Tensile Instability in a Thick Elastic Body

A range of instabilities can occur in soft bodies that undergo large deformation. While most of them arise under compressive forces, it has previously been shown analytically that a tensile instability can occur in an elastic block subjected to equitriaxial tension. Guided by this result, we conducted centimeter-scale experiments on thick elastomeric samples under generalized plane strain conditions and observed for the first time this elastic tensile instability. We found that equibiaxial stretching leads to the formation of a wavy pattern, as regions of the sample alternatively flatten and extend in the out-of-plane direction. Our work uncovers a new type of instability that can be triggered in elastic bodies, enlarging the design space for smart structures that harness instabilities to enhance their functionality.

Figure 1

Force-stretch bifurcation diagram for (a) a cube subjected to triaxial tension, (b) a square under plane strain conditions subjected to biaxial tension, and (c) a biaxially stretched cross-shaped sample under plane strain conditions. The solid and dashed lines represent stable and unstable load paths, respectively. The contours show the maximum in-plane principal strain. Note that (a) and (b) were obtained analytically, while (c) was obtained using finite element analysis.

Figure 2

(a) Experimental setup to subject our thick cross-shaped sample to biaxial tension. (b) Top and bottom views of the deformed sample at $u/W=2.90$. (c) Cross-sectional views of the sample at $u/W=0$, 0.72, 1.45, 2.17, and 2.90 obtained using a micro-CT (computed tomography) x-ray imaging machine. (d) Relation between the normalized displacement $u/W$ applied to the sample and the stretches ${\lambda }_1$ and ${\lambda }_2$ of the diagonals located at the center of the sample. The results were obtained by manually processing the images and averaging five individual measurements (the error bars indicate the standard deviation) (scale bars 10 mm).

Figure 3

(a) Numerical snapshots of the sample, (b) stretches that define the deformation of the diagonals of the central domain (${\lambda }_1$ and ${\lambda }_2$) along the depth of the sample, and (c) out-of-plane stretch of the center line (${\lambda }_3$) along the depth of the sample, at $u/W=0$, 0.96, 1.91, 2.90. (d) Schematic of the bilayer under generalized plane strain conditions used in our analytical model. (e) Deformed states of the analytical model at $F/(\mu LD)=3/2$ and $5/2$ for $D=2L$.

Figure 4

Deformation of a mechanical metamaterial comprising a square array of circular pores subjected to equibiaxial tension. Similar to the case of the cross-shaped sample shown in Fig. 1, an instability is triggered at $u/W\approx 2$ resulting in a checkerboard pattern of pores with two different sizes. The contours represent the maximum in-plane strain ${\epsilon }_{\max }$.