Void formation and roughening in slow fracture

Slow crack propagation in ductile, and in certain brittle materials, appears to take place via the nucleation of voids ahead of the crack tip due to plastic yields, followed by the coalescence of these voids. Postmortem analysis of the resulting fracture surfaces of ductile and brittle materials on the $\mu \mathrm{m}\text{−}\mathrm{mm}$ and the nm scales, respectively, reveals self-affine cracks with anomalous scaling exponent $\zeta \approx 0.8$ in 3 dimensions and $\zeta \approx 0.65$ in 2 dimensions. In this paper we present an analytic theory based on the method of iterated conformal maps aimed at modelling the void formation and the fracture growth, culminating in estimates of the roughening exponents in 2 dimensions. In the simplest realization of the model we allow one void ahead of the crack, and address the robustness of the roughening exponent. Next we develop the theory further, to include two voids ahead of the crack. This development necessitates generalizing the method of iterated conformal maps to include doubly connected regions (maps from the annulus rather than the unit circle). While mathematically and numerically feasible, we find that the employment of the stress field as computed from elasticity theory becomes questionable when more than one void is explicitly inserted into the material. Thus further progress in this line of research calls for improved treatment of the plastic dynamics.

Figure 1

The fracture scenario suggested in [12]. This scenario had been documented in detail in corrosive glass fracture, and also more recently in the fracture of paper [13].

Figure 2

A forward direction profile of the hydrostatic tension $P$ in units of ${\sigma }_{\mathrm{Y}}$. On the crack $P=\sqrt{3}∕2$ and it attains a maximum of $\sqrt{3}$ on the yield curve. The threshold line indicates a value of $P_c$ such that $\sqrt{3}∕2<P_c<\sqrt{3}$. The typical length ${\xi }_c$ is shown. Other directions exhibit qualitatively similar profiles.

Figure 3

Panel (i): the tip of a straight crack and the yield curve in front of it. Panel (ii): three probability distribution functions calculated for the configuration in (i). The abscissa is $\theta$, the angle measured from the crack tip as seen in panel (i). The ordinate is the normalized probability (per unit $\theta$) to grow in the $\theta$ direction. The distributions are symmetric and wide enough to allow deviations from the forward direction. For all the curves ${\sigma }_{\mathrm{Y}}∕{\sigma }^{\infty }=6$. For curve (a) $p\left(\theta \right)\propto \exp \left[(P-P_c)\right]-1$ and $P_c∕{\sigma }^{\infty }=8$, for curve (b) $p\left(\theta \right)\propto \exp \left[0.2(P-P_c)\right]-1$ and $P_c∕{\sigma }^{\infty }=6$, and for curve (c) $p\left(\theta \right)\propto P-P_c$ and $P_c∕{\sigma }^{\infty }=6$.

Figure 4

The step size obtained using pdf (6) (see Sec. 2), vs the crack length. Note that these results agree with Eq. (5), i.e., the step size grows linearly with the crack length. The units here are the “bump” radius which is introduced explicitly in Eq. (47).

Figure 5

Two typical cracks generated with our model. Note the different scales of the abscissa and ordinate, and that the lower crack had been translated by $-300$. The upper crack exhibits two decades of self-affine scaling with a Hurst exponent 0.64. The lower crack has smaller standard deviation and therefore a shorter scaling range. Nevertheless it appears that in its shorter scaling range it exhibits an exponent that is very close to the upper crack.

Figure 6

Panel (i): the tip of a “rough” crack and the yield curve in front of it. Panel (ii): three probability distribution functions calculated for the configuration in panel (i). The abscissa is $\theta$, the angle measured from the crack tip as seen in panel (i). The ordinate is the normalized probability (per unit $\theta$) to grow in the $\theta$ direction. The pdf’s are those used in Fig. 3, using the same parameters. Note the upward preference in all the pdf’s due to the broken symmetry.

Figure 7

Calculation of the anomalous roughening exponent. The slopes of the dotted lines are 0.64 for the upper plot (curve a) and 0.68 for the lower (curve b). Note that the initial scaling with slope 1 is relevant for length scales smaller than ${\xi }_c$. This scaling is unphysical, resulting from our algorithm that connects the crack tip to a void by a straight line.

