Helmholtz free energy stretching

Consider a film of liquid on a wire balance as shown in Fig. 1. A force is applied to a frictionless, movable wire, stretching the film, which adheres to the wire. The process occurs at constant temperature and volume of the liquid. The reversible work done as the movable wire moves a distance dx (at constant T and V) is equal to the increase in the Helmholtz free energy of the system ... 

Here we begin with a sample of rabber having initial dimensions l, I2, I3. We deform it by an amount A/, A/2, A/3 and define the stretch (ratio) in each direction as A, = (/, -I- A/,)//, = ///,. The purpose of Finite Elasticity Theory has been to relate the deformations of the material to the stresses needed to obtain the deformation. This is done through the strain energy density function, which we will describe using the Valanis-Landel formalism as IT(A, A2, A3). Importantly, as we will see later, this is the mechanical contribution to the Helmholtz free energy. Vala-nis and Landel assumed [60] that the strain energy density function is a separable function of the stretches A, ... 

From the thermodynamic law (1.12), the Helmholtz free energy at a constant temperature is given by the work dR done for stretching the end vector from 0 to R. By the relation (1.11), the partition function is given by... 

---