different cycle numbers for a specimen cycled at a strain amplitude of 0.006 and a strain rate of 0.1 s- '.
tude increases with increasing number of cycles (strain hardening) as the loops become more "pointed" at higher cycles. The general shape of the internal stress variables a and r during cyclic deformation is established in the first few cycles and then it changes only slightly to reach saturated values. This is shown in Fig. 7 which presents a plot of tensile peak flow stress, kinematic back stress and isotropic drag stress vs. number of cycles for copper specimens subjected to cyclic straining under test conditions shown in Table 2. The stresses increase non-linearly with increasing number of cycles until saturation is observed, beyond which the shapes and sizes of the loops remain constant. Complete saturation is reached between 50-100 cycles for all the cases (Figs. 7(a)-(f)). In addition, the saturation is reached simultaneously in all the measured stresses, suggesting that saturation cycling is essentially a steady state. The increase in the stresses during cyclic hardening are associated with the formation and growth of dislocation tangles and dislocation cells [18]. The effect of strain rate on kinematic back stress is more pronounced than on isotropic drag stress. The isotropic drag stress is, however, the predominant
component of the flow stress at the strain rates and strain amplitudes utilized in the experiments. Figure 8 shows plots of experimental tensile peak flow stress oexp and calculated tensile peak flow stress trea~ taken as a sum of corresponding kinematic back stress and isotropic drag stress as explained in Section 2.1 and establishes the validity of eqn. (1). Figures 9 and 10 show the plots of kinematic back stress at and isotropic drag stress Kt in tension and kinematic back stress a c and isotropic drag stress r c in compression vs. number of cycles to determine the stress asymmetry during cyclic hardening and saturation for specimens cycled to saturation. In Figs. 9 and 10, the stresses in compression are plotted on the tensile axis for comparison. No significant stress asymmetry was observed in isotropic drag stress r between tension and compression (Fig. 9). The kinematic back stress a, however, tended to be about 10 MPa higher in tension than in compression (Fig. 10). 4.2. Within-cycle data
Figure 11 shows the experimental stress-strain curve for the first quarter-cycle in tension and first compressive reversal for a specimen cycled
190 2OO
2:0C ~-~ =0.003 ~~ = O.I/s
150 ~ I
~
fO0 50 0
o"
15C
A
g
K
~
50
I i I i I IO0 300 500 Number of Cycles (o)
fso ? o--
. t2
0 0
I i I i I IO0 300 500 Number of Cycles (d)
AE T " 0.002 ;{ = O.OI/s
u) (n
150
(1)
100
#
50
2001 I - ~ = 0.005; ~ =O.O05/s 150~,¢-- :
:
o"
4)-
(I
a) O0
I J I I [ I00 300 500 Number of Cycles (b)
0~ 0
I I I I I I00 300 500 Number of Cycles (e)
=0.01/s "•=0.005;
- ~ =0.005; ~ =O.I/s
150 I00
lOOt
50
50~
0 0
I I I I I I00 300 500 Number of Cycles (c)
u
0
*
150
3
1
50 d
# O'col (: a + K)
O~
o O'ex p I i I i I I00 300 500 Number of Cycles (o)
0 ¢D t. .6.. cO 4) Q.
200
0
0
0
ZOO
- O.005;~, • -
150
150 A
I00
I00
50
50
I
i
I
I
a00
I
500
Number of Cycles (b)
[
I , I , I I00 300 500 Number of Cycles (c)
0
- ~ - O.003; ~ = O.O05/s
0 f00
,0.005; ~ • O.OOS/s
I
I 500 Number of Cycles (d)
0 100
I
I
O. I / s
I
300
Fig. 8. Tensile peak flow stresses (experimental) and as a sum of corresponding kinematic back stresses, and isotropic drag stresses (calculated) at different strain amplitudes and strain rates. The inset in (a) shows the corresponding symbols.
200
~00
~
200 [ y
- O.OI/s
I OOI A o Q.
:¢ 200
-,~- 0.003;
o"
lOG
K
~.-~a.-- a
0
200 - ~ =0.003 ; ~ = O.O05/s
0-G
j),._
OL~ I J I , I 0 I00 300 5OO Number of Cycles (f)
Fig. 7. Tensile peak stresses: flow stress, kinematic back stress and isotropic drag stress v s . number of cycles for specim e n s cycled at different amplitudes and strain rates.
at a strain rate of 0.1 s-x and a strain amplitude Ae/2 of 0.003. The stresses reach their maximum values at maximum applied strain, thus acting to lower the yield stress in the reverse direction. Figures 12(a) and 12(b) shows plots of o, a and r vs. Ae for the above specimen during cyclic hardening (second cycle) and saturation (515th cycle) respectively. The isotropic drag stress curves remain relatively "flat" at strain limits in the earlier stage of hardening within the cyclic loop (Fig. 12(a)). The back stress, on the contrary, is dependent upon the strain within the loop. As the cells develop and become well defined, the drag stress shows slight variation within the loop at higher cycles (Fig. 12(b)). There is considerable variation in back stress, within the cycle, at high cycles. The internal variables attain a maxima at the maximum peak stress.
5. Discussion The method of measurement and the results of cyclic tests can be discussed in relation to KWL [12] method if one assumes that the peak tensile and compressive stress values in a particular loop are of the same magnitude. In fact, the kinematic back stress and isotropic drag stress measurements at the tensile and compressive strain limits on comparison with the measurements carried out in ref. 12 reveal equivalent results (Fig. 13). The microstructural back stress Ob and the kinematic back stress a are equal and given by the expression Orb= O~ = ( O p t - -
oso)/2
(9)
Similarly, the microstructural effective stress and isotropic drag stress are given by oe = K =(Op, + Osc)/2
(10)
The results imply that the yield surface expands and contracts as the effective stresses increase and decrease within the primary hysteresis loop. Similarly, the yield surface translates to and fro by an amount given by the microstructural back stress which changes sign as the direction of straining is reversed. The shape of the a
191
200 150 --
.
r--T
,,
50
; 0
T=Temperoture C :Compression I i I i I I00 300 500 Number of Cycles (o)
¢_2oo~
20
50;
I , I I I I00 300 500 Number of Cycles Id)
0i
0
J~
u
o
t~-
~
t
~
~-T,_
O~ 0
1 I I i I iO0 300 500 Number of Cycles (d)
60t
40
o m
r':
Lc-0-
6o
T
40-
T=Temp eroture C: Compression ,
I t I t I I00 300 500 Number of O/des (a)
0
o fl_ :~
:L
150
~
f-T
I--T
I00
I00
60
40~n~
150 ^
0
6o
200
~ ~
f-T tc
20
50
4o 20 1
_ t--C
11.
