Strain-Hardening Modulus of Cross-Linked Glassy Poly(methyl methacrylate)

Transcription

1 Strain-Hardening Modulus of Cross-Linked Glassy Poly(methyl methacrylate) MICHAEL WENDLANDT, THEO A. TERVOORT, ULRICH W. SUTER Department of Materials, ETH Zurich, Zurich 8093, Switzerland Received 1 October 2009; revised 5 January 2010; accepted 25 January 2010 DOI: /polb Published online in Wiley InterScience ( ABSTRACT: The strain hardening modulus, defined as the slope of the increasing stress with strain during large strain uniaxial plastic deformation, was extracted from a recently proposed constitutive model for the finite nonlinear viscoelastic deformation of polymer glasses, and compared to previously published erimental compressive true stress versus true strain data of glassy crosslinked poly(methyl methacrylate) (PMMA). The model, which treats strain hardening predominantly as a viscous process, with only a minor elastic contribution, agrees well with the erimentally observed dependence of the strain hardening modulus on strain rate and crosslink density in PMMA, and, in addition, predicts the well-known decrease of the strain hardening modulus in polymer glasses with temperature. General scaling aspects of continuum modeling of strain hardening behavior in polymer materials are also presented Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 48: , 2010 KEYWORDS: mechanical properties; non-crystalline polymers; shear activation volume; work hardening; yielding INTRODUCTION A characteristic feature of the mechanical behavior of most ductile polymer glasses at large deformations in uniaxial deformation, is the so-called strain hardening response: after yielding and potential strain softening, the total stress increases upon increasing strain. It is now generally accepted that the long-chain nature of polymers plays an important role in strain hardening. During plastic deformation, the covalent chains orient, resulting in anisotropic materials with enhanced properties in the orientation direction. The mechanisms of this orientation process, however, have not yet been fully resolved. Recent erimental evidence suggests that strain hardening of ductile polymer glasses increases with increasing entanglement or crosslink density (In the following, the terms entanglement and crosslink density always refer to the values of the corresponding melt), while it decreases with increasing temperature. 1 5 Also the influence of strain rate on strain hardening still evokes controversy in literature. While some studies, including the present one, report a strain-rate dependent strain hardening, 6 9 most do not, 2 4, 10, 11 possibly due to the relatively small effect of strain rate on strain hardening compared to the total stress for many polymer glasses. 8 Consequently, a large number of molecular and continuum theories have been proposed that describe the erimentally observed strain hardening of polymer glasses by a strain-rate independent elastic contribution to external stress. In addition, erimentally, strain hardening increases with increasing crosslink or entanglement density. 2, 4, However, attempts to relate the strain hardening of polymer glasses to the mechanical response of a rubber-elastic network have failed to agree with erimental 4, 13, 14 results, with the main issues being: the observed decrease of strain hardening with temperature instead of an ected increase, the observation that erimental strain hardening is much larger than what would be ected from even the highest entanglement densities found in the melt (e.g., in polycarbonate 12 ), and the observed strain-rate dependence of strain hardening. 6 9 Assuming that the total strain hardening is only partially the result of an elastic stress response, and partially that of a strain-dependent viscous contribution due to strain-retarded segmental motion, as recently proposed in a nonlinear viscoelastic continuum model by Wendlandt et al., 8 offers a alternative view that could circumvent the aforementioned difficulties. In this contribution, we intend to demonstrate the ability of this new model to describe the strain hardening modulus of crosslinked glassy poly(methyl methacrylate) (PMMA) as a function of strain rate and crosslink density. It will be shown that the newly proposed continuum model also predicts the often observed decrease of the strain hardening modulus with temperature. In addition, we will discuss general scaling aspects of continuum modeling of strain hardening in polymer solids. Remarkably, in the case of semi-crystalline polymer solids, which are not the focus of this study, qualitatively similar correlations between strain hardening and temperature, crosslink or entanglement density have been found erimentally (see e.g., refs ). There are strong indications that strain Correspondence to: T. A. Tervoort ( theo.tervoort@mat.ethz.ch) Journal of Polymer Science: Part B: Polymer Physics, Vol. 48, (2010) 2010 Wiley Periodicals, Inc INTERSCIENCE.WILEY.COM/JOURNAL/JPOLB

