^{1}, R. L. Moorcroft

^{1}, R. G. Larson

^{2}and M. E. Cates

^{3}

### Abstract

Glassy polymers show “strain hardening”: at constant extensional load, their flow first accelerates, then arrests. Recent experiments under such loading have found this to be accompanied by a striking dip in the segmental relaxation time. This can be explained by a minimal nonfactorable model combining flow-induced melting of a glass with the buildup of stress carried by strained polymers. Within this model, liquefaction of segmental motion permits strong flow that creates polymer-borne stress, slowing the deformation enough for the segmental (or solvent) modes then to re-vitrify. Here, we present new results for the corresponding behavior under step-stress shear loading, to which very similar physics applies. To explain the unloading behavior in the extensional case requires introduction of a “crinkle factor” describing a rapid loss of segmental ordering. We discuss in more detail here the physics of this, which we argue involves non-entropic contributions to the polymer stress, and which might lead to some important differences between shear and elongation. We also discuss some fundamental and possibly testable issues concerning the physical meaning of entropic elasticity in vitrified polymers. Finally, we present new results for the startup of steady shear flow, addressing the possible role of transient shear banding.

M.E.C. is funded by a Royal Society Research Professorship. This work was funded in part by Engineering and Physical Sciences Research Council (U.K.) EPSRC(GB) EP/E030173 and EPSRC/E5336X/1. The research leading to these results has received funding to S.M.F. from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-13)/ERC Grant Agreement No. 279365. R.G.L. is partially supported from National Science Foundation (NSF) under Grant No. DMR 0906587. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF.

I. INTRODUCTION

II. STRAIN HARDENING

III. DUMB-BELL FLUIDITY MODEL

IV. ELONGATIONAL LOADING

V. ELONGATIONAL UNLOADING

VI. SOME ISSUES OF PRINCIPLE

VII. SHEAR DEFORMATION: STEP-STRESS

VIII. SHEAR STARTUP

IX. CONCLUSION

##### C21D1/02

## Figures

Solid curves: local strain , ^{ 8 } reduced relaxation time τ(*t*)/*t* _{ w } and tensile stresses of the polymer (p) and solvent (s) during loading of an infinite uniform cylinder. Here, *t* _{ w } = τ(0^{−}) is the age of the system when the experiment begins. Parameters are *G* ^{ s }/*G* ^{ p } = 8.5, μ = 12.5, *t* _{ w }/τ_{0} = 10^{4}, τ_{0} = 6 s; applied force/initial area *f* = 2.7*G* _{p}. (The curve for *T* ^{ p }, in red, initially lies below *T* ^{ s } but crosses it during strain hardening.) The unload results for the basic model (θ = 1) is shown dashed; the solid curve after unload has θ = 0.1. The horizontal axis is marked both in dimensionless model units (top) and real time (converted using τ_{0}), bottom. (As explained in Ref. ^{ 23 } , the numerical solver introduces, in lieu of inertia, a small additional fluid viscosity η_{ n } = 0.05*G* ^{ p }τ_{0} into Eq. (1) , whose magnitude has negligible influence on these plots.)

Solid curves: local strain , ^{ 8 } reduced relaxation time τ(*t*)/*t* _{ w } and tensile stresses of the polymer (p) and solvent (s) during loading of an infinite uniform cylinder. Here, *t* _{ w } = τ(0^{−}) is the age of the system when the experiment begins. Parameters are *G* ^{ s }/*G* ^{ p } = 8.5, μ = 12.5, *t* _{ w }/τ_{0} = 10^{4}, τ_{0} = 6 s; applied force/initial area *f* = 2.7*G* _{p}. (The curve for *T* ^{ p }, in red, initially lies below *T* ^{ s } but crosses it during strain hardening.) The unload results for the basic model (θ = 1) is shown dashed; the solid curve after unload has θ = 0.1. The horizontal axis is marked both in dimensionless model units (top) and real time (converted using τ_{0}), bottom. (As explained in Ref. ^{ 23 } , the numerical solver introduces, in lieu of inertia, a small additional fluid viscosity η_{ n } = 0.05*G* ^{ p }τ_{0} into Eq. (1) , whose magnitude has negligible influence on these plots.)

Schematic evolution of the relaxation time τ(*t*) in a sample with τ(0^{−}) = *t* _{ w }—which is its age or “waiting time” in our model—subjected later to a step strain causing a sudden drop in τ. In the simple aging picture, τ(*t*) rebuilds from this point with the same slope as before (upper curve). However, if aging and rejuvenation are factorable, the slope of the curve drops by the same factor as τ does (lower curve).

Schematic evolution of the relaxation time τ(*t*) in a sample with τ(0^{−}) = *t* _{ w }—which is its age or “waiting time” in our model—subjected later to a step strain causing a sudden drop in τ. In the simple aging picture, τ(*t*) rebuilds from this point with the same slope as before (upper curve). However, if aging and rejuvenation are factorable, the slope of the curve drops by the same factor as τ does (lower curve).

