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Editorial Manager(tm) for Mechanics of Time-Dependent Materials Manuscript Draft Manuscript Number: Title: Uniaxial compression of a glassy epoxy resin: rate-free yield stress and volumetric instability Article Type: Original Research Keywords: Mechanical characterization; Strain measurement; Epoxy resins; Strength; Viscoelasticity; Viscoplasticity. Corresponding Author: Dr. Lorenzo Bardella, Ph.D. Corresponding Author's Institution: University of Brescia First Author: Lorenzo Bardella, Ph.D. Order of Authors: Lorenzo Bardella, Ph.D.; Andrea Belleri, Ph.D. Abstract: We report the results of uniaxial compressive tests on a DGEBA epoxy resin at room temperature, well below its glass transition. Within the strain rate range (1.E-6, 2.E-3) 1/s we confirm the linear trend relating the logarithm of the strain rate to the strength, as already observed by Mayr et al. (1998) [A.E. Mayr, W.D. Cook, and G.H. Edward. "Yielding behaviour in model epoxy thermosets - I. Effect of strain and composition". Polymer 39(16):3719-3724, 1998] for the same epoxy resin. Instead, for strain rates below 1.E-6 1/s, we found a negligible rate-dependence, as our data indicate a lower limit of the strength, of about 87 MPa, which may be recognised as the true, or rate-free yield stress for the epoxy under study. On the basis of these results, we propose how to extend the nonlinear viscoelastic model put forward in [L. Bardella. "A phenomenological constitutive law for the nonlinear viscoelastic behaviour of epoxy resins in the glassy state". Eur. J. Mech. A-Solid. 20(6):907-934, 2001] in order to account for the viscoplastic regime of deformation. Moreover, our measurements show a volumetric instability, allowed by the free lateral expansion, not ascribable to any macroscopic structural effect; such a behaviour has never been reported in the literature, to the best of our knowledge.

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Uniaxial compression of a glassy epoxy resin: rate-free yield stress and volumetric instability Lorenzo Bardellaa,∗ and Andrea Bellerib a

DICATA, University of Brescia Via Branze, 43 — 25123 Brescia, Italy b

Department of Design and Technology, University of Bergamo Viale Marconi, 5 — 24044 Dalmine (BG), Italy

Abstract We report the results of uniaxial compressive tests on a DGEBA epoxy resin at room temperature, well below its glass transition. Within the strain rate range 1.E-6÷2.E-3 s−1 we confirm the linear trend relating the logarithm of the strain rate to the strength, as already observed by Mayr et al. [23] for the same epoxy resin. Instead, for strain rates below 1.E-6 s−1 , we found a negligible rate-dependence, as our data indicate a lower limit of the strength, of about 87 MPa, which may be recognised as the true, or rate-free yield stress for the epoxy under study. On the basis of these results, we propose how to extend the nonlinear viscoelastic model put forward in [3] in order to account for the viscoplastic regime of deformation. Moreover, our measurements show a volumetric instability, allowed by the free lateral expansion, not ascribable to any macroscopic structural effect; such a behaviour has never been reported in the literature, to the best of our knowledge. Keywords: Mechanical characterization; Strain measurement; Epoxy resins; Strength; Viscoelasticity; Viscoplasticity.

1

Introduction

We present the results of uniaxial compressive tests carried out on a DGEBA (diglycidyl ether of bisphenol A) epoxy resin with the purpose of investigating some fundamental features of its viscoplastic regime of deformation at room temperature, well below its glass transition temperature (Tg ). This work aims at extending what has been done since [2] on the characterisation of the nonlinear viscoelastic behaviour of the same epoxy resin. Two main results, reported in next section 2, have been obtained. The first one is peculiar of the viscoplastic deformation range and regards the strain rate dependence of the strength. In fact, beside having observed, as already done by other investigators (e.g., Cherry and Thomson [8], Mayr et al. [23], and Iwamoto et al. [18]), a linear relation between the strength and the logarithm of the strain rate for a certain range of the latter, Corresponding author: [email protected] ∗

Tel.:

+39-030-3711238;

1

fax:

+39-030-3711213.

E-mail address:

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we have also found that for extremely low strain rates (lower than 1.E-6 s−1 ) the strength does not decrease below a certain value, which is here referred to as the rate-free strength, or rate-free yield stress, as strength and yield stress are close to each other for this polymer. This result is going to be crucial for the extension to the viscoplastic range of the nonlinear viscoelastic model proposed in [3] for this epoxy resin. The main idea of such an extension will be discussed in section 3 and developed in the near future. About this, let us just aniticipate that we wish to describe the viscoelastic and viscoplastic behaviour of thermoset polymers such as epoxy resins by keeping these two types of deformation distinguished. On the contrary, most of the literature available is concerned with models where the time-dependent deformations are accounted for all at once through just one (tensorial) variable governed by an endochronic rule. This rule may be based on micromechanical models such as those of Argon [1] or Eyring [12]. Regarding the modelling, beside the difficulty in distinguishing between viscoelastic and viscoplastic deformations, it is even less clear how to describe the different mechanical behaviours of different kind of polymers below their Tg , such as thermoplastics and thermosets. Some contribution in this direction (Haward and Thackray [15], Boyce et al. [7], and Wu and van der Giessen [35]) aims at unifying the constitutive modelling of such materials by assuming that any glassy polymer can exhibit large inelastic strains if it overcomes two physically distinct sources of resistance. In particular, first, the network must be stressed to allow molecular chains to rotate; then, after molecular alignment has occurred, another internal resistance to flowing arises, called orientational hardening, which produces the so-called locking, typical of rubbers, in the stress-strain curve. This picture gives an insight into a temperature-dependent material behaviour often prevented from by brittleness and, in some cases, far beyond the strain range of engineering interest. In fact, this is the case of the epoxy resin under study, for which the room temperature is far below the Tg , so that there is not enough ductility to enter a deformation regime where the marcomolecules are aligned. Instead, the focus of this research is a detailed characterisation of the time-dependent mechanical behaviour for deformations up to those corresponding to the material strength (i.e., the first yield), at about 10% of total strain. The second new main result obtained is related to the viscoelastic range. At stress values 10÷20% lower than the strength, we have systematically observed a transversal strain increase in magnitude in such a way as to make the incremental bulk modulus negative. For the reasons explained in subsection 2.3, this volumetric instability, un reported in the literature to the best of our knowledge, does not seem to be related to any barrelling or structural effect, while seems to be an intrinsic feature of this polymer and, perhaps, related to some sort of micro-instability of the chemical network.

