the elasticity tensor
The elasticity tensor is the complete rulebook that tells a material how much stress to develop for a given strain -- the 3D, fully directional generalization of a spring constant. Where a spring answers stretch with one number k, a real solid can respond differently to being pulled along the grain versus across it, or to shear versus compression. The elasticity tensor encodes every one of those directional stiffnesses in a single object.
It is a rank-four tensor C_ijkl that linearly maps the strain tensor to the stress tensor: sigma_ij = C_ijkl epsilon_kl (summed over k and l). In three dimensions that is naively 3^4 = 81 numbers, but symmetries cut this down fast. Because stress and strain are each symmetric, C is symmetric in its first pair and second pair of indices (C_ijkl = C_jikl = C_ijlk), leaving 36; and because there exists a stored elastic-energy function, C is symmetric under exchange of the two pairs (C_ijkl = C_klij), leaving at most 21 independent constants for the most general (triclinic) crystal. Higher symmetry cuts further -- a cubic crystal has 3, and a fully isotropic material has just 2.
You meet the elasticity tensor whenever a material's stiffness depends on direction: the grain of wood, the layering of composites, the crystalline axes of a semiconductor wafer, the fiber orientation in bone. Its inverse, mapping stress back to strain, is the compliance tensor. For the common idealization of an isotropic solid, all 21 constants collapse to the two Lamé parameters.
Wood is strongly anisotropic: it is far stiffer along the grain than across it, so its elasticity tensor needs several distinct constants. Steel, by contrast, is well modeled as isotropic, so its entire elastic behavior is captured by just two numbers.
A rank-4 tensor with up to 21 independent constants (2 if isotropic) links stress to strain.
The reduction to 21 constants assumes a strain-energy function exists (hyperelasticity); the further reduction to 2 assumes isotropy. Neither is universal -- crystals, wood, and fiber composites genuinely need more, and treating them as isotropic gives wrong stiffnesses.