Dynamic fragmentation of brittle materials: analytical mechanics-based models

Abstract Two analytical mechanics-based models of dynamic fragmentation in brittle materials are proposed and solved to predict fragment size and time to fragmentation onset in terms of fundamental material properties and the applied strain rate. Previous widely adopted analytical models of dynamic fragmentation are based on relatively simple energy balance arguments, and assume that the fragmentation event occurs instantaneously. The present models account for the actual time-varying dynamic deformation that occurs prior to fragmentation onset. One of the models treats the fragmenting material as initially flaw-free, and determines the minimum fragment size predicted by a dynamic instability analysis. The second model accounts for initial flaw spacing (which may correlate physically with, for example, grain size), and a dynamic instability analysis is employed to determine which flaws become critical. The fragment size predictions of the present models and two previous energy-based models are found to agree at extremely high strain rates (≈5×10 7 /s for dense alumina), but the present, more realistic analysis indicates that the regime of validity of the energy-based models is rather restricted. The predictions of the present models are also shown to agree with those of a recent numerical finite element simulation of dynamic fragmentation which applies to a lower strain rate regime. Comparisons of the two new models show that if a material contains initial flaws whose spacing is smaller than the predicted fragment size of an equivalent “unflawed” material, the fragment size of the preflawed material will be smaller in general, but usually not as small as the initial flaw spacing. The analysis also permits determination of the evolution of the strain rate distribution in a prospective fragment before and after fragmentation initiation; results are presented for some example cases. Finally, closed-form analytical results are derived for minimum fragment size and time to fragmentation for strain rates in the quasi-static regime; these show the fragment size to be independent of strain rate in this regime, and the time to fragmentation initiation to be inversely proportional to the strain rate.