In this analysis, the red blood cell is idealized as a spherical liquid drop, encapsulated by a thin membrane. By transforming G. I. Taylor's equations for the surface loading on a liquid drop with no membrane in Couette flow, spherical loading components are calculated for a drop with a membrane. A solution developed by Fliigge for unsymmetrical loading of spherical shells is then used to derive the complete membrane stress pattern for spherical, membrane-encapsulated, liquid drops in couette flow. The maximum distortion energy is calculated for any given shear rate, and using the relation, a critical shear stress for short duration failure is found as a function of the critical distortion energy. Viscoelastic solutions based on the maximum normal strain failure theory and the maximum stress failure theory are then developed in order to predict the entire critical shear stress vs. time curve for hemolysis. Using data supplied by Rand's micropipette experiments on red cell membrane strength, the theories are then shown to roughly predict the entire couette flow hemolysis curve for times up to 1 seconds. Thus, the theory in this analysis provides a much greater understanding of the mechanism causing red blood cell damage in artificial heart valves, heart-lung machines, artificial kidneys, etc. In addition, due to the model cell's physical generality, the theory is applicable to other biological and non-biological systems.