Shakedown of elastic-perfectly plastic materials with temperature-dependent elastic moduli

Peigney, Michaël

doi:10.1016/j.jmps.2014.06.008

Cited by 26 publications

(32 citation statements)

References 24 publications

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“…(9) to be cast in the form (10) [22]. In the general case when L(t) is not radial, then the properties of Eq.…”

Section: Evolution Equation For the Stress Fieldmentioning

confidence: 99%

“…Condition (ii) sets restriction on the time variations of the elastic moduli. Setting such a restriction is necessary to obtain global convergence results: One can indeed find counterexamples in which condition (i) is fulfilled and some solutions of (9) do not become elastic in the large time limit [22].…”

Section: Shakedown Theoremmentioning

confidence: 99%

“…(19). The inequality (19) is obtained by performing minor modifications to a similar result obtained in [22] for perfect plasticity, hence we only sketch the main steps of the proof and refer to [22] for details. Recalling that d(L(t)ρ * (t))/dt = 0 and setting H(t) = − 1 2 τ(t),L 0 (t)τ(t) , some straightforward manipulation shows thatḟ (t) = τ(t),η(t) + H(t)…”

Section: Variation Of the Elastic Energymentioning

confidence: 99%

“…The main difficulty is that the proof used in the original Melan's theorem -as well as in most of its knows extensions -crucially relies on some contraction properties that are lost when the elastic moduli are allowed to vary in time. A shakedown theorem has recently been proposed for elastic-perfectly plastic materials with time-periodic elastic properties [22]. The statement and proof of that theorem differ significantly from the case of constant material properties.…”

Section: Introductionmentioning

confidence: 99%

“…In this communication, the theorem of [22] is presented and improved upon. The extension to elastic-viscoplastic materials is discussed.…”

Section: Introductionmentioning

confidence: 99%

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On Shakedown of Elastic-Plastic Bodies With Temperature Dependent Properties

Peigney¹

2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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Abstract. For elastic-perfectly plastic structures under prescribed loading histories, the wellknown Melan's theorem gives a sufficient condition for shakedown to occur, i.e. for the evolution to become elastic in the large-time limit. The original Melan's theorem rests on the assumption that the material properties remain constant in time, independently on the applied loading. This communication addresses the long standing issue of extending Melan's theorem to temperaturedependent (or time-dependent) elastic moduli. The main motivation is to extend the range of applications of Melan's theorem to thermomechanical loading histories with large temperature fluctuations: In such case, the variation of the elastic properties with the temperature cannot be neglected. Some recent results obtained in perfect plasticity and viscoplasticity are presented and discussed.

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“…(9) to be cast in the form (10) [22]. In the general case when L(t) is not radial, then the properties of Eq.…”

Section: Evolution Equation For the Stress Fieldmentioning

confidence: 99%

Section: Shakedown Theoremmentioning

confidence: 99%

Section: Variation Of the Elastic Energymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…In this communication, the theorem of [22] is presented and improved upon. The extension to elastic-viscoplastic materials is discussed.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On Shakedown of Elastic-Plastic Bodies With Temperature Dependent Properties

Peigney¹

2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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Shakedown and Safety Assessment

Zouain

2017

Encyclopedia of Computational Mechanics Second Edition

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The main subjects of this chapter are the variational formulations for the computation of a safety factor in shakedown analysis, the numerical procedures available to implement this analysis, and an overview on the wide field of applications and conceptual extensions. In structural engineering and solid mechanics, the word shakedown , introduced in the 1940s by Prager, became, since then, synonymous with 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 with inelastic strains). Concerning the failure analysis of ductile structures, the terms alternating plasticity (AP) and incremental collapse (IC), besides plastic collapse , are also widely used to identify failure modes under variable loadings. A structure undergoes AP when the fluctuating loading program produces, after any arbitrarily large time, some plastic deformation as well as a subsequent vanishing of the net plastic deformation. This induces failure due to low‐cycle fatigue. Likewise, the structure fails by IC when plastic deformations accumulate 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 chapter with direct methods that are based solely on the knowledge of a range of load variations (or a reference loading, in limit analysis). It is worth emphasizing that the differences between incremental and direct methods go beyond the computational advantages frequently obtained with the latter ones. In fact, the fundamental questions about whether or not critical loads or cycles do exist, independently from loading histories, can only be answered by the theories of direct methods. The mathematical foundations of shakedown theory strongly rest on optimization theory and the related areas of convex analysis and mathematical programming. This chapter contains simplified derivations of the static, kinematic, and mixed variational principles of shakedown theory by means of the basic techniques of convex analysis. Moreover, numerical procedures suitable to solve the shakedown analysis problem are discussed here in the framework of mathematical programming techniques combined with common procedures in the field of finite element methods. Applications in fundamental examples are then presented, namely (i) a restrained block under thermomechanical loadings; (ii) the classical Bree problem of a tube under variable pressure and temperature, with a more realistic model and including an exact solution; (iii) a plate with a circular hole subjected to variable tractions; (iv) a plate with simulated imperfections showing critical mechanisms with strongly localized deformations; and (v) a pipe elbow under internal pressure and in plane bending moment.

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On Cyclic Steady States and Elastic Shakedown in Diffusion-Induced Plasticity

Peigney

2020

Direct Methods

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This chapter is devoted to media in which plasticity and diffusion are coupled, such as electrode materials in lithium ion batteries. We present some recent results on the large time behavior of such media when they are submitted to cyclic chemo-mechanical loadings. Under suitable technical assumptions, we notably show that there is convergence towards a cyclic steady state in which the stress, the plastic strain rate, the chemical potential and the concentration of guest atoms are all periodic in time (with the same period as the applied loading). A special case of interest is that of elastic shakedown, which corresponds to the situation where the medium behaves elastically in the large time limit. We present general theorem that allow one to construct both lower and upper bounds of the set of loadings for which elastic shakedown occurs, in the spirit of Melan and Koiter theorems in classical plasticity. An illustrative example-for which all the relevant calculations can be done in closed-form-is presented.

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Shakedown of elastic-perfectly plastic materials with temperature-dependent elastic moduli

Cited by 26 publications

References 24 publications

On Shakedown of Elastic-Plastic Bodies With Temperature Dependent Properties

On Shakedown of Elastic-Plastic Bodies With Temperature Dependent Properties

Shakedown and Safety Assessment

On Cyclic Steady States and Elastic Shakedown in Diffusion-Induced Plasticity

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