Deformation and failure maps for PMMA in uniaxial tension

Uniaxial tensile tests are performed on a polymethyl methacrylate (PMMA) grade over a range of temperatures near the glass transition and over two decades of strain rate. Deformation maps are constructed for Young's modulus, flow strength, and failure strain as a function of temperature for selected strain rates. The glassy, glass transition and rubbery regimes are identified, and constitutive relations are calibrated for the modulus and flow strength within each regime.


Introduction
The linear, amorphous version of the polymer polymethyl methacrylate (PMMA) is employed in a variety of technical products ranging from transparent windshields to nanocellular polymeric foams [1,2]. Commonly, these components are formed in the hot state at a temperature close to the glass transition temperature Tg. However, measurements of the tensile stress-strain response of PMMA near Tg are limited. Our aim is to identify the regimes of deformation and fracture for a commercial PMMA grade near the glass transition. Existing constitutive models are calibrated and an emphasis is placed on the magnitude of the failure strain as a function of temperature and strain rate.
Gilbert et al. [3] have identified four constitutive regimes for a linear, amorphous polymer under uniaxial tension. These regimes are the (1) glassy, (2) glass transition, (3) rubbery, and (4) viscous flow regimes in sequence of increasing temperature T, as normalized by the glass transition temperature Tg, or of decreasing strain rate ε , see Figure 1a. The typical stress versus strain response associated with each regime is illustrated in Figure 2a (nominal tensile stress S versus nominal tensile strain e) and Figure 2b (true tensile stress σ versus logarithmic or true 2 tensile strain ε ). The deformation mechanism map of Gilbert et al. [3] made use of data in the literature for various PMMA grades as insufficient comprehensive data exists for a single grade. A series of simplified constitutive laws were fitted to the experimental modulus data. In a subsequent study, Ahmad et al. [4] reported similar plots for the rate and temperature dependent flow strength y σ by an analogous approach, see Figure 1b.
Few data exist in the literature for the true tensile failure strain f ε of PMMA as a function of temperature and strain rate. A possible trend is depicted in Figure 1c, as based on Cheng et al. [5], Smith [6] and Vinogradov et al. [7]. Cheng et al. [5] measured f ε in the glassy regime, Smith [6] elucidated the effect of rate and temperature on f ε for cross-linked elastomers near the glass transition temperature, and Vinogradov et al. [7] investigated the effect of strain rate ε on f ε for monodisperse, amorphous polymer melts.
The deformation mechanism maps of Gilbert et al. [3] and of Ahmad et al. [4] relied upon limited measurements in the glass transition and rubbery regimes. Moreover, they needed to calibrate their models to test data for various polymer grades with, for instance, different molecular weights and weight distributions. This motivates the current paper: our aim is to construct experimentally validated deformation maps in terms of the modulus E and the flow strength y σ as a function of temperature T and strain rate ε of a single commercial PMMA grade close to the glass transition. A failure envelope for the tensile failure strain as a function of temperature and strain rate is also obtained.
The present study is structured as follows. First, the deformation mechanism regimes and associated constitutive laws of Figure 1 are reviewed for a linear, amorphous polymer such as PMMA in terms of the underlying molecular deformation processes. Tensile tests on a commercial grade of PMMA are then detailed. Finally, the constitutive laws are calibrated from the measured stress-strain curves and deformation and failure maps are constructed. 3

A brief review of deformation mechanisms
Linear, amorphous polymers (such as PMMA, polycarbonate or polystyrene) exhibit a range of deformation mechanisms, depending upon temperature (relative to Tg) and strain rate. These include the glassy regime, glass transition, rubbery, viscous flow and decomposition regime in order of increasing temperature (or decreasing strain rate). For each regime (excluding decomposition), constitutive laws can be stated for the modulus E (at vanishing strain 1 ), tensile flow strength y σ and tensile failure strain f ε . We consider each regime in terms of molecular deformation mechanism and state constitutive laws for E, y σ , and f ε . Later in the paper, the measured tensile stress-strain response of the PMMA is used to calibrate these constitutive equations.

