Elsevier

Polymer

Volume 54, Issue 15, 8 July 2013, Pages 3949-3960
Polymer

Nonlinear stress relaxation in an epoxy glass and its relationship to deformation induced mobility

https://doi.org/10.1016/j.polymer.2013.05.034Get rights and content

Abstract

The mobility evolution in an epoxy glass during constant strain rate uniaxial deformation was examined via the stress relaxation response observed at various locations along the stress–strain curve, including the pre-yield, yield, and post-yield regions. Experiments were performed in the temperature range from Tg-30 °C to Tg-5 °C in both uniaxial extension and compression. At the faster loading strain rates the initial rate of stress relaxation is the slowest in the pre-yield region and the fastest in the post-yield region. However, at the slower loading strain rate the ordering is unexpectedly reversed so that the initial rate of stress relaxation is the fastest in the pre-yield region and the slowest in the post-yield region. These findings challenge the key assumption made in most existing constitutive models for glassy polymers that yield is the straightforward consequence of deformation induced increase in the mobility.

Introduction

Yield is the key feature of the stress–strain behavior of glassy polymers in a constant strain rate deformation, and has been reported for uniaxial, multi-axial, and shear deformations [1], [2], [3], [4], [5] as well as recently for a dilatationally dominated longitudinal deformation [6]. The yield stress is typically defined as the point on the stress–strain curve where the tangent modulus becomes zero, although in some cases yield is only identified with an abrupt change in slope of the stress–strain curve. Depending upon the thermal history used to form the glass, yield may be followed by a post-yield softening region, where the magnitude of the stress overshoot and subsequent post-yield softening increase as the sub-Tg aging is increased [7], [8]. For a material that has been rapidly quenched into the glassy state the stress–strain curve may not exhibit a maximum, but only a change in the tangent modulus [9]. As the deformation proceeds a post-yield flow regime occurs, where the tangent modulus is nearly zero. If the deformation is continued to even higher strains the flow regime often exhibit the strain-hardening, where the tangent modulus increases with increasing strain [1], [7].

The development of a constitutive model to describe yield and other features of the deformation behavior of glassy polymers has been one of the long standing objectives in polymer science and engineering. A number of nonlinear viscoelastic [10], [11], [12], [13] and nonlinear viscoplastic [14], [15], [16] constitutive models for describing glassy polymers have been proposed, where the key idea is the assumption that the deformation enhances the mobility of the material (or equivalently that the deformation decreases the viscosity in a viscoplastic model). As a consequence of this deformation induced increase in mobility (or decrease in viscosity), the material is eventually unable to support any further increase in stress and begins to flow, resulting in yield. In these constitutive models the mobility is controlled by the material time, t*, where the mapping between laboratory time t and material time t* is governed by the log a shift factor; specifically, dt*/dt = 1/a. This class of models is sometimes referred to as ‘material clock’ models. If t* is significantly longer than the laboratory time, t, the material rapidly relaxes and the mechanical response begins to approach that of a rubber (or a viscous liquid for thermoplastics); alternatively, if t* is much shorter than t there is little relaxation and material responds like a solid. A number of proposals have been made concerning the physical quantity that controls log a, including free volume [17], [18], [19], strain [20], stress [13], configurational entropy [21] and configurational internal energy [10], [22]. An important distinction is in how material time models describe the dependence on thermal and deformation history. For example, the strain clock models assume that the shift factor log a depends on the current value of strain; thus, there is no history dependence in log a, since the strain is externally controlled. On the other hand, the free volume models assume that log a depends on the current specific volume, but since the specific volume does depend on history there is implicit history dependence in log a. The thermo-viscoelastic model of Caruthers and coworkers [10], [11] assumes that log a explicitly depends on the thermal and deformation histories, where the expression for log a includes convolution integrals. The viscoplastic class of constitutive models is based on the pioneering work of Eyring [23], where the mobility enhancement comes about via a strong, usually exponential, dependence of the relaxation time(s) on the current stress [14], [15].

