General results on the flow of chopped fiber compounds in compression molding

Abstract

Two main theorems are proved from the governing differential equations for the flow of sheet molding compounds in compression molding. The first states that among all possible velocity distributions which satisfy the incompressibility condition and the kinematic boundary condition at a fixed edge, the actual solution minimizes the instantaneous rate of work by the molding press. The second is a general representation theorem for the velocity solution in terms of two scalar potentials. One of these potentials satisfies Laplace's equation and the second satisfies Helmholtz's equation. Each of these theorems is applied to problems of practical interest. The variational theorem is used to obtain a simple approximate solution for the flow front progression in a rectangular charge which, for a limited range of parameters, agrees remarkably well with previous numerical solutions of the exact equations. The representation theorem is used to examine the form of the solution in the important practical limit of a thin cavity. This limit is a singular perturbation of the governing differential equations, with a boundary layer at the flow front. Nevertheless, it is possible to prove that the flow front progression in thin cavities depends only on the outer solution, which can be expressed in terms of just one scalar potential.