content_image/295508.jpg content_image/295509.jpg
where E$_{eff}$ is the effective elastic modulus of the sample. 〈 . . . 〉 denotes the spatial average along the x axis. The external strain e$_{ext}$ , which is negative, is changed at a constant rate, and the concomitant change in the plastic strain e follows from the yield condition, 

where Σ$_{S}$ evolves according to 

<img src='content_image/295508.jpg'>

<img src='content_image/295509.jpg'>

From these equations, a necessary condition for structural softening and strain localization derives as aε$_{S}$ ≤ Σ$_{S}$ / (Σ$_{R}$ + Σ$_{S}$) . This condition (for derivation and further discussion, see supplementary material) is tantamount to the requirement that the strain softening processes due to shear deformation must outweigh the competing process of densification-induced hardening. 

To solve the system of integrodifferential equations for e , s and Σ$_{S}$ , we start from initial conditions s = e = 0 , Σ$_{S}$ = Σ 0 $_{S}$. In each time step we change e$_{ext}$ according to the imposed strain rate and evaluate s and e self-consistently from Eqs. 7 and 8 with the boundary condition ∂$_{x}$e (0) = ∂$_{x}$e ( L ) = 0 . By comparing with the previous time step we determine the local plastic strain rates ˙ e , evaluate the changes in Σ$_{S}$ from Eq. 9, and repeat until a given end strain is reached. 

## 4 Results 

## 4.1 Experimental Observations 

Fig. 1 shows a snapshot of the patterns of axial strain ϵ$_{xx}$ ( x, y ) and strain rate ˙ ϵ$_{xx}$ ( x, y ) emerging during a compression test. The plots use Eulerian coordinates, i.e., each volume element is shown at its position in the laboratory frame. It is seen that compaction proceeds in a strongly heterogeneous manner, as a region of increased density at the top of the sample is separated from an uncompacted region at the bottom by a sharp moving front where the strain rate is concentrated. This front is perpendicular to the compression direction. Owing to the intrinsic structural heterogeneity of the snow microstructure, the deformation state behind the front is not completely homogeneous. To facilitate comparison with our model, which provides an effective one-dimensional description, in the following we average the strain and strain rate fields over the y direction. 

To visualize band motion, we use space-time plots where we plot the y -averaged strain rate ˙ e$_{tot}$ ( x, t ) = 〈 ˙ ϵ$_{xx}$ ( x, y, t ) 〉 y in colorscale as a function of x and t (Fig. 2). In these plots we use Lagrangian x coordinates. 

