Some numerical simulations of pseudoelastic hysteresis in shape memory alloys

Recently, Fedelich and Zanzotto have developed a model for the nonisothermal pseudoelastic behaviour of a shape memory material and have conducted some numerical simulation experiments. We present a different method for the numerical solution and discuss it in comparison with their results.     

A Preisach type model for temperature driven hysteresis memory erasure in shape memory materials

We establish the well-posedness and thermodynamic consistency of a variational inequality modeling temperature-induced memory erasure in shape memory materials. It is shown that the input–output operator is continuous with respect to uniform convergence.     

Coupled thermomechanical dynamics of phase transitions in shape memory alloys and related hysteresis phenomena

In this paper the nonlinear dynamics of shape memory alloy phase transformations is studied with thermomechanical models based on coupled systems of partial differential equations by using computer algebra tools. The reduction procedures of the original model to a system of differential-algebraic equations and its solution are based on the general methodology developed by the authors for the analysis of phase transformations in shape memory materials with low dimensional approximations derived from center manifold theory. Results of computational experiments revealing the martensitic-austenitic phase transition mechanism in a shape-memory-alloy rod are presented. Several groups of computational experiments are reported. They include results on stress- and temperature-induced phase transformations as well as the analysis of the hysteresis phenomenon. All computational experiments are presented for Cu-based structures.     

On non-monotonic rate dependence of stress hysteresis of superelastic shape memory alloy bars

This paper presents a simple thermo-mechanical model to explain and quantify the observed strain-rate dependence of the stress hysteresis of shape memory alloys (SMAs) bars/strips during stress-induced forward/reverse phase transition with latent heat release/absorption. By solving the convective heat transfer equation and employing the temperature dependence of the SMA’s transformation stresses, we are able to prove that the stress hysteresis depends non-monotonically on the applied strain rate with a peak appearing at an intermediate strain rate. We further showed that such a non-monotonic rate dependence is governed by the competition of phase-transition time (or latent-heat release/absorption time) and the time of heat exchange with the environment, and that the hysteresis peak is achieved when the two time scales become comparable. A bell-shaped scaling law of the rate dependence is derived, agreeing quantitatively well with the results of experiments.     

Existence and stability of limit cycles in a delayed dry-friction oscillator

Periodic, chaotic, chattering, and bifurcation behavior are fundamental consequences of the nonsmooth nature of systems with dry friction. This work is concerned with the analysis of a single degree of freedom system which is additionally damped by a delayed dry friction device. We get a complete set of closed-form expressions to describe the dynamics of the delay-induced phenomena exhibited by the system. The conditions to determine the existence and stability of limit cycles are clearly defined. This analysis is addressed in the context of both classic stability theory for nonlinear systems and the qualitative theory of Piecewise Smooth Dynamical Systems. Through exhaustive numerical simulations the effectiveness of the set of closed-form expressions is confirmed. Excellent agreement was found between the numerical and analytical results.     

Ferroelasticity of materials with a mechanical shape memory

It is extremely difficult to take into account the variety of aspects of the behavior of solids when they are deformed. Hence, when designing constructions, simplifying models are introduced which take into account only the most important properties of the materials in each specific case (creep, aftereffect, plasticity, etc.), and the corresponding phenomenological theories are employed [1], In this paper we attempt to construct the fundamentals of a theory which describes the phenomenon of ferroelasticity due to the behavior of thermoelastic martensite, first discovered in [2], The apparatus of this theory will be necessary when designing self-recovering constructions which can be manufactured from materials with a mechanical shape memory.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 122–126, May–June, 1979.     

Thermomechanical modeling of nonlinear internal hysteresis due to incomplete phase transformation in pseudoelastic shape memory alloys

This paper presents a thermomechanical model for pseudoelastic shape memory alloys (SMAs) accounting for internal hysteresis effect due to incomplete phase transformation. The model is developed within the finite-strain framework, wherein the deformation gradient is multiplicatively decomposed into thermal dilation, rigid body rotation, elastic and transformation parts. Helmholtz free energy density comprises three components: the reversible thermodynamic process , the irreversible thermodynamic process and the physical constraints of both. In order to capture the multiple internal hysteresis loops in SMA, two internal variables representing the transition points of the forward and reverse phase transformation, \(\phi _s^f\) and \(\phi _s^r\), are introduced to describe the incomplete phase transformation process. Evolution equations of the internal variables are derived and linked to the phase transformation. Numerical implementation of the model features an Euler discretization and a cutting-plane algorithm. After validation of the model against the experimental data, numerical examples are presented, involving a SMA-based vibration system and a crack SMA specimen subjected to partial loading–unloading case. Simulation results well demonstrate the internal hysteresis and free vibration behavior of SMA.

