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Effects of stress triaxiality ratio on the heat build-up of

polyamide 11 under loading

C Ovalle, G. Boisot, L. Laiarinandrasana

To cite this version:

C Ovalle, G. Boisot, L. Laiarinandrasana.

Effects of stress triaxiality ratio on the heat

build-up of polyamide 11 under loading.

Mechanics of Materials, Elsevier, 2020, 145, pp.103375.

�10.1016/j.mechmat.2020.103375�. �hal-03167535�

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Effects of stress triaxiality ratio on the heat build-up of

PolyAmide 11 under loading

C. Ovallea,∗, G. Boisotb, L. Laiarinandrasanaa

aMINES ParisTech, PSL University, Centre des mat´eriaux (CMAT), CNRS UMR 7633,

BP 87 91003 Evry, France

bARKEMA Cerdato / Route du Rilsan / 27470 Serquigny, France

Abstract

Adiabatic heat build-up in polymers attributed to the conversion of plastic work into dissipative heat is a well known phenomenon. The temperature in the vicin-ity of a crack tip due to heat build-up may be exceptionally high so that it can lo-cally reach the glass transition temperature Tg, even though the ambient testing temperature is lower than Tg. A significant alteration of the local material re-sponse and damage mechanisms is then induced. By simultaneous measurement of temperature during experimental tests under quasi-static loading and for a wide range of stress triaxiality ratios, a Gurson-Tvergaard-Needleman based thermo-mechanical constitutive model, integrating temperature-dependent co-efficients, has been developed. Predictive capabilities of the proposed thermo-mechanical model to simulate the isothermal behaviour of PolyAmide 11 (PA11) have led to adiabatic simulations, to account for the heat build-up highlighted experimentally, of ductile crack extension. The model parameters were identi-fied using experimental data obtained from PA11 samples with a given stress triaxiality ratio. Predicted evolutions given by the proposed constitutive model for other stress triaxiality ratios and geometries are found to be in good agree-ment with experiagree-mental data.

Keywords: Polymers, Damage mechanics, Fracture mechanics, Finite Element analysis, Thermo-mechanical coupling, Heat build-up.

Corresponding author

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1. Introduction

Polymers are frequently and increasingly used in industrial applications un-der different strain rates, temperatures and loading conditions. It is well known that the fracture behaviour of polymeric materials depends significantly on time and temperature effects; however, depending on loading conditions, fracture can

5

be brittle, semi-ductile or ductile.

Many studies have dealt with the effects of temperature and strain rate on the mechanical behaviour of polymers such as their effects on the yield, strain softening and hardening of the material [1, 2, 3, 4, 5]; furthermore, both temperature and strain rate were used in the constitutive modelling of such

10

effects. Among these studies great interest has been focused on yield stress. An increasing yield stress with decreasing temperature or an increasing strain rate has been revealed. Indeed, since the seventies, many researchers, for instance Bauwens and co-worker’s [6, 7, 8, 9] have studied both the temperature and strain-rate dependent yield stress of polymers. On the other hand, G’Sell and

15

Jonas [1] used a non-linear visco-elastic representation of strain hardening to model the effects of both temperature and strain rate; in fact, the rate effect was modelled with a low strain rate sensitivity and a high strain hardening strain-proportional coefficient. Later, Arruda et al. [4] investigated the effects of the strain rate, temperature and thermo-mechanical coupling on the large

20

deformation response of PMMA. Modelling was carried out by means of a finite strain constitutive model developed by Boyce et al.[10] for the study of glassy polymers, an Argon-based model [11] to account for the rate and temperature effects over the isotropic resistance to chain segment rotation and, finally, an additional temperature dependency based on the study of Farren and Taylor

25

[12].

Another kind of study involving temperature dealt with the heat build-up of polymers during cyclic or dynamic loading. Early in the last century, Taylor and co-workers [12, 13] identified the conversion of a fraction of the

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mechani-cal energy into heat. Decades later, the dependence of the plastic work fraction

30

with strain and strain-rate was shown[14, 15]. Thus, under adiabatic conditions, a rise of temperature is expected as the heat cannot flow out. Sometimes, a sig-nificant temperature rise [16] can be encountered in the case of cyclic [17, 18] and dynamic loading [19, 20]. The temperature field at the crack tip has been the subject of many investigations [21, 22, 23, 24, 25, 26, 27]; heat generation

35

caused by plastic work can be very high: a temperature rise of 130K in iron was measured by Weichert and Sch¨onert [23] and a temperature rise of 450K in PMMA was measured by Fox and Fuller [28]. Thus, determining the tem-perature field at the crack tip becomes a key data in the investigation of crack propagation. By using thermocouples, D¨oll [22] measured experimentally the

40

heat at a propagating crack tip and found that an increasing heat generation due to plastic work was related with an increasing crack speed. Zehnder and Rosak [29] developed an experimental technique which enabled the temperature field during a dynamic crack propagation to be assessed. On the other hand, many studies have focused on the modelling of the heat build-up temperature

45

at a crack tip [23, 30, 31]. The investigations of Williams [21], Fuller et al. [30] and D¨oll [32] which highlighted a considerable temperature rise at the crack tip dealt with a running crack and not with a stationary crack. However, in-vestigations carried out by Rittel [26] for the case of dynamic experiments on PolyCarbonate (PC) fatigue pre-cracked samples showed a temperature rise of

50

70◦C at the notch tip.

The mechanical behaviour and damage of semi-crystalline polymers such as PolyAmide 11 (PA11) and PolyVinyliDene Fluoride (PVDF) was numerically modelled [33] by a Gurson-based model [34], initially developed for metallic ma-terials it was extended by Tvergaard and Needleman [35, 36, 37], and is known

55

as the GTN model. Further to the work of Challier et al. [33], Laiarinandrasana et al. [38] extended the model to take temperature into account.

In this paper, the developed thermo-mechanical Gurson-Tvergaard-Needleman constitutive model, integrating temperature-dependent coefficients, was imple-mented in an in-house finite element code Zset [39]. By simultaneous

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surement of temperature, obtained by an InfraRed (IR) camera, during experi-mental tests under quasi-static loading on gradually increasing stress triaxiality ratio samples, the model parameters were identified at several temperatures and compared numerically with experimental data. Afterwards, a continuous depen-dence of the model parameters with temperature was deduced. The predictive

65

capability of the proposed thermo-mechanical model to simulate the isothermal behaviour of PA11 led to adiabatic simulations accounted for the heat build-up temperature highlighted experimentally. Predicted evolutions given by the pro-posed model for other stress triaxiality ratios and geometries are found to be in good agreement with experimental data.

