A cubic elastomer is compressed by a steel piston in a defor

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 = ao + beta P, where a0 is the initial mold size and beta 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.

Solution

Relationship between Displacement and Pressure

Consider a monochromatic plane sound wave traveling down the length of a tube of gas of ambient density 0 at a phase speed s.

Find an expression for the relationship between the amplitude of the molecular displacement and that of the pressure oscillations. As is proven below in three different ways, the answer is

                                               P = ₀ v_s v_mol

where P is the gauge pressure and _mol is the longitudinal molecular speed due to the sound wave (i.e., averaged over all molecules within a thin transverse slice so that their random thermal motions can be neglected). Since the maximum longitudinal speed is equal to A where A is the displacement amplitude of the molecules from their (average) equilibrium position and is the angular frequency of the sound, we conclude that the pressure amplitude is

                                                      P _max = ₀v _s A

Suppose the sound wave is generated by a piston which fits snugly in the left end of a long tube of cross-sectional area S, which is open at the right end to the ambient surroundings, as sketched in Fig. 1 above. Initially, let the piston be in static equilibrium. Now consider applying a rightward force F to the piston. This will generate a compression whose wavefront will move some distance dx to the right of the original position of the piston during a small interval of time dt. Since this wavefront moves at the speed of sound, it follows that dx =dt V_s . The molecules themselves, however, move at the lower speed mol imparted by the piston. Since the distance from the piston to the wavefront is differentially small, we can assume that all of the molecules in this volume have the same speed. Clearly, the total mass of gas moving to the right is dm =S dx 0 , with a corresponding linear momentum of

dp = ₀ v_s S   dt V _{mol .}

But by Newton’s second law, this must result from a pressure in excess of ambient of

                                      P= F/ S = 1/*S( dp/ dt )= ₀ v_s v_mol