The glassy regime and the glass transition regime
The small strain, viscoelastic response The glassy regime exists at temperatures below the glass transition, 0 < T/Tg < 1. Following Yannas and Luise [8] and Gilbert et al. [3], we shall assume that the glassy modulus Eg decreases in an almost linear fashion with increasing temperature due to the increase in vibrational energy of chain segments to give [3,8]: where g E and WLF η are defined in Eq. (0) and Eq. (0), respectively.

The large strain, viscoplastic response
The dependence of the plastic response in the glassy regime upon temperature is more complex, and can be sub-divided into two regimes, 0 < T/Tg < 0.8, and 0.8 < T/Tg < 1. Typically, when a linear, amorphous polymer is subjected to uniaxial tension at T/Tg < 0.8, the stress versus strain response is almost linear and brittle fracture intervenes (at a true strain below 0.1) by the formation of an unstable craze or by fast fracture from a pre-existing flaw [12,18]. Deformation occurs by the stretching and bending of secondary (van der Waals) bonds, resulting in a low sensitivity to both strain rate and temperature by secondary relaxation mechanisms [3].
For 0.8 < T/Tg < 1, shear yielding follows an initial elastic response. Typically, yield is accompanied by a load drop and by the development of a neck. High strain rate sensitivity reduces the degree of neck formation and orientation hardening leads to neck propagation along the length of the specimen [13][14][15]. After the neck has propagated along the entire length of the sample, the load increases again until failure intervenes. 5 Near the glass transition temperature (T/Tg ≈ 1), the van der Waals bonds melt and polymer chain segments can reptate past each other [3]. The stress versus strain response is highly rate and temperature sensitive at temperatures near Tg. At temperatures above Tg, there ceases to be a load drop at yield, and no neck is formed.
It is widely accepted that plastic yielding of linear, amorphous polymers is governed by the thermally activated motion of molecular chain segments [16,17]. The time and temperature dependence of the flow strength may be described by a modified version of the Eyring model which was originally developed for the viscous flow of liquids [18][19][20][21]. The Ree-Eyring model assumes that an applied stress σ (with a strict definition to be made precise below) reduces the energy barrier for vibrating chain segments to jump forward and increases the barrier to jump backwards. Between the secondary β transition and the glass transition temperature Tg, the von Mises, effective plastic strain rate e ε is related to σ via a single transition process such that [17]: where 0 ε is a reference strain rate, q is an activation energy, v is an activation volume, and k . Note that h σ is equal in magnitude but opposite in sign to the pressure p. A common assumption is to assume that the stress measure σ , which activates plastic flow, is given by: where α is a pressure sensitivity index taken to be a material constant, see for example Ward [22]. Thus, the tensile yield strength ty σ > 0 reads: While the compressive yield strength cy σ < 0 reads: Consequently, the ratio cy ty σ |/σ | is: For PMMA, measurements of this ratio imply α 0.4  [4,20]. The consensus of experimental evidence suggests that plastic flow of PMMA is close to incompressible [23]. Consequently, for the isotropic case we can assume that: . We shall assume that Eq. (0) describes the large strain viscoplastic response of PMMA in the glassy regime and the glass transition regime.

Failure strain
Within the plastic yielding regime, 0.8 < T/Tg < 1, large plastic strains are achievable prior to ductile fracture. Failure is by the stable growth of cavities (known as 'diamonds') originating from surface defects or crazes until they reach a critical size [24]. A few experimental studies report the brittle and ductile tensile failure strain of PMMA as a function of temperature and strain rate in the glassy regime [5,16]. To a first approximation, the dependence of f ε upon T/Tg is governed by two linear relations, one for brittle elastic behaviour (small slope) and one for failure at large plastic strains (large slope), see Figure 1c. The experimental data of Cheng et al. [5] suggest that ductility is relatively insensitive to strain rate in the glassy regime. 7 The small strain, viscoelastic response Linear, amorphous polymers of moderate to high molecular weight typically exhibit a rubbery plateau at a temperature regime just above the glass transition; the extent of the rubbery plateau depends on the molecular weight (distribution) of the polymer [25]. Within this rubbery regime, most of the secondary bonds are broken and long range sliding of polymer chains is prevented by chain entanglements. The entanglement points act as physical cross-links governing the polymer's constitutive state in a similar fashion to chemical cross-links of rubbers. Instead of making use of entropy-based models that relate the rubbery modulus to temperature (see Treloar et al. [26]), we find that the following empirical relation is adequate to describe the rate and temperature sensitivity of the modulus in the rubbery regime:

The rubbery regime
Here, R α is a temperature coefficient, n is a rate sensitivity index and 0 R E and R ε refer to a reference modulus and strain rate, respectively.