2 2.1

Uniaxial compressive tests Specimen preparation

The epoxy resin tested has been uniformly produced by stochiometrically mixing the resin (DGEBA DER 332 produced by Dow Chemical) and the hardener (DDM 32950 produced by Fluka) in the ratio of 3.51:1 by weight and by allowing the system to cure for 24 hours into a oven kept under vacuum at 60◦ C, thus obtaining a glass transition temperature Tg ≈ 170◦ C, very close to the stabilised Tg for this thermoset, usually below 180◦ C. The 2

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density of the cured system is about 1.16 g/cm3 . The effect of a post-curing of 24 hours at 180◦ C has been shown to be unimportant for the epoxy mechanical properties [2]. In most cases, the tests were performed on cylindrical samples of height h = 75 mm and diameter φ = 30 mm. When necessary, as specified later in the paper, we employed either longer cylindrical samples of height h = 100 mm or dog bone specimens. The geometry of the specimens is reported in figure 1. Each sample was instrumented with four strain gauges of the type 1-LD20-6/120, produced by Hottinger Baldwin Messtechnik, which can accurately measure direct deformations up to 10% either in tension or in compression and have measuring grid of dimensions 6 mm × 2.8 mm, nominal resistance equal to 120 Ω, and a gauge factor of about 2.10 (the precise value of this last datum is dependent upon the batch). All the strain gauges were applied to the specimens by means of the one-component adhesive Z70, produced by Hottinger Baldwin Messtechnik, as well. The four strain gauges were applied at half height of the samples in such a way that both longitudinal and transversal strains were recorded twice, each strain measured on two diametrally opposite points of the specimens. This allows us to check the presence of unwanted bending due to imperfectly centered loading, and, if this is the case, to mitigate its effect (or to eliminate it within the linear elastic range) by averaging the two analogous measures. The longitudinal Cauchy stress σ is computed by means of the relation: σ=

4P + εt )2

πφ2 (1

(1)

in which P is the axial force and εt is the transversal strain. All the tests were conducted at room temperature (≈ 23◦ C, anyway far below the Tg ) and, in most cases, by prescribing a crosshead constant displacement rate by means of a servo-hydraulic Instron testing machine (model 1274) at the Laboratory for Tests on Materials Pietro Pisa of the Faculty of Engineering of Brescia. Thin sheets of teflon were inserted between the loading platens and the specimen bases, in such a way as to minimise the friction and reduce the barrelling as much as possible.

2.2

Longitudinal stress-strain behaviour

By means of some of the tests reported henceforth and many others carried out also to investigate the viscoelastic behaviour, the elastic constants were computed in [2] by linear regression on the set of data referred to the longitudinal strain ranging from 0 to ±0.004, in such a way as to minimise viscous effects. The new tests performed within this research have confirmed the elastic constant values of about 2800 MPa for the Young modulus and 0.41 for the Poisson ratio. Behaviour at yield Figures 2–4 compare the loading part of most of the stress-strain curves obtained with to investigate the strength dependence on the strain rate, the latter computed by dividing the prescribed crosshead speed of the testing machine to the initial specimen height. We have investigated the material behaviour for strain rates ranging from 1.6667E-7 s−1 to 1.6667E-3 s−1 . Only a few of these tests were conducted to failure, as most samples were unloaded to measure the residual deformation. In figure 2 we have 3

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Figure 1: Geometry of the specimens employed in most of the tests.

collected the results of a significant part of these type of tests carried out on the epoxy resin DGEBA DER 332 at the Pietro Pisa laboratory since [2]; while this figure has just the purpose of giving an idea of the strain rate effect, figure 3 provides the detail about the instant in which the stress reaches the strength. In figure 3, the stress-strain curves whose legend description contains “from disp” actually corresponds to tests in which the strain gauges data were not available for the whole test, so that we have computed the strain by dividing the prescribed shortening displacement to the specimen height, whereas the stress was computed from equation (1) by assuming a constant lateral contraction ratio −εt (t)/εl (t) = 0.41, where εl is the longitudinal strain; 1 it is straightforward to verify that this assumption has a negligible influence on the stress values. Also, of course, for such tests we have checked that the recorded strain data was sufficiently close to that computed from the displacement. The aging and postcuring (also indicated in the legend of figure 3) do not seem to significantly affect the material behaviour, as the stress-strain curves are almost unchanged whether the samples underwent these processes or not. In figure 4 we have selected the results of the tests in which the loading modality turned out to be very close to the ideal uniaxial case and all the measurements were not affected by any problem; in other words, this is the case of those tests in which for each couple of 1

Because of our uniaxial test modalities, consisting in imposing a constant strain rate, we can in fact compute the time-dependent lateral contraction ratio, that is the negative of the ratio between the measured transversal and longitudinal strains, −εt (t)/εl (t). By choosing this terminology, we cling to the view of Tschoegl et al. [32], who reserve the term time-dependent Poisson’s ratio for −εt (t)/εl (t) in a uniaxial relaxation test. It is worth noting that because of the nonlinearity, we cannot use the standard formulæ based on the correspondence principle (see, e.g., Tschoegl et al. [32], Lakes and Wineman [19], and Hilton [16] ) in order to obtain the time-dependent Poisson’s ratio from our measurments of −εt (t)/εl (t). Also, let us add that some investigators suggest different definitions of time-dependent Poisson’s ratio (see, e.g., Hilton [16]).

