Analytical study on torsion of shape-memory-polymer prismatic bars with rectangular cross-sections M. Baghani ⇑ School of Mechanical Engineering, College of Engineering, University of Tehran, P.O. Box 11155-4563, Tehran, Iran article info Article history: Received 29 May 2013 Received in revised form 18 November 2013 Accepted 29 November 2013 Keywords: Shape memory polymers Rectangular bars Torsion Analytical solution Finite element abstract In this paper, the response of shape memory polymer (SMP) bars with rectangular cross- sections under torsional loadings is analytically studied. To this end, we first reduce the recently proposed small-strain 3D phenomenological constitutive model for SMPs to the shear case. Then, an analytical solution for torsional response of SMP rectangular bars in a full cycle of stress-free strain recovery is derived. We also implement the 3D constitutive equations in a finite element program and simulate a full cycle of stress-free strain recovery of a rectangular SMP bar. Analytical and numerical results are then compared showing that the analytical solution gives, besides the global load–deflection response, accurate stress distributions in the cross-section of the rectangular SMP bar. Some case studies are also presented to show the validity of the presented analytical method. Results are compared with the experimental data recently reported in the literature which showing an agree- ment between the predicted results and experiments. The analytical solution can also be used for analysis of helical springs in which both the curvature and pitch effects are neg- ligible. This is the case for helical springs with large ratios of mean coil radius to the cross- sectional equivalent radius (spring index) and also small pitch angles. Using this solution simplifies the analysis of the helical springs to that of the torsion of a straight bar with rect- angular cross-section. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction Since the first observation of the shape memory effect in some polymers, research on SMPs has been an active field. Re- cently, manufacturing of SMP devices, has considerably increased, thanks to their unique ability in recovering large stored strains (Lendlein et al., 2009; Lendlein & Langer, 2002; Liang, Rogers, & Malafeew, 1997; Barot, Rao, & Rajagopal, 2008). SMPs have been researched, developed, and utilized in a wide range of applications such as advanced technologies in the aerospace, medical, microelectromechanical systems (MEMS) and oil exploration industries (Díaz Lantada et al., 2010; Monkman, 2000; Ghosh, Reddy, & Srinivasa, 2012). Compared to other smart materials such as shape memory alloys, SMPs have ability of large elastic deformation, low energy consumption for shape programming, potential biocompatibility, low cost, low density, biodegradability and excellent manufacturability (Baghani, Naghdabadi, & Arghavani, 2012b, 2012a; Bel- oshenko, Varyukhin, & Voznyak, 2005; Cheng & Li, 2008; Jarali, Raja, & Upadhya, 2010; Saı  , 2010; Ghosh & Srinivasa, 2011). Despite of all advantages mentioned in the above, the much lower stiffness of un-reinforced SMPs prevents them from practical applications in cases where a large recovery stress is required (e.g., as actuators). To overcome such disadvantage, different reinforced-SMP composites have been developed and utilized (Li & Wang, 2011; Xu & Li, 2010). 0020-7225/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijengsci.2013.11.016 * Tel.: +98 21 6111 4020. E-mail address: baghani@ut.ac.ir International Journal of Engineering Science 76 (2014) 1–11 Contents lists available at ScienceDirect International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci