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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3. Simulation of post-curing SMAC plate shape using a thermodynamically consistent behaviour of SMA wires and a numerical procedure to identify optimal design 3.1. Constitutive Model of SMA Several SMA constitutive models have been proposed and developed over the past few decades to describe the SMA phase transformation phenomenon. A constitutive model for SMA is available as a user-defined material model (UMAT) for the ABAQUS™ 6.13 software in Lagoudas (2003). This particular constitutive model is rigorously developed to ensure that the key constitutive relations and evolution equations are thermodynamically consistent. A detailed discussion of the model derivation process can be found in Qidwai and Lagoudas (2000). 3.2. Geometrically nonlinear lamination theory modelling of the SMAC plate Equations relating the deformation of the SMAC plate to external loads and constituents behavior were developed within the lamination theory, differently from the previous section where the model was solved only by finite element analysis. Since displacements may be large in the snap-through process, the model to predict the cured shape is based on a nonlinear extension of the classical laminated plate theory with approximate midplane strain functions and nonzero in-plane shear strain. The resulting constitutive equation of the laminate is, in matrix notation: = + , = + (2) where N, M are in plane forces and moment, respectively, Tot and k membrane strain and curvatures, and A, B, D the extensional, bending-extension coupling and bending stiffness terms. The additional SMA layer is accommodated at the mid-plane of the composite host plate, assuming that epoxy resin fills in the space within wires such that even and an equal number of composite plies are placed on the top and bottom surface of epoxy/SMA layer as shown in Figure 1a. This approach requires a volume fraction method to combine the properties of the SMA and epoxy. Accordingly, the thermo-elastic properties of an epoxy/SMA layer are written as

[ 11 = + 22 = /( + ) 12 = + 12 = /( + ) 11 = + 22 = /( + )

(3)

21 = 12 22 11 ] ,

where the “m” and “s” subscripts stand for the composite matrix and SMA fibers, respectively , and the value of is = 2 (4) while n, and stands for number of SMA wires, the diameter of SMA wire and epoxy/SMA ply thickness