the stress tensor

The stress tensor describes the internal forces a material transmits across itself -- how hard neighboring bits of a solid push and shear on each other, per unit area. Imagine slicing an object with an imaginary plane: the material on one side exerts a force on the other. That force per area is the traction, and it depends on the orientation of the cut. The stress tensor is the single object that gives the traction on a cut of any orientation.

Its component sigma_ij is the i-th component of the force per unit area acting on a surface whose outward normal points in the j direction. Cauchy's relation packages this: the traction vector on a surface with unit normal n is t_i = sigma_ij n_j. The diagonal components are normal stresses (tension when positive, compression when negative), and the off-diagonal components are shear stresses. Balancing angular momentum on an infinitesimal element (with no internal body-couples) forces the stress tensor to be symmetric, sigma_ij = sigma_ji. The equation of motion of a continuum is Cauchy's momentum equation, rho a_i = partial sigma_ij / partial x_j + f_i, where f is the body force per volume.

Stress has units of pressure (pascals). For a fluid at rest it reduces to a pure pressure, sigma_ij = -p delta_ij, with no shear; a solid, unlike a fluid, can sustain static shear stress, which is why solids hold their shape. The stress tensor is the language of strength of materials and structural engineering -- yield, fracture, and buckling are all statements about when stress exceeds what a material can bear.