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Cyclic loading and fatigue in ice
Cold Regions Science and Technology, 4 (1981) 41-53 Elsevier Scientific Publishing Company - Printed in The Netherlands
41
C Y C L I C L O A D I N G A N D F A T I G U E IN ICE Malcolm Mellor and David Cole Cold Regions Research and Engineering Laboratory, Hanover, NH (U.S.A.)
(Received April 21, 1980;acceptedin revisedform May 22, 1980)
ABSTRACT
INTRODUCTION
Isotropic polycrystalline ice was subjected to cyclic loading in uniaxiai compression at-5°C, with stress limits 0 - 2 and 0 - 3 MPa, and frequencies in the range 0.043 to 0.5 Hz. Stress-strain records showed hysteresis loops progressing along the strain axis at non-uniform rates. The effective secant modulus, which was about half the true Young's modulus, decreased during the course o f a test. The elastic strain amplitude and the energy dissipated during a loading cycle both increased with increase o f time and plastic strain. Strain-time records gave mean curves which were identical in form to classical constant-stress creep curves, with a small cyclic aiternation o f recoverable strain about the mean curve. The inflection point o f the "creep curve", marking the transition from strain hardening to strain softening, occurred at a plastic strain o f l% (+-0.1%), which is about the same as the "ductile failure strain" found in constant stress creep tests and in constant strain-rate tests on ice o f the same type at the same temperature. The dissipation o f strain energy up to this "failure point" was much higher for the cyclic tests than for corresponding quasi-static tests - 100 to 600 kPa {or kN-m/m a) in comparison to about 30 kPa. The number o f cycles taken to reach the "failure point" was o f no direct significance, varying greatly with stress amplitude and with frequency. The results o f the tests suggest that maximum resistance under compressive cyclic loading occurs at an axial plastic strain o f about 1%, which is essentially the same as the failure strain for ductile yielding under constant stress and under constant strain-rate.
As far as we know, there have been no studies on failure* under cyclic loading, or on fatigue failure, for ice. Vibrational methods have been employed to study the complex modulus and complex compliance (see, for example, Mellor, 1975), but in most of these experiments the stress and strain amplitudes have been very small, linear viscoelasticity has been assumed, and neither failure nor permanent deformation have been considerations. Vinson (1978) subjected ice to low frequency cyclic straining, with confining pressure and fairly large strains, but he limited the number of cycles to less than 20 for any given set of conditions. Some years ago, one of us cycled uniaxial compressive stress on ice specimens, tracing out stress/strain curves from zero stress up to a moderately high stress and back down to zero, with a low frequency alternation (<0.1 Hz). The trace on the X-Y plotter showed a hysteresis loop for each cycle, with progressive displacement along tile strain axis after each cycle. The displacement per cycle decreased progressively during the rather brief test, until eventually the hysteresis loop appeared to be tracking over the same path repeatedly, although it was recognized that *"Failure" is an arbitrary term. For present purposes w e define failure, or incipient failure, as the limit c o n d i t i o n where there is a transition from strain-hardening to strainsoftening, i.e. the point at which the ratio of stress to strainrate is a maximum. Failure strain is thus given by the inflection point of a complete creep curve for constant stress, or by the peak of a complete stress/strain curve for c o n s t a n t strain-rate. Beyond this strain, the material becomes progressively weaker.
0165-232X]81[0000-0000/$02.50 © 1981 Elsevier Scientific Publishing Company
42 there was probably still an imperceptible progression. This was incidental to the work then in hand, the testing machine was an obsolescent type, incapable of suitable automatic control, and nothing more was done on repetitive loading. Since that time, constant-stress creep tests and constant strain-rate strength tests on polycrystalline ice have led us to the conclusion that the failure strain for creep rupture, at least in uniaxial compression, has remarkably little variation with stress, loading rate, input strain energy, and probably temperature (a more detailed description, with caveats, is given by Mellor, 1980). If this is true, the implications for ice mechanics could be far-reaching, since it opens up the possibility of specifying failure criteria rather simply in terms of strain, instead of in very complicated terms of stress. In considering the possible limitations of a strain failure criterion, one question that came to mind was whether fatigue failure, or failure after repetitive loading, would occur at the level of permanent strain (approximately 1% axial strain) found to define ductile failure under constant stress and under constant strain-rate. Because of the energy dissipation in each loading cycle (as given by the area of the stress/strain hysteresis loop), it seemed that the total dissipation of strain energy would be much greater for failure under cyclic loading. To explore some of these questions, a small experimental program was undertaken, and results are given here. The work is of a preliminary nature and more detailed tests, including conventional fatigue tests, are planned.
Tests were made in uniaxial compression, with stress cycled sinusoidally between zero and some selected value (roughly 30% of the uniaxial compressive strength for moderately high loading rates). Load was applied by a closed loop electro-hydraulic testing machine, which has a high capacity servo-valve to provide rapid hydraulic response. The actuator was programmed to vary load according to a haversine function, with preset load limit and preset frequency. Axial strain was measured by a displacement transducer mounted across the bonded end caps of the specimen. Force and axial displacement were recorded as functions of time on a strip chart recorder, and stress/strain loops were recorded on an X-Y plotter. To provide data for a parallel study, acoustic emissions from the specimen were monitored by transducers set close to, but not in contact with, the specimen. The narrow gap between the ice and the acoustic transducer was filled with silicone grease (holes were cut in the rubber membrane that surrounded the specimen to prevent evaporation). The specimen was housed in an environmental-control cabinet, with cabinet temperature controlled by a recirculated air refrigeration system to -5 ° -+0.2°C. At the end of each test the specimen was obviously deformed, but still intact. The axial plastic strain of 7% or more produced "barreling", and the surfaces had characteristic bumps and valleys representing outthrust grains and depressed grain boundaries ("alligator skin" surface). In the specimens that had been subjected to the highest stresses, internal microcracks were clearly visible. The used specimens were finally put into storage at -30°C for possible later study.
TEST PROCEDURE
PROCEDURE FOR DATA ANALYSIS
Specimens of polycrystalline ice were prepared by methods described earlier (Cole, 1979). The specimens were right circular cylinders, 50.8 mm diameter by 127 mm long. End caps of fiber-based phenolic resin were bonded on as an integral part of the specimen during the moulding process. The faces of the end caps were faced-off in a lathe so as to pluck the textile fabric and produce a "hairy" surface. Specimens were homogeneous and initially isotropic, with a mean grain size of 1.2 mm. The electrical conductivity of melt water from test specimens was 4 /aS.
in this preliminary analysis, the data are interpreted in a certain way so as to answer particular questions. The output from the test equipment is a continuous chart record of displacement as a function of time (an approximately sinusoidal wave with progressively increasi~ag mean strain), and a chart record of force as a function of displacement (hysteresis loops). The general characteristics of these records are shown in Figure 1. From these records, we obtain: (1) accumulated strain as a function of time and number of load cycles (creep curves), (2) the strain energy
43
dissipation per cycle, and the accumulated strain energy dissipation, (3) the amplitude or recoverable strain per cycle, and the effective secant modulus for each cycle. The stress o varies with time t in accordance with the haversine function, i.e.
