PHYSICAL REVIEW E 82, 041803 ͑2010͒
Viscoplasticity and large-scale chain relaxation in glassy-polymeric strain hardening
Robert S. Hoy* and Corey S. O’Hern Department of Mechanical Engineering and Materials Science, Yale University, New Haven, Connecticut 06520-8286, USA and Department of Physics, Yale University, New Haven, Connecticut 06520-8120, USA ͑Received 2 April 2010; revised manuscript received 6 August 2010; published 13 October 2010͒ A simple theory for glassy-polymeric mechanical response that accounts for large-scale chain relaxation is presented. It captures the crossover from perfect-plastic response to Gaussian strain hardening as the degree of polymerization N increases, without invoking entanglements. By relating hardening to interactions on the scale of monomers and chain segments, we correctly predict its magnitude. Strain-activated relaxation arising from the need to maintain constant chain contour length reduces the characteristic relaxation time by a factor ϳ⑀˙N during active deformation at strain rate ⑀˙. This prediction is consistent with results from recent experiments and simulations, and we suggest how it may be further tested experimentally.
DOI: 10.1103/PhysRevE.82.041803 PACS number͑s͒: 61.41.ϩe, 62.20.FϪ, 81.40.Lm, 83.10.Rs
I. INTRODUCTION hardening is thus cast as plastic flow in a medium where the effective flow stress increases with large-scale chain orienta- Developing a microscopic analytic theory of glassy- tion. Relaxation of chain orientation is treated here as inher- polymeric mechanical response has been a long-standing ently strain-activated and coherent, i.e., cooperative along challenge. Plasticity in amorphous materials is almost always the chain backbone. viscoplasticity, i.e., plasticity with rate dependence. Many Our theory predicts a continuous crossover from perfect recent studies have focused on plasticity in metallic or col- plasticity to “Gaussian” ͑Neohookean͓͒11͔ strain hardening loidal glasses. Relative to these systems, polymer glasses as the degree of polymerization N increases. The latter form possess a wider range of characteristic length and time scales is predicted when chains deform affinely on large scales and because of the connectivity, uncrossability, and random- corresponds to the limit where the system is deformed faster walk-like structure of the constituent chains. These alter the than chains can relax. A key difference from most previous ͓ ͔ mechanical properties significantly 1 ; for example, a theories is that instead of invoking entanglements, we relate uniquely polymeric feature of plastic response is massive the time scale for large-scale chain relaxation to the seg- strain hardening beyond yield. mental relaxation time ␣. This is consistent with ͑i͒ the pic- The relationships between polymeric relaxation times on ture that stress arises predominantly from local plasticity different spatial scales are fairly well understood for melts ͓12͔, ͑ii͒ recent dielectric spectroscopy experiments indicat- ͓ ͔ ͓ ͔ 2 but much less so for glasses. Recent experiments 3–5 , ing connections between relaxations on small and large ͓ ͔ ͓ ͔ simulations 5–7 , and theories 8,9 have all shown that lo- scales ͓13͔, and ͑iii͒ recent NMR experiments ͓14͔ that have ͑ ͒ cal segment-level relaxation times in polymer glasses de- found the effective “constraint” density for deformed glasses ͑ crease dramatically under active deformation especially at is much larger than the entanglement density measured in the ͒ yield and increase when deformation is ceased. Analysis of melt. By relating strain hardening to interactions on the scale these phenomena has focused on stress-assisted thermal ac- of monomers and segments, we make a prediction of its tivation of the local relaxation processes, but it is likely that magnitude that is consistent with the underlying glassy phys- other structural relaxation processes at larger scales or of ics. We test our predictions using coarse-grained molecular- ͑ ͒ different e.g., strain-activated character are also important dynamics simulations, and in all cases find ͑at least͒ semi- in determining the mechanical response. Improved under- quantitative agreement. standing of the concomitant scale-dependent relaxation is The rest of this paper is organized as follows. In Sec. II necessary to better understand polymeric plasticity and ma- we motivate and develop the theory for mechanical response, terial failure. However, theoretical prediction of large-scale make predictions that illustrate the effect of coherent relax- relaxation in deformed polymer glasses is still in its infancy; ation in constant-strain-rate deformation and constant-strain most treatments evaluate mechanical response “neglecting relaxation experiments, and test these using simulations. In the effect of the ongoing structural relaxation during the ex- Sec. III, we summarize our results, place our work in the ͓ ͔ periment” 10 . context of recent theories and experiments, discuss how the In this paper, we develop a theory that treats large-scale model could be more quantitatively tested experimentally, relaxation of uncross-linked chains during active deforma- and conclude. tion. Stress in the postyield regime is assumed to arise from local plastic rearrangements similar to those which control plastic flow; as strain increases, these increase in rate with II. THEORY AND SIMULATIONS the volume over which they are correlated. Polymeric strain A. Background Consider a bulk polymer sample deformed to a macro- *[email protected] scopic stretch ¯. Classical rubber elasticity relates the de-
1539-3755/2010/82͑4͒/041803͑10͒ 041803-1 ©2010 The American Physical Society ROBERT S. HOY AND COREY S. O’HERN PHYSICAL REVIEW E 82, 041803 ͑2010͒
TABLE I. Functional forms for strain hardening assuming affine sess a chain-level stretch ¯ ͓Fig. 1͑a͔͒ which describes the eff constant-volume deformation by a stretch . deformation of chains on large scales; for example, its zz ͗ / 0͘ component is Rz Rz , where Rz is the z component of the Deformation mode ¯ g͑͒ ˜g͑͒ 0 rms end-to-end distance Rc and Rz is its value in the unde- −1/2 2 / / ͑2 / ͒ Uniaxial x = y = z −1 1 3 +2 formed glass. The deformation of well-entangled chains is 1 ¯ ¯ Plane strain =−1, =1 2 − 1 / 3͑2 +1+1/ 2 ͒ consistent with an affine deformation, eff= , while for un- x z y 2 entangled systems the deformation is subaffine ͓21͔. Hereon, we drop tensor notation, e.g., ¯→, but all quantities remain ͑¯͒ 1 ͑2 2 2͓͒ ͔ tensorial. crease in entropy density to ˜g = 3 x + y + z 15,16 . Phenomenological “Neohookean” theories assume a strain A key insight is that the evolution of stress in polymer energy density of the same form. Both approaches give an glasses is controlled by the stretch eff of chains on scales comparable to their radius of gyration and only indirectly by ln͑¯͒ϵg͑¯͒. All results ץ/g͑¯͒˜ץassociated ͑true͒ stress ϰ . Well below the glass transition temperature T , stress is in this paper are presented in terms of true stresses and g well described ͓22,23͔ by strains. Stress-strain curves in well-entangled polymer glasses are often fit ͓11͔ by ͑¯͒= +G g͑¯͒, where is ͑¯͒ 0 ͑ ͒ ͑ ͒ 0 R 0 = 0 + GRg eff , 1 comparable to the plastic flow stress and G is the flow R 0 ͓ ͔ strain hardening modulus. Because of this, strain hardening where GR is the value of GR in the long-chain limit 24 . ͑ ͒ has traditionally been associated ͓17͔ with the change in