A quasi-static algorithm that includes effects of characteristic time scales for simulating failures in brittle materials

Abstract

When the brittle heterogeneous material is simulated via lattice models, the quasi-static failure depends on the relative magnitudes of $T_{\mathrm{elem}}$ , the characteristic releasing time of the internal forces of the broken elements and $T_{\mathrm{lattice}}$ , the characteristic relaxation time of the lattice, both of which are infinitesimal compared with $T_{\mathrm{load}}$ , the characteristic loading period. The load–unload (L–U) method is used for one extreme, T_elem ≪ T_lattice, whereas the force–release (F–R) method is used for the other, T_elem ≫ T_lattice. For cases between the above two extremes, we develop a new algorithm by combining the L–U and the F–R trial displacement fields to construct the new trial field. As a result, our algorithm includes both L–U and F–R failure characteristics, which allows us to observe the influence of the ratio of $T_{\mathrm{elem}}$ to $T_{\mathrm{lattice}}$ by adjusting their contributions in the trial displacement field. Therefore, the material dependence of the snap-back instabilities is implemented by introducing one snap-back parameter γ. Although in principle catastrophic failures can hardly be predicted accurately without knowing all microstructural information, effects of γ can be captured by numerical simulations conducted on samples with exactly the same microstructure but different γs. Such a same-specimen-based study shows how the lattice behaves along with the changing ratio of the L–U and F–R components.