plastic strain ~p is also assumed invariant with time. Thus the variation of strain e through a limited number o f stress cycles n can be described in terms o f the time t from the start o f the first o f these n stress cycles: e = ~1 emax [1 - cos(cot--6)] + ~-pt
a = (Omax/2)(1 - cos cot) = (Omax/2)(1 - c o s
27r/t)
- f i -1 emax [1 - cos(cot-6)] + ~-~ t
(1)
where amax is the maximum stress, co is the angular velocity, and f is the frequency. The variation o f strain e with time t is complicated, but an approximate description can be given for a limited number o f cycles b y making steady-state assumptions. The initial conditions at t = 0 are ignored because the transient phase relationship between stress and strain is difficult to define at the beginning o f a test. The range o f recoverable strain emax and the phase lag 6 are assumed invariant with time, and the mean rate o f
where epl is the plastic strain accumulated in one cycle and T is the wave period. To facilitate illustration o f eqn. (2) in a diagram, epl can be expressed as a fraction, or multiple, o f emax, such that epl/ emax = k and: e t emax = ]1 [1 - c o s ( c o t - 6 ) ] + k--T
cot
- i ' [1-cos(cot-~)] + / c - 27r -
point of Inflecl~on
' '
c ic
~Meono 5trointroinRRile roinin t, Time
Area Gives Energy Dissipated m One Cycle
Strain Accumulated in One Cycle
(3)
Fig. 1. (at. Simplified representation of a strain/time record, showing the cyclic alternation of strain and the accumulation of plastic strain with time. The time scale is condensed, while the cyclic strain amplitude and the wavelength are exaggerated. (b) Diagram showing part of a stress/strain record. The loop described by a stress]strain cycle gives the energy dissipation for the cycle, and the progression of successive loops along the strain axis represents the accumulation of plastic strain.
Recovetoble Strain Amplitude
E
(2)
i. 11
/
ZAccumulated Plastic Strain After
n Cycles
Slope Gives Secant Modulus
Strain
44 The strain/time and stress/strain characteristics described by eqn. (1) and eqn. (3) are illustrated in Fig. 2 by taking as convenient values/5 = Ir/24 = 7°30 ', k = 0.1, so that G
- { ( 1 - c o s wt) Gmax (4) - ~ 1-cos
emax
t---
*
24
10 21r
In the foregoing equations the dissipative component of the recoverable anelastic strain is described solely by the lag angle 8, while the irrecoverable viscous creep is described by epl or ~p. If 8 = 0 and epl = 0, the material is purely elastic, with emax = Omax/E, where E is Young's modulus. With/5 :/: 0 and epl :/: 0, an effective secant modulus E s is defined by the slope of a line joining the upper and lower limits of a
(Tma* 0 4
7r
[
I 2Tr
3~r
I 4~
5~
I STr
08
Ep
04
~ot
!
lot
hysteresis loop; E s is the reciprocal of the change in e[emax between successive points where tr = 0 and 17 = O ' m a X .
In the cyclic load "creep curves", the behaviour is qualitatively identical to the behaviour displayed in conventional constant-stress creep curves, i.e. creep deceleration (strain hardening) followed by a transition to creep acceleration (strain softening). In accordance with the procedure adopted for the interpretation of conventional complete creep curves (see Mellor, 1980), the inflection point of the cyclic load "creep curves" is used to define failure (~p = 0), which in this case could perhaps be called fatigue failure. The inflection point is best found by replotring the mean strain rate for strain accumulation (slope of the "creep curve") against time and number of cycles, or alternatively against accumulated strain (Figs. 6 and 7). The area of a stress/strain hysteresis loop gives a measure of the strain energy dissipated in one loading cycle. Since the shape of the hysteresis loop changes only slowly after the first few cycles, representative measurements of area can be made for sets of loops that are essentially identical. The strain energy dissipated up to the failure point (i.e. the inflection point of the "creep curve") is the summation of hysteresis losses, which includes the progressive displacement effect. The phase lag/5 was not monitored systematically, and it proved difficult to measure /5 reliably from occasional scale expansions on the strip charts (see Fig. 3,b,c,d) because of imperfections in the pen alignments. Measurements of/5 were therefore obtained from the discontinuous stress/strain records by comparing times for maximum stress and maximum cyclic strain.
RESULTS
o
o~ A_ o8
,!o '
,!~
~mox
Fig. 2. Illustration of the steady-state stress/strain/time relationships described by eqns. (1-4). (a) alternation of stress and strain with time (or angle), (b)stress/strain hysteresis loops.
A representative sample of strain/time records is shown in Fig. 3, "while examples of stress/strain records are given in Fig. 4. Details of all the tests that were properly completed are given in Table 1 (three additional tests were terminated prematurely by power failure, recorder problems, or equipment uncertainties).
45
Fig. 3. Representative strip-chart records giving deformation and load as functions of time. These results are for test FT-30-2, with peak stress 3 MPa and frequency 0.077 Hz: (a) start of the test, (b) after 1% plastic strain and approximately 200 cycles, (c) after 3.5% plastic strain and 540 cycles, (d) after 6.8% plastic strain and 750 cycles.
TABLE 1 Cyclic loading tests on polycrystaUing ice at -5°C Test no.
Maximum axial stress Omax (MPa)
Frequency (Hz)
Minimum mean creep rate ~min (s-l)
Number of cycles to ~-min
Time to ~min (s)
Total strain at ~-min (%)
Strain energy per unit volume to ~min (kN-m/m3=-kPa)
FT-20-1 FT-20-2 FT-20-3 FT-25-3 FT-30-1 FT-30-2 FT-30-3
2.0 2.0 2.0 2.52 3.0 3.0 3.0
0.043 0.077 0.50 0.10 0.043 0.077 0.50
1.24 8.56 6.93 1.51 3.01 2.95 2.89
286 548 5,000 600 96 168 1,035
6,580 7,120 10,000 6,000 2,210 2,180 2,070
1.05 1.08 1.13 1.07 0.90 0.93 0.90
118 149 628 190 96.4 136 306
X 10 -~ × 10 -7 X 10 -T X 10 -~ X 10 -~ X 10 -~ X 10 -n
~,~
l ~ - ~ Q ~ ::,~o
Fig. 4. Typical records of load versus deformation, taken directly from the X-Y plotter; (a) sequence of hysteresis loops at the start of a test, showing initial transients, (b) loops near the plastic strain that correspond to minimum mean strain rate, (c)loops recorded near the end of the test, at comparatively high plastic strains.
............ ~
.............. :~..
~
4~
47
Figure 5 gives "creep curves" that plot the accumulated creep strain (summation of the residual strain at the end of each load cycle) against time and against the number of cycles. The amplitude of recoverable strain oscillations during load cycling is also indicated. Figure 6 gives plots of the mean strain rate
12F ZO~ ~
Z
'
against time, with indication of the number of cycles. Values of ~ are measured from chart records such as those shown in Fig. 3, and ~ at any time is the slope of the "creep curve", as illustrated in Fig. 5. The minimum on a curve of this type is taken as the indicator of "failure", i.e., it gives the minimum value of ~ for given stress conditions. Thus Fig. 6 gives time-to-failure and the number of cycles for failure.
7
/ -E~
~
I01~ i~
'
$~r(IAmpl#ude m
o5r 0
q
.
//f
f ~'~"8 ~ ~-~ l
02
04
Time(s)
5xlO"
"~
06
,
[
08
IOMO4
.
/
FT20-2
Z/O'077Hz FTgO'I oo4~H
/
I
--
i
/
FT20-3
~
FT25-3
/
/ ///4
2,000 cycle;/ /.cycles//
4~
(G~..o=.2.52 M
!
~
StrQin 0'0Hz / Amplitude / /I,000 /
[
2O0O
oooc ce,
2,000 cycles//
~
~
~
/ ( _ .
