The Stress Principle of Euler and Cauchy

Near the end of an earlier blog - the one called "vectors," I almost inadvertently mentioned something so important that Truesdell referred to it as "the defining principle of continuum mechanics." He called it The Stress Principle of Euler and Cauchy. Note that it is a basic assumption about the nature of continuous media, rather than something that can be derived.

This principle states that there is a set of stress vectors acting on the surface of any region in a material which completely represent the forces which are exerted on the region by the material outside.

We can put this statement in the form of the equation below. The left hand side (LHS) represents the volume integral of the i th component of the force density acting on each point inside the volume, and the right hand side (RHS) is the integral of the i th component of the stress vector acting on the surface of V.

The mass density of the material at each point is given by the Greek letter rho, the force density f is the force per unit mass, and the stress vector is the force per unit area acting on the surface of V.

Since the region of of interest in the material can be any shape or size, including arbitrarily small, it follows from this equation that we don't need to know the forces inside the region in order to calculate anything. We only need to know the forces on its surface.

The integral on the LHS represents the total force due to the exterior material which acts upon the interior material. Similarly, the interior material acts on the exterior, with the same total force, because we are considering bodies which are in equilibrium.

By equilibrium, we mean the following. When we first apply some combination of constant forces to a body, it moves - it changes its shape. When the movement stops, we say the body is in equilibrium. The vector sum of the external forces applied to the body are exactly balanced by the reactive forces of the body onto whatever is applying the external forces, and a similar statement applies to each region of the body we choose.

Note that this force balancing only applies to elastic materials, like rubber. For example, if you pulled hard enough on the ends of a length of aluminum rod, it would be permanently deformed. You could hold the rod without pulling on it at all, and it would still be deformed.

On the other hand, a long piece of aluminum (or copper, or steel) wire can be held horizontally at one end, and moved up and down vertically at the other end without any permanent deformation, as long as the distance traveled by the free end is small enough compared with the wire's length. So if you let go of the free end, the wire will return to its initial position.

This observation about the balance of interior and exterior forces also applies to the RHS of the equation, but it requires some further discussion, along with a drawing, so I'll save that for the next blog.