1. A cubic elastomer is compressed by a steel piston in a deformable mold placed on a steel table, as shown in the figure below. The mold is initially cubic, tightly fitting with the elastomer block. The mold will expand along lateral directions under the inner pressure, P. It can be assumed that the relationship between the size of the mold and the pressure is linear: a = a₀ + β.P, where a₀ is the initial mold size and β is a system constant. Assume that the lateral surfaces remain planar during deformation.

Derive, in the framework of rubber elasticity, the relationship between the piston pressure (P) and the piston displacement, d. Clearly state all your assumptions.

2. Some shape-memory polymers (SMP) have relatively low glass transition temperatures, T_g. As glass transition occurs, the volume/shape of the SMP changes significantly. Typically, a shape-memory experiment consists of the following steps:

(1) as T > T_g, the SMP has a specified shape ("A"); (2) the temperature is lowered to below T_g; (3) the SMP is deformed to another shape ("B"); (4) the temperature is increased back to above T_g; it can be observed that the shape of SMP changes from "B" back to "A". The polymer under consideration is a thermoplastic.

(a) Do an independent literature research; describe in detail why the thermoplastic polymer has such a shape memory characteristic.

(b) If after the 4th step, the temperature is changed to below T_g again, will the shape of SMP change from "A" to "B" again? Why?

3. Two amorphous linear polymers have high number densities of pendant groups. Assume that the pendant groups of polymer "A" are C₄ groups (i.e. each pendant group has 4 C-C bonds); the pendant groups of polymer "B" are C16 groups (i.e. each pendant group has 16 C-C bonds). A C₄ group can be described as -CH₂-CH₂-CH₂- CH₃; a C₁₆ group is of a similar structure but four times longer. Everything else of the two polymers, e.g. the degree of polymerization, is the same.

(a) Which one tends to have a lower glass transition temperature, Tg? Why?

(b) Which one tends to have a lower melting point, Tm? Why?

4. A constitutive equation is the governing equation that relates stress, σ(t), to strain, ε(t), where t is time. For instance, for a linear elastic material, the constitutive equation is the Hooke's law: σ(t) = E.ε(t), where E is the Young's modulus.

(a) Derive the constitutive equation of the two-parameter Voigt model, in which a spring and a dashpot are placed in parallel with each other. Denote the spring constant as k and the dashpot viscosity as η.

(b) In a constant-stress-rate (CSR) test, a tensile stress is applied and it is increased at a constant rate, R_σ; i.e. σ(t) = R_σt. If a polymer can be characterized by the two- parameter Voigt model, derive its strain, ε(t), for the CSR test. Hint: Ordinary differential equation (ODE) may be solved either directly or by using the Laplace Transform method.