Velocity gradient of single crystals deforming by basal glide

The velocity gradient of single crystals, deforming "only" by basal glide, can give us a pretty good idea of what type of deformation to expect for a given fabric.  This is especially true in the case of the homogenous stress assumption, in which case each crystal is completely independent of it's neighbors.

If we take cone fabric as an example, which has all the crystals within the cone angle alpha, we can visualize that distribution on an equal area plot as a circle growing from the center out, towards 90 deg which is random, isotropic, fabric.  Then the deformation, using the homogeneous stress assumption, is simply the average deformation rate within that circle.

Taking the uniaxial compression gradient as an example, we can immediately see that only the diagonal terms will be non-zero; the center-symmetric average of L13, for instance, clearly is 0 for all cone angles.  At the same time it is clear that non-vertical fabric can cause fairly unexpected deformation patterns.

In uniaxial compression

The velocity gradient tensor, Lcij(q, f), of a single crystal in uniaxial compression. The crystal is deforming only by glide in the basal plane. Each plot shows a different component Lcij, with the Lc11-component in the top left corner and Lc33 in the bottom right corner.

Good quality PDF version of figure and caption

In pure shear

The velocity gradient tensor, Lcij(q, f), of a single crystal in pure shear. The crystal is deforming only by glide in the basal plane. Each plot shows a different component Lcij normalized by A s3 , with the Lc11-component in the top left corner and Lc33 in the bottom right corner.

Good quality PDF version of figure and caption

 

In simple shear

The velocity gradient tensor, Lcij(q, f), of a single crystal in simple shear. The crystal is deforming only by glide in the basal plane. Each plot shows a different component Lcij, with the Lc11-component in the top left corner and Lc33 in the bottom right corner.

Good quality PDF version of figure and caption