Weakly Nonlinear Theory of Dynamic Fracture

The common approach to crack dynamics, linear elastic fracture mechanics, assumes infinitesimal strains and predicts a $r^{-1/2}$ strain divergence at a crack tip. We extend this framework by deriving a weakly nonlinear fracture mechanics theory incorporating the leading nonlinear elastic corrections that must occur at high strains. This yields strain contributions “more divergent” than $r^{-1/2}$ at a finite distance from the tip and logarithmic corrections to the parabolic crack tip opening displacement. In addition, a dynamic length scale, associated with the nonlinear elastic zone, emerges naturally. The theory provides excellent agreement with recent near-tip measurements that cannot be described in the linear elastic fracture mechanics framework.

Figure 1

Top: Measured $u_x(r,0)$ (circles) fitted to the $x$ component of Eq. (3) (solid line) for (a) $v=0.20c_s$ with $K_I=1070\mathrm{Pa}\sqrt{m}$, $T=-3150\mathrm{Pa}$, and $B=18$; (b) $v=0.53c_s$ with $K_I=1250\mathrm{Pa}\sqrt{m}$, $T=-6200\mathrm{Pa}$, and $B=7.3$; (c) $v=0.78c_s$ with $K_I=980\mathrm{Pa}\sqrt{m}$, $T=-6900\mathrm{Pa}$, and $B=26$. Bottom: Corresponding measurements of ${\epsilon }_{yy}(r,0)={\partial }_y{u}_y(r,0)$ (circles) compared to the theoretical nonlinear solution [cf. Eq. (3)] with no adjustable parameters (solid lines); $K_I$, $T$, and $B$ are taken from the fit of $u_x(r,0)$. LEFM predictions (analysis as in 4) were added for comparison (dashed lines).

Figure 2

Measured crack-tip profiles [${\varphi }_y(r,\pm \pi )$ vs ${\varphi }_x(r,\pi )$] (circles). Shown are the parabolic LEFM best fit (dashed line) and the profiles predicted by the second order nonlinear corrections (solid line). (a) $v=0.2c_s$ and (b) $v=0.53c_s$. $T$ and $B$ are as in Fig. 1. In contrast to the $\sim 20\%$ discrepancy in values of $K_I$ obtained in 4, the respective values $K_I=1170\mathrm{Pa}\sqrt{m}$ and $K_I=1300\mathrm{Pa}\sqrt{m}$ correspond to within 9% and 4%, respectively, of $K_I$ obtained from $u_x(r,0)$ using the nonlinear theory, cf. Fig. 1.