William Toh School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, 639798, Singapore e-mail: wtoh1@e.ntu.edu.sg Zhiwei Ding Institute of High Performance Computing, 1 Fusionopolis Way, #16-16 Connexis, 138632, Singapore e-mail: dingzw@ihpc.a-star.edu.sg Teng Yong Ng Mem. ASME School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, 639798, Singapore e-mail: mtyng@ntu.edu.sg Zishun Liu 1 Mem. ASME International Center for Applied Mechanics, State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, No. 28, West Xianning Road, Xi’an Shaanxi 710049, China e-mail: zishunliu@mail.xjtu.edu.cn Wrinkling of a Polymeric Gel During Transient Swelling When exposed to an external solvent, a dry polymeric network imbibes the solvent and undergoes large deformation. The resulting aggregate is known as a hydrogel. This swel- ling process is diffusion driven and thus results in differential swelling during transient swelling. When subjected to external geometrical constraints, such as being rigidly fixed or attachment to a compliant substrate, wrinkles have been shown to appear due to mechanical instabilities. In the case of free swelling, there are no external constraints to induce the instabilities accounting for wrinkling patterns. However, during the transient swelling process, the swelling differential between the gel on the exterior and the interior causes compressive stresses and gives rise to mechanical instabilities. It is also observed that the time dependence of the swelling profile causes the wrinkles to evolve with time. In this work, we investigate this interesting phenomenon of transient wrinkle mode evolu- tion using the finite element and state-space methods. From our simulations and predic- tion, we find that there is an inverse relation between critical wave number and time, which has earlier been observed in experiments. [DOI: 10.1115/1.4030327] Keywords: surface wrinkling, polymeric gel, finite element method, state-space method 1 Introduction When a dry polymer network comes into contact with a solvent, it imbibes the solvent and swells, forming an aggregate known as a hydrogel. Hydrogels may swell to many times their initial vol- umes, with reports showing a volume change of up to 1000 times the dry volume. Due to this complex large deformation process, swelling induced instability is a commonly observed phenomenon. The instability of gel deformation causes wrinkling or buckling, which has traditionally been deemed as an undesirable trait in structures. However, there has been significant recent interest in harnessing the instabilities for potential applications, such as gear formation, soft electronics, actuators, and tunable adhesion [1]. The buckling of gels can be broadly classified into two types: bulk buckling and localized surface buckling [2]. Many studies have been carried out on the instability mechanisms that a hydro- gel experiences during the swelling process. Tanaka et al. [3] reported that even in the absence of external constraints, a gel may still experience wrinkling during the transient process of swelling due to the differential swelling that occurs between the surface of the gel and the inner layers. The wrinkles that appear are reported to grow with time as solvent molecules diffuse within the gel [4,5]. Although abundant literature on the analysis of swel- ling induced instabilities of gels is available, we note that most of these works involve static analyses, such as of bilayer structures [6], thin gel structures [7–9], functionally graded materials [10], and creasing [11]. Relatively few works have focused on the analysis of surface instability that a hydrogel experiences during the transient swel- ling process. Finite element models for simulating the transient swelling kinetics of hydrogels have been developed [12–15], but there is minimal utilization of the models for the investigation of the surface wrinkling process. In this paper, we investigate the evolution of wrinkle patterns that develop over the course of tran- sient swelling of the gel numerically by a recently developed finite element subroutine for the swelling kinetics of a polymeric gel [13], and semi-analytically by linear perturbation stability analysis solved with the state-space method [16]. 2 Theory In this section, we describe briefly the inhomogenous large deformation kinetic theory of polymeric gels and its implementa- tion into commercial finite element software, ABAQUS, through user-defined subroutines. The kinetic theory of a swelling poly- meric gel couples the equilibrium theory of a gel swelling under external loads with diffusion kinetics of a solvent within a solid. 2.1 Inhomogeneous Large Deformation of a Polymeric Gel. The state of deformation of a gel is characterized by the deformation gradient F iK defined as the partial derivative of the current state x i with respect to the reference state X K at any time t, F iK X; t ð Þ¼ @x i X; t ð Þ @X K (1) In equilibrium, the external mechanical and chemical loads are balanced by the change in free energy. Writing in a conservative form, this energy conservation is expressed as [17–20] 1 Corresponding author. Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received January 22, 2015; final manuscript received April 2, 2015; published online April 30, 2015. Editor: Yonggang Huang. Journal of Applied Mechanics JUNE 2015, Vol. 82 / 061004-1 Copyright V C 2015 by ASME Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 10/08/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use