Geometric control of failure behavior in perforated sheets

Adding perforations to a continuum sheet allows new modes of deformation, and thus modifies its elastic behavior. The failure behavior of such a perforated sheet is explored, using a model experimental system: a material containing a one-dimensional array of rectangular holes. In this model system, a transition in failure mode occurs as the spacing and aspect ratio of the holes are varied: rapid failure via a running crack is completely replaced by quasistatic failure, which proceeds via the breaking of struts at random positions in the array of holes. I demonstrate that this transition can be connected to the loss of stress enhancement, which occurs as the material geometry is modified.

Figure 1

When the geometry of the 1D metamaterial is changed, the failure mode transitions from a running crack to random breaking; the strut breaking order as well as the total failure time, $\tau$ (time interval between first and last break) changes markedly. Numbering and color both indicate strut breaking order; the color map runs from red to blue. (a) When the struts composing the material are small and spaced relatively closely, failure proceeds via a propagating crack: defined as ordered breaking of struts, which occurs at speeds comparable to material sound speeds. Here, $\tau =149.3\mu \mathrm{s}$. (b) A transition in failure mode occurs when the struts become long and narrow, or spaced far apart from each other. In this regime, failure occurs via the breaking of struts in a nonordered fashion, and the breaking rate is set by the pulling rate, i.e., failure is quasistatic. Here, $\tau =1.58$ s.

Figure 2

(a) Imaging studies were conducted using a custom-built apparatus. The samples are held fixed at one boundary, while the other boundary is displaced at a constant rate. Enlargement indicates the parameters that characterize the hole size ($s,l$) and spacing ($d$). (b) Force-displacement curves for two sample geometries, $d$ = 1 mm, $s$ = 2 mm, $l$ = 2 mm (left) and $d$ = 1 mm, $s$ = 2.5 mm, $l$ = 5 mm (right). The lines show the data obtained from the experimental samples, while the open symbols show the results of finite-element calculations. The good agreement between measurements and calculations demonstrates that the samples behave in a nearly linear fashion.

Figure 3

Transition in failure mode. (a) As $l/d$ increases, the failure time $\tau$ increases dramatically. Data for two displacement rates are shown. Tests were performed at constant $s/d$ = 3. (b) The fraction of nonadjacent breaks, $f$, increases with $l/d$ as well. (c) $f$ vs $s/d$; tests were performed at constant $l/d$ = 2. As $s/d$ increases, the fraction of nonadjacent breaks, $f$, increases dramatically. (d) All of the data in (b) (circles) and (c) (triangles) can be collapsed by plotting $f$ as a function of $\frac{l+cs}{d}$, where $c=0.25\pm 0.1$.

Figure 4

Finite-element calculations of the stress enhancement due to a broken strut. (a) Stress on each strut (${\sigma }_{yy}$) vs strut number; each point represents the total stress on a single strut. The inset shows a visualization of the stress field in the sample (${\sigma }_{yy}$); see the color bar for scale. (${\sigma }_{yy}$ represents the stress averaged over the strut surface.) (b) Stress per strut (${\sigma }_{yy}$) vs strut position after the center strut is broken. There is an enhancement of ${\sigma }_{yy}$ adjacent to the break location. The inset shows a visualization of the stress field in the sample (${\sigma }_{yy}$); see the color bar for scale. (c) and (d) Normalized stress enhancement, $\Sigma$, vs strut number, for various values of (c) $s/d$ and (d) $l/d$. The calculations are for a center-broken strut; points represent the average over the left and right sides of the sample. $\Sigma$ is a strong function of material geometry. (e) and (f) The maximum stress enhancement, ${\Sigma }_0$, vs (e) $s/d$ and (f) $l/d$. Data are shown for both crack geometries, i.e., a center crack (filled symbols) and an edge crack (open symbols). In all cases, ${\Sigma }_0$ decreases dramatically as either $s/d$ or $l/d$ is increased. Insets show the width of the stress enhancement, $\overline{\Sigma }$. $\overline{\Sigma }$ remains relatively constant as $s/d$ and increases as $l/d$ is increased.

Figure 5

The stress enhancement data [Figs. 4 and 4] can be collapsed for both crack geometries by plotting ${\Sigma }_0$ as a function of $\frac{l+cs}{d}$. (a) ${\Sigma }_0$ data from the center-crack geometry. The data were collapsed using $c_1=0.8\pm 0.1$. (b) ${\Sigma }_0$ data from the edge-crack geometry. The data were collapsed using $c_2=1.3\pm 0.1$.

Figure 6

${\Sigma }_0$ vs crack length, at low $l/d$ and $s/d$ (black circles), high $l/d$ (red $\times$'s), and high $s/d$ (blue triangles). Only the first 80% of breaks are shown. In all cases after an initial increase, ${\Sigma }_0$ plateaus as more and more struts are broken. Furthermore, at high $s/d$ or $l/d$, ${\Sigma }_0$ always remains far below the value observed when $s/d$ and $l/d$ are small.

Figure 7

(a) Crack velocity vs $s/d$. An unperforated material is represented by $s/d=0$. There are two dynamical regimes: $s/d<1$, where the velocity increases with increasing $s/d$, and $s/d>1$, where the velocity is independent of $s/d$. The arrow indicates the Rayleigh wave velocity in acrylic, 1017 m/s. (b) The simple model of Eq. (2) does not agree with the data, as indicated by the dashed line. The fit was obtained by forcing the model to agree with the points where $s/d<$ 1. (c) Crack velocity vs strut aspect ratio, $l/d$. The crack velocity is independent of strut aspect ratio (when $l/d\lesssim 5$). These tests were conducted for fixed $s/d=3$. (d) Crack velocity vs $d$, at a fixed value of $s/d=3$. The crack velocity is independent of $d$; the dashed line represents the mean value of $v$. (e) Crack velocity vs $s/d$ for three different materials: acrylic (blue circles), delrin (red open squares), and impact-modified acrylic (green asterisk). (f) The same data, with the crack velocity normalized by the Rayleigh wave velocity in each material, $c_R$. The nondimensionalized data collapse, and thus they do not show any dependance on material properties.