70

The paper is organized as follows. In Section 2 the experimental methods, modelling and simulation methods are described. Results of experimental and numerical simulations with available experimental observations are presented and discussed in Section 3. Finally, we close the paper with concluding remarks.

2. Methods

75

2.1. Experiments 2.1.1. Materials

The material under study is a Rilsan polyamide supplied by ARKEMA. The main material characteristics of the semi-crystalline polymer, provided by the manufacturer, are summarized in Table 1. The PolyAmide (PA11) exhibited

80

a crystallinity index of about 20-25%. The glass transition temperature was estimated by Differential Scanning Calorimetric (DSC) technique to be 50◦C, and confirmed by Dynamic Mechanical Analysis (DMA). An initial amount of porosity of 1% was identified through Scanning Electron Microscopy (SEM) observations on cryo-fractured surfaces. The porosity was calculated as the

85

area fraction of voids. Under loading, the material whitening was assumed to be related with the growth of the initial amount of porosity[40].

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Melting point 184-188◦C Index of crystallinity 20-25% Glass transition temperature 50◦C Young modulus (at room temperature) 1500 MPa

Poisson ratio 0.42

Table 1: Main physico-chemical properties of the PA11.

2.1.2. Samples

In order to study the stress triaxiality ratio effects on the thermo-mechanical response of PA11, three geometries were used: Uniaxial Tensile (UT), Notched

90

Tensile (NT) and Single Edge Notch Bending (SENB) samples. UT flat samples were machined from 6 mm thickness extruded sheets. A gauge length of 100 mm and a cross section of 10 × 4 mm2 were used.

Circumferentially NT samples were machined from a PA11 pipe with an inner diameter of 90 mm and an outer diameter of 100 mm. Samples had a

95

length of 85 mm, a diameter of 7.2 mm, a minimum section diameter of 4 mm and a notch radius ρ =[4, 1.6, 1.2, 0.8] mm (samples were then referenced as NTρ), see Fig. 1a. According to the Bridgman formula [41], which assumes a perfect-plastic and isochoric behaviour, the maximum initial stress triaxiality ratio τσ located at the centre of the minimal cross section is:

100 τσ= σm σeq = 1 3 + log(1 + φ0 4ρ) (1)

where σm= 13trace(˜σ) is the mean stress, ˜σ is the stress tensor, σeq is the von Mises equivalent stress and φ0 is the initial diameter. If ρ → 0, i.e. in the case of a sharp notch, Eq.1 is obviously not valid due to a geometric singularity. If ρ → ∞, the geometry of the sample tends to an uniaxial tensile sample and then τσ = 13. According to Eq.1, for the aforementioned NTρ samples, the

105

maximum initial stress triaxiality ratio, for ρ =[4, 1.6, 1.2, 0.8] mm, is equal to τσ =[0.43, 0.54, 0.60, 0.69], respectively. Previous studies on PolyVinyliDene Fluoride (PVDF) by Challier et al. [33] and Laiarinandrasana et al. [38, 42]

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a) NTρ sample b) SENB sample

Figure 1: Sample geometries

showed that the Bridgman formula could be used as a first approximation in the case of polymeric materials subjected to moderate strains.

110

As for NT samples, SENB samples were machined from a PA11 pipe. Sam-ples had a length L = 80 mm, width W = 10 mm and thickness B = 10 mm. A 2.5 mm notch depth was machined in such a way that pre-crack depth ratio a0/W = 0.25. The first two millimetres were machined whilst the final 0.5 mm were obtained by pushing a razor blade into the material, see Fig. 1b.

Howe-115

verBesides, the Bridgman formula 1 seems to provide a poor approximation of the stress field of sharply notched samples, associated with a high triaxiality ratio, because of the extreme strain gradients at the notch tip.

2.1.3. Testing methods

Temperature-dependent mechanical characterization. As a standard approach

120

to investigate changes in the Young’s modulus Dynamic Mechanical Analysis (DMA) was performed on UT samples. Experimental tests were carried out on a MTS tensile machine for a wide range of temperatures, from -70◦C to 90◦C, and three frequencies f=[0.05, 0.5, 5]Hz.

Mechanical tests on NT4samples were carried out using a MTS machine

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vided with a high temperature environmental chamber. A displacement-control function was imposed with two crosshead speeds [0.05, 3] mm.s−1 for a wide range of temperatures [0, 10, 20, 30, 40, 50, 60, 70, 80, 100]◦C. The feedback re-sponses from the MTS load cell and the Linear Variable Differential Transformer (LVDT) displacement actuator were collected. Both the diametrical reduction

130

of the minimum cross section and the Notch Opening Displacement (NOD) were recorded respectively by means of a strain gauge at the notch root and a second one positioned symmetrically between both sides of the notch root, see Fig. 1a. The net strain was then defined as ∆h/h0where ∆h was the NOD measured by the axial gauge and h0= 6.4 mm was the initial notch height. Additionally, the

135

diametrical strain was defined as ∆φ/φ0 where ∆φ was the diameter evolution measured by the radial strain gage and φ0 was the initial diameter of the min-imum cross section. The net stress consisted of the ratio between the applied load F and the initial minimal cross section of the sample Sn = π(φ0/2)2.

Temperature field measurement. Tensile tests were conducted in an Instr¨on

test-140

ing machine provided with a temperature environmental chamber at 0◦C, load and displacement versus time being recorded during the tests. The low temper-ature level was attained by automatically-adjustable nitrogen flow. UT samples were tested at three strain rates [0.1, 0.01, 0.001] s−1whereas NT4samples were tested at two crosshead speeds [0.05, 3] mm.s−1. In the case of NT4samples, the

145

reduction of the minimum section was also measured by use of a strain gauge at the notch root. Three point bending tests on SENB samples were conducted at two constant deflection speeds [50, 100] mm.min−1. Furthermore, the Crack Opening Displacement (COD) was monitored by a MTS extensometer.

