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 ¯␭ g͑␭͒ ˜g͑␭͒ 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 / 3͑␭2 +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 ˜g͑␭͒ and g͑␭͒ 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 = <R /R0> eff z z Equation ͑2͒ is a standard “fading memory” form implying −1 (b) Stress chains “forget” their large-scale orientation at a rate ␶ .In ␶ ⑀ other words, is the time scale over which eff will relax ⑀ ͑ toward its “equilibrium” value eff=0 we assume that chains MacroscopicStretch λ−λ are not cross-linked͒. ⑀ eff MolecularStretch In this formulation eff corresponds to the strain in the “spring” in Fig. 1͑b͒. However, the stress we will associate 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 .