4

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analogous strain gauges we measured about the same deformation on each strain gauge. Henceforth, we will refer to such tests as to the “most accurate data”; we note that one of these tests, that conducted at the slowest rate (1.6667E-7 s−1 ), shows an anomalous non-monotonic behaviour after the strength is reached. Figure 5 reports the measured strength as a function of the strain rate, the latter in a logarithmic scale. On the basis of our previous results [2] (showing no irreversible deformation in cyclic tests with peak stress quite close to the strength), we adopt the usual convention by which the strength is assumed to coincide with the yield stress (see also Liang and Liechti [22]). Our main new result is the yield stress plateaux value ≈ 87 MPa reached as the strain rate decreases. This suggests that such a value is the true, or rate-free yield stress, while higher values obtained at higher strain rates are apparent yield stress values, affected by the viscosity of the resin. Also, we observe a linear trend that can be extrapolated between log ε˙l and the strength within the range ε˙l ∈ (1.E-6 s−1 , 2.E-3 s−1 ): by linear regression and considering only the “most accurate data” we evaluate a slope equal to 7.01 MPa/log[s−1 ] if referred to the Cauchy stress, or equal to 8.39 MPa/log[s−1 ] if referred to the nominal stress. The former slope is that of the fitting indicated in figure 5. Instead, by considering all the measured strength values within the range ε˙l ∈ (1.E-6 s−1 , 2.E-3 s−1 ), the slope, referred to the Cauchy stress, becomes 7.24 MPa/log[s−1 ]. A similar linear relation has also been observed by other researchers. Mayr et al. [23] found, for the same epoxy resin as that concerned here, an approximately linear behaviour with slope 8.25 MPa/log[s−1 ] within the range ε˙l ∈ (1.67E-5 s−1 , 1.67 s−1 ), by compressing cylindrical specimens of height equal to 16 mm and diameter equal to 10 mm and computing the nominal stress. Regarding different epoxy resins, Cherry and Thomson [8] were among the first investigators who found a linear relation between the strength and log ε˙l , in their case within the range ε˙l ∈ (1.E-4 s−1 , 1.E-2 s−1 ) and for five different epoxy resins. Recently, also Iwamoto et al. [18] confirmed that linear relation for two epoxy resins within the range ε˙l ∈ (1.E-4 s−1 , 1. s−1 ) (with slopes of about 5 and 9 MPa/log[s−1 ]), while they found a largely higher-than-linear behaviour for extremely fast loading, with log ε˙l > 1.E+2; the results of Iwamoto et al. are based on the compression of very short cylindrical specimens, of height equal to 7 mm and diameter equal to 14 mm. On the viscoelastic effect on the stress In order to give a further insight on the ground of the definition of the rate-free yield stress, we have carried out an extremely slow cyclic compressive test on the dog bone specimen of figure 1, at a strain rate equal to 1.0741E-7 s−1 . By assuming that the area included between different loading and unloading paths of the stress-strain curve be proportional to the viscosity, the idea was to check whether at such a low strain rate the time-dependence disappears. As shown in figure 6, the answer is negative, even though the viscosity observed in the stress-strain curve obtained from the extremely slow cyclic test (continuous curve) is quite small with respect to that observed in a much faster cyclic test (dashed plot), the latter being performed until a stress value quite close to the strength, in order to better highlight the viscous behaviour before plasticity. Also, let us observe that the cyclic test might be a quite severe test for our purpose of finding a rate at which viscosity becomes totally negligible; in fact, some investigators (e.g., Xia et al. [36]) believe that the loading inversion somehow regenerate 5

the viscosity.

Longitudinal stress [MPa]

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-115 -110 -105 -100 -95 -90 -85 -80 -75 -70 -65 -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0

0

-0.01

-0.02

-0.03

-0.04

-0.05

-0.06

-0.07

-0.08

-0.09

-0.1

Longitudinal strain

Figure 2: Uniaxial compressive tests: stress-strain curves obtained by varying the applied longitudinal strain rate from 1.6667E-7 s−1 to 1.6667E-3 s−1 .

6

Longitudinal Cauchy stress [MPa]

-110

Longitudinal Cauchy stress [MPa]

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-105

v = 1.6667E-7 s

-1

v = 2.7728E-6 s

-1

v = 2.7407E-7 s (from disp) -100

v = 4.1853E-7 s v = 1.1519E-6 s

v = 6.4815E-6 s

-1

v = 1.7741E-5 s

-1

v = 3.3333E-5 s

-1 -1 -1 -1

-95 -90 -85 -80

-105 -100 -95 -90 v = 7.5778E-5 s -85

v = 1.6667E-4 s

-1

v = 8.6296E-4 s

-1

-1

-1

v = 1.6667E-4 s (aged) -80

-1

v = 1.6667E-3 s (aged, postcured, from disp)

-1

v = 1.6667E-4 s (aged)

-1

v = 1.6667E-3 s (aged, postcured, from disp) v = 1.6667E-3 s

-1

-1

v = 4.4444E-4 s -75 -0.05 -0.055 -0.06 -0.065 -0.07 -0.075 -0.08 -0.085 -0.09 -0.095

Longitudinal strain

-0.055 -0.06 -0.065 -0.07 -0.075 -0.08 -0.085 -0.09 -0.095 -0.1

Longitudinal strain

Figure 3: Uniaxial compressive tests: stress-strain curves at various strain rates. Detail of the plots of previous figure 2.