The surface temperature field was measured during deformation by an

In-150

frared Camera (IR) interfaced with a computer which allowed the storage of the temperature field during the test. Stored images with a video frequency of 1 Hz could then be post-processed to determine the temperature evolution in the test sample. In order to detect the infrared radiation, as samples were boxed in the closed environmental chamber at the testing temperature, the door of the

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environmental chamber was left opened during the tests. In spite of the external surrounding atmosphere, the testing temperature was assured by means of an enveloping surrounding nitrogen atmosphere. Moreover, tests run were short in duration.

2.2. Modelling and simulation

160

2.2.1. Heat dissipation modelling

During deformation, an important issue when studying the temperature rise, due to the thermo-mechanical coupling, is related to the heat transfer mecha-nisms in the sample. Indeed, the heat dissipation can vary from adiabatic at high strain rates to nearly isothermal at low strain rates. Deformation and

165

temperature fields were studied by Lai and van der Giessen [43, 44] for these two extreme cases. Thus, stress and temperature fields at the crack tip are not only influenced by temperature-dependent plasticity but, in addition, by the heat transfer due to thermal conduction in the process zone. Both isothermal and adiabatic mechanisms become thus special cases with infinite or zero

con-170

ductivity, respectively. From a one-dimensional argument, Basu and van der Giessen [27] defined a non-dimensional parameter κ to assess the conductivity effect:

κ = λt0 ρcvL2

0

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where λ is the thermal conductivity coefficient, t0is a characteristic time scale, ρ is the mass density for a unit reference volume, cvis the specific heat at constant

175

volume and L0is a characteristic length scale associated with the problem. For κ  1, simple isothermal calculations are relevant, whilst 1  κ, adiabatic conditions were assumed for the calculations. According to Basu and van der Giessen [27], if the chosen time scale was the time to reach a particular value of the stress intensity factor KI then t0was directly related to the applied loading

180

rate as t0= KI/ ˙KI. When Eq. 2 was applied in the present study, the κ value suggested a significant affect of the heat transfer mechanisms. Nevertheless, in the present paper, attention has been paid only to the two extreme cases,

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i.e. isothermal and adiabatic simulations. Indeed, applying the local approach to fracture with adiabatic conditions accounting for heat transfer mechanisms

185

appears quite complex.

2.2.2. GTN model

The initial model proposed by Gurson [34, 45] is a micromechanically based model which was phenomenologically extended by Tvergaard and Needleman [35, 36, 37] and the GTN model. The modified GTN model proposed in this

pa-190

per has been developed by Boisot and co-workers following previous studies on PolyVinyliDene Fluoride (PVDF) by Challier et al. [33] and Laiarinandrasana et al. [38, 42]. The GTN model deals with the Gurson’s yield criterion and takes into account the isotropic hardening, the strain rate sensitivity and the coalescence of voids. Moreover, the model satisfies the thermodynamical

frame-195

work proposed by Lemaˆıtre and Chaboche [46, 47] as discussed by Besson and Guillemer-Neel [48]. A damage-dependent effective stress σ∗is implicitly defined by Besson et al. [49] as follows:

Φ(˜σ, f∗, σ∗) = σ 2 eq σ2 ∗ + 2q1f∗cosh q2 2 σkk σ∗  − 1 − (q1f∗)2= 0 (3) where σkk is the trace of the stress tensor ˜σ. f∗ is a function of porosity f , where ft=t0 = f0 , introduced by Tvergaard and Needleman [37] to represent 200

the void coalescence. q1 and q2 are model damage parameters introduced by Tvergaard and Needleman [36, 37].

Besides, the yield surface Φ reads:

Φ = σ∗− R(p) (4)

with R(p) the flow stress of the matrix material expressed such that:

R = R0+ Q(1 − e−bp) + A(e−Bp− 1) (5) where R0, Q, b, A and B are material parameters to be identified and p is

205

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the initial hardening stage supposed to be isotropic whereas the third term A(e−Bp− 1) allows the rheo-hardening associated to the large stretching of the fibrils to be simulated [33, 38, 42].

The normality rule, i.e. the maximum plastic dissipation rate, yields the

210

expression of the viscoplastic strain rate tensor:

˜ ˙ εp= (1 − f ) ˙p∂Φ ∂ ˜σ = (1 − f ) ˙p ∂σ∗ ∂ ˜σ (6)

and ˙p is obtained by using the matrix viscoplastic Norton law:

˙ p = Φ

K n

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where K and n are material parameters to be identified. K denotes the strain rate intensity and n the non-linearity index of the viscoplastic behaviour.

The evolution of porosity follows the mass balance equation:

215

˙

f = (1 − f ) ˙εkk (8)

where ˙εkkis the trace of the strain rate tensor ˜˙ε. On the other hand, to account for the decreasing of the void growth with elongation of the cavities [50], q2 is defined as a function of the maximum principal plastic strain p1as follows:

q2= (M − m)tanh(υ(pt− p1) + 1)

2 + m (9)

where the coefficients M , m, υ and ptare material parameters to be identified. M is the initial value of q2 whereas m is the final value. In addition, pt is the

220

transition’s maximum principal plastic strain and υ is the transition rate.

2.2.3. Simulation model

The aforementioned modified thermo-mechanical GTN model was imple-mented in an in-house FE code Zset [39]. Integration of the constitutive model was achieved with an implicit scheme and the consistent tangent matrix was

de-225

picted by the Simo and Taylor [51] description. An updated Lagrangian finite strain formulation associated with the Jaumann stress rate was used [52].

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UT and SENB samples were meshed using quadratic elements whereas NTρ samples were meshed using axisymmetrical quadratic elements with reduced integration.

230

3. Results and discussion

3.1. Experimental results

In a first section, the temperature effects on the mechanical response of PA11 are discussed: Young’s modulus, peak stress, stress and strain at failure. The stress triaxiality ratio effect on the adiabatic heat build-up temperature due to

235

thermo-mechanical coupling is detailed in a second section.

3.1.1. Temperature-dependent response

Observations on UT samples. In Fig. 2, the temperature-dependent Young’s modulus E at different frequencies, from the DMA test, is shown. As only two frequency decades were considered, a slight dependence of the Young’s modulus

240

with the strain rate was generally observed; however, for temperatures below -50◦C and between 0◦C and 60◦C a significant dependence was highlighted. Indeed, near the glass transition temperature Tg ≈ 50◦C, a difference of 200-300 MPa between 0.05 Hz and 5 Hz was revealed.

