Theory of Plasticity and Strain Hardening of Glassy Polymers

We extend a theory for the deformation of glassy polymers based on the heterogeneous nature of the dynamics up to the strain-hardening regime. We attribute the latter to the increase of free-energy barriers for α-relaxation as a consequence of local orientation of monomers. The free-energy barriers are set on a scale ξ ≈ 5 nm or of about N c ∼ 1000 monomers which are involved in the α-relaxation mechanism. The variation of the local free-energy barriers is given by the expression Δ F ( σ ∼ , q ∼ ) = Δ F 0 − ξ 3 σ ∼ : σ ∼ 2 G 0 ′ + μ̃ 2 · N c · T r ( q 2 ∼ ) , where ΔF 0 is the free-energy barrier per monomer in the glassy state, typically ∼40–45k B T g for an aged polymer, q ∼ is the local order parameter (nematic in nature) whose distribution is computed during the course of deformation, σ ∼ is the local stress, and G 0′ is the bulk glassy modulus. μ̃ 2 is an energy scale of the typical order of 0.2–0.3k B T g. The second term is negative and is responsible for yielding and the onset of plastic flow. The third one is positive and becomes important after a large deformation has significantly oriented the chains on the scale of the monomers. It may overcompensate the decrease of the free-energy barriers due to the increasing stress and is responsible for strain hardening. Since the contribution of the stress to the reduction of the free-energy barrier between stress softening and the deformation λ ∼ 2 is of the order of −5k B T g, the contribution which leads to strain hardening, μ̃ 2 · N c · T r ( q 2 ∼ ) , is found to be of the order of 10 k B T g, which corresponds to an increase of the order of 0.01k B T g per monomer. This order of magnitude is compatible with the calculated values of the order parameter q ∼ 0.3 in the direction of tension during our simulations as well as that measured by Vogt et al. (1990) by NMR. We calculate the evolution dynamics of the order parameter q ∼ . Its dynamics is controlled by a driving force due to the local stress and a relaxation process due to rotational diffusion. The latter is entropic in nature and may be very slow in glassy polymers. We compare the predictions of our model to recent experimental results regarding the evolution of both the dominant relaxation time under applied strain and the width of the relaxation times distribution up to a large deformation amplitude, and more specifically their non-monotonic behavior.