Issue 29

A Fortini et alii, Frattura ed Integrità Strutturale, 29 (2014) 74-84; DOI: 10.3221/IGF-ESIS.29.08 82 where f (T)=(A f -T)/(A f -A s ), and  sr (-h/2)=0.2 is the value taken by the single-variant martensite distribution at the point z=-h/2 at the end of the uniform bending at low temperature (  sr (-h/2)=0.2 for the parameters listed in Tab. 2, see Fig. 5). Thus, assuming that z 0  -h/2) as ܶ ⟶ A f and using (6), one obtains the following asymptotic behavior of ߯ሺܶሻ  near A f :   0 ( ) 2 ( / 2) ( )      f L sr f s A T T h h A A     (7) One could tentatively approximate the curvature evolution with a cubic curve having minimum at T=A s (and the minimum value is ߯ ௥ ) passing through the point (A f , ߯ ଴ ) and having tangent line at T=A f given by (7). For the material parameters of the alloy studied in this paper, the result turns out to be in acceptable agreement with the exact curvature evolution in almost all the range of temperatures (A s , A f ), as shown in Fig. 7. So far only the first free recovery upon heating has been considered, and the results have been obtained under the assumption that the beam recovers completely its original curvature    at T=A f . However, the experimental data described in the previous sections indicates that this does not occur, and that the curvature recovered at the end of heating decreases with the number of cycles. This could be attributed to the presence of residual martensite in the parent phase at macroscopic free stress state, due to accumulation of plastic dislocations. Experimental evidence of residual strain due to accumulated martensite has been reported in [23, 29]. To phenomenologically describe the accumulation of martensite, one could assume that  sr (- h /2) depends upon the number ݊ of cycles as follows: 0 (1 / ) 0 ( / 2; ) (1 )       n n sr h n e    (8) where ߦ ଴ = 0.2 is the value calculated for the first cycle, ݊ ଴ and ߦ ஶ  are two material parameters. Substituting (8) into (7) gives an equation for the curvature evolution near A f with the number of cycles. Fig. 9 shows the plot of the curvature at T= 101 °C as a function of ݊ obtained with the following values of the material parameters: ݊ ଴ =4.75 and ߦ ஶ  =  0.32  These values have been chosen so as to fit the experimental data at high temperature (points labeled with the circles in Fig. 9) as close as possible. Fig. 9 shows also the numerical results obtained using the model proposed in [25], which is adopted to reproduce the experimental data during the training and in particular, the evolution of the curvature at high temperature during the cycles. Figure 9 : Evolution of the recovered curvature with the number of cycles. Triangles: recovered curvatures measured at high temperature. Squares: recovered curvatures measured at low temperature after cooling. Continuous line: theoretical evolution based upon the phenomenological assumption (8). Dotted line: simulated evolution based on the model proposed in [25].