Introduction

Reference Nixon, Schulson, Murthy, Connor and BrebbiaNixon and Schulson (1986) showed that, at −10°C, the fracture toughness (K_c ) of fresh-water ice increased as the loading rate decreased below . Above this loading rate the toughness was constant. They explained this transition by considering the size of the creep zone at the crack tip. By analogy with the American Society for Testing and Materials (1981) fracture-toughness testing code for metals, they defined the transition loading rate at which plane-strain conditions were violated as that loading rate at which the size of the crack-tip creep zone (Reference Riedel and RiceRiedel and Rice, 1980) exceeded l/50th of the notch depth (N.B. in a different sample geometry, another sample dimension may be critical in this regard). The value of thus calculated was in good agreement with the observed behavior of K_c with loading rate (Reference Nixon, Schulson, Murthy, Connor and BrebbiaNixon and Schulson, 1986, 1987). Ralston (personal communication) suggested that at the lowest loading rate a notch-strengthening process may be occurring. Notch strengthening implies that the nominal section fracture stress across a notched sample is greater than that across an unnotched sample loaded under similar conditions (i.e. φ_nn > テ_un;sww Fig. 1).

Notch strengthening is well documented for metals and normally arises because the triaxial stress at the notch tip suppresses yield until the axial stress across the notch is considerably greater than the yield stress, σ_y. From slip-line field theory, a value of 3σ_y is generally taken as the maximum possible notch yield stress, though Reference OrowanOrowan (1945) has shown that the maximum value of this “plastic constraint factor” is 2.57. However, if the notch stress, while still too low to cause yielding in the notch, exceeds the brittle-fracture stress (σ_f ) of the material, then brittle fracture may occur. In such cases a material is termed notch brittle. In steel one finds that above a certain temperature the material is notch tough because both σ_ν and ЗОу (the unnotched and notched yield stresses) arc less than σ_f. As the temperature decreases below this temperature, the steel becomes notch brittle (that is σ_y is less than σ_f, but 3σ_y is greater than σ_f) until, at some still lower temperature, it is fully brittle.

Fig. 1. Notched and unnotched samples.

The presence or otherwise of a notch-strengthening effect in ice is of more than academic interest. It is often noted that sea ice has many defects (e.g. cracks, polynas, pores, etc.) and the implication is drawn that the defected ice should be weaker than unflawed laboratory ice. Recent work (Reference Schulson, Hoxie and NixonSchulson and others, 1989) shows that short cracks have no effect on the tensile strength, and that even long cracks which have blunted appear not to weaken the material. The effect of existing cracks on the strength of ice is thus far from clear. A study of notch-strengthening effects may help to clarify this complex problem. The purpose of this paper is to describe experiments designed to investigate the notch-strengthening effect in ice and to explain the results obtained in a micromechanical context.

Experimental Procedure

Cylindrical samples of randomly oriented polycrystalline fresh-water ice were made using the procedure described by Reference LeeLee and others (1984).Specimens made in this manner had an average density of 916 kg m⁻³ (i.e. 0.11% porosity) at −°C.Melt-water conductivity of the ice was 8.8×10⁻⁶mho/cm, also at −5°C.The ice appeared to be very clear. What few bubbles were found tended to be along the central axis of the sample, as might be expected fromthe freezing process. Because it has been shown that both the fracture toughness (Reference Nixon and SchulsonNixon and Schulson, 1988) and the tensile strength(Reference Lee and SchulsonLee and Schulson, 1988) of fresh-water ice are grain-size dependent, tests were performed on samples with average grain-size (as measured by the linear intercept method) between 7.3 and 2.2 mm. Tensile tests were performed on a closed loop servo-hydraulic testing machine placed in a cold room with a temperature of −10.0 ±0.2°C. All tests were performed in load control adjusted so that the stress rate in the unnotched samples, σ_un, was equal to the nominal notch stress rate in the notched samples . It should be noted that tests were not run in strain control (as is normal practice) because strain, and thus strain-rate, varies significantly with position in the notched samples. The notched and unnotched samples had an initial diameter of 91 mm. In the notched samples, this was reduced at the notch to 72 mm. The notch half-angle was 22° (see Fig. 1). Sample lengths between end caps were 254 ± 3.2 mm, and the notches were placed within 13.2 mm of the sample midpoint. Samples were notched to within 0.25 mm of their final depth on a lathe in a cold room at −14°C with a shaped notching tool. These blunt notches were then sharpened to their final depth with a razor blade mounted in a special holder for use on the tool post of the lathe. Tests on notched samples were performed between 12 and 16 h after notching. Prior to testing, samples were stored in insulated boxes at the test temperature. Since tests lasted typically 3 h, all samples were wrapped in plastic “clingfilm” to avoid sublimation during testing. Strain was measured by means of two extensometers, as used by Reference LeeLee and others (1984).