The large strain, viscoplastic response
Ahmad et al. [4] ommit the presence of the rubbery plateau between the glassy and viscous flow regimes to describe y σ as a function of rate and temperature. As the width of the rubbery plateau is governed by the average molecular weight (distribution), this approach may suffice for grades of low molecular weight. In contrast, commercial linear amorphous polymers typically possess a high molecular weight, and an extensive rubbery regime on a y σ versus T/Tg plot is anticipated above the glass transition. Here, we assume a linear elastic stress-strain behaviour to represent the rubbery response of PMMA. For the strength versus temperature plot it is necessary to introduce a reference strain ref ε such that the 'strength' is given by: with R E governed by Eq. (0).
Failure strain 8 In contrast to the abundance of rate and temperature dependent constitutive relations for E and y σ , data on the effect of rate and temperature on the tensile failure strain f ε of linear, amorphous polymers in the glass transition and rubbery regime are more scarce. Smith [6] investigated the ultimate tensile strength and failure strain of cross-linked amorphous elastomers such as styrene-butadiene rubbers (SBR) near their glass transition and demonstrated the applicability of a time-temperature superposition for the tensile failure strain.
Now consider the ductility in the vicinity of the rubber to viscous flow transition. Vinogradov et al. [7] found that the tensile failure strain f ε of polymer melts went through a minimum at T far above Tg and then underwent a steep increase as the viscous flow regime is approached.
The dependence of f ε upon T/Tg for a commercial linear, amorphous polymer may exhibit a similar trend: a local maximum is attained in the rubbery regime, followed by a dip as the polymer transitions from the rubbery to the viscous state, see Figure 1c. However, insufficient measurements are reported in the literature for firm conclusions to be drawn. For instance, tensile tests on some commercial grades of PMMA and polystyrene melts show that the failure strain is independent of strain rate [27]. Fundamental models on the magnitude of failure strain as a function of molecular weight (distribution) in the rubbery state also appear to be lacking [28,29].

Failure strain
It is broadly accepted that the rupture strain of a viscous melt is governed by the surface tension-driven instability of Rayleigh-Plateau [36,37].

More sophisticated three dimensional constitutive models
Boyce and co-workers [14,[38][39][40][41] gave a more sophisticated three dimensional theoretical framework to simulate the stress-strain response of linear, amorphous polymers over the entire glass-to-rubber transition regime. They assume that the polymer has the character of a parallel combination of an intermolecular and molecular network resistance. Palm et al. [42] employed the Boyce et al. [43] framework originally developed for PETG to describe the constitutive behaviour of PMMA around Tg for uniaxial compression. The three dimensional models of G'Sell and Souahi [44] and Dooling et al. [45] comprise the few experimentally validated frameworks that focus on the constitutive response of PMMA loaded in uniaxial tension around and above its glass transition. These three-dimensional large strain constitutive models give good curve fits to uniaxial compressive and tensile test data. However, they make use of a large number of fitting constants and there are few non-proportional, multi-axial tests available to calibrate the full three-dimensional response. Moreover, these models were developed to predict the deformation response, but give little insight into failure.
The Tg (= 116 C) was measured by Differential Scanning Calorimetry (DSC) with a heating 10 rate of 10 C min -1 . The molecular weight distribution of the samples was measured via Gel Permeation Chromatography 2 (GPC) with tetrahydrofuran as the eluent and detection by refractive index. The molecular weight distribution is shown in Figure 3. The weight-average molecular weight Mw equals 3 580 000 g mol -1 , while the number-average molecular weight Mn equals 607 000 g mol -1 , resulting in a polydispersity index Mw/Mn equal to 5.9.

