Modeling of Multiscale Fatigue Crack Growth: Nano/Micro and Micro/Macro Transitions

Abstract

Nanocracks less than 10^–2 mm have been detected to run at about 0.15–9 × 10^–6 mm/s in stainless steels that undergo intergranular stress corrosion cracking (IGSCC). Macrocracks propagate in cm/s with more than an order of magnitude difference in velocity. The difference in the rate of energy dissipated due to damage by cracking obviously cannot be overlooked in the development of multiscale crack models, particularly if the design were to depend on virtual testing rather than physical testing. If the loading rate is not low enough to take advantage of the nanostructure, then the added reinforcement may not serve a useful purpose. In other words, micro and/or macro reinforcements should suffice for structures that entail only fast energy release. The consideration of nanoscale in addition to micro and macro is a relatively new development that has received much attention. Although triple scaling appears to be a refinement of dual scaling, the scale range accuracy is actually determined by the two nearest scales such as nano–micro or micro–macro. A triple nano–macro scale model may be too coarse and may result into the insertion of a meso- or the micro-scale. This in effect returns to the treatment of two dual scales consisting of the nano–micro and micro–macro as mentioned earlier. It can thus be said that multiscaling constitutes a series of dual-scale models, each having a well-defined range of accuracy. Hidden subtleties can arise from the underlying physics of damage when the scale range is changed. A case in point is the 1/r^0.5 behavior of a classical macrocrack and the double singular behavior of the microcrack having a weak singularity of 1/r^0.15 and a strong singularity of 1/r^0.75. The physics of nanocracking can introduce additional alterations.

A triple scale model involving nano-, micro-, and macro-cracks will be developed using a new paradigm for evaluating the explicit time history of the internal structure degradation process of solids from a knowledge of three micro/macro parameters $\upmu_{\textrm{micro}}^{\ast}$, $\upsigma_{\textrm{micro}}^{\ast}$, and ${\textrm{d}}_{\textrm{micro}}^{\ast}$for connecting the quantities at the micro- and macro-scale and another three nano/micro parameters $\upmu_{\textrm{nano}}^{\ast}$, $\upsigma_{\textrm{nano}}^{\ast}$, and ${\textrm{d}}_{\textrm{nano}}^{\ast}$for connecting the quantities at the nano- and micro-scale by using only the properties of the undamaged material at time t = 0. Since there can be no valid counterparts, say micro or nano, to specimen data collected at the macro-scale, to which lower-scale material property data can be related but not determined separately. The present analysis will entail the inverse approach of specifying a unique solution for nonlinear problems and finding the conditions to which the solution will satisfy. This corresponds to specifying the critical crack growth states and determining the evolution of the microstructure and nanostructure material degradation process.

Crack growth rate da/dN models are constructed from range of micro/macro energy density factor $\Delta {\textrm{S}}_{{\textrm{micro}}}^{{\textrm{macro}}}$ and range of nano/micro energy density factor $\Delta {\textrm{S}}_{{\textrm{nano}}}^{{\textrm{micro}}}$ to reflect the scale transitory behavior as the crack grows through from nano- to micro- to macro-size, that is, a_nano→ a_micro→ a_macro. The multiscale scheme is applied to fatigue crack growth data for pre-cracked 2024-T3 and 7075-T6 aluminum panels. Assumptions are made for obtaining the time-dependent character of ($\upmu_{\textrm{micro}}^{\ast}$, $\upsigma_{\textrm{micro}}^{\ast}$, ${\textrm{d}}_{\textrm{micro}}^{\ast}$) and ($\upmu_{\textrm{nano}}^{\ast}$, $\upsigma_{\textrm{nano}}^{\ast}$, ${\textrm{d}}_{\textrm{nano}}^{\ast}$) from which the fictitious nano and micro material properties and their corresponding geometric and restraining characteristics are found. Nanocrack growth rates are found to be in the range of 10^–7 to 10^–9 mm/s for 2024-T3 and 7075-T6 aluminum with initial crack size of a _o = 10^–4 mm. They reached their critical conditions about 5–8 years at which the nanocracks become microcracks. The segment transition is assumed to simplify the discussion. The same procedure can be used to predict the transition of microcracks to macrocracks and finally the instability of macrocracking. Starting from the initial defect of a _o = 1.8 mm growing at an approximate rate of 10^–6 mm/s for 18 years, macrocracking instability can be reached although this terminal condition has been set arbitrarily. It is not a restriction of the approach. The end life can be preset with the appropriate crack growth rate to match the design conditions.