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# Journal Club for September 2017: Some recent developments in constitutive modeling of glassy polymers

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The physics of glassy materials is a fascinating area of research. On one hand, the statistical mechanics understanding of their behavior is an active and exciting area of research. On the other hand, it is still quite challenging to develop and calibrate predictive constitutive models that reproduce all the observed behaviors. Many of the physical aspects the thermo-mechanics and ageing behavior of glasses are what actually make their constitutive modeling complex. The objective of this short post is to mention some of these aspects and point out a number of traditional and newer approaches to develop models that qualitatively and quantitatively match their response. The topic has garnered numerous review articles in the past few years. Therefore, this post can only touch on an ad-hoc and limited selection of work in this area and is not meant to be exhaustive by any stretch.

A short and accessible description of many important features of glassy polymers may be found in [1]. The first such feature that is invariably discussed in literature has to do with the apparent persistent memory in structural recovery. This is observed both in volume and enthalpy relaxations [2]–[5]. In simplest cases of volume tests, the samples were “equilibrated” to various initial temperatures and then T-jumped to the same final temperature. While the final volume at the target temperature is the same, the instantaneous time scale, *τeff*, of the approach to this value is a decidedly dependent on the initial temperature (and history), no matter how close to the final temperature (i.e. no matter how small *δ=V/Vf*-1). Even in multi-step processes, the material “remembers” the original temperature and *τeff*, eventually reverts to the trace associated with the last aged temperature. On the other hand, the complexity of the heat capacity trace, *Cp*, measured in standard DSC tests, and their dependence on the time of ageing [4], has been instrumental in developing early constitutive models for glasses such as KAHR model [3], and Narayanaswamy [6] and Moynihan et al. [7] models. These models all make use of the reduced time concept, which simply enters the formulation via a scaling of the actual time through shift function a. The shift function is defined as the ratio of the relaxation time at each instant to that at the reference state, and the current reduced time is defined as the integral of this scaled increment throughout the history of the specimen. Such formulations have been successful in qualitatively reproducing the significant memory effects discussed above both for volume relaxation asymmetry of approach and the peak in sub *Tg* heat capacity due to ageing [5].

The situation gets more complicated when mechanical stress is present as discussed in [8]. It was hypothesized that mechanical stress “rejuvenates” and cancels the effects of ageing. This idea naturally leads to introduction a (or enhancement of the) time shift associated with the applied stress. However, careful consideration by Lee and McKenna [9], [10] showed that this effect may not be as simply modeled, in the sense that the mechanical deformations may not fully and directly affect the underlying glass structure for higher values of stress. Nevertheless material clock models have been applied to separate phenomena with varying levels of success. One more recent example is the TVEM model [11] which uses a material clock based on “configurational energy”.

However, many phenomena observed in nonlinear deformation of glasses are not easily and consistently reproducible by any such models. Persistent heterogeneity in glasses and its evolution and interplay with the external conditions and forces were discussed by Ediger [12]. While it was understood that mobility is enhanced under stress, the post-yield effects on segmental dynamics in creep experiments showed dramatically higher effects than low stress values and was shown to be more pronounced in low mobility regions, leading to a more homogeneous material [13]. Based on these observations and needs, Caruthers and colleagues have proposed a stochastic approach in a series of recent publications. The effects of dynamic heterogeneity in their stochastic constitutive model (SCM) affects the overall response of glasses in a macroscopically observable way, distinct from mean field expectations. In their work, they use a partial distribution function for relaxation times of an ensemble of meso-domains, and develop its evolution equations as well as the macroscopically observable quantities due to the external forces as well as stochastic local fluctuation in the heterogeneous domains [14]. This approach has had great success in producing post-yield softening [14], stress-memory effect [15], volume relaxation [16], and nonlinear creep, particularly tertiary creep [17]. All these phenomena have presented significant challenges both in physical description and constitutive modeling and the success of this approach is quite encouraging and illuminating.

