The linear Elasticity complex: a natural formulation

Romain Lloria (LMPS) , Boris Kolev (LMPS)

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classification 🧮 math.DG physics.class-ph

keywords complexelasticityformulanaturalrecoveraccountallowscompatibility

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The pith

The linear elasticity complex is reformulated as a natural modification of the de Rham complex for symmetric tensors, with an integration formula for displacement from strain and a dual complex for stress potentials in 2D and 3D.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

In linear elasticity, the strain tensor must satisfy compatibility conditions to arise from a displacement field, known as Saint-Venant's conditions. The paper uses a generalized differential complex from Dubois-Violette and Henneaux, which adjusts the standard de Rham complex to handle the symmetry of tensor indices. This creates a more natural formulation of the elasticity complex. They supply an explicit integrating formula to recover the displacement vector from the strain tensor, similar to the Poincaré formula for integrating differential forms. They also define a Hodge star operator suited to this symmetric setting and construct a dual complex. This dual complex enables recovery of stress potentials, which are particularly useful in two and three dimensions for expressing the stress field in terms of potential functions. The overall approach treats the mathematical structure of elasticity in a more intrinsic geometric manner.

Core claim

We reformulate the Elasticity complex and Saint-Venant's compatibility condition using the generalized differential complex of Dubois-Violette-Henneaux.

Load-bearing premise

That the generalized differential complex of Dubois-Violette-Henneaux applies directly and naturally to the index-symmetric tensors appearing in linear elasticity without introducing inconsistencies or requiring substantial additional structure.

read the original abstract

We reformulate the Elasticity complex and Saint-Venant's compatibility condition using the generalized differential complex of Dubois-Violette-Henneaux. This is just a slight and natural modification of the de Rham complex to take account of the index symmetry of the tensors involved. An integrating formula to recover the displacement from the strain and similar to the Poincar{\'e} formula is provided. Finally, a Hodge star operator and a dual complex is introduced, which allows to recover stress potentials in dimensions 2 and 3.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of an existing generalized differential complex to the symmetric-tensor setting of elasticity; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption The generalized differential complex of Dubois-Violette-Henneaux exists and can be applied to symmetric tensor fields in the elasticity context.
Invoked as the foundation for the reformulation of the elasticity complex and compatibility conditions.

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