Observations on NT4samples. Net stress-strain curves for different testing

tem-245

peratures, ranging from 0◦C to 100◦C, and a crosshead speed of 0.05 mm.s−1 are shown in Fig. 3a. Fig. 3b displays the diametrical strain versus net strain curve of the minimum cross section. The corresponding results in the case of uniaxial tensile tests on NT4 samples with a crosshead speed of 3 mm.s−1 are shown in Fig. 3c and d. NT4 samples generally exhibit an approximately linear

250

response prior to the peak stress. At the end of the stress softening, commonly associated with the re-necking of the notched zone, rheo-hardening before fail-ure is seen [33, 38, 53]. In Fig. 3a and Fig. 3c, for 0.05 mm.s−1 and 3 mm.s−1 respectively, it is shown that an increasing testing temperature resulted in a decrease of both the peak stress and the net stress at failure. Moreover, the

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Figure 2: Temperature-dependent Young’s modulus at different frequencies.

decrease of the peak stress with temperature was associated with the wipe-out of the stress softening at higher temperatures. In the case of the tensile test at 0.05 mm.s−1, the knee disappeared at a temperature of about 50-60◦C whereas at 3 mm.s−1, it disappeared at a temperature of about 70-80◦C.

Fig. 3b and Fig. 3d display the evolution of the diametrical strain of the

260

minimum cross section with respect to the net strain for a wide range of test-ing temperatures. A non-linear diametrical strain response is shown; indeed, a considerable diametrical strain increase with the net stress softening, at low tem-perature, was observed. A temperature-independent diametrical strain plateau, before failure, of about 0.075 was estimated. In addition, a significant

crosshead-265

speed effect over the diametrical strain evolution was revealed, i.e. a higher crosshead speed seems to be associated with an earlier re-necking. Indeed, a higher diametrical strain evolution seemed to be related with a higher crosshead speed; however, the diametrical strain plateau, before failure, appeared to be a time-independent variable.

270

Fig. 4a and Fig. 4b illustrate respectively the evolution of the peak stress and the net stress at failure with the testing temperature. An increase of tem-perature was related to a non-linear decrease of the peak stress; above Tg, the material was in a rubbery state and became softer and more flexible. Moreover,

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0.05 mm.s−1

3 mm.s−1

a) b)

c) d)

Figure 3: Tensile tests on NT4 samples for different temperatures at 0.05 mm.s−1: a) net

stress-strain, b) diametrical strain-strain, and at 3 mm.s−1: c) net stress-strain, d) diametrical

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a) b)

Figure 4: Temperature-dependent response at two crosshead speeds [0.05 and 3] mm.s−1: a)

peak stress, and b) net stress at failure.

an increase on temperature tended to minimize the effect of the crosshead speed

275

on the peak stress; this means that, as soon as the temperature reached Tg, the peak stresses at 0.05 mm.s−1 and 3 mm.s−1 became similar. As for the peak stress, a non-linear decrease of the stress at failure was associated with an in-creasing temperature; however, the temperature-dependent net stress at failure seemed to be crosshead-speed independent. In addition, it appeared that the

280

peak-failure stress relation, i.e. the peak stress over the stress at failure, was less than 1 for 0.05 mm.s−1, whereas for 3 mm.s−1 it went from a higher to a lower value than 1 with increasing temperature; it seemed to be equal to 1 at a temperature between 10-20◦C.

The net strain at failure, from Fig. 3a and Fig. 3c, is reported on Fig. 5.

285

In line with the Young’s modulus, the strain at failure showed an S-shaped response. At low temperature, a slight reduction in strain at failure of about 0.24 was estimated, whereas a considerable strain at failure of about 0.5 was observed at high temperature. In Fig. 5, a linear evolution of the strain at failure with temperature, near the glass transition temperature, is illustrated; moreover,

290

the slope of the aforementioned linear evolution seemed to be crosshead-speed independent. The strain at failure response revealed that the material evolved to a rubbery state with an increase of the testing temperature.

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Figure 5: Temperature-dependent net strain at failure at two crosshead speeds [0.05 and 3] mm.s−1.

3.1.2. Adiabatic heat build-up

The adiabatic heating due to plastic deformation at 0◦C was investigated

295

by measuring changes of surface temperature during deformation for different stress triaxiality factor conditions: UT, NT4 and SENB samples.

Observations on UT samples. The rate-dependent evolution of the engineering stress with the engineering strain at 0◦C is shown in Fig. 6. The engineering stress was defined as the ratio between the applied load F and the initial cross

300

section of the sample S0 = 10 × 4 mm2, whereas the engineering strain was defined as ∆L/L0 where ∆L was the displacement measured by an axial gauge and L0= 100 mm was the gauge length. UT samples exhibited an initial non-linear elastic behaviour accompanied by rate-dependent yielding until stress softening for an applied engineering strain of about 0.25, related to the necking

305

of the sample. Before failure, a strain-rate independent stress plateau of about 40 MPa was observed; indeed, the stress plateau was commonly associated with necking propagation. However, for the highest strain rate, i.e. 0.1 s−1, the engineering stress-strain curve exhibited a negligible stress plateau and failure seemed to occur just after stress softening of the sample. Besides, contrary

310

to the mechanical response of NT4 samples, peak stress and rheo-hardening was not exhibited with UT samples; however, similar to NT4 samples, stress

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Figure 6: Tensile tests on UT samples at 0◦C for different strain rates.

softening was observed. Moreover, the stress plateau was observed only on UT samples.

The temperature rise at the surface of the UT sample at 0.001 s−1, assessed

315

by the IR camera, is shown in Fig. 7. Initially the maximum temperature at the surface was about 6◦C, see a in Fig. 7b, where the sample was in an horizontal position; indeed, a small increase of temperature of 0.1◦C was observed during the elastic deformation. Before necking, a negligible increase on temperature of 4◦C due to thermo-mechanical coupling can be seen (b in Fig. 7b). In c

320

(Fig. 7), the formation of necking and a considerable temperature rise of 12◦C can be seen to have occurred; consequently, the heat build-up could have been associated with the dissipative plastic work during necking. Before failure (d in Fig. 7), a slight increase of 0.7◦C, compared to the temperature at the onset of the plateau, can be observed to have occurred; however, the maximum

temper-325

ature was located in one shoulder of the neck, where failure finally appeared. Moreover, a significant difference of 12.7◦C between the two shoulders was seen; indeed, the surface temperature field highlights the heterogeneous distribution of temperature during necking propagation.