2 ARTICLE hardening in such multi-phase systems is dominated by the amorphous phase. 15 The conclusions for glassy polymer solids drawn in this study might, therefore, be of benefit, particularly for the understanding of the contribution of the glassy amorphous phase to strain hardening in semi-crystalline polymer solids below the glass-transition temperature. CONTINUUM MODELING OF STRAIN HARDENING Recently, we proposed a three-dimensional nonlinear, viscoelastic constitutive relation for polymer glasses 8 (hereafter referred to as the WTS model ), which is based on the threedimensional compressible Leonov model. In the WTS 18, 19 model, the total Cauchy stress tensor is decomposed into a driving viscoelastic stress due to segmental motion and a deviatoric elastic neo-hookean strain hardening stress. In the limiting case of large post yield stresses and strains in uniaxial deformation, the total true uniaxial stress σ in deformation direction is a function of the total true strain ε, the true strain rate ε, and the absolute temperature T as: σ(ε, ε, T) = 3 k B TV(ε) 1 ( α + β + ln ε) + G nh (T) g(ε) }{{ T }}{{} viscous stress neo-hookean stress (1) with α = ΔH 0 (2) k B β = ln(2 A 0 3) (3) g(ε) = (2ε) ( ε) (4) = λ 2 λ 1 (5) where G nh (T) is a (possibly temperature dependent) shear modulus that characterizes the elastic neo-hookean stress contribution G nh (T)g(ε) to the total strain hardening response, k B is Boltzmann s number, A 0 is a constant pre-onential factor involving the fundamental vibration energy, ΔH 0 represents the height of the energy barrier for plastic flow at zero equivalent stress, and λ = (ε) is the draw ratio. V(ε) is a strain-dependent shear activation volume for plastic flow, defined as: 8 V(ε) = ( ) dσ(ε, ε, T) 1 3 k B T (6) dln ε It should be noted that the activation volume does not correspond to the size of a plastic event. It is not even a real volume in the physical sense, but merely a material parameter with the dimension volume that measures how stress reduces the effective free-energy barrier for segment mobility. A small activation volume (compared to kt) indicates that stress-induced mobility only commences at high stress levels. Therefore, reducing the activation volume upon plastic deformation, leads to an increasing post yield flow stress with strain that manifests itself as strain hardening. An interesting feature of this approach is that the use of a nonconstant strain-dependent activation volume simultaneously captures the strain-rate dependence at large deformations and, in addition, leads to a significantly increasing viscous contribution to the strain hardening response. This was also realized by Buckley, 20 who employed an anisotropic activation volume to describe the finite-strain behavior of glassy polymers. Experimentally it has often been observed that the total strain hardening response of polymer solids during monotonous loading in uniaxial deformation appears similar to an elastic neo-hookean response that is a function of the crosslink or entanglement density and temperature. 2, 4, 12 A common way to characterize strain hardening is, therefore, given by the so-called strain hardening-modulus G R defined as G R (ε, ε, T) = dσ(ε, ε, T) dg(ε) The above definition of G R can be used to extract a strain hardening modulus from the WTS model, even though most of the stress-upswing is of a viscous origin. 8 Using eqs 1 and 7, we can identify the viscous and elastic neo-hookean contribution to the total strain hardening modulus G R in the WTS model as: G R (ε, ε, T) = ( 3 k B T dv(ε) 1 α ) dg(ε) T + β + ln ε }{{} viscous strain hardening + G nh (T) }{{} neo-hookean strain hardening Note that in the WTS model the total strain hardening response is generally not elastic, let alone neo-hookean, due to the additional viscous contribution to strain hardening. However, eq 8 would exhibit neo-hookean like strain hardening during monotonous uniaxial loading if V(ε) is constant, or when V(ε) 1 is linear in g(ε). In the latter case G R (ε, ε, T) would equal G R ( ε, T), whereas a constant activation volume would result in G R (ε, ε, T) = G R (T) = G nh (T). A more detailed discussion of the shape of V(ε) will