Theory for the strain hardening of glassy polymers

We propose a model for the strain-hardening of glassy polymers. 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  of 5 nm or of N_c = 1000 monomers.

The variation of the free energy barriers as a function of the applied stress and of th monomer orientations is a sum of two terms: one which scales like -σ² where σ 

is the local stress, and one which scales like Tr(q²), where q is a local nematic order parameter (tensorial in nature) the distribution of which we calculate. 

The first 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 and is responsible for strain hardening.

During these various regimes, the α-relaxation process remains an activated process with bariers which evolve from 45 k_BT down to 35 k_BT in the stress oftening 

regime, and then increase up to 40 k_BT in the strain hardening regime. 

The first contribution tdue to the stress to the change of free energy barrier varies instantaneously with the stress, whereas the second contribution due to chain orientation 

relaxes very slowly upon cessation of the applied deformation for instance. This feature is responsible

for specific memory effects such as the Bauschinger effect.