The effects of the strain rate on the heat build-up of the UT sample for three

330

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a) b) a b c d

Figure 7: Tensile test on a UT sample at 0◦C for a strain rate of 0.001 s−1: a) engineering

stress-strain curve, and b) temperature field evolution.

a) b) c) d) × a ×b ×A × B

Figure 8: Thermo-mechanical response of a UT sample in uniaxial tension at 0◦C for different strain rates: a) 0.001 s−1, b) 0.01 s−1, c) 0.1 s−1, and d) temperature field before and after

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increase of temperature with deformation was displayed. As mentioned above, regarding 0.001 s−1 and 0.01−1, Figs. 8a and 8b respectively, it is clearly illus-trated that the temperature rose significantly at first during the stress softening stage, when the necking appeared and developed, and subsequently a second

335

time during necking propagation before failure, located in a specific point in the extended neck. Moreover, the experiments show that the temperature in-creased with increasing strain rate; indeed, at a low strain rate, i.e. 0.001 s−1, the temperature rise at necking was about 10◦C, whereas at a higher strain rate, i.e. 0.01 s−1, the temperature rise reached 30◦C. Regarding the highest

340

strain rate, i.e. 0.1 s−1, only one sudden rise of temperature at the surface can be seen in Fig. 8c; however, a significant temperature rise of about 70◦C is displayed, i.e. a surface temperature higher than Tg. On the other hand, a marked temperature rise could be associated with sample failure (Fig. 8d). An important difference for the maximum temperature in the surface between

345

the temperature field before and after failure has been highlighted, e.g. a tem-perature rise, concerning the neck-propagation temtem-perature, of about 31.7◦C is shown for a strain rate of 0.01 s−1; furthermore, the results show that the failure temperature increased with increasing strain rate. In conclusion, it is possible to establish that the temperature rise was directly related to the work

350

of plastic deformation; a higher temperature rate was developed during the high plasticity development phase, i.e. necking and failure.

Observations on NT4 samples. The net stress-strain and the diametrical strain-strain behaviour superposed with the temperature evolution of the minimum cross section of the NT4 sample, for a crosshead speed of 0.05 mm.s−1, are

355

shown respectively in Fig. 9a and Fig. 9b. As expected, an initial slight increase of temperature, before the peak stress, can be seen. A substantial linear increase of temperature occurred once the peak stress was reached, i.e. adiabatic heating occurred during the re-necking of the NT4 sample, in a similar way as necking induced a significant rise of temperature in the case of UT samples. Moreover,

360

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a b c d e f h a b c d e f h

a) b)

c)

Figure 9: Thermo-mechanical response of a NT4 sample under uniaxial tension at 0◦C for

0.05 mm.s−1: a) net stress-strain vs temperature, b) diametrical strain-strain vs temperature,

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formation, independently of the strain rate, probably linked to the greater mass of the NT4samples. Afterwards, during the rheo-hardening phase, the temper-ature rose non-linearly to attain a plateau value of about 15◦C, at the end of the rheo-hardening. Before failure, a sudden increase of temperature was observed.

365

A similar observation was reported by Rittel [26] who investigated the response of PMMA and PC at the relatively high strain rates, 7000 s−1 and 6500 s−1. The history-dependence of the heat-build up temperature field is illustrated in Fig. 9c. The temperature increased especially within the necking zone; further-more, the sudden temperature rise before failure was confined to the cracking

370

region, from f to h in Fig. 9c. In line with [54]:

∆T =3α

ρc∆p (10)

where the rate of change of the hydrostatic pressure ∆p is related to the rate of change of temperature ∆T , the experimental results confirmed that the increase of the heat build-up temperature with the stress triaxiality ratio, as a high hydrostatic pressure is commonly associated with a high stress triaxiality ratio.

375

In Eq. 10, α is the linear expansion coefficient, ρ is the mass density and c is the heat capacity coefficient. However, the temperature field appeared to highlight the onset of failure as a sudden temperature rise was observed, before failure, without any visible damage, see f in Fig. 9c. Consistent with Fuller et al. [30], similar observations for different fast crack speeds on PMMA where discussed.

380

Besides, as for UT samples, a significant difference for the maximal temperature before and after failure was shown.

In the case of a higher crosshead speed, i.e. 3 mm.s−1, the net stress-strain and the diametrical stress-strain-stress-strain behaviour superposed with the increase of temperature of the minimum cross section of the NT4 sample are shown

385

respectively in Fig. 10a and Fig. 10b. Concerning the increase of temperature at the surface, little information was obtained as a consequence of the test duration and the infrared camera video frequency: the sample was broken after 4 s. The last temperature field before failure (b in Fig. 10c) seemed to highlight the

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a b c a b c

a) b)

c)

Figure 10: Thermo-mechanical response of a NT4 sample under uniaxial tension at 0◦C for

3 mm.s−1: a) net stress-strain vs temperature, b) diametrical strain-strain vs temperature, and c) temperature field evolution.

considerable rise of temperature in the necking zone; moreover, the temperature

390

rise before failure almost reached the material Tg. However, a slight difference for the maximum temperature before and after failure was revealed, from b to c in Fig. 10c. As for the UT samples, the maximum temperature seemed to increase with increasing crosshead speed.

Observations on SENB samples. Let us now focus on a fracture mechanics

ge-395

ometry involving a higher local stress triaxiality ratio in a pre-cracked SENB sample under 3 point bending at 0◦C. The IR camera was set up to measure the temperature field at the crack tip during deformation. In Fig. 11a both the load-COD and heat build-up temperature curves for a deflection speed of 50 mm.min−1 are shown. Experimental results highlight stable crack

propaga-400

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see a in Fig. 11b, followed by a linear increase with the COD; indeed, before crack propagation, a significant temperature rise of 17.4◦C was observed, see c in Fig. 11b. In b (Fig. 11b), the maximum temperature was located at the crack tip whereas, immediately after the crack initiation (c in Fig. 11b), the

405

maximum temperature was located ahead of the crack tip. From Guo et al. [55], who investigated the competition between shear yielding and crazing on semi-crystalline polymers, the temperature field appears to be explained by the local stress distribution and the cumulated plastic deformation; furthermore, location of maximum values of both variables were related with the onset of

410

crack nucleation in a similar way as the location of the highest temperature was associated with crack nucleation in UT and NT4 samples. A quantitative way to estimate the distance x between the notch tip and the crack nucleation as x/R ≈ 0.6, where R is the notch radius, was proposed by Guo et al. [55]; then, in our case for R ≈ 0.2 mm, the aforementioned expression returns x ≈ 0.12

415

mm which seems in good agreement with experimental observations. A sim-ilar observation was reported by Laiarinandrasana et al. [56] on CT samples inspected by laminography. During crack propagation, the temperature at the crack tip slowly decreased; however, the highest temperature was located in-variably a small distance ahead of the crack tip, see d in Fig. 11b. For a higher

420

deflection speed, i.e. 100 mm.min−1, the temperature rise at the onset of the crack propagation was about 20◦C, lower than the material Tg.

