Neutral Holes: Theory and Design

As previously demonstrated it is possible to leave the geometry of an opening unspecified and invert the standard problem in elasticity to solve for shapes which achieve an optimum design condition on the final stress state. This fundamental strategy is applied to the problem of reinforced holes to determine optimum liner shape and stiffness properties. The design specification used is that the neutral liner completely eliminates any perturbation in the field giving no stress concentration whatsoever. The general theory is developed and the resulting expressions for equilibrium and compatibility are, in fact, not limited to the neutral condition but are valid for any thin reinforcement and, therefore, are fundamental to the general interaction problem. Closed form solutions are obtained for both the circular and the membrane shapes for a variety of free fields. New neutral liner designs are demonstrated for two basic situations important in practice: a) Circular holes in deviatoric and biaxial fields (flexural liner), and b) the membrane deloid shape and liner for the gradient field with an isotropic component. The second case is particularly useful in that it allows a design within a bending field and closely resembles the geostatic field condition for shallow pipes and tunnels.