7

Longitudinal Cauchy stress [MPa]

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-112 -108 -104 -100 -96 -92 -88 -84 -80 -76 -72 -68 -64 -60 -56 -52 -48 -44 -40 -36 -32 -28 -24 -20 -16 -12 -8 -4 0

-1

v = 1.6667E-7 s

-1

v = 1.1519E-6 s

-1

v = 6.4815E-6 s

-1

v = 1.7741E-5 s

-1

v = 8.6296E-4 s

-1

v = 1.6667E-3 s

0

-0.01

-0.02

-0.03

-0.04

-0.05

-0.06

-0.07

-0.08

-0.09

-0.1

Longitudinal strain

Figure 4: Uniaxial compressive tests: stress-strain curves at various strain rates, including only the “most accurate data”.

8

-112 -110 -108

Most accurate data Fitting

-106 -104 Strength [MPa]

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-102 -100 -98 -96 -94 -92 -90 -88 -86 1e-07

0.0001 1e-06 1e-05 -1 Strain rate computed from the imposed shortening rate [s ]

0.001

Figure 5: Uniaxial compressive tests: yield stress as a function of the strain rate.

9

Longitudinal Cauchy stress [MPa]

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-95 -90 -85 -80 -75 -70 -65 -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0

-1

v = 1.0741E-7 s

-1

v = 1.6667E-3 s

0

-0.005 -0.01 -0.015 -0.02 -0.025 -0.03 -0.035 -0.04 -0.045 -0.05 -0.055 -0.06 Longitudinal strain

Figure 6: Uniaxial compressive cyclic tests at two very different strain rates.

10

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2.3

Volumetric instability

In our compressive tests we have systematically observed a volumetric instability occurring before the strength is reached, at stress values 10%÷20% lower than the strength. More specifically, around that stress level, we macroscopically observe a quite large increase in the transversal strain rate ε˙t with respect to the imposed longitudinal strain rate ε˙l . This leads to a positive increase of the volumetric strain rate (ε˙l + 2ε˙t )/3 (expansion) in spite of the applied compression (with pressure equal to σ/3), which corresponds to a negative incremental bulk stiffness. The first time this occurs is taken here as the starting point of a volumetric instability. In figure 7, such an instant is clearly shown for each test by means of a hollow circle. We observe from figure 7 that before instability occurs the bulk behaviour is almost linear until a stress value quite close to that corresponding to volumetric instability, in most cases significantly lower than the strength. Note that the quite large variation of εt due to volumetric instability is numerically unimportant in the Cauchy stress computation (see equation (1)). Figure 8 shows a typical picture taken with the Scanning Electron Microscope (SEM) on a polished cross section of a tested specimen. The visible wrinkles are exclusively due to the SEM preparation process of the specimen; this leaves out the possibility of motivating the volumetric instability with the nucleation of microcracks with faces parallel to the loading axis (a sort of diffused crazing allowed by the free lateral expansion, albeit the hydrostatic stress is compressive). Hence, from the SEM analysis we could not find any indication on the origin of the volumetric instability, which might be ascribed to some peculiar stochastic behaviour of the chemical network, perhaps possible after an appropriate stress level allows the macromolecules to overcome an energy barrier and change conformation. Also, we think this phenomenon is not a barrelling mode of instability because (i) we have minimised the friction at the sample bases by means of sheets of teflon, (ii) nothing close to barrelling is visible at naked eye during the tests, (iii) it always occurs, also for the slowest tests, in which the tangential stress due to friction, if any, should relax, so that they should hardly promote the barrelling, (iv) the slenderness of the samples employed should be large enough to avoid barrelling, as analytically found by Del Piero and Rizzoni [10] for hyperelastic materials. Finally, note that the curves of figures 2–4 do not give any advise of the instability occurrence. The same observation holds in analysing the force-displacement curves, reported in figure 9 for what concerns the “most accurate data”. This last figure will be further commented later, while now we wish to try to get more insight on the volumetric instability. 2.3.1

Analysis of the imperfectness of loading uniaxiality

In order to analyse the tests, we define an overall measure called the loading eccentricity e as: !$ % $ (1) % " (1) (2) 2 " ε − ε(2) 2 ε − ε t t l + (2) e = 2# l(1) (2) (1) (2) εl + εl εt + εt

11

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(i)

where εα (α = l, t i = 1, 2) refers to the i strain gauge applied in such a way as to measure a deformation α. Of course, e = 0 corresponds to the ideal uniaxial stress state, while e %= 0 accounts for many sources of imperfection, such as the actual loading eccentricity, the imperfect normality to the loading axis and planarity of the sample bases, and so on. Figure 10 shows, for the “most accurate data”, the loading eccentricity as a function of the applied shortening displacement. As already noticed, for these tests the difference in the deformation measures recorded by the strain gauges is quite limited. Here, it is also clear that such a difference remains small at incipient instability (indicated by the hollow circles in figure 10). The difference increases at high stresses because of the nonlinear behaviour (while at the beginning of each test the eccentricity given by equation (2) is obviously an inaccurate measure of the closeness to ideal uniaxiality of the loading modality). For what concerns the volumetric instability, the most interesting aspect is that, because of the eccentricity, in general, one obtains quite different curves of the type plotted in figure 7 depending on the strain gauges measures used in order to compute the volumetric strain θ. So far we have always employed the averages of analogous strain gauges (i.e., θ = εl + 2εt , but things may change if one evaluates the volumetric strain by just choosing one of the two strain gauges for the longitudinal and transversal strains (so that, (1) (1) (1) (2) other four estimates of the volumetric strain are possible: θ = εl + 2εt , θ = εl + 2εt , (2) (1) (2) (2) θ = εl + 2εt , θ = εl + 2εt .) In some tests, also among the “most accurate data”, this procedure may in fact lead to a volumetric strain-pressure curve where at a pressure level similar to that leading to instability in terms of average strains one does not see a maximum anymore, as the volumetric strain keeps increasing in modulus, even more than linearly. This is an indication of the difficulty of measuring and analysing this behaviour and, of course, it is due to the eccentricity. However, the test which has the smallest eccentricity at incipient instability (i.e., the magenta curve in figure 10, related to a strain rate equal to 8.6296E-4 s−1 ) does not exhibit this phenomenon, and all the volumetric strain-pressure curves are qualitatively similar, as shown in figure 11. This seems an indication of the fact that the volumetric instability has a micromechanical nature, probably related to the behaviour of the chemical network, and it should not be due to a structural macroscopic phenomenon. 2.3.2

Would it be possible to define a macroscopic parameter driving the instability?

