A Free Energy Model for the Inner Loop Behavior of Pseudoelastic Shape Memory Alloys
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A Free Energy Model for the Inner Loop Behavior of Pseudoelastic Shape Memory Alloys Olaf Heintze and Stefan Seelecke Dept. Mech. & Aero. Eng. Campus Box 7910, 3211 Broughton Hall, North Carolina State Univ., Raleigh, NC 27695-7910 ABSTRACT The paper presents a free energy model for the pseudoelastic behavior of shape memory alloys. It is based on a stochastic homogenization process, which uses distributions in energy barriers and internal stresses to represent effects typically encountered in polycrystalline materials. This concept leads to a realistic desription of the rate-dependent inner loop behavior, but is characterized by rather long computation times. This is prohibitive in regard to a potential implementation into other numerical codes, such as finite element or optimal control programs or a Matlab/Simulink environment. For this purpose a parameterization method is introduced, which is derived from the concept of a representative single crystal. The approach preserves the desirable properties of the original formulation, at the same time reducing the numerical effort significantly. Finally, we show that the method can reproduce the experimentally observed behavior accurately over a large range of strain rates including the minor loop behavior. INTRODUCTION Early applications of SMA were pipe couplings and circuit breakers acting autonomously to changes in the environmental temperature. The range and type of applications has changed to more complex and sophisticated devices and actuation mechanisms in micro- and macroscale extending from medical to aerospace engineering. If one wants to make an efficient use of these novel applications and successfully improve their design, one needs to be able to model the complex, non-linear hysteretic and thermomechanically-coupled material behavior of shape memory alloys as it is determined for the device in use. Although the modelling of perfect single crystal SMA is well understood, it is based on the assumption of a perfectly homogeneous material, which is not typical and realistic. Therefore, we need focus on an extension to a realistic polycrystal material behavior. One concept was presented, e.g., by Lu and Weng in [14], [15]. A similar approach has been taken by Huang and Brinson [10], who first illustrate the procedure for a fictitious two-variant material and later extend it to a typical 24 variant case. Bo and Lagoudas [3, 4, 5, 6] also present a paper on a polycrystalline model, consisting of four parts. Lagoudas and Entchev continued this work in [13] and further applied this model to porous SMAs in [7]. This type of modelling is based on a more detailed study of the underlying microstructure in contrast to so-called Preisach models often used by hysteresis researchers with a mathematical background. Khan et al [12] present a Preisach model polycrystalline pseudoelastic response that can model minor hysteresis loops, but does not consider the strain rate effect. Massad and Smith describe in [16] and [17] the more complex grain structure of polycryst
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