Drying patterns: Sensitivity to residual stresses

Volume alteration in solid materials is a common cause of material failure. Here we investigate the crack formation in thin elastic layers attached to a substrate. We show that small variations in the volume contraction and substrate restraint can produce widely different crack patterns ranging from spirals to complex hierarchical networks. The networks are formed when there is no prevailing gradient in material contraction, whereas spirals are formed in the presence of a radial gradient in the contraction of a thin elastic layer.

Figure 1

(Color online) Four different stages in the evolution of a hierarchical crack network. The total uniform contraction was 9% and cracks were nucleated inside domains whenever the maximum principal stress exceeded ${\sigma }_c=0.85$ in arbitrary units. The substrate restraining force was drawn from a uniform distribution with 10% disorder and a mean of unity.

Figure 2

(Color online) The distribution density of $\left|\chi -1/2\right|$ for simulations with 5%,10%, and 15% disorder on the substrate restraining. The distributions are averaged over domains formed at the sixth generation of cracks (average of ten runs). No significant variation is seen at these fairly low levels of disorder. The black line on top represents a best fit with an exponential distribution $\text{exp}\left(-\left|\chi -1/2\right|/\alpha \right)$, where $\alpha =0.03$. Inset: for the same data, the cumulative distributions of the domain areas. The dashed line on top is an estimate of the distributions considering the individual domain divisions to be uncorrelated (see text).

Figure 3

Simulation of spiral cracks for various values of the material contraction $\beta$ and the rotation angles. In all the panels, the crack was initiated at the center and was allowed to propagate until it reached the outer boundary. Note that the smaller the shear stress is, the more pronounced the spiraling is.

Figure 4

(Color online) The relation between $\kappa$ and the rotation angle for four different values of the material contraction $\beta$. $\kappa$ is computed as the exponent of a best fit to a logarithmic spiral, $r\left(\varphi \right)=r_0{e}^{\kappa \varphi }$. The rotation angle is the prefactor ${\theta }_0$ used in the expression for the relative rotation between the substrate and the thin film. Fluctuations are due to numerical inaccuracies. The inset shows a data collapse of $\kappa \beta$ for the same curves and is in agreement with the simple scaling form $\kappa \sim {\theta }_0/\beta$. The errors in determining $\kappa$ can be estimated from the fit to the logarithmic spiral and are of the order of 2–4%. Note the systematic bend in the lowest graph which may be due to nonlinear effects.