en- Equation 1 shows that predicting eff is a critical com- ponent of the correct theory for the mechanics of uncross- tropy of an affinely deformed entangled network, with GR linked polymer glasses. However, to our knowledge, no assumed to be proportional to the entanglement density e. Forms for ˜g͑͒ and g͑͒ for the most commonly imposed simple microscopic theory that predicts the functional form deformation modes are given in Table I; Fig. 1͑a͒ depicts the of eff in glasses has been published. Most viscoelastic and “uniaxial” case. viscoplastic constitutive models that describe strain harden- There are, however, many problems with the entropic de- ing ͑e.g., Refs, ͓25,26͔͒ decompose into rubber-elastic and scription ͓18–20͔. One is that chains in uncross-linked plastic parts or use other internal state variables but do not glasses will not, in general, deform affinely at large scales explicitly account for eff or the N dependence of non- ͓ ͔ comparable to the radius of gyration. Rather, they will pos- affine relaxation 27–32 . In this paper we do so; eff is treated as a mesoscopic ͑chain-level͒ internal state variable (a) λ = Macroscopic Stretch ͓27,28͔. λ−1/2 −1/2 B. Maxwell-like model for eff R0 λ z 1 We now develop a predictive theory for eff. The problem is most naturally formulated in terms of true strains ⑀ 0 eff λeffRz ͑ ͒ ⑀ ͑͒ =ln eff and =ln . We postulate a simple Maxwell-like λ ⑀ model for the relaxation of eff. The governing equation for λ = Chain-level Stretch eff the model shown in Fig. 1͑b͒ is Nonaffine deformation: ⑀˙ = ⑀˙ − ⑀ /. ͑2͒ λ =λ eff eff eff λ =
λ with increasing ͉⑀ ͉ is viscoelastoplastic. Here, ⑀ is an ͑in
eff eff eff principle͒ microreversible strain corresponding to chain ori- entation; it is not an elastic strain in the macroscopic sense of ͓ ͔ ⑀ ϵ⑀ ⑀ shape recoverability of a bulk sample 33 . pl − eff cor- responds to the “dashpot” strain used in many constitutive models; it is plastic in the sense of being both microirrevers- FIG. 1. ͑Color online͒ Schematic of our model. ͑a͒ is the ible and macroirreversible. macroscopic stretch ͑observed on the scale of the experimental Maxwell-like models have been used to describe polymer sample if deformation is homogeneous͒, while eff is the large-scale viscoelasticity for more than half a century, and complicated chain stretch. Constant-volume uniaxial deformation is depicted. ͑b͒ ladder models were developed ͑e.g., Ref. ͓34͔͒ because of The spring-dashpot model for the nonaffine chain response is de- the inadequacy of earlier single rate models. However, we scribed in Eq. ͑2͒. will provide evidence below that the correct choice of meso-
041803-2 VISCOPLASTICITY AND LARGE-SCALE CHAIN… PHYSICAL REVIEW E 82, 041803 ͑2010͒
͑ ⑀ ͒ variable i.e., eff or eff restores the applicability of a single a Bauschinger-effect experiment ͓36,37͔, and the sign of ⑀˙ is ͑ ͒ ⑀ albeit N-dependent relaxation-time model, at least for opposite that of eff, active deformation may not produce monodisperse systems. coherent relaxation.
D. Bead-spring simulations C. Coherent strain-activated relaxation Some of the arguments made in Secs. II B and II C were The next step in constructing a useful microscopic theory heuristic, and several assumptions were made, so it is impor- is prediction of . If one supposes that chains under active tant to compare the theoretical predictions with results from deformation at strain rate ⑀˙ relax coherently and that the simulations. For example, observations of sharp changes in relaxation is strain activated, is reduced by a factor of N⑀˙ segmental relaxation times with strain and significant dy- relative to its quiescent value. namical heterogeneity ͓4,5,7͔ during deformation are seem- We assume that relaxation on large scales is coupled to ingly at odds with our postulated single constant relaxation segmental relaxation. For an a priori unspecified relaxation time . Further, entangled chains may not be able to relax dynamics in the quiescent state, coherently if the entanglements concentrate stress, so the re- ␥ duction of in actively deformed entangled systems may be ϳ N ␣ ͑incoherent͒, ͑3͒ weaker than predicted above. where ␣ is the “alpha” or segmental relaxation time and ␥ is The basic ideas presented above can be tested using unspecified and may be N dependent. molecular-dynamics simulations of the Kremer-Grest bead- 2 ͓ ͔ Gaussian polymers have chain statistics defined by Rc spring model 38 . All MD simulations were performed us- ϵ 2 2 LAMMPS ͓ ͔ Rs N, where Rs is the squared statistical segment length. ing 39 . Polymer chains are formed from N mono- Mathematically, this gives the identity mers of mass m. All monomers interact via the truncated and ͕͑ / ͒12 ͑ / ͒6 shifted Lennard-Jones potential ULJ=4u0 a r − a r t. ͑4͒ ͓͑ / ͒12 ͑ / ͒6͔͖ץ/R2 ץ t = Nץ/R2ץ c s − a rc − a rc . Here, rc =1.5a. Covalently bonded While Eq. ͑4͒ surely oversimplifies the physics of glasses monomers additionally interact via the finitely extensible ͑ ͒ ͑ 2 / ͒ ͓ ͑e.g., it does not hold in quiescent systems because R2 is nonlinear elastic FENE potential UFENE=− kR0 2 ln 1 c ͑ / ͒2͔ ͓ ͔ / 2 stationary͒, nonetheless it suggests an associated relaxation − r R0 ; the canonical 38 values k=30u0 a and R0 rate under active deformation that is N times larger ͑or a time =1.5a are employed. All quantities are expressed in terms of N times smaller͒ than the value in the quiescent state, the intermonomer binding energy u0, monomer diameter a, ͱ 2 / and characteristic time LJ= ma u0. The equilibrium cova- ϳ ␥−1 ͑ ͒ ͑ ͒ N ␣ coherent . 5 lent bond length is l0 =0.96a and the Kuhn length in the melt state is l =1.8a. We postulate that Eq. ͑4͒ becomes valid in actively deformed K All systems have N chains, with N NӍ2.5ϫ105. Peri- glasses; segmental rearrangements become coherent because ch ch odic boundary conditions are applied along all three direc- rearrangements that restore R2 toward its initial value domi- c tions of the simulation cell, which has periods L ,L ,L nate over those which do not. In practice, coherent relaxation x y z along the x,y,z directions. Melts are equilibrated ͓41͔ and is forced by the stiffness of the covalent backbone bonds, ͑ ˙ / ͒ which have ͑nearly͒ constant length l and so maintain rapidly quenched kBT=−0.002u0 LJ into glasses at T 0 / Ӎ ͑nearly͒ constant chain contour length L=͑N−1͒l . =0.2u0 kB 0.6Tg. Uniaxial-stress compressive deforma- 0 ͓ ͔ ⑀ Recent work assists in hypothesizing a more specific re- tions are then imposed. 22 A constant true strain rate ˙ ˙ / −5/ / 0 laxation dynamics for actively deformed systems. Hoy and =Lz Lz =−10 LJ is applied, with =Lz Lz . A Langevin Robbins ͓23͔ showed that chains in not-too-densely ͓35͔ en- thermostat with damping time 10 LJ is used to maintain T, tangled model polymer glasses well below Tg orient indepen- and a Nose-Hoover barostat with damping time 100 LJ is dently of one another during active deformation. The behav- used to maintain zero pressure along the transverse direc- ior observed was that of individual chains coupled to a tions. The values of ͉⑀˙͉ and T employed here lie within “mean-field” glassy medium. If relaxations on chain and seg- ranges shown ͓42,43͔ to reproduce many experimental trends mental scales are tightly coupled and chains relax indepen- ͓1͔, such as logarithmic dependence of on ⑀˙ and linear dently of one another, ␥=2 is predicted ͓2͔. Then, scaling of the hardening modulus with the flow stress.
