Effects of nonuniform segment deformation on the constitutive relation of polymeric solids

A polymeric solid is modeled as a network of beads and strings. Beads in the polymer are divided into bead groups represented by the segment end points. Based on the equation of motion for each bead in the system, a macroscopic equation of motion for the polymer is derived. Velocity fluctuations of beads result in a pressure, the isotropic component of the stress, in the polymer. Interaction forces transmitted through segments connected to bead groups are the major source of the stress in the polymer. The tendency of segments to achieve their minima in Helmholtz free energy results in thermal elasticity of the polymer. Compared to the time scale of these segment interactions, other interactions among beads groups are short in time, therefore, result in a viscous stress. The motion of an average segment is modeled as an elastic spring immersed in a viscous fluid. The inertia of the segment is neglected because the viscous force is much larger than the inertial force. The governing equation for the deformation of the average segment is found to be a diffusion equation representing the balance between the viscous force and the elastic force in the segment. If the time scale of the macroscopic deformation is short compared to that of the deformation diffusion, the application of forces at both ends results in nonuniform deformation of the segment, which diffuses from the ends toward the center. This diffusion leads to relaxation of the macroscopic stress which we represent by a history integral. The kernel in the integral is asymptotic to 1/sqrt[t] for a short time t, and is asymptotic to an exponentially decaying function for a long time t, in which t is the elapsed time from initiation of the deformation. This theoretically predicted kernel is observed in experiments conducted at constant temperature.