Elasticity is the part of deformation a material takes back. Load it, and it changes shape; unload it, and it returns. The relationship between the load and the shape change is linear for small strains, and the constant of proportionality is the stiffness tensor.
In a crystal that constant depends on direction, so an elastic answer is never a single number. It also propagates: an elastic disturbance travels as a wave, and its speed is set by the same tensor. This is why elasticity is the bridge between a laboratory measurement on a crystal and a seismic observation of the Earth - the two are the same physics at different scales.
Below, the speed of a compressional wave through olivine, in every direction.
plottingConvention.default('y↑→x');
cs = crystalSymmetry('mmm',[4.7646 10.2296 5.9942],'mineral','Olivine');
C = stiffnessTensor.load(fullfile(mtexDataPath,'tensor','Olivine1997PC.GPa'),cs);
% a velocity is a stiffness divided by a density, so the density is required
C = addOption(C,'density',3.355);
% compressional wave velocity over all directions
[vp,~,~] = velocity(C);
plot(vp,'complete','upper')
mtexColorbar('title','v_p in km/s')
The fastest direction is close to thirty per cent faster than the slowest, in a single crystal. In a rock whose olivine grains have been aligned by flow, that anisotropy survives the averaging and becomes measurable hundreds of kilometres away.
Note that the density had to be supplied. A stiffness on its own does not determine a speed - the wave equation divides it by density - so a velocity computed from a tensor carrying no density is not in km/s and is not a velocity.
Three waves, not one
An isotropic solid carries two kinds of elastic wave: one compressional and one shear, the latter with any polarisation you like. Anisotropy breaks that tie. In a crystal there are three waves in every direction - one roughly compressional and two shear waves with definite and mutually perpendicular polarisations - and the two shear waves generally travel at different speeds.
This splitting is the most useful thing in the chapter, because it is directly observable. A shear wave entering an aligned rock leaves as two pulses separated in time, and the size and orientation of that separation is a measurement of the texture along the path.
"Roughly compressional" is meant literally: away from symmetry directions the particle motion of the fast wave is not exactly along the propagation direction. The pages below keep track of the polarisation as well as the speed for this reason.
Where to start
Isotropic Theory is the familiar case, and worth reading first even if your material is anisotropic - it fixes what the usual moduli mean and gives the reference the anisotropic results are compared against.
Anisotropic Theory is the general case: Hooke's law with the full tensor, the directional Young's modulus, and how symmetry reduces the number of independent constants.
Wave Velocities computes the three waves and their polarisations, and is where the splitting above is made quantitative.
CPO Seismic Properties is the whole chain end to end: measured orientations, an averaged tensor for the aggregate, and the seismic anisotropy that follows. It is the natural place to see how much of the chapter fits together.
Next
The tensor machinery underneath is Tensors, including the averaging schemes that turn single-crystal constants into aggregate ones. The orientation distributions those averages need come from ODF or from EBSD. Deformation that is not recovered on unloading is Plasticity.