~
I0,000 ~ [I
Io,
.,
, io
2
l
Time(s)
5
r
~
,
~ ....... 4
"
050/.~/_
2,000 cycles
4//~
I
r
cycles T=-5°C
Jxlo4
'
SlrolnAmplt~d~ FT30-3
0
]
2
I
3
Time (s)
T30=30MPe T=-5*C
F'T30-2 0043H O077Hz
O'ree•
400 cNles r w ¢ycle~. 700( # Strmn
-~
~,~,% ~
41104
1
'FT303 050 Hz
o
FT 20-3
/
/
~; 2
o
, o
. . . . . . .
0
FT20-2 ,
FT25-3
0.soHz
o6p
/ FT 20-1 O.043Hz
/
'
,o
~¢~
A~p~ //
cycles
700 cycles
°077.,
g ,2
CrY"=30 MPa T=-5°C ] 8
FT30-3 050Hz cyck
5O0O
i
I
Time (s)
400 c les yc
FTSO-2
1
L I0
•
08 Z
t J 12x rO~
Fig. 5. Creep curves showing the accumulated plastic strain as a function of time. The magnitude of the total amplitude of recoverable cyclic strain is indicated, as axe the numbers of stress cycles: (a) Tests at 0 - 2 MPa, (b) Tests at 0 - 3 MPa.
n
i 2
i
I 4
l
I 6 T~me l=)
E
I 8
t
I K)
l
12~103
Fig. 6. Curves giving the mean creep rate as a function of time: (a) tests at 0 - 2 MPa, (b) tests at 0 - 3 MPa.
48 Mean strain rate can also be plotted against accumulated strain, as in Fig. 7, to give a direct indication o f failure strain. The recoverable strain amplitude for one load cycle did not vary much during the course of a test, but there was usually a monotonic increase of strain amplitude with time and number of cycles• In Fig. 8 the data for all tests are combined by normalizing strain amplitude with respect to the amplitude at the failure stage, i.e. at ~-min.
141
I
i
°t
-o
~'~:-~"
~
" O
ul _~ 0 E ~ Z
D
.
(o) (~) (n) (.) (.) (,) (,)
r-~ S
•
0.8~
<
060
•
I
Sample Frequency (Hz) FT20- I 0043 FT20-2 0077 FT20- 5 050 FT25-5 010 FTSO- I 0045 FT30- 2 0077 FT30-5 050
,I J
J
i
q
I
2
£plEf~ Normalized
6 Plastic
8
10
Strain
, x l 0 -6 '
I
f
I
W
I
i
Fig• 8. Amplitude of the recoverable elastic strain per cycle plotted against the accumulated plastic strain. The values of strain amplitude A have been normalized with respect to the value A. occurring at the point where the mean creep strain rate is a minimum (~min)- The values of accumulated plastic strain ep have been normalized with respect to the "failure strain" ef, measured where ~min occurs.
~mQx=2.0 MPO FT20-1
T=-5*C
0
0
~
FT 2 0 - 2 0 . 0 7 7 Hz
I 2
_J I , 4 A c c u m u l a t e d Creep Strain (%)
I 6
,
20x106
k
o
/
2
/
/F730-2
4 6 Accumulated Creel) Strain (%)
q
e
to
Fig. 7. Mean creep rate as a function of accumulated plastic strain: (a) tests at 0 - 2 MPa, (b) tests at 0-3 MPa.
The energy dissipation per cycle also increased progressively during the course of a test, but the variation was greater than the variation o f recoverable strain amplitude (since the energy dissipation includes the non-recoverable or permanent strain)• Figure 9 combines data for all the tests by normalizing with respect to the energy dissipation per cycle at ~min. Both the cyclic strain amplitude and the energy dissipated per cycle increased with the stress amplitude (or peak stress), but tests were made at only two stress levels (with one exception). Variation of frequency at a given stress level did not have much effect on cyclic strain amplitude over the range covered in the tests, and the energy dissipation per cycle did not appear to vary systematically with frequency. Measured values of the lag angle ~ were plotted against total plastic strain, and lines drawn through the data points are shown in Fig. 10. The actual points, which have been omitted for clarity, scattered about the lines'by up to +2 ° .
49
2.4
'
e (o) (~) (o) (-) (.) (*) (,)
,O
2.0
=*, I.aJ
._= 1.6 o
I
I
I
'
Sample Frequency (Hz_) FT20-1 0.043 FT20-2 0.077 FT20-3 0.50 FT25-5 0.10 FT:30- I 0.045 FTS0-2 0.0?7 FT30-3 0.,50
I
l
1
'
l
o
o
c* o
(/)
z~o
,o
o
1.2 ..a
d
N
o.87
~*0.4 J 2
0
I
I 4
L
1 6
Ep/Ef. Normalized Plastic Strain
I
] 8
I
1
I0
I'1
115
Fig. 9. Energy dissipation per cycle plotted against accumulated plastic strain. Values o f dissipated energy w are normalized with respect to the value w , occurring at the point where mean creep strain rate is a minimum (~min)- Values o f accumulated plastic strain ep are normalized with respect to the "failure strain" ef, measured where ~min occurs.
45 °
I
I
i
I
I
l
]
I
r
40*
35 °
~
30 °
3 ~d 25 o
20 o
15
i
I 2
i
I
J
4
Accumulated
I 6
Strain
i
I
R
8
(%)
Fig. 10. Summary o f lag angle data.
DISCUSSION The main purpose of the experiment was to find how the failure strain, as defined by the transition from strain hardening to strain softening, might be affected by load cycling and by large variations in the
total dissipation of strain energy prior to failure. The simple answer appears to be that, for the range of conditions in the investigation, these things have no significant effect on the failure strain. The failure strain for low frequency cyclic loading in uniaxial compression at -5°C had a mean value of 1.01%, with a standard deviation of 0.10%. The failure strain appeared to decrease as the maximum stress increased, although the evidence is scant: at 2.0 MPa the mean failure strain was 1.09%, at 2.52 MPa the single value was 1.07%, and at 3.0 MPa the mean value was 0.91%. In the tests with amax = 2.0 MPa the failure strain appeared to increase slightly with increase of frequency, but there was no systematic variation with frequency in the tests at Omax = 3.0 MPa. In conventional quasi-static compression tests on ice of the same type at -5°C (not yet published), we have obtained mean values of failure strain that are quite close to those given above. For constant load creep tests, with stresses in the range 1.25-3.68 MPa, the mean failure strain was 0.96%, with a standard deviation of 0.14%. For tests under constant displace-
50 ment rate, with strain rates in the range 1.64 X 10 - 6 5.1 × 10 -s s -t, the mean failure strain was 1.05%, with a standard deviation of 0.11%. In these tests there was no significant correlation between failure strain and applied stress; or between failure strain and applied strain rate. In the cyclic load tests, the dissipated strain energy per unit volume up to the failure point ranged from about 1 0 0 - 6 0 0 kPa (i.e. kN-m/ma). By contrast, the strain energy dissipated up to the failure point in the quasi-static tests was much lower. In the constant stress creep tests, the strain energy dissipation was about 31 kPa, while in the constant strain rate tests it was about 26 kPa. All o f these tests are assumed to be isothermal, with absorbed strain energy eventually being dissipated by heat loss from the specimen. If, by contrast, the tests were to be made under adiabatic conditions, the absorbed strain energy would presumably raise the temperature of the specimen and ultimately cause internal melting. The creep curves obtained by plotting accumulated strain against time have the same form as conventional creep curves for constant stress tests. This is perhaps not so surprising, since cyclic tests can be regarded as creep tests in which there is oscillation about an effective mean stress. In attempting comparison between conventional creep curves and curves for cyclic loading, the effective mean stress cannot be taken as the arithmetic mean stress, since the stress/ strain-rate relation is known to be strongly non-linear. We have therefore taken values of ~min and compared them with a stress/strain-rate curve representing the
relations for @/O)min at -5°C, as derived from data for conventional quasi-static tests. Results of the comparison are shown in Table 2, and it seems that values of ~min for the cyclic creep curves correspond to the minimum strain rates o f conventional creep curves for a constant stress oe equal to about 73% of the maximum stress Omax applied in the cyclic tests. Following a procedure suggested to us by Samuel Colbeck, it can be shown that this ratio Oe/Omax is what might be expected for plastic straining according to a power relation ~ = o n , where n is approximately 4. Assuming that the anelastic strains balance out during a complete loading cycle in the vicinity of e~min, the mean plastic strain rate for one cycle is co -
27r
2~r/w f ~ dt
(5)
0
where ~ is the instantaneous plastic strain rate and co is the angular frequency. If the stress o varies cyclically according to 1
o =" ~ Omax (1 - cos co t)
(6)
and strain rate is related to stress by (7)
= Ao n
then 2----~"
f
(1 - cos 60 t ) n
Effective stress for comparison with quasi-static tests Frequency
emin
(Hz)
(s-1 )
Equivalent stress from conventional tests, Oeff
Ratio of effective static stress to maximum cyclic stress, Oe/Omax
(MPa)
FT-20-1 FT-20-2 FT-20-3 FT-25-3 FT-30-1 FT-30-2 FT-30-3
0.043 0.077 0.50 0.10 0.043 0.077 0.50
1.24 8.56 6.93 1.51 3.01 2.95 2.89
X I0-' X X X X X X
I0 -~ I0 -~ 10-' I0-6 I0-' 10-'
(8)
For n = 3, the value of the integral is 5 z r / w . For. n = 4,
TABLE 2
Test no.