Solid curves: shear strain γ, reduced relaxation time τ(*t*)/*t* _{ w } and shear stresses of the polymer (p) and solvent (s) during shear loading. Parameters as in Fig. 1 , with a matched ratio of shear stress to solvent yield stress. (The curve for Σ^{ p }, in red, initially lies below Σ^{ s } but crosses it during strain hardening.) The unload results for the basic model (θ = 1) is shown dashed; the solid curve after unload has θ = 0.1.

Solid curves: shear strain γ, reduced relaxation time τ(*t*)/*t* _{ w } and shear stresses of the polymer (p) and solvent (s) during shear loading. Parameters as in Fig. 1 , with a matched ratio of shear stress to solvent yield stress. (The curve for Σ^{ p }, in red, initially lies below Σ^{ s } but crosses it during strain hardening.) The unload results for the basic model (θ = 1) is shown dashed; the solid curve after unload has θ = 0.1.

As in Fig. 3 but with shear stress increased by a factor 2.76. (This roughly matches the depth of the minimum in τ(*t*) to the elongational data of Fig. 1 .)

Startup of steady shear at applied strain rate for a system with the parameters of Fig. 1 . (Top panel) Total shear stress (solid) and the polymer (dotted) and solvent (dashed) contributions when the flow is imposed to be spatially uniform. (Within the resolution of the plot, the total stress calculated allowing for inhomogeneity is indistinguishable.) (Second panel) Total first normal stress difference *N* _{1} (solid) and polymer (dotted) and solvent (dashed) contributions under the same flow. (Third panel) The “degree of banding” (found by subtracting the smallest from the largest shear rate present at any time) for a 2D run with heterogeneity allowed. (A small diffusivity was added to the governing equations for all stress components and for τ, and the system was initialized with a small spatially varying noise; see Ref. ^{ 58 } .) (Bottom panel) Snapshots of the strain rate as a function of position *y* in the flow gradient direction with symbols identifying strain values as in the middle panel.

Startup of steady shear at applied strain rate for a system with the parameters of Fig. 1 . (Top panel) Total shear stress (solid) and the polymer (dotted) and solvent (dashed) contributions when the flow is imposed to be spatially uniform. (Within the resolution of the plot, the total stress calculated allowing for inhomogeneity is indistinguishable.) (Second panel) Total first normal stress difference *N* _{1} (solid) and polymer (dotted) and solvent (dashed) contributions under the same flow. (Third panel) The “degree of banding” (found by subtracting the smallest from the largest shear rate present at any time) for a 2D run with heterogeneity allowed. (A small diffusivity was added to the governing equations for all stress components and for τ, and the system was initialized with a small spatially varying noise; see Ref. ^{ 58 } .) (Bottom panel) Snapshots of the strain rate as a function of position *y* in the flow gradient direction with symbols identifying strain values as in the middle panel.

(Upper panel) Shear stress as a function of time in shear startup for the fluidity model without (lower curve) and with polymer. (*G* ^{ p }/*G* ^{ s } = 0, 1/8.5, respectively.) (Dashed regions) Transient instability as found by linear stability analysis. ^{ 58 } Dotted curve is total stress allowing for inhomogeneous flow. (Middle panel) The resulting absolute value of shear rate variations with polymer (multiply bumped curve with added symbols) and without (singly cusped curve). ^{ 58 } (Lower panel) Snapshots of strain rate profile with polymer present, at strain points identified by the symbols as in middle panel. Note the strong strain inhomogeneity (shear banding) at strains just beyond the stress overshoot. Parameter values *G* _{ s }, τ_{0} = 1, μ = 12.5, *t* _{ w } = 10^{8}, . Other parameters as in Fig. 1 and Ref. ^{ 58 } .

(Upper panel) Shear stress as a function of time in shear startup for the fluidity model without (lower curve) and with polymer. (*G* ^{ p }/*G* ^{ s } = 0, 1/8.5, respectively.) (Dashed regions) Transient instability as found by linear stability analysis. ^{ 58 } Dotted curve is total stress allowing for inhomogeneous flow. (Middle panel) The resulting absolute value of shear rate variations with polymer (multiply bumped curve with added symbols) and without (singly cusped curve). ^{ 58 } (Lower panel) Snapshots of strain rate profile with polymer present, at strain points identified by the symbols as in middle panel. Note the strong strain inhomogeneity (shear banding) at strains just beyond the stress overshoot. Parameter values *G* _{ s }, τ_{0} = 1, μ = 12.5, *t* _{ w } = 10^{8}, . Other parameters as in Fig. 1 and Ref. ^{ 58 } .

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