“Deborah numbers”, coupling multiple space and time scales and governing damage evolution to failure

Two different spatial levels are involved concerning damage accumulation to eventual failure. This can entail sample size l (~cm) to characteristic microdamage size c*(~μm). Associated are three physical processes with three different rates, namely macroscopic elastic wave velocity a, nucleation and growth rates of microdamage n_N* and V*.

It is found that the trans-scale length ratio c*/L does not directly affect the process. Instead, two independent dimensionless numbers: the trans-scale one De*=ac*/(lV*) including the length ratio and the intrinsic one D*=nN * c*5 /V* including mesoscopic parameters only, play the key role in the process of damage accumulation to failure.

The above implies that there are three time scales involved in the process: the macroscopic imposed time scale t_im = l /a and two meso-scopic time scales, nucleation and growth of damage, t_N=1/(n*_Nc^*4)and t_V=c*/V*. Clearly, the dimensionless number De*=t_V/t_im refers to the ratio of microdamage growth time scale over the macroscopically imposed time scale. So, analogous to the definition of Deborah number as the ratio of relaxation time over external one in rheology. Let De be the imposed Deborah number while De represents the competition and coupling between the microdamage growth and the macroscopically imposed wave loading. In stress-wave induced tensile failure (spallation) De* < 1, this means that microdamage has enough time to grow during the macroscopic wave loading. Thus, the microdamage growth appears to be the predominate mechanism governing the failure.

Moreover, the dimensionless number D* = t_V/t_N characterizes the ratio of two intrinsic mesoscopic time scales: growth over nucleation. Similarly let D* be the “intrinsic Deborah number”. Both time scales are relevant to intrinsic relaxation rather than imposed one. Furthermore, the intrinsic Deborah number D* implies a certain characteristic damage. In particular, it is derived that D* is a proper indicator of macroscopic critical damage to damage localization, like D* ~ (10^-3~10^-2) in spallation. More importantly, we found that this small intrinsic Deborah number D* indicates the energy partition of microdamage dissipation over bulk plastic work. This explains why spallation can not be formulated by macroscopic energy criterion and must be treated by multi-scale analysis.