Age-Dependent Modes of Extensional Necking Instability in Soft Glassy Materials

We study the instability to necking of an initially cylindrical filament of soft glassy material subject to extensional stretching. By numerical simulation of the soft glassy rheology model and a simplified fluidity model, and by analytical predictions within a highly generic toy description, we show that the mode of instability is set by the age of the sample relative to the inverse of the applied extensional strain rate. Young samples neck gradually via a liquidlike mode, the onset of which is determined by both the elastic loading and plastic relaxation terms in the stress constitutive equation. Older samples fail at smaller draw ratios via a more rapid mode, the onset of which is determined only by the solidlike elastic loading terms (though plastic effects arise later, once appreciable necking develops). We show this solidlike mode to be the counterpart, for elastoplastic materials, of the Considère mode of necking in strain-rate-independent solids.

Figure 1

Necked profiles. Extension rate shown by arrow in Fig. 2. Top: young sample, $t_w={10}^2$. Bottom: old, $t_w={10}^4$.

Figure 2

Fluidity model in protocol (a). Solid lines: draw ratio at which the radius of the filament at its thinnest point has fallen to 35%, 30%, $\dots$, 15% of the initial filament radius (curves upwards). Data are shown for three different initial sample ages (panels downwards). Initial aspect ratio $\mathrm{\Lambda }=2$. Dotted lines: Considère criterion [22]. Dashed lines: counterpart of the Considère criterion for elastoplastic materials. Parameters: $\mu =0.1$ and $\eta =0.01$. Arrow shows $\overline{\dot{\epsilon }}$ for Fig. 1.

Figure 3

SGR model in protocol (a). Solid contour lines upward show where initial area perturbations have increased in amplitude by successive powers of 1.4. The initial area profile was seeded such that $A=1+\xi \cos (2\pi u)$, where $\xi =3\times {10}^{-4}$. Dotted line: Considère criterion. Dashed line: counterpart criterion for elastoplastic material. Parameters: $x=0.3$, $\lambda =1.5$, $t_w={10}^2$.