To summarize, the experimental results reveal a temperature rise due to heat conversion of plastic work within the material. Furthermore, the heat build-up was dependent on the deformation/displacement rate, stress triaxiality ratio

425

and geometry, e.g. a bulkier sample was linked with a higher temperature rise. Note that, during the testing conditions, the glass transition temperature Tg = 50◦C was not attained; however, a temperature increment can be expected to modify the thermo-mechanical response of the material. On account of the temperature increment, the proposed thermo-mechanical model, based on a

430

previously developed GTN model [53], takes into account the heat build-up due to the thermo-mechanical coupling.

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×a

×b

×

c ×d

a) b)

Figure 11: Thermo-mechanical response of a SENB sample under 3 point bending at 0◦C for

50 mm.min−1: a) load-COD vs temperature, and b) temperature field evolution.

3.2. Modelling and simulation results

Material coefficients involved in Eqs. 5, 7 and 9 have to be determined as well as the elastic properties: the apparent modulus of elasticity E and Poisson ratio

435

ν. For the sake of simplicity, the Poisson ratio is assumed to be temperature-independent and equal to 0.42 [53]. Many details of the optimization procedure at 0◦C have already been described in [53], nevertheless a brief account is given here. The elasto-viscoplastic parameters have been adjusted from both UT sample engineering stress-strain curves and NT4 sample net stress-diametrical

440

strain of the minimal cross section curves. Moreover, the damage parameters optimization was carried out through Scanning Electron Microscopy (SEM) ob-servations of microtome-cut surfaces and the assessment of the void volume fraction through image analysis. Damage parameters of Eq. 9 can be depicted by comparing the numerical void volume fraction, given by the GTN model,

445

with aforementioned experimental observations.

The optimization of the temperature-dependent material parameters was carried out by considering solely the tensile response of NT4 samples because, in the present research work, no uniaxial tensile tests on UT samples were performed for temperatures ranging from 0◦C to 100◦C. Moreover, damage

pa-450

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Figure 12: Experimental (symbol) and numerical (line) net stress-diametrical strain of the minimal cross section curves of a NT4 sample under uniaxial tensile loading at 0.05 mm.s−1

for three temperatures [0, 50 and 100]◦C.

q2

q1 M m pt υ 0.85 0.3 0.925 6 2

Table 2: Values of the damage parameters for the PA11 at 0◦C.

independent of temperature, i.e. the values identified at 0◦C prevailed, see Table 2.

Table 3 summarizes the results of the material parameter optimization pro-tocol. Values are given for a wide range of temperatures from 0◦C to 100◦C,

455

i.e. below and above the glass transition temperature Tg≈ 50◦C. A continuous variation of each coefficient was imposed with respect to temperature during the optimization protocol.

In Fig. 12 the experimental and numerical net stress-diametrical strain of the minimal cross section of NT4samples below, near and above the glass transition

460

temperature, respectively [0, 50 and 100]◦C, are shown. A good agreement between experimental and numerical engineering stress-diametrical strain can be observed; therefore, the distribution of the strain and stress tensors at the minimum cross section is expected to be accurate. Consequently, the local parameters can be estimated by the Finite Element method (FE).

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Figure 13: Numerical contour maps of the maximum principal plastic strain (above) and the void volume fraction (below) at failure for 3 temperatures: a) 0◦C, b) 50C, and c) 100C.

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E K n R0 Q b A B MPa MPa.s−n - MPa MPa - MPa -0◦C 1500 38.5 10 10 25 30 7.9 3 10◦C 1500 38 10 10 20 30 7.9 3 20◦C 1500 38.5 10 10 18 20 7.9 3 30◦C 1000 35 10 8 18 10 7.9 3 40◦C 600 30 10 7 15 10 7.9 3 50◦C 500 30 10 7 14 10 7.9 3 60◦C 300 25 10 7 12 10 5.9 3.7 70◦C 200 25 10 7 5 10 5.9 4.1 80◦C 200 25 10 4 3 10 3.9 4.2 100◦C 200 25 10 2 2 10 3 4.2

Table 3: Optimized material parameters from 0◦C to 100C.

Fig. 13 displays the numerical contour maps of the maximum plastic strain p1 and void volume fraction f at the onset of failure on the deformed meshes of the NT4 sample. As expected [53], maximum values were found near the minimum cross section of the sample; however, a non-linear evolution of both the maximum plastic strain and the void volume fraction, from the centre to the

470

surface, was identified. Moreover, the maximum plastic strain was located at the surface whereas the maximum void volume fraction was located at the centre of the NT4 sample. Besides, in line with Boisot and co-workers [53], Fig. 13 clearly shows a fracture criterion, i.e. in such a situation, failure occurred by critical plastic strain since the void volume fraction remained relatively small.

475

3.3. Determination of the Taylor-Quinney factor

Once each material parameter had been described as a function of temper-ature, FE simulations could be used to estimate, initially, the Taylor-Quinney factor and then the study of the adiabatic heating associated with the conversion of the plastic work.

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λ ρ cp W.m−1.K−1 kg.m−3 J.kg−1.K−1

0.3 1.03×103 2.35×106

Table 4: Thermal and mechanical properties of the PA11.

In order to study the temperature contour maps, the conversion of the plastic work into heat had to be quantified. In case of metallic materials, the fraction of plastic work converted to heat, given by the Taylor-Quinney factor [13], is estimated at about 0.9. In the case of glassy polymers, Rudnev et al. [57] showed experimentally that the fraction of plastic work converted into heat was in the

485

range of 0.6-0.8, depending on the chemical structure of the macromolecules. According to Godovsky [58], the remaining energy may be associated with the appearance of internal stresses. The temperature rise due to the conversion of plastic work into heat is evaluated by means of the following expression:

∆T = β 1 ρcp

Z

σeqδεeq (11)

where β is the fraction of plastic work converted into heat, cp is the specific

490

heat capacity and δεeq is the plastic strain increment. Both values of ρ and cp are summarized in Table 4.

