Stress
When a generic load is applied to an object, stress in any point P of the object can be expressed through a matrix of 9 components. We can imagine to consider a little cube of matter around point P, with the normals to the faces oriented as the axis x, y, z of the cartesian coordinate system. Components σx, σy, σz of the matrix are the normal stresses, acting on the faces of normal x, y, z respectively, while components τxy, τxz, τyx, τyz, τzx, τzy are the tangential or shear stresses: the first subscript refers to the direction of the normal to the face on which the stress acts, the second subscript refers to the direction of the component (Figure 1).

Figure 1: Cartesian stress components

It can be shown that the stress matrix is symmetric, meaning that τxy = τyx, τxz = τzx, τyz = τzy.
When these stress components associated with three mutually perpendicular faces at a point are known, the stress vector associated with any face through the same point with a generic orientation can be calculated. Moreover for each state of stress one set of three mutually perpendicular planes can be determined on which the shear stresses are all zero. The normal stresses σ1, σ2, σ3 corresponding to the principal directions are known as principal stresses.

Strain
Strain in a point P of an object subjected to a system of loads can be expressed through a 3x3 matrix, exactly as explained above for stress.

Components εx, εy, εz on the diagonal of the matrix are the normal strain components are γxy, γxz, γyx, γyz, γzx, γzy are the shear strain components in the xy, xz, yx, yz, zx, zy planes. It can be shown that also the strain matrix is symmetric.

Relations between stress and strain
In a perfectly elastic material the relation between stress and strain is expressed throug the Hooke's law. For an isotropic material, which is a material that has the same elastic properties in all directions, the Hooke's law is expressed through the following six equations:

εx = [σx - ν(σy + σz)] / E	γxy = τxy / G
εy = [σy - ν(σz + σx)] / E	γyz = τyz / G
εz = [σz - ν(σx + σy)] / E	γzx = τzx / G

where E is the Young's modulus, G is the shear modulus and ν is the Poisson's ratio. The shear modulus G can be expressed as: G = E / [ 2 (1 + ν) ].