Variation of Stress Within a Body

In a stressed body, the components of stress generally vary from point to point. These variations are not arbitrary; they are governed by Newton’s second law of motion. By applying this law to an infinitesimal element, we can find the governing equations for the variation of stress within a body.

Consider an element with dimensions , , and . Stress components act on each face of this element. On each face of this cuboidal element, the stress components may differ from those on the opposite face. For example, if the xx-component of stress on one face is , then on the opposite face it will be taken as ​.

Let be the body force per unit mass. For example, if we consider only the gravity, then , where is the gravitational acceleration. Therefore, the -component of the force due to body force is where is the mass-density at the point.

Now, summing the forces acting in the xxx-direction due to stresses on all faces and the body force, and applying Newton’s second law gives After simplification and dividing both sides by the volume , we obtain: In the limit as , , and , the finite difference ratios become partial derivatives: By repeating this process for the y and z directions ( and ), we arrive at the full set of equations: These three equations can be written concisely as: This is also commonly written in vector or tensor notation as: which means This equation is known as Cauchy’s First Law of Motion.