? n
0
t~_~,
,
,
I00 300 500 Number of Cycles (b)
02 0
200
20°
150
150
I t I I I I00 300 5OO Number of Cycles (e)
0
. 0
I i I J I K)O 300 50O Number of Cycles (b)
60
60
r--T 40
..__c_J_ I00
I00
5O
5O
,~
I i I I I I00 300 500 Number of Cycles (c)
0
0
,.~..~,
T
C
20
0
O~L t R I , I 0 I00 300 500 Number of Cycles (e)
I t i t I I0 20 30 4 0 50 60 Number of Cycles (f)
O~ 0
2O
1 I I , I I00 30O 5O0 Number of Cycles (c)
0
0
I i t I I I0 20 30 40 50 6 0 Number of Cycles (f)
Fig. 9. Isotropic drag stress asymmetry vs. number of cycles for specimens cycled at different strain amplitudes and strain rates: T, tension; C, compression.
Fig. 10. Kinematic back stress asymmetry vs. number of cycles for specimens cycled at different strain amplitudes and strain rates: T, tension; C, compression.
number of cycles data (Fig. 7) in the present work is in agreement with the results of KWL [12] for cyclic deformation of single crystals of copper as well as recent results obtained by Dickson e t al. [13] and Wang and Laird [19] on cyclic deformation of polycrystalline copper. The present experiments as well as the data of the above workers indicates a rapid increase in internal stress variables during the period of cyclic hardening followed by a saturation in the values. Figure 14 shows a schematic diagram of flow stress and internal stresses within the stress-strain hysteresis loop based upon the experimental data. The diagram shows a higher stress value for the applied stress and kinematic back stress in tension compared with compression. The isotropic drag stress value is about the same in tension and compression. The microstructural back stresses are at least a quarter of
the applied stresses. A characteristic dislocation movement within the hysteresis cycle results in the above internal stress field. More specifically, the isotropic drag stress, being symmetric in tension and compression, could be due to the presence of well-defined dislocation walls in cellular configurations containing a high density of immobile dislocations. Asymmetric back stresses similarly arise from the anisotropy in the pile-up characteristics of mobile dislocations in the cell interior. In addition, a stronger dependence of strain rate on back stress compared with drag stress is also indicative of its direct relationship to the mobile dislocation density. A welldefined dislocation cell structure was observed during later stages of hardening and saturation for all the experimental conditions. Cyclic evolution of a and r are related directly to the parameters describing the dislocation structure such as
a n d u vs.
192 o"
IOO 50-
Yt ,OP t
Stress (MPa)
0 - 50 -
- too
I
I
E
- 0.004 -0.002 0.0 0.002 0.004 Strain Interval (a) Yc , O'st I00 50
Stress
(MPo)
yc ,O'pc
-
_j
0
Fig. 13. Comparison of the technique of measurement of kinematic back stress and isotropic drag stress in the present work with the KWL [12] method.
-50 o - I00
I
I
I
-0.004-0.0(~ 0.0 Q002 0.004 Strain I n t e r v a l (b)
Fig. 11. Stress-strain data for (a) the first quarter-cycle in tension and (b) the first compressive reversal, for specimen cycled at a strain rate of 0.1 s -~ and a strain amplitude of 0.006.
i
~
~,,..........--~Crt Applied stress (0"]
y
Isotropic drag
~-S~:;~--ac KC
~
Stress
(MPo)
stress (K)
Qt Kinematicbackstress (~)
O" = Q+K t
AE -
-
at > Qc Kt = Kc
K
Fig. 14. Schematic representation of the cyclic flow stress and internal stresses within the hysteresis loop.
o
-50 i -I00 I
I
-O.OOS-0.003 0 0.003 0.005 Sfrain (0)
~$tress
I -O.OOS-0.003 0 0.003 0.005 Strain
(b) Fig. 12. Tensile flow stress, back stress and drag stress v s . strain for specimens cyclically strained at a strain rate of 0.1 s -l and a strain amplitude of 0.006: (a) second cycle; (b) 515th cycle.
mobile dislocation density in the ceil interior, immobile dislocation density in the cell walls, average cell diameter and average ceil width [18]. A stress asymmetry was observed in the experimental plots of a t a n d a c vs. number of cycles (Fig. 10). Identical stress asymmetry was found in the applied cyclic stress in tension and in compression. No significant asymmetry existed in the isotropic drag stress component (Fig. 9). The results therefore indicate that the asymmetry in the applied flow stress of the cycle is due to the kinematic back-stress component. A Bauschinger effect in the flow stress of copper has been observed by several workers [20-22]. However, no such observation has been made for the backstress component of the flow stress in the past. The results can be rationalized on a microstructural basis as follows. If it is assumed that the cell walls harden more rapidly than the cell interior
193
during cyclic hardening, then on unloading from a tensile cycle the cell interior will be placed further into compression than the cell walls compared with the case when the cell walls and cell interior harden at the same rate. The large residual compressive stress in the cell interior would produce a smaller value of kinematic back stress. Steady state values of a, a and r depend upon the total strain amplitude although the values, particularly a, increase with increasing strain rate. Asymmetry in the flow stress and kinematic back stress also increased slightly with strain rate. The relative contribution of kinematic back stress to the applied stress also increased slightly with increasing strain rate. This is perhaps due to a large width of dislocations in f.c.c, metals and the consequent small effect of thermal activation in the glide of dislocations at room temperature.
