Issue 29

A. Fortini et alii, Frattura ed Integrità Strutturale, 29 (2014) 74-84; DOI: 10.3221/IGF-ESIS.29.08 79 Figure 4 : Two-way recoverable shape change versus TWSME cycles. C ONSTITUTIVE MODELLING he thermomechanical behaviour of the SMA strip in bending is simulated by using the results obtained in [20], which are briefly reviewed in this section. In [20] a one-dimensional phenomenological model is adopted, based on external control variables, the stress σ and the temperature T, and on internal variables, which are the single- variant martensite, multi-variant martensite and austenite volume fractions. The phase production processes that are considered during bending and free shape recovery under heating are the following: during bending at low temperature, multi-variant martensite transforms into single-variant martensite; during shape recovery upon heating, both multi-variant and single-variant martensite volume fractions transform to austenite. Each phase production is detailed by kinetic equations describing the evolution of the phase fractions during transformation in terms of the current values of the stress and the temperature. The kinetic equations are assumed to be linear for simplicity and the reader is referred to [20] for further details. The stress σ is related to the strain ߳ via the following constitutive Eq. (4):     0        L s L s E if E otherwise         (4) Here ߳ ௅ >0 is the maximum strain achievable by the transformation of multi-variant into single-variant (detwinning), ܧ is the elastic modulus, assumed to take the same value, 28423 MPa, for all the three phases and ξ s is the single-variant martensite volume fraction. The behavior is assumed symmetric in traction and compression. The first bending at low temperature (T<M f ) of the NiTi strip is modeled as the uniform bending of a beam entirely made of multi-variant martensite. The beam is taken to be l =107 mm long, and it has the same rectangular cross-section of the NiTi strip, with thickness h = 0.8 mm and width b = 7 mm. Two opposite couples are applied to the ends of the beam, to bend it from the initial memorised curvature   = 23.63 * 10 -3 mm -1 to the final curvature reached after elastic springback  r = 0.00 m -1 , corresponding to a flat shape. Using the Euler-Bernoulli theory as done in [20], it can be shown that during bending single-variant forms initially at the outer and inner fibers of the cross-section and the transformation spreads inside the beam as the value of the applied couples is further increased. The residual stress and single-variant martensite distributions, calculated using the material data listed in Tab. 2, are piecewise linear and they are shown in Fig. 5. In the following,  will denote the distance from the neutral axis to the level at which the residual single-variant fraction  sr vanishes. For the free shape recovery under heating, a semi-analytical solution is proposed in [20] and here applied for the data of the alloy studied in this paper. The solution, valid under the assumption of a uniform temperature distribution throughout the cross-section, is based on the existence of transformation fronts, nucleating from the points where the residual stress vanishes and evolving with the temperature. Indeed, the kinetic relations for the transformation of multi and single-variant martensite into austenite considered in [20] impose that points at zero stress transform first. There are two points at which the residual stress vanishes: one is the origin, and the other is calculated at the distance z 0 (A s ) = 0.27 mm from the origin. The single-variant martensite is absent at the origin (cfr. Fig. 5), thus from the origin only a front nucleates, that one for -5 0 5 10 15 20 25 30 0 5 10 15 20 25 30 χ [mm -1 ]*10 -3 TWSME cycles χ hot χ cold Desired hot shape Desired cold shape T