Crack-tip process zone as a bifurcation problem

Stress concentration at a crack tip generates a solid structural transformation in its vicinity, the process zone. We argue that its formation represents a local phase transition described by a multicomponent order parameter. We derive a system of equations describing the dynamics of the order parameter driven by an inhomogeneous, time-dependent stress field in the solid and show that it exhibits a bifurcation. The latter corresponds to the emergence of a process zone characterized by the distribution of the order parameter localized in the vicinity of the crack tip. The emergence temperature $T_{*}$ considerably differs from the temperature of the bulk phase transformation $T_{\text{c}}$. We demonstrate that $T_{*}$ exhibits a universal behavior $T_{*}-T_{\text{c}}\sim K_I^{4/3}$, in terms of the stress intensity factor $K_I$, and that the zone universally vanishes upon achieving a critical velocity. These facts together give rise to a universal dynamic phase diagram.

Figure 1

Schematic view of the process zone (a) with the size $R_Z$ located at the crack tip (b) and embedded in the mother phase matrix (c).

Figure 2

(a) Example of the potential $U^{ij}({\rho }_x,{\rho }_y)=U({\rho }_x,{\rho }_y){\delta }^{ij}$ in the elastically isotropic and purely dilatant case. (b) The eigenfunction ${\psi }_{*}({\rho }_x,{\rho }_y)$ as the functions of the Cartesian coordinates ${\rho }_x=\rho cos\left(\theta \right)$ and ${\rho }_y=\rho sin\left(\theta \right)$. (i) indicates the crack-tip position.

Figure 3

Potential $U_1({\rho }_x,{\rho }_y)$ (a) and the eigenfunction ${\psi }_{*}({\rho }_x,{\rho }_y)$ (b) at the tip of the crack with the shear spontaneous strain at $Q_1>0$. (i) indicates the crack-tip position.

Figure 4

The eigenfunctions for the case of the ${\mathrm{BaTiO}}_{\text{3}}$. (a) Shows the function ${\psi }_{*}^1({\rho }_x,{\rho }_y)$, while (b) represents ${\psi }_{*}^2({\rho }_x,{\rho }_y)$. (i) indicates the position of the crack tip.

Figure 5

Phase diagram of the process zone in the plane $(K_I,T)$ exhibits two regions: I with no zone at the tip and II where the crack is dressed. (a)–(c) Show the phase diagram at $0\le K_I\le 2K_{\mathrm{IC}}$ with $\nu =0.3$, 0.5, and 0.9 correspondingly. The dotted-dashed line separates the phase diagram part with the motionless crack (left) from that with the propagating one (right). (d)–(f) Show the phase diagram for a propagating crack at $K_{\mathrm{IC}}\le K_I<\infty$, as a function of $1-K_{\mathrm{IC}}^2/K_I^2$. $M$ indicates the point of maximum of the zone emergence line (if any).

Figure 6

Dependence of the process zone size on the stress intensity factor. Parameters of the formula (76) correspond to the estimates given above.

Figure 7

Seach for the bifurcation point ${\lambda }_{*}$ of the equation (A1). Dots show the results of the numeric simulation, while the solid line represents the fitting.

Figure 8

Search for the bifurcation point ${\lambda }_{*}$ of Eq. (B4). Dots and squares show the results of the numeric simulation of the integrals $j^1$ and $j^2$ correspondingly, while the dashed and solid lines represent their fitting.

Figure 9

Convergence control of the numerical process shows the dependence of the integrals $j^i\left({\lambda }_0\right)$ on the dimensionless simulation time ${\tau }_{max}$. Disks show the behavior of $j^1\left({\lambda }_0\right)$ and squares that of $j^2\left({\lambda }_0\right)$. The inset demonstrates the behavior of $j^1\left({\lambda }_0\right)$ in more fine details within the interval $2500\le {\tau }_{max}\le 7000$.