d t
0
1.56 1.46
1.49 1.68 2.19 2.16 2.15
0.78 0.73 0.75 0.67 0.73 0.72 0.72
51
the value is 35 zr/4co. If ~- for cyclic loading up to amax is equal to ~ for creep under constant stress oe, then Aw(omax~ A on = - - ~ - - - ~ )
f27r/to( 1 - c o s w t ) n d t .
(9)
0
With
n = 3
ae/Crmax = 0.679
(10)
With n = 4 Oe/Omax = 0.723
(11)
The latter value is almost exactly equal to the mean of the values in Table 2. Traditional descriptions of fatigue failure are given in terms of the number of stress cycles up to a failure point, which is typically identified with fracture of the specimen. The number of cycles to cause failure is recognized to be a function of stress or strain amplitude, or of the mean stress, but frequency of cycling is not normally considered. The present tests are significantly different from typical fatigue tests on manufactured materials, since there is no question of stress being below the elastic limit, there is no external tensile stress, and the failure is ductile. For ice under these conditions, the number of cycles to failure is an unsatisfactory description of fatigue life. With a stress range of 2.0 MPa, the number of cycles to failure varied by a factor of 17 as frequency changed by a factor of 12. With a stress range of 3.0 MPa, the number of cycles to failure varied by a factor of 11 as frequency changed by a factor of 12. In both cases the number of cycles increased as frequency increased, and the number of cycles needed to reach the failure point was greater at the lower stress level. The variation of recoverable strain amplitude per cycle with the progressive accumulation of plastic strain is not established very clearly in Fig. 8. The data are badly scattered, but they do exhibit a general trend, the strain amplitude increasing as the test
progresses and plastic strain accumulates. In other words, the effective modulus appears to decrease as plastic strain increases. Fig. 11 shows this more directly by plotting the secant modulus for one cycle against the accumulated plastic strain. The apparent modulus tends to decrease with the time and net strain. The initial values are about half the value for the true Young's modulus, as measured by the initial tangent modulus in rapid tests, or else by high frequency vibrational methods that impose small strain amplitudes. Values from the 3 MPa tests appear to be lower than those from corresponding 2MPa tests (discounting results from test FT-20-3), which is in accord with Vinson's (1978) finding that the effective modulus decreased with increasing strain amplitude. The normalized data showing the variation of absorbed strain energy with the net plastic strain define a fairly clear trend. The ice becomes more dissipative, or "lossy", as the test progresses. The increase of hysteresis loss as the material strain-softens is quite orderly for plastic strains above 1%, but the data are more confused between 0 and 1% plastic strain. It is too early to decide whether the suggestion of a "hump" around 0.3-0.5% plastic strain in Fig. 9 is a real feature, but it might be noted in passing that some quasi-static tests display an initial yield point in this range of strain, even while the general trend is for continued strain-hardening. The lag angle ~ is related to the viscosity of the ice, and in broad terms it is expected to increase as viscous behaviour becomes increasingly significant in relation to elastic behavior. In Fig. 10 the 6 values for tests up to 3 MPa are systematically higher than those for tests up to 2 MPa. In the 2 MPa tests, 8 increases as frequency decreases, and in the 3 MPa tests the same overall trend occurs. An increase in ~ with increasing stress and decreasing frequency is what might be expected for a material with nonlinear viscous behaviour. The overall increase of 6 with increasing plastic strain, seen in all tests, is another indication that the ice tends to become more dissipative as strain accumulates.
52
'
I
I
'
'
I
i
~5
FT20-3
~E ~4
._>"
FT2~2-
o
F.,3
to be caused by bonding of end cops
Believed imperfect
m
FT 20-1
l
I
J
I
4 Strain (%)
Accumulated
i
I
l
6
I
I
l
I
I
F T 30-3
FT 30-2
o
FT 30-I
= o
I 2
,
I I 4 Accumulated Strain (%)
I 6
l
I 8
Fig. 11. Secant modulus as a function of accumulated plastic, strain: (a) tests at 0-2 MPa, (b) tests at 0-3 MPa.
CONCLUSION
Cyclic loading tests yield interesting information on the viscoelastic properties o f ice, and on failure criteria. An extension of such tests to cover tensile loading and tension/compression alternation would be worthwhile.
Perhaps the most interesting outcome o f the present tests is the finding that ductile failure in compression, as defined by a transition from strain hardening to strain softening, occurs at an approximately constant level of plastic strain, irrespective of the number of stress cycles needed to reach that strain, and apparently irrespective of the strain energy
53
absorbed and dissipated up to that point. It is particularly significant that this "failure strain" (about 1% axial) is approximately equal to the ductile failure strain found in quasi-static tests under constant stress and constant strain rate. If tests for other loading conditions and stress states produce similar results, then it seems possible that fatigue life for ice (and conceivably for some other materials) might be defined and monitored very simply in terms of accumulated plastic strain. The results of cyclic loading tests are directly relevant to some practical problems, including flexure o f ice plates by waves or wheel loads, and cyclic thermal straining of ice.
REFERENCES Cole, D.M., (1979), Preparation of polycrystalline ice specimens for laboratory experiments, Cold Regions Sci. Technol., 1(2) 153-159. Mellor, M., (1975), A review of basic snow mechanics, Proc. Grindewald Syrup., Internat. Comm. on Snow and Ice, IAHS-IUGG, IAHS Publication (114): 251-291. Mellor, M., (1980), Mechanical properties of polycrystaUine ice, Proc. Copenhagen Symp., Internat. Union Theor. Appl. Mech., Springer-Verlag, New York, pp. 117-245. Vinson, T.S., (1978), Dynamic behavior of ice under cyclic axial loading, J. Geotech. Div., Proc. Am. Soc. Cir. Eng., 104(GT7): 801-814.