2 ϳ N ␣ ͑incoherent͒, E. Chain conformations under deformation and constant-strain relaxation ϳ ͑ ͒ ͑ ͒ N ␣ coherent . 6 Constant-strain-rate deformation and constant-strain re- Within the framework of Eqs. ͑3͒, ͑5͒, and ͑6͒, when re- laxation are two of the most commonly performed mechani- laxation is strain activated, both ␣ ͓8͔ and will be reduced cal experiments. In a constant-strain rate experiment, assum- by a factor ϳ⑀˙, so the overall reduction in during active ing is independent of ⑀, i.e., assuming polymer glasses are deformation scales as N⑀˙. Note that the scaling analysis pre- linearly viscoplastic, the solution to Eq. ͑2͒ is ͓44͔ sented above does not account for additional changes in ␣ ⑀ ͑⑀͒ = ⑀˙͓1 − exp͑− ⑀/⑀˙͔͒ ϵ ⑀˙͓1 − exp͑− t/͔͒. ͑7͒ arising from other causes, e.g., increased mobility associated eff ⑀ ⑀0 ⑀0 with yield. Also note that the above arguments assume eff If a system is deformed to a strain and effective strain eff and ⑀˙ have the same sign. If deformation is reversed, e.g., in and then deformation is ceased, Eq. ͑2͒ has the solution
041803-3 ROBERT S. HOY AND COREY S. O’HERN PHYSICAL REVIEW E 82, 041803 ͑2010͒
1.0 1.0
0.8 0.8
0.6 0.6 eff eff 0.4 0.4
0.2 0.2
0.0 0.0 0120.5 1.5 0.0 0.5 1.0 1.5 2.0 ~ ~t t ͑ ͒ ⑀ FIG. 2. ͑Color online͒ Compressive deformation followed by FIG. 3. Color online eff vs ˜t for uniaxial compression fol- relaxation at constant strain. Curves from top to bottom are for N lowed by constant-strain relaxation; comparison of theory and =500, N=40, N=10, N=5, and N=3.˜t=͉⑀˙͉t is time scaled by the simulation results. Symbols from top to bottom are bead-spring Յ strain rate applied for 0Յ˜tՅ1. For˜tϽ1 curves show predictions of simulation results for N=500, 36, 18, and 12. For˜t 1, lines are fits ͑ ͒ Ն Eq. ͑7͒, while for˜tϾ1 curves and symbols show predictions of Eq. to Eq. 7 ; the fit values of are given in Table II. For˜t 1 the lines ␥ ␥ ͑ ͒ ͑8͒. Solid curves assume ϰN −1 during deformation and N at are given in Eq. 8 ; no further fitting is employed, and the values of ␥ ⑀−1 constant strain, while symbols assume ϰN −1 at all ˜t. Here, ␥=2. from Table II are multiplied by N˙ , consistent with the transition from coherent to incoherent relaxation assumed to occur upon ces- sation of deformation. ⑀ ⑀0 ͑ /͒ ͑ ͒ eff = eff exp − t . 8 dynamical homogeneity and narrowing of the relaxation In this case, our model predicts slowdown in relaxation upon −1 spectrum under flow observed in similar models in Refs. cessation of deformation; the in Eq. ͑8͒ is N⑀˙ times larger ͓ ͔ ͑ ͒ 7,47 . In developing our theory for the mechanical response, than the in Eq. 7 . we will assume for simplicity that is indeed independent of Figure 2 shows theoretical predictions of Eqs. ͑7͒ and ͑8͒ ⑀ ⑀ ͑ ͒ . for evolution of eff t in systems compressively strained to After cessation of deformation, simulation results in Fig. ⑀ ͑ Ͻ Ͻ⑀−1͒ =−1.0 at constant rate for 0 t ˙ and then allowed to 3 suggest that Eqs. ͑2͒ and ͑8͒ give a qualitatively valid ⑀ relax at constant strain. Results are plotted against ˜t= ˙t, description of large-scale chain relaxation at constant strain. where ⑀˙ is the strain rate applied during compression. For the ͉⑀ ͉ ͑ ␦ ␦Ӷ ͒ The small initial decreases in eff at times ˜t=1+ , 1 purpose of contrast, the symbols show predictions assuming are larger than the theoretical predictions of Eq. ͑8͒ for small ͑ that relaxation remains coherent i.e., increases by only a N. This may be attributable to chain end or segmental- factor ⑀˙−1͒ after deformation is ceased. N=500 chains orient relaxation effects not included in our model. At larger˜t, how- nearly affinely during strain, ⑀ Ӎ⑀˙t, while short chains ori- eff ever, it is clear that the increase in upon cessation of de- ent much less. The solid lines are far more consistent with formation is qualitatively captured. both experiments ͓3–5͔ and simulations ͓6,7͔, which show relaxation slows dramatically upon cessation of active defor- F. Microscopic theory for viscoplastic stress mation. ⑀ ͑ ͒ Here, we will derive a theory of Neohookean viscoplas- Figure 3 shows simulation data for eff t under the same ticity for the N-dependent stress-strain response. The work to procedure of compression followed by constant-strain relax- ˜Յ ͑ ͒ ation. Solid lines for t 1 are fits to Eq. 7 . Values for TABLE II. Values of obtained by fitting ⑀eff͑t͒ to Eq. ͑7͒͑with from these fits are given in Table II. The data are quantita- ␥=2͒ as a function of chain length N for flexible bead-spring ϰ ␥−1 ␥ tively consistent with N and =2. In particular, values chains. For these systems, Ne Ӎ85 ͓45͔. Statistical uncertainties on for /͑N−1͒ are nearly constant. For this model, careful to- are estimated at 5% . The factor of N−1 in the rightmost column pological analyses and rheological simulations have shown arises because in a discrete-bead model, relaxation is associated ͓ ͔ Ӎ ⑀ 45,46 that the entanglement length is Ne 85, so the trend with the number of covalent bonds rather than monomers. Note ˙ −5/ ⑀ ϳ spans the range from unentangled to well-entangled chains. =10 LJ so ˙ is on the order of unity for chains with N 10. The effect of entanglements is more consistent with an in- / /͓͑ ͒ ͔ crease in the prefactor of /͑N−1͒ than a change in ␥. N LJ N−1 LJ As shown in Table II, the agreement of with the predic- ϫ 5 ϫ 4 Ӎ ⑀˙−1 ͓ ͔ 12 1.15 10 1.05 10 tion ␣ 0.1 under active deformation 8 , where we 5 4 ␥−1 18 1.78ϫ10 1.05ϫ10 identify ␣ =/͑N−1͒ , is quantitative. The data in Fig. 3 5 4 are consistent with our assumption that remains constant 36 4.49ϫ10 1.09ϫ10 during deformation; only small deviations from the fits to 71 7.19ϫ105 1.03ϫ104 Eq. ͑7͒ are apparent on the scale shown. Given the large 107 1.12ϫ106 1.05ϫ104 variation in stress as strain increases ͑see Sec. II G͒,itis 250 2.76ϫ106 1.11ϫ104 ⑀ remarkable that a single relaxation rate theory fits eff so 500 7.14ϫ106 1.43ϫ104 well. However, this is consistent with the enhancement of