The metastable state of glasses is the underlying factor leading to the rich features of their macroscopic behavior. The pronounced need to include dynamic (evolving) heterogeneity to arrive at a constitutive model that qualitatively matches the thermo-mechanics and ageing of glassy polymers agrees with some of the recent statistical mechanics insights in glass forming liquids [18]. A dynamics approach to this phase transition suggests that while spatial pair distribution functions in such systems may be indistinguishable between equilibrium and non-equilibrium phases, the Van Hove’s self-correlation (or intermediate scattering) function [19] for the non-equilibrium phase is non-zero for all times. In other words, the creation of such disordered metastable states, their distinction from equilibrium states (liquid), and their slow ageing characteristics may be explained by considerations of dynamic heterogeneity. The confluence and compatibility of these theoretical observations with the recent success of the stochastic constitutive modeling approach is encouraging that similar tools may be applied to other challenging problems in this area in the future.

[1] G. B. McKenna, “On the physics required for prediction of long term performance of polymers and their composites,” JOURNAL OF RESEARCH-NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, vol. 99, pp. 169–169, 1994.

[2] A. J. Kovacs, “Transition vitreuse dans les polymères amorphes. Etude phénoménologique,” in Fortschritte Der Hochpolymeren-Forschung, Berlin, Heidelberg: Springer Berlin Heidelberg, 1964, pp. 394–507.

[3] A. J. Kovacs, J. J. Aklonis, J. M. Hutchinson, and A. R. Ramos, “Isobaric volume and enthalpy recovery of glasses. II. A transparent multiparameter theory,” Journal of Polymer Science Part B: Polymer Physics, vol. 17, no. 7, pp. 1097–1162, 1979.

[4] A. R. Berens and I. M. Hodge, “Effects of annealing and prior history on enthalpy relaxation in glassy polymers. 1. Experimental study on poly (vinyl chloride),” Macromolecules, vol. 15, no. 3, pp. 756–761, 1982.

[5] I. M. Hodge and A. R. Berens, “Effects of annealing and prior history on enthalpy relaxation in glassy polymers. 2. Mathematical modeling,” Macromolecules, vol. 15, no. 3, pp. 762–770, 1982.

[6] O. Narayanaswamy, “A model of structural relaxation in glass,” Journal of the American Ceramic Society, vol. 54, no. 10, pp. 491–498, 1971.

[7] C. T. Moynihan et al., “Structural relaxation in vitreous materials,” Annals of the New York Academy of Sciences, vol. 279, no. 1, pp. 15–35, 1976.

[8] L. C. E. Struik, “Physical aging in plastics and other glassy materials,” Polymer Engineering & Science, vol. 17, no. 3, pp. 165–173, Mar. 1977.

[9] A. Lee and G. Mckenna, “Effect of crosslink density on physical ageing of epoxy networks,” Polymer, vol. 29, no. 10, pp. 1812–1817, Oct. 1988.

[10] A. Lee and G. B. McKenna, “The physical ageing response of an epoxy glass subjected to large stresses,” Polymer, vol. 31, no. 3, pp. 423–430, 1990.

[11] J. M. Caruthers, D. B. Adolf, R. S. Chambers, and P. Shrikhande, “A thermodynamically consistent, nonlinear viscoelastic approach for modeling glassy polymers,” Polymer, vol. 45, no. 13, pp. 4577–4597, Jun. 2004.

[12] M. D. Ediger, “Spatially heterogeneous dynamics in supercooled liquids,” Annual review of physical chemistry, vol. 51, no. 1, pp. 99–128, 2000.

[13] H.-N. Lee et al., “Molecular mobility of poly(methyl methacrylate) glass during uniaxial tensile creep deformation,” Journal of Polymer Science Part B: Polymer Physics, vol. 47, no. 17, pp. 1713–1727, Sep. 2009.

[14] G. A. Medvedev and J. M. Caruthers, “Development of a stochastic constitutive model for prediction of postyield softening in glassy polymers,” Journal of Rheology, vol. 57, no. 3, pp. 949–1002, May 2013.

[15] J. W. Kim, G. A. Medvedev, and J. M. Caruthers, “The response of a glassy polymer in a loading/unloading deformation: The stress memory experiment,” Polymer, vol. 54, no. 21, pp. 5993–6002, Oct. 2013.