41
C Y C L I C L O A D I N G A N D F A T I G U E IN ICE Malcolm Mellor and David Cole Cold Regions Research and Engineering Laboratory, Hanover, NH (U.S.A.)
(Received April 21, 1980;acceptedin revisedform May 22, 1980)
ABSTRACT
INTRODUCTION
Isotropic polycrystalline ice was subjected to cyclic loading in uniaxiai compression at-5°C, with stress limits 0 - 2 and 0 - 3 MPa, and frequencies in the range 0.043 to 0.5 Hz. Stress-strain records showed hysteresis loops progressing along the strain axis at non-uniform rates. The effective secant modulus, which was about half the true Young's modulus, decreased during the course o f a test. The elastic strain amplitude and the energy dissipated during a loading cycle both increased with increase o f time and plastic strain. Strain-time records gave mean curves which were identical in form to classical constant-stress creep curves, with a small cyclic aiternation o f recoverable strain about the mean curve. The inflection point o f the "creep curve", marking the transition from strain hardening to strain softening, occurred at a plastic strain o f l% (+-0.1%), which is about the same as the "ductile failure strain" found in constant stress creep tests and in constant strain-rate tests on ice o f the same type at the same temperature. The dissipation o f strain energy up to this "failure point" was much higher for the cyclic tests than for corresponding quasi-static tests - 100 to 600 kPa {or kN-m/m a) in comparison to about 30 kPa. The number o f cycles taken to reach the "failure point" was o f no direct significance, varying greatly with stress amplitude and with frequency. The results o f the tests suggest that maximum resistance under compressive cyclic loading occurs at an axial plastic strain o f about 1%, which is essentially the same as the failure strain for ductile yielding under constant stress and under constant strain-rate.
As far as we know, there have been no studies on failure* under cyclic loading, or on fatigue failure, for ice. Vibrational methods have been employed to study the complex modulus and complex compliance (see, for example, Mellor, 1975), but in most of these experiments the stress and strain amplitudes have been very small, linear viscoelasticity has been assumed, and neither failure nor permanent deformation have been considerations. Vinson (1978) subjected ice to low frequency cyclic straining, with confining pressure and fairly large strains, but he limited the number of cycles to less than 20 for any given set of conditions. Some years ago, one of us cycled uniaxial compressive stress on ice specimens, tracing out stress/strain curves from zero stress up to a moderately high stress and back down to zero, with a low frequency alternation (<0.1 Hz). The trace on the X-Y plotter showed a hysteresis loop for each cycle, with progressive displacement along tile strain axis after each cycle. The displacement per cycle decreased progressively during the rather brief test, until eventually the hysteresis loop appeared to be tracking over the same path repeatedly, although it was recognized that *"Failure" is an arbitrary term. For present purposes w e define failure, or incipient failure, as the limit c o n d i t i o n where there is a transition from strain-hardening to strainsoftening, i.e. the point at which the ratio of stress to strainrate is a maximum. Failure strain is thus given by the inflection point of a complete creep curve for constant stress, or by the peak of a complete stress/strain curve for c o n s t a n t strain-rate. Beyond this strain, the material becomes progressively weaker.
0165-232X]81[0000-0000/$02.50 © 1981 Elsevier Scientific Publishing Company
42 there was probably still an imperceptible progression. This was incidental to the work then in hand, the testing machine was an obsolescent type, incapable of suitable automatic control, and nothing more was done on repetitive loading. Since that time, constant-stress creep tests and constant strain-rate strength tests on polycrystalline ice have led us to the conclusion that the failure strain for creep rupture, at least in uniaxial compression, has remarkably little variation with stress, loading rate, input strain energy, and probably temperature (a more detailed description, with caveats, is given by Mellor, 1980). If this is true, the implications for ice mechanics could be far-reaching, since it opens up the possibility of specifying failure criteria rather simply in terms of strain, instead of in very complicated terms of stress. In considering the possible limitations of a strain failure criterion, one question that came to mind was whether fatigue failure, or failure after repetitive loading, would occur at the level of permanent strain (approximately 1% axial strain) found to define ductile failure under constant stress and under constant strain-rate. Because of the energy dissipation in each loading cycle (as given by the area of the stress/strain hysteresis loop), it seemed that the total dissipation of strain energy would be much greater for failure under cyclic loading. To explore some of these questions, a small experimental program was undertaken, and results are given here. The work is of a preliminary nature and more detailed tests, including conventional fatigue tests, are planned.
Tests were made in uniaxial compression, with stress cycled sinusoidally between zero and some selected value (roughly 30% of the uniaxial compressive strength for moderately high loading rates). Load was applied by a closed loop electro-hydraulic testing machine, which has a high capacity servo-valve to provide rapid hydraulic response. The actuator was programmed to vary load according to a haversine function, with preset load limit and preset frequency. Axial strain was measured by a displacement transducer mounted across the bonded end caps of the specimen. Force and axial displacement were recorded as functions of time on a strip chart recorder, and stress/strain loops were recorded on an X-Y plotter. To provide data for a parallel study, acoustic emissions from the specimen were monitored by transducers set close to, but not in contact with, the specimen. The narrow gap between the ice and the acoustic transducer was filled with silicone grease (holes were cut in the rubber membrane that surrounded the specimen to prevent evaporation). The specimen was housed in an environmental-control cabinet, with cabinet temperature controlled by a recirculated air refrigeration system to -5 ° -+0.2°C. At the end of each test the specimen was obviously deformed, but still intact. The axial plastic strain of 7% or more produced "barreling", and the surfaces had characteristic bumps and valleys representing outthrust grains and depressed grain boundaries ("alligator skin" surface). In the specimens that had been subjected to the highest stresses, internal microcracks were clearly visible. The used specimens were finally put into storage at -30°C for possible later study.
TEST PROCEDURE
PROCEDURE FOR DATA ANALYSIS
Specimens of polycrystalline ice were prepared by methods described earlier (Cole, 1979). The specimens were right circular cylinders, 50.8 mm diameter by 127 mm long. End caps of fiber-based phenolic resin were bonded on as an integral part of the specimen during the moulding process. The faces of the end caps were faced-off in a lathe so as to pluck the textile fabric and produce a "hairy" surface. Specimens were homogeneous and initially isotropic, with a mean grain size of 1.2 mm. The electrical conductivity of melt water from test specimens was 4 /aS.
in this preliminary analysis, the data are interpreted in a certain way so as to answer particular questions. The output from the test equipment is a continuous chart record of displacement as a function of time (an approximately sinusoidal wave with progressively increasi~ag mean strain), and a chart record of force as a function of displacement (hysteresis loops). The general characteristics of these records are shown in Figure 1. From these records, we obtain: (1) accumulated strain as a function of time and number of load cycles (creep curves), (2) the strain energy
43
dissipation per cycle, and the accumulated strain energy dissipation, (3) the amplitude or recoverable strain per cycle, and the effective secant modulus for each cycle. The stress o varies with time t in accordance with the haversine function, i.e.