041803-4 VISCOPLASTICITY AND LARGE-SCALE CHAIN… PHYSICAL REVIEW E 82, 041803 ͑2010͒ deform the system is broken into two components corre- for the radius of gyration Rg of Gaussian polymers, Rg /ͱ sponding to an isotropic resistance to flow and an anisotropic =Rc 6. resistance to chain orientation. This general approach has We will now associate polymeric strain hardening with ͓ ͔ p been employed many times before, e.g., in Refs. 17,48 . W and replace by eff in order to calculate an N dependent Here we present a microscopic picture that quantitatively as- p. We postulate that Wp is controlled by the increase in V;in sociates the Gaussian or Neohookean strain hardening with other words, strain hardening occurs because the volume V the energy dissipated in local segmental hops and quantita- controlling Wp increases faster than it can relax ͓51͔. Studies tively associates the smaller hardening observed for shorter of bidisperse mixtures have provided strong evidence that ͑ ͒ chains to strain-activated relaxation of large-scale chain ori- the evolution of eff and hence strain hardening can be ͑ ͒ entation i.e., eff . understood in terms of single chains interacting with a glassy From the second law of thermodynamics, the work W mean field ͓23͔, so it is natural to assume the plastic events required to deform a polymer glass is W͑⑀͒=⌬E͑⑀͒+⌬Q͑⑀͒, are “unary” ͑in the sense that Ն2-chain effects are unimpor- ͒ p where E is the internal energy and Q is the portion of work tant . W will then scale linearly with cr, where converted into heat, including both dissipative and entropic ͱ /͑ ͒͑͒ terms. Without loss of generality, this can be rewritten as cr = 6 l0 NRc 11 ͑⑀͒ ͑⑀͒ ͑⑀͒ W =W1 +W2 , where W2 is the viscoplastic compo- is the density of coherently relaxing contours and is mono- nent of the work. W1 captures “everything else,” such as mer number density. Then, incrementally, elastic terms preyield and ͑in principle, though they are not ⌬ p Ӎ͑ / 3͒⌬͑ ͒ ͑ / 3͒ −1 ⌬͑ 2͒/ ͑ ͒ treated in this paper͒ other effects such as softening, anelastic W u0 a crV = u0 a N l0 Rc 6. 12 energetic stresses, and entropic stresses. In experiments, duc- 2 ͑ ͒ ͑ ͒ Recall Rc =3l0lKNg˜ eff , where lK is here the equilibrium tile deformation of glassy polymers occurs at nearly constant Kuhn length in the undeformed glass. For a deformation in- volume ͓1͔; thus, we assume constant-volume deformation crement ⌬¯, and treat W1 and W2 as intensive quantities. 2 Both rubber elasticity and Neohookean elasticity associate ⌬͑R ͒ =2l l N⌬͓˜g͉͑͒ ͔, ͑13͒ c 0 K eff strain hardening with W1. However, experiments and simu- The term in parentheses indicates .⑀ץ/g˜ץlations ͓19,22͔ have shown that W is the dominant term for where g͑⑀͒=͑3/2͒ 2 strains ranging from the beginning of the plastic flow regime that the difference is evaluated at = eff. to the onset of dramatic “Langevin” ͓17͔ hardening. The lat- Combining Eqs. ͑12͒ and ͑13͒ gives ter occurs at very high strains for most synthetic polymers ⌬Wp = ͑u /a3͒l2l ⌬͓˜g͑ ͔͒/2. ͑14͒ ͓11͔ and has been associated with the increase in energy 0 0 K eff arising from chain stretching between entanglements ͓22,49͔. This result combined with Eq. ͑9͒ gives a prediction for the Simulations have provided strong evidence that in this re- stress, gime, W is closely connected with the same local interchain 2 u plastic events that control the flow stress ͓22͔. These events ͑⑀ ͒ 0 ͓ ͑ ⑀/⑀ ͒ 2 ͉ ͑⑀ ͉͒/ ͔ ͑ ͒ eff = 3 1 − exp − y + l0lK g eff 2 , 15 ϳ / 3 a have a characteristic energy density u0 a , where u0 is the ͑ ͒ energy scale of secondary i.e., noncovalent interactions The absolute value ͉g͑⑀ ͉͒ appears because we have so far ͓ ͔ eff 50 . Here, we treat the regime where W2 dominates and treated as positive. neglect W1, i.e., we make the approximation W=W2. Equation ͑15͒ has several interesting features, which we W may be further broken down into “segmental” and now relate to previous models. First, it predicts that p ͑at s p s “polymeric” terms: W=W +W , where W accounts for the fixed strain͒ increases with increasing l . This is consistent p K plastic flow stress in the absence of hardening and W ac- with the well-established result that straighter chains are counts for the viscoplastic component of strain hardening. ͓ ͔ harder to plastically deform 1,11 . However, the power of lK Since this paper focuses on strains well beyond yield, for on which p depends is sensitive to our theoretical assump- convenience we choose a standard viscous-yield term ͓ ͑ ͔͒ tions, specifically the definition of cr Eq. 11 . Other defi- nitions can produce G ϰl3/2 or l3 , but choosing which one is u R K K Ws = 0 ͓⑀ + ⑀ exp͑− ⑀/⑀ ͔͒, ͑9͒ “best” ͓52͔ requires greater knowledge of the variation in a3 y y ͑ ͒ / local plasticity at a microscopic level with lK l0 than is ͑ ͒ ⑀ currently available. We test Eq. 15 using molecular dynam- where y is the yield strain. ͑ ͒ ics MD simulations in Sec. II G. While only one value of ⑀ץ/ ץ The stress is given by = W . It can similarly be writ- / ͑ ͒ lK a is considered here, it is large enough 1.8 that visco- ten as a sum of segmental and polymeric contributions, 3 plastic contributions to scaling as lK would be much larger s p s p .than contributions scaling as lK 10͒͑ .⑀ץ/ Wץ + ⑀ץ/ Wץ = = + Second, for long chains, our theory predicts little relax- ͓ ͔ ϰ ͑ Reference 22 showed that Rp, where Rp is the rate per ation during deformation, and a strain hardening modulus ͒ Ӎ 2 unit strain of plastic events identified by local rearrange- GR l0lK. This value is much closer to the effective con- ments. The natural correlation length scale for the local plas- straint density measured in ͑NMR͒ experiments of deformed ͓ ͔ tic rearrangements, since according to the above arguments glassy samples 14 than to the entropic prediction GR ͓ ͔ they are coherent, is Rc, and the associated volume is V = ekBT. Increasing lK also increases e 53 ; this makes re- 3 / 3/2 ͱ =Rc 6 . Here, factors of 6 arise from the standard relation lating hardening to entanglement a subtle problem. However,
