On the derivation of the equilibrium equations in terms of displacement

TL;DR

This paper investigates errors in deriving equilibrium equations via displacement, revealing that variations in displacement order can alter stress states and that classical equations may not always be equivalent, with implications for elastomer wave velocities.

Contribution

It identifies errors in the traditional derivation of equilibrium equations in terms of displacement and clarifies their limitations in elastomer stress analysis.

Findings

Errors exist in the derivation of equilibrium equations in displacement form.

Lame-Navier equations are not always equivalent to stress-based equilibrium equations.

Elastomer wave velocity depends on thickness.

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.