Mechanical Engineering, Washington University, St. Louis) for his help in conducting the cyclic tests and many suggestions. Appreciation is expressed to the Department of Mechanical Engineering, Washington University, for providing a fellowship during my stay at the University. References
5 6 7
6. Conclusions (1) Kinematic back stress and isotropic drag stress increase rapidly during cyclic hardening and approach saturation at large cycles. (2) A Bauschinger effect exists in the kinematic back stress and cyclic flow stress within the cycle and from cycle to cycle. The magnitude of the Bauschinger effect depends upon the strain rate and strain amplitude. (3) Continuum internal variables as measured by the controlled unloading-reloading method used in the present work and the microstructural variable measurement by the KWL [12] method yield values of the same order of magnitude and a similar shape of evolution of the stresses. (4) The nature asymmetry in a and a (the Bauschinger effect) can be explained on the basis of dislocation dynamics in the cell interior.
8
9 10 11
12 13 14
15 16 17 18 19 20
Acknowledgments I am indebted to Dr. E. J. Tuegel, Material and Process Development, McDonnell Aircraft Company, St. Louis (formerly with Department of
21 22
E. Krempl, Acta Mech., 69 (1987) 25. E. J. Tuegel, Res Mech., 22 (1987) 65. E. T. Onat, Rep. ORNL-SUB-3863-2, November 1976 (Yale University) (Oak Ridge National Laboratory Contract). E. Orowan, in G. Rassweiler and W. Grube (eds.), Internal Stresses and Fatigue in Metals, Elsevier, Amsterdam, 1959, p. 59. W. G. Johnston and J. J. Gilman, J. Appl. Phys., 30 1959) 129. J. C. Gibeling and W. D. Nix, Mater. Sci. Eng., 45 1980) 123. H. Conrad, J. Met., 16 (1964) 582. J. T. Michalak, Acta Metall., 13 ( 1965) 213. J. C. Tsou and D. J. Quesnel, Mater. Sci. Eng., 56 1982) 289. J. Polak, M. Klesnil and J. Helesic, Kovovd Mater., 17 (1979) 476. W. B. Jones, R. W. Rohde and J. C. Swearengen, in R. W. Rohde and J. C. Swearengen (eds.), Mechanical Testing for Deformation Model Development, ASTM Spec. Tech. Publ. 765, 1982, p. 301. D. Kuhlmann-Wilsdorf and C. Laird, Mater. Sci. Eng., 37 (1979) 111. J. I. Dickson, J. Boutin and L. Handfield, Mater. Sci. Eng., 64 (1984) L7. S. Cescotto and F. A. Leckie, in C. S. Desai and R. H. Gallagher (eds.), Proc. Int. Conf. on Constitutive Laws for Engineering Materials: Theory and Application, Tucson, AZ, 1983, Elsevier, New York, 1983, p. 105. T. C. Lowe and A. K. Miller, Scr. Metall., 17 (1983) 1177. ASTM Stand. E466-82, 3.01 (1986) 571. C. S. Smith and L. Guttman, J. Met., 5 (1953) 81. M. Jain, submitted to J. Mater. Res. Z. Wang and C. Laird, Mater. Sci. Eng., 101 (1988) L 1. I. Gokyu and T. Kishi, Proc. 13th Jpn. Congr. on Materials Research--Metallic Materials, 1970, Japan Institute of Metals, Tokyo, 1970, p. 125. R. C. Daniel and G. T. Home, Metall. Trans., 2 (1971) 1161. O. B. Pedersen, L. M. Brown and W. M. Stobbs, Acta Metall., 29(1981) 1843.
183
Evolution of Internal Stress Variables during Cyclic Deformation of Copper MUKESHJAIN Departmentof MechanicalEngineering, Universityof Manitoba, Winnipeg,Manitoba R3T 2N2 (Canada) (ReceivedJanuary 10, 1990)
Abstract
Continuum mechanical and microstructural theories of cyclic plasticity are analysed with regard to the development of internal stress variables and their techniques of measurement. Annealed polycrystalline commercial copper specimens are deformed in strain-controlled low cycle fatigue at various strain rates and strain amplitudes. The internal stresses are measured by unloading and reloading the specimens within the primary hysteresis loop by assuming that the strength of the material is composed of an isotropic stress component and a kinematic stress component. The evolution of internal stress variables as a function of strain rate, strain amplitude and number of cycles is determined. A number of features such as stress asymmetry in cyclic flow stress, back stress and the Bauschinger effect are observed and their dependence on strain rate and strain amplitude during cyclic hardening and saturation is determined. The results are discussed in terms of dislocation dynamics in the dislocation cell interior during fatigue. 1. Introduction
The concept of internal stresses has been employed both in viscoplastic models of continuum mechanics as well as in microscopic theories of deformation. In the continuum approach, the internal stresses are often based on the assumption of the existence of a yield surface. The two commonly used internal stress variables are kinematic back stress and isotropic drag stress [1]. These two quantities are chosen because of their usefulness when working with multiaxial states of stress and strain. The evolution of these variables along with flow law and kinetic equations constitute a complete set of equations defining a viscoplastic problem. In the microscopic approach, the two variables often chosen are microstructural 0921-5093/90/$3.50
back stress and effective stress. Although few predictive models based on the evolution of microstructural stress variables exist, the variables are empirically related to the microstructural quantities such as grain size, dislocation density, dislocation velocity and dislocation cell size. In the following, the continuum and microscopic variables are described with particular attention devoted to the terminology and the different methods of measurements of internal stress variables. Modification of a recent technique [2] utilized in the measurement of kinematic back stress and isotropic drag stress in the present work is described. The experimental results of these measurements during cyclic deformation at various strain amplitudes and strain rates are presented. The technique of measurement and the results are compared with the microstructural approaches of other workers. Qualitative similarities are discussed in relation to yield surface motion and dislocation dynamics within the cell interior during cyclic deformation.
