the strain tensor
The strain tensor measures how much a material has been locally stretched and sheared -- the geometry of deformation, stripped of any rigid motion. Draw a tiny grid on a rubber sheet and pull: some squares elongate, some shear into parallelograms. The strain tensor is the local recipe for that distortion at every point, telling you the fractional change in length along each direction and the change in angle between directions.
It is built from the displacement field u(x), the vector by which each material point moves. In the small-deformation (linear) theory the strain tensor is the symmetric part of the displacement gradient: epsilon_ij = (1/2)(partial u_i / partial x_j + partial u_j / partial x_i). The diagonal components epsilon_xx, epsilon_yy, epsilon_zz are normal strains -- fractional stretches along each axis -- and the off-diagonal components are shear strains, measuring the tilting of originally perpendicular lines. The antisymmetric part of the same gradient is a local rigid rotation and carries no deformation, which is exactly why strain uses only the symmetric part. Strain is dimensionless, and its trace epsilon_kk equals the fractional change in volume (the dilatation).
The strain tensor is the input to elasticity: pair it with the stress tensor through Hooke's law and you can predict how any solid bends, stretches, and vibrates. It appears in bridge and aircraft design, in seismology, and wherever materials deform. The linear form above is an approximation valid only for small gradients of displacement; for large deformations (rubber, biological tissue) you need the full finite-strain (Green-Lagrange) tensor, which keeps quadratic terms.
A steel rod one meter long stretched by 0.5 millimeter has an axial normal strain of epsilon = (0.0005 m)/(1 m) = 5 x 10^-4, or 0.05 percent. That single dimensionless number, plugged into Hooke's law, gives the enormous internal stress the steel is carrying.
Normal strain is fractional stretch; the strain tensor collects stretches and shears at a point.
Strain measures deformation only -- a body translated or rotated rigidly has zero strain even though every point moved. This is guaranteed by taking only the symmetric part of the displacement gradient; the antisymmetric part is the discarded rotation.