It is important to note that deformation induced changes in the relaxation time, or equivalently in the plastic viscosity, are not necessarily needed to effect a yield-like response. The stress–strain curve for a linear viscoelastic material deformed at a constant strain rate ε˙ is given byσ(ε)=ε˙i=1nEiτi(1exp(εε˙τi))where σ is the axial stress, ε is the axial strain and the Ei's are spectral components of the viscoelastic Young's modulus corresponding to the relaxation times τi. Although the stress–strain response does not exhibit a maximum or post-yield softening, there is a flow stress (which can be identified as the yield stress) given byσF=ε˙i=1nEiτiwhere the τi’s contain the temperature dependence via τi = a(T)τi0. If the spectrum {Ei,τi} is determined from an independent linear viscoelastic experiment, then the yield stress predicted by Eqn. (2) is much larger than the experimentally determined yield stress. Also, the linear viscoelastic response given in Eqn. (2) predicts a linear dependence of the yield stress on the strain rate, but the data show that the yield is a linear function of the logarithm of the strain rate [24]. These deficiencies in the linear viscoelastic description are the reasons for introducing a deformation induced mobility into the non-linear viscoelastic and viscoplastic constitutive models even though yield response itself does not require deformation induced mobility.

What is the evidence supporting deformation induced changes in mobility. First, there is direct evidence of deformation induced changes in mobility that includes the recent work of Ediger and coworkers on the re-orientation dynamics of photoactive dye molecules dispersed in a polymeric matrix during creep [25], [26] as well as solid state NMR data [27]. Second, there is indirect evidence needed for the viscoelastic description of the nonlinear relaxation response of glassy polymers as described in the previous paragraph. As an example, in a previous communication [9] we have argued that stress relaxation experiment provides information on the state of mobility in a deforming polymer glass. A lightly cross-linked PMMA was isobarically cooled into the glass and then deformed in uniaxial extension at a constant strain rate to a predetermined strain, where the strain was then held constant and the stress relaxation response was monitored. A relaxation time was extracted from the short time stress relaxation response, where it was shown that the relaxation time decreased as the strain was increased in the pre-yield region, but became essentially constant at yield and in the post-yield flow region. As the deformation temperature was increased from Tg-30 °C to Tg-10 °C the relaxation time became less strain dependent, where extrapolation of the data to Tg indicated that the deformation would not affect the relaxation time. The data in the Lee et al. [9] communication are consistent with the basic idea used in the current collection of nonlinear viscoelastic [10], [11], [12], [13] and nonlinear viscoplastic [14], [15], [16] constitutive models that mobility is increased as the material is deformed.

In this communication we will report on more extensive study of the effect of deformation on mobility in a glassy polymer in the Tg region for both uniaxial extension and compression, where the mobility will be determined from the stress relaxation response as in the previous study of Lee et al. [9] These measurements will be done as a function of (i) temperature, (ii) strain rate during loading, and (iii) the time of sub-Tg annealing. The data will clearly show that the simple picture of mobility increase as the deformation is increased from the pre-yield to post-yield flow region does not occur for all thermal/deformation conditions. The implications of this data on what should be an appropriate log a model as well as the appropriate structure of nonlinear viscoelastic and viscoplastic constitutive models will be discussed.

Section snippets

Experimental

The material system used in this study was a neopentyl glycol diglycidyl ether epoxy resin, DGENG, (Miller-Stephenson Chemical Co.) cured with 4,4′-methylenedianiline, MDA, (Sigma Aldrich). The chemical structures are shown in Fig. 1. All the chemicals were used as received without any additional purification. The procedure for producing the cured resin is as follows: DGENG is first heated to 60 °C and evacuated for 1 h in a vacuum oven to remove any volatiles. The 44′-MDA curing agent is then

Results

The primary objective of this study is to perform stress relaxation experiments at various points on the stress–strain curve of a glassy polymer, including the pre-yield, yield, and post-yield regions. Of particular interest are (i) the difference in the stress relaxation responses at different points along the stress–strain curve and (ii) the effect of the loading strain rate on the stress relaxation response. The various testing conditions for the DGENG-44′MDA epoxy are summarized in Table 1,

Discussion

The conventional view of the relationship between mobility and yield in glassy polymers is straightforward. Specifically, when a constant strain rate deformation is applied, the mobility increases and at some point becomes so high that the material effectively begins to deform like a viscous liquid (or a rubber for a crosslinked material), where this transition in macroscopic mobility effects yield in the stress–strain response. In the class of nonlinear viscoelastic constitutive models that

Conclusions

The evolution of mobility in a glassy polymer undergoing uniaxial deformation was studied via stress relaxation experiments carried out at various points along the stress–strain curve, including the pre-yield, yield, and post-yield regions. It was experimentally observed that the relationship between yield and mobility is not straightforward, where under some conditions there is a decrease in mobility with increasing deformation–a result that represents a serious challenge to the material clock

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