1.1. Description of stress variables The physical basis of the continuum internal stress variables--kinematic back stress and isotropic drag stress--is macroscopic. No specific microstructural feature or quantity is usually emphasized with regard to the origin of these variables. Instead, existence of a yield surface (or elastic region) is hypothesized. The isotropic drag stress r accounts for isotropic expansion (or contraction) of the yield surface. Similarly, the translation of the yield surface (or kinematic hardening) is accounted for by kinematic back stress a. The shape of the yield surface is taken to be a sphere and the evolution of plastic state is considered in terms of the evolution of these two variables. In the uniaxial state, the centre and radius of the yield surface (or elastic region) are © Elsevier Sequoia/Printed in The Netherlands
184
compressive strain~ limit /
(tensile strain ~, limit /
( no strain) o-2 (-~yl
, (0~ =0"t ) deformation
(--0 i "0"c ) :
axis
( nc'Kc)--/-/ T
I
Ilsotmpic yield surface)
!
-~(
j
+a(
~'~'-'~-- (Or, Kt) I
z_ Expandedand translated yield surfaces
Fig. 1. Yield surface description of stress states in a symmetrical strain-controlled cyclic test: t, tension; c, compression. Ot = tzt + Kt; t r C = Ore + K c.
often taken as kinematic back stress and isotropic drag stress variables respectively [2, 3]. The isotropic stress variable associated with the plastic strain rate tensor has its end points lying on the yield surface (Fig. 1). The kinematic stress variable relates the centre of the yield surface, at a certain cyclic plastic strain, to the initial state of the material where the origin of the coordinate system is assumed. Therefore, in general, one can follow the evolution of two precisely defined stress variables which terminate at the centre and radius of a continuously evolving yield surface. The applied stress is taken to be the sum of kinematic back stress and isotropic drag stress. As the applied stress attains a peak tensile value, corresponding to the tensile strain limit, the yield surface expands and translates in the positive tensile direction. On stress reversal, it translates in the reverse direction and contracts to a smaller value during unloading to zero load. It expands again to the maximum value at the peak compressive stress corresponding to the compressive strain limit. This to-and-fro motion of the yield surface and its contraction and expansion within the loop bring the material to a steady state. In the steady state, the size of the yield surface becomes constant at any given position along the hysteresis loop. The microstructural variables--back stress and effective stress--are defined on the basis of dislocation behaviour during plastic deformation. Orowan [4] proposed that plastic strain rate is related to the mobile dislocation density Pm and velocity vm by a relationship of the form, ~p -- ~bPm'ffm, where ~ is a geometrical factor and b the Burgers vector of the dislocations. The empirical relationship between vm and the stress oe acting on a dislocation proposed by Johnston and Gilman [5] is expressed by v m= Bo~m, where
1 w ill
O"
n
0n
n
!o o .o I
Distance Fig. 2. A schematic representation of the microstructural internal stress distribution in a material with a well-developed cell structure. (From ref. 7.)
B and m are constants inherent to the material. The stress o e acting on the dislocations, commonly referred to as the effective stress, is rarely equivalent to the applied stress o and a longrange back stress, oh, resists the motion of dislocations (or frictional stress). The back stress o b opposing the movement of dislocations is generally referred to as the microstructural back stress but other terms such as internal stress and threshold stress have been mentioned in the literature [6]. In the present work, two variables--microstructural back stress and microstructural effective stress--are used based on their microstructural origin. These variables evolve during plastic deformation as various mobile dislocations, and immobile dislocation configurations such as tangles, forests and cells interact with one another by processes of multiplication, annihilation, immobilization and remobilization. A schematic diagram of the microstructural back stress and effective stress distribution in a material with a well-developed cell structure is shown in Fig. 2 [7].
185
2. M e t h o d s o f m e a s u r e m e n t o f internal state variables in cyclic state 2.1. Microstructural internal stresses
Different experimental techniques have been used for measuring the microstructural internal stresses during uniaxial cyclic deformation [8-13]. The measurements are usually made of only one of the two internal stress variables and the other is determined from the assumption that their sum is equal to the applied cyclic stress, i.e. a=a+
~c
(1)
One of the older techniques of measurement of microstructural effective stress involves short strain rate change tests within the primary hysteresis loop [8]. The strain rate is decreased by a certain fraction of the outer loop strain rate for a short time until a steady state value of the flow stress is achieved. The strain rate is again increased to the base value and the cyclic deformation is continued. The effective stress is determined by the following equation: (~b/~o)l/m* o ~ - . . . . . 1/m*-1 A o ~eb/en)
(2)
where Ao, eb and gn are the stress change during the strain rate jump, the base strain rate, and the jump strain rate respectively. The constant m* is taken to be the dislocation velocity-strain exponent. The microstructural back stress ob is determined by substituting the effective stress from the flow stress. However, the procedure involves considerable uncertainty in the determination of m*. A more commonly used technique for determining the microstructural back stress is to perform a stress relaxation test to exhaustion [9]. Since the plastic strain rate is directly proportional to the stress rate during a stress relaxation test, a zero stress rate corresponds to a zero plastic strain rate and the microstructural back stress is taken as the measured rate of exhaustion. A more recent method propOsed by Polak et al. [10] is the stress dip technique where no extrapolation to zero strain rate is needed. At the saturation point on the loop when the straining is reversed, the stress is decreased by a constant stress value and is held constant. During this time the strain vs. time dependence is followed and the average strain rate is evaluated. The plot of mean relaxation rate vs. stress decrement is obtained
and the critical stress drop for which zero strain rate results is evaluated. Long times are, however, required to obtain an accurate estimate of the back stresses, in the stress relaxation and stress dip tests. Another method for determining the microstructural back stress is the strain transient dip test [11]. In this method, the load on a creeping specimen is rapidly reduced and the subsequent strain rate behaviour is monitored. The microstructural back stress is equal to the reduced stress that results in a zero creep rate immediately after the load reduction. The technique is difficult to apply at room temperature owing to limited resolution at low strain rates. Also, there is recovery of the microstructure if the tests are conducted at high temperatures. In addition, a large number of tests are required to determine the evolution of internal stresses as a function of number of cycles. A general drawback of the stress relaxation, stress dip technique and strain transient dip tests is the difficulty in the interpretation of the transient strain rate behaviour. A relatively simple technique for the measurement of microstructural back stress and effective stress which consists of analysing the recorded cyclic stress-strain hysteresis loops is the method of Kuhlmann-Wilsdorf and Laird (KWL) [12]. At the start of plastic tensile deformation the back stress which was generated in the preceding compressive half-cycle acts in the same direction as the applied tensile stress. Therefore in the presence of the effective stress, as, can be written as as, = Oe,- Obo
(3)
and is shown in Fig. 3. At the end of the forward O"
/ ~
/ Compression
% "
~"
,,apt
_
•
%t + ~ c
l
|
"1
Fig. 3. Schematic diagram of a typical cyclic stress-strain hysteresis loop.
