Strain Hardening in Polymer Glasses: Limitations of Network Models

Abstract

Simulations are used to examine the microscopic origins of strain hardening in polymer glasses. While traditional entropic network models can be fit to the total stress, their underlying assumptions are inconsistent with simulation results. There is a substantial energetic contribution to the stress that rises rapidly as segments between entanglements are pulled taut. The thermal component of stress is less sensitive to entanglements, mostly irreversible, and directly related to the rate of local plastic arrangements. Entangled and unentangled chains show the same strain hardening when plotted against the microscopic chain orientation rather than the macroscopic strain.The stress needed to deform a polymer glass increases as the strain rises. This strain hardening plays a critical role in stabilizing polymers against strain localization and fracture, and reduces wear [1]. While models have had some success in fitting experimental data, fundamental inconsistencies in fit parameters and trends imply that our understanding of the microscopic origins of strain hardening is far from complete.Most theories of strain hardening [2,3] are based on rubber elasticity theory [4]. These entropic network models assume that polymer glasses behave like crosslinked rubber, with the number of monomers between crosslinks equal to the entanglement length N e . The increase in the stress σ due to deformation by a stretch tensorλ is attributed to the decrease in entropy as polymers are stretched between affinely displaced entanglements. One finds [3]where σ 0 is the yield or plastic flow stress, G R is the strain hardening modulus, L −1 is the inverse Langevin function, g(λ) describes the entropy reduction for Gaussian chains, and L −1 (h)/3h corrects for the finite length of segments between entanglements. The value of N e enters in h, which is the ratio of the Euclidean distance between entanglements to the contour length.Stress-stretch curves for a wide variety of glassy polymers can be fit to Eq. 1, but the fit parameters are not consistent with the microscopic picture underlying entropic network models [5]. For example, values of N e from fitting h may be several times smaller than those obtained from the plateau modulus G 0 N [3]. Entropic network models predict G R ≈ G 0 N near T g , but measured G R are about 100 times larger [6]. G R also rises as T decreases [6,7], while any entropic stress must drop to zero as T → 0 [5]. Recent work [7,8] shows that changes in G R correlate with those in the plastic flow stress. Indeed entire stress-stretch curves collapse when normalized by σ 0 [7]. This is not expected from entropic models, where σ 0 is treated as an independent parameter arising from local plasticity. A more conceptual difficulty in entropic models is that, unlike rubber, glasses are not able to dynamically sample chain conformations.In this Letter we use simulations to examine the microscopic origins of strain hardening. While our results for the total stress can be fit to Eq. 1, network models are not consistent with ...

Deforming glassy polystyrene: Influence of pressure, thermal history, and deformation mode on yielding and hardening

“…The importance of mechanical history is recently observed. 27 Simulations on chemically detailed [25][26][27] and model polymers 16,17 showed that strain hardening can even occur for chains below the entanglement length contributing once more to the need for a view different from the rubber-elasticity theory of the physical mechanism behind strain hardening.…”

“…For some mechanical properties such as the Young modulus, the yield peak, and the strain-hardening modulus this has been successful, as being illustrated by numerous studies on various polymer models, such as on bead-spring models [6][7][8][9][10][11][12][13] and on bead spring with bond-angle-potential models. [14][15][16][17] More chemically realistic MD simulations of polymers have been carried out on amorphous polyethylene ͑PE͒, [18][19][20][21][22][23] PS, [24][25][26][27] and PC. 24,[26][27][28] Also other simulation techniques are applied to study the deformation of polymers, such as Monte Carlo algorithms or variants of energyminimization methods for PE-alike, 29 polypropylene, 30,31 poly͑oxypropylene͒, 32 PC, [33][34][35] and PE.…”

Simple Model for the Deformation-Induced Relaxation of Glassy Polymers

“…For simplicity we treat the dumb-bells initially as purely elastic elements -as is valid in the molten state, where the elasticity is of entropic origin [2]. However, we later return to discuss the true nature of the polymer stress in polymeric glasses which is not solely entropic in character [24,28]. This might seemingly call into question our time-scale separation into "polymeric" and "solvent" degrees of freedom; however recent careful experiments [30] and modelling [29] give clear evidence for such a separation.…”

“…Despite recent efforts [11,22,24,[27][28][29], creating a complete theory of rheological aging in polymer glasses remains a formidable task. Here we show that a minimal model, combining just two key elements of any such theory (nonfactorable aging/rejuvenation, and the strain dependence of polymer-borne stresses), semiquantitatively explains many of the results reported in [3].…”

Elasticity and plasticity in stiff and flexible oligomeric glasses

“…The microscopic origin of strain hardening was studied using molecular dynamic simulations in Ref. [7,8], finding that the origin of this phenomenon is related to plastic rearrangements of the monomers. This also leads to short-range ordering.…”