PSI - Issue 24

Alessandro Pirondi et al. / Procedia Structural Integrity 24 (2019) 455–469 Author name / Structural Integrity Procedia 00 (2019) 000 – 000 7 respectively. Material parameters 11 , 22 , 12 , υ s 12 , α s 11 , α s 22 , denote therefore the orthotropic values of the in plane elastic moduli, Poisson's ratio and thermal expansion coefficients of the SMA/epoxy ply, while those with subscripts m and s denote the isotropic matrix and SMA wire values, respectively. The change in Young's modulus of SMA with phase transformation is reflected in Eqn. (3), hence the stiffness of the laminate depends on the martensite volume fraction and, in turn, on stress making the problem nonlinear also in the constitutive behaviour. However, for the number of wires that can be embedded in practice, as considered in this paper, the value of makes the contribution of SMA wires to the stiffness of the laminate very small. Based on this, it is possible to assume that the change in the due to phase transformation can be neglected and can be set to a value equal to that at the end of the transformation, simplifying the solution of the optimization problem. The constitutive equations of off-axis stress strain relationship for epoxy/SMA ply with arbitrary wire orientation is therefore: { } = [ ̅̅̅̅ 11 ̅̅̅̅ 12 ̅̅̅̅ 16 ̅̅̅̅ 12 ̅̅̅̅ 22 ̅̅̅̅ 26 ̅̅̅̅ 16 ̅̅̅̅ 26 ̅̅̅̅ 66 ] {{ } − { x 2 } ∆ } − { 2 Ө 2 Ө Ө Ө } . (5) where ̅̅̅̅ are the ply elastic constants and Ө is the orientation of wires with respect to the reference system. The appearance of the Martensite Volume Fraction (MVF), , and the Transformation Strain (TRNS), , in the stress strain relation for the epoxy/SMA ply in Equation (26) manifest the presence of the SMA wires on the resultant force developed inside the epoxy/SMA ply. Hence, the total forces per unit length of the SMAC can be expressed as: { } = { } − { } , (6) where stands for the resultant force in the epoxy/SMA ply which can be calculated by using Eqn. (5) into the the laminate stiffness terms in Eqn. (2). The linear superposition principle can be used to calculate the total internal force based on the assumption that the stiffness of the epoxy/SMAC ply is constant (see previous discussion). 3.3. Optimization of SMAC An optimization technique for the design of bistable laminates was presented by Betts (2012) where a laminate of arbitrary orthogonal layup having two stable shapes with equal and opposite curvatures was considered with the following design rules: 1. Even number of ply groups, where those below the laminate midplane are rotated 90° with respect to the corresponding ones above the midplane (see Figure 5a); 2. Square plate of edge length ; 3. All plies are of the same material. The formulation of the optimization problem maximizes the bending stiffness in a given direction of loading while at the same time the bending stiffness in the direction of snap to the second stable configuration is minimized (see Fig. 5b). The objective function that minimizes the ratio of bending stiffness in two chosen direction of the laminate is represented by Equation (7), where 1 is the direction of low bending stiffness, 2 is the direction of high bending stiffness and is the differential operator, 461