A steady-state modeling framework incorporating the Kuroda–Tvergaard model: demonstrated on single crystal crack growth

A new modeling framework for analyzing steady-state elastic–viscoplastic single crystal problems at micron scale is presented. The model takes as offset the higher-order gradient plasticity theory by Kuroda and Tvergaard (J Mech Phys Solids 54:1789–1810 2006), and it is strongly inspired by an existing steady-state framework for conventional elastic–viscoplastic materials. Details on the modeling framework are laid out, and the numerical challenges and stability issues are discussed. Subsequently, the modeling framework is demonstrated on Mode I steady-state crack growth in HCP single crystals, where the impact of size effects on the active plastic zone that surrounds the crack tip is investigated. The focus is on the plastic zone shape and size, as well as its influence on the macroscopic fracture toughness. In this way, the chosen benchmark problem for the new modeling framework serves as an extension to the conventional study of the corresponding problem without size effects presented in Juul et al. (J Mech Phys Solids 101:209–222 2017). The tendency is that the plastic zone is smeared out (seen as less distinct features) when increasing the material rate sensitivity. This is in-line with already published work in the literature. It is shown that the size effect has a limited effect on the qualitative features of the plastic zone, whereas it clearly suppresses the magnitude of the features due to the added gradient hardening effect. Moreover, the hardening effect tends to lower the shielding ratio, and hence a key result is that single crystal materials appear less crack resistant than conventional studies suggest at the micron scale.