Davide Bernardini

Thermomechanical modelling, nonlinear dynamics and chaos in shape memory oscillators

A constitutive model for the restoring force in pseudo-elastic shape memory oscillators is proposed. The model is developed in a thermomechanical framework and allows one to predict the temperature variations that typically arise in shape memory materials under dynamical loading. A peculiar feature of the model is that all the constitutive equations follow from two basic ingredients, the free energy and the dissipation functions, through the restrictions imposed by the balance equations, instead of being directly postulated as in standard internal variable formulations. The model is then implemented and employed to systematically characterize the nonlinear dynamic response of the oscillator. It turns out that non-regular responses occur around the jumps between different branches of frequency - response curves. The features of the response and the modalities of transition to chaos are described mainly by means of bifurcation diagrams. The effect of the main model parameters (pseudo-elastic loop shape and thermal effects) on the dynamics of the system is also investigated.

Models for one-variant shape memory materials based on dissipation functions

Some simple models for the macroscopic behavior of shape memory materials whose microstructure can be described as a mixture of two phases are derived on the basis of a free energy and a dissipation function. Keeping a common expression for the free energy, each model is based on a different expression for the dissipation function. Temperature-induced as well as isothermal, adiabatic and convective stress-induced transformations are studied. Attention is paid to closed form solutions, comparison among the models and parameter identification.

Non-isothermal oscillations of pseudoelastic devices

Abstract The non-linear dynamic response of a pseudoelastic oscillator embedded in a convective environment is studied taking into account the temperature variations induced, during oscillations, by the latent heat of transformation and by the heat exchange with the surroundings. The asymptotic periodic response under harmonic excitation is characterized by frequency–response curves in terms of maximum displacement, maximum and mean temperature. The periodic thermomechanical response is computed by a multi-component harmonic balance method implemented within a continuation algorithm that enables to trace out multivalued frequency–response curves. The accuracy of the results is checked by comparison with the results of the numerical integration of the basic equations governing the dynamics of the system. The response is investigated for various excitation amplitude levels and in various material parameters ranges. The resulting picture of the mechanical response shows, in some cases, features similar to other hysteretic oscillators, while, in other cases, points out peculiar behaviors. It turns out that the temperature variations induced by the phase transformations influence the mechanical response and that the results obtained under the simplifying assumption of isothermal behavior can be rather different from those obtained in a fully thermomechanical setting.

On the macroscopic free energy functions for shape memory alloys

Abstract Most of the models for the macroscopic behaviour of shape memory alloys (SMA) rely upon the assumption of equal material properties of the phases while, on the contrary, experiments show significant differences. On the basis of the variational formulation of the problem governing the behaviour of a linear elastic heterogeneous material with prescribed eigenstrains, macroscopic free energies for SMA are defined taking into account the phase heterogeneity. The general structure and the dependence on the macroscopic state variables of such functions are discussed and formal expressions in terms of proper concentration tensors given. In the case of an underlying two-phase microstructure exact connections between the quantities that determine the free energies (macroscopic transformation strain, interaction energy, effective thermal expansion tensor) and the effective elastic compliance are derived. Estimates of the SMA macroscopic free energies based on Reuss, Voigt and Hashin-Shtrikman bounds for the effective elastic moduli are explicitly calculated and compared in the specific case of a NiTi alloy.

Identification of regular and chaotic isothermal trajectories of a shape memory oscillator using the 0-1 test

Shape memory oscillators are thermomechanical hysteretic systems that, in a wide range of model parameters, can exhibit complex non-periodic non-linear dynamic responses under the excitation of a periodic force. In this study, the statistical 0–1 test based on the asymptotic properties of a Brownian motion chain is applied to periodic and non-periodic isothermal trajectories to examine the type of motion. The analysis is based on the computation of the control parameter K that approaches asymptotically 0 or 1 for regular and chaotic motions, respectively. The presented approach is independent of the integration procedure, being based on the characteristic sampling distance between the points of the analysed time series. The numerical results show that the test is able to unambiguously distinguish periodic from chaotic trajectories.