Special care must be taken when working with the Taylor-Quinney factor linked with a continuum damage framework such as the GTN model. Indeed, damage induces an increase of the void volume fraction which should lead to

495

a decrease of the Taylor-Quinney factor; as a consequence, the Taylor-Quinney factor should be described as a function of the void volume fraction. However, the adiabatic heating is computed in the Zset FE code in such a way that, in Eq. 11, the cp term is described as a function of (1 − f ).

In order to estimate the Taylor-Quinney factor, the measured temperature

500

rise at the surface has been compared with the numerical adiabatic result in the case of the tension test of a NT4sample at 0◦C for 0.05 mm.s−1. Fig. 14 shows the evolution-in-time of the heat build-up temperature at the surface for both

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Figure 14: Experimental and numerical temperature rise with respect to time for different values of β

experimental and numerical tests parametrized by the Taylor-Quinney factor value. For β = 0.9 and β = 0.6 − 0.8, corresponding values of metals and glassy

505

polymers, the temperature rise was overestimated whereas a good agreement is reached for a value of about 0.2, i.e. approximately only 20% of the plastic work seems to be converted into heat. The remaining 80% of energy could induce the appearance of residual stresses and phase modification. Similar results, in the case of elastomeric materials, have been highlighted [59, 60, 61]. Note that as

510

the convection and radiation phenomena were not taken into account, the value of 0.2 stands for a lower bound. On the other hand, the thermal expansion of the PA11 under investigation was estimated to be about 5 × 10−5 K−1; as a consequence, a thermal strain of about 0.001 for a temperature rise of 20◦C could be expected. Therefore, thermal expansion will be neglected in the following.

515

3.4. Simulations results 3.4.1. NT samples

The developed thermo-mechanical GTN constitutive model, integrating tem-perature dependent coefficients, has been compared with experimental data from NT samples with different notch radii [0.8, 1.2, 1.6 and 4] mm at 0◦C

520

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Figure 15: Experimental (symbol) and numerical (line) engineering stress-diametrical strain of the minimum cross section curves of NT samples under uniaxial tensile loading at 0◦C for 0.05 mm.s−1. Adiabatic simulations accounting for the heat build-up.

decrease of the notch radius would be associated with an increase of the stress triaxiality ratio; therefore, by working on several notch radii, the effects of the stress triaxiality ratio on the adiabatic heating were investigated. Experimental and adiabatic numerical global curves, i.e. net stress versus diametrical strain

525

of the minimum cross section, are displayed in Fig. 15. The results show that the model captured, in a satisfactory manner, the stress triaxiality ratio effect so that further analysis on the local contour maps could be carried out.

In Fig. 16, the numerical contour maps illustrate the distribution of the void volume fraction at the failure onset for the four investigated geometries. As

530

shown by several authors, Challier et al. [33], Laiarinandrasana et al. [38, 42] and Boisot et al. [53], the damage by void growth clearly depended on the stress triaxiality ratio, associated with the notched root radius of the NT sample. As predicted by Boisot et al [53], the higher void volume fraction was located in the centre of the NT samples, independently of notch root radius. A higher

535

void volume fraction of about 22% was predicted for the lower notch root radius (ρ = 0.8), related to a higher stress triaxiality ratio, whereas for the higher notch root radius (ρ = 4), a lower void volume fraction of about 4.7% was expected. Moreover, at a low stress triaxiality ratio, i.e. for a notch radius of

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Figure 16: Void volume fraction numerical predictions at the failure onset of NT samples with different notch radii [0.8, 1.2, 1.6 and 4] mm at 0◦C for 0.05 mm.s−1. Adiabatic simulations accounting for the heat build-up.

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Figure 17: Temperature contour numerical predictions at the failure onset of NT samples with different notch radii at 0◦C for 0.05 mm.s−1: a) NT4, b) NT1.6, c) NT1.2, and d) NT0.8

sample. Adiabatic simulations accounting for the heat build-up.

4 mm, the maximum void volume fraction was quite low and localized within

540

the whole ligament, induced by necking propagation. Besides, an increase of the stress triaxiality ratio was associated with an increase of the damage by void growth, as well as a more pronounced localization of the maximum void volume fraction. Indeed, the maximum void volume fraction was located exclusively at the centre of the cross section, related to the stress distribution induced by the

545

plasticity.

Numerical predictions of the temperature contour map at the failure onset are displayed in Fig. 17; in addition, the highest heat build-up temperature, for each geometry, is reported. It appears that a higher curvature radius, i.e. lower stress triaxiality ratio, resulted in a higher surface temperature. In fact,

550

a low triaxiality ratio was linked with high plasticity and, as a consequence, a higher heat build-up due to the adiabatic conversion of the plastic work. A temperature decreasing gradient, from the surface to the centre of the notch root section, was generally observed, i.e. the maximum temperature was expected at the surface whereas the minimum temperature was expected at the centre

555

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seem to associate a low stress triaxiality ratio with a lower temperature gradient within the ligament. Indeed, the initial stress triaxiality ratio calculated with the Bridgman formula [41], which was already low (0.56 for a NT4 sample), tended to decrease towards the stress triaxiality ratio of a UT sample, i.e. 0.33,

560

within the ligament at high strain [53]. On the other hand, within the NT0.8 sample, the temperature distribution in the ligament was more complex. The maximum temperature was still located at the surface of the sample but another significant heat source was located at the centre, i.e. the minimum temperature at the notch root section seems to be located between the surface and the centre.

565

As the necking did not propagate, the ligament was still under a triaxial stress state. Besides, an important temperature in the centre of the notch root section was commonly associated with a low heat conductivity of polymers. Similar results, for filled rubbers, have been predicted by Ovalle et al. [61, 62, 63], Behnke et al. [64] and, more recently, by Guo et al. [65].

570

As predicted, evolutions given by the proposed thermo-mechanical GTN model, for different NT samples have been found to be in good agreement with the experimental data, the capability of the proposed model to simulate and predict the thermo-mechanical response of a fracture mechanics geometry, in-volving a much higher stress triaxiality ratio such as SENB samples under 3

575

point bending, has been evaluated.

