Bending models for superelastic shape memory alloy laminated composite cantilever beams with elastic core layer

Free vibration characteristics of sectioned unidirectional/bidirectional functionally graded material cantilever beams based on finite element analysis

“…The equations of the shape functions are obtained by using the following assumed forms of the slope and displacements [29] :…”

Modeling the behavior of bilayer shape memory alloy/functionally graded material beams considering asymmetric shape memory alloy response

“…The formulation of the volume fraction z as function of the coordinate y , measured with respect to the neutral axis within the transformed region, is assumed to obey the following linear relation (Viet et al, 2018b)…”

“…The stress thresholds for the beginning and the end of reverse transformation at a point where a volume fraction z = z * ei ± was achieved during loading are calculated by letting z equal z * ei ± and 0, respectively (Viet et al, 2018b), in equation (1). Then the following relations are obtained…”

“…The reverse phase transformation first occurs at transverse line 4. Afterwards, in order to identify the order of reverse phase transformation start for lines 1 , 2 , 3 , 5 , and 6, the method similar to that described in literature done by Viet et al (2018b) is adopted. That is, the threshold values for curvatures are first determined; subsequently, the corresponding moment and tip loads are computed via an appropriate moment equation in the elastic unloading stage.…”

“…In the case considered here, the order from first to last is given as 4 → 2 → 5 → 3 → 6 → 1 . For the sake of brevity, we represent beam’s phase structure at the sub-stage level in which only line 4 having reverse phase transformation, subsequent structures are similar to that described in work done by Viet et al (2018b).…”