Introduction

The ability to predict natural fragmentation of metals is of critical importance in a range of industrial and military applications. As such, there is a large body of research dedicated to understand the factors and relationships that link the material properties and imposed conditions (e.g. strain rate) to the size and velocity of fragments produced in a natural process. However, this remains a challenging problem, since the natural fragmentation process is stochastic on the macro-scale and occurs very rapidly, making reproducible experiments difficult to perform.

The natural fragmentation problem is simplified by removing the complicating effect of realistic geometries. Given its simplicity, the expanding cylindrical ring is a popular test that produces natural fragmentation. A typical cylindrical ring test consists of a thin walled (\(\simeq 1\,\)mm), short (\(\simeq 1\,\)mm) cylinder of diameter \(< 30\,\)mm [1]. These rings are then expanded at high strain rate, with the force driving the expansion coming from explosives or magnetic induction. The magnetic induction method is preferred to provide a reproducible and consistent expansion velocity. Magnetic methods can produce radial ring expansion velocities up to 300 m s\(^{-1}\) [1]. Providing the expansion of the ring can be measured accurately and the driving impulse is well known, a ring expansion test can be used to determine both the high strain rate constitutive behaviour (stress–strain response) and the fragmentation behaviour [1, 3].

The classical model by Mott [4] predicts that there is a characteristic distribution of fragment sizes after high strain rate fracture of a ring. In Mott’s analysis, the material property that controls the fragment size distribution is the scatter in the local fracture strains and it is thus a statistical model. Other models for fragmentation consider fragmentation based on the energy required to generate the necessary fractures [1]. Although based on different theory, both approaches can give similar predictions, which are in reasonable agreement with experiment [1].

Grady has reviewed the results of several ring fragmentation studies and compared them to theory [1]. One intriguing plot is presented in this review (Fig. 8.11 [1]) in which the number of fragments for two distinctly different alloys collapse onto a single curve when plotted as a function of the expansion kinetic energy. However, as Grady notes, this observation for only two materials from two studies could certainly be fortuitous and warrants further study.

The objective of the present analysis is to perform this further study by compiling data from a wide range of ring fragmentation experiments and investigating whether simple scaling to specific kinetic energy still holds. The ability of such scaling to predict the number of fragments has been compared to that of the statistical and energetic fragmentation models. This in turn can focus efforts to develop better models for natural fragmentation.

Methodology

Fragmentation data have been gathered for a wide range of alloys from previous literature studies of ring expansion. Only data for which specimens conform to the ideal Mott ring geometry, and for which reliable measurements were made of the expansion velocity and final fragment number, were considered. A number of ring expansion studies reported in the literature did not meet all of these requirements and therefore were excluded from the analysis (e.g. [5] in which the number of fragments was not reported or [6] in which only the sum of necks and fractures is reported).

The alloys included, sources of the data, and relevant physical and mechanical properties are shown in Table 1. The fragmentation fracture toughness \(K_f\), a property discussed in detail later, was not reported for Mild Steel ES, Al-6061-O, and AA-5052-O. For the aluminium alloys, it was assumed the same value as for Al-1100-O could be applied, which is not unreasonable given that all these materials are in a soft temper state and have a similar elongation to failure. For Mild Steel ES and AZ31B the values used were the plane strain fracture toughness \(K_{1c}\). It can be seen that the alloys cover a wide range of strengths (a factor 20 difference from weakest to strongest), densities (a factor 7 difference between least and most dense), and elongations to failure (a factor 2 difference between the lowest and highest values). They therefore encompass the range of alloys of practical interest.

Table 1 Density (\(\rho\)) and mechanical properties (yield stress, Y, strain to failure \(\epsilon _f\), fragmentation toughness \(K_f\)) of alloys for which data has been obtained from listed reference
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Results and Discussion

A plot of the data showing the number of fragments as a function of the radial expansion strain rate imposed in the test is shown in Fig. 1. To produce this plot, the velocity at the point of fragmentation was identified from the literature studies and converted to strain rate based on the ring geometry. The number of fragments was taken directly from each paper. Since there is only one experiment for each condition, it is not possible to determine reliable error bars, However, the considerable scatter in the results that reflects the stochastic nature of the fragmentation process gives an indication of the uncertainty in fragment number for a given strain rate, which is typically around 10% from a mean value (determined by comparing similar conditions).

A number of general observations can be made from this plot. Firstly, the number of fragments increases as the strain rate increases, which is expected from classical fragmentation models [7]. Secondly, there is a systematic difference between the number fragments that becomes apparent as the strain rate increases. The highest number of fragments is produced in the highest density alloy and the lowest number is produced in the lowest density alloy.