follow in the results and discussion section. Finally it should be noted that, in the special case of crosslinked polymers, the activation volume V(ε) can also depend on crosslink density μ as discussed later. To compare the strain hardening response of the WTS model to other approaches that have been published, eq 1 is rewritten as: with σ(ε, ε, T) = S 0 (ε, T) S 1 ( ε, T) }{{} viscous stress + S 2 (ε, T) }{{} neo-hookean stress (7) (8) (9) S 0 (ε, T) = 3 k B TV(ε) 1 (10) S 1 ( ε, T) = α + β + ln ε T (11) S 2 (ε, T) = G nh (T) g(ε) (12) In the limit of large strains, Hope and Ward 21 and Botto et al. 22 suggested the same form as in eq 9, while most of the published continuum models for polymer glasses comprise additive contributions that depend either on strain rate or 2 4, 10, 11 total strain exclusively, e.g., σ(ε, ε, T) = S 0 ( ε, T)+S 1 (ε, T). STRAIN-HARDENING MODULUS OF PMMA, WENDLANDT, TERVOORT AND SUTER 1465

3 JOURNAL OF POLYMER SCIENCE: PART B: POLYMER PHYSICS DOI /POLB TABLE 1 Materials and Testing Conditions Grade ρ e (1/nm 3 ) μ (1/nm 3 ) ε (1/s) T ( C) PMMAc , 0.31, 0.63, ,3 10 4,10 3,3 10 3, ρ e and μ are the entanglement 25 and chemical crosslink density, respectively, as determined in the melt, ε is the true compressive strain rate, and T is the temperature during the test. Further details have been published elsewhere. 8 In the latter case, a change in strain rate would result in a vertical shift of the stress strain curve without any impact on the strain hardening. However, it is the existence of terms depending in a non additive way on strain rate and total strain, e.g., multiplicative, as the viscous contribution in eq 9, that yields a strain-rate dependent strain hardening. In other words, a strain-rate dependent strain hardening implies that the total stress response increases stronger with ongoing strain than a neo-hookean stress response only. This is supported by eriments of Botto et al., 22 who found that the yield stress of previously drawn glassy PMMA increases stronger with the amount of pre drawing than the so-called shrinkage stress. The latter occurs after unloading when the sample is held at constant length and heated above the glass transition temperature T g. Botto et al. suggested that the shrinkage stress is generated by an entropic molecular network that has been deformed and frozen in during drawing below T g and is released upon heating above T g. Thus, these authors concluded that the viscous term of the stress response is a function of strain, which indicates that there is a contribution to strain hardening additional to that from an entropic network. Recently, Hoy and Robbins published an interesting series of molecular simulation studies on strain hardening of polymer glasses 9, 13, 23, 24 in which the following stress strain relationship in the strain hardening regime was discussed: with σ(ε, ε, T) = S I (ε, T) S II ( ε, T) (13) S I (ε, T) = F(ε) (14) S II ( ε, T) = σ( ε 0, T) + k B TV 1 ln ε (15) Here F(ε) is a dimensionless function of strain, ε 0 is a reference strain rate and V is a constant activation volume similar to the strain-dependent activation volume in eq 1 at the yield point. A dependence of stress on entanglement or crosslink density has been omitted in eq 13 for the sake of simplicity, although it has been thoroughly discussed in the publication of Hoy and Robbins. 9 Further, S II equals the yield stress of rejuvenated material. Using eq 7, the strain hardening modulus of a multiplicative model as in eq 13 scales with the flow stress S II as G R (ε, ε, T) = S II ( ε, T) df(ε) dg(ε) (16) Plotting σ/s II against strain for different strain rates, should, therefore, result in a strain-rate independent mastercurve, that is, however, neither observed erimentally 14 nor has it been shown to hold strictly in simulations of Hoy and Robbins, particularly at large strains. 24 Inspection of eqs 9 12 shows that the multiplicative form of eq 13 is a special case of the general ression given by eq 1 with V(ε) 1 g(ε) and F(ε) g(ε). RESULTS AND DISCUSSION The definition of the strain hardening modulus, eq 7, was used to re-interpret previously published erimental data of compressive true stress vs. true strain for crosslinked PMMA. 8 A summery of the material and testing conditions is given in Table 1. Note that for the linear PMMA used in this study, no influence of polydispersity or molecular weight on stress strain behavior was found in the range of 1.02 < M W /M N < 3 and 300M e > M W > 5M e respectively, 8 where M e is the entanglement molecular weight of the corresponding melt. In