plastic strain ~p is also assumed invariant with time. Thus the variation of strain e through a limited number o f stress cycles n can be described in terms o f the time t from the start o f the first o f these n stress cycles: e = ~1 emax [1 - cos(cot--6)] + ~-pt
a = (Omax/2)(1 - cos cot) = (Omax/2)(1 - c o s
27r/t)
- f i -1 emax [1 - cos(cot-6)] + ~-~ t
(1)
where amax is the maximum stress, co is the angular velocity, and f is the frequency. The variation o f strain e with time t is complicated, but an approximate description can be given for a limited number o f cycles b y making steady-state assumptions. The initial conditions at t = 0 are ignored because the transient phase relationship between stress and strain is difficult to define at the beginning o f a test. The range o f recoverable strain emax and the phase lag 6 are assumed invariant with time, and the mean rate o f
where epl is the plastic strain accumulated in one cycle and T is the wave period. To facilitate illustration o f eqn. (2) in a diagram, epl can be expressed as a fraction, or multiple, o f emax, such that epl/ emax = k and: e t emax = ]1 [1 - c o s ( c o t - 6 ) ] + k--T
cot
- i ' [1-cos(cot-~)] + / c - 27r -
point of Inflecl~on
' '
c ic
~Meono 5trointroinRRile roinin t, Time
Area Gives Energy Dissipated m One Cycle
Strain Accumulated in One Cycle
(3)
Fig. 1. (at. Simplified representation of a strain/time record, showing the cyclic alternation of strain and the accumulation of plastic strain with time. The time scale is condensed, while the cyclic strain amplitude and the wavelength are exaggerated. (b) Diagram showing part of a stress/strain record. The loop described by a stress]strain cycle gives the energy dissipation for the cycle, and the progression of successive loops along the strain axis represents the accumulation of plastic strain.
Recovetoble Strain Amplitude
E
(2)
i. 11
/
ZAccumulated Plastic Strain After
n Cycles
Slope Gives Secant Modulus
Strain
44 The strain/time and stress/strain characteristics described by eqn. (1) and eqn. (3) are illustrated in Fig. 2 by taking as convenient values/5 = Ir/24 = 7°30 ', k = 0.1, so that G
- { ( 1 - c o s wt) Gmax (4) - ~ 1-cos
emax
t---
*
24
10 21r
In the foregoing equations the dissipative component of the recoverable anelastic strain is described solely by the lag angle 8, while the irrecoverable viscous creep is described by epl or ~p. If 8 = 0 and epl = 0, the material is purely elastic, with emax = Omax/E, where E is Young's modulus. With/5 :/: 0 and epl :/: 0, an effective secant modulus E s is defined by the slope of a line joining the upper and lower limits of a
(Tma* 0 4
7r
[
I 2Tr
3~r
I 4~
5~
I STr
08
Ep
04
~ot
!
lot
hysteresis loop; E s is the reciprocal of the change in e[emax between successive points where tr = 0 and 17 = O ' m a X .
In the cyclic load "creep curves", the behaviour is qualitatively identical to the behaviour displayed in conventional constant-stress creep curves, i.e. creep deceleration (strain hardening) followed by a transition to creep acceleration (strain softening). In accordance with the procedure adopted for the interpretation of conventional complete creep curves (see Mellor, 1980), the inflection point of the cyclic load "creep curves" is used to define failure (~p = 0), which in this case could perhaps be called fatigue failure. The inflection point is best found by replotring the mean strain rate for strain accumulation (slope of the "creep curve") against time and number of cycles, or alternatively against accumulated strain (Figs. 6 and 7). The area of a stress/strain hysteresis loop gives a measure of the strain energy dissipated in one loading cycle. Since the shape of the hysteresis loop changes only slowly after the first few cycles, representative measurements of area can be made for sets of loops that are essentially identical. The strain energy dissipated up to the failure point (i.e. the inflection point of the "creep curve") is the summation of hysteresis losses, which includes the progressive displacement effect. The phase lag/5 was not monitored systematically, and it proved difficult to measure /5 reliably from occasional scale expansions on the strip charts (see Fig. 3,b,c,d) because of imperfections in the pen alignments. Measurements of/5 were therefore obtained from the discontinuous stress/strain records by comparing times for maximum stress and maximum cyclic strain.
RESULTS
o
o~ A_ o8
,!o '
,!~
~mox
Fig. 2. Illustration of the steady-state stress/strain/time relationships described by eqns. (1-4). (a) alternation of stress and strain with time (or angle), (b)stress/strain hysteresis loops.
A representative sample of strain/time records is shown in Fig. 3, "while examples of stress/strain records are given in Fig. 4. Details of all the tests that were properly completed are given in Table 1 (three additional tests were terminated prematurely by power failure, recorder problems, or equipment uncertainties).
45
Fig. 3. Representative strip-chart records giving deformation and load as functions of time. These results are for test FT-30-2, with peak stress 3 MPa and frequency 0.077 Hz: (a) start of the test, (b) after 1% plastic strain and approximately 200 cycles, (c) after 3.5% plastic strain and 540 cycles, (d) after 6.8% plastic strain and 750 cycles.
TABLE 1 Cyclic loading tests on polycrystaUing ice at -5°C Test no.
Maximum axial stress Omax (MPa)
Frequency (Hz)
Minimum mean creep rate ~min (s-l)
Number of cycles to ~-min
Time to ~min (s)
Total strain at ~-min (%)
Strain energy per unit volume to ~min (kN-m/m3=-kPa)
FT-20-1 FT-20-2 FT-20-3 FT-25-3 FT-30-1 FT-30-2 FT-30-3
2.0 2.0 2.0 2.52 3.0 3.0 3.0
0.043 0.077 0.50 0.10 0.043 0.077 0.50
1.24 8.56 6.93 1.51 3.01 2.95 2.89
286 548 5,000 600 96 168 1,035
6,580 7,120 10,000 6,000 2,210 2,180 2,070
1.05 1.08 1.13 1.07 0.90 0.93 0.90
118 149 628 190 96.4 136 306
X 10 -~ × 10 -7 X 10 -T X 10 -~ X 10 -~ X 10 -~ X 10 -n
~,~
l ~ - ~ Q ~ ::,~o
Fig. 4. Typical records of load versus deformation, taken directly from the X-Y plotter; (a) sequence of hysteresis loops at the start of a test, showing initial transients, (b) loops near the plastic strain that correspond to minimum mean strain rate, (c)loops recorded near the end of the test, at comparatively high plastic strains.
............ ~
.............. :~..
~
4~
47
Figure 5 gives "creep curves" that plot the accumulated creep strain (summation of the residual strain at the end of each load cycle) against time and against the number of cycles. The amplitude of recoverable strain oscillations during load cycling is also indicated. Figure 6 gives plots of the mean strain rate
12F ZO~ ~
Z
'
against time, with indication of the number of cycles. Values of ~ are measured from chart records such as those shown in Fig. 3, and ~ at any time is the slope of the "creep curve", as illustrated in Fig. 5. The minimum on a curve of this type is taken as the indicator of "failure", i.e., it gives the minimum value of ~ for given stress conditions. Thus Fig. 6 gives time-to-failure and the number of cycles for failure.
7
/ -E~
~
I01~ i~
'
$~r(IAmpl#ude m
o5r 0
q
.
//f
f ~'~"8 ~ ~-~ l
02
04
Time(s)
5xlO"
"~
06
,
[
08
IOMO4
.
/
FT20-2
Z/O'077Hz FTgO'I oo4~H
/
I
--
i
/
FT20-3
~
FT25-3
/
/ ///4
2,000 cycle;/ /.cycles//
4~
(G~..o=.2.52 M
!
~
StrQin 0'0Hz / Amplitude / /I,000 /
[
2O0O
oooc ce,
2,000 cycles//
~
~
~
/ ( _ .