In order to analyse the results, we find it useful to define the deviatoric-volumetric ratio ψ=

εl − εt εl + 2εt

(3)

In figures 12 and 13 we have plotted ψ as a function of the transversal strain εt and the applied stress σ, respectively, together with the instability points determined in figure 7. Also, in figures 12 and 13, as a reference, we have plotted, in a thick dashed curve, the constant value that ψ would assume in linear elasticity, i.e., ψlin ela =

1+ν , 1 − 2ν

12

(4)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

with ν the Poisson ratio; by taking ν = 0.41 [2], one obtains ψlin ela = 7.83. The fact that all the curves drift away from ψlin ela since the very beginning of the tests is due to the (nonlinear) viscoelasticity which characterises the deviatoric behaviour (as the volumetric behaviour is almost linear for most of the test — see figure 7). Neither εt nor σ are good macroscopic variables to describe the instability occurrence, even though the latter seems to be better since its values at incipient instability turn out to be less scattered than the corresponding values of εt . The main problem is that the values of σ for different tests do not exhibit any reasonable trend with respect to the loading rate. We notice that establishing which state variable drives the onset of instability is made more difficult by its probable dependence on the loading rate. 2.3.3

Further study of the interaction between volumetric instability and viscoelasticity

In order to get more insight on the volumetric instability we have carried out a further compressive test on a longer specimen, of height equal to 100 mm instead of 75 mm, so that barrelling is even less likely to occur. The sample was compressed at a displacement rate equal to 0.2 mm/min, corresponding to a strain rate of 3.3333E-5 s−1 . Then, after the onset of instability, we have changed the test modality: first we have blocked the displacement and let the sample relax (in fact, the stress level reached at the end of the first loading ramp was quite high, about 82 MPa), then, after the stress decreased to the value of about 76 MPa, we have imposed a constant force and let the sample creep. The three stages of loading are represented in figure 14, while figure 15 shows the volumetric strain as a function of the time. From this figure it is evicted that neither in the relaxation stage nor in the creep stage the volumetric strain reverts the tendency acquired after instability. In particular, it is noticeable that in the creep stage the increase of longitudinal strain continues to be less than double than that of the transversal strain. Hence, the volumetric instability is related to the nonlinear viscoelastic behaviour. Finally we note that, even though the volumetric strain keeps decreasing its magnitude, the creep stage does not seem to lead to a sudden failure.

13

Pressure [MPa]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

-36 -34 -32 -30 -28 -26 -24 -22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0

-1

v = 1.6667E-7 s

-1

v = 1.1519E-6 s

-1

v = 6.4815E-6 s

-1

v = 1.7741E-5 s

-1

v = 8.6296E-4 s

-1

v = 1.6667E-3 s

-8.0e-04

-1.6e-03

-2.4e-03 -3.2e-03 Volumetric strain

-4.0e-03

-4.8e-03

-5.6e-03

Figure 7: Uniaxial compressive tests: pressure σ/3 as a function of the volumetric strain εl + 2εt .

14

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Figure 8: Uniaxial compressive tests: SEM picture on a polished cross section of the specimen tested at strain rate equal to 1.7741E-05 s−1 .

15

Force [kN]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

-88 -84 -80 -76 -72 -68 -64 -60 -56 -52 -48 -44 -40 -36 -32 -28 -24 -20 -16 -12 -8 -4 0

-1

v = 1.6667E-7 s

-1

v = 1.1519E-6 s

-1

v = 6.4815E-6 s

-1

v = 1.7741E-5 s

-1

v = 8.6296E-4 s

-1

v = 1.6667E-3 s

0

-1

-2

-3 -4 -5 -6 Applied shortening displacement [mm]

-7

-8

-9

Figure 9: Uniaxial compressive tests: force-displacement curves of the tests with “most accurate data”.

16

0.26 -1

0.24

v = 1.6667E-7 s

0.22

v = 6.4815E-6 s

-1

v = 1.1519E-6 s

-1 -1

v = 1.7741E-5 s

0.2

-1

v = 8.6296E-4 s

-1

v = 1.6667E-3 s

0.18 Loading eccentricity

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 -1

-2

-3 -4 -5 Applied shortening displacement [mm]

-6

-7

Figure 10: Uniaxial compressive tests: the loading eccentricity as a function of the applied displacement.

17

Pressure [MPa]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

-36 -34 -32 -30 -28 -26 -24 -22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0

average measures

-8.0e-04

-1.6e-03

-2.4e-03 -3.2e-03 Volumetric strain

-4.0e-03

-4.8e-03

Figure 11: Uniaxial compressive test at strain rate equal to 8.6296E-4 s−1 : pressure as a function of volumetric strain computed by combining the recorded measures in different ways.

18

Deviatoric-volumetric ratio

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0

-1

v = 1.6667E-7 s

-1

v = 1.1519E-6 s

-1

v = 6.4815E-6 s

-1

v = 1.7741E-5 s

-1

v = 8.6296E-4 s

-1

v = 1.6667E-3 s

0

0.004

0.008

0.012

0.016

0.02 0.024 0.028 Transversal strain

0.032

0.036

0.04

0.044

0.048

Figure 12: Uniaxial compressive tests: the deviatoric-volumetric ratio (εl − εt )/(εl + 2εt ) as a function of the transversal strain.

