Shakedown theory applied to structural analysis and high-cycle fatigue modelling

This presentation deals with the foundations and some applications of the theory of shakedown analysis.

In structural engineering and solid mechanics, the word shakedown became synonymous of elastic adaptation in the presence of variable loadings, a safe stabilized phenomenon, often due to some limited plastic deformation (or equivalently, due to the residual stress distribution associated to inelastic strains).

Concerning the failure analysis of ductile structures, the terms: alternating plasticity and incremental collapse, besides plastic collapse, are widely used to identify failure modes under variable loadings. A structure undergoes alternating plasticity, when the fluctuating loading program produces some plastic deformation in each cycle althoughthe net plastic deformation per cycle is zero. This induces failure due to low cycle fatigue. Likewise, the structure fails by incremental collapse when plastic deformations accumulates in the form of a compatible strain distribution that leads to excessive inelastic deformation.

Shakedown analysis allows working under the realistic assumption that only the range of variable loadings is known, unlike the usual prescription of a particular loading history. That is, we deal in this paper with direct methods that are based solely in the knowledge of a range of load variations (or a reference loading, in limit analysis).
We briefly present the derivation of statical, kinematical and mixed variational principles of the shakedown theory by means of the basic techniques of convex analysis. Numerical procedures suitable to solve the shakedown analysis problem are also discussed in the framework of mathematical programming techniques combined with common procedures in the field of finite element methods. Aplications of shakedown analysis for the safety assessment of structures are then reported. Finally, we present a model for high-cycle fatigue based on the phenomenological description of infinite life endurance as an elastic accommodation process produced at the scale of the representative volume of the material.