There is one data-point for the U–6 wt% Nb alloy that is a clear outlier (circled in red). This point corresponds to a different heat treatment variant of the U–6 wt% Nb alloy, discussed in detail elsewhere [7]. This heat treatment produces a high fraction of brittle \(\alpha\) phase, and in this condition the failure of the specimens was observed to occur with very little ductility but instead by cleavage fracture [7]. The number of fragments in this case lies well outside the general trend-line for this alloy in a more ductile state, even accounting for scatter.

Fig. 1
figure 1

Measured number of fragments as a function of strain rate for Mott ring data compiled from the literature (see Table 1)

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Grady [1] demonstrated that by plotting the number of fragments as a function of specific kinetic energy, the data for the Al-1100-O and Cu-OFHC alloys clustered around a single curve. The specific kinetic energy is given by:

$$\begin{aligned} T = \frac{1}{2}\rho u.^2 \end{aligned}$$
(1)

where \(\rho\) is the alloy density and u the radial expansion velocity.

Fig. 2
figure 2

a Measured number of fragments as a function of specific kinetic energy for Mott ring data compiled from the literature (see Table 1), b zoomed into the lower strain rate range

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This analysis is now extended to all the available data from the literature, and the resulting figure is shown in Fig. 2a. The previously noted scatter in the data notwithstanding, this is consistent with Grady’s finding [1], showing the results when plotted in this form are scattered approximately around a single curve. At higher specific kinetic energy, a systematic difference is apparent between mild steel and U 6 wt% Nb data, although the highest strain rate measurement, which was for mild steel, lies close the the extrapolation of a best fit line from the U 6 wt% Nb. More data are required in the higher specific kinetic energy regime (\(>1\times 10^8\,\)J m\(^{-3}\)) to draw any clear conclusions about how well this data conforms to a simple kinetic energy fit.

Grady [1] showed the functional form of the best fit line is:

$$\begin{aligned} N_T = \left( {\frac{T}{\tau }}\right) ^\frac{1}{2} \end{aligned}$$
(2)

where \(N_T\) is the number of fragments per length and \(\tau\) is a fitting parameter. Setting the constant \(\tau\) to provide the best fit (\(\tau = 694\,\)KJ m\(^{-3}\)) gives the line shown in Fig. 2a. It can be seen that all the data are scattered around this line, and all points fall within or close to the upper and lower bound lines drawn at \(1.5N_T\) and \(0.5N_T\).

Since much of the data is clustered at the lower strain rate range, this portion of the graph has been magnified in Fig. 2b. The scatter is large, but the fit of Eq. 2 remains approximately valid over this narrower range. Notably, the systematic difference between different alloys is largely eliminated when plotting the number of fragments as a function of specific kinetic energy.

This simple analysis based only on the kinetic energy can be compared with the other models used to rationalize fragmentation data. As discussed in the introduction, there are two distinct theoretical approaches. The method of Mott [4] is based on the assumption that the number of fragments is controlled by the statistical distribution of fracture strains around the ring. Further details of the Mott model are given elsewhere [4, 7]. The Mott model gives the number of fragments as:

$$\begin{aligned} N_S = \left( \frac{\rho \dot{\epsilon }^2}{2\pi Y}\frac{n}{\sigma } \right) ^{\frac{1}{2}} \end{aligned}$$
(3)

where Y is the yield stress of the alloy and \(n/\sigma\) are the parameters that define the hazard function (distribution of fracture strains [1, 4]). Since the latter are not known a priori, this equation cannot usually be used to make a forward prediction of the fragment number. However, if it is assumed that the hazard function is identical for all alloys, Eq. 3 can be directly applied and used to predict the total fragment number, which can be compared with a prediction based on the simple kinetic energy approach. Taking the standard deviation in strain-to-fracture at high strain rate as 0.05 (a value shown to be reasonable for copper [1]), the predictions are as shown in Fig. 3a.

Fig. 3
figure 3

Measured number of fragments as a function of strain rate with predicted results from a the statistically based fragmentation model (lines), and b the energy based fragmentation model (lines)

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It can be seen that this analysis does not represent the underlying trends in the data well. Although the number of fragments predicted in each case is correct to within a factor of 2, the relative number of fragments for each material at a given strain rate is wrong. For example, the U 6 wt% Nb alloy is predicted to produce the fewest fragments, whereas in practice it produces the most. This could be corrected by assuming this alloy has a different distribution of fracture strains, but then the model is not predictive. In the Mott analysis, assuming the distribution of fracture strains is invariant between alloys, the only material parameter controlling the number of fragments is \(\rho /Y\). Values for this ratio are presented in Table 1. The highest density material (U 6 wt% Nb) also has the highest yield strength, so that the \(\rho /Y\) ratio is not maximized for this material. Rather, this ratio is greatest for the Cu–OFHC material that has a relatively high density but low strength, which explains why the greatest number of fragments are predicted for this alloy using the statistical model, which is not consistent with observations.