CHAOS ROBUSTNESS AND STRENGTH IN THERMOMECHANICAL SHAPE MEMORY OSCILLATORS PART II: NUMERICAL AND THEORETICAL EVALUATION

In this two-part paper the problem of evaluating robustness and strength of chaos in thermomechanically-based Shape Memory Oscillators (SMOs) is addressed. In Part I, several tools for the theoretical prediction of the main features of the pseudoelastic loops of Shape Memory Devices (SMDs) have been proposed. In this Part II, the Method of Wandering Trajectories (MWT), that has already been validated as a systematic, yet affordable, numerical tool for the evaluation of the chaotic response of SMOs, is enhanced by complementing it with a quantitative indicator of chaoticity: the maximum value of the displacement normalized separation over a fixed time interval. The method is used to compute numerical 3D behavior charts for several model parameters sets, properly selected in order to highlight the influence of various model parameters on the nonlinear dynamical response of SMOs. From the numerical analyses it turns out that two main aspects of the SMD behavior do influence the robustness and strength of chaos, namely the hardening of the pseudoelastic plateaus and the area of the hysteresis loop, both of them being meaningfully affected by the model mechanical and thermal parameters. The numerical results are also interpreted by means of the theoretical indicators discussed in Part I, which provide a reliable framework for the prediction of the main features of dynamic response before actual computation of the trajectories.

CHAOS ROBUSTNESS AND STRENGTH IN THERMOMECHANICAL SHAPE MEMORY OSCILLATORS PART I: A PREDICTIVE THEORETICAL FRAMEWORK FOR THE PSEUDOELASTIC BEHAVIOR

In this two-part paper the problem of evaluating robustness and strength of chaos in thermomechanically-based Shape Memory Oscillators (SMO) is addressed. In the first part, a theoretical analysis of the main features of the pseudoelastic loops exhibited by the underlying Shape Memory Devices (SMD) is accomplished with the aim to establish a predictive framework for accompanying numerical investigations. The analysis is based on the evaluation of suitable synthetic indicators of the SMD behavior that can be computed from the model parameters before the computation of SMO actual trajectories, and provide information about the hysteresis loops and their dependence on temperature variations. By means of such indicators, a detailed analysis of the influence of thermomechanical coupling on the rate-dependent mechanical response is presented. It is shown that a careful interpretation of the synthetic indicators permits to obtain a reasonable estimation of the influence of various model parameters on the hysteresis loop area and slopes of the pseudoelastic plateaus, that are the main global aspects influencing the occurrence of chaotic responses. In the second part, the theoretical predictions based on the synthetic indicators will be exploited to interpret the results of a systematic numerical investigation based on an enhanced version of the Method of Wandering Trajectories.

Models of hysteresis in the framework of thermomechanics with internal variables

Abstract The Ziegler–Green–Naghdi approach to the thermomechanics with internal variables is used to model the hysteretic behavior of a one-dimensional system with a single internal variable. In this framework, models of hysteresis arise from proper assumptions on the dissipation function. Two examples from elastoplasticity and pseudoelastic behavior of shape memory alloys are presented.

Characterizing the nonlinear behavior of a pseudoelastic oscillator via the wavelet transform

The nonlinear dynamics of a shape memory oscillator (SMO) subjected to an ideal or nonideal excitation is studied. The restoring force of the oscillator is provided by a shape memory device (SMD), described by a thermomechanical model capable of reproducing the hysteretic behavior via the evolution of a suitable internal variable. Due to nonlinearities in the model, the SMO can exhibit periodic or non-periodic behaviors. The effects of the external sources on the response of SMO are studied through the scalogram analysis of continuous wavelet transform by using a new measure, called the Scale index (Benitez R, Bolos VJ and Ramirez ME. A wavelet-based tool for studying non-periodicity. Comput Math Appl 2010; 60: 634–641). Numerical results show that the Scale index can successfully detect the behavior of the system when the signal is periodic or nonperiodic, and distinguish between them in a way consistent with the indications provided by the alternative 0-1 test.

A Multifield Theory for the Modeling of the Macroscopic Behavior of Shape Memory Materials

The macroscopic behavior of shape memory materials is modeled within the framework of multifield theories. Two scalar fields and a second-order tensor field are used as descriptors of the relevant microstructural phenomena. In this way it is possible to allow for pseudoelasticity and shape memory effect, as well as low and high temperature reorientation of Martensitic variants. The general aspects of the theory are discussed paying special attention to the treatment of balance equations and to the exploitation of the constitutive structure, which is characterized by the prescription of a response function for the entropy production. An example of an explicit model is also given.