19

Deviatoric-volumetric ratio

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0

-1

v = 1.6667E-7 s

-1

v = 1.1519E-6 s

-1

v = 6.4815E-6 s

-1

v = 1.7741E-5 s

-1

v = 8.6296E-4 s

-1

v = 1.6667E-3 s

0

-10

-20

-30

-40 -50 -60 -70 Longitudinal stress [MPa]

-80

-90

-100

-110

Figure 13: Uniaxial compressive tests: the deviatoric-volumetric ratio (εl − εt )/(εl + 2εt ) as a function of the stress.

20

-64 -60 -56 -52 -48 -44 -40 Force [kN]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

-36 -32 -28 -24 -20 -16 -12 -8 -4 0

0

1000

2000

3000 4000 Time [s]

0 -0.5 -1 -1.5 -2 -2.5 -3 -3.5 -4 -4.5 -5 -5.5 Displacement [mm]

5000

Figure 14: Uniaxial compressive test, at strain rate of 3.3333E-5 s−1 within the first loading ramp: compressive force as a function of both the time and the displacement.

2.4

Residual strains after unloading

For all the “most accurate data” tests and a few others, we have also measured the residual deformation after unloading. For instance, in figure 9 we have also reported the data regarding the unloading, always conducted at a displacement rate of 0.75 mm/min, that is a strain rate of 1.6667E-4 s−1 , assuming that the sample remains in contact with the moving platen. The relevant results are reported in figure 16, by which it is evicted that (i) after 5 years the longitudinal strain is still recovering in the specimens which experienced a faster loading and (ii) the amount of residual longitudinal strain is inversely proportional to the loading rate. This second observation gives ground to the modelling by which the lower the rate of deformation the higher the amount of (visco)plasticity developed. For what concerns the transversal strain, we observed an almost complete recovery, which means that the transversal plastic deformation is almost negligible and it makes it questionable the usual assumption of traceless plastic strain.

3

A major open issue: the development of a viscoelasticviscoplastic model

The results of the tests described in the previous section, together with others formerly obtained in [2], will be used in the near future in order to develop a phenomenological model for the viscoelastic and viscoplastic mechanical behaviour of epoxy resins in the glassy state. The goal is to obtain a model in which the viscoelastic and viscoplastic 21

-3.6e-03 -3.3e-03 -3.0e-03 -2.7e-03 -2.4e-03 Volumetric strain

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

-2.1e-03 -1.8e-03 -1.5e-03 -1.2e-03 -9.0e-04 -6.0e-04 -3.0e-04 0

1000

2000

3000 Time [s]

4000

5000

6000

Figure 15: Uniaxial compressive test at 3.3333E-5 s−1 : volumetric strain as a function of the time.

22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

deformation regimes are distinguished, the latter being activated on the basis of an appropriate criterion. Since we consider the mechanical behaviour of epoxy resins far below their Tg , attention will be restricted to small strains and rotations. Here, we wish to sketch the main idea, which consists in properly extending the model developed in [3], a phenomenological law capable to describe some of the main features of the pre-yield nonlinear viscoelastic behaviour of epoxy resins.

3.1

Summary of the capability of the viscoelastic model proposed in [3]

The phenomenological constitutive law is based on the rheological model depicted in figure 17 (hereafter labeled with the acronym SNS, for Standard Nonlinear Solid), formerly exploited by Haward and Thackray [15] in order to describe the uniaxial behaviour of thermoplastic polymers. In our case, the elements consist of a linear elastic spring connected in series with a nonlinear spring in parallel with a dashpot, based on the Eyring equation [12] and extended to account for triaxial stress states. This model governs the deviatoric part of the constitutive law, while the volumetric behaviour is assumed to be linear elastic. It has been shown that this model can reasonably describe creep and cyclic compressive tests on the epoxy resin DGEBA here under study [3]. In particular, the proposed model is able to predict the flex characterising the unloading part of stress–strain curves in uniaxial cyclic tests (see, e.g., figure 6), until stress values quite close to the strength, at which yield starts. The main lack of the model seems to be its tendency to anticipate the plateaux strain value of creep tests. Good agreement with experimental data has also been found for the viscoelastic cyclic behaviour and the brittle fracture within the viscoelastic range of syntactic foams (i.e., particulate composites whose filler consists of glassy hollow microspheres), described by means of finite element micromechanical models involving the constitutive law here concerned to describe the triaxial stress state of the epoxy matrix [3, 5]. The model capability is dependent both on how the Eyring dashpot has been extended for triaxial stress states and on the adopted SNS rheological model. The physical picture relies on the belief that the instantaneous linear elastic behaviour is determined by weak van der Waals intermolecular forces, while the viscous deformation involve the strong bonds of the chemical network (Oleinik [24]). This is incorporated by the rheological model of figure 17, while other equally complex rheological models, as the so-called “PoyntingThomson” (where a Maxwell rheological model is put in parallel with a spring), do not cling to this view and, in fact, have been shown to be unable to describe the epoxy nonlinear viscoelasticity (see, e.g., [4]). 2 The physical picture behind the choice of the SNS rheological model has been confirmed by the identification of its material parameters [25]. First of all, the initial shear stiffness of the nonlinear spring turned out to be equal to 7042 MPa, much higher than the linear elastic shear modulus, equal to 1004 MPa. Second, one of the parameters characterising 3 the Eyring dashpot, the so-called activation volume, was identified to be 1536 ˚ A , almost exactly equal to the value one obtains by resorting to its physical meaning. In fact, Mayr 3 et al. [23], for the same epoxy, calculated an activation volume equal to 1500 ˚ A based on 2

The phenomenological constitutive laws based on the Poynting-Thomson rheological model, or, in general, on many Maxwell-like models connected in parallel, are indeed quite popular, also because their easy implementation in standard (compatible) finite element codes.