041803-5 ROBERT S. HOY AND COREY S. O’HERN PHYSICAL REVIEW E 82, 041803 ͑2010͒
͑ ͓ ͔͒ considerable evidence e.g., 9,19,23,36,37,54,55 suggests that chain orientation and local secondary interactions which 3.0
ϳ͑ ϳ ϳ ͒ act over scales l0 a lK are the true controlling factors for at least during the initial stages of hardening. Finally, 2.0 Haward postulated that GR arises from the constraints im- ∗ posed by the mesh of uncrossable chains ͓11͔; our argument
−σ that hardening scales with cr is consistent with this hypoth- esis. 1.0
G. Predictions for stress-strain curves 0 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 In Eq. ͑15͒ we have assumed that all contributions to ͑ ͒͑ / 3͒ scale with a single energy density i.e., stress u0 a . This is an approximation since flow and hardening stresses are FIG. 4. ͑Color online͒ Stress-strain curves predicted in Eqs. ͑16͒ ء typically linearly rather than directly proportional ͓56͔. How- and ͑17͒ with A=1. − and −⑀ are shown because stress and strain ever, the “constant offset” term in this linear relationship is are negative for compression. Curves from top to bottom are for →ϱ often fairly small compared to the linear term ͑especially for N=500, N=10, N=5, and N=3. Predictions in the N limit are T well below T ͓23,56͔͒, so the approximation is reasonable. not distinguishable from N=500 predictions on the scale of this g ⑀ ␥−1 ⑀ Therefore, we associate u /a3 with and scale it out. plot. Solid curves assume ˙ =BN , while symbols assume ˙ 0 flow BN␥ ␥ ͓ ͔ B This gives = . In both cases, =2, and as suggested in Ref. 8 =0.1. ͑⑀ӷ⑀ ͒ ͑⑀ ͒ The large-strain y mechanical response predicted by A eff , ͑16͒ our model varies continuously from perfect-plastic flow = ͒ ⑀͑ء ͔ →͒⑀͑ءeff u /a3 ͓ 0 a constant flow to networklike polymeric response 2 ͑⑀͒/ ͔ ⑀˙ ϱ →͒⑀͑ء͓ where A is a prefactor arising from our neglect of prefactors 1+ l0lKg 2 as varies from zero to ͑equivalently, as N increases͒. In between these limits, the in the above analysis. Note that all “thermal” aspects of our ͑ response is “polymeric viscoplasticity” or more accurately theory are implicitly wrapped into A and ␣. In both real and ⑀ “viscoelastoplasticity” since we treat eff as microreversible simulated systems flow and A are approximately propor- ͓ ͔͒ ͑ / ͓͒ ͔ 61 . However, since our model does not treat stress relax- may be regarded as a ءtional to 1−T Tg 12,20,42,57 , so this scaling should re- ation after cessation of deformation, move much of the T dependence. The merits of “multiplica- ͑reduced͒ orientation-dependent plastic flow stress. ͑ ͒ tive” forms like Eq. 16 for predicting stress have been We now compare theoretical predictions for stress-strain ͓ ͔ discussed recently in Refs. 37,58 . More sophisticated mod- curves to results from bead-spring simulations. In Fig. 5, els ͑e.g., Ref. ͓9͔͒ explicitly treat the variation of A with ⑀˙ dashed lines show predictions of Eqs. ͑16͒ and ͑17͒ with A and T and/or the variation of ␣ with , T, and local micro- =0.43, values of taken from Table II, and the same values ͑ ͒ ͉⑀ ͉ structure. Here, however, A and for fixed N ␣ are treated of y , l0, and lK as in Fig. 4. The value of A obtained in Fig. ͓ ͔ ͑ / ͒ as numerical constants 59 . 5 is comparable to 1−T Tg . Stress-strain curves from ,ءTo predict the N dependence of ͑⑀͒, an analytic form for simulations are shown as solid lines. Panel ͑a͒ shows ء͓ ءQ ͒ ͑ ͒⑀͑ ⑀ ͔͒⑀͑ ⑀͓ g eff is obtained by plugging the solution for eff while panel b shows its dissipative component = ءQ ͔ ͓͔⑀ץ/ ץ3͒−1 / ͑ from Eq. ͑7͒ into the form of g͑⑀͒ for uniaxial deformation − u0 a U 22 . While this definition of includes ͓Table I, with exp͑⑀͒=͔, entropic terms, these are very small, on the order of 1% of the total stress at this T ͓22͔. ͑⑀ ͑⑀͒͒ ͕ ⑀͓ ͑ ⑀/⑀͔͖͒ g eff = exp 2˙ 1 − exp − ˙ In both panels, the correct trends are predicted, and quan- ϳ ͑ − exp͕− ⑀˙͓1 − exp͑− ⑀/⑀˙͔͖͒. ͑17͒ titative agreement is within 20%. Very short chains N =4͒ show nearly perfect-plastic flow, while longer chains in show strain hardening similar to results analyzed in many ͒⑀͑ءFigure 4 shows predictions of Eqs. ͑16͒ and ͑17͒ for ͉⑀ ͉ uniaxial compression at various N, with A=1, y =0.02 , previous studies. The quantitative differences at small strains −3 =1.0a , l0 =0.96a, and lK =1.8a; the latter three are chosen arise primarily from our oversimplified treatment of yield to match the flexible bead-spring model employed in the ͓62͔. At large strains, comparing panels ͑a͒ and ͑b͒,itis simulations. Solid lines assume ϰN␥−1 with ␥=2 as dis- apparent that differences between predictions of Eqs. ͑16͒ cussed above, while symbols assume that coherent chain re- and ͑17͒ and simulation results arise largely from energetic laxation is not important and ϰN␥. The solid lines are quali- terms associated with strain hardening, i.e., covalent bond tatively consistent with simulations ͓22,57͔, while the energy and additional plastic deformation arising from chain symbols are inconsistent. Both show increasing strain hard- stretching between entanglements. Experiments on amor- ening with increasing N, but in the latter case hardening in- phous polymer glasses ͑e.g., ͓19͔͒ have shown that a similar creases much faster and saturates at a much lower value of N percentage of the total stress is associated with energetic than is realistic. For example, the ϰN␥ predictions for N terms, strain softening, etc., so the agreement between theo- =40 and N=500 are indistinguishable on the scale of the retical predictions and bead-spring results is satisfactory plot. Results for tension are not presented here because our given the simplicity of our model. In panel ͑b͒, the “spike” in model includes no asymmetry between tension and compres- simulation results at small strains for NϾ4 reflects yield and sion ͓60͔. subsequent strain softening.