the following, all presented values of G R were determined as the slope of the true stress true strain curve at an arbitrary reference true compressive strain of ε = 0.7. Using other strains in the large strain regime [ 0.5 < ε < 0.8] to evaluate the slope of the stress strain curve, result in similar values for G R and lead to the same conclusions. Table 2 summarizes the fit parameters obtained from a least squares fit of the WTS model to erimental stress strain data of chemically crosslinked PMMA at 22 C. Note that fitting the pre- and post-yield isothermal stress response at different strain rates gives A = A 0 (ΔH 0 /k b T) as a fit parameter rather than A 0 or ΔH 0 directly (The parameters A 0 and ΔH 0 can be obtained from stress strain data at different temperatures: dσ/dt = 3k B V(ε, μ) 1 (β + ln ε) gives A 0 with the knowledge of V(ε, μ) and finally ΔH 0 from A. Alternatively, a fit of the post yield isothermal strain hardening response with eq 1, would give (α/t + β) as a fit parameter, and A 0 and ΔH 0 as described above.). 8 The correct fit parameters for other selected polymers that appear in ref. 8 are listed in Table 3. The values in Table 3 differ from the data in the original Tables 2 and 3 in ref. 8, which are flawed due to an erroneous handling of the hydrostatic stress contribution. The quality of the fits and any conclusions drawn in ref. 8, however, were not affected. Gratifyingly, for all polymers studied, also the corrected fit of the neo- Hookean contribution to the strain hardening response was always found to be much smaller than the total erimental strain hardening modulus. For linear PMMA, the newly found fit for the neo-hookean contribution to the strain hardening response is G nh = 10.3 MPa. This corresponds well to the values found for crosslinked PMMA, indicating that the neo-hookean contribution is independent of crosslink density INTERSCIENCE.WILEY.COM/JOURNAL/JPOLB

4 ARTICLE TABLE 2 Fit Parameters Obtained from a Least Squares Fit of the WTS Model to Experimental Stress Strain Data of Chemically Crosslinked PMMA at 22 C μ (1/nm 3 ) G nh (MPa) A (10 10 s) a (nm 3 ) b (nm 3 ) The constants a and b (eq 17) describe the strain dependence of the shear activation volume. A plot of the strain hardening modulus versus average chemical crosslink density for PMMA is given in Figure 1. The data of Figure 1 suggest, to a first approximation, a linear relation between the strain hardening modulus G R and the crosslink density μ. However, surprisingly, performing a least-square fit of the WTS model to the erimental stress-strain data, it was found that not G nh, but V(ε) depends on the crosslink density (see Table 2 and ref. 8). At first sight it might appear counterintuitive that the crosslink density only influences V(ε) and not the (rubber) elastic network modulus G nh. However, this result is in agreement with a recent solid-state NMR study of Wendlandt et al., 26 where it was found that segmental orientation appears to be generated by a rubber-elastic network with a constant network density that is much higher than either the entanglement or crosslink densities used in their study. The crosslink density, and hence the theoretical modulus of this rubber-elastic network was derived from the average number of Kuhn segments per elastically active chain n, the material s density, the molar mass of one statistical Kuhn segment, and the functionality of the network as shown in ref. 26 Assuming now that a Kuhn segment in glassy PMMA consists of two monomeric units, and using the value for n as determined from solid state NMR eriments, 26 leads to a modulus of G 13 MPa. This value compares favorably to the value of the neo-hookean contribution to strain hardening of this study of G nh MPa. Also a recent neutron scattering study of PMMA by Casas et al. 27 indicates the presence of a glassy apparent network TABLE 3 Corrected Fit Parameters Obtained from a Least Squares Fit of the WTS Model to Experimental Stress Strain Data of Selected Polymers at 22 C (shown in ref. 8) Material G nh (MPa) A (10 10 s) a (nm 3 ) b (nm 3 ) PMMA PPO PC PS < In the original publication, a negative offset in the calculated stress due to an erroneous handling of the hydrostatic stress contribution, lead to wrong fit parameters, see Tables 2 and 3 therein, which, however, did not affect the quality of the fits or any conclusions drawn in ref. 8. FIGURE 1 Experimental strain hardening modulus G R as a function of crosslink density μ for chemically crosslinked PMMA, measured at different true strain rates as indicated in the graph. The straight