23

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

the DGEBA chemical structure. Also, Tcharkhtchi et al. [31] found an activation volume 3 v = 1517 ˚ A for an epoxy DGEBA cured with norbornene anhydride. 3 Finally, in order to appreciate the capability of the nonlinear viscoelastic model developed in [3], let us add that it can be shown that this model can qualitatively describe some peculiar responses of glassy polymers, whose description has been proposed by some investigators as a test in order to select good models. Two of these peculiar responses are (i) the fast recovery after creep, by which after some creep the unloading leads to an instantaneous recovery larger than the initial instantaneous elastic deformation, and (ii) the multiaxial relaxation test (see, e.g, Sweeney and Ward [30]). More details on this will be given in a forthcoming paper.

3.2

Other proposals, with reference to the Eyring law

One of the most exploited theory to deal with nonlinear viscoelasticity is the Schapery integral approach [26]. Schapery [27] has provided a thermodynamically consistent link between his integral approach and the differential approach. Unfortunately, the connection with the differential approach exploited within this research, involving nonlinear elements such as the Eyring dashpot, would require the introduction of heavy assumptions. Ward and coauthors (see, for instance, Sweeney and Ward [30], and references therein) developed the so-called “two-process (Leeds) model”, a uniaxial law based on a rheological model consisting of two Maxwell-like models connected in parallel, involving two linear elastic spring and two Eyring-like dashpots. This constitutive law has been used in order to predict viscous deformations (without distinction between elastic and plastic regimes) for polymers like polyethylene subjected to creep, relaxation, and multistep recovery. One of the main differences between the model proposed in [3] and many others available in the literature exploiting the Eyring law consists of the different use of the Eyring dashpot: in [3] it is employed to describe viscoelastic deformations, while usually it is used to describe the evolution of viscoplastic deformations (see, e.g., Haward and Thackray [15] and Hasan and Boyce [14]). Let us specify that the term viscoplastic is in this last context used to mean a theory in which plastic strains are constantly accumulated. As pointed out in Bardenhagen et al. [6], the term viscoplastic should be more appropriately used, as done henceforth, for models in which plastic strains develop only depending on a yield criterion, at a different deformation regime from that ruling the viscoelastic deformations. Furthermore, in the theories characterised by two distinct deformation regimes, the material parameters governing the yield criterion can be dependent on the strain rate, which is the case of the so-called rate-dependent plasticity. In this context, Ward [33], Lesser and Kody [21], Mayr et al. [23], and Cook et al. [9], among other authors, have proposed to correlate the yield stress to the applied strain rate through an Eyring equation, in some cases (for instance, Ward [33] and Lesser and Kody [21]) extended to account for the pressure. In the following, we wish to propose an alternative way to describe the rate dependence of the apparent yield stress. 3

Let us stress the fact that such an encouraging confirmation regarding the optimum values of the material parameters is lacking if one bases the modelling on the equally complex Poynting-Thomson rheological model [4].

24

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

3.3

Possible extensions of the model [3] to the viscoplastic regime

Pressure dependence The need to account for the pressure dependence of the inelastic part of the mechanical behaviour of polymers has been extensively documented in the literature. As examples for what concerns epoxy resins, Wronsky and Pick [34] proposed pyramidal yield criteria, while Hu et al. [17] have shown that a compressive hydrostatic stress component increases the stiffness and the strength of the epoxy resin EPON 828. As quoted in Argon [1], Sternstein and Ongchin [28] were the first ones to show that (generic) polymers obey a pressure dependent von Mises yield criterion. Many attempts are reported in the literature which aim at modifying the von Mises criterion in such a way as to properly account for the hydrostatic stress component (see, e.g., Swadener and Liechti [29], Ward [33], Lesser and Kody [21], Duckett et al. [11], and Frank and Brockman [13]). In the viscoelastic range, the dependence on the first stress invariant is also expected to help in explaining the different behaviour in uniaxial tension and compression of epoxy resins. In order to account for pressure dependence of inelastic deformations, we will extend the model developed in [3] by first of all introducing a pressure dependence into the Eyring equation, as, for instance, propose in Boyce et al. [7] among other authors, i.e., by assuming that the material parameter called the energy barrier can be linearly related to the pressure through a further material constant which plays the role of a volumetric activation volume. Description of viscoplastic deformations We propose to extend our model as represented in the rheological scheme of figure 18. This is motivated by the belief that plasticity, when it occurs, it should affect the behaviour of the main backbone chains, modelled by the nonlinear spring. We may adopt for the friction element just an inviscid plasticity criterion, as this seems to be enough for this rheological model to qualitatively predict the yielding behaviour of epoxy resins, because the time and pressure dependences are “indirectly” provided by the dashpot. As a reinterpretation of the so-called “overstress models”, let us observe that this model predicts an apparent yield stress, increasing with the strain rate, always higher than (or, for extremely slow prescribed monotonic strain, equal to) a true, or rate-free yield stress, i.e., that characterising the time-independent friction element. Also, accordingly to what observed in our tests (see, e.g., subsection 2.4) the proposed extension is able to qualitatively model the fact that the lower the imposed strain rate, the higher the plastic content of the total deformation. On the contrary, in the literature, usually, the apparent yield stress of epoxy resins is described through rate-dependent criteria. For instance, Duckett et al. [11], Lee [20], Lesser and Kody [21], Cook et al. [9], and Mayr et al. [23] employ the Argon, Eyring, Robertson, Bowden, and Ward theories in this sense.

