Genesis, Degeneracy and Progress of Clapeyron Theorem in Linear Elasticity

Document Type : Research Paper

Authors
Giovanni Romano 1
Raffaele Barretta 2
Marina Diaco 2
1 Alma Mater and past professorship: University of Naples Federico II, Naples, Italy
2 Department of Structures for Engineering and Architecture, University of Naples Federico II, Via Claudio, 21 - 80125 Naples, Italy
10.22055/jacm.2025.48656.5401
Abstract
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.
Keywords
Standing structural models
Virtual power and work principles
Linear elasticity
Elastic potential
Thermodynamic consistency
Subjects

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Journal of Applied and Computational Mechanics
Volume 12, Issue 2
Spring 2026
Pages 624-631