lines are linear fits to the data. Also shown is the neo-hookean contribution to the strain hardening modulus, G nh, that follows from fitting the erimental stress-strain curves at different strain rates with the WTS model. with a mesh-density much smaller than the entanglement density. Finally, a constant elastic contribution as suggested by Figure 1 is also in agreement with erimental studies by Oleinik, 28 who observed that the stored energy upon unloading of crosslinked glassy epoxy networks is independent of crosslink density. All the aforementioned results indicate the existence of local steric hinderance, that acts as a crosslinkdensity independent generator for segmental orientation during plastic deformation, and that results in a neo-hookean rubber-elastic contribution to the strain hardening response, characterized by a much higher network density than the crosslink or entanglement density of the corresponding melt. If the true neo-hookean contribution characterized by G nh is constant (see Fig. 1) then, according to eq. 1, the observed dependence of the total strain hardening modulus G R on crosslink density must result from a dependence of the viscous contribution dv(ε, μ) 1 /dg(ε) on μ. Figure 2 shows a plot of the strain-dependent term of strain hardening G R, i.e., dv(ε, μ) 1 /dg(ε) versus μ, which follows a linear relation to a first approximation. In a preceding publication, 8 we proposed the following empirical functional form for V(ε, μ): V(ε, μ) mod = a(μ) b(μ) I B II B (17) Here, a(μ) and b(μ) are functions of the crosslink density μ, and I B and II B are invariants of the isochoric Cauchy-Green strain tensor B. 8 Experiments (not shown here) support the assumption that a(μ) and b(μ) can be approximated by linear functions of μ, 8 which would be in line with the data shown in Figure 2. Figure 3 shows the variation of the strain hardening G R with true strain rate. According to eq 8, strain hardening should STRAIN-HARDENING MODULUS OF PMMA, WENDLANDT, TERVOORT AND SUTER 1467

5 JOURNAL OF POLYMER SCIENCE: PART B: POLYMER PHYSICS DOI /POLB FIGURE 2 dv (ε, μ) 1 /dg(ε) at reference true strain ε = 0.7 as a function of the crosslink density μ of chemically crosslinked PMMA. The straight line is a linear fit to the data. depend linearly on the logarithmic true strain rate, which is in good agreement with erimental data. Following the approach of eq 16, the strain hardening modulus would be proportional to the yield stress. Experimental data of PMMA (this study) and other glassy polymers 14 at large strains do not support this assumption satisfactorily. Instead, they support a linear rather than a proportional dependence of strain hardening on the yield stress, as also predicted by the WTS model (see Fig. 4). Figure 5 shows the total stress response recorded for crosslinked PMMA at different strain rates, normalized by the yield stress, which deviate at large strains as predicted by the WTS model [see Fig. 5(b,c)]. Hence, also the condition for rewriting eq 9 into a purely multiplicative approach as in eq 13, i.e., V(ε, μ) 1 g(ε), is violated as visualized FIGURE 4 Strain hardening modulus as a function of the yield stress. Data were calculated at different temperatures and true strain rates using the WTS model with the fit parameters shown in Table 2. in Fig. (5d). Obviously, V(ε, μ) 1 is not proportional to g(ε), as dv(ε, μ) 1 /dg(ε) still shows a significant dependence on strain. In contrast, the proposed strain dependence of eq 17 is closer to the erimental data as dv(ε, μ) 1 /dv(ε, μ) 1 mod shows much less variation with strain [solid curve in Fig. (5d)]. The favorable comparison between erimental finite stress strain behavior and the WTS-model depicted in Figure 5, demonstrates that a decreasing activation volume with strain is able to accurately capture the strain hardening response in glassy PMMA. A decreasing activation volume indicates an increasing effective free-energy barrier for segmental mobility, for which it is difficult to provide a detailed molecular description. Nevertheless, the increasing stress that is needed for molecular mobility, implies that the size and complexity of plastic micro-events increase with strain, which has indeed been observed in molecular simulation studies. 24 Recently, the decrease in segmental mobility with strain during strainhardening was also directly observed erimentally for glassy PMMA, using an optical photobleaching technique. 