186
cycle, the microstructural back stress Oh, generated in this tensile deformation will be in a direction opposite to the tensile deformation (the subsubscripts t and c refer to tension and compression respectively). Hence, the maximum applied tensile stress Op, becomes ap = oe,+ Ob,
(4)
Similarly, for the compressive half-cycle the appropriate equations are osc = aeo- oh,
(5)
and ape = o~o+ Oh0
(6)
Equations (3)-(6) under the conditions of symmetry of stresses in tension and compression (i.e. Oe=Oeo=oe; O'bt=~be=O'b; ~¢pt=O'pc = Otp; as, = as0 = as) yield Oe= (Op + as)/2
(7)
a b = (Op - os)/2
(8)
The above technique, although utilized by many in recent years, however, remains unsuitable for the measurement of internal stresses within the hysteresis loop. In addition, the assumptions with regard to the symmetry will not hold for the materials exhibiting the Bauschinger effect. Also, as pointed out by Dickson et al. [13], no account is taken of the stress relaxation at the tensile and compressive peak stresses. Dickson et al. [13] suggested a variation of the previous technique involving a stress dip to zero stress relaxation without performing the actual relaxation. The portion of the hysteresis loop where the slope agrees with the experimental elastic modulus is determined by taking the midpoint of this portion and then measuring the effective stress as equal to the difference between the preceding peak stress and the midpoint of the elastic region. This technique, however, accounts for the stress relaxation at the tensile and compressive strain limits but suffers from the other limitations of the KWL [12] method. TABLE 1
2.2. Measurement techniques for continuum mechanical stress variables Several different techniques have been employed for the measurement of kinematic back stress and isotropic drag stress during cyclic deformation [2, 3, 14]. Cescotto and Leckie [14] have utilized creep tests at several stress levels, monotonic tension tests at high strain rates and the stable hysteresis loops from completely reversed cyclic tests at several strain amplitudes to determine the changes in kinematic back stress and isotropic drag stress. A large number of tests are required to determine the evolution of a and r. Controlled unloading within the primary loop has been recently utilized to determine the elastic range in real time [3, 15]. The technique, first suggested by Onat [3], involves measurement of yield strengths in tension and compression by unloading the specimen at predetermined strain points within the primary hysteresis loop. At a specified strain offset during elastic unloading, as determined in real time, the loading is stopped and the material is reloaded (Fig. 4). In this way, small load reversals are used to locate continuously the elastic range of a material during cyclic plastic deformation. The kinematic back stress and isotropic drag stress are then taken as the centre and radius of the elastic range. This method overcomes the limitations of the techo-
I Fig. 4. Schematic of unloading-reloading procedure within the primary hysteresis loop.
As-received chemical composition, mechanical properties and grain size of copper specimens
Composition (wt.%)
0.2% initial oy
Fe
Zn
Pb
Sn
P
Cu
(MPa)
0.002
0.0027
0.0024
<0.01
0.0008
>99.98
25.26
Ours (MPa)
(/tin)
218
55
Grain size
187
niques mentioned earlier. A single specimen provides accurate and consistent data on the evolution of kinematic back stress and isotropic drag stress variables as a function of number of cycles. A modification of the method [3, 15] was utilized in the present experiments and is discussed in more detail in the next section.
3. Experimental conditions 3.1. Specimen The chemical composition and initial mechanical properties of the as received commercially pure (99.99%) polycrystalline copper used in the present work are shown in Table 1. The material was received in the form of bars 12.7 mm in diameter. The test specimens were machined out from the bars according to ref. 16. The specimens were cylindrical with a uniform reduced test section of 25.5 mm length and 6.35 mm in diameter. The specimens were annealed in a muffle furnace in air atmosphere at a temperature of 375 + 2 °C for 1 h. As-received-and-annealed specimens were cold mounted, polished, etched and observed by optical microscopy. Also, foils were made and observed by transmission electron microscopy (TEM) to ensure complete recrystallization of grains. The resulting average grain size, determined by the linear intercept method of Smith and Guttman [17], was found to be 55 pm. The specimens were mechanically polished with 0.3 p m alumina particles suspended in water until bright smooth gauge sections free' of any machining marks were obtained. 3.2. Measurement of kinematic back stress and isotropic drag stress To evaluate the kinematic back stresses and isotropic drag stresses during cyclic straining, high precision is required in the recording of the hysteresis loops and in the measurement of the elastic modulus (experimental) applicable to each unloading-reloading loop within a specific hysteresis loop. This was accomplished by a computerized materials testing system. The specimens were subjected to strain-controlled uniaxial cyclic loading in a microprocessor-based closed-loop servo-controlled electrohydraulic material test machine of 20 kin s-1 capacity. An IBM PC AT was utilized for the real-time control of the test machine as well as for data acquisition, storage and analysis. The system controller con-
stantly compared the command from the computer with the feedback from the transducer and provided a correctional signal to a servo valve, permitting a precisely controlled test to match the desired command. The computer interface allowed close monitoring of the loading, unloading and reloading within a particular hysteresis loop. The experimental procedure for the measurement of state variables--kinematic back stress and isotropic drag stress--was as follows. The procedure involved measuring the dynamic yield stress Y~ in tension and the dynamic yield stress Yc in compression at various predetermined strain levels within a completely reversed uniaxial cyclic stress-strain loop after a certain number of specified cycles. At a predetermined strain, the cyclic strain history was interrupted and the specimen was unloaded until the material yielded in the reverse direction. In order for subsequent measurements on the same specimen to be valid, it was imperative that the unloading be stopped immediately after the yield criterion was reached. Thus the determination of reverse yielding Ychad to be made in real time. This was possible with the data acquisition and analysis system attached to the testing machine. As the unloading began, the computer determined the elastic unloading line. The elastic modulus was calculated