Temperature of emergence of a transformational process zone at a crack tip

We analyze the emergence of a process zone of improper ferroelastic class. We derive expressions for the zone emergence temperature and the size of the incipient zone in terms of experimentally accessible physical parameters. Our numerical estimates suggest that the region of existence of the process zone typically occupies a considerable part of the phase diagram.

Figure 1

Schematic view of the process zone (2) at the crack tip (a) embedded into the matrix (1). The local axes $(x^{\prime },y^{\prime })$, as well as the tangential unit vector $\mathbf{t}$ and normal unit vector $\mathbf{n}$, are shown. (b) indicates the phase interface.

Figure 2

The left-hand part of Eq. (12) above the bifurcation is shown against $R$ (a). The bifurcation takes place as soon as the line hits the horizontal axis at $R=R_{*}$ (b), the value $R_{*}$ being the zone size in the emergence point. The further increase of the stress intensity factor $K_I$ or the temperature decrease moves the line downwards (c), followed by the increase of the zone size $R_1$.

Figure 3

The plane $(K_I/K_{IC},T_{*}/T_{*\text{c}})$ is divided into two parts: I, where no zone exists at the crack tip, and II, where the crack has the transformational process zone. These regions are separated by the zone emergence line.

Figure 4

Schematic view of the process zone at the crack tip in a solid in a form of a solid cylinder (a) and discrete droplets (b). (i) Indicates the solid, (ii) schematically shows the crack tip, (iii) and (iv) show the process zone. In the case of the droplet, its height $h$ is below the value (8).

Figure 5

(a) Schematic view of a soft (i) and pronounced (ii) first-order transition. In the latter case, the order-parameter jump at the transition temperature $T_0$ is close to its saturation value ${\eta }_{\text{s}}$. (b) The behavior of the $\eta =\eta \left(x^{\prime }\right)$ (j) and its derivative ${(\partial \eta /\partial x^{\prime })}^2={\left[(n·\mathbf{\nabla })\eta \right]}^2$ normal to the phase interface (jj) in the vicinity of the phase boundary in the case of the pronounced first-order transition.