Optimal spacing and penetration of cracks in a shrinking slab

A method based on energy minimization is used to determine the spacing and penetration of a regular array of cracks in a slab that is shrinking due to a changing temperature field. The results show a range of different crack propagation behavior dependent on a single dimensionless parameter, being the ratio of the slab thickness and a characteristic length for the material. At low parameter values the minimum energy state can be achieved by continually adding more cracks until a steady state is achieved. At higher values, a minimum crack spacing is reached at finite time, beyond which the cracks are constrained to propagate with the minimum spacing. In the latter case, the uniform propagation is potentially unstable to a spatial period doubling, leading to increasingly complex crack penetration patterns. The energy minimization combined with the period doubling instability provides a means of determining the minimum energy state of cracks for all time. The problem considered here can be seen as a paradigm for cracking phenomena that occur on a large range of scales, from planetary to microscopic.

Figure 1

Geometry of a slab with an array of regularly spaced cracks, propagating from the top bounding surface. Dashed lines denote the representative geometry used in the finite element calculations.

Figure 2

Boundary condition specification. The solid lines are actual surfaces of the slab and the dashed lines are symmetry planes.

Figure 3

Contour plots of $E^{\prime }(l,p)$ for various times (marked on each graph) when $L∕L_c=6$. The square dot in 3 of the graphs indicates the location $l_0∕L,p_0∕L$ which minimizes $E^{\prime }$. Note that the contour spacing is small near the minimum compared to the remainder of the graph.

Figure 4

Graphs of $l_0∕L$ (solid line) and $p_0∕L$ (dashed line), versus $t$ for $L∕L_c=6$.

Figure 5

Graphs of $l_0∕L$ (solid line) and $p_0∕L$ (dashed line), versus $t$ for $L∕L_c=2$.

Figure 6

Graphs of $l_0∕L$ (solid line) and $p_0∕L$ (dashed line), versus $t$ for $L∕L_c=16$. The inset shows an expanded view of the behavior at small time.

Figure 7

Graph of $p_m$ vs $t$ for cracks with spacing $2l_{0m}$, when $L∕L_c=16$. The solid line is for the case when all cracks propagate together, while the dashed line is when every second crack has stopped, after $t_{m1}$. The dotted line is a graph of $p_0$ versus $t$.

Figure 8

Graph of ${\partial }^2{S}^{\prime }∕\partial p^2$ vs $t$ for $L∕L_c=16$. The solid line is for equal length cracks and the dotted line for every second crack stopped, $t>t_{m1}$.

Figure 9

Schematics of the final crack states for different $L∕L_c$, whose value is shown next to each configuration.

Figure 10

Graphs of $l∕L_c$ and $p∕L_c$ vs ${(L∕L_c)}^{2}t$ for varying $L∕L_c$. The values of $L∕L_c$ are shown beside each curve. A logarithmic scale has been used to clearly illustrate the small time behavior.

Figure 11

Schematics of the final crack states for different $L∕L_c$, representing different slab thickness, whose value is shown next to each configuration.