On shakedown of elastic-plastic bodies with temperature-dependent properties

International audienceFor elastic-perfectly plastic structures under prescribed loading histories, the well-known 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 temperature-dependent (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

A Direct Method For Predicting The High-Cycle Fatigue Regime In SMAs: Application To Nitinol Stents

International audienceIn metal fatigue, it is common practice to distinguish between high-cycle fatigue (i.e. failure occurring after 10000-100000 cycles) and low-cycle fatigue. For elastic-plastic materials, there is an established correlation between fatigue and energy dissipation. In particular, high-cycle fatigue occurs when the energy dissipation remains bounded in time. Although the physical mechanisms in shape-memory alloys (SMAs) differ from plasticity, the hysteresis that is commonly observed in the stress-strain response shows that some energy dissipation occurs. It can be reasonably assumed that situations where the energy dissipation remains bounded are the most favorable for fatigue durability. In this communication, we present a direct method for determining if the energy dissipation in a SMA structure (submitted to a prescribed loading history) is bounded or not. That method is simple to use and could be relevant for the design of SMA systems with high durability requirements, such as stents. Some numerical results are presented for the design of Nitinol stents and compared with experimental results from the literature

Improved bounds on the energy-minimizing strains in martensitic polycrystals

International audienceThis paper is concerned with the theoretical prediction of the energy-minimizing (or recoverable) strains in martensitic polycrystals, considering a nonlinear elasticity model of phase transformation at finite strains. The main results are some rigorous upper bounds on the set of energy-minimizing strains. Those bounds depend on the polycrystalline texture through the volume fractions of the different orientations. The simplest form of the bounds presented is obtained by combining recent results for single crystals with a ho-mogenization approach proposed previously for martensitic polycrystals. However, the polycrystalline bound delivered by that procedure may fail to recover the monocrystalline bound in the homogeneous limit, as is demonstrated in this paper by considering an example related to tetragonal martensite. This motivates the development of a more detailed analysis, leading to improved polycrystalline bounds that are notably consistent with results for single crystals in the homogeneous limit. A two-orientation polycrystal of tetragonal martensite is studied as an illustration. In that case, analytical expressions of the upper bounds are derived and the results are compared with lower bounds obtained by considering laminate textures

On the energy-minimizing strains in martensitic microstructures-Part 2: Geometrically linear theory

International audienceThis paper addresses the theoretical prediction of the quasiconvex hull of energy-minimizing (or stress-free) strains that can be realized by martensitic microstructure. Polyconvexification and related notions are used to derive some upper bounds (in the sense of inclusion). Lower bounds are obtained from lamination techniques. The geometrically linear setting (infinitesimal strains) is considered in the present Part 2. Three-, four-, and twelve-well problems are considered. In particular, the structure of the set of energy-minimizing strains in cubic to monoclinic transformations is investigated in detail. That investigation is notably supported by three-dimensional vizualisations obtained by considering four-well restrictions

Shakedown theorems and asymptotic behaviour of solids in non-smooth mechanics

International audienceThe work presented in this communication is at the crossroad between direct methods and non-smooth mechanics. One objective is to derive shakedown theorems in situations where constraints are prescribed on the internal variables. A motivation is the study of shape memory alloys (SMAs) : in those materials, inelastic deformation occurs as the result of a solid/solid phase transformation between different crystallographic structures. Much effort has been devoted to developing constitutive laws for describing the behaviour of SMAs. The phase transformation is typically tracked by an internal variable which - depending on the complexity of the material model - may be scalar or vectorial. A fundamental observation is that, in most of SMA models, that internal variable must comply with some a priori inequalities, resulting from the mass conservation in the phase transformation process. As a consequence, the internal variable is constrained to take values in a set T that is not a vectorial space. The presence of such constraints constitutes a crucial difference with standard plasticity models, and calls for special attention when the structural evolution problem is considered. Non-smooth mechanics offer a sound mathematical framework for handling constraints on state variables. This communication is devoted to studying the asymptotic behaviour (i.e. as time tends to infinity) of solids in the framework of non-smooth mechanics

