IOP PUBLISHING SMART MATERIALS AND STRUCTURES Smart Mater. Struct. 21 (2012) 045014 (11pp) doi:10.1088/0964-1726/21/4/045014 A semi-analytical study on helical springs made of shape memory polymer M Baghani 1 , R Naghdabadi 1 ,2 and J Arghavani 1 ,3 1 Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran 2 Institute for Nano-Science and Technology, Sharif University of Technology, Tehran, Iran 3 Department of Mechanical Engineering, Golpayegan University of Technology, Golpayegan, Iran E-mail: naghdabd@sharif.edu Received 2 August 2011, in final form 30 January 2012 Published 21 March 2012 Online at stacks.iop.org/SMS/21/045014 Abstract In this paper, the responses of shape memory polymer (SMP) helical springs under axial force are studied both analytically and numerically. In the analytical solution, we first derive the response of a cylindrical tube under torsional loadings. This solution can be used for helical springs in which both the curvature and pitch effects are negligible. This is the case for helical springs with large ratios of the mean coil radius to the cross sectional radius (spring index) and also small pitch angles. Making use of this solution simplifies the analysis of the helical springs to that of the torsion of a straight bar with circular cross section. The 3D phenomenological constitutive model recently proposed for SMPs is also reduced to the 1D shear case. Thus, an analytical solution for the torsional response of SMP tubes in a full cycle of stress-free strain recovery is derived. In addition, the curvature effect is added to the formulation and the SMP helical spring is analyzed using the exact solution presented for torsion of curved SMP tubes. In this modified solution, the effect of the direct shear force is also considered. In the numerical analysis, the 3D constitutive equations are implemented in a finite element program and a full cycle of stress-free strain recovery of an SMP (extension or compression) helical spring is simulated. Analytical and numerical results are compared and it is shown that the analytical solution gives accurate stress distributions in the cross section of the helical SMP spring besides the global load–deflection response. Some case studies are presented to show the validity of the presented analytical method. (Some figures may appear in colour only in the online journal) 1. Introduction Since the first observation of the shape memory effect in some polymers, research on shape memory polymers (SMPs), known as a class of smart materials, has been an active field [1–3]. SMPs have been researched, developed, and utilized in a wide range of applications such as advanced technologies in aerospace, medical and oil exploration industries. Compared to other smart materials such as shape memory alloys, SMPs have the ability of large elastic deformation, low energy consumption for shape programming, potential biocompatibility, low cost, low density, biodegradability and excellent manufacturability. Moreover, they have a promising future for application in sensors, actuators and smart devices [4–8]. The energy absorbing springs used in seal and backup rings for the oil and gas industry were historically made from ferrous metals. Over time, metals not only corrode but also score the backup ring shaft as the seal wears away. Therefore, polymeric materials are a class of potential substitutes for metallic components. Among different polymeric materials, SMPs are a potential candidate for substituting metallic materials. SMPs are capable of memorizing a permanent shape and recovering their initial shape. This property is useful and desirable for the design of springs in a wide range of applications. The necessity of obtaining an accurate analytical and numerical solution for SMP devices, besides their new emerging applications, motivated the authors to seek analytical and numerical solutions for SMP springs. Although 1 0964-1726/12/045014+11$33.00 c  2012 IOP Publishing Ltd Printed in the UK & the USA