at the start of the test using the least-squares procedure and the stressstrain data from the initial loading. As the unloading continued, the computer calculated the strain offset from the elastic line for pairs of stress-strain values collected during the unloading. When the critical strain offset of 0.05% was met or exceeded, the unloading was stopped and the specimen reloaded until the 0.05% offset yield strength was once again reached. The cyclic strain history was then resumed. The above procedure was repeated at the next strain where the yield stress was to be measured. The kinematic back stress and isotropic drag stress variables were measured at the end of each strain increment. For uniaxial loading, kinematic back stress is simply given by ( Yt + Yc)/2 while isotropic drag stress is equal to ( Yt - Yc)/2. The test history of any specimen consisted primarily of three basic sets as shown schematically in Fig. 5. The first set consisted of the very first cycle of tensile straining and compressive reversal (Fig. 5(a)). The small unloading-reloading loops within the first cycle indicate the addi-
188
-
I
-0.005 -0.003
0.0
ODO~
0.005
Strain (a)
I00
50 Stress (MPo) o
~:/~
//////7//
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Strain
Strain
(b)
(c)
I t 0.003 0 . 0 0 5
Fig. 5. Experimental test history showing unloading-reloading loops used for the measurement of kinematic back stress and isotropic drag stress: (a) first tensile quarter-cycle and first compressive half-cycle; (b) completely reversed second or a subsequent loop where the measurements were made; (c) completely reversed loop without measurements.
TABLE 2 tests
Summary of experimental conditions for cyclic
Strain amplitude Ae/2
(s- 1)
Strain rate ~
Number of cycles N
0.003 0.003 0.003 0.003 0.005 0.005 0.005
0.005 0.01 0.01 0.1 0.005 0.01 0.1
509 20 515 515 509 34 509
tional path required to estimate the yield stresses in tension and compression at the unloading point. Subsequent cycles were divided into two sets depending on whether or not the measurements of yield stress were made within the hysteresis loop (Figs. 5(b) and 5(c)). The tensile part of the first block of history, which was smaller, was divided into five increments. The strain range for subsequent measurement cycles was divided into seven increments. Measurements of Yt and Y~ were made on the first four cycles of the history and thereafter about ten measurements were made in the fatigue-hardening region and saturation region. The test was interrupted in
some cases during cyclic hardening or saturation and the specimen was removed to observe the dislocation struciures by T E M [18]. The entire test was conducted at a constant total strain amplitude and a constant strain rate with the exception of a usually fast strain rate (~=0.2 s -1) for the small unloading-reloading loops within a completely reversed large loop. The loading waveform was a fully reversed triangular wave for the large outer loop. 4. Results
4.1. Cycle-to-cycle data Table 2 summarizes the various specimen details, and experimental conditions. The specimens were cycled in the range 1-515 cycles at strain amplitudes Ae/2 of 0.3% and 0.5% and strain rates of 0.005, 0.01 and 0.1 s -1. All the specimens exhibited initial hardening and a tendency to saturation during which the hardening rate approached zero. Figure 6 shows the experimental plot of loading-unloading loops within the cyclic strain interval at a constant total strain amplitude Ae/2 of 0.3% at different numbers of cycles (Figs. 6(a)-(c)). The stress ampli-
189 200 I00 Stress
-
0
'¢'~'~" ~ ' ~ "
-100 -
200 I I - 0 . 0 0 4 - 0 . 0 0 2 0.0 0 0 0 2 0004 Stroin
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200
200
I00 Stress (MPa)
o
IOO -
~
-
~//
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- 200
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1
- 0 . 0 0 4 - 0 0 0 2 0.0 0.002 0 . 0 0 4
Strain (c) Cycle No. II
Fig. 6. Experimental plot of tensile half-cycle at three different cycle numbers for a specimen cycled at a strain amplitude of 0.006 and a strain rate of 0.1 s- '.
tude increases with increasing number of cycles (strain hardening) as the loops become more "pointed" at higher cycles. The general shape of the internal stress variables a and r during cyclic deformation is established in the first few cycles and then it changes only slightly to reach saturated values. This is shown in Fig. 7 which presents a plot of tensile peak flow stress, kinematic back stress and isotropic drag stress vs. number of cycles for copper specimens subjected to cyclic straining under test conditions shown in Table 2. The stresses increase non-linearly with increasing number of cycles until saturation is observed, beyond which the shapes and sizes of the loops remain constant. Complete saturation is reached between 50-100 cycles for all the cases (Figs. 7(a)-(f)). In addition, the saturation is reached simultaneously in all the measured stresses, suggesting that saturation cycling is essentially a steady state. The increase in the stresses during cyclic hardening are associated with the formation and growth of dislocation tangles and dislocation cells [18]. The effect of strain rate on kinematic back stress is more pronounced than on isotropic drag stress. The isotropic drag stress is, however, the predominant
component of the flow stress at the strain rates and strain amplitudes utilized in the experiments. Figure 8 shows plots of experimental tensile peak flow stress oexp and calculated tensile peak flow stress trea~ taken as a sum of corresponding kinematic back stress and isotropic drag stress as explained in Section 2.1 and establishes the validity of eqn. (1). Figures 9 and 10 show the plots of kinematic back stress at and isotropic drag stress Kt in tension and kinematic back stress a c and isotropic drag stress r c in compression vs. number of cycles to determine the stress asymmetry during cyclic hardening and saturation for specimens cycled to saturation. In Figs. 9 and 10, the stresses in compression are plotted on the tensile axis for comparison. No significant stress asymmetry was observed in isotropic drag stress r between tension and compression (Fig. 9). The kinematic back stress a, however, tended to be about 10 MPa higher in tension than in compression (Fig. 10). 4.2. Within-cycle data
Figure 11 shows the experimental stress-strain curve for the first quarter-cycle in tension and first compressive reversal for a specimen cycled
190 2OO
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0 100
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Fig. 8. Tensile peak flow stresses (experimental) and as a sum of corresponding kinematic back stresses, and isotropic drag stresses (calculated) at different strain amplitudes and strain rates. The inset in (a) shows the corresponding symbols.
200
~00
~
200 [ y
- O.OI/s
I OOI A o Q.
:¢ 200
-,~- 0.003;
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OL~ I J I , I 0 I00 300 5OO Number of Cycles (f)
Fig. 7. Tensile peak stresses: flow stress, kinematic back stress and isotropic drag stress v s . number of cycles for specim e n s cycled at different amplitudes and strain rates.
at a strain rate of 0.1 s-x and a strain amplitude Ae/2 of 0.003. The stresses reach their maximum values at maximum applied strain, thus acting to lower the yield stress in the reverse direction. Figures 12(a) and 12(b) shows plots of o, a and r vs. Ae for the above specimen during cyclic hardening (second cycle) and saturation (515th cycle) respectively. The isotropic drag stress curves remain relatively "flat" at strain limits in the earlier stage of hardening within the cyclic loop (Fig. 12(a)). The back stress, on the contrary, is dependent upon the strain within the loop. As the cells develop and become well defined, the drag stress shows slight variation within the loop at higher cycles (Fig. 12(b)). There is considerable variation in back stress, within the cycle, at high cycles. The internal variables attain a maxima at the maximum peak stress.
5. Discussion The method of measurement and the results of cyclic tests can be discussed in relation to KWL [12] method if one assumes that the peak tensile and compressive stress values in a particular loop are of the same magnitude. In fact, the kinematic back stress and isotropic drag stress measurements at the tensile and compressive strain limits on comparison with the measurements carried out in ref. 12 reveal equivalent results (Fig. 13). The microstructural back stress Ob and the kinematic back stress a are equal and given by the expression Orb= O~ = ( O p t - -
oso)/2
(9)
Similarly, the microstructural effective stress and isotropic drag stress are given by oe = K =(Op, + Osc)/2
(10)
The results imply that the yield surface expands and contracts as the effective stresses increase and decrease within the primary hysteresis loop. Similarly, the yield surface translates to and fro by an amount given by the microstructural back stress which changes sign as the direction of straining is reversed. The shape of the a
191
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.
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t~-
~
t
~
~-T,_
O~ 0
1 I I i I iO0 300 500 Number of Cycles (d)
60t
40
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6o
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40-
T=Temp eroture C: Compression ,
I t I t I I00 300 500 Number of O/des (a)
0
o fl_ :~
:L
150
~
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I00
I00
60
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0
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200
~ ~
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20
50
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_ t--C
11.
? n
0
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,
,
I00 300 500 Number of Cycles (b)
02 0
200
20°
150
150
I t I I I I00 300 5OO Number of Cycles (e)
0
. 0
I i I J I K)O 300 50O Number of Cycles (b)
60
60
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..__c_J_ I00
I00
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5O
,~
I i I I I I00 300 500 Number of Cycles (c)
0
0
,.~..~,
T
C
20
0
O~L t R I , I 0 I00 300 500 Number of Cycles (e)
I t i t I I0 20 30 4 0 50 60 Number of Cycles (f)
O~ 0
2O
1 I I , I I00 30O 5O0 Number of Cycles (c)
0
0
I i t I I I0 20 30 40 50 6 0 Number of Cycles (f)
Fig. 9. Isotropic drag stress asymmetry vs. number of cycles for specimens cycled at different strain amplitudes and strain rates: T, tension; C, compression.
Fig. 10. Kinematic back stress asymmetry vs. number of cycles for specimens cycled at different strain amplitudes and strain rates: T, tension; C, compression.
number of cycles data (Fig. 7) in the present work is in agreement with the results of KWL [12] for cyclic deformation of single crystals of copper as well as recent results obtained by Dickson e t al. [13] and Wang and Laird [19] on cyclic deformation of polycrystalline copper. The present experiments as well as the data of the above workers indicates a rapid increase in internal stress variables during the period of cyclic hardening followed by a saturation in the values. Figure 14 shows a schematic diagram of flow stress and internal stresses within the stress-strain hysteresis loop based upon the experimental data. The diagram shows a higher stress value for the applied stress and kinematic back stress in tension compared with compression. The isotropic drag stress value is about the same in tension and compression. The microstructural back stresses are at least a quarter of
the applied stresses. A characteristic dislocation movement within the hysteresis cycle results in the above internal stress field. More specifically, the isotropic drag stress, being symmetric in tension and compression, could be due to the presence of well-defined dislocation walls in cellular configurations containing a high density of immobile dislocations. Asymmetric back stresses similarly arise from the anisotropy in the pile-up characteristics of mobile dislocations in the cell interior. In addition, a stronger dependence of strain rate on back stress compared with drag stress is also indicative of its direct relationship to the mobile dislocation density. A welldefined dislocation cell structure was observed during later stages of hardening and saturation for all the experimental conditions. Cyclic evolution of a and r are related directly to the parameters describing the dislocation structure such as
a n d u vs.
192 o"
IOO 50-
Yt ,OP t
Stress (MPa)
0 - 50 -
- too
I
I
E
- 0.004 -0.002 0.0 0.002 0.004 Strain Interval (a) Yc , O'st I00 50
Stress
(MPo)
yc ,O'pc
-
_j
0
Fig. 13. Comparison of the technique of measurement of kinematic back stress and isotropic drag stress in the present work with the KWL [12] method.