Specimen geometries and testing procedures
Dogbone specimens were machined from the PMMA sheets as shown in Figure 4, and were adhered 3 to aluminum alloy tabs (35 mm x 26 mm x 1.4 mm). The specimens were pin-loaded and dots of diameter equal to 1 mm were painted onto the gauge section of the specimens with white acrylic paint. Their relative displacement was tracked by a video camera during each test.
The average nominal strain e in the gauge section of the specimen was measured from the axial separation between the two pairs of horizontally aligned white dots. Assuming uniform deformation in the gauge section, a measure for the true strain ε is obtained via: The dogbone specimens were tested in a servo-hydraulic test machine in displacement control, and the test temperature was set by making use of an environmental chamber with feedback temperature control and fan-assisted air circulation. The specimen's temperature was independently monitored by a placing a thermocouple against it. The front face of the dogbone specimens was viewed through a transparent window of the environmental chamber. The cross-head displacement, tensile load P, air chamber temperature, and sample temperature were simultaneously monitored and logged on a PC. Assuming uniform and incompressible deformation in the gauge section, the true stress is computed by: where A and 0 A refer to the current and nominal cross-sectional area of the gauge, respectively.
2 GPC was performed with an Agilent Technologies PL GPC220 (USA) with a nominal flow rate equal to -5 1.67 10 l/s  at a testing temperature equal to 30 C. 3 Tabs were glued with Loctite EA 9497 high temperature resistant structural bonding epoxy (Henkel, Germany). The adhesive was allowed to cure at room temperature for at least 48 hours.
Tests were performed for temperatures ranging from 80 C to 185 C with three cross-head velocities (vch = 5 mm s -1 , vch = 0.5 mm s -1 and vch = 0.05 mm s -1 ) per testing temperature. The tests were terminated at the point of specimen failure or when the maximum actuator displacement (100 mm) was reached. The latter constraint corresponds to a true strain of approximately unity within the gauge section of the dogbone specimens. Upon recognizing that higher failure strains occur when the test temperature is at or above the glass transition temperature, additional tests were performed using an alternative hourglass-shaped specimen used to probe the failure strain at temperatures near Tg (in the range of 115 C to 190 C) following the practice of Hope et al. [46].

Strain measurements for the hourglass specimens
Consider the geometry of the hourglass specimen as shown in Figure 5. Recall that the loading direction is aligned with the 3-direction, the thickness direction with the 2-direction, and the transverse direction is aligned with the 1-direction. The axial strain can be measured by the change in shape of the dots at mid-section (longitudinal measurement method). However, this method is prone to scatter, and the alternative transverse gauge-based measurement method of Hope et al. [46] was used in order to reduce this measurement scatter: (i) An average measure for the transverse true tensile strain 1 ε at the minimum section of hourglass specimen is obtained from the spacing w between the pair of white marker dots.
(ii) Assume that the through-thickness strain 2 ε equals the transverse strain 1 ε . This was verified by post-failure examination of the thickness and width of the hourglass specimens.
Then, the true tensile strain is related to the current spacing w and initial spacing 0 w by:

Results and discussion
True tensile stress versus true tensile strain curves are reported for a temperature T in the range Gilmour et al. [48]. In contrast, the plastic Poisson's ratio p ν , defined as the ratio of the transverse to axial value of the true plastic strain, was almost constant at 0.45 -0.5 for 80 C  T  185 C, implying that plastic flow is almost incompressible in nature [23].