Figure 8

Illustration of how the conformal map $\Phi \left(\omega \right)$ operates. The unit circle is mapped to $C_1$ and the inner circle to $C_2$. The point $a$ is mapped to infinity.

Figure 9

Illustration of how the conformal map ${\varphi }_1$ operates. ${\varphi }_1$ maps the annulus into the exterior of the unit circle and a circle on its right. The unit circle boundary is mapped to itself and the inner circle of radius $\rho$ is mapped to the circle on the right with radius $(x_1-x_2)∕2$.

Figure 10

Illustration of how the conformal map $\Phi \left(\omega \right)$ operates. ${\varphi }_1$ maps the annulus onto the exterior of two circles, ${\varphi }_2$ rotates the whole plane and ${\varphi }_3$ maps the left circle into the desired crack shape, leaving the right circle almost unchanged in its shape.

Figure 11

Example of how to construct the conformal mapping along a line.

Figure 12

The configuration of a line with a circle. The comparison of results exhibited in Fig. 13 refers to this configuration.

Figure 13

$K_{\mathrm{I}}$ for the configuration shown in Fig. 12. Note that the results are normalized by the stress intensity factor with no void, i.e., $K_{\mathrm{IO}}={\sigma }^{\infty }\sqrt{\pi a}$. The stress intensity factor was calculated near the point $A$ (i.e., at the tip which is close to the void). The results were calculated for $a∕(d-R)=0.4$, varying $d∕R$ in the range [0.1, 0.7]. Isida’s results (b) are shown in squares, and ours (a) in circles.

Figure 14

(Color online) Panel (i): field lines of $\sqrt{J_2}$, cf. Eq. (1), given in units of ${\sigma }^{\infty }$. The bold line is the yield curve (i.e., $\sqrt{J_2}∕{\sigma }^{\infty }={\sigma }_{\mathrm{Y}}∕{\sigma }^{\infty }=12$). The crack covers the $x$ interval $[-{10}^4,1.01\times {10}^4]$ and the bump radius is $\sqrt{{\lambda }_0}=2$. The loading is mode I with ${\sigma }_{yy}\left(\infty \right)={\sigma }^{\infty }$. Panel (ii): field lines of the hydrostatic pressure, $P\equiv \frac{1}{2}\mathrm{Tr}\mathbf{\sigma }$ for the same crack as the figure above, in units of ${\sigma }^{\infty }$. The bold line is the yield curve from the previous figure. A reasonable range for $P_c$ in such a configuration is $12⩽P_c∕{\sigma }^{\infty }⩽20$, in accordance with $(\sqrt{3}∕2){\sigma }_{\mathrm{Y}}<P_c<\sqrt{3}{\sigma }_{\mathrm{Y}}$ (see Sec. 2A). Note that $P_c$ in this range ensures forward growth in the next step.

Figure 15

(Color online) The same crack as in Fig. 14 with the addition of a void on the yield curve.Panel (i): field lines of $\sqrt{J_2}$ in units of ${\sigma }^{\infty }$. The bold line is the yield curve with the same value of ${\sigma }_{\mathrm{Y}}$ as in Fig. 14 panel (i). The dotted line is the yield curve from Fig. 14 panel (i), i.e., the yield curve for the same crack with no void. The addition of the void creates a very local perturbation of the yield curve. Panel (ii): field lines of the hydrostatic pressure, $P\equiv \frac{1}{2}\mathrm{Tr}\mathbf{\sigma }$ in units of ${\sigma }^{\infty }$. The bold line is the yield curve from the figure above. Again the perturbation of the field lines with respect to Fig. 14 panel (ii) is very localized.

Figure 16

An example of a conformal mapping of the annulus onto two circular holes. In this example $x_1=1.1$ and $x_2=3.1$. Note that the unit circle is mapped onto itself, while the $\rho$ circle in the $\omega$ plane is mapped onto the circle on the right in the $z$ plane.