Stress Function Formulation

Abstract

Newton’s law of gravitation states that two heavy bodies attract each other with a force proportional to the inverse square of their distance — thus it is essentially a vector theory, being concerned with forces. However, the idea of a scalar gravitational potential can be introduced by defining the work done in moving a unit mass from infinity to a given point in the field. The principle of conservation of energy requires that this be a unique function of position and it is easy to show that the gravitational force at any point is then proportional to the gradient of this scalar potential. Thus, the original vector problem is reduced to a problem about a scalar potential and its derivatives.

In general, scalars are much easier to deal with than vectors. In particular, they lend themselves very easily to coördinate transformations, whereas vectors (and to an even greater extent tensors) require a set of special transformation rules (e.g. Mohr’s circle).

In certain field theories, the scalar potential has an obvious physical significance. For example, in the conduction of heat, the temperature is a scalar potential in terms of which the vector heat flux can be defined. However, it is not necessary to the method that such a physical interpretation can be given. The gravitational potential can be given a physical interpretation as discussed above, but this interpretation may never feature in the solution of a particular problem, which is simply an excercise in the solution of a certain partial differential equation with appropriate boundary conditions. In the theory of elasticity, we make use of scalar potentials called stress functions or displacement functions which have no obvious physical meaning other than their use in defining stress or displacement components in terms of derivatives.