-50 o - I00
I
I
I
-0.004-0.0(~ 0.0 Q002 0.004 Strain I n t e r v a l (b)
Fig. 11. Stress-strain data for (a) the first quarter-cycle in tension and (b) the first compressive reversal, for specimen cycled at a strain rate of 0.1 s -~ and a strain amplitude of 0.006.
i
~
~,,..........--~Crt Applied stress (0"]
y
Isotropic drag
~-S~:;~--ac KC
~
Stress
(MPo)
stress (K)
Qt Kinematicbackstress (~)
O" = Q+K t
AE -
-
at > Qc Kt = Kc
K
Fig. 14. Schematic representation of the cyclic flow stress and internal stresses within the hysteresis loop.
o
-50 i -I00 I
I
-O.OOS-0.003 0 0.003 0.005 Sfrain (0)
~$tress
I -O.OOS-0.003 0 0.003 0.005 Strain
(b) Fig. 12. Tensile flow stress, back stress and drag stress v s . strain for specimens cyclically strained at a strain rate of 0.1 s -l and a strain amplitude of 0.006: (a) second cycle; (b) 515th cycle.
mobile dislocation density in the ceil interior, immobile dislocation density in the cell walls, average cell diameter and average ceil width [18]. A stress asymmetry was observed in the experimental plots of a t a n d a c vs. number of cycles (Fig. 10). Identical stress asymmetry was found in the applied cyclic stress in tension and in compression. No significant asymmetry existed in the isotropic drag stress component (Fig. 9). The results therefore indicate that the asymmetry in the applied flow stress of the cycle is due to the kinematic back-stress component. A Bauschinger effect in the flow stress of copper has been observed by several workers [20-22]. However, no such observation has been made for the backstress component of the flow stress in the past. The results can be rationalized on a microstructural basis as follows. If it is assumed that the cell walls harden more rapidly than the cell interior
193
during cyclic hardening, then on unloading from a tensile cycle the cell interior will be placed further into compression than the cell walls compared with the case when the cell walls and cell interior harden at the same rate. The large residual compressive stress in the cell interior would produce a smaller value of kinematic back stress. Steady state values of a, a and r depend upon the total strain amplitude although the values, particularly a, increase with increasing strain rate. Asymmetry in the flow stress and kinematic back stress also increased slightly with strain rate. The relative contribution of kinematic back stress to the applied stress also increased slightly with increasing strain rate. This is perhaps due to a large width of dislocations in f.c.c, metals and the consequent small effect of thermal activation in the glide of dislocations at room temperature.
Mechanical Engineering, Washington University, St. Louis) for his help in conducting the cyclic tests and many suggestions. Appreciation is expressed to the Department of Mechanical Engineering, Washington University, for providing a fellowship during my stay at the University. References
5 6 7
6. Conclusions (1) Kinematic back stress and isotropic drag stress increase rapidly during cyclic hardening and approach saturation at large cycles. (2) A Bauschinger effect exists in the kinematic back stress and cyclic flow stress within the cycle and from cycle to cycle. The magnitude of the Bauschinger effect depends upon the strain rate and strain amplitude. (3) Continuum internal variables as measured by the controlled unloading-reloading method used in the present work and the microstructural variable measurement by the KWL [12] method yield values of the same order of magnitude and a similar shape of evolution of the stresses. (4) The nature asymmetry in a and a (the Bauschinger effect) can be explained on the basis of dislocation dynamics in the cell interior.
8
9 10 11
12 13 14
15 16 17 18 19 20
Acknowledgments I am indebted to Dr. E. J. Tuegel, Material and Process Development, McDonnell Aircraft Company, St. Louis (formerly with Department of
21 22
E. Krempl, Acta Mech., 69 (1987) 25. E. J. Tuegel, Res Mech., 22 (1987) 65. E. T. Onat, Rep. ORNL-SUB-3863-2, November 1976 (Yale University) (Oak Ridge National Laboratory Contract). E. Orowan, in G. Rassweiler and W. Grube (eds.), Internal Stresses and Fatigue in Metals, Elsevier, Amsterdam, 1959, p. 59. W. G. Johnston and J. J. Gilman, J. Appl. Phys., 30 1959) 129. J. C. Gibeling and W. D. Nix, Mater. Sci. Eng., 45 1980) 123. H. Conrad, J. Met., 16 (1964) 582. J. T. Michalak, Acta Metall., 13 ( 1965) 213. J. C. Tsou and D. J. Quesnel, Mater. Sci. Eng., 56 1982) 289. J. Polak, M. Klesnil and J. Helesic, Kovovd Mater., 17 (1979) 476. W. B. Jones, R. W. Rohde and J. C. Swearengen, in R. W. Rohde and J. C. Swearengen (eds.), Mechanical Testing for Deformation Model Development, ASTM Spec. Tech. Publ. 765, 1982, p. 301. D. Kuhlmann-Wilsdorf and C. Laird, Mater. Sci. Eng., 37 (1979) 111. J. I. Dickson, J. Boutin and L. Handfield, Mater. Sci. Eng., 64 (1984) L7. S. Cescotto and F. A. Leckie, in C. S. Desai and R. H. Gallagher (eds.), Proc. Int. Conf. on Constitutive Laws for Engineering Materials: Theory and Application, Tucson, AZ, 1983, Elsevier, New York, 1983, p. 105. T. C. Lowe and A. K. Miller, Scr. Metall., 17 (1983) 1177. ASTM Stand. E466-82, 3.01 (1986) 571. C. S. Smith and L. Guttman, J. Met., 5 (1953) 81. M. Jain, submitted to J. Mater. Res. Z. Wang and C. Laird, Mater. Sci. Eng., 101 (1988) L 1. I. Gokyu and T. Kishi, Proc. 13th Jpn. Congr. on Materials Research--Metallic Materials, 1970, Japan Institute of Metals, Tokyo, 1970, p. 125. R. C. Daniel and G. T. Home, Metall. Trans., 2 (1971) 1161. O. B. Pedersen, L. M. Brown and W. M. Stobbs, Acta Metall., 29(1981) 1843.