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5 1.5 (a) 4
3 /a na 1.0 22 ∗ D 2
−σ 1 0.5 0 0.4 0.6 0.8 1.0 λ 0 FIG. 6. ͑Color online͒ Nonaffine displacement at low T in sys- 1.5 (b) tems of long chains. Circles show data from bead-spring simula- ͑ / ͒ tions with N=500 and T=0.01u0 kB . The solid line shows a fit to 2 / 2 ͓ ͑͒ ͔͓ ͑ ͔͒ Dna a =C S −1 see Eq. 18 with C=7.39. Note that the 2 maximum value of Dna is well below the squared tube diameter 1.0 2 ∗ d /a2 ϳ100 ͓2,46͔, consistent with the picture that entanglements
Q T do not control the mechanical response of these systems. −σ 0.5 scales with Wp at large strains. This form gives slightly less good fits to our data, but in any case the principle shown in ͓ ͔ 0 Fig. 6 is the same as outlined in Ref. 63 ; for long chains, 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 nearly affine deformation at large scales ͑in our language, Ӎ ͒ ͑ eff increasingly drives nonaffine displacements i.e., ͒ .plastic activity at smaller scales, leading to strain hardening ͔ءQ ͒ ͑ ء ͒ ͓͑ ͒ ͑ FIG. 5. Color online Stress-strain curves a ; b from 2 bead-spring simulations ͑solid lines͒ and theoretical predictions This effect weakens and values for both and Dna decrease ͑ ͒ ͑dashed lines͒ in Eqs. ͑16͒ and ͑17͒. Values of N from top to bottom for large strains with decreasing N. are 500, 36, 18, 12, and 4. The top 4 are from the same simulations shown in Fig. 3 and use the fit values of ͑Table II͒, while the III. DISCUSSION AND CONCLUSIONS bottom theory curve assumes ⑀˙=0.1͑N−1͒. Note that stresses and strains are negative in compression. We derived a simple theory for polymeric strain harden- ing based on the notion that the increase in stress beyond H. Nonaffine displacement and plastic deformation yield amounts to an increase in the “flow” stress in an in- creasingly anisotropic viscoplastic medium. In the strain We have argued above that coherent relaxation is driven hardening regime, long and short chains relax on large scales by the stiff covalent bonds and the need to maintain constant via similar mechanisms ͑ϰ⑀˙−1N͒ and longer chains show chain contour length L. Since long chains have ⑀˙ӷ1 and greater hardening because they cannot relax on large length deform affinely on the end-to-end scale, they must deform scales over the time scale of the deformation. In practice, this nonaffinely on smaller scales to maintain chain connectivity. is due to the high interchain friction in the glassy state. It is interesting to relate this nonaffine deformation to the We theoretically predicted and provided evidence using chain stretching that would occur if deformation was affine simulations that coherent chain relaxation driven by resis- on all scales and L was not constrained, i.e., the limit of a tance to chain contour length increase is a key factor in the Gaussian coil with zero spring constant ͓2͔ embedded in a large-strain mechanical response of uncross-linked polymer deforming medium. For uniaxial tension or compression the glasses. Coherent relaxation reduces the dominant ͑chain 2 ϳ ͑͒ nonaffine displacement should be given by Dna S −1, scale͒ relaxation time by a factor N⑀˙ during active defor- with mation. Further, we claim that the increase in relaxation times when deformation is ceased is at least partially associ- L͑͒ 1 sin−1͑ͱ1−3͒ S͑͒ϵ = ͩ + ͪ. ͑18͒ ated with the fact that relaxation need no longer be coherent. ͱ 4 L͑1͒ 2 − Our results are consistent with many previous simulations and experiments and semiquantitatively capture the increase Figure 6 shows data for the squared nonaffine displacement in strain hardening as chain length increases. 2 ͗͑ជ ¯ជ ͒2͘ of monomers, Dna= r− r0 , where the monomer posi- Much recent work has emphasized the predominantly vis- ͕ ͖ ¯ ͕ ͖ tions are rជ at stretch and rជ0 in the initial state, for N cous or viscoelastic nature of polymeric strain hardening, at / =500. Data for T=0.01u0 kB are shown to minimize the ther- least prior to entanglement stretching. Recent experiments 2 ͑͒ ͓ ͔ mal contribution to Dna.AfittoS −1 is also displayed. and modeling 37,54,58,64 have provided strong support to There is qualitative agreement at large strains ͑Ӷ1͒, and the notion that entanglements play only a secondary role in 2 the underestimation of Dna at smaller strains is largely attrib- glassy-polymeric strain hardening, at least for the majority of utable to smaller scale ͑i.e., incoherent͒ plasticity on scales synthetic polymers and in the weak hardening regime. Hine approaching the monomer diameter a. Another reasonable and co-workers ͓54,64͔ also emphasized the role of meltlike 2 ͑͒ 2 form for fitting to Dna is ˜g , which would suggest Dna relaxation mechanisms in the marginally glassy state.