~
I0,000 ~ [I
Io,
.,
, io
2
l
Time(s)
5
r
~
,
~ ....... 4
"
050/.~/_
2,000 cycles
4//~
I
r
cycles T=-5°C
Jxlo4
'
SlrolnAmplt~d~ FT30-3
0
]
2
I
3
Time (s)
T30=30MPe T=-5*C
F'T30-2 0043H O077Hz
O'ree•
400 cNles r w ¢ycle~. 700( # Strmn
-~
~,~,% ~
41104
1
'FT303 050 Hz
o
FT 20-3
/
/
~; 2
o
, o
. . . . . . .
0
FT20-2 ,
FT25-3
0.soHz
o6p
/ FT 20-1 O.043Hz
/
'
,o
~¢~
A~p~ //
cycles
700 cycles
°077.,
g ,2
CrY"=30 MPa T=-5°C ] 8
FT30-3 050Hz cyck
5O0O
i
I
Time (s)
400 c les yc
FTSO-2
1
L I0
•
08 Z
t J 12x rO~
Fig. 5. Creep curves showing the accumulated plastic strain as a function of time. The magnitude of the total amplitude of recoverable cyclic strain is indicated, as axe the numbers of stress cycles: (a) Tests at 0 - 2 MPa, (b) Tests at 0 - 3 MPa.
n
i 2
i
I 4
l
I 6 T~me l=)
E
I 8
t
I K)
l
12~103
Fig. 6. Curves giving the mean creep rate as a function of time: (a) tests at 0 - 2 MPa, (b) tests at 0 - 3 MPa.
48 Mean strain rate can also be plotted against accumulated strain, as in Fig. 7, to give a direct indication o f failure strain. The recoverable strain amplitude for one load cycle did not vary much during the course of a test, but there was usually a monotonic increase of strain amplitude with time and number of cycles• In Fig. 8 the data for all tests are combined by normalizing strain amplitude with respect to the amplitude at the failure stage, i.e. at ~-min.
141
I
i
°t
-o
~'~:-~"
~
" O
ul _~ 0 E ~ Z
D
.
(o) (~) (n) (.) (.) (,) (,)
r-~ S
•
0.8~
<
060
•
I
Sample Frequency (Hz) FT20- I 0043 FT20-2 0077 FT20- 5 050 FT25-5 010 FTSO- I 0045 FT30- 2 0077 FT30-5 050
,I J
J
i
q
I
2
£plEf~ Normalized
6 Plastic
8
10
Strain
, x l 0 -6 '
I
f
I
W
I
i
Fig• 8. Amplitude of the recoverable elastic strain per cycle plotted against the accumulated plastic strain. The values of strain amplitude A have been normalized with respect to the value A. occurring at the point where the mean creep strain rate is a minimum (~min)- The values of accumulated plastic strain ep have been normalized with respect to the "failure strain" ef, measured where ~min occurs.
~mQx=2.0 MPO FT20-1
T=-5*C
0
0
~
FT 2 0 - 2 0 . 0 7 7 Hz
I 2
_J I , 4 A c c u m u l a t e d Creep Strain (%)
I 6
,
20x106
k
o
/
2
/
/F730-2
4 6 Accumulated Creel) Strain (%)
q
e
to
Fig. 7. Mean creep rate as a function of accumulated plastic strain: (a) tests at 0 - 2 MPa, (b) tests at 0-3 MPa.
The energy dissipation per cycle also increased progressively during the course of a test, but the variation was greater than the variation o f recoverable strain amplitude (since the energy dissipation includes the non-recoverable or permanent strain)• Figure 9 combines data for all the tests by normalizing with respect to the energy dissipation per cycle at ~min. Both the cyclic strain amplitude and the energy dissipated per cycle increased with the stress amplitude (or peak stress), but tests were made at only two stress levels (with one exception). Variation of frequency at a given stress level did not have much effect on cyclic strain amplitude over the range covered in the tests, and the energy dissipation per cycle did not appear to vary systematically with frequency. Measured values of the lag angle ~ were plotted against total plastic strain, and lines drawn through the data points are shown in Fig. 10. The actual points, which have been omitted for clarity, scattered about the lines'by up to +2 ° .
49
2.4
'
e (o) (~) (o) (-) (.) (*) (,)
,O
2.0
=*, I.aJ
._= 1.6 o
I
I
I
'
Sample Frequency (Hz_) FT20-1 0.043 FT20-2 0.077 FT20-3 0.50 FT25-5 0.10 FT:30- I 0.045 FTS0-2 0.0?7 FT30-3 0.,50
I
l
1
'
l
o
o
c* o
(/)
z~o
,o
o
1.2 ..a
d
N
o.87
~*0.4 J 2
0
I
I 4
L
1 6
Ep/Ef. Normalized Plastic Strain
I
] 8
I
1
I0
I'1
115
Fig. 9. Energy dissipation per cycle plotted against accumulated plastic strain. Values o f dissipated energy w are normalized with respect to the value w , occurring at the point where mean creep strain rate is a minimum (~min)- Values o f accumulated plastic strain ep are normalized with respect to the "failure strain" ef, measured where ~min occurs.
45 °
I
I
i
I
I
l
]
I
r
40*
35 °
~
30 °
3 ~d 25 o
20 o
15
i
I 2
i
I
J
4
Accumulated
I 6
Strain
i
I
R
8
(%)
Fig. 10. Summary o f lag angle data.
DISCUSSION The main purpose of the experiment was to find how the failure strain, as defined by the transition from strain hardening to strain softening, might be affected by load cycling and by large variations in the
total dissipation of strain energy prior to failure. The simple answer appears to be that, for the range of conditions in the investigation, these things have no significant effect on the failure strain. The failure strain for low frequency cyclic loading in uniaxial compression at -5°C had a mean value of 1.01%, with a standard deviation of 0.10%. The failure strain appeared to decrease as the maximum stress increased, although the evidence is scant: at 2.0 MPa the mean failure strain was 1.09%, at 2.52 MPa the single value was 1.07%, and at 3.0 MPa the mean value was 0.91%. In the tests with amax = 2.0 MPa the failure strain appeared to increase slightly with increase of frequency, but there was no systematic variation with frequency in the tests at Omax = 3.0 MPa. In conventional quasi-static compression tests on ice of the same type at -5°C (not yet published), we have obtained mean values of failure strain that are quite close to those given above. For constant load creep tests, with stresses in the range 1.25-3.68 MPa, the mean failure strain was 0.96%, with a standard deviation of 0.14%. For tests under constant displace-
50 ment rate, with strain rates in the range 1.64 X 10 - 6 5.1 × 10 -s s -t, the mean failure strain was 1.05%, with a standard deviation of 0.11%. In these tests there was no significant correlation between failure strain and applied stress; or between failure strain and applied strain rate. In the cyclic load tests, the dissipated strain energy per unit volume up to the failure point ranged from about 1 0 0 - 6 0 0 kPa (i.e. kN-m/ma). By contrast, the strain energy dissipated up to the failure point in the quasi-static tests was much lower. In the constant stress creep tests, the strain energy dissipation was about 31 kPa, while in the constant strain rate tests it was about 26 kPa. All o f these tests are assumed to be isothermal, with absorbed strain energy eventually being dissipated by heat loss from the specimen. If, by contrast, the tests were to be made under adiabatic conditions, the absorbed strain energy would presumably raise the temperature of the specimen and ultimately cause internal melting. The creep curves obtained by plotting accumulated strain against time have the same form as conventional creep curves for constant stress tests. This is perhaps not so surprising, since cyclic tests can be regarded as creep tests in which there is oscillation about an effective mean stress. In attempting comparison between conventional creep curves and curves for cyclic loading, the effective mean stress cannot be taken as the arithmetic mean stress, since the stress/ strain-rate relation is known to be strongly non-linear. We have therefore taken values of ~min and compared them with a stress/strain-rate curve representing the
relations for @/O)min at -5°C, as derived from data for conventional quasi-static tests. Results of the comparison are shown in Table 2, and it seems that values of ~min for the cyclic creep curves correspond to the minimum strain rates o f conventional creep curves for a constant stress oe equal to about 73% of the maximum stress Omax applied in the cyclic tests. Following a procedure suggested to us by Samuel Colbeck, it can be shown that this ratio Oe/Omax is what might be expected for plastic straining according to a power relation ~ = o n , where n is approximately 4. Assuming that the anelastic strains balance out during a complete loading cycle in the vicinity of e~min, the mean plastic strain rate for one cycle is co -
27r
2~r/w f ~ dt
(5)
0
where ~ is the instantaneous plastic strain rate and co is the angular frequency. If the stress o varies cyclically according to 1
o =" ~ Omax (1 - cos co t)
(6)
and strain rate is related to stress by (7)
= Ao n
then 2----~"
f
(1 - cos 60 t ) n
Effective stress for comparison with quasi-static tests Frequency
emin
(Hz)
(s-1 )
Equivalent stress from conventional tests, Oeff
Ratio of effective static stress to maximum cyclic stress, Oe/Omax
(MPa)
FT-20-1 FT-20-2 FT-20-3 FT-25-3 FT-30-1 FT-30-2 FT-30-3
0.043 0.077 0.50 0.10 0.043 0.077 0.50
1.24 8.56 6.93 1.51 3.01 2.95 2.89
X I0-' X X X X X X
I0 -~ I0 -~ 10-' I0-6 I0-' 10-'
(8)
For n = 3, the value of the integral is 5 z r / w . For. n = 4,
TABLE 2
Test no.