29 A notable feature of the WTS model is the dependence of strain hardening on temperature. Assuming that V(ε, μ) is temperature invariant, it follows from eq 1 that the total stress at large deformations will depend on temperature as: σ(ε, ε, μ, T)/T = ( α ) 3 k B V(ε, μ) 1 T + β + ln ε + G nh(t) } {{ } viscous term g(ε) } T {{} neo-hookean term (18) Using eq 8, the temperature dependence of the strain hardening modulus G R is then given by: FIGURE 3 Strain hardening modulus G R as a function of logarithmic true strain rate for chemically crosslinked PMMA. Data points have been recorded at different crosslink densities as indicated in the graph. The straight lines are linear fits to the data. dg R dt = dv(ε, μ) 1 3 k B (β + ln ε) + dg nh(t) dg(ε) }{{}} dt {{} viscous term neo-hookean term (19) 1468 INTERSCIENCE.WILEY.COM/JOURNAL/JPOLB

6 ARTICLE FIGURE 5 True stress normalized by the yield stress. (a) Normalized erimental data of crosslinked PMMA with μ = 0.11/nm 3. (b) Least squares fit of the model of this study to the erimental data in (a). 8 (c) Comparison between model fit and eriment. Note that, the hatched areas of the graphs were not taken into account during fitting as the model describes the mechanical behavior of fully rejuvenated materials. 8 (d) Variation of dv (ε, μ) 1 /dg(ε) and dv (ε, μ) 1 /dv (ε, μ) 1 with true strain. mod Assuming a rubber-elastic neo-hookean network response, G nh should be positive proportional to the absolute temperature. Nevertheless, if the decrease of the viscous part of strain hardening with temperature is greater than the increase of the neo-hookean part in eq 19, the total strain hardening will still decrease with temperature. In contrast, assuming a constant activation volume would always result in an increasing strain hardening with temperature, which is in disagreement with 1 4, 31 numerous published erimental data. The simulations shown in Figure 6 at higher temperatures than 22 C have been conducted using the fit parameters for room temperature data (see Table 2), and selecting ΔH 0 = 173 kj/mol, a value that is in agreement with reported temperature dependence of the compressive yield stress of PMMA given by Bauwens-Crowet. 32 In addition, inspired by standard rubber elasticity, G nh was assumed to be positive proportional to the absolute temperature. As shown in Figure 6(a,c), the WTS model indeed predicts a realistic decrease of stress and strain hardening with increase in temperature as described in literature. Unfortunately, no erimental data of the investigated specimen was available for a quantitative validation. Note that, according to the WTS model, the deviation of stress strain curves recorded at different strain rates and normalized by the yield stress becomes more significant with increasing temperature as shown in Figure 6(b). Figure 6(d) shows in more detail the predicted ratio between the strain hardening and the yield stress σ y, which would have no strain-rate dependence if the strain hardening would scale proportionally to the yield stress. In Figure 7, the contributions to strain hardening are shown licitly for crosslinked and linear PMMA in the subfigures (a) and (b), respectively. Indeed, assuming that G nh (T) is positive proportional to T, leads to a decrease of the total strain hardening with increasing temperature, while the viscous contribution vanishes when approaching the glass transition temperature. However, the elastic neo-hookean contribution of the glassy apparent network does not decrease when approaching the glass transition zone, which is unrealistic as the total strain hardening modulus should transfer to the rubber elastic modulus of the crosslinked network, which is lower than G nh,abovet g. In Figure 7(b) erimental strain hardening moduli as a function of temperature for linear PMMA obtained by Kierkels et al. 30 are also shown. Gratifyingly, these values seem to match with the prediction of the total strain hardening of this study at temperatures far below T g. However, when approaching T g, the erimental values of Kierkels closely follow the viscous contribution of the presented model, suggesting that for linear PMMA, G nh should vanish upon approaching T g in contrast to a linear increase STRAIN-HARDENING MODULUS OF PMMA, WENDLANDT, TERVOORT AND SUTER 1469