3.4.2. SENB samples

In previous studies, under isothermal conditions, the GTN model was suc-cessfully used by Challier et al. [33] and Laiarinandrasana et al. [38, 42] to simulate, in fully 3D, the crack propagation on SENB samples of PVDF in 3

580

point bending tests. In the present paper, focus is set only on the temperature rise at the crack propagation initiation during adiabatic simulations, which was expected to be significantly more important during the deformation of the PA11 under study.

In Fig. 18, the load versus Crack Opening Displacement (COD) experimental

585

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numer-Figure 18: Experimental and numerical load-COD curves of a 3 point bending test at 0◦C for 50 mm.min−1up to crack initiation. Adiabatic simulations accounting for the heat build-up.

ical simulation prediction. A good agreement was found up to the peak load, corresponding to the crack propagation initiation. Predicted contour maps of the maximum principal stress σ11, void volume fraction fg and temperature evolution T in the crack tip at the crack initiation are shown in Fig. 19. The

590

capability of adiabatic simulations to predict the temperature rise is shown. In line with [55], a decreasing void volume fraction, from the notch centre to the surface, was found, i.e. the higher void volume fraction was located at the centre of the notch related to a higher stress triaxiality ratio. Besides, contrary to the predictions on NT samples, where a higher triaxiality ratio was associated with

595

a higher void volume fraction and a lower heat build-up temperature, SENB sample predictions revealed an important temperature rise in the notch centre, decreasing through the thickness.

The distribution of the maximum principal stress in the uncracked liga-ment prior to the crack initiation on a SENB sample simulation at 0◦C for

600

50 mm.min−1 is illustrated in Figure 20. Results from the isothermal and adi-abatic simulations are reported. As expected, the highest maximum principal stress was located in the centre of the notch, ahead of the crack tip, decreas-ing through the ligament ordinate; then a higher temperature rise was expected ahead of the crack notch, through the direction of the crack growth, see Fig. 19d.

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Figure 19: Predicted contour maps at the peak load: a) Maximum principal stress, b) void volume fraction, c) temperature through the thickness direction, and d) temperature through the crack growth direction. Adiabatic simulations accounting for the heat build-up.

Figure 20: Opening stress as a function of the ligament ordinate at crack propagation initia-tion: comparison between isothermal and adiabatic data.

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On the other hand, a superposition of both isothermal and adiabatic curves is clearly revealed. As a consequence, the temperature rise during deformation, due to the thermo-mechanical coupling, was not so high and, therefore, isother-mal considerations were sufficient at the macroscopic scale.

Summary and conclusions

610

In this paper, stress triaxiality ratio effects on the adiabatic heating of a PA11 under static and quasi-static loading at 0◦C have been investigated. By simultaneous measurement of the temperature during the displacement-imposed experiments, the effects of temperature were studied.

Experimental results showed an enhancement of ductility with temperature.

615

An increase of temperature tended to reduce the peak stress and the net stress at failure but increased the strain at failure. Moreover, the value of the dia-metrical strain of the minimal cross section seemed not to be influenced by the temperature, as failure always occurred for a reduction of about 45%.

At 0◦C, a significant heat build-up temperature due to adiabatic heating may

620

rise within the material depending on deformation rate/displacement speed and stress triaxiality ratio, i.e. loading conditions and geometry of the sample. In-deed, it seems that a higher curvature radius, i.e. lower stress triaxiality ratio, was related to a higher surface temperature; furthermore, a low triaxiality ratio was linked with high plasticity and, as a consequence, a higher heat build-up

625

due to the adiabatic conversion of the plastic work. On the other hand, the tem-perature always started rising during enhanced plasticity stages such as necking in the case of Uniaxial Tensile samples, re-necking for the Notched Tensile sam-ples or crack propagation initiation regarding the Single Edge Notched under Bending samples. The temperature rise could reach 50◦C, corresponding to the

630

glass transition temperature of the material under study, mainly just prior to failure as sudden temperature rise occurs.

Consequently, a comprehensive experimental work has been dedicated to the development of a thermo-mechanical constitutive model based on the

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Gurson-Tvergaard-Needleman model integrating temperature-dependent coefficients. To

635

this end, the set of material coefficients obtained at 0◦C in a previous study [53] were extended to different temperatures, ranging from 0◦C to 100◦C. From then on, material coefficients became functions of the temperature, assuming a constant evolution with temperature. For each investigated temperature, good agreement between experimental and numerical engineering stress versus

dia-640

metrical strain curves were obtained. Moreover, at high temperature, it was shown that damage by void growth was reduced and failure could occur by critical plastic strain [53].

Predictive capabilities of the proposed thermo-mechanical model to simulate the isothermal behaviour of PA11 led to adiabatic simulations to account for

645

the heat build-up observed experimentally. Results of the adiabatic simulations are encouraging. The model was found to predict the temperature-dependent response of PA11 during isothermal and adiabatic simulations. Moreover, the capabilities of the thermo-mechanical model to predict the heat build-up tem-perature for different stress triaxiality ratios were shown. However, the

super-650

position of isothermal and adiabatic simulation results on a SENB sample at 0◦C for 50 mm.min−1revealed that the heat build-up temperature, occurring at crack initiation, was not significant enough; therefore, isothermal considerations were sufficient. Besides, different conditions (e.g. higher COD speed) can result in a significant heat build-up temperature effect which may be accounted for

655

by the thermo-mechanical model in order to predict the mechanical behaviour. This topic will be studied in a future study.

Finally, the results highlighted the importance of adiabatic heating occurring in a semi-crystalline polymer. It was revealed that accounting for temperature effects due to heat dissipation from plastic work may provide a tool to further

660

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Acknowledgements

The authors wish to extend their gratitude to G. Hochstetter (ARKEMA Cerdato) and C. Fond (Institut Charles Sadron/Institut de M´ecaniques des Flu-ides et des SolFlu-ides) for their inestimable contribution to this research work.

665

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Figure

Figure 1: Sample geometries
Figure 2: Temperature-dependent Young’s modulus at different frequencies.
Figure 3: Tensile tests on NT 4 samples for different temperatures at 0.05 mm.s −1 : a) net stress-strain, b) diametrical strain-strain, and at 3 mm.s −1 : c) net stress-strain, d) diametrical strain-strain.
Figure 4: Temperature-dependent response at two crosshead speeds [0.05 and 3] mm.s −1 : a) peak stress, and b) net stress at failure.
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