Stress-strain curves
The true stress versus true strain responses of the dogbone specimens are plotted in Figure 6a for  strain responses shown in Figure 6a for 0.91  T/ Tg  0.97 therefore give the overall structural response of the specimen rather than the local material response. We shall make later use of the modulus and initial peak strength, and not discuss further the neck development. See, for example, Wu and van der Giessen [49] for an analysis of necking of polymers in the presence of strain softening. 13 Now consider the response at higher temperatures, in the range of T = 120 C to T = 155 C, (corresponding to 1.01  T/Tg  1.1). The tensile response in this regime is almost independent of temperature and strain rate, characteristic of rubberlike elasticity, see Figure 6b. The temperature and rate sensitivity increases again for the highest testing temperatures (T = 170 C and T = 185 C, i.e. 1.14  T/ Tg  1.18), indicating the transition to a viscous deformation regime.
Additional insight is obtained by interrupting selected tensile tests prior to failure, and then fully unloading them. The loading-unloading stress-strain curves are shown in Figure 7. For T/Tg = 0.94 (within the glassy regime), elastic unloading occurs in the manner of an elastoviscoplastic solid, with a substantial remnant strain at zero load. When the temperature is increased to T/Tg = 1.06, the elastic rubbery regime is entered and the unloading curve is almost coincidental with the loading curve; marginal hysteresis occurs and the remnant strain upon unloading is close to zero. Finally, at T/Tg = 1.16, the viscous regime is just entered and unloading is accompanied by remnant strain.

Modulus as a function of rate and temperature
The dependence of the modulus E upon T/Tg is given in Error! Reference source not found.a for three nominal strain rates. Note that E is measured as the secant modulus based on a true Bauwens-Crowet [20]) are summarized in Table 2. The fitted value for the activation volume is in the same order of magnitude as the ones reported in the literature [22]. This volume is about 5 to 10 times larger than the estimated volume of a monomer unit which is in agreement with the idea that shear yielding is governed by the collective movement of numerous chain segments [16].

Failure strain as a function of rate and temperature
The measured true tensile failure strain f ε is plotted 5 versus T/Tg in Error! Reference source not found.a for two nominal strain rates ( the peak in load [22]. This distinct yield point disappears when the polymer enters the glass transition regime. The flow strength in the glass transition and rubbery regime is therefore defined as the value of the true stress at a reference true strain equal to 0.05. 5 The specimen did not fail at T = 185 C at the maximum actuator displacement for -2 -1 × e = 5.9 10 s . 15 minimum section of the hourglass-shaped specimen is larger than the elongation of the dots.
We conclude that the transverse gauge gives a more accurate value for the local failure strain.
The transverse gauge-based failure envelope for The failure strain versus T/Tg fits (Eq. (0) and Eq. (0)) are compared with the failure strain measurements reported by Cheng et al. [5] for PMMA in the glassy regime and Smith [46] for SBR in the glass transition and rubbery regime in Figure 9Error! Reference source not found.c. The dependency of strain rate on the measured failure strain in the (plastic yielding) glassy and glass transition regime is found to be small, in agreement with the data reported by Cheng et al. [5]. The f ε measurements for the glass transition and rubbery regime are in relatively good agreement with the f ε versus T/Tg data reported for a cross-linked SBR elastomer [50]. The failure strain envelope above Tg of the linear, amorphous, high molecular weight PMMA is therefore found to be similar to the failure strain envelope above Tg of a chemically cross-linked polymer. 16

Concluding remarks
The tensile response of a commercial polymethyl methacrylate (PMMA) grade has been characterized over a range of temperatures near the glass transition and over two decades of strain rate via a series of uniaxial tensile tests. Modulus, flow strength and failure strain are plotted as a function of temperature via deformation and failure maps for selected strain rates.
Fitted constitutive relations in terms of modulus and flow strength are reported for three identified constitutive regimes: the glassy, glass transition, and rubbery regime. The models can be generalized in a straightforward fashion to three dimensional behaviour assuming isotropy. It is emphasized that the constitutive descriptions (including failure) are needed in order to model the processing of PMMA close to the glass transition temperature. Failure is of prime importance in processes ranging from melt blow moulding to solid-state foaming. Figure 1 -Constitutive trends from the literature for a linear, amorphous polymer in uniaxial tension at temperatures close to its glass transition: (a) vanishing strain modulus as a function of T/Tg for two distinct strain rates [3], (b) flow strength as a function of T/Tg for two distinct strain rates [4], and (c) true tensile failure strain as a function of T/Tg for two distinct strain rates [5][6][7].      0)) along with the reported experimental f ε versus T/Tg data by Cheng et al. [5] for PMMA in the glassy regime and Smith [50] for SBR in the glass transition and rubbery regime.