Nonlinear dynamics and chaos in shape memory alloy systems Marcelo A. Savi Universidade Federal do Rio de Janeiro, COPPE – Department of Mechanical Engineering, 21.941.972 Rio de Janeiro, RJ, Brazil article info Article history: Received 3 December 2013 Received in revised form 8 May 2014 Accepted 6 June 2014 Available online 21 June 2014 Keywords: Smart material systems Shape memory alloys Nonlinear dynamics Chaos Bifurcations Non-smooth systems abstract Smart material systems and structures have remarkable properties responsible for their application in different fields of human knowledge. Shape memory alloys, piezoelectric ceramics, magnetorheological fluids, and magnetostritive materials constitute the most important materials that belong to the smart materials category. Shape memory alloys (SMAs) are metallic alloys usually employed when large forces and displacements are required. Applications in aerospace structures, rotordynamics and several bioengineering devices are investigated nowadays. In terms of applied dynamics, SMAs are being used in order to exploit adaptive dissipation associated with hysteresis loop and the mechanical property changes due to phase transformations. This paper presents a general overview of nonlinear dynamics and chaos of smart material systems built with SMAs. Oscillators, vibration absorbers, impact systems and structural systems are of concern. Results show several possibilities where SMAs can be employed for dynamical applications. & 2014 Elsevier Ltd. All rights reserved. 1. Introduction Nature should be the essential inspiration for researchers and engineers that try to develop systems and structures. The main inspirational point is certainly the adaptive behavior that provides the self-regulation ability. Through the history, human technology is always related to different materials and it is possible to recognize ages defined by some material invention: stone and metal, for instance. Recently, smart materials should be identified as the stimulus of a new age. Basically, smart materials have a coupling between mechanical and non-mechanical fields that confers the material a special kind of behavior. In this regard, it is possible to imagine numerous applications due to the coupling of fields that usually are not connected. The smart material age tries to exploit the idea to construct systems and structures with adaptive behavior that have the ability to change properties due to environmental changes and repairing themselves when necessary. Among many possibilities, smart materials can be classified according to the different field couplings. Nowadays, the most used materials are the shape memory alloys, the piezoelectric materials, the magnetostrictive materials and the electro- and magneto-rheological fluids. These materials have the ability of changing their shape, stiffness, among other properties, through the imposition of temperature or stress, electrical or electro- magnetic fields. Smart materials are usually employed as sensors and actuators in smart structures. The choice of proper material for each application depends on many factors and two design drivers need to be highlighted [32]: the actuation energy density; and the actuation frequency. Shape memory alloys (SMAs) present a mechanical-temperature coupling in such a way that they have the ability to recover a shape previously defined, when subjected to an appropriate thermome- chanical loading process. SMA application is usually associated with high force–displacement and low frequency. The remarkable prop- erties of SMAs are related to phase transformations responsible for different thermomechanical behaviors of these alloys. Basically, two different phases are possible in SMAs: austenite and martensite. Austenitic phase is stable at high temperatures and stress-free state presenting a single variant. On the other hand, martensitic phase is stable at low temperature in a stress-free state, being related to numerous variants. Phase transformation may be induced either by stress or by temperature. SMA thermomechanical behavior is very complex being represented by different phenomena. Pseudoelasti- city, shape memory effect, two-way shape memory effect, transfor- mation induced plasticity are some examples of important aspects of the thermomechanical behavior of SMAs. The macroscopic behavior of SMAs can be expressed by stress– strain curves, Fig. 1. Pseudoelasticity happens at high tempera- tures, where the austenitic phase is stable for a stress-free state. Fig. 1a shows a typical stress–strain curve of the pseudoelastic behavior. A mechanical loading causes an elastic response until a critical stress value is reached, point A, when the martensitic transformation (austenite-detwinned martensite) arises, finish- ing at point B. For higher stress values, SMA presents a linear elastic response. During unloading process, the sample presents an elastic recovery (B-C). From point C to D one can note the reverse martensitic transformation (detwinned martensite-austenite). Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/nlm International Journal of Non-Linear Mechanics http://dx.doi.org/10.1016/j.ijnonlinmec.2014.06.001 0020-7462/& 2014 Elsevier Ltd. All rights reserved. E-mail address: savi@mecanica.ufrj.br International Journal of Non-Linear Mechanics 70 (2015) 2–19