Genesis, Degeneracy and Progress of Clapeyron Theorem in Linear Elasticity

About two centuries have passed since Émile Clapeyron conceived a simple formula for expressing the energy stored in a linearly elastic structure which is standing during the dynamic process of loading from an initial to a final state of static equilibrium. Standing means displacement and deformation fields are small enough to allow for a geometric linearised theory to be applicable in structural formulations. Clapeyron’s theorem is often introduced in an over-restricted context. A formulation in which essential assumptions are explicitly stated and redundant elements are eliminated is the prerequisite for detecting range of validity and possible extensions. Physical evidence and notion of virtual work clarify the claim of thermodynamic paradox concerning a gap between mechanical work done by loading forces and storage of elastic energy in the structure. Beauty and power of Clapeyron’s theorem, as equality based on variational conditions of equilibrium and on linear elasticity, are thus recovered. The standard result is extended for application to structural models, including kinematic constraints, prestress, impressed distortion and is reshaped with a formulation of incremental elasticity based on the notion of elastic state and pertinent linearised approximation.