Cauchy formula in matrix form

The stress vector t is now written as a 3-component column vector. The equation tells how to write three equations for the components of t, by multiplying the stress matrix by the unit vector n.

The multiplication is done as follows (see equations below): the first component of t is given by multiplying the "11" component of sigma by the first component of n, etc.

Note how the first subscript of sigma corresponds to the t subscript, and the second subscript of sigma matches the n subscript.

Each component of the stress vector t depends on three components of the stress tensor, and on the three components of the surface normal vector n. Recall that t is the force applied at each point on the surface of a chosen region of the body by the surrounding material, and that n is the unit vector normal to that surface at the same point.

Put another way, here's the remarkable thing about Cauchy's theorem: it means that for every possible surface through a given point in the body - and their are an infinite number of such surfaces - all you need are the components of the stress tensor at that point. Then you can calculate the stress vector for any surface through that point, converting its normal vector into the stress vector with the Cauchy formula above.

The first, or x, component of t represents the force in the x-direction. If positive, the x component points out from the surface at that point; i.e., it is a tensile force. If negative, the x component points into the region enclosed by the surface, and is therefore compressive.

Similarly, the y and z components of t are in the y and z directions.

Note that the x component of t depends not only on the component of the normal vector n in the x direction, but also on the y and z components of n, and similarly for the y and z components of t.

The stress matrix element that corresponds to the x component of t and the x component of n is the -11- or -xx- element, and similarly for the y and z components. Later we will see why these components of the stress tensor are called normal stresses. The off-diagonal components are the shear stresses.