Past the elastic limit a crystal does not spring back. It deforms by sliding: whole blocks of lattice shear over one another along particular planes and in particular directions. Which planes and which directions is not arbitrary - they are the most densely packed ones, where the atoms have least distance to travel - and a plane together with a direction in it is called a slip system.
This is why plasticity belongs in a texture toolbox. Slip systems are fixed in the crystal, so how easily a grain deforms depends on how the grain is oriented relative to the load. Deformation is anisotropic for the same reason stiffness is, and it feeds back: sliding rotates the lattice, so deforming a material changes its texture, which changes how it deforms next.
Below is a cubic crystal with its slip planes drawn inside it.
plottingConvention.default('y↑→x');
cs = crystalSymmetry('m-3m','mineral','Aluminium');
cS = crystalShape.cube(cs);
sS = slipSystem.fcc(cs);
plot(cS,'faceAlpha',0.2)
hold on
plot(cS,sS,'faceColor','red')
hold off
A face-centred cubic metal slips on the {111} planes along 110 directions: four planes with three directions in each, so twelve slip systems - or twenty-four if the two senses of shear are counted separately, which is what symmetrise returns.
Which system slips
A slip system responds only to the part of the applied stress that shears it. Resolve the stress onto the slip plane and along the slip direction, and the result is the resolved shear stress; the factor relating it to the applied stress is the Schmid factor, which depends purely on geometry and ranges from zero to one half.
Schmid's law says slip begins when the resolved shear stress reaches a critical value, so the system with the largest Schmid factor goes first. This is a good first answer and a poor last one: it treats each grain as though it were alone, when in reality a grain is wedged among neighbours that constrain it.
The same bounds as elasticity, for the same reason
Predicting how a polycrystal deforms means deciding what every grain has in common. The Taylor model assumes all grains undergo the same strain, which needs five independent slip systems per grain and generally overestimates the strength. The Sachs model assumes all grains feel the same stress, lets each slip on its best system alone, and underestimates it.
These are the same two assumptions that bound elastic averages, in plastic dress, and neither is true. Real grains compromise, and the models below differ mainly in how they let them.
Where to start
Slip Systems defines them for the common structures. Schmid Factor covers the geometry above and is where most single-grain reasoning happens.
Then the polycrystal models. Taylor Model is the equal-strain end, Single Slip Model the equal-stress end - it is also where the Sachs model is treated, since the physics is the same. Taylor Hex applies the Taylor model to hexagonal metals, where the slip systems have very different strengths and the model is more delicate. Slip Transmission asks whether slip in one grain can continue into its neighbour, which is the constraint the models above approximate away.
Dislocation Systems, GND and WBV approach the same material from the other side. A gradient of orientation inside a grain cannot exist without dislocations of one sign left over, so measuring the gradient measures those geometrically necessary dislocations. This is one of the few places where an orientation map yields a defect density directly.
Texture Evolution closes the feedback loop described at the start, and Lankford computes the anisotropy coefficient that sheet forming is judged by. VPSC reads results from the widely used visco-plastic self-consistent code.
Next
Deformation that is recovered is Elasticity, and the tensors both use are Tensors. The orientation changes inside grains that GND analysis needs are measured in EBSD. Deformation twinning is a second mechanism besides slip and is treated with Grain Boundaries.