[16] G. A. Medvedev and J. M. Caruthers, “Predictions of Volume Relaxation in Glass Forming Materials Using a Stochastic Constitutive Model,” Macromolecules, vol. 48, no. 3, pp. 788–800, Feb. 2015.

[17] G. A. Medvedev and J. M. Caruthers, “Stochastic model prediction of nonlinear creep in glassy polymers,” Polymer, vol. 74, pp. 235–253, Sep. 2015.

[18] D. Chandler and J. P. Garrahan, “Dynamics on the Way to Forming Glass: Bubbles in Space-Time,” Annual Review of Physical Chemistry, vol. 61, no. 1, pp. 191–217, Mar. 2010.

[19] L. Van Hove, “Correlations in space and time and Born approximation scattering in systems of interacting particles,” Physical Review, vol. 95, no. 1, p. 249, 1954.

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## Comments

## Dear Alireza,

Dear Alireza,

Thanks for this nice review on the modeling glassy polymers. Indeed, the behaviors of glassy polymers show complex dependence on temperature, rate, thermal history as well as mechanical deformation, which raises a great challenge for developing constitutive models to describe them.

In addition to the above-mentioned approaches, another approach based on effective temperature has also been developed to describe the nonequilibrium behaviors of amorphous solids. The concept effective temperature originates from Tool’s fictive temperature [1]. By introducing the configurational entropy as a thermodynamic conjugate variable to the effective temperature, a fully consistent nonequilibrium thermodynamic framework has been built [2,3]. Based on the works of the physical community, we have developed a finite deformation continuum mechanics version of the effective temperature theory [4-6]. More importantly, we show that the effective temperature theory can quantitatively capture the dependence of stress response on temperature, strain rate, physical aging, and mechanical pre-deformation [4]. The effective temperature theory can also describe the experimentally measured DSC results, including an increase of enthalpy overshoot with aging, a decrease of enthalpy overshoot with mechanical deformation and the appearance of an undershoot of deformed glassy polymers [6].

Currently, we are also using the effective temperature to describe the mechanical instability of polymers, such as necking and shear banding. Hope we can obtain some preliminary results in the very near future.

Rui

Reference:

[1] A.Q. Tool, 1946. “Viscosity and extraordinary heat effects in glass”. J. Am. Ceram. Soc. 29, 240.

[2] T.M. Nieuwenhuizen, 1998. “Thermodynamics of the glassy state: effective temperatures an additional system parameter”. Phys. Rev. Lett.80, 5580.

[3] E. Bouchbinder, J.S. Langer, 2009. “Nonequilibrium thermodynamics of driven amorphous materials .II. effective-temperature theory”. Phys. Rev. E. 80, 031132.

[4] R. Xiao, T. D. Nguyen, 2015. “An effective temperature theory for the nonequilibrium behavior of amorphous polymers”, J. Mech. Phys. Solids, 82, 62.

[5] R. Xiao, T. D. Nguyen, 2016. “A thermodynamic modeling approach for dynamic softening in glassy amorphous polymers”, Extreme Mech. Lett., 8, 70.

[6] R. Xiao, G. Ghazaryan, T. A. Tervoort, T. D. Nguyen, 2017. “Modeling the energy storage and structural evolution during finite viscoplastic deformation of glassy polymers ”, Phys. Rev. E., 95, 063001.

## Dear Rui

Dear Rui,

Many thanks for your contribution to this journal club issue and I look forward to your future contributions as delineated in your post. The use of modified internal variables have been a very powerful appropach to developing constitutive models for polymers. Thanks for bringing to attention your work and classical work of Tool and others. The furhter challenge perhaps lies in modeling physical phenomena for which a transition of length scales (call it effective field theory applied to thermo-mechanics of nano-heterogeneous metastable media) masks the observed macro-scale phenomena which can be naturally explained by a stochastic description. My hope is that this very limited journal club post attracts the attention to such situations.

Regards,

Alireza.