Results

Test results are presented in Table I. For the most part there is little scatter in the results, the exception being the strengths of the notched samples of 3.5 mm grain-size. The unnotched samples showed increasing non-linear load-extension behavior as the grain-size decreased. The larger-grained (d > 3 mm) notched samples showed no non-linear behavior, while for the notched samples with grain-size less than 3 mm some ductility was evident. Typical load-extension curves for the tests are shown in Figure 2. Figure 3 shows the failure loads for notched and unnotched specimens as a function of grain-size. From this it is clear that a greater load is required to break an unnotched specimen than a notched specimen. However, when failure stress, rather than failure load, is considered (see Fig. 4), a different trend emerges. It can be seen that there is a definite notch-strengthening effect at the larger grain-sizes and that it diminishes as the grain-size decreases and becomes negligible for the finest-grained aggregates. A significant result, discussed below, is that while all unnotched samples had, after failure, other cracks present away from the fracture path, none of the notched samples had any remnant cracks.

TABLE 1. Notched and unnotched tensile strengths of fresh-water ice

Fig. 2. Load-cxtension curves: (a) large-grained, notched; (h) small-grained, notched; (c) large-grained, unnotched; (d) small-grained, unnotched.

Fig. 3. Failure load vs grain-size.

Discussion

It should be noted that ductility in the samples for this study was less than in samples tested under strain-rate control by Reference Lee, Currier, Lim and SchulsonLee and Schulson (1988) at similar rates of loading. The reason for the reduced sample extension at failure in these tests, as compared with those of Lee and Reference Schulson, Lim and LeeSchulson (1988), lies in the fact that the tests herein were performed under “load control”. By using the load feed-back control(the load was increased at a constant rate; see above), the rate of straining increases within the sample as soon as non-linear load-extension behavior begins, which promotes earlier fracture than in constant strain-rate tests. Thus, while the fracture stress is similar to those of Lee and Schulson, the fracture strains are somewhat less (Figure 5 shows a stress-strain curve from Lee and Schulson’s work for comparison with Figure 2).

Fig. 4. Failure stress vs grain-size.

Fig. 5. Stress—strain curve from constant strain-rate test (Reference Lee and SchulsonLee and Schulson. 1988).

As noted by Reference Schulson, Lim and LeeSchulson and others (1984), in low strain-rate tension tests on ice , the onset of non-linear stress-strain behavior is linked with the initiation of cracks. Under such strain-rates, cracks initiate and further stress must be applied for those cracks to propagate. The failure mode in such low strain-rate tests can be clearly identified as one of propagation of pre-existing cracks. Two observations support this. First, in this study and others (Reference Schulson, Lim and LeeSchulson and others, 1984; Lee, unpublished), remnani cracks were observed in samples tested at low rates, clearly indicating that cracks had initiated prior to final failure. Secondly, in a number of individual tests, cracks observed prior to failure lay on the actual failure surface. These two sets of observations, further supported by theoretical discussion (Reference Schulson, Lim and LeeSchulson and others, 1984), appear to indicate conclusively that at low strain-rates failure in tension is by crack propagation rather than crack initiation. However, at higher strain-rates (), again under tension, the stress to cause crack initiation is greater than that to cause the initiated cracks to propagate, and thus fracture occurs immediately upon crack initiation. The process of crack initiation was presumably one of microplasticity (Reference Schulson, Lim and LeeSchulson and others, 1984; Reference Lee, Currier, Lim and SchulsonLee and Schulson, 1988), possibly involving the piling up of dislocations at grain boundaries.

In the vicinity of a notch, such plastic flow will be suppressed because of the triaxial stress state there. In terms of principal stresses, σ₁, σ₂, σ₃ (with σ₁ > σ₂ > σ₃), as σ₂and σ₃ become non-zero and positive σ₁ must increase to a higher value than in the unnotched case for plastic flow to occur. Of course, if the notch is sharp enough to act as a crack then initiation is not needed. However, in low-rate tests, because of crack-tip creep and the test time involved, we may expect significant crack-tip blunting. In fact, a simple calculation (see Appendix) of the crack-tip creep-zone size (Reference Riedel and RiceRiedel and Rice, 1980) shows that for the notched sample the creep zone develops fully across the sample during the tests. Hence, if notch strengthening does occur, we would expect to find it most prevalent at low rates and would observe two effects. First, the nominal fracture stress of notched samples will be greater than those of unnotched samples. Secondly, if plotted against (grain-size)^−½, the notched ice strengths should have a typical Hall-Petch dependence on grain-size:

(1)

where σ₀ and k: are constants, in constrast with the unnotched low-rate strengths (cf. Reference Lee and SchulsonLee and Schulson, 1988) which are expressed as:

(2)

where K is also a constant directly related to the fracture toughness of the ice. Further, we would expect the values of σ_nn at a given grain-size to be between two and three times the value of the stress at which crack nucleation commences (causing non-linear stress-strain behavior) in an unnotched sample of the same grain-size loaded under similar conditions.