4

Concluding remarks

Uniaxial compressive tests on an epoxy resin DGEBA in the glassy state have allowed us to establish an important material parameter that we have called the rate-free strength, 25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

and which corresponds to the minimum compressive strength observable, as it is affected by the least, almost negligible, amount of viscosity. Since for the tested epoxy resin, as for many other polymers, the strength is almost coincident with the yield stress, the minimum compressive strength observable is also the rate-free yield stress. We would find it worth to perform experimental tests involving multiaxial stress states, at very low strain rates, in such as way as to establish a rate-free yield criterion, which could be plugged, as put forward in section 3, into the nonlinear viscoelastic model proposed in [3] in order to extend it to account for viscoplastic deformations. This would constitute a different approach from that usually followed which attempts to identify yield criteria on the basis of experimental results without taking care of the effect of the viscosity (see, for instance, the work of Wronski and Pick [34], where, in uniaxial compression, the imposed strain rate is equal to 1.125E-04 s−1 , quite high with respect to the value of about 1.E-06 s−1 below which we could measure the rate-free yield stress). We have also observed a volumetric instability, occurring within the viscoelastic deformation regime, which seems to be triggered by some micromechanical phenomena peculiar of the epoxy chemical network. Further studies are needed in order to clarify and describe this behaviour.

Acknowledgments Work done within a research project financed by the Italian Government (Ministero dell’Istruzione, dell’Universit`a e della Ricerca, MIUR). The authors wish to thank Mr. Alessandro Rivetti for the assistance with the Scanning Electron Microscope.

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[8] B. W. Cherry and K. W. Thomson. The fracture of highly crosslinked polymers. Part 1: Characterization and fracture toughness. J. Mater. Sci., 16:1913–1924, 1981. [9] W. D. Cook, A. E. Mayr, and G. H. Edward. Yielding behaviour in model epoxy thermosets — II. Temperature dependence. Polymer, 39(16):3725–3733, 1998. [10] G. Del Piero and R. Rizzoni. Weak local minimizers in finite elasticity. J. Elast., 93:203–244, 2008. [11] R. A. Duckett, A. Rabinowitz, and I. M. Ward. The strain-rate, temperature and pressure dependence of yield of isotropic poly(methylmethacrylate) and poly(ethylene terephthalate). J. Mater. Sci., 5:909–915, 1970. [12] H. Eyring. Viscosity, plasticity and diffusion as examples of absolute reaction rates. J. Chem. Phys., 4:283–291, 1936. [13] G. J. Frank and R. A. Brockman. A viscoelastic-viscoplastic constitutive model for glassy polymers. Int. J. Solids Struct., 38:5149–5164, 2001. [14] O. A. Hasan and M. C. Boyce. A constitutive model for the nonlinear viscoelastic viscoplastic behavior of glassy polymers. Polym. Eng. Sci., 35:331–344, 1995. [15] R. N. Haward and G. Thackray. The use of a mathematical model to describe isothermal stress-strain curves in glassy thermoplastics. Proc. R. Soc. London A, A302:453–472, 1968. [16] H. H. Hilton. The elusive and fickle viscoelastic poisson’s ratio and its relation to the elasticviscoelastic correspondence principle. J. Mech. Mater. Struct., 4(7-8):1341–1364, 2009. [17] Y. Hu, Z. Xia, and F. Ellyin. Mechanical behaviour of an epoxy resin under multiaxial loadings. Part I: Experimental study. Polym. Polym. Compos., 8(1):11–18, 2002. [18] T. Iwamoto, T. Nagai, and T. Sawa. Experimental and computational investigations on strain rate sensitivity and deformation behavior of bulk materials made of epoxy resin structural adhesive. Int. J. Solids Struct., 47:175–185, 2010. [19] R. S. Lakes and A. Wineman. On Poisson’s ration in linearly viscoelastic solids. J. Elast., 85:45–63, 2006. [20] S. M. Lee. Plastic deformations in epoxy resins. In R. Dickie, S. Labana, and R. Bauer, editors, Cross-linked Polymers: Chemistry, Properties, and Applications. ACS Symposium series no 367, pages 136–144. American Chemical Society, Washington D.C., 1988. [21] A. J. Lesser and R. S. Kody. A generalized model for the yield behavior of epoxy networks in multiaxial stress states. J. Polym. Sci. Pol. Phys., 35:1611–1619, 1997. [22] Y.-M. Liang and K. M. Liechti. On the large deformation and localization behavior of an epoxy resin under multiaxial stress states. Int. J. Solids Struct., 33(10):1479–1500, 1996. [23] A. E. Mayr, W. D. Cook, and G. H. Edward. Yielding behaviour in model epoxy thermosets — I. Effect of strain and composition. Polymer, 39(16):3719–3724, 1998. [24] E. F. Oleinik. Epoxy-aromatic amine networks in the glassy state: structures and properties. Adv. Polym. Sci., 80:49–99, 1986. [25] M. Prandini and L. Bardella. Identification of a constitutive model for epoxy resins. In Proceedings of the XIII Italian Congress on Computational Mechanics (GIMC 2000), pages 125–131, 2000. ISBN 88-86524-47-1. [26] R. A. Schapery. On the characterization of nonlinear viscoelastic materials. Polym. Eng. Sci., 9:295–310, 1969. [27] R. A. Schapery. Nonlinear viscoelastic and viscoplastic constitutive equations based on thermodynamics. Mech. Time-Depend. Mat., 1:209–240, 1997.

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28

0.015

Residual longitudinal strain after unloading

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

0.014

v = 1.6667E-7 s

0.013

v = 2.7407E-7 s

-1 -1 -1

v = 4.1853E-7 s

0.012

-1

v = 1.1519E-6 s

-1

0.011

v = 2.7728E-6 s

-1

v = 6.4815E-6 s

0.01

-1

v = 7.5778E-5 s

-1

0.009

v = 8.3111E-5 s

0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0

10

100 Days

1000

Figure 16: Uniaxial compressive tests: residual longitudinal deformation after unloading.

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Figure 17: The SNS rheological model adopted in [3].

Figure 18: The rheological model in which plasticity is accounted for.

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