A semi-analytical solution for bending response of SMA composite beams considering SMA asymmetric behavior

Abstract

This paper develops a novel and robust semi-analytical model to investigate the bending response of an SMA composite beam during a loading-unloading cycle based on an improvement of Brinsons model for an Euler–Bernoulli beam. The improvement of Brinsons model is taken into account to consider the asymmetric behavior of SMAs in tension and compression. In the proposed semi-analytical solution first a linear distribution of strain along the cross section is assumed and then employing Newton–Raphson method, an iterative numerical procedure is implemented to satisfy the classical equilibrium equations. The proposed method in this paper is applicable to any Euler–Bernoulli beam with any material and layup as well as any cross sectional geometry (with a plane of symmetry). To validate the results of the proposed model a homogeneous SMA beam is studied and the results are compared to the results of a 2D finite element (FE) solution. The results of the bending response of the beam are reported for both the symmetric and asymmetric behavior and also for both FE and semi-analytical solution. The results for both approaches are generally in good agreement with each other, and a maximum error of 3.6% is observed between FE and semi-analytical solution. Further, a Bi-layer composite beam consisting of two different SMAs is studied. Accordingly, Load-deflection, moment-curvature and also variation of the location of the neutral axis with respect to the center line of the cross-section is investigated. In one of the case studies there is 15% error in the prediction of the symmetric model with respect to the asymmetric model which is a more actual prediction, proves the importance of employing the asymmetric model. Moreover, for the same case the maximum deviation of the neutral axis with respect to the centroid is 17% for the asymmetric model and 24% for the symmetric model. Furthermore, hysteresis inner loops for the Bi-layer composite beam is investigated. The proposed formulation is appropriate for design or optimization of smart structures consisting of multilayer SMA components under bending where a large number of simulations are necessary.