It is also noteworthy that this analysis produces no better prediction of the fragment number than the simple kinetic energy based method. It could be made accurate by tuning the assumed standard deviation in local strain-to-failure at the strain rate of interest for each material separately, but this would require calibration to a large number of experiments. It is therefore not predictive without experimental calibration for each material case.

The second theoretical approach is the energy based theory of Grady [12] and Kipp and Grady [13]. In this model, the energy dissipation associated with growing the fracture and the associated delay time is critical in controlling the fragment size distribution. The material parameter controlling the energy dissipation to grow a fracture is identified as the fragmentation toughness, given the symbol \(K_f\) to recognise its relationship to the more well known quasi-static fracture toughness \(K_c\). As with the statistical model, this parameter is not usually known a priori and has to be fitted to data from fragmentation experiments. However, it has been shown that in some cases, \(K_f\) and \(K_{1c}\) are similar (within a factor of 2) [7], so that in the absence of better data, the quasi-static fracture toughness may provide a useful approximation. Data for \(K_f\) (or \(K_{1c}\) where \(K_f\) is unknown) are reported in Table 1.

The energy based model gives the number of fragments per unit length for a given material and strain rate as [7]

$$\begin{aligned} N_E = \left( \frac{\rho c \dot{\epsilon }}{\sqrt{12}K_f}\right) ^{\frac{2}{3}} \end{aligned}$$
(4)

where c is the velocity of sound in the ring, given by \(\sqrt{E/\rho }\) with E being the Young’s modulus. Using the data in Table 1, Eq. 4 was applied to calculate the total number of fragments for the alloys of interest in this study, and the results are plotted in Fig. 3b.

It can be seen that this model has given predictions that are in much better agreement with observation than those obtained with the statistical (Mott) model. In particular, the larger number of fragments observed in the U 6 wt% Nb alloy at a given strain rate is correctly captured. Of course, one reason for this better fit is that the critical material parameter (\(K_f\)) is different for each alloy, so that the fracture behaviour becomes alloy sensitive. However, even in the case where \(K_f\) is approximated by \(K_{1c}\) (Mild Steel ES), the agreement remains good, at least until the highest strain rate. Although the fit is clearly better than in the case of the statistical model, it is noteworthy than the maximum error in the number of fragments predicted is still greater than the simple kinetic energy fit. For example, for Cu OFHC, the energy based model underestimates the total number of fragments by a factor of 2 for some strain rates. This fit could of course be improved by refining \(K_f\) for that particular material, but this makes a forward prediction for a new material impossible.

It should be noted that both the statistical and energy models are physically based, and seek to explain how the observed number of fragments relates to the underlying controlling mechanisms, whereas no mechanism is inferred from the kinetic energy fit. However, the confirmation of the original observation by Grady [1] that a simple relationship exists between the number of fragments and kinetic energy in the ring suggests that details of the local material properties (flaw distribution, microstructure etc.) are far less important that the overall energy driving the fragmentation process.

Finally, the outlying point for the U 6 wt% Nb alloy may be noted, where the number of fragments observed was far greater than any of the models predict. As mentioned, this point corresponds to material heat treated into a much more brittle state, where low ductility fracture was observed. As the material becomes inherently more brittle, its failure becomes more sensitive to small flaws and the scatter in failure strains in expected to be much larger (of the order of 20% rather than less than 5% [4]). Plastic relaxation at crack tips is more difficult than in a ductile metal, driving high levels of stress concentration. In such cases, it may no longer be reasonable to ignore the local variations within the material. The analysis here therefore appears valid only if there is sufficient ductility to reduce the effect of initial defects so that inertial effects dominate. If the material fails without significant plastic deformation, this analysis will probably underestimate the number of fragments considerably.

Conclusions

The available data in the literature on fragmentation of Mott rings has been compiled for a range of very different alloys, ranging from a strong and high density U 6 wt% Nb alloy to a weak, low density aluminium alloy (1100–O). The data have been used to confirm an initial suggestion by Grady [1] based on a much more limited dataset; namely that the number of fragments for these disparate alloys collapses onto a single curve of best fit when plotted as a function of the kinetic energy of the ring. The following conclusions may be drawn from this work.

  1. 1.

    Although the experimental data have a high degree of scatter, a simple fit of fragment number to kinetic energy provides a remarkably good representation. The best fit relation scales with the square root of the kinetic energy.

  2. 2.

    The simple kinetic energy analysis provides a comparable fit to the fragmentation data than a classical physical analysis based on either the statistical or energetic fragmentation theory, at least for the case where accurate material data are not available for the physical models.

  3. 3.

    One datapoint does not fit any of the models, and this corresponds to the U 6 wt% Nb alloy heat treated to a more brittle condition. This implies the current analysis is only valid in the case of ductile metals, where there is significant plasticity associated with failure.