7 JOURNAL OF POLYMER SCIENCE: PART B: POLYMER PHYSICS DOI /POLB FIGURE 6 Experimental and calculated strain hardening of crosslinked PMMA with μ = (a) Experimental and calculated total true stress at 22 C (red and black solid lines) and calculated total true stress at 50 C (dotted blue lines) using ΔH 0 = 173 kj/mol. (b) Calculated total true stress response at 22 C (solid black lines) and 50 C (dotted blue lines) normalized by the yield stress. (c) Calculated total yield stress normalized by temperature as a function of logarithmic strain rate. (d) Ratio of strain-hardening modulus and yield stress of calculated stress-strain curves at different true strain rates as a function of temperature. FIGURE 7 (a) Experimental (black symbols) and calculated (T > 22 C) strain hardening modulus G R at true strain rate of ε = as a function of temperature for cross-linked PMMA. The solid line at T > T g show the modulus of the crosslinked affine network above T g calculated from the crosslink density using a linear temperature dependence G nh = G aff = 2 μ k b T. (b) Experimental data from Kierkels (black symbols) 30 and the calculated strain hardening modulus G R at true strain rate of ε = 0.01 as a function of temperature for linear PMMA INTERSCIENCE.WILEY.COM/JOURNAL/JPOLB

8 ARTICLE with T as we have assumed for chemically crosslinked PMMA as shown in Figure 7(a). Another important aspect that has not been addressed in this publication is the three dimensional behavior of the constitutive model. In this contribution, constitutive modeling and erimental verification is limited to non-reversal uniaxial compression. However, the neo-hookean character of strain hardening has been verified erimentally also in shear, and uniaxial tension deformation, 12 which would suggest that an ression for V(ε) that would mimic three-dimensional neo-hookean behavior more accurately than eq 17 might be preferable. CONCLUSIONS The strain hardening modulus derived from a new threedimensional nonlinear viscoelastic model for the finite deformation behavior of glassy polymers, was compared to the erimental strain hardening response of crosslinked glassy PMMA. In the new constitutive model, strain hardening only has a minor elastic component and mainly results from a viscous stress contribution that depends in a multiplicative non additive manner on strain and strain rate, by means of a straindependent activation volume. 8 According to the WTS model, the activation volume, which measures how stress reduces the effective free-energy barrier for segment mobility, decreases with increasing strain. This suggests that the segmental mobility decreases in the strain-hardening regime as an increasing stress level is required to maintain stress-induced plastic flow. Confronting the model with erimental stress strain data for crosslinked PMMA, it was found that the elastic part of the strain hardening response was essentially neo-hookean with a strain rate and crosslink-density independent modulus G nh that is much smaller than the total strain hardening modulus. The major part of the strain hardening response was found to be of viscous origin, characterized by a strain and crosslink-density dependent activation volume for plastic flow. The new model agrees well with the erimentally found linear increase of strain hardening with crosslink density and strain rate and realistically predicts the well-known decrease of strain hardening with increase in temperature. A simpler form of the continuum model that omits the additive elastic neo-hookean contribution to strain hardening could not describe the finite deformation behavior correctly at large compressive strains. Only the introduction of the neo-hookean term resulted in good agreement with erimental largestrain data of glassy PMMA as a function of crosslink density. Clearly, using a strain hardening modulus G R to characterize the strain hardening response is to some extend misleading, as it suggests that the upswing in stress is fully elastic. Upon application of the WTS model to the strain hardening of PMMA, it is evident that only a small part of the stress could be identified as strain-rate independent. In the WTS model, most of the strain hardening has a viscous origin that gives rise to a neo-hookean like response in non reversal uniaxial compression. It is interesting to note that the erimentally observed development of segmental orientation in crosslinked PMMA 26 is in good agreement with the orientation that would be generated by deformation of a glassy apparent Gaussian network with modulus G nh, as described above. A Matlab package WTS for the implementation of the continuum model of Wendlandt, Tervoort, and Suter is available free of charge via the Internet at matlabcentral/fileexchange/. The authors are grateful to Leon Govaert for fruitful discussions and support and to Dirk Senden for bringing to our attention an error in our previous publication, leading to the new fit parameters in Table 3. 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