Energy minimizing strains in martensitic microstructures

This communication is concerned with the theoretical prediction of the energy-minimizing (or stress-free) strains that can be realized by martensitic microstructures. Polyconvexification and related notions are used to derive some upper bounds (in the sense of inclusion) on the set of energy-minimizing strains. Lower bounds are obtained from lamination techniques. Three-, four-, and twelve-well problems are considered. In particular, the structure of the set of energy-minimizing strains in cubic to monoclinic transformations is investigated in detail

On shakedown of shape memory alloys with permanent inelasticity

International audienceShape memory alloys (SMAs) offer interesting perspectives in various fields such as aeronautics, robotics, biomedical sciences, or structural engineering. The distinctive properties of those materials stem from a solid/solid phase transformation occuring at a microscopic level. Modeling the rather complex behavior of SMAs is a topic of active research. Lately, SMA models coupling phase-transformation with permanent inelasticity have been proposed to capture degradation effects which are frequently observed experimentally for cyclic loadings. In this paper, the classical static and kinematic shakedown of plasticity theory are extended to such material models. Those results gives conditions for the energy dissipation to remain bounded, and might be relevant for the fatigue design of SMA systems

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

International audienceFor elastic-perfectly plastic structures under prescribed loading histories, the celebrated Melan's theorem gives a sufficient condition for the evolution to become elastic in the large-time limit. That situation, classically referred to as shakedown, is associated with the intuitive idea that the plastic strain tends to a limit as time tends to infinity. The Melan's theorem has the distinctive property of being path-independent, i.e. independent on the initial state of the structure. Regarding fatigue design, shakedown corresponds to the most beneficial regime of high-cycle fatigue, as opposed to the regime of low-cycle fatigue which typically occurs if the plastic strain does not converge towards a stabilized value.This communication addresses the extension of Melan's theorem to situations in which the elastic moduli are fluctuating in time. Such time fluctuations may result from significant variations of the temperature. In a lot of practical situations, structural elements are indeed submitted to thermomechanical loading histories in which variations of the temperature are large enough for the temperature dependence of the material not to be negligible. It has been conjectured that Melan's theorem could be extended to temperature dependent (or time-dependent) elastic moduli , but no theoretical result is available. This communication aims at providing results in that direction, with a special emphasis on time-periodic variations. If Melan's condition is satisfied, we show that shakedown indeed occurs provided the time fluctuations of the elastic moduli satisfy a certain condition (which in particular is fulfilled if the time fluctuations are not too large). We provide a counterexample which shows that setting such a constraint on the elastic moduli is necessary to reach path-independent theorems as proposed. A simple mechanical system is studied as an illustrative example

study of energy harvesting from traffic-induced bridge vibrations

International audienceThis communication is concerned with energy harvesting of traffic-induced vibrations in bridges. In situ measurements of vibrations in a pre-stressed concrete bridge are presented and analyzed. Using those results, a prototype of cantilever piezoelectric harvester was designed, tested, and modeled. Even though the considered bridge vibrations are characterized by small amplitude and a low frequency (i.e. below 15 Hz), it is shown that mean power of the order of 0.03 mW can be produced, with a controlled voltage between 1.8 and 3.6 V. A simple theoretical model is proposed for predicting the delivered power in terms of traffic intensity. That model shows good agreement with the experimental results and leads to a simple but effective design rule for piezoelectric harvesters to be used on bridges

A Direct Method For Predicting The High-Cycle Fatigue Regime In Shape Memory Alloys: Application To Nitinol Stents

International audienceIn fatigue design of metals, it is common practice to distinguish between high-cycle fatigue (occurring after 10000-100000 cycles) and low-cycle fatigue. For elastic-plastic materials, there is an established correlation between fatigue and energy dissipation. In particular, high-cycle fatigue occurs when the energy dissipation remains bounded in time. Although the physical mechanisms in Shape Memory Alloys (SMAs) differ from plasticity, the hysteresis observed in the stress-strain response shows that some energy dissipation occurs, and it can be reasonably assumed that situations where the energy dissipation remains bounded is the most favorable for fatigue design. We present a direct method for determining if the energy dissipation in a SMA structure is bounded or not. That method relies only on elastic calculations, thus bypassing incremental nonlinear analysis. Moreover, only a partial knowledge of the loading (namely the extreme values) is needed. Some results related to Nitinol stents are presented