  Constitutive modeling of dielectric elastomers has been of long standing
interest in mechanics. Over the last two decades rigorous constitutive models
have been developed that couple the electrical response of these polymers with
large deformations characteristic of soft solids. A drawback of these models is
that unlike classic models of rubber elasticity they do not consider the
coupled electromechanical response of single polymer chains which must be
treated using statistical mechanics. The objective of this paper is to compute
the stretch and polarization of single polymer chains subject to a fixed force
and fixed electric field using statistical mechanics. We assume that the
dipoles induced by the applied electric field at each link do not interact with
each other and compute the partition function using standard techniques. We
then calculate the stretch and polarization by taking appropriate derivatives
of the partition function and obtain analytical results in various limits. We
also perform Markov chain Monte Carlo simulations using the Metropolis and
umbrella sampling methods, as well as develop a new sampling method which
improves convergence by exploiting a symmetry inherent in dielectric polymer
chains. The analytical expressions are shown to agree with the Monte Carlo
results over a range of forces and electric fields. Our results complement
recent work on the statistical mechanics of electro-responsive chains which
obtains analytical expressions in a different ensemble.