041803-7 ROBERT S. HOY AND COREY S. O’HERN PHYSICAL REVIEW E 82, 041803 ͑2010͒
The present work, which focuses on relaxation mecha- most accurate at low temperatures. We expect it to break nisms, is consistent with these trends. We quantitatively re- down concurrently with the validity of the mean-field ͓ ͔ lated strain hardening to local plasticity at the segmental independent-chain-relaxation behavior 23 of eff as T → scale and showed that the power-law dependence of the Tg. large-scale chain relaxation time on N during active defor- An ideal approach would explicitly account for both ther- ␥−1 mation, ϰN , is consistent with ␥=2, the same value as mally and strain-activated relaxation without sacrificing sim- the Rouse model for unentangled polymer melts ͓2,65,66͔. plicity. For example, stress-assisted rearrangements should Another “Rouselike” aspect is the apparent tight coupling of reduce as ͉͉ increases, and thermal activation should pro- to a single microscopic relaxation time ␣. duce logarithmic corrections to ⑀˙−1 scaling. Combining our eff can now be accurately measured in scanning near- theory for large-scale chain relaxation with a more sophisti- field optical microscopy ͑SNOFM͒ experiments ͓67͔, which cated theory for ͑e.g., Refs. ͓8͔ or ͓48͔͒ might be fruit- ⑀ Ͻ⑀ flow have shown that eff for entangled chains deformed ful. Modern techniques in the nonequilibrium thermodynam- slightly above Tg. Analogous studies, well below Tg, could ics of internal state variables ͓13,28͔ may also prove useful be performed to test the theory developed here; modern in this effort. neutron-scattering techniques might be employed for the Clearly, further work is necessary to quantitatively predict same purpose. Additionally, deformation calorimetry ͑DC͒ stress-strain and stress-relaxation curves. In particular, im- experiments ͓68͔ can be performed to better understand the proved understanding of the effects of chain stiffness is re- dissipative contribution to the stress. We are not aware of any quired. It seems certain that microscopic structural detail at studies in which the SNOFM or DC methods have been ap- the Kuhn scale ͑e.g., chemistry-dependent effects͒ exerts sig- plied to study strain hardening in polymer glasses. nificant influence on segmental relaxation processes, and it is Our model is minimal. It cannot quantitatively predict probable that these effects couple to chain-scale relaxation stress-strain curves either at small or at very large strains ͑see, e.g., Refs. ͓6,13,40,56,63,67,68͔͒. Studies using more because it neglects energetic components of stress and also chemically realistic models or real polymers would be wel- strain softening, which play important roles in glassy poly- come. mer mechanics. However, these limitations do not violate the spirit of our modeling effort, which was to illustrate the role of coherent relaxation in controlling large-scale chain con- ACKNOWLEDGMENTS formations and influencing stress in actively deforming poly- Mark O. Robbins contributed significantly to this project, mer glasses. with numerous helpful discussions. Kenneth S. Schweizer, The theory presented here serves as a complement to a Grigori Medvedev, Kang Chen, Daniel J. Read, Robert A. recent microscopic theory by Chen and Schweizer ͓9͔, which Riggleman, and Edward J. Kramer also provided helpful dis- also provides a unified description of plastic flow and strain cussions and K. S. provided the original concept for eff. hardening in polymer glasses. However, Ref. ͓9͔ assumes Gary S. Grest provided equilibrated N=500 states. The au- ¯ ¯ eff= , is based on an extension of liquid state theory to the thors are grateful to the organizers of the KITP Glasses ’10 glassy state ͓8͔, and ͑formally͒ breaks down at T=0. In con- conference. Support from NSF Awards No. DMR-0520415 trast, the theory developed here assumes that relaxation is ͑R.S.H.͒ and No. DMR-0835742 ͑R.S.H. and C.S.O.͒ is dominated by strain-activated processes and so should be gratefully acknowledged.
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͑2005͒. ͓42͔ J. Rottler and M. O. Robbins, Phys. Rev. E 68, 011507 ͑2003͒. ͓19͔ O. A. Hasan and M. C. Boyce, Polymer 34, 5085 ͑1993͒. ͓43͔ M. O. Robbins and R. S. Hoy, J. Polym. Sci., Part B: Polym. ͓20͔ H. G. H. van Melick, L. E. Govaert, and H. E. H. Meijer, Phys. 47, 1406 ͑2009͒. Polymer 44, 2493 ͑2003͒. ͓44͔ Eq. ͑7͒ predicts “saturation” of orientation and hardening at ͓ ͔ ⑀ տ⑀sat ⑀ ͑ ⑀ ͑⑀͒ 21 M. Dettenmaier, A. Maconnachie, J. S. Higgins, H. H. Kaush, large strains eff eff= ˙ . It can be rewritten as eff ͑ ͒ ⑀sat͓ ͑ ⑀/⑀sat͔͒ ͒ and T. Q. Nguyen, Macromolecules 19, 773 1986 . = eff 1−exp − eff . Our theory will break down in the ͓ ͔ ͑⑀sat͒ϳ / 22 R. S. Hoy and M. O. Robbins, Phys. Rev. Lett. 99, 117801 limit where chains are pulled taut, i.e., ˜g eff Nl0 lK. How- ͑2007͒; Phys. Rev. E 77, 031801 ͑2008͒. ever, this limit is difficult to reach at experimentally accessible ͓23͔ R. S. Hoy and M. O. Robbins, J. Chem. Phys. 131, 244901 strains. ͑2009͒. ͓45͔ R. S. Hoy, K. Foteinopoulou, and M. Kröger, Phys. Rev. E 80, ͓24͔ Very short chains do not significantly orient, and ͑if brittle 031803 ͑2009͒. fracture does not intervene͒ oligomeric glasses show nearly ͓46͔ R. Everaers, S. K. Sukumaran, G. S. Grest, C. Svaneborg, A. perfect-plastic flow, similar to that observed in atomic glasses Sivasubramanian, and K. Kremer, Science 303, 823 ͑2004͒; ¯ ¯ ͓22,57͔; Eq. ͑1͒ still applies in this case because effӍI and J.