PSI - Issue 24

Alessandro Pirondi et al. / Procedia Structural Integrity 24 (2019) 455–469 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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The out-of-plane deflection (see Fig. 5c), expressed by Eqn. (9), is applied as a constraint for the optimization, ( , ) = 1 2 ( 2 + 2 + ) = − 2 2 = − , = − 2 2 = − , = −2 2 = − , (9) For a deflection constraint, the out-of-plane displacement between stable configurations I and II was used (see Figure 1b). The deflection between the two states, at = = /2 , can be expressed as: = I − II = 1 8 ( + + ) 2 − 1 8 ( + + ) 2 , (10) The curvatures , and can be expressed in terms of the first state (I) alone, as the two states have equal and opposite curvatures. Therefore, the equation can be rewritten in terms of and only, = 1 4 ( + ) 2 , (11) A minimum deflection requirement > ̅ is used to set a restriction on the allowable design space. The inequality was solved as a part of the numerical solution of the optimization problem. It is noted that, as a matter of fact, Eqn. (11) incorporates the bistability constraint, + > 0 . The constrained optimization formulation is therefore established as follows: { min Ө 1 ,Ө 2 , , (x) ∶ Ratio of bending stiffness in two given directions φ 1 φ 2 Subject to { Bistability constraint: + > 0 Deflection constraint: = ( 1 + 1 ) 2 4 (12) The optimization problem is solved using MATLAB’s sequential quadratic programming (SQP) method, fmincon. Furthermore, due to the presence of multiple global and local minima in the optimization domain, MATLAB’s MultiStart algorithm is used to generate potential initial points that are uniformly distributed within the domain in order to identify all minima. 3.4. Boundary conditions and material properties In this work, two (n = 2) ply groups are considered, each one comprising two plies (see Figure 1a). The SMA ply contains virtually four SMA wires of 0.1mm diameter and length equal to L. The thickness of the SMA ply is set to = 0.2 mm, while the thickness, t, and the length, L, of the laminate are design parameters alike the orientation angles of the two plies,  1 and  2 . The SMAC plate is considered to be fixed at its center when the temperature changes from cure (453K) to ambient (293K). The direction of low bending stiffness 1 is set alternatively to 45° or 75° and the direction of high bending stiffness 2 is set based on the orientation of SMA wires, that is 0°. The material properties used in this work are shown in Table 1-3.