d t
0
1.56 1.46
1.49 1.68 2.19 2.16 2.15
0.78 0.73 0.75 0.67 0.73 0.72 0.72
51
the value is 35 zr/4co. If ~- for cyclic loading up to amax is equal to ~ for creep under constant stress oe, then Aw(omax~ A on = - - ~ - - - ~ )
f27r/to( 1 - c o s w t ) n d t .
(9)
0
With
n = 3
ae/Crmax = 0.679
(10)
With n = 4 Oe/Omax = 0.723
(11)
The latter value is almost exactly equal to the mean of the values in Table 2. Traditional descriptions of fatigue failure are given in terms of the number of stress cycles up to a failure point, which is typically identified with fracture of the specimen. The number of cycles to cause failure is recognized to be a function of stress or strain amplitude, or of the mean stress, but frequency of cycling is not normally considered. The present tests are significantly different from typical fatigue tests on manufactured materials, since there is no question of stress being below the elastic limit, there is no external tensile stress, and the failure is ductile. For ice under these conditions, the number of cycles to failure is an unsatisfactory description of fatigue life. With a stress range of 2.0 MPa, the number of cycles to failure varied by a factor of 17 as frequency changed by a factor of 12. With a stress range of 3.0 MPa, the number of cycles to failure varied by a factor of 11 as frequency changed by a factor of 12. In both cases the number of cycles increased as frequency increased, and the number of cycles needed to reach the failure point was greater at the lower stress level. The variation of recoverable strain amplitude per cycle with the progressive accumulation of plastic strain is not established very clearly in Fig. 8. The data are badly scattered, but they do exhibit a general trend, the strain amplitude increasing as the test
progresses and plastic strain accumulates. In other words, the effective modulus appears to decrease as plastic strain increases. Fig. 11 shows this more directly by plotting the secant modulus for one cycle against the accumulated plastic strain. The apparent modulus tends to decrease with the time and net strain. The initial values are about half the value for the true Young's modulus, as measured by the initial tangent modulus in rapid tests, or else by high frequency vibrational methods that impose small strain amplitudes. Values from the 3 MPa tests appear to be lower than those from corresponding 2MPa tests (discounting results from test FT-20-3), which is in accord with Vinson's (1978) finding that the effective modulus decreased with increasing strain amplitude. The normalized data showing the variation of absorbed strain energy with the net plastic strain define a fairly clear trend. The ice becomes more dissipative, or "lossy", as the test progresses. The increase of hysteresis loss as the material strain-softens is quite orderly for plastic strains above 1%, but the data are more confused between 0 and 1% plastic strain. It is too early to decide whether the suggestion of a "hump" around 0.3-0.5% plastic strain in Fig. 9 is a real feature, but it might be noted in passing that some quasi-static tests display an initial yield point in this range of strain, even while the general trend is for continued strain-hardening. The lag angle ~ is related to the viscosity of the ice, and in broad terms it is expected to increase as viscous behaviour becomes increasingly significant in relation to elastic behavior. In Fig. 10 the 6 values for tests up to 3 MPa are systematically higher than those for tests up to 2 MPa. In the 2 MPa tests, 8 increases as frequency decreases, and in the 3 MPa tests the same overall trend occurs. An increase in ~ with increasing stress and decreasing frequency is what might be expected for a material with nonlinear viscous behaviour. The overall increase of 6 with increasing plastic strain, seen in all tests, is another indication that the ice tends to become more dissipative as strain accumulates.
52
'
I
I
'
'
I
i
~5
FT20-3
~E ~4
._>"
FT2~2-
o
F.,3
to be caused by bonding of end cops
Believed imperfect
m
FT 20-1
l
I
J
I
4 Strain (%)
Accumulated
i
I
l
6
I
I
l
I
I
F T 30-3
FT 30-2
o
FT 30-I
= o
I 2
,
I I 4 Accumulated Strain (%)
I 6
l
I 8
Fig. 11. Secant modulus as a function of accumulated plastic, strain: (a) tests at 0-2 MPa, (b) tests at 0-3 MPa.
CONCLUSION
Cyclic loading tests yield interesting information on the viscoelastic properties o f ice, and on failure criteria. An extension of such tests to cover tensile loading and tension/compression alternation would be worthwhile.
Perhaps the most interesting outcome o f the present tests is the finding that ductile failure in compression, as defined by a transition from strain hardening to strain softening, occurs at an approximately constant level of plastic strain, irrespective of the number of stress cycles needed to reach that strain, and apparently irrespective of the strain energy
53
absorbed and dissipated up to that point. It is particularly significant that this "failure strain" (about 1% axial) is approximately equal to the ductile failure strain found in quasi-static tests under constant stress and constant strain rate. If tests for other loading conditions and stress states produce similar results, then it seems possible that fatigue life for ice (and conceivably for some other materials) might be defined and monitored very simply in terms of accumulated plastic strain. The results of cyclic loading tests are directly relevant to some practical problems, including flexure o f ice plates by waves or wheel loads, and cyclic thermal straining of ice.
REFERENCES Cole, D.M., (1979), Preparation of polycrystalline ice specimens for laboratory experiments, Cold Regions Sci. Technol., 1(2) 153-159. Mellor, M., (1975), A review of basic snow mechanics, Proc. Grindewald Syrup., Internat. Comm. on Snow and Ice, IAHS-IUGG, IAHS Publication (114): 251-291. Mellor, M., (1980), Mechanical properties of polycrystaUine ice, Proc. Copenhagen Symp., Internat. Union Theor. Appl. Mech., Springer-Verlag, New York, pp. 117-245. Vinson, T.S., (1978), Dynamic behavior of ice under cyclic axial loading, J. Geotech. Div., Proc. Am. Soc. Cir. Eng., 104(GT7): 801-814.