     

Multistability in a three-dimensional oscillator: tori,resonant cycles and chaos

The emergence of multistability in a simple three-dimensional autonomous oscillator is investigated using numerical simulations, calculations of Lyapunov exponents and bifurcation analysis over a broad area of two-dimensional plane of control parameters. Using Neimark–Sacker bifurcation of 1:1 limit cycle as the starting regime, many parameter islands with the coexisting attractors were detected in the phase diagram, including the coexistence of torus, resonant limit cycles and chaos; and transitions between the regimes were considered in detail. The overlapping between resonant limit cycles of different winding numbers, torus and chaos forms the multistability.     

Transitions in a noisy birhythmic vibro-impact oscillator with improved memory damping regime

Nonlinear Dynamics - We study the critical transition problem for a stochastically forced birhythmic vibro-impact (BVI) model with improved memory damping regime (IMDR) effect. This paper then...     

Chaos control of a nonlinear oscillator with shape memory alloy using an optimal linear control: Part II: Nonideal energy source

In Part II of this two-part work, we apply an optimal linear control technique to suppress chaotic behavior in a SMA oscillator, driven by a DC motor with limited power supply (Nonideal sources). Nonideal sources of vibrations of structures are those whose characteristics are coupled to the motion of the structure. Thus, in this case, an additional equation of motion is written, related to the motor rotation. Special attention is focused into the resonance region, when the nonideal excitation frequency is near the natural frequency of the SMA oscillator. Numerical results are presented, which show the efficiency of the method in resonance region. The ideal case, when we do not consider the interaction between an energy source and structure was considered in Part I of this paper.     

Chaos in a shape memory two-bar truss

The study of the structural response of two-bar trusses may be very helpful to understand some of the main stability characteristics of framed structures, as well as of flat arches and of many other physical phenomena associated with bifurcation buckling. This article is concerned with the dynamic response of a shape memory two-bar truss, which is an interesting example of a structural system that exhibits both kinematic and constitutive non-linearities. A polynomial constitutive model is assumed to describe the behavior of the shape memory bars. Free and forced responses are investigated. Numerical simulations show that the system can easily reach a chaotic response.     

Chaos control of a nonlinear oscillator with shape memory alloy using an optimal linear control: Part I: Ideal energy source

In this paper, an optimal linear control is applied to control a chaotic oscillator with shape memory alloy (SMA). Asymptotic stability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function, which can clearly be seen to be the solution of the Hamilton–Jacobi–Bellman equation, thus guaranteeing both stability and optimality. This work is presented in two parts. Part I considers the so-called ideal problem. In the ideal problem, the excitation source is assumed to be an ideal harmonic excitation.     

Torsional vibration of a shape memory wire

The paper studies the rotational motion of a rigid mass suspended by a shape memory wire. The wire represents a torsional spring, which causes the mass to oscillate. At the same time, however, the wire induces damping through the hysteretic phase transitions that it undergoes during the oscillation. Due to the strong temperature dependence of the wire's thermomechanical properties, a complex behavior can be observed. The oscillation is shown to differ substantially depending on whether the wire is quasiplastic (low temperature) or pseudoelastic (high temperature). This provides a mechanism for an active control of the system. By an appropriate heating strategy, it is possible to compensate for the damping, and the oscillation can thus be stabilized. Received May 7, 1997     

Micromechanical prediction of the two-way shape memory effect in shape memory alloy composites

The two-way shape memory effect in monolithic shape memory alloys has been widely investigated both theoretically and experimentally. In the present study, this effect is analyzed for shape memory alloy composites by employing a micromechanical model. To this end, the responses of polymeric matrix and metal matrix unidirectional composites with embedded shape memory alloy fibers are determined. For the polymeric matrix composite, the effect of axial, transverse and shear loadings as well as the fiber volume fraction on the resulting two-way shape memory behavior are studied. The local distributions of stresses among the shape memory alloy fiber and epoxy matrix in the low- and high-temperature shapes of the composite are also investigated. Two training procedures that generate the two-way shape memory effect in the metal matrix composite are offered. The present analysis shows that the two-way shape memory effect in the chosen type of metal matrix composite is not as useful as in the polymeric matrix one. Finally, for a polymeric matrix composite that is subjected to a transverse normal loading, the effect of imperfect bonding between the shape memory alloy fibers and the neighboring matrix is investigated.     

Determination of limit cycles for a modified van der Pol oscillator

The determination of amplitude and period of limit cycles is a crucial question in non-linear mechanics. In this paper, a van der Pol oscillator containing a periodic potential is considered. Amplitude and period of limit cycles are calculated by He’s variational method and Krylov–Bogoliubov–Mitropolsky (KBM) method.