On the derivation of the equilibrium equations in terms of displacement

Abstract

The study shows that errors exist in the derivation of equilibrium equations in terms of displacement. It is discovered that when the equilibrium equations in terms of displacement are derived, the variation of the differential order of displacement may cause the variation of the stress state in an elastomer. For plane stress problems, the Lame-Navier equations are not equivalent to the equilibrium equations described with stress. By submitting the displacement field of the well-known issue of a rectangular beam purely bent into the Lame-Navier equations, the conclusion is confirmed. It is also revealed that the velocity of longitudinal wave in an elastomer is affected by its thickness.