-X. Hou, C. Svaneborg, R. Everaers, and G. S. Grest, Phys. ͑ ͒ g͑¯I͒=0. Rev. Lett. 105, 068301 2010 . ͓ ͔ ͓25͔ L. Anand and M. E. Gurtin, Int. J. Solids Struct. 40, 1465 47 M. Warren and J. Rottler, Phys. Rev. Lett. 104, 205501 ͑ ͒ ͑2003͒; J. M. Caruthers, D. B. Adolf, R. S. Chambers, and P. 2010 . ͓ ͔ Shrikhande, Polymer 45, 4577 ͑2004͒; R. B. Dupaix and M. 48 M. C. Boyce, D. M. Parks, and A. S. Argon, Mech. Mater. 7, ͑ ͒ C. Boyce, Mech. Mater. 39,39͑2007͒. 15 1988 . ͓ ͔ ͑ ͒ ͓26͔ M. Wendlandt, T. A. Tervoort, and U. W. Suter, Polymer 46, 49 C. 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An important ͓ ͔ ͑⑀͒ lent to transient network elasticity, which predicts 32 a g difference between the approaches is that in the Eyring-like ͑ ͒ similar to Eq. 17 . models, V is a parameter with the dimensions of volume rather ͓ ͔ ͑ ͒ 28 M. B. Rubin, Int. J. Solids Struct. 31, 2635 1994 . than a physical volume. ͓ ͔ ͑ ͒ 29 D. J. Brown and A. H. Windle, J. Mater. Sci. 19, 1997 1984 ; ͓52͔ Similar ambiguities arise in theoretical prediction of the melt G. R. Mitchell, D. J. Brown, and A. H. Windle, Polymer 26, 0 plateau modulus GN from chain microstructure: W. W. Graess- 1755 ͑1985͒. ley and S. F. Edwards, Polymer 22, 1329 ͑1981͒. ͓30͔ H. L. Cheng, J. Wang, and Z. P. Huang, Mech. Time-Depend. ͓53͔ L. J. Fetters, D. J. Lohse, S. T. Milner, and W. W. Graessley, Mater. 14, 261 ͑2010͒. Macromolecules 32, 6847 ͑1999͒. ͓31͔ C. Miehe, S. Göktepe, and J. Méndez Diez, Int. J. Solids ͓54͔ P. J. Hine, A. Duckett, and D. J. Read, Macromolecules 40, Struct. 46, 181 ͑2009͒. 2782 ͑2007͒. ͓32͔ M. S. Green and A. V. Tobolsky, J. Chem. Phys. 14,80 ͓55͔ Fitting experimental curves to GR =ekBT produces values of ͑1946͒. e about 100 times larger ͑near Tg͓͒18,20,57͔ than values mea- ͓33͔ Restoration of ⑀eff toward its equilibrium value ͑zero͒ can, in sured in melt-rheological experiments. principle, drive shape recovery in a stress-relaxation experi- ͓56͔ L. E. Govaert, T. A. P. Engels, M. Wendlandt, T. A. Tervoort, ment; motions decreasing ͉⑀eff͉ correspond roughly to the “re- and U. W. Suter, J. Polym. Sci., Part B: Polym. Phys. 46, 2475 verse shear transformations” ͓12͔ studied by many authors. ͑2008͒. However, treatment of such effects is beyond the scope of this ͓57͔ R. S. Hoy and M. O. Robbins, J. Polym. Sci., Part B: Polym. ͑ ͒ work. In practice, the time scale of relaxation of ͉⑀eff͉ after a Phys. 44, 3487 2006 . ͓ ͔ large deformation would be very long if T is well below Tg. 58 M. Wendlandt, T. A. Tervoort, and U. W. Suter, J. Polym. Sci., ͓34͔ B. Gross and R. M. Fuoss, J. Polym. Sci. 19,39͑1956͒. Part B: Polym. Phys. 48, 1464 ͑2010͒. ͓ ͔ Շ ͓ ͔ ͑ ͒ 35 In practice, systems are “densely” entangled if Nel0 10lK 59 R. B. Dupaix and M. C. Boyce, Polymer 46, 4827 2005 . ͓22͔. ͓60͔ This asymmetry has been treated in other work ͑see, e.g., Refs. ͓36͔ T. Ge and M. O. Robbins, J. Polym. Sci., Part B: Polym. Phys. ͓42,48͔͒ and could, in principle, be treated by modifying Ws 48, 1473 ͑2010͒. appropriately. ͓37͔ D. J. A. Senden, J. A. W. van Dommelen, and L. E. Govaert, J. ͓61͔ Senden et al. ͓37͔ also treated glassy polymers as Neohookean Polym. Sci., Part B: Polym. Phys. 48, 1483 ͑2010͒. viscoelastoplastics using a continuum model with a phenom- ͓38͔ K. Kremer and G. S. Grest, J. Chem. Phys. 92, 5057 ͑1990͒. enological mixing of reversible and irreversible contributions ͓39͔ S. Plimpton, J. Comput. Phys. 117,1͑1995͒. to the stress. ͓40͔ A. V. Lyulin and M. A. J. Michaels, J. Non-Cryst. Solids 352, ͓62͔ Interestingly, our theoretical stress-strain curves are remark- 5008 ͑2006͒. ably similar to those in preoriented systems where deformation ͓41͔ R. Auhl, R. Everaers, G. S. Grest, K. Kremer, and S. J. is applied along the preorientation direction: see, e.g., Fig. 1͑b͒ Plimpton, J. Chem. Phys. 119, 12718 ͑2003͒. of Ref. ͓36͔ and Fig. 3͑b͒ of Ref. ͓37͔.
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͓63͔ B. Vorselaars, A. V. Lyulin, and M. A. J. Michels, Macromol- other experimental studies have systematically examined the ecules 42, 5829 ͑2009͒. variation of large-strain glassy response with N, but recent data ͓64͔ D. S. A. de Focatiis, J. Embery, and C. P. Buckley, J. Polym. for quiescent dynamics near Tg ͓13͔ suggest that the presence ͑ ͒ Sci., Part B: Polym. Phys. 48, 1449 2010 ; K. Nayak, D. of absence of any large-N crossover in ␥ may be chemistry Read, T. C. B. McLeish, P. Hine, and M. Tassieri, J. Polym. dependent. Sci., Part B: Polym. Phys. ͑to be published͒. ͓66͔ L. E. Govaert and T. A. Tervoort, J. Polym. Sci., Part B: ͓65͔ Our results do not exclude the possibility of a crossover to Polym. Phys. 42, 2041 ͑2004͒. higher ␥ for chains with NӷN . Govaert and Tervoort ͓65͔ e ͓67͔ T. Ube, H. Aoki, S. Ito, J. Horinaka, and T. Takigawa, Polymer presented data for NӷN systems that suggest hardening and e ͑ ͒ melt relaxation vary similarly with N ͑i.e., both are dominated 48, 6221 2007 ; T. Ube, H. Aoki, S. Ito, J. Horinaka, T. Taki- ͑ ͒ by large-scale relaxation͒. Although data were interpreted in gawa, and T. Masuda, ibid. 50, 3016 2009 . ͓ ͔ terms of relaxation of the entanglement network, the results are 68 G. W. Adams and R. J. Ferris, J. Polym. Sci., Part B: Polym. not inconsistent with ours because ␥ was not measured. Few Phys. 26, 433 ͑1988͒.
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