Figure 6 shows the test results plotted against (grain-size)^−½. Also shown are the tensile test results obtained by Lee and Schulson (1988) for fresh-water ice strained at a strain-rate of 10₋₇ s⁻¹ at a temperature of −10°C. As can be seen, the unnotched samples show good agreement with the results of Lee and Schulson, as would be expected. However, the notched samples exhibit higher nominal strengths at large grain-sizes. The line of best fit for the notched samples is shown (correlation coefficient, r² = 0.82) and it has a large non-zero intercept giving values for Equation (1) of σ₀ = 0.80 MPa and k = 0.33 MPa mm^½. It is possible that the trends of strength with grain-size observed herein result from samples having too few grains across a diameter to exhibit true polycrystalline behavior. The question of how many grains are required across a sample diameter for a “good” test is one generating much debate at present, and no unambiguous and generally applicable answer has yet been obtained. In this study, it is not felt that the sample diameter/grain-size ratio is too small. Whilst for large-grained specimens this ratio is ∼10 for the notched and ∼13 for the unnotched samples, for the fine-grained specimens the ratios are ∼33 and ∼41, respectively. Assuming the fine-grained samples have a sufficiently high specimen size/grain-size ratio, it seems unlikely that the trends observed in fine-grained samples consistent with Equations (1) and (2) would continue in the large-grained samples as a result of too few grains being present in the sample. Further, if we compare the notched strengths with the stress for a 0.005% strain offset obtained from Lee (unpublished), we find (see Fig. 7) that the notched strengths are about two times greater than these offset strengths. The use of a 0.005% offset can be justifed because the initiation of the first cracks coincides with the onset of non-linear behavior on the stress-strain curve, and 0.005% strain gives the smallest discernible offset. If the failure of the notched samples were controlled by initiation of cracks from the notch root, we would expect the initiation to occur when the stress state local to the notch was suitable for microplasticity to occur (assuming, as noted above, that this is the crack-initiation mechanism). The shape of the notch causes a triaxial stress state at the notch which inhibits plastic flow and thus the stress to cause plastic flow in the notch region would be two to three times as high as in an unnotched sample. The fact that the notched strengths are approximately twice the offset stresses measured by Lee (unpublished) appears to provide qualitative support for the notion that the notch-strengthening effect observed in these tests is indeed due to notch constraint inhibiting initiation at the blunted notch tip. Thus fracture was controlled by the initiation of a crack from the notch root, which immediately propagated. In constrast, in the unnotched tests a number of cracks was initiated before the stress increased to the point at which one of those cracks propagated. This model is supported by the observation that there were no remnant cracks observed in the broken notched samples, a result which contrasts with the remnant cracks (up to 20 in a sample) found both in the broken unnotched samples in these tests and in the low-rate tension tests performed by Lee and Schulson (1988).

Fig. 6. Failure stress vs (grain-size)^−½ .

Fig. 7. Failure stress (notched) and unnotched “yield stress” (from Lee. unpublished) vs (grain-size)^−½.

Conclusions

The following conclusions may be drawn:

1. At low loading rates and large grain-sizes (d ≥ 5mm), polycrystalline fresh-water ice does exhibit notch-strengthening behavior (at −10°C).

2. This notch-strengthening effect diminishes and disappears as the grain-size is decreased below 3 mm.

3. The notch-strengthening effect is consistent with a model in which the presence of the notch inhibits crack initiation and causes fracture to be controlled by the stress at which a crack initiates at the notch root, while for unnotched samples the fracture stress is that to cause one among several previously initiated cracks to propagate.

Acknowledgements

The authors would like to thank Dr T.D. Ralston for useful discussions. The work was supported by the U.S. Army Research Office, contract no. DAAG29–84-K-0031, and was perfomed in the Ice Research Laboratory (IRL) of the Thayer School of Engineering, Dartmouth College. The IRL is a co-operative university/industry/government laboratory supported by the following organizations: Amoco Production Company, Exxon Production Research Company, Mobil Research and Development Corporation, Shell Development Company, Standard Oil Company, the U.S. Army Research Office, the U.S. Office of Naval Research, the U.S. National Science Foundation, the Maritime Administration, the Minerals Management Service, and U.S. Coastguard. This support is gratefully acknowledged.

Appendix

Reference Riedel and RiceRiedel and Rice (1980) gave an expression for the maximum size of the crack-tip creep zone, which, taking the value of the creep exponent n from the power-law creep equation:

(A1)

to be n = 3 (Reference Barnes, Tabor and WalkerBarnes and others, 1971), gives the value of the creep-zone radius, r _cr to be:

(A2)

where E (Young’s Modulus) = 11.8 GPa (Reference Gammon, Kiefte, Clouter and DennerGammon and others, 1983); B (at −10°C) = 1.62 × 10⁻⁷ s⁻¹ MPa⁻³ (Reference Barnes, Tabor and WalkerBarnes and others, 1971); K (rate of increase of K₁) = K_c = 120 kpThis gives a value of r_cr of

Since the creep zone will grow from all around the notch, this result for the value of r_cr shows that at failure the whole of the region within the notch is within the creep zone (i.e. 2r _cr > D_n.