J Elast (2010) 99: 39–73 DOI 10.1007/s10659-009-9229-x Equilibrium States of Mechanically Loaded Saturated and Unsaturated Polymer Gels Hua Deng · Thomas J. Pence Received: 9 July 2009 / Published online: 3 December 2009 © Springer Science+Business Media B.V. 2009 Abstract By a gel we mean a system of crosslinked polymer chains mixed together with a low molecular weight liquid. The polymer and liquid components mix in definite propor- tions as determined primarily by entropic and enthalpic effects. Swollen gels in equilibrium with a surrounding fluid bath in the absence of mechanical load are often described by a generalized Flory-Huggins equation. In this paper we consider the connection between such a treatment and the broader hyperelastic theory that treats the effect of mechanical loading in deforming the gel. A change in the mechanical loading will generally alter the proportion of liquid in the mixture, leading to either fluid loss (swelling reduction) or fluid gain (swelling increase). In such a case the gel reestablishes equilibrium only when the relative motion of the liquid through the polymer has ceased and processes have come to rest. Such processes are inherently dissipative. Our objective is to study how such reestablished equilibria de- pend upon mechanical load. For quasi-static loadings that give fluid gain, we then consider a situation in which the amount of available fluid is limited. In this case, increasing quasi- static loading may reach a point at which no additional fluid is available for uptake into the gel system. The associated equilibrium then transitions from a state of liquid saturation to a state in which the gel is no longer saturated. We first consider this quasi-static transition in the context of homogeneous deformation where an appropriate hyperelastic analysis shows that the equilibrium mechanical response is inherently stiffer after loss of saturation. We then consider such a transition in the context of inhomogeneous deformation by studying the boundary value problem of an everted tube subject to an axial load. Loss of saturation again leads to an inherently stiffer quasi-static response. Keywords Hydrogels · Free swelling · Flory-Huggins · Hyperelasticity · Eversion · Liquid saturated gels Mathematics Subject Classification (2000) 74A20 · 74B20 · 74F10 H. Deng · T.J. Pence ( ) Mechanical Engineering, Michigan State University, East Lansing, MI 48824-1226, USA e-mail